CRANFIELD UNIVERSITY SUNAN CHUMALEE ROBUST GAIN-SCHEDULED H CONTROL FOR UNMANNED AERIAL VEHICLES SCHOOL OF ENGINEERING. PhD THESIS

Transcription

1 CRANFIELD UNIVERSITY SUNAN CHUMALEE ROBUST GAIN-SCHEDULED H CONTROL FOR UNMANNED AERIAL VEHICLES SCHOOL OF ENGINEERING PhD THESIS Supervisor: Dr James F. Widborne June 21

2 This page intentionally contains only this sentence.

3 CRANFIELD UNIVERSITY SCHOOL OF ENGINEERING PhD THESIS SUNAN CHUMALEE Robust gain-scheduled H control for unmanned aerial vehicles Supervisor: Dr James F. Widborne June 21 c Cranfield University 21. All rights reserved. No part of this publication may be reproduced without the written permission of the copyright owner.

4 This page intentionally contains only this sentence.

5 Abstract This thesis considers the problem of the design of robust gain-scheduled flight controllers for conventional fixed-wing unmanned aerial vehicles (UAVs. The design approaches employ a linear parameter-varying (LPV control technique, that is based on the principle of the gain-scheduled output feedback H control, because a conventional gain-scheduling technique is both expensive and time-consuming for many UAV applications. In addition, importantly, an LPV controller can guarantee the stability, robustness and performance properties of the closed-loop system across the full or defined flight envelope. A flight control application problem for conventional fixed-wing UAVs is considered in this thesis. This is an autopilot design (i.e. speed-hold, altitude-hold, and heading-hold that is used to demonstrate the impacts of the proposed scheme in robustness and performance improvement of the flight controller design over a fuller range of flight conditions. The LPV flight controllers are synthesized using single quadratic (SQLF or parameterdependent (PDLF Lyapunov functions where the synthesis problems involve solving the linear matrix inequality (LMI constraints that can be efficiently solved using standard software. To synthesize an LPV autopilot of a Jindivik UAV, the longitudinal and lateral LPV models are required in which they are derived from a six degree-of-fredoom (6-DOF nonlinear model of the vehicle using Jacobian linearization. However, the derived LPV models are nonlinearly dependent on the time-varying parameters, i.e. speed and altitude. To obtain a finite number of LMIs and avoid the gridding parameter technique, the Tensor-Product (TP model transformation is applied to transform the nonlinearly parameter-dependent LPV model into a TP-type convex polytopic model form. Hence, the gain-scheduled output feedback H control technique can be applied to the resulting TP convex polytopic model using the single quadratic Lyapunov functions. The parameter-dependent Lyapunov functions is also used to synthesize another LPV controller that is less conservative than the SQLF-based LPV controller. However, using the parameter-dependent Lyapunov functions involves solving an infinite number of LMIs for which a number of convexifying techniques exist, based on an affine LPV model, for obtaining a finite number of LMIs. In this thesis, an affine LPV model is converted from the nonlinearly parameter-dependent LPV model using a minimum least-squares method. In addition, an alternative approach for obtaining a finite number of LMIs is proposed, by simple manipulations on the bounded real i

6 lemma inequality, a symmetric matrix polytope inequality form is obtained. Hence, the LMIs need only be evaluated at all vertices. A technique to construct the intermediate controller variables as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter is also proposed. The time-varying real parametric uncertainties are included in the system statespace model matrices of an affine LPV model as a linear fractional transformation (LFT form in order to improve robustness of the designed LPV controllers in the presence of mismatch uncertainties between the nonlinearly parameter-dependent LPV model and the affine LPV model. Hence, a new class of LPV models is obtained called an uncertain affine LPV model which is less conservative than the existing parameter-dependent linear fractional transformation model (LPV/LFT. New algorithms of robust stability analysis and gain-scheduled controller synthesis for this uncertain affine LPV model using single quadratic and parameter-dependent Lyapunov functions are proposed. The analysis and synthesis conditions are represented in the form of a finite number of LMIs. Moreover, the proposed method is applied to synthesize a lateral autopilot, i.e. heading-hold, for a bounded flight envelope of the Jindivik UAV. The simulation results on a full 6-DOF Jindivik nonlinear model are presented to show the effectiveness of the approach. ii

7 Acknowledgements Since the 1 st of July, 27 (that was a sunny July day and was also the first day of my Ph.D. research, it has been three years with fulfilling a lots of adventure, challenge, enjoyment, happiness and great experience. I have been particularly fortunate during my research to have had the opportunity to collaborate with a broad range of School of Engineering and students. I would like to take this opportunity to acknowledge the help and support of a few. First and foremost, I would like to express my deep gratitude to my research advisor, Dr. James F. Whidborne, who motivate me with his enthusiasm, guide me with valuable suggestions, and support me throughout the research project. He helped me to expand my knowledge through self-learning and provided me the opportunity to work in this exciting field. His keen insights and clear guidance gave me great encouragement to carry out this research and made this thesis possible. I am also grateful to my thesis review panel, Dr James Whidborne (supervisor, Dr Alastair Cooke (appropriate academic panel member selected by the supervisor and Dr Peter Sherar (research degree committee panel member, for their valuable comments and suggestions. My research and study would not have been possible without financial support from the Royal Thai Air Force. There are a number of people, family and friends that I am grateful and deserve to be a part of my success. I would first like to thank my parents, Chiang Chumalee, and, Laor Chumalee, and my sister, Porntip Chumalee, for their ever-present support and encouragement during my study. Special thanks to Dr. Stephen Carnduff for his helpful advice on my first journal article. I would also like to thank all my colleagues in the Dynamics, Simulation and Control Group, especially Naveed Rahman, Naseem Akhtar, Stuart Andrews, Peter Thomas, Deborah Saban, Rick Drury, Pierre-Daniel Jameson, Mudassir Lone and Ken Lai. They are always open for discussions on my research struggles. Their friendship and support also sustained me through many challenging occasions, and helped me to move forward. I also would like to give special thanks to the Thai community in Cranfield; they are supportive throughout my study. My time here at Cranfield has been memorable and valuable. I have enjoyed every aspect of this country. Once again, I would like to thank, everybody who helped me finish this research and made this Cranfield experience a most pleasurable one. iii

8 Last but not the least, I would like to thank my wife, Parichat, and my kids, Sirada and Sirasa, for their patience, understanding and support during the last three years. Their love was worth far more than any degree. Sunan Chumalee June, 21 iv

9 Contents Contents v List of Figures xi List of Tables xv Notation xvii List of Acronyms xxi 1 Introduction Control Design Considerations Aircraft Mathematical Nonlinear Model Model Uncertainties, Disturbances, and Sensor Noises Model Nonlinearities Sensor Limitations Actuator Limitations Conventional Gain Scheduling Aims & Objectives Publications Thesis Outline Preliminaries Introduction v

10 Contents 2.2 Mathematical Preliminaries Vector, Matrix, Signal, and System Norms Singular Value Decomposition Linear Fractional Transformation Linear Matrix Inequalities The S-Procedure Useful Tools Basic LPV Models Literature Review LPV Systems Theory Methods for Deriving LPV models Jacobian Linearization State Transformation Function Substitution Summary of Three Derivation Methods LPV Models for Controller Synthesis Grid LPV Model Affine LPV Model TP Convex Polytopic Model LPV/LFT Models Stability Analysis of LPV Systems Robustness Analysis using SQLF Robustness Analysis using Small Gain Theorem Controller Synthesis for LPV Systems Gain-Scheduled Controller Design using SQLF Gain-Scheduled Controller Design via LFT Numerical Example 45 vi

11 Contents 4.1 Jacobian Approach State Transformation Approach Function Substitution Approach Mismatch Uncertainty Pole Placement Approach Conclusion Longitudinal LPV Autopilot Design: A TP Approach Jacobian-Based Longitudinal LPV Model Longitudinal TP Convex Polytopic Model Gain-Scheduled H Autopilot Design Nonlinear Simulation Results Conclusion Longitudinal LPV Autopilot Design: A PDLF Approach Stability Analysis using PDLF Controller Synthesis using PDLF Numerical Example Longitudinal Affine LPV Model Gain-Scheduled H Autopilot Design Nonlinear Simulation Results Conclusion Robust Lateral LPV Autopilot Design Jacobian-Based Lateral LPV Model Stability Analysis of Uncertain Affine LPV Systems Robustness Analysis using SQLF Robustness Analysis using PDLF Controller Synthesis for Uncertain Affine LPV Systems vii

12 Contents Gain-Scheduled Controller Design using SQLF Gain-Scheduled Controller Design using PDLF Numerical Example Lateral Uncertain Affine LPV Model Robust Gain-Scheduled H Autopilot Design Nonlinear Simulation Results Conclusion Conclusions Conclusions & Discussions Main Contributions Further Work A Aircraft Nonlinear Model 157 A.1 Reference Frames & Sign Conventions A.2 Aircraft Equation of Motion A.3 Jindivik Nonlinear Mathematical Model A.3.1 Aerodynamic Force and Moment Models A.3.2 Thrust Model A.3.3 Sensor Model A.3.4 Actuator Model A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model B UAV aerodynamic model identification from a racetrack manoeuvre 181 B.1 Introduction B.2 Flight System Configuration B.2.1 Flight Control Computer B.2.2 Avionic Instrumentation B.2.3 Radio Telemetry viii

13 Contents B.2.4 Racetrack manoeuvre B.3 Aircraft Parameter Estimation B.3.1 Model Postulation B.3.2 Flight Data Post-Processing B.3.3 Equation-error Method B.3.4 Results B.4 Design, HIL simulation and flight test B.4.1 PID Autopilot Design B.4.2 Hardware-In-the-Loop (HIL Simulation B.4.3 Flight Test B.5 Conclusions C Explicit Controller Formulas for PDLF-based Gain-Scheduled H Synthesis 25 Bibliography 29 ix

14 This page intentionally contains only this sentence.

15 List of Figures 1.1 The Jindivik UAV Block diagrams structures of linear fractional transformation Block diagrams structure of an LPV/LFT model ( Block diagrams structures of LPV/LFT closed-loop systems Open-loop dynamic step response of the system (4.1 at t= LPV system: H mixed S/KS synthesis problem Singular value of S and KS over θ Θ (with frozen θ H norm of W 1 S and W 2 KS over θ Θ (with frozen θ Nonlinear step response from -3 to of the Jacobian-based LPV controller ( Nonlinear step response from to of function substitution-based and state transformation-based LPV controllers Nonlinear step response from to of new Jacobian-based LPV controller with the original nonlinear plant Block diagram structure of the gain-scheduled pole placement controller Pole placement controller design in MATLAB Simulink environment Nonlinear step response from to with the original nonlinear plant Control input to the original nonlinear plant A comparison between the simulated thrust and the estimated thrust X q and α trim are nonlinearly dependent on speed and altitude xi

16 List of Figures 5.3 The open-loop characteristic of transfer function u(s of the Jindivik rpm(s longitudinal nonlinearly parameter-dependent LPV model The cno type convex weighting functions in one-dimensional parameter, i.e. w n,j ( pn (t, of the longitudinal TP polytopic model The cno type convex weighting functions in two-dimensional parameters, i.e. w a ( p(t, of the longitudinal TP polytopic model The weighted open-loop interconnection for the longitudinal TP convex polytopic plant model Singular value of S and KS over θ Θ (with frozen θ H norm of W 1 S and W 2 KS over θ Θ (with frozen θ The transient performance of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 56 ft/s and altitude = 1, ft The stability and robustness properties of the closed-loop system were achieved over the defined flight envelope The transient performance of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 56 ft/s and altitude = 1, ft The desired performance and robustness objectives are achieved across the defined flight envelope The open-loop characteristic of transfer function v(s δ r(s of the Jindivik lateral LPV model Structure comparisons of uncertain affine LPV and uncertain LPV/LFT [9] models Structure comparisons of uncertain affine LPV and LPV/LFT [9] closed-loop systems Nonlinear step response from -3 to of the LPV ctroller with the original nonlinear plant The variation of a nonlinear Y δr and an affine Y δr with speed and altitude The variation of δ Yδr and δ Yδr (normalize with speed and altitude.. 14 xii

17 List of Figures 7.7 The weighted open-loop interconnection for the lateral uncertain affine LPV plant model The transient performance of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 464 ft/s and altitude = 7,5 ft The rate one turn of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 56 ft/s and altitude = 1, ft A.1 Reference axes & sign conventions [58] A.2 The Jindivik nonlinear model with its autopilot in MATLAB Simulink environment [41] A.3 Aircraft open-loop dynamic responses to.1 degree elevator step A.4 Aircraft open-loop dynamic responses to 5 RPM engine speed step 177 A.5 Aircraft open-loop dynamic responses to.1 degree aileron step A.6 Aircraft open-loop dynamic responses to.1 degree rudder step B.1 RTAF aerial target B.2 SWSDC flight controller B.3 Flight system configuration B.4 Flight data (racetrack pattern for identification B.5 Static thrust measurement result B.6 Power spectral densities B.7 Data compatibility analysis B.8 Estimation of aerodynamic coefficients B.9 Validation of aerodynamic coefficients B.1 Structure design of the autopilot B.11 Real-time hardware-in-the-loop simulation environment B.12 Validation of the PID autopilot through flight test xiii

18 This page intentionally contains only this sentence.

19 List of Tables 3.1 Comparison between three methods for deriving LPV models Simulation data of Jindivik s engine Jindivik s engine thrust coefficients Stability and control derivative data (longitudinal mode of Jindivik nonlinear model Jindivik s longitudinal aerodynamic coefficients Numerical comparisons of LPV synthesis techniques; an (X(θ, Y (θ case Performance γ comparison for different cases of (X(u, h, Y (u, h Computational time of Âk(θ, ˆBk (θ, Ĉ k (θ and D k (θ Stability and control derivative data (lateral mode of Jindivik nonlinear model Jindivik s lateral aerodynamic coefficients Performance γ comparison for different cases of (X(u, h, Y (u, h A.1 The Jindivik UAV specifications A.2 Mode characteristics of the determined longitudinal and lateral dynamics B.1 RTAF aerial target specifications B.2 Estimated instrumentation error parameters B.3 Parameter correlation coefficient matrix (coefficients of drag, lift, and pitching moment xv

20 List of Tables B.4 Parameter correlation coefficient matrix (coefficients of sideforce, rolling, and yawing moments B.5 Parameter correlation coefficient matrix (engine speed coefficient B.6 The Details of Vehicle Instrumentation B.7 The Identification Results xvi

21 Notation N, R, and C sets of natural, real, and complex numbers C m n set of m n complex matrix X set of all possible decisions in an optimization problem s set of feasible decisions: s X end of proof belong to for all subset := defined as (a, b {x R : a < x < b} (a, b] {x R : a < x b} I n n n identity matrix I i index upper bound [a ij ] a matrix with a ij as its ith row and jth column element diag(a 1,..., a n an n n diagonal matrix with a i as its i-th diagonal element A T and A transpose and complex conjugate of a matrix A A 1 inverse of a matrix A A + Moore-Penrose pseudo-inverse of a matrix A λ(a eigenvalue of a matrix A σ(a and σ(a the largest and the smallest singular values of a matrix A A spectral norm of a matrix A: A = σ(a L m 2 set of m-dimensional vector valued, measurable, square integrable functions with norm 2 [23] δ normalized time-varying real parametric uncertainties γ closed-loop quadratic H performance [1] ω n natural frequency θ time-varying parameters ˆθ structured block-diagonal real uncertainty θ normalized time-varying parameters ξ damping ratio sup{f} the supremum or least upper bound of a set F inf{f} the infimum or greatest lower bound of a set F G(t u(t convolution integral of g(t and u(t, t R F l (M, Q lower LFT F u (M, Q upper LFT xvii

22 Chapters 5-B only C D, C L C Di C DCL >C Lcrit aerodynamic drag and lift coefficients in stability axes induced drag coefficient drag coefficient due to the lift coefficient being greater than the critical lift coefficient C DM drag coefficient due to Mach number C DRe Reynolds number dependent profile drag coefficient C Ds drag coefficient increment due to the deflection of spoilers C Duc drag coefficient increment due to the extension of undercarriage C DW, C YW aerodynamic drag and side force coefficients in wind axes C DZ Reynolds number constant profile drag coefficient C Lcrit critical lift coefficient C Ls lift coefficient increment due to the deflection of spoilers C LT tailplane lift coefficient C Lwb wing-body lift coefficient C X, C Y, C Z aerodynamic force coefficients C Xwb, C Zwb wing-body combination coefficient of axial and normal forces C l, C m, C n aerodynamic moment coefficients C m1/4 quarter chord pitching moment coefficient D int engine intake drag I x, I y, I z, I xz mass moments of inertia, slug-ft 2 K ff1 flexibility factor applied to the rigid wing-body combination lift curve slope K ff2 flexible factor applied to the rigid body rate of change of downwash with respect to angle of attack L, M, N rolling, pitching, and yawing moments L aero, M aero, N aero aerodynamic moments M Mach number N nd non dimensional engine speed P atmospheric pressure P intratio intake pressure ratio P ratio pressure ratio P rec engine pressure recovery Re Reynolds number S wing surface area, ft 2 S T tailplane surface area, ft 2 T engine thrust, lb f or outside air temperature, kelvin T G, T Gnd gross and non dimensional gross thrusts T wl height of the tailplane from the waterline V airframe velocity, ft/s V δar aileron reversal speed, ft/s X, Y, Z axial, side, normal forces X aero, Y aero, Z aero aerodynamic forces a speed of sound wing-body combination lift curve slope of the flexible aircraft a 1wb xviii

23 a 1T tailplane lift curve slope a 2T elevator effectiveness a x, a y, a z body axis translational axial, lateral, and normal accelerations, ft 2 /s b, c wingspan and mean aerodynamic chord, ft c 1/4wl height of the quarter chord from the waterline g acceleration due to gravity, assume constant = ft/s 2 h aerodynamic centre position as a percentage of the mean aerodynamic chord h cg centre of gravity position as a percentage of the mean aerodynamic chord h wl height of the centre of gravity from the waterline k TG gross thrust factor k Prec intake pressure recovery factor k h altitude correction factor kṁ mass flow coefficient l T tail moment arm, ft lag T tailplane angle of attack lag angle due to downwash m aircraft mass, slug ṁ engine mass flow p, q, r body axis roll, pitch, and yaw angular rates, rad/s q dynamic pressure, lb f /ft 2 rpm, N 1 engine speed u, v, w body axis translational axial, lateral, and normal velocities, ft/s x E, y E, h longitude, latitude, and altitude, ft α, β angle of attack and sideslip, rad α R, α T tailplane rigging and angle of attack angle, rad α w wing setting angle, rad α wb wing-body combination angle of attack, rad β pg Prandtl-Glauert factor β pg1 Prandtl-Glauert factor at the critical mach number δ a, δ e, δ f, δ r, δ th aileron, elevator, flap, rudder, and throttle deflections, rad η dynamic viscosity φ, ψ roll, pitch, and yaw angles, rad θ pitch angle (rad or unknown parameters vector ρ air density, slug/ft 3 ρ sea level air density, =.2377 slug/ft 3 ε downwash angle, rad Subscripts, trim bias and trim values p, q, r, v, α, α derivative with respect to indicated quantity β, δ a, δ e, δ r xix

24 This page intentionally contains only this sentence.

25 List of Acronyms 6-DOF ACE ADC CAS CFD DCM GPS HIL HOSVD ISP LFT LFR LMI LPV LQE LQG LQR LTI NDI PDLF PID PSD PWM RFI RTAF SAS SDRE SLPV SQLF SVD SWSDC TP UAV Six Degree-of-Freedom Asynchronous Communication Element Analog-to-Digital Converter Control Augmentation System Computational Fluid Dynamics Direction Cosine Matrix Global Positioning System Hardware-In-the-Loop Higher Order Singular Value Decomposition In-System Programmable Linear Fractional Transformation Linear Fractional Representation Linear Matrix Inequality Linear Parameter-Varying Linear Quadratic Estimator Linear Quadratic Gaussian Linear Quadratic Regulator Linear Time-Invariant Nonlinear Dynamic Inversion Parameter-Dependent Lyapunov Function Proportional, Integral and Derivative control Power Spectral Density Pulse-Width Modulation Radio Frequency Interference Royal Thai Air Force Stability Augmentation System State Dependent Riccati Equation Switching Linear Parameter-Varying Single Quadratic Lyapunov Function Singular Value Decomposition Science and Weapon System Development Center Tensor-Product Unmanned Aerial Vehicle xxi

26 This page intentionally contains only this sentence.

27 Chapter 1 Introduction Unmanned Aerial Vehicles (UAVs could be defined as power-driven aircraft, other than model aircraft, that are designed to fly without a human operator on board. Typically, UAVs are used primarily to avoid putting persons at risk, or for missions where the task is better suited to a machine, e.g. D 3 missions - Dirty, Dull, Dangerous. To date, UAVs have been used in both military and civilian applications. Some military type applications include reconnaissance, surveillance (air, land, maritime, border patrol, and drug interdiction. In addition, some civilian type applications include surveillance, hydro-line inspections, water resources management, flood damage, and city mapping. UAVs have shown potential as being strong effective platforms for supporting both military and civilian applications. The increasing requirements on the capabilities of UAVs means that there is considerable ongoing research. Advanced control methodologies, such as optimal control, robust control, nonlinear control, intelligent control, etc., that guarantee the stability, robustness and performance properties of the closed-loop system are some of the very interesting active research areas for UAVs. Traditionally, automatic flight control systems can be categorised according to their function as; (i Stability Augmentation System (SAS is designed in order to improve flying characteristics (damping ratio and natural frequencies of an aircraft to achieve an acceptable requirement level of flying qualities standards, e.g. MIL-F-8785C, (ii Control Augmentation System (CAS is additionally designed to provide a specific type of response to the pilot s input, e.g. pitch rate commands, (iii Autopilot system is a pilot relief task that is fully automatic control systems, e.g. airspeed hold, altitude hold, heading hold, and turn coordination. Since the 195 s, classical controller design techniques such as root-locus, Bode and Nyquist plots, and frequency response analysis have been successfully and popularly implemented in the automatic flight control systems for the aerospace industry. These design techniques are well understood, clearly visible, highly structured system, and highly amenable to implementation. However, classical control techniques are limited when the controllers for multivariable systems with high internal coupling are to be designed [21]. More detail of the limitations of classical control 1

28 1.1 Control Design Considerations techniques are described in the next section, where common control design considerations of flight control systems are also outlined. Sections 1.2 and 1.3 give the motivations and aim & objectives, respectively. The publications and thesis outline are briefly summarized in sections 1.4 and 1.5, respectively. 1.1 Control Design Considerations Aircraft Mathematical Nonlinear Model Before a flight controller can be designed, an aircraft mathematical nonlinear model has to be determined. Traditionally, the model parameters are determined using wind tunnel tests by measuring the aerodynamic forces and moments introduced on an aircraft. Recently, computational fluid dynamics (CFD methods are becoming important. Furthermore, the aircraft moments of inertia are calculated and the aircraft engine model is determined from experimental data. However, these standard processes are both expensive and time-consuming and may not be affordable or practicable for many UAV applications [31, 32, 36] Model Uncertainties, Disturbances, and Sensor Noises In practice, most flight control design techniques require a linearized model of an aircraft s dynamics about some trim condition. An aircraft linear model is typically either derived from a six degree-of-freedom (6-DOF nonlinear model [33, 68] or determined from experimental measurement by parameter identification methods [28, 36, 53, 57, 78]. Having an accurate linear model, a flight controller can be successfully designed. However, the linear model will never be an entirely accurate representation of aircraft flight because the true aircraft dynamic model parameters are not exactly known and the aircraft dynamics are non-linear. Often this is primarily because there are characteristics which cannot easily be modelled; this is especially the case under extremely aggressive manoeuvring of aircraft where highly nonlinear flight regimes yield unsteady and nonlinear aerodynamic effects which are hard and complicated to model [52]. In addition, there are unmodelled dynamics and/or inaccuracies in the model parameters hence there will always be modelling inaccuracies in aircraft dynamic models [52]. Moreover, there will always be (usually uncertain external disturbance and sensor noises influencing the dynamic behavior of the aircraft. The model uncertainties, disturbances, and sensor noises are another practical problem for flight control design. 2

29 1.1 Control Design Considerations Model Nonlinearities In general, most real plants are nonlinear; this is especially the case for aircraft where their dynamic characteristics vary, following some time-varying parameters. Chumalee and Whidborne [33] showed that, when an aircraft is about a wings level and constant altitude and airspeed flight condition, its dynamic characteristics are nonlinearly dependent on attitude and velocity. In addition, for some UAV missions such as targeting, deception, electronic warfare and offensive operations, UAVs are often required to operate over a full flight envelope at various attitudes and velocities in which the model nonlinearities are another common problem for flight control design Sensor Limitations An automatic flight control system requires some data feedbacks in order to stabilize the aircraft and achieve asymptotic tracking of a known reference trajectory. Practically, these data feedbacks are passed to the automatic flight controller via flight sensors. Therefore, the sensor limitations, e.g. maximum measurement value or maximum update rate, are another actual problem for flight control design. For example, suppose the Crossbow VG4CD is assumed to be used in a flight control system for measuring attitude roll and pitch angles, roll, pitch, and yaw rates, and X, Y, and Z body axis accelerations. In addition, the pitch and roll rates are assumed to be data feedbacks for closed-loop control. Since the maximum roll, pitch, and yaw rates of the Crossbow VG4CD are 1 /s, the automatic flight controller has to be designed not to manoeuvre the aircraft with pitch and roll rates that are greater than 1 /s Actuator Limitations An automatic flight control system controls an aircraft via actuators, hence actuator saturation (position and rate and actuator time lag present another common problem for flight control design. For example, suppose the Futaba S926 (high-torque airplane servo is assumed to be used as the actuators in a flight control system. Since the time lag of Futaba S926 is.2 s, the automatic flight controller has to be designed not to control the aircraft with a frequency that is greater than 8 Hz (-3 db bandwidth. In addition, the ratio of elevator deflection to Futaba S926 deflection is assumed to be one to three (1:3. Since the speed of Futaba S926 is 315 /s, the automatic flight controller has to be also designed not to control the aircraft with an excessive gain that make elevator deflection s speed greater than 15 /s. 3

30 1.2 Conventional Gain Scheduling 1.2 Conventional Gain Scheduling A common approach in industry to handle the nonlinear property of the aircraft is by means of gain-scheduling. The conventional approach is to design a local linear time invariant (LTI controller for each member of a set of operating conditions that cover the whole of the flight envelope. As the operating conditions change, a global controller of the closed-loop system is determined on-line and in flight by interpolating the gain values of each local LTI controller (that are within the varied operating conditions according to the current value of the scheduling parameters. Experimentally, this design approach has been successfully and popularly implemented in many engineering applications (e.g. submarines, engines, aircraft, etc. in order to cover the entire operating range of system plants but, theoretically, it comes with no guarantees on the robustness, performance, or even nominal stability of the overall gain scheduled design [9]. The cost-effective development of many UAV applications is very significant but the conventional gain-scheduling technique is both expensive and time-consuming. In addition it cannot guarantee the stability, robustness and performance properties of the closed-loop system [9]. Hence this design approach is less suitable for a UAV application. Based on these shortcomings of the conventional gain-scheduling technique, two challenging motivation problems of this thesis are how to design a single robust gain-scheduled flight controller for conventional fixed-wing UAVs for which (i the designed controller can operate in a fuller range of flight conditions and (ii the designed controller can guarantee the stability and robustness properties of the closed-loop system. 1.3 Aims & Objectives An advanced robust gain-scheduling technique, namely linear parameter-varying (LPV control [91] that is based on the principle of the H control [47, 14], can be used to handle uncertainties and nonlinearities of a nonlinear plant model. Importantly, an LPV controller theoretically guarantees stability, robustness, and performance properties of the closed-loop system [8, 23, 97]. Therefore, the motivation problems listed in the previous section can be solved using an LPV control technique with the mixed-sensitivity criterion [2, 33, 47]. This will provide a controller with good command following (i.e. small tracking error, good disturbance attenuation, low sensitivity to measurement noise, reasonably small control efforts, and that is robustly stable to additive plant perturbations. To design a single robust gainscheduled flight controller using an LPV control technique for which the designed controller satisfies the two motivation problems is the main aim of this thesis. Moreover, an interesting flight control application problem for conventional fixedwing UAVs is considered in this thesis. This is an autopilot design (i.e. speed-hold, altitude-hold, and heading-hold and is also a main objective of this thesis in order 4

31 1.4 Publications to demonstrate the impacts of the proposed scheme in robustness and performance improvement of the flight controller design over a fuller range of flight conditions. The effectiveness of the proposed methods in designing an LPV autopilot is verified and validated through a 6-DOF nonlinear model of the Jindivik UAV, shown in Figure 1.1, that has been developed by Fitzgerald [41] in the MATLAB Simulink environment. 1.4 Publications The publications of this thesis are the following: Conference papers S. Chumalee and J. F. Whidborne. Pole Placement Controller Design for Linear Parameter Varying Plants. Proceedings of the UKACC International Conference on Control 28, Manchester, UK, September 28. S. Chumalee and J. F. Whidborne. Experimental Development of an UAV Nonlinear Dynamic Model. Proceedings of the 24th Bristol International Unmanned Air Vehicle Systems (UAVS Conference, Bristol, UK, March 29. S. Chumalee and J. F. Whidborne. LPV Autopilot Design of a Jindivik UAV. AIAA Guidance, Navigation, and Control Conference and Exhibit, Chicago, Illinois, Aug. 29. S. Chumalee and J. F. Whidborne. Identification and Control of RTAF Aerial Target. Proceedings of European Control Conference 29, Budapest, Hungary, Aug. 29. Journal articles S. Chumalee and J. F. Whidborne. UAV Aerodynamic Model Identification from a Racetrack Manoeuvre. J. Aerospace Engineering, Proc.IMechE Vol. 224 Part G, pages , 21. S. Chumalee and J. F. Whidborne. Gain-scheduled H Autopilot Design via Parameter-Dependent Lyapunov Functions. Journal of Guidance, Control, and Dynamics, 21. (submitted. S. Chumalee and J. F. Whidborne. Robust Flight Control for Uncertain Affine Linear Parameter-Varying Models. International Journal of Control, 21. (submitted. 5

32 1.5 Thesis Outline 1.5 Thesis Outline This thesis is organized as follows: Chapter 2 provides a brief overview of research in LPV control techniques. The relevant mathematical background is also briefly summarized. Chapter 3 outlines theory involved in LPV systems is which includes (i methods for deriving LPV models, i.e. Jacobian linearization, state transformation and function substitution, (ii LPV models, i.e. grid LPV model, affine LPV model and TP convex polytopic model, (iii parameter-dependent linear fractional transformation models (LPV/LFT, (iv stability analysis of LPV systems, i.e. Lyapunov-based stability analysis and small gain theorem, and (v controller synthesis for LPV Systems, i.e. bounded real lemma, gain-scheduled H control and gain scheduling via LFT. Chapter 4 illustrates the implementation of LPV systems theory for the nonlinear control problem via a simple numerical example [61] that is known to cause difficulties for LPV controllers. The parameters variation of the example [61] is cancelled using the gain-scheduled pole placement state feedback. For the example, the approach yields reliable closed-loop stability and good closed-loop transient performance of the system because it makes the nonlinear plant appear to be an LTI plant, hence well-developed LTI tools can be applied. In chapter 5, a longitudinal nonlinearly parameter-dependent LPV model is derived from a 6-DOF nonlinear dynamic model using Jacobian linearization. To synthesize an LPV controller with a finite number of LMIs and avoid the gridding technique, the TP model transformation is employed in order to transform a given nonlinearly parameter-dependent LPV model into a TP convex polytopic model form. Having determined the longitudinal TP polytopic model, the H gain-scheduling control that is proposed by Apkarian et al. [1] can immediately be applied to the resulting TP polytopic model to yield an LPV autopilot that guarantees the stability and robustness properties of the closed-loop system. In chapter 6, the longitudinal nonlinearly parameter-dependent LPV model is converted into an affine LPV model using the minimum least-squares method [58]. Based on this longitudinal affine LPV model, another LPV controller is synthesized with a finite number of LMIs using a new parameter-dependent Lyapunov functions approach. An existing PDFL approach, that is based on a multi-convexity method [11], is given in Appendix C for which an improvement of the parameterdependent Lyapunov-based stability and performance analysis from the proposed method can be compared with those from a multi-convexity approach [11]. In chapter 7, a lateral nonlinearly parameter-dependent LPV model is derived from a 6-DOF nonlinear dynamic model using Jacobian Linearization. A lateral affine LPV model is obtained using the minimum least-squares method [58] in a similar approach to Chapter 6. The time-varying real parametric uncertainties are included in the system state-space model matrices in an LFT form in order to guarantee closed-loop stability and improve transient performance in presence of the mismatch 6

33 1.5 Thesis Outline Figure 1.1: The Jindivik UAV uncertainties between the lateral nonlinearly parameter-dependent LPV model and the affine LPV model. Based on the proposed uncertain affine LPV model, a robust lateral LPV controller is synthesized with a finite number of LMIs using parameterdependent Lyapunov functions. Chapter 8 summarizes the discussions and main contributions of the thesis, and suggestions for future work directions are given. Appendix A briefly summarizes an aircraft mathematical model; this is especially applicable for the Jindivik nonlinear model [41] from which the mathematical modelling of aerodynamic forces and moments, thrust, sensors, and actuators are presented. In addition, two major lateral and longitudinal modes of the vehicle about wings level and constant altitude 1, ft and airspeed 56.3 ft/s straight flight condition are also presented. In appendix B, an ordinary piloted manoeuvre and off-trim condition flight data (racetrack manoeuvre of a Royal Thai Air Force (RTAF aerial target was studied and identiflied in order to estimate the aerodynamic coefficients of the vehicle. Only two flight tests had to be undertaken. The first flight test was done to record flight data by controlling the aerial target manually. The second flight test was done to validate the proportional, integral and derivative (PID autopilot. As shown by the flight test results of the PID autopilot, the identified 6-DOF non-linear model was sufficiently reliable and accurate for the design of a satisfactory control system. This appendix demonstrates the usefulness of system identification techniques for UAV control system design and development in a cost-effective manner. Appendix C briefly summarizes an existing parameter-dependent Lyapunov functions approach for synthesizing a PDLF-based LPV controller using a multi-convexity method [11]. In addition, the controller is constructed using an explicit controller formulas [43]. 7

34 This page intentionally contains only this sentence.

35 Chapter 2 Preliminaries 2.1 Introduction Advanced control methodologies can be roughly classified according to their objective nature as: (i optimal control is developed to achieve certain optimal performance (i.e. minimizing a quadratic cost function, (ii robust control is developed to handle system plants subject to uncertainties, disturbances, and noise measurements with high performances, (iii nonlinear control is developed to handle nonlinear systems with high performances, and (iv intelligent control is developed to handle systems with unknown dynamic models of system plants. Although advanced control methodologies have shown potential in the field of improving robustness, better performance, de-coupling control and simplifying the design process, some of them do not yet have the maturity required for industrial application [67]. From an industrial point of view, the desirable features of a very good flight controller are simplicity, transparency, quality, accuracy, reliability, generality and implementability [67]. The optimal and robust control methods are fairly well-studied, however, their performances would be conservative when UAVs have to operate in a nonlinear flight regimes with a wide range of attitudes and velocities. Nonlinear control methods may be suitable for designing a flight controller in this thesis but they are generally very complex. The intelligent control would not be also suitable because it is hard to guarantee the robustness and performance properties of the closed-loop system in presence of model uncertainties and nonlinearities, disturbances, and noises. In addition, intelligent control typically requires a lot of training data and it is actually not very transparent. Moreover, an intelligent control can be quite hard to implement due to its complexity of the developed control algorithm. In this thesis, an LPV control approach [91] is selected for designing a flight controller becase it can handle uncertainties and nonlinearities of a nonlinear plant model. Importantly, an LPV controller can guarantees stability and robustness properties of the closed-loop system [8, 23, 97]. 9

36 2.2 Mathematical Preliminaries 2.2 Mathematical Preliminaries This section briefly summarizes useful mathematical background that will be used throughout this thesis Vector, Matrix, Signal, and System Norms The material in this sub-section is essentially taken from Zhou et at. [14] and Gu et at. [47]. Let a vector x = [x 1,..., x n ] T C n, then a real valued function x p is the vector p-norm of x defined as ( n 1/p x p := x i p, for 1 p (2.1 x 1 := i=1 n x i, for p = 1 (2.2 i=1 x 2 := n x i 2, for p = 2 (2.3 i=1 x := max 1 i n x i, for p = (2.4 A norm of a vector is a measure of the vector length. When p = 2, x 2 is the Euclidean norm of a vector x (or the Euclidean distance of a vector x from the origin. It shall be denoted by x := x 2. Let a matrix A = [a ij ] C m n and a vector x C n, the matrix norm induced by a vector p-norm is defined by Ax p A p := sup, for 1 p (2.5 x x p m A 1 := max a ij, for p = 1, column sum (2.6 1 j n i=1 A 2 := λ max (A A, for p = 2 (2.7 n A := max a ij, for p =, row sum (2.8 1 i m j=1 When p = 2, A 2 is the spectral norm (i.e. largest singular value of a matrix A. It shall be denoted by A := A 2. 1

37 2.2 Mathematical Preliminaries The p-norm of a vector signal x(t = [x 1 (t,..., x n (t] T, t R, is defined by x p := x 1 := x 2 := ( i=1 i=1 1/p n x(t dt p, for 1 p (2.9 n x(t dt, for p = 1 (2.1 i=1 n x 2 (tdt, for p = 2 (2.11 x := sup x(t, for p = (2.12 t R Note that, given a set of real numbers F, sup(f is the supremum or least upper bound of a set F is defined to be the smallest real number that is greater than or equal to every number in F while inf(f is the infimum or greatest lower bound of a subset F is defined to be the biggest real number that is smaller than or equal to every number in F. Let ˆx(jω be the Fourier transform of x(t where ω is the real frequency variable in radians per unit time, the frequency domain 2-norm is defined by 1 ˆx 2 := ˆx 2π (jωˆx(jωdω (2.13 Note that, by Parseval s identity, x 2 = ˆx 2. That is 1 x 2 (tdt = 2π ˆx (jωˆx(jωdω (2.14 The normed spaces, consisting of a vector signal x(t = [x 1 (t,..., x n (t] T, t R with finite norm, is defined by ( 1/p n L n p(r := x(t : x p = x(t dt p <, for 1 p (2.15 i=1 L n 2(R := x(t : x 2 = n x(t 2 dt <, for p = 2 (2.16 i=1 For a stable LTI system G : L m 2 (R L n 2(R, the H norm of G(s is given by G = sup σ{g(jω} (2.17 ω R where G(s or G(jω C n m is the transfer function matrix of G and σ( is defined in the next sub-section. In addition, let u(t L m 2 and y(t L n 2 be the input 11

38 2.2 Mathematical Preliminaries and output vectors of system G respectively, the induced L 2 -norm or L 2 -gain of the system G is given by G i2 = sup u L m 2 G u 2 u 2 = sup u L m 2 y 2 u 2 (2.18 where G(t u(t denotes the convolution integral of G(t and u(t. That is y(t = G(t u(t = t g(t τu(τdτ (2.19 Note that, for LTI systems, the H norm is equal to the induced L 2 -norm or L 2 -gain (i.e. the maximal gain of the system [37, page 75]. That is G = sup σ{g(jω} = sup w R u L m 2 y 2 u 2 ( Singular Value Decomposition A singular value decomposition (SVD is a very useful tool to measure the size of a matrix for which the corresponding singular vectors are good indications of strong/weak input/output directions. Lemma (Singular Values and Eigenvalues [37] Given a complex matrix A C m n, the set of singular values of A is denoted by {σ i (A} which equals the k largest square roots of the eigenvalues λ of A A where k = min{m, n}. That is σ i (A = λ(a A, i = 1, 2,..., k (2.21 Normally, the singular values are ordered as σ i σ i+1 Hence: σ(a = σ 1 (A = sup x C n Ax 2 x 2 = A (2.22 σ(a = σ k (A = inf x C n Ax 2 x 2 (2.23 Theorem (Singular Value Decomposition [14, Theorem 2.11] Given a complex matrix A C m n, there exist two unitary matrices U = [ u 1 u 2... u m ] C m m V = [ v 1 v 2... v m ] C m m such that A = UΣV (

39 2.2 Mathematical Preliminaries 1 1 u l 2 2 (a Lower LFT structure (b Upper LFT structure Figure 2.1: Block diagrams structures of linear fractional transformation where [ ] Σ1 Σ = R m n σ 1... σ 2... Σ 1 = Rk k... σ k and σ 1 σ 2 σ k, k = min{m, n} Linear Fractional Transformation Linear Fractional Transformations (LFTs are a powerful and flexible approach to represent uncertainty in matrices and systems. Definition [14] Let M be a complex matrix partitioned as [ ] M11 M M = 12 C (p 1+p 2 (q 1 +q 2 M 21 M 22 A lower LFT with respect to l C q 2 p 2, shown in Figure 2.1 a, that a mapping F l (M, l : C q 2 p 2 C p 1 q 1 is defined by F l (M, l := M 11 + M 12 l (I M 22 l 1 M 21 (2.25 where the inverse (I M 22 l 1 exists. An upper LFT with respect to u C q 1 p 1, shown in Figure 2.1 b, that a mapping F u (M, u : C q 1 p 1 C p 2 q 2 is defined by F u (M, u := M 22 + M 21 u (I M 11 u 1 M 12 (2.26 where the inverse (I M 11 u 1 exists. 13

40 2.2 Mathematical Preliminaries Linear Matrix Inequalities The following definitions are required. Definition (Convex sets [89] A set s in a linear vector space is said to be convex if x 1, x 2 s then {x := αx 1 + (1 αx 2 } s for all α (, 1 Definition (Convex combinations [89] Let s be a subset of a vector space. The point n x := α 1 x 1 + α 2 x α n x n = α i x i (2.27 is called a convex combination of x 1,..., x n s if α i for i = 1,..., n and n i=1 α i = 1 Definition (Convex hull [89] The convex hull Co{s} of any subset s X is the intersection of all convex sets containing s. If s consists of a finite number of elements, then these elements are referred to as the vertices of Co{s}. Definition (Affine functions [89] A function f : s T is affine if for all x 1, x 2 s and α R f(αx 1 + (1 αx 2 = αf(x 1 + (1 αf(x 2 (2.28 Definition [1] A matrix polytope is defined as the convex hull of a finite number of matrix vertices N i with the same dimensions. { r } r Co {N 1, N 2,..., N r } := α i N i : α i, α i = 1 (2.29 i=1 Definition [34] Given a matrix M R p p, (i M is a negative definite symmetric matrix, i.e. M = M T <, if X T MX < for all nonzero vector X R p. (ii M is a positive definite symmetric matrix, i.e. M = M T >, if X T MX > for all nonzero vector X R p. A linear matrix inequality is an affine function [89] mapping F : R n R m m that is expressed of the form F (x = F + x 1 F 1 + x 2 F x n F n n = F + x i F i < (2.3 i=1 where x = [x 1,..., x n ] T R n is a vector of n real numbers called the decision variables. F,..., F n R m m are real symmetric matrices, i.e. F j = F T j, for j =,..., n. F (x is a negative definite symmetric matrix hence, by Definition 2.2.9, u T F (xu < for all nonzero vector u R m. This is equivalent to the condition that all eigenvalues λ(f (x are negative or equivalently, the maximal eigenvalue λ max (F (x <. 14 i=1 i=1

41 2.2 Mathematical Preliminaries Normally the variable x in (2.3, which we are interested in, is composed of one or more matrices whose columns have been stacked as a vector [49]. That is, F (x = F (X 1, X 2,..., X p where X i R q i r i is a matrix, p i=1 q i r i = n, and the columns of all the matrix variables are stacked up to form a single vector variable. Hence, (2.3 can be modified further as [49]: F (X 1, X 2,..., X p = F + G 1 X 1 H 1 + G 2 X 2 H G p X p H p (2.31 p = F + G i X i H i < (2.32 i=1 where F, G i R m q i, H i R r i m are given matrices and the X i are the matrix variables which we seek. Corollary Let s := {x F (x < } be the set of feasible solutions to the LMI F (x < then s is convex. Proof. We show that if x 1, x 2 s then {x := αx 1 + (1 αx 2 } s for all α (, 1. By Definition 2.2.7, F (x = F (αx 1 + (1 αx 2 = αf (x 1 + (1 αf (x 2. In addition, α >, (1 α >, F (x 1 <, and F (x 2 <, it is obvious that αf (x 1 + (1 αf (x 2 < for all α (, 1, therefore we have {x := αx 1 + (1 αx 2 } s for all α (, The S-Procedure The LPV control constraint often come with some quadratic inequalities (or quadratic function being negative whenever some other quadratic inequalities are all negative. The several quadratic inequalities can be combined into one single inequality (generally with some conservatism using the S-procedure [25]. Let F,..., F p be quadratic functions of the variable x R n : where T i = T T i. Consider F i (x := x T T i x + 2u T i x + v, i =,..., p, F (x for all x such that F j (x, j = 1,,..., p. (2.33 It is obvious that if there exist τ 1,..., τ p such that for all x, F (x p τ i F i (x (2.34 i=1 then (2.33 is satisfied. 15

42 2.2 Mathematical Preliminaries Useful Tools Lemma (Congruence Transformation [51, page 399] Given a negative definite symmetric matrix M = M T R p p, M < then for another real matrix T R p q such that rank(t T MT = rank(t = q, the following inequality holds: Note that, based on rank(t = q, we have q [1, p] N. T T MT < (2.35 Lemma (Schur complement [44] Let a partitioned symetric matrix [ ] P M N = M T R p p, Q N is a negative definite symmetric matrix if and only if Q < where P MQ 1 M T is the Schur complement of Q. P MQ 1 M T < (2.36 Proof. [89] Let a non-singular matrix T R p p and u = T v where [ ] I T = Q 1 M T, (2.37 I We have u T Nu < for all nonzero vector u R p for which this is equivalent to v T T T NT v < for all nonzero vector v R p. Computing T T NT : [ ] P MQ T T NT = 1 M T (2.38 Q Obviously, v T T T NT v < if and only if (2.36 is satisfied. Lemma [82] Given a pair of positive definite symmetric matrices (X, Y R p p. Then there exists matrices X 2, Y 2 R p m and X 3, Y 3 R m m, where m is a positive integer, such that X 3 = X3 T, [ ] [ [ ] X X2 X X2 Y Y2 X T 2 X 3 >, and X T 2 X 3 ] 1 = if and only if X Y 1, and rank(x Y 1 m. Y T 2 Y 3 (2.39 Lemma (Projection lemma [44] Given an inequality problem of the form Ψ + Q T K T P + P T KQ < (2.4 where Ψ R m m is a symmetric matrix, Q and P are matrices with column dimension m. Let Q and P be any matrices whose columns form bases of the null spaces of Q and P respectively; the above problem is solvable for a matrix K of compatible dimensions if and only if Q T ΨQ <, P T ΨP < (

43 2.3 Basic LPV Models Lemma (Finsler s Lemma [49] For some real number σ R, (2.4 is equivalent to two inequalities Ψ + σq T Q < (2.42 Ψ + σp T P < ( Basic LPV Models In nature, most real plants are nonlinear. A general nonlinear model can be written in the form. ẋ (t = f(x (t, u (t y (t = g(x (t, u (t (2.44 where t R is the time, x(t = [x 1 (t,..., x p (t] T R p is the state vector, u(t = [u 1 (t,..., u m (t] T R m is the control input vector, y(t = [y 1 (t,..., y q (t] T R q is the measurement output vector, f( and g( are continuous mapping functions: R p R m R p, and R p R m R q, respectively. Having linearized (2.44 using Jacobian method about one equilibrium point, an LTI model is obtained in which it can be written as a state-space system of the form (see sub-section 3.1.1: ẋ (t = Ax(t + Bu(t where A R p p, B R p m, C R q p, and D R q m. y (t = Cx(t + Du(t (2.45 In general, most real plants have more than one equilibrium point. At each equilibrium point, some operating parameters could be selected as time-varying parameters. When the time-varying parameters vary slowly, (2.45 become an LPV model for which it can be written in the form (see sub-section 3.1.1: ẋ(t = A ( θ(t x(t + B ( θ(t u(t y(t = C ( θ(t x(t + D ( θ(t u(t (2.46 where A(, B(, C(, and D( are known functions of time-varying parameters, θ(t = [θ 1 (t,..., θ n (t] T R n, and are continuous mapping matrix functions: R n R p p, R n R p m, R n R q p, and R n R q m, respectively. The time variation of each of the parameters θ(t is not known in advance, but can be measured in real-time and lies in some set bounded by known minimum and maximum possible values, i.e. θ 1 (t [θ 1, θ 1 ], θ 2 (t [θ 2, θ 2 ],..., θ n (t [θ n, θ n ]. It can be seen that two interesting features of an LPV model are (i an LPV model can represent nonlinear dynamic characteristics of an original nonlinear model better than an LTI model because it uses the time-varying parameters θ(t to capture the dynamic characteristics of the original nonlinear model and (ii an LPV model is still a linear system, whose state-space descriptions are functions of time-varying parameters θ(t. Hence the single quadratic or parameter-dependent Lyapunov functions can be used to prove the stability of LPV models. 17

44 2.4 Literature Review 2.4 Literature Review Comprehensive overviews of research in gain-scheduled control techniques can be found in [87] and [62]. Although most real plants are nonlinear, they can often be modelled as an LPV plant model [13, 39, 8] or characterized as a linear fractional transformation (LFT [9, 29, 81, 98] or as a linear fractional representation (LFR [86, 96] for gain-scheduled control synthesis and analysis purpose. An LFT (or LFR gain-scheduled controller is often synthesized via the scaled smallgain theorem [81] or scaled bounded real lemma [9, 96]. Two advantages of the parameter-dependent plant having LFT (or LFR parameter dependency are (i the existence of an LFT (or LFR gain-scheduled controller that is fully characterized by a finite number of LMIs [96, 98], and (ii its favourable LFT (or LFR structure offers obvious advantages in reducing computational burden and ease of controller implementation [9, 83]. However, in the LFT (or LFR formulation, the variations of parameters are allowed to be complex, thus conservatism is introduced when the scheduled parameters are real [1, 6, 83]. An LPV plant model was first introduced by Shamma and Athans [91] whereby its dynamic characteristics vary, following some time-varying parameters whose values are unknown a priori but can be measured in real-time and lie in some set bounded by known minimum and maximum possible values. An algebraic manipulation method, e.g. Jacobian linearization [39, 66, 8], state transformation [13, 92], or function substitution [94], etc., is normally used to derive an LPV model from the original nonlinear model. Moreover, in the literature, there are several different varieties of LPV models, e.g. the grid LPV model [39, 66, 1, 11], the affine LPV model [8, 7, 1] (or polytopic LPV model, the tensor-product (TP convex polytopic model [15, 16, 18], etc., these have been introduced for the analysis and gain-scheduled control synthesis which is usually based on single quadratic Lyapunov function [1, 23] or parameter-dependent Lyapunov functions (e.g. parameter-dependent [8, 39, 1, 11], affine parameter-dependent [45], piecewise-affine parameter-dependent [63, 64], blending parameter-dependent [94], multiple parameter-dependent Lyapunov functions [65, 66], etc.. A grid LPV model was introduced by Becker and Packard [23], whereby system state-space model matrices are functions of the scheduled parameters at all grid points over the entire parameter spaces and an affine LPV model, introduced by Apkarian et al. [1], the system matrices are known functions and depend affinely on the parameters that vary in a polytope of vertices. The TP convex polytopic model has been recently proposed by Baranyi [15] for transforming a given LPV model, whose system matrices are nonlinearly dependent on the parameters, into a convex polytopic model. It uses a higher order singular value decomposition in order to decompose a given N-dimensional tensor into a full orthonormal system in a special ordering of higher order singular values which express the rank properties of the given LPV model for each element of the parameter vector in the L 2 -norm [15, 16, 18]. Hence, the TP-type convex polytopic model is obtained, where the parameterdependent weighting functions of the LTI vertice components of the polytopic model 18

45 2.4 Literature Review are one-dimensional functions of the elements of the parameter vector [15, 16, 18]. Using single quadratic Lyapunov functions, for both the affine LPV models [1] and the TP convex polytopic models [33] cases, a finite number of LMIs need only be evaluated at all vertices while, for the grid LPV models [99] case, an infinite number of LMIs have to be evaluated at all points over the entire parameter space in order to determine a pair of positive definite symmetric matrices (X, Y. However, in practice, the symmetric matrices (X, Y can be determined from a finite number of LMIs by gridding the entire parameter space with a non-dense set of grid points. Having determined the symmetric matrices (X, Y, a more dense grid points set can be tested with these determined symmetric matrices (X, Y to check whether the LMIs are satisfied [1, 11]. If not, this process is repeated with a denser grid until the symmetric matrices (X, Y, that satisfy the LMIs for all points over the entire parameters space, are obtained [64, 1, 11]. Hence, the result of heuristic gridding technique is unreliable and the analysis result is dependent on choosing the gridding points [96]. In addition, for a grid LPV model case, the resulting gain-scheduled controller has high computational on-line complexity at the gainscheduling level [99] while, for the other two cases, the gain-scheduled controller is constructed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter [1, 33]. In general, the single quadratic Lyapunov function is more conservative than the parameter-dependent Lyapunov function when the parameters are time-invariant or slowly varying [45]. Moreover, when the parameters have a large variation, the piecewise-affine parameter-dependent [63, 64], blending parameter-dependent [94], and multiple parameter-dependent Lyapunov functions [65, 66] are less conservative than the parameter-dependent Lyapunov function. This is because an LPV model with a large parameter variation can be modeled as a switching linear parametervarying (SLPV system that can be discontinuous along the switching surface by dividing the entire parameters spaces into parameters subsets that are small variation regions. Hence, solving LMIs with the parameters subsets, the performance measure (γ can be improved. Moreover, the sufficient conditions to guarantee the stability of the SLPV systems in terms of the dwell time and the average dwell time have been provided in [12]. However, using parameter-dependent [8, 39, 1, 11], blending parameter-dependent [94], and multiple parameter-dependent Lyapunov functions [65, 66], an infinite number of LMIs have to be evaluated at all points over the entire parameters space. Moreover, the resulting gain-scheduled controller requires more complex on-line computations at the gain-scheduling level. 19

46 This page intentionally contains only this sentence.

47 Chapter 3 LPV Systems Theory An LPV control technique is motivated by the shortcomings of the conventional gain-scheduling technique. Since Shamma and Athans [91] introduced LPV plant models, the LPV control approach has developed quite rapidly. There are three common approaches that are normally used to derive an LPV model from an original nonlinear model. These are (i Jacobian linearization [39, 66, 8], (ii state transformation [13, 92], and (iii function substitution [94]. Section 3.1 gives a brief overview of these three approaches. Details of three types of LPV models and details of parameter-dependent linear fractional transformation models (LPV/LFT are also provided in sections 3.2. In addition, the theory involved in stability analysis (i.e. Lyapunov-based stability analysis and small gain theorem and controller synthesis for LPV Systems (i.e. bounded real lemma, gain-scheduled H control and gain scheduling via LFT are presented in sections 3.3 and 3.4, respectively. 3.1 Methods for Deriving LPV models An LPV model is of great importance for synthesizing an LPV controller. The following sub-sections summarize the common methods for deriving a reliable LPV model from an original nonlinear model Jacobian Linearization Jacobian linearization is a well known method which is usually used to linearize an nonlinear ordinary differential equation about a specific operating condition, called an equilibrium point (or trim point. Consider an equilibrium point at t i in the time space of a general nonlinear model of the form (2.44, the equilibrium state vector x trim, equilibrium input vector u trim, 21

48 3.1 Methods for Deriving LPV models and equilibrium output vector y trim can be defined as shown below, x trim = x (t i u trim = u (t i y trim = g(x trim, u trim (3.1 where f(x trim, u trim =. Consider a small perturbation about (x trim, u trim, and y trim. The deviation variables can be defined as shown below, in order to measure the difference. δx (t = x (t x trim δu (t = u (t u trim (3.2 Substitute x (t = x trim + δx (t and u (t = u trim + δu (t into (2.44, we get ẋ (t = f(x trim + δx (t, u trim + δu (t Appling the Taylor expansion to (3.3 gives y (t = g(x trim + δx (t, u trim + δu (t (3.3 f(x trim + δx (t, u trim + δu (t = f(x trim, u trim + δx (t + δu (t + 2δx (t f(x (t, u (t u (t + x(t=xtrim u(t=u trim f(x (t, u (t δu (t x (t f(x (t, u (t x (t { 1 δx 2 (t 2 f(x (t, u (t 2! x (t 2 f(x (t, u (t u (t x(t=xtrim u(t=u trim } + δu 2 (t 2 f(x (t, u (t u (t 2 + x(t=xtrim u(t=u trim g(x trim + δx (t, u trim + δu (t = g(x trim, u trim + δx (t + δu (t + 2δx (t g(x (t, u (t u (t + x(t=xtrim u(t=u trim g(x (t, u (t δu (t x (t + δu 2 (t 2 g(x (t, u (t u (t 2 g(x (t, u (t x (t { 1 δx 2 (t 2 g(x (t, u (t 2! x (t 2 g(x (t, u (t u (t x(t=xtrim u(t=u trim } x(t=xtrim u(t=u trim + (3.4 22

49 3.1 Methods for Deriving LPV models By neglecting second and higher order terms, approximations of (3.4 are f(x (t, u (t f(x trim + δx (t, u trim + δu (t =f(x trim, u trim + δx (t x (t + δu (t f(x (t, u (t u (t g(x trim + δx (t, u trim + δu (t =g(x trim, u trim + δx (t Equation (3.5 can be rewritten as ẋ (t = δx (t y (t y trim = δx (t f(x (t, u (t x (t g(x (t, u (t x (t + δu (t g(x (t, u (t u (t + δu (t x(t=xtrim u(t=u trim x(t=xtrim u(t=u trim + δu (t x(t=xtrim u(t=u trim g(x (t, u (t x (t x(t=xtrim u(t=u trim x(t=xtrim u(t=u trim (3.5 x(t=xtrim u(t=u trim f(x (t, u (t u (t g(x (t, u (t u (t x(t=xtrim u(t=u trim (3.6 x(t=xtrim u(t=u trim Finally, (3.6 can be rewritten in a standard state-space of the form (LTI model. δẋ (t = Aδx (t + Bδu (t δy (t = Cδx (t + Dδu (t (3.7 where δẋ (t = ẋ (t, δx (t = x (t x trim, δu (t = u (t u trim, δy (t = y (t y trim, (x trim, u trim, y trim are at an equilibrium point, and f(x (t, u (t f(x (t, u (t A = B = x (t x(t=xtrim u (t x(t=xtrim u(t=u trim C = g(x (t, u (t x (t x(t=xtrim u(t=u trim D = g(x (t, u (t u (t u(t=u trim x(t=xtrim u(t=u trim It is noted that Jacobian linearization gives a single LTI model about a given trim point (a small perturbation condition. However, most actual plants have more than one trim point. The set of trim points typically varies, following the operating condition. At each trim point, some operating paramters could be selected as timevarying parameters, e.g. Mach number, altitude, dynamic pressure, or angle of attack in case of aircraft. When the time-varying parameters are fixed, (3.7 is an LTI model. However, when the time-varying parameters vary slowly over the entire parameters space, (3.7 becomes an LPV model. 23

50 3.1 Methods for Deriving LPV models State Transformation Shamma and Cloutier [92] proposed the state transformation method to derive a quasilinear parameter varying (quasi-lpv of a missile nonlinear model. An LPV model is said to be a quasi-lpv model if its state vector can be partitioned into a scheduling state vector and a non-scheduling state vector. In many applications (e.g. turbofan engine [13], missile [92], etc., the nonlinear model of (2.44 can be rewritten as: ] [ẋ1 (t = ẋ 2 (t [ ] [ ] A11 (x 1 (t A 12 (x 1 (t x1 (t + A 21 (x 1 (t A 22 (x 1 (t x 2 (t [ ] B1 (x 1 (t u(t + B 2 (x 1 (t [ ] K1 (x 1 (t K 2 (x 1 (t (3.8 where x 1 (t = [x 11 (t,..., x 1p1 (t] T R p 1 is the scheduling state vector and is also the the time-varying parameters, x 2 (t = [x 21 (t,..., x 2p2 (t] T R p 2 is the nonscheduling state vector where p = p 1 + p 2. A 11 (, A 12 (, A 21 (, A 22 (, B 1 (, B 2 (, K 1 (, and K 2 ( are continuous mapping functions: R p 1 R p 1 p 1, R p 1 R p 1 p 2, R p 1 R p 2 p 1, R p 1 R p 2 p 2, R p 1 R p1 m, R p 1 R p2 m, R p 1 R p 1, and R p 1 R p 2, respectively. Suppose there exist continuously differentiable functions x 2eq (x 1 (t and u eq (x 1 (t that are continuous mapping functions: R p 1 R p 2 and R p 1 R m, respectively, such that for every x 1 (t, [ ] = [ A11 (x 1 (t ] [ A 12 (x 1 (t A 21 (x 1 (t A 22 (x 1 (t Subtracting (3.9 from (3.8 obtains x 1 (t x 2eq (x 1 (t ] + [ ] B1 (x 1 (t u B 2 (x 1 (t eq (x 1 (t + ẋ 1 (t = A 12 (x 1 (t[x 2 (t x 2eq (x 1 (t] + B 1 (x 1 (t[u(t u eq (x 1 (t] [ ] K1 (x 1 (t K 2 (x 1 (t (3.9 ẋ 2 (t = A 22 (x 1 (t[x 2 (t x 2eq (x 1 (t] + B 2 (x 1 (t[u(t u eq (x 1 (t] (3.1 Differentiating x 2eq (x 1 (t with respect to time t gives d dt x 2eq(x 1 (t = ẋ 2eq (x 1 (t = x 2eq(x 1 (t ẋ 1 (t x 1 (t = x 2eq(x 1 (t {A 12 (x 1 (t[x 2 (t x 2eq (x 1 (t] + B 1 (x 1 (t[u(t u eq (x 1 (t]} x 1 (t (3.11 Subtracting (3.11 from (3.1 obtains ẋ 2 (t ẋ 2eq (x 1 (t =[A 22 (x 1 (t x 2eq(x 1 (t A 12 (x 1 (t][x 2 (t x 2eq (x 1 (t] x 1 (t + [B 2 (x 1 (t x 2eq(x 1 (t B 1 (x 1 (t][u(t u eq (x 1 (t] x 1 (t (

51 3.1 Methods for Deriving LPV models Hence, (3.1 can be rewritten as a quasi-lpv model in a state-space equation of the form [ ] [ ] [ ] [ ] ξ1 (t A12 (ξ = 1 (t ξ1 (t B1 (ξ + 1 (t v(t (3.13 ξ 2 (t à 22 (ξ 1 (t ξ 2 (t B 2 (ξ 1 (t where ξ 1 (t = x 1 (t, ξ 2 (t = x 2 (t x 2eq (x 1 (t, v(t = u(t u eq (x 1 (t and à 22 (ξ 1 (t = A 22 (ξ 1 (t x 2eq(ξ 1 (t A 12 (ξ 1 (t ξ 1 (t B 2 (ξ 1 (t = B 2 (ξ 1 (t x 2eq(ξ 1 (t B 1 (ξ 1 (t ξ 1 (t Function Substitution Tan et al. [95] have introduced this method to linearize a missile nonlinear model around a single equilibrium point. Unlike Jacobian linearization, the principle of this method is to select a suitable single equilibrium point such that the original nonlinear model can be rewritten in a quasi-lpv model form. Suppose a nonlinear model in (2.44 can be rewritten as (3.8. Let x 1r = [x 1r1,..., x 1rp1 ] T R p 1, x 2r = [x 2r1,..., x 2rp2 ] T R p 2, and u r = [u r1,..., u rm ] T R m be a selected equilibrium point. Define x 1 (t = x 1 (t+x 1r, x 2 (t = x 2 (t+x 2r, and u(t = ũ(t+u r. Substituting into (3.8, we have [ ] [ ] ] [ ] [ ] x 1 (t A11 (x = 1 (t A 12 (x 1 (t [ x1 (t B1 (x + 1 (t F1 (x ũ(t+ 1 (t, x 1r, x 2r, u r x 2 (t A 21 (x 1 (t A 22 (x 1 (t x 2 (t B 2 (x 1 (t F 2 (x 1 (t, x 1r, x 2r, u r (3.14 where F 1 ( and F 2 ( are continuous mapping functions: R p 1 R p 1 R p 2 R m R p 1 and R p 1 R p 1 R p 2 R m R p 2, respectively, and [ ] [ ] [ F1 (x 1 (t, x 1r, x 2r, u r A11 (x = 1 (t A 12 (x 1 (t x1r F 2 (x 1 (t, x 1r, x 2r, u r A 21 (x 1 (t A 22 (x 1 (t x 2r ] + [ ] [ ] B1 (x 1 (t K1 (x u B 2 (x 1 (t r + 1 (t K 2 (x 1 (t (3.15 Given a suitable single equilibrium point (x 1r, x 2r, u r, the goal of this method is to decompose functions F 1 ( and F 2 ( into functions E 1 ( and E 2 ( that are linear in x 1 (t and are continuous mapping functions: R p 1 R p 1 p 1 and R p 1 R p 2 p 1 such that for all x 1 (t, [ E1 (x 1 (t E 2 (x 1 (t ] ] [ x1 (t x 2 (t [ ] F1 (x 1 (t, x 1r, x 2r, u r F 2 (x 1 (t, x 1r, x 2r, u r (3.16 The decomposition of functions E 1 ( and E 2 ( can be posed as an optimization problem, for example, as a minimum least-squares problem [58] that minimizes the sum of squared differences between the true functions (F 1 (, F 2 ( and the approximate functions (E 1 (, E 2 (. That is to minimize ɛ T ɛ/2 where [ ] F1 (x ɛ = 1 (t, x 1r, x 2r, u r E 1 (x 1 (t x 1 (t (3.17 F 2 (x 1 (t, x 1r, x 2r, u r E 2 (x 1 (t x 1 (t 25

52 3.1 Methods for Deriving LPV models Table 3.1: Comparison between three methods for deriving LPV models Method Advantages Disadvantages Method Advantages Disadvantages Method Advantages Disadvantages Jacobian linearization (i Applicable to a general class of nonlinear models. (ii Simple structure model. (i Only accurately represent the original nonlinear dynamics about the neighborhood of a set of equilibrium points. (ii The time-varying parameters must vary slowly. State Transformation (i Exactly represent the original nonlinear dynamics. (i Applicable to the only special form of nonlinear models (ii Complex structure model. Function Substitution (i Can represent the original nonlinear dynamics over a non-trim region. (i Applicable to the only special form of nonlinear models (ii How to find a suitable equilibrium point is still not established. (iii Complex structure model. For example, Shin et al. [94] proposed the function substitution method to derive a quasi-lpv model of the F-16 aircraft that can cover the aircraft non-trim region. Having determined functions E 1 ( and E 2 ( and substituted them back into (3.14, we have a quasi-lpv model as [ ] x 1 (t = x 2 (t [ ] ] A11 (x 1 (t + E 1 (x 1 (t A 12 (x 1 (t [ x1 (t + A 21 (x 1 (t + E 2 (x 1 (t A 22 (x 1 (t x 2 (t [ ] B1 (x 1 (t ũ(t (3.18 B 2 (x 1 (t Summary of Three Derivation Methods A summary of the advantage and disadvantage of these three methods, i.e. Jacobian linearization, state transformation, and function substitution, is presented in Table 3.1. It can be seen from Table 3.1 that both state transformation-based and function substitution-based LPV models can represent nonlinear dynamic characteristics of the original nonlinear model better than Jacobian linearization-based LPV model. However, the state transformation approach assumes that there exist equilibrium function for the non-scheduling states, x 2eq (x 1 (t, and the control input, u eq (x 1 (t. Unfortunately, this is not always the case. In addition, the function substitution approach lacks a theoretical method of obtaining a suitable equilibrium point. 26

53 3.2 LPV Models for Controller Synthesis 3.2 LPV Models for Controller Synthesis Having derived a parameter-dependent model from an original nonlinear model, the resulting model can often be grouped into three LPV model types for gain-scheduled control analysis and synthesis purposes. The following subsections outline these three model types that include the grid LPV model [39, 66, 1, 11], the affine LPV model [7, 8, 1] (or polytopic LPV model, and the tensor-product (TP convex polytopic model [15, 16, 18] Grid LPV Model A resulting model, that is derived from an original nonlinear model using an algebraic manipulation method, is often nonlinearly dependent on the time-varying parameters in which an infinite number of LMIs is obtained when an LPV controller is synthesized based on this nonlinearly parameter-dependent LPV model. To obtain a finite number of LMIs, Becker and Packard [23] have introduced a grid LPV model which can be written as a state-space system of the form (2.46 where the system matrix, S : R n R (p+q (p+m, is a function of the scheduled parameters at all grid points over the entire parameter spaces and can be writen as S ( θ(t [ ( ( ] A θ(t B θ(t = C ( θ(t D ( θ(t ( Affine LPV Model An alternative LPV model that yields a finite number of LMIs when synthesizing an LPV controller is an affine LPV model that has been introduced by Apkarian et al. [1]. The affine LPV model can also be written as a state-space system of the form (2.46, but where the system matrix, S ( θ(t, is assumed to depend affinely on the time-varying parameters. That is S ( θ(t = S + θ 1 (ts θ n (ts n (3.2 where θ(t lies in a polytope Θ, θ(t Θ, Θ = [θ 1, θ 1 ] [θ 2, θ 2 ]... [θ n, θ n ], n is the total number of θ(t, and [ ] Ai B S i = i, i =,..., n (3.21 C i D i and S, S 1,..., S n are known fixed matrices. The system matrix, S ( θ(t, can also be written as a convex combination of the matrix vertices (see definition as S ( θ(t } = Co {Ŝ1, Ŝ2,..., Ŝr = α 1 Ŝ 1 + α 2 Ŝ α r Ŝ r r ] [Âj ˆBj = α j Ŝ j, Ŝ j = (3.22 ˆDj j=1 Ĉ j 27

54 3.2 LPV Models for Controller Synthesis where r = 2 n is the total number of vertices, Ŝj are the LTI system matrices at each vertex, α j [, 1], and r j=1 α j = 1. S i, i =,..., n map to Ŝj, j = 1,..., r as Ŝ 1 1 θ 1 θ 2... θ n 1 θ n Ŝ 2 1 θ 1 θ 2... θ n 1 θ n S Ŝ 3 = 1 θ 1 θ 2... θ n 1 θ S 1 n ( S n 1 θ 1 θ 2... θ n 1 θ n Ŝ r Following Pellanda et al. [83, Algorithm 3.1], in order to compute α i, we first compute the normalized co-ordinates α θi = θ i θ i (t θ i θ i, i = 1,..., n (3.24 Then, for each vertex Θ j, j = 1,..., r, the corresponding polytopic co-ordinates are calculated by { n α θi, if θ α j = α θi, α θi = i is a co-ordinate of Θ j ; ( α θi, if θ i is a co-ordinate of Θ j. i= TP Convex Polytopic Model Baranyi [15] has introduced a TP model transformation to transform a given parameterdependent model to a convex polytopic model. The resulting model is called a TP polytopic model that also yields a finite number of LMIs when synthesizing an LPV controller. The TP model transformation is an automatically executable numerical method and has three key steps. The first step is the discretization of the given system matrix over a huge number of points. The discretized points are defined by a dense hyper-rectangular grid of the parameters. The second step extracts the LTI vertex systems from the discretized systems using a higher order singular value decomposition (HOSVD to decompose a given n-dimensional tensor into a full orthonormal system in a special ordering of higher order singular values which express the rank properties of the given parameter-dependent model for each element of the parameter vector in the L 2 -norm. The third step defines the continuous weighting functions to the LTI vertex systems. This sub-section only outlines a brief overview of the TP polytopic model; for further details refer to [15, 16, 18]. Although the TP polytopic model cannot be written in an affine combination because it is not a type of affine LPV model, following [15, 16, 18], the TP polytopic model can also be written in a convex combination of the matrix vertices as ( ẋ(t y(t = R a=1 w a (θ(ts a ( x(t u(t (3.26 where R = I 1 I 2 I n = n I n is the total number of vertices, n is the total number of the parameters vector, I i, i = 1,..., n, is the index upper bounds of the 28

55 3.2 LPV Models for Controller Synthesis weighting functions used in the i-th dimension of the parameter vector, θ(t Θ, and w a (θ(t = w 1,j1 (θ 1 (t w 2,j2 (θ 2 (t w n,jn (θ n (t = w i,j (θ i (t, j = 1,..., I i, i = 1,..., n (3.27 n ( Aa B S a = S i1 i 2 i n = a R (p+q (p+m (3.28 C a w a : R n R is a continuous weighting function, w i,j (θ i (t is the j-th one variable weighting function defined on the i-th dimension of Θ, and θ i (t is the i-th element of the parameter vector θ(t, S a is an LTI vertex systems, a = ordering (i 1 i 2 i n. The goal of the TP model transformation is to determine the LTI vertex systems, S a, and the weighting functions, w i,j (θ i (t, such that the system matrix, S ( θ(t, in (3.19 is given for any grid points over the entire parameter spaces and can be expressed as the combination of the vertex system matrices, S a, and the weighting functions, w a ( θ(t, which are nonlinearly dependent on the scheduled parameters, S( θ(t S ( θ(t D a R ( w a θ(t Sa, θ(t Θ (3.29 a=1 R ( w a θ(t Sa ɛ (3.3 a=1 Here, ɛ symbolizes the approximation error. Furthermore, the convex combination of the LTI vertex systems is ensured by the condition, i [1, n], j [1, I i ], θ i (t : w i,j ( θi (t [, 1] (3.31 i [1, n], θ i (t : I i j=1 w i,j ( θi (t = 1 (3.32 a [1, R], θ(t : ( w a θ(t [, 1] (3.33 θ(t : R ( w a θ(t = 1 (3.34 Hence, S ( θ(t is within the convex hull of the LTI vertex systems S a for θ(t Θ. For convenience, in the following sections, we will henceforth often drop the dependence on t. a= LPV/LFT Models An alternative approach for gain-scheduled control analysis and synthesis is to characterize a nonlinear model as a parameter-dependent linear fractional transformation 29

56 3.2 LPV Models for Controller Synthesis model (LPV/LFT from which the LFT gain-scheduled controller is often synthesized via the scaled small-gain theorem [81] or scaled bounded real lemma [9, 96]. Assume the system (2.46 depends affinely on the scheduled parameters θ (see [81] for other cases. We have A(θ = A + θ 1 A 1 + θ 2 A θ n A n B(θ = B + θ 1 B 1 + θ 2 B θ n B n C(θ = C + θ 1 C 1 + θ 2 C θ n C n D(θ = D + θ 1 D 1 + θ 2 D θ n D n (3.35 Define normalized time-varying parameters θ i [ 1, 1], i = 1,..., n as [83] where θ i = θ i T i S i (3.36 T i = θ i + θ i 2 S i = θ i θ i 2 (3.37 (3.38 Substitute (3.36 into (3.35, we get A(θ = [A + T 1 A T n A n ] + θ 1 (S 1 A θ n (S n A n B(θ = [B + T 1 B T n B n ] + θ 1 (S 1 B θ n (S n B n C(θ = [C + T 1 C T n C n ] + θ 1 (S 1 C θ n (S n C n D(θ = [D + T 1 D T n D n ] + θ 1 (S 1 D θ n (S n D n (3.39 Based on the LFT technique [14], the scheduled parameters θ in (2.46 can be separated from the system state-space model matrices (3.39 as ẋ A B θ1 B θ2... B θn B x z θ1 C θ1 D θθ11 D θθ12... D θθ1n D θ11 w θ1 z θ2 C θ2 D θθ21 D θθ22... D θθ2n D θ12 w θ2 =... z θn C θn D θθn1 D θθn2... D θθnn D θ1n w θn y C D 1θ1 D 1θ2... D 1θn D u w θ1 θ 1 I s1 z θ1 w θ2. = θ2 I s2 z θ2.. w θn θn I sn z θn (3.4 3

57 where w θi, z θi R s i and [ ] Si A i S i B i = S i C i S i D i A = A + B = B + C = C + D = C Stability Analysis of LPV Systems n T i A i i=1 n T i B i i=1 n T i C i i=1 n T i D i i=1 [ Bθi D 1θi] [Cθi D θ1i ] (3.41 Note that, D θθii is introduced in order that (3.4 is in a general state-space equation form. With notation Equation (3.4 can be rewritten as w θ = [ w T θ 1 w T θ 2 w T θ n ] T z θ = [ z T θ 1 z T θ 2 z T θ n ] T B θ = [ B θ1 B θ2 B θn ] C θ = [ ] Cθ T 1 Cθ T 2 Cθ T T n D θθ11 D θθ12... D θθ1n D θθ21 D θθ22... D θθ2n D θθ =. D θθn1 D θθn2... D θθnn D θ1 = [ D T θ1 1 D T θ D T θ1 n ] T D 1θ = [ D 1θ1 D 1θ2... D 1θn ] ẋ = Ax + B θ w θ + Bu z θ = C θ x + D θθ w θ + D θ1 u y = Cx + D 1θ w θ + Du (3.42 w θ = ˆθz θ (3.43 where w θ, z θ R s, s = s 1 +s 2 + +s n, ˆθ = diag( θ 1 I s1, θ 2 I s2,..., θ n I sn, and ˆθ 1. The system (3.43 is called a parameter-dependent linear fractional transformation model (LPV/LFT. 3.3 Stability Analysis of LPV Systems Robust stability analysis is one of the most important issues in control-systems design because it is a useful tool for control engineers to validate and guarantee the 31

58 3.3 Stability Analysis of LPV Systems stability property of the closed-loop system in the presence of perturbations and uncertainty in the parameters of the system plant. The following lemma is required. Lemma [34] Given a symmetric matrix polytope, M(θ(t R p p, for which M(θ(t = m i=1 α im i, where α i is determined using (3.24 and (3.25, is a negative definite symmetric matrix for all possible parameter trajectories, M(θ(t <, θ Θ, if and only if M i <, i = 1,..., m. Proof. Sufficiency: Since α i [, 1] for i = 1,... m and m i=1 α i = 1 then there is always at least one i such that α i >. Thus M i < for all i implies m i=1 α im i < and hence M(θ < for all θ Θ. Necessity: From (3.24 and (3.25, for all j there exists a θ Θ such that there is an α j = 1 and α i = for i = 1,... m, i j. Hence for all j there exists a θ Θ such that M(θ = M j and so it is necessary that M j < for all j Robustness Analysis using SQLF Consider the state-trajectories of system (2.46 with the control input vector u identically zero. ẋ = A(θx (3.44 Definition (Quadratic Stability [5, 19, 23, 26, 96] The system (3.44 is said to be quadratically stable if there exists a quadratic Lyapunov function V (x = x T P x whose derivative is negative, d/dt ( V (x <, along all state trajectories. Note that, d/dt ( V (x = x T [A T (θp + P A(θ]x. The above definition is equivalent to the following proposition. Proposition The system (3.44 is quadratically stable whenever there exists a positive definite symmetric matrix P R p p such that the following LMI conditions hold P > (3.45 A(θ T P + P A(θ <, θ Θ (3.46 Obviously, an inequality (3.46 yields an infinite number of LMIs. However, in practice, a finite number of LMIs can be obtained by gridding the entire parameter space with non-dense set of grid points (a grid LPV model approach [39, 66, 1, 11]. An alternative approach [33] is to transform the system (3.44, that is nonlinearly dependent on θ, into a TP convex polytopic model using the TP model transformation [15, 16, 18]. Hence, A(θ can be written as a convex combination of the matrix vertices in a similar manner to S(θ in (3.29 as A(θ = w 1 (θa 1 + w 2 (θa w R (θa R (

59 3.3 Stability Analysis of LPV Systems where w i (θ [, 1] and R i=1 w i(θ = 1. Substituting (3.47 into (3.46, we get w 1 (θ[a T 1 P + P A 1 ] + w 2 (θ[a T 2 P + P A 2 ] + + w R (θ[a T RP + P A R ] <, θ Θ (3.48 By Lemma 3.3.1, solving the above inequality for a positive definite symmetric matrix P need only be done at all vertices. Hence, we get the following proposition. Proposition Assume the system (3.44 is a TP convex polytopic system, then the system (3.44 is quadratically stable whenever there exists a positive definite symmetric matrix P such that the following LMI conditions hold P > (3.49 A T a P + P A a <, a = 1, 2,..., R (3.5 In addition, suppose the system (3.44 depends affinely on the scheduled parameters θ, A(θ can be written as a convex combination of the matrix vertices in a similar manner to S(θ in (3.22 as A(θ = α 1  1 + α 2  α r  r (3.51 Then, Proposition is also applicable to the affine LPV systems by replacing A a, a = 1, 2,..., R with Âj, j = 1, 2,..., r in ( Robustness Analysis using Small Gain Theorem Assume the system (3.44 depends affinely on θ. Moreover, having separated θ from A(θ, the system (3.44 can be written in a similar manner to (3.43 as ẋ = Ax + B θ w θ z θ = C θ x + D θ w θ w θ = ˆθz θ (3.52 where w θ, z θ ˆθ 1. R s, s = s 1 + s s n, ˆθ = diag( θ1 I s1, θ 2 I s2,..., θ n I sn, and Consider the system (3.52, define a transfer function matrix M(s = D θ + C θ ( si A 1 Bθ C s s, by the Nyquist and small-gain theorem [14], the system (3.52 is quadratically stable if and only if I M(sˆθ(s and I ˆθ(sM(s are nonsingular. This is equivalent to the following theorem. Theorem (Small Gain Theorem, [14, Theorem 9.1] Suppose ˆθ(s and M(s are stable and let γ >. Then the interconnected system (3.52 shown in Figure 3.1 is well-posed and internally stable for all ˆθ(s C s s with ˆθ 1/γ if and only if M < γ 33

60 3.3 Stability Analysis of LPV Systems θ θ Figure 3.1: Block diagrams structure of an LPV/LFT model (3.52 Note that ˆθ is a structured uncertainty. By Theorem 3.3.5, the necessary condition to make the system (3.52 quadratically stable is M γ that is equivalent to the induced L 2 -norm (or L 2 -gain of the operator mapping the disturbance signal w θ into the error signal z θ of the system (3.52 is bounded by γ (i.e. z θ 2 w θ 2 where, for this case, γ must be < 1. Based on single quadratic Lyapunov functions, M γ if and only if there exists P = P T such that P >, d ( x T P x + zθ T z θ γ 2 wθ T w θ <, θ Θ (3.53 dt Inequality (3.53 leads to the well-known bounded real lemma[1] inequality A T P + P A P B θ Cθ T Bθ T P γi DT θ < (3.54 C θ D θ γi The robust stability requirement is that γ < 1. However there generally exist an infinite number of the factor matrices pairs (B θ, C θ in which only some factor matrices pair give γ < 1. Instead of searching for such a factor matrix pair manually, by introducing a scaling matrix L 1/2, we can select any factor matrix pair for which γ will alway be < 1 if the system (3.52 is quadratically stable and the factor matrix pair can be determined using singular value decomposition, see sub-section L 1/2 denotes the unique positive definite square root of L L θ. The set of L θ is defined as { L θ = L > : Lˆθ = ˆθL, } θ Θ R s s (3.55 Therefore, (3.52 can be modified further to ẋ = Ax + B θ L 1 2 ẃθ ź θ = L 1 2 Cθ x + L 1 2 Dθ L 1 2 ẃθ ẃ θ = ˆθź θ (3.56 where ź θ = L 1/2 z θ and w θ = L 1/2 ẃ θ. With parameters in (3.56, (3.54 becomes a scaled bounded real lemma [9], L 1/2 M(sL 1/2 < γ. A T P + P A P B θ Cθ T Bθ T P γl DT θ < (3.57 C θ D θ γl 1 34

61 3.4 Controller Synthesis for LPV Systems Rearranging (3.57 using the Schur complement (Lemma , we get the following proposition. Proposition The system (3.52 with ˆθ < 1/γ is quadratically stable along all possible parameter trajectories, θ Θ, if and only if the following LMI condition hold for some positive definite symmetric and scaling matrices (P, L: ( A T P + P A + Cθ T LC θ P B θ + Cθ T LD θ Bθ T P + DT θ LC θ γ 2 L + Dθ T LD < (3.58 θ Note that the minimization of γ can be achieved heuristically or by a simple grid search. 3.4 Controller Synthesis for LPV Systems In the previous section, a sufficient condition to guarantee the stability property of the LPV closed-loop system has been presented in which the analysis conditions can be represented in the form of a finite number of LMIs. Next, we consider the problem of designing a gain-scheduled output feedback H control with guaranteed L 2 -gain performance for a class of affine LPV systems for which the proposed techniques in the previous section can be directly extended to synthesizing a gain-scheduled H controller. In addition, the proposed techniques in this section are also applicable to the TP Convex Polytopic LPV model and can be further modified for the grid LPV models. The material in this section is derived directly from [1, 8, 44] and [43]. Consider a given affine LPV plant model with state-space realization ẋ = A(θx + B 1 (θw + B 2 u z = C 1 (θx + D 11 (θw + D 12 u y = C 2 x + D 21 w (3.59 where x R p is the state vector, w R m 1 is the generalized disturbance vector, u R m 2 is the control input vector, z R q 1 is the controlled variable or error vector, y R q 2 is the measurement output vector, θ Θ, and continuous mapping matrix functions A : R n R p p, B 1 : R n R p m 1, C 1 : R n R q1 p and D 11 : R n R q 1 m 1. The assumptions on the plant are as follows [1]: (i D 22 =, (ii (B 2, C 2, D 12, D 21 are parameter-independent (constant matrices, and (iii the pairs (A(θ, B 2 and (A(θ, C 2, are quadratically stabilizable and quadratically detectable over Θ respectively. If assumption (ii is not satisfied, the computation for a problem solution requires solving an infinite number of LMI constraints, and is therefore not easily tractable [23]. However, the constant matrices restrictions can be overcome by pre-filtering of the control inputs u and/or post-filtering the measured outputs y; for further details refer to [1]. A loop-shifting argument suffices to overcome the 35

62 3.4 Controller Synthesis for LPV Systems D 22 = restriction, see [47, pages 44-45]. In addition, the quadratic stabilizability of (A(θ, B 2 over Θ in the assumption (iii means the existence of a matrix X > such that N T [A T (θx + XA(θ]N <, θ Θ (3.6 where N denote the null space of B2 T [1]. Note that the A(, B 1 (, C 1 ( and D 11 ( matrices can also be written as a convex combination of the matrix vertices in a similar manner to (3.51: A(θ B 1 (θ B 2 C 1 (θ D 11 (θ D 12 C 2 D 21 = r i=1 α i  i ˆB1i B 2 Ĉ 1i ˆD11i D 12 C 2 D 21 (3.61 Theorem (Bounded real lemma [1, 44, 88] Given an LTI system G(s and a state-space realization G(s = D + C ( si A 1 B. The following statements are equivalent: (i A is stable and G(s < γ (ii The existence of a positive definite symmetric matrix P such that A T P + P A P B C T B T P γi D T C D γi < The bounded real lemma can be extended to LPV systems in conjunction with the notion of quadratic H performance [1]. Definition (Quadratic H performance [1, 23] An LPV system of the form (2.46 has quadratic H performance γ if and only if the existence of a positive definite symmetric matrix P such that A(θ T P + P A(θ P B(θ C T (θ B T (θp γi D T (θ < C(θ D(θ γi for all admissible parameter trajectories. Then, the system (2.46 is quadratically stable and ensures the induced L 2 -norm of the operator mapping the disturbance signal w into the controlled signal z is bounded by γ Gain-Scheduled Controller Design using SQLF The gain-scheduled output feedback H control problem using single quadratic Lyapunov functions is to compute a dynamic affine LPV controller, K(θ, with state-space equations ẋ k = A k (θx k + B k (θy u = C k (θx k + D k (θy (

63 3.4 Controller Synthesis for LPV Systems which stabilizes the closed-loop system, (3.59 and (3.62, and minimizes the closedloop quadratic H performance (Definition t1 t1 z T zdt γ 2 w T wdt, t 1 (3.63 along all possible parameter trajectories, θ Θ. Note that A and A k have the same dimensions, since we restrict ourselves to the full-order case. With the notation ( Ak (θ B K(θ = k (θ r = α C k (θ D k (θ i K i (3.64 i=1 ( Aki B K i = ki, i = 1, 2,..., r (3.65 C ki D ki where r is the total number of vertices and α i is determined using (3.24 and (3.25. The closed-loop system, (3.59 and (3.62, is described by the state-space equations [ ] [ ] ẋ x = A ẋ cl (θ + B k x cl (θw k [ ] x z = C cl (θ + D cl (θw (3.66 x k where [ ] A(θ r A cl (θ = + BK(θC = α i  cli p p i=1 ] [Âi  cli = + BK i C p p [ ] B1 (θ r B cl (θ = + BK(θD 21 = α i ˆBcli i=1 [ ] ˆB1i ˆB cli = + BK i D 21 C cl (θ = [ C 1 (θ ] r + D 12 K(θC = α i Ĉ cli Ĉ cli = [ Ĉ 1i ] + D 12 K i C D cl (θ = D 11 (θ + D 12 K(θD 21 = i=1 r i=1 α i ˆDcli ˆD cli = ˆD 11i + D 12 K i D 21 (3.67 and [ ] [ ] B2 Ip B =, C = I p C 2 D 12 = [ [ ] ] D 12, D21 = D 21 (

64 3.4 Controller Synthesis for LPV Systems Based on the single quadratic Lyapunov functions V (x = x T P x, there is an LPV controller K(θ of the form of (3.62 that stabilizes the closed-loop system, (3.59 and (3.62, and ensures the induced L 2 -norm of the operator mapping the disturbance signal w into the controlled signal z is bounded by γ along all possible parameter trajectories if and only if there exists P = P T such that [45] P >, d ( x T P x + z T z γ 2 w T w <, θ Θ (3.69 dt Inequality (3.69 leads to the well-known bounded real lemma [1] inequality A T cl (θp + P A cl(θ P B cl (θ Ccl T (θ Bcl T (θp γi DT cl (θ < (3.7 C cl (θ D cl (θ γi Substituting (3.67 in (3.7, we get r  T cl i P + P Âcl i P ˆB cli Ĉcl T i α i ˆB cl T i P γi ˆDT cli < (3.71 i=1 Ĉ cli ˆDcli γi Inequality (3.71 can be also rewritten as (see [44] r α i (Ψ cli + Q T Ki T P cl + PclK T i Q < (3.72 i=1 where ] T ] [ ] [Âi [Âi ˆB1i P + P P p p p p [Ĉ1i ] T [ ] Ψ cli = T ˆB1i P γi ˆDT 11 i [Ĉ1i ] ˆD11i γi Q = [ ] C, D 21, (p+q2 q 1 P cl = [ ] B T P, (p+m2 m 1, D12 T (3.73 (3.74 (3.75 Having determined the quadratic Lyapunov variable P R 2p 2p, the system matrix vertices K i of the LPV controller K(θ for each vertex Θ i, i = 1,..., r, can be determined from (3.72 that is an LMI in K i. By Lemma 3.3.1, the LMIs (3.72 need only be evaluated at all vertices. Alternatively, a more efficient explicit scheme for determining K i is given in [43]. Knowing K i, the controller system matrices A k (θ,..., D k (θ can be computed on-line in real-time using (3.64 with an instantaneous measurement value of θ. To determine the quadratic Lyapunov variable P, we have to define a structure of P. Although the exact structure of P is still not certain, a typical structure of P is suggested in [1, 23, 43, 66, 96] and [11] for which, in this thesis, the structure of P is taken from [66, 96] and [11] as [ P = X ( X Y 1 ( X Y 1 X Y 1 ] [ ] Y Y, P 1 = Y ( X Y 1 1 XY 38 (3.76

65 3.4 Controller Synthesis for LPV Systems where a pair of positive definite symmetric matrices (X, Y R p p, X Y 1, and rank(x Y 1 p [82]. By Lemma , LMIs (3.72 are solvable for K i if and only if there exist a pair of positive definite symmetric matrices (X, Y satisfying the following LMIs: r i=1 r i=1 α i ( T NX I α i ( T NY I  T i X + XÂi X ˆB 1i Ĉ T 1 i ˆB T 1 i X γi ˆDT 11i Ĉ 1i ˆD11i γi  i Y + Y ÂT i Y ĈT 1 i ˆB1i Ĉ 1i Y γi ˆD11i ˆB T 1 i ˆDT 11i γi ( NX I ( NY I ( X I I Y < (3.77 < (3.78 > (3.79 where N X and N Y denote bases of the null spaces of [C 2, D 21 ] and [B T 2, D T 12], respectively. Note that, (3.79 ensures X, Y > and X Y 1. By Lemma 3.3.1, (3.77 (3.79 need only be evaluated at all vertices. Hence we get the following theorem. Theorem (Convex solvability conditions [1] There exists an LPV controller K(θ guaranteeing the closed-loop system, (3.59 and (3.62, quadratic H performance γ along all possible parameter trajectories, θ Θ, if and only if the following LMI conditions hold for some positive definite symmetric matrices (X, Y, which further satisfy Rank(X Y 1 p: ( NX I ( NY I where i = 1, 2,..., r T T  T i X + XÂi X ˆB 1i Ĉ T 1 i ˆB T 1 i X γi ˆDT 11i Ĉ 1i ˆD11i γi  i Y + Y ÂT i Y ĈT 1 i ˆB1i Ĉ 1i Y γi ˆD11i ˆB T 1 i ˆDT 11i γi ( NX I ( NY I ( X I I Y < (3.8 < (3.81 > (3.82 Note that, when the parameters θ are time-invariant or slowly varying, the conservatism of the above theorem can be reduced using parameter dependent Lyapunov functions in which the improved theorem is presented in section Gain-Scheduled Controller Design via LFT Consider a given affine LPV plant model of the form (3.59 for which, based on the LFT technique [14], the scheduled parameters θ can be separated from the system 39

66 3.4 Controller Synthesis for LPV Systems matrices in a similar manner to (3.43 as ẋ = Ax + B θ w θ + B 1 w + B 2 u z θ = C θ x + D θθ w θ + D θ1 w + D θ2 u z = C 1 x + D 1θ w θ + D 11 w + D 12 u y = C 2 x + D 2θ w θ + D 21 w w θ = ˆθz θ (3.83 where w θ, z θ R s, s = s 1 + s s n, ˆθ = diag( θ1 I s1, θ 2 I s2,..., θ n I sn, and ˆθ 1. Note that, unlike the gain-scheduled controller design using SQLF case, the method in this sub-section is not applicable to a problem where the vectors w and z are not the same dimension, i.e. m 1 q 1 [9, 24]. Consistently with (3.83, we seek an LPV controller such that (i the closed-loop system is internally stable for all parameter trajectories, θ Θ, and γ 2ˆθT ˆθ 1 and (ii the induced L 2 -norm of the operator mapping the disturbance signal into the controlled signal is bounded by γ [9]. Note that, the controller is defined to have the same dependency on θ as the plant because it can use the available information of θ to adjust its dynamic to the current plant dynamic on-line in real-time. This LPV controller can be written as a state-space system of the form: [ ] [u ẋ k ] [ A k ] [ Bk1 B kθ ] [y x k ] = Ck1 Dk11 D k1θ ũ ỹ C kθ D kθ1 D kθθ ỹ = ˆθũ (3.84 Note that A and A k have the same dimensions. Actually, the above controller is given in a lower LFT with respect to ˆθ, F l (K, ˆθ, in which θ plays the role of scheduling variable and gives the rule for updating the controller state space matrices based on the measurements of θ. Equivalently, (3.84 can be further written as [9] where ẋ k = A k (θx k + B k (θy u = C k (θx k + D k (θ (3.85 A k (θ = A k + B kθ Λ θ C kθ B k (θ = B k1 + B kθ Λ θ D kθ1 C k (θ = C k1 + D k1θ Λ θ C kθ D k (θ = D k11 + D k1θ Λ θ D kθ1 Λ θ = ˆθ(I D kθθ ˆθ 1 and assume the inverse (I D kθθ ˆθ 1 exists θ Θ with ˆθ 1/γ; for further details refer to [9]. To apply the small gain theorem for this problem, all parameterdependent components ˆθ, that enter both the plant (3.83 and the controller (3.84, 4

67 3.4 Controller Synthesis for LPV Systems have to transfer into a single uncertainty block, shown in Figure 3.2. Introducing the augmented plant [ ] [ ẋ A Bθ B 1 B2 ] x z θ I w θ z θ [ z = C θ D θθ D θ1 D θ2 w θ ] [ C 1 ] [ D 1θ D 11 ] D [ 12 ] w [ ] y C2 D2θ D 21 u ỹ I ũ [ ] ] ] wθ [ˆθ [ zθ = w θ ˆθ (3.86 z θ The closed-loop system, (3.86 and (3.84 shown in Figure 3.2, is described by the state-space equations x cl A cl B θcl L 1 2 B 1cl x cl z Θ = L 1 2 C θcl L 1 2 D θθcl L 1 2 L 1 2 D θ1cl w Θ z C 1cl D 1θcl L 1 2 D 11cl w where [ x x cl = x k w θ = ˆΘz θ (3.87 ], z Θ = [ zθ z θ ], w Θ = [ wθ w θ ] ] [ˆθ, ˆΘ = ˆθ, L L Θ L is a scaling matrix. The set of L Θ is defined as { [ ] } L1 L L Θ = L = 2 L T > : L 2 L ˆΘ = ˆΘL, θ Θ R 2s 2s ( and [ ] [ ] A [ A k ] [ Bk1 B kθ ] A cl = + BKC, K = Ck1 Dk11 D k1θ p p C kθ D kθ1 D kθθ [ ] [ ] Bθ B1 B θcl = + BKD θ21, B 1cl = + BKD 121 [ ] C θcl = + D C θ θ12 KC, C 1cl = [ C 1 ] + D 112 KC [ ] [ ] D θθcl = + D D θ12 KD θ21, D θ1cl = + D θθ D θ12 KD 121 θ1 D 1θcl = [ D 1θ ] + D112 KD θ21, D 11cl = D 11 + D 112 KD 121 (3.89 with I p C = C 2, D θ21 = D 2θ, D 121 = D 21, I [ ] [ ] B2 I B =, D I p θ12 =, D D θ2 112 = [ D 12 ] 41

68 3.4 Controller Synthesis for LPV Systems Based on the single quadratic Lyapunov functions V (x = x T P x, there is a controller K, (3.84, that stabilizes the closed-loop system, (3.86 and (3.84, and ensures the induced L 2 -norm of the operator mapping the disturbance signal into the controlled signal is bounded by γ along all possible parameter trajectories if and only if there exists P = P T such that [45] P >, d ( x T P x + (z T dt Θz Θ + z T z γ 2 (wθw T Θ + w T w <, θ Θ (3.9 Inequality (3.9 leads to the scaled bounded real lemma [9] inequality A T cl P + P A cl P B θcl P B 1cl Cθ T cl C1 T cl Bθ T cl P γl Dθθ T cl D T 1θ cl B1 T cl P γi Dθ1 T cl D11 T cl C θcl D θθcl D θ1cl γl 1 < (3.91 C 1cl D 1θcl D 11cl γi Inequality (3.91 can be also rewritten as (see [44] Ψ cl + Q T K T P cl + P T clkq < (3.92 where [ ] T [ ] [ ] [ ] [ ] T A A Bθ B1 [C1 P + P P P ] T p p p p C θ [ ] T [ ] T Bθ [ ] T P γl D1θ D [ ] θθ Ψ cl = T [ ] T B1 P γi D11 D T [ ] [ ] [ ] θ1 γl C θ D θθ D 1 [ θ1 C1 ] [ ] D 1θ D 11 γi (3.93 Q = [ ] C, D δ21, D 121, (p+q2 (q 1 +s (3.94 P cl = [ ] B T P, (p+m2 (m 1 +s, Dδ T 12, D1 T 12 (3.95 Having determined the quadratic Lyapunov variable P R 2p 2p and the scaling matrix L L Θ, the LPV controller K can be determined from (3.92 that is an LMI in K. Knowing A k, B k1,..., D kθθ, the controller system matrix A k (θ,..., D k (θ can be computed on-line in real-time using (3.85 with an instantaneous measurement value of θ. To determine the quadratic Lyapunov variable P and the scaling matrix L, we have to define a structure of P and L. The structure of P is defined as (3.76. Although the exact structure of L is still not certain, a typical structure of L is suggested in [9, 24] and [81] for which, in this thesis, the structure of L follows the structure of P which is taken from [66, 96] and [11] as [ L3 J3 1 ( L L = 3 J3 1 ( L 3 J3 1 L 3 ], L 1 = 42 [( L3 J 1 3 1L3 ] J 3 J 3 J 3 J 3 (3.96

69 3.4 Controller Synthesis for LPV Systems (a LPV/LFT control structure T T (b Augmented structure Figure 3.2: Block diagrams structures of LPV/LFT closed-loop systems 43

70 3.4 Controller Synthesis for LPV Systems Obviously, based on (3.96, L 3, J 3 L θ is a necessary condition to make L L Θ. In addition, L 3 J3 1, and Rank(L 3 J3 1 s [82]. By Lemma , the following theorem is obtained. Theorem There exists an LPV controller K guaranteeing the closed-loop system, (3.86 and (3.84, quadratic H performance γ along all possible parameter trajectories, θ Θ with ˆθ 1/γ, if and only if the following LMI conditions hold for some positive definite symmetric matrices (X, Y R p p and ( L 3, J 3 L θ, which further satisfy Rank(X Y 1 1 p and Rank( L 3 J 3 s. ( NX I ( NY I T T A T X + XA XB θ XB 1 C T θ C T 1 B T θ X L 3 D T θθ D T 1θ B T 1 X γi D T θ1 D T 11 C θ D θθ D θ1 J 3 C 1 D 1θ D 11 γi AY + Y A T Y C T θ Y C T 1 B θ B 1 C θ Y J 3 D θθ D θ1 C 1 Y γi D 1θ D 11 B T θ D T θθ D T 1θ L 3 B T 1 D T θ1 D T 11 γi ( NX I ( NY I < (3.97 < (3.98 ( X I > (3.99 I Y ( L3 γ > (3.1 γ J3 where L 3 and J 3 are equal to γl 3 and γj 3 respectively, N X and N Y denote bases of the null spaces of [C 2, D 2θ, D 21, ] and [B2 T, Dθ2 T, DT 12, ] respectively. Note that, (3.99 and (3.1 ensure X, Y >, X Y 1 and L 3, J 3 >, L 3 J3 1, respectively. 44

71 Chapter 4 Numerical Example This chapter aims to illustrate the implementation of LPV systems theory for the nonlinear control problem which include methods for deriving an LPV model from a nonlinear model, analysis and synthesis probelms of gain-scheduled output feedback H controller design. To demonstrate the method, we explicitly consider the example of Leith and Leithead [61]. In this example it was shown that for an LPV model derived from the Jacobians, a common approach, with an LPV controller synthesized using the method of Apkarian et al. [1], is unstable when applied to the original nonlinear plant [3, 61]. Consider the nonlinear plant example taken from [61] ẋ 1 (t = x 1 (t + r(t ẋ 2 (t = x 1 (t x 2 (t x 2 (t 1 y(t = x 2 (t (4.1 where t R is time, both x 1 (t, x 2 (t R are the state, r(t, y(t R are the control input and the measurement output, respectively. The control requirement is to design an output-feedback controller which ensures a step response settling time of less than 2 seconds with zero steady-state error [61]. Step inputs with different amplitudes are applied to the system (4.1, at t= where an initial condition of the system is x 1 ( = and x 2 ( = 3.16, in order to investigate the open-loop dynamic step response, shown in Figure 4.1. It can be seen that the system (4.1 is an open-loop stable system and its step response is similar to a first-order transfer function with a varying time constant (time lag and a varying low frequency gain, following its output y(t. Moreover, a set of equilibrium points of the system can be calculated by setting ẋ 1 = ẋ 2 = in (4.1. Then, the results are x 1trim = r trim = { x 2 2 trim + 1, if x 2trim ; 1 x 2 2 trim, if x 2trim <. y trim = x 2trim (4.2 45

72 4.1 Jacobian Approach 4.1 Jacobian Approach Applying the Jacobian linearization method to (4.1 by using (3.7, we get [ ] A = ( x 1 (t+r(t x 1 (t (x 1 (t x 2 (t x 2 (t 1 x 1 (t [ ] 1 = 1 2 x 2trim [ B = [ ] 1 = C = D = ( x 1 (t+r(t r(t (x 1 (t x 2 (t x 2 (t 1 r(t [ x2 (t x 1 (t = [ 1 ] [ x2 (t r(t ] ] ] x 2 (t x 2 (t x 1 (t=x 1trim x 2 (t=x 2trim r(t=r trim x 1 (t=x 1trim x 2 (t=x 2trim r(t=r trim ( x 1 (t+r(t x 2 (t (x 1 (t x 2 (t x 2 (t 1 x 2 (t x 1 (t=x 1trim x 2 (t=x 2trim r(t=r trim x 1 (t=x 1trim x 2 (t=x 2trim r(t=r trim = [ ] (4.3 where (x 1trim, x 2trim, r trim, and y trim is one point in a set of equilibrium points (4.2, therefore a Jacobian-based LTI model can be written [ ] [ ] [ ] [ ] δẋ1 (t 1 δx1 (t 1 = + δr(t δẋ 2 (t 1 2 x 2trim δx 2 (t δy(t = [ 1 ] [ ] δx 1 (t (4.4 δx 2 (t where δx 1 (t = x 1 (t x 1trim, δx 2 (t = x 2 (t x 2trim, δr(t = r(t r trim, and δy(t = y(t y trim. Note that (4.2 shows that x 1trim, x 2trim and r trim are dependent on y trim. With x 2trim fixed, (4.4 is an LTI model. However, as x 2trim varies slowly over the defined parameter space, (4.4 becomes an LPV model. That is ] [ ] [ ] [ ] [ṅ1 1 n1 1 = + r ṅ 2 1 2θ n 2 y = [ 1 ] [ ] n 1 (4.5 n 2 where n 1 (t = x 1 (t x 1trim, n 2 (t = x 2 (t x 2trim, and θ = x 2trim = y(t is arbitrarily defined from to 1. As a result of the Jacobian linearization method, the dynamic characteristics of the system (4.5 vary, following its output, x 2trim = y trim shown in (4.2, if the original plant (4.1 is operating about the neighborhood of a set of equilibrium points (4.2. However, x 2trim is not equal to y trim whenever the original plant is operating in a region faraway from its equilibrium points. 46

73 4.1 Jacobian Approach r(t steps from to 2.5 r(t steps from to 5. r(t steps from to 7.5 r(t steps from to 1. Y (units Time (s Figure 4.1: Open-loop dynamic step response of the system (4.1 at t= Remark An LPV controller, that is synthesized based on the Jacobian-based LPV model (4.5, adjusts its dynamics to the current plant dynamics using instantaneous measurement values of y(t. But, whenever the original plant is not about an equilibrium condition, the true current dynamic of the original plant does not follow the value of y(t. This means that the LPV controller will adjust its dynamic to the wrong plant dynamic. This is a usual problem for Jacobian linearization method. Having determined a Jacobian-based LPV model, a Jacobian-based LPV controller can be synthesized using the method of Apkarian et al. [1] with the criterion W 1S W 2 KS < 1 (4.6 as shown in Figure 4.2, where the performance weighting functions W 1 and robustness weighting functions W 2 taken from [61] are.5 W 1 (s = s +.2 W 2 (s =.2s s + 1 (4.7 Using MATLAB routines ltisys, we get W 1 and W 2 in the state-space equation of 47

74 4.1 Jacobian Approach W 1 z 1 W 2 z 2 w θ + - e K(θ r G(θ y w r P(θ LPV plant G(θ + y - W 1 W 2 z 1 z 2 e θ K(θ LPV controller Figure 4.2: LPV system: H mixed S/KS synthesis problem the form ẋ w1 (t =.2x w1 (t +.5e(t z 1 (t = x w1 (t ẋ w2 (t = 1x w2 (t + 4r(t z 2 (t = 5x w2 (t +.2r(t (4.8 From (4.5 and (4.8, in addition, e(t = w(t y(t but y(t = n 2 (t, hence e(t = w(t n 2 (t. Finally, we get the minimal realization of the augmented plant P (θ as ṅ 1 (t = n 1 (t + r(t ṅ 2 (t = n 1 (t 2θn 2 (t ẋ w1 (t =.2x w1 (t +.5w(t.5n 2 (t ẋ w2 (t = 1x w2 (t + 4r(t z 1 (t = x w1 (t z 2 (t = 5x w2 (t +.2r(t Equation (4.9 can be rewritten further as e(t = w(t n 2 (t (4.9 48

75 4.1 Jacobian Approach ṅ ṅ 2 1 2θ n 1 ẋ w n 2 [ ẋ w2 ] = [ ] 1 [ 4 x w1 ] [ ] z1 1 x w2 [ ] [ z 2 ] [ 5 ].2 [ w ] r e 1 1 (4.1 Note that, the augmented plant P (θ (4.1 can also be determined using MATLAB routines sysic, iconnect or sconnect. Solving LMIs in Theorem by using a MATLAB Robust Control Toolbox function [12], mincx, we get an LPV controller with a quadratic H performance γ =.1211 as shown below: where 1 y α 1 =, α 2 = y 1 1 ẋ k = (α 1 A k1 + α 2 A k2 x k + B k (w y r = C k x k ( e e e e + 3 A k1 = e e e e e e e e e e e e e e e e + 3 A k2 = 4.246e e e e e e e e e e e e + 3 B k = e e e e + C k = [ e e e e + 3 ] The matrices are different from those presented in [61]. Also note that the LPV controller presented in [61] is actually open-loop unstable. The controller presented above is open-loop stable for all y [, 1]. In addition, the synthesizing scheme from Theorem has been implemented in a MATLAB Robust Control Toolbox function as hinfgs [12]. To confirm that the mixed-sensitivity criterion (4.6 is achieved, the singular values of the transfer matrices S and KS are computed over θ Θ (with frozen θ, and are shown in Figure 4.3. Obviously, the singular values of S and KS are shaped and bound by W 1 and W 2 respectively. In addition, Figure 4.4 shows that [W1 S, W 2 KS] T < γ. Hence, the mixed-sensitivity criterion (4.6 is satisfied. But, the simulation results that are presented in Figure 4.5 still show the closed-loop instability problem described in [61]. It can be seen that the closed-loop system is 49

76 4.2 State Transformation Approach stable when the LPV controller is applied to the LPV model for a step response that is a change in demand from -3 units to units. However, when the same LPV controller is applied to the original nonlinear plant, the nonlinear closed-loop system appears to be unstable. In order to investigate this closed-loop instability with more information, different LPV controller synthesis methods or different deriving LPV model techniques should be also employed. Unfortunately, the Theorem can not be applied because of q 1 m 1. Moreover, in order to derive a TP convex polytopic model of the system (4.1, the TP model transformation toolbox [17] has been used to apply a TP transformation to (4.4 by selecting x 2trim as a scheduling parameter. It turns out that the resulting TP polytopic model is identical to the Jacobian-based LPV model in ( State Transformation Approach According to (4.2, we can define r eq (x 2 = x 1eq (x 2 = x 2 x The equation (4.1 can be rewritten in the form of equation (3.8 as ] [ ] [ ] [ ] [ ] [ẋ1 1 x1 1 = + r + ẋ 2 1 x 2 x 2 1 y = [ 1 ] [ ] x 1 (4.12 x 2 Since r eq (x 2 and x 1eq (x 2 is a continuously differentiable function, such that for every x 2, (4.12 is in equilibrium points ẋ =. We get [ ] [ ] [ ] [ ] [ ] 1 x1eq(x = r 1 x 2 x 2 eq (x y eq (x 2 = [ 1 ] [ ] x 1eq (x 2 (4.13 x 2 Subtracting (4.13 from (4.12 obtains ẋ 1 = x 1 + x 1eq (x 2 + r r eq (x 2 ẋ 2 = x 1 x 1eq (x 2 y y eq (x 2 = (4.14 Then, y = y eq (x 2 = x 2. Moreover, differentiating x 1eq (x 2 = x 2 x 2 +1 with respect to t gives d dt x 1 eq (x 2 = ẋ 1eq (x 2 = ( x 2 x d x 2 dt x 2 = 2 x 2 [x 1 x 1eq (x 2 ] (4.15 Subtracting (4.15 from (4.14 obtains ẋ 1 ẋ 1eq (x 2 = [ 1 2 x 2 ][x 1 x 1eq (x 2 ] + [r r eq (x 2 ] (4.16 5

77 4.2 State Transformation Approach 4 2 Singular Values (S sensitivity (/W1 performance bound Frequency (rad/sec (a σ(s 25 2 Singular Values (KS control sensitivity (/W2 robustness bound Frequency (rad/sec (b σ(ks Figure 4.3: Singular value of S and KS over θ Θ (with frozen θ 51

78 4.2 State Transformation Approach (a W 1 S (b W 2 KS Figure 4.4: H norm of W 1 S and W 2 KS over θ Θ (with frozen θ 52

79 4.2 State Transformation Approach 1.5 Response with Jacobian-based LPV plant Response with original nonlinear plant Time (s Figure 4.5: Nonlinear step response from -3 to of the Jacobian-based LPV controller (4.11 Hence, (4.14 can be rewritten as a state-space equation of the form ] [ẋ1 ẋ 1eq(x 2 = ẋ 2 [ 1 2 x2 1 y = [ 1 ] [ x 1 x 1eq (x 2 x 2 ] [ ] x1 x 1eq(x 2 + x 2 ] [ ] 1 (r r eq (x 2 (4.17 Subsequently, (4.17 can be rewritten as an LPV model of the form ] [ ] [ ] [ṅ1 1 2 n2 n1 = + ṅ 2 1 n 2 y = [ 1 ] [ ] n 1 n 2 [ ] 1 u (4.18 where n 1 = x 1 x 1eq (x 2 = x 1 x 2 x 2 1, n 2 = x 2, and u = r r eq (x 2 = r x 2 x 2 1. The state transformation-based LPV plant presented above is identical to that presented in [93]. Applying hinfgs with the same weighting function, the state transformation-based LPV controller for this LPV model is obtained with γ = Figure 4.6 shows the simulation results of the state transformationbased LPV controller. The closed-loop instability does not occur. 53

80 4.3 Function Substitution Approach.5 State transformation-based LPV controller responses with nonlinear plant Function substitution-based LPV controller responses with nonlinear plant Time (s Figure 4.6: Nonlinear step response from to of function substitution-based and state transformation-based LPV controllers 4.3 Function Substitution Approach Substitute x 1 = δx 1 + x 1r, x 2 = δx 2 + x 2r, r = δr + r r, and y = δy + y r in equation (4.1, we get ẋ 1 = d dt δx 1 + d dt x 1r = δx 1 + = δx 1 + δr + [ x 1r + r r ] ẋ 2 = d dt δx 2 + d dt x 2r = δx 2 + = δx 1 + [x 1r δx 2 + x 2r (δx 2 + x 2r 1] y = δy + y r = δx 2 + x 2r (4.19 where (x 1r, x 2r, r r, and y r is one trim point in a set of the equilibrium points. Selecting a trim point as (x 1r = 1, x 2r =, r r = 1, and y r =, equation (4.19 can be rearranged as an state-space equation of the form [ ] [ ] [ ] δẋ1 1 δx1 = + δẋ 2 1 δx 2 δx 2 δy = [ 1 ] [ ] δx 1 δx 2 [ ] 1 δr (4.2 54

81 4.3 Function Substitution Approach Furthermore, (4.2 can be rewritten in an LPV model of the form ] [ ] [ ] [ ] [ṅ1 1 n1 1 = + u ṅ 2 1 n 2 n 2 y = [ 1 ] [ ] n 1 n 2 (4.21 where n 1 = δx 1, n 2 = δx 2, and u = r 1. The function substitution-based LPV plant presented above is identical to that presented in [93]. Using hinfgs with the same weighting function, the function substitution-based LPV controller for this LPV model is obtained with γ = Figure 4.6 shows the simulation results of the state transformation-based LPV controller. The closed-loop instability does not occur. According to Figures 4.5 and 4.6, in this particular example, we make two assumptions that (i function substitution and state transformation methods give an LPV plant model that more accurately represents the nonlinear plant than the Jacobian linearization method, and (ii there is a mismatch uncertainty between the Jacobian-based LPV model and the original nonlinear model. However, both state transformation-based and function substitution-based LPV models are identical to the original nonlinear model as shown below. First, we show that the state transformation-based LPV model is identical to the original nonlinear model. By substituting r(t = u(t + x 2 (t x 2 (t + 1 x 1 (t = n 1 (t + x 2 (t x 2 (t + 1 x 2 (t = n 2 (t (4.22 in (4.1. A new nonlinear equation can be obtained in the form. ṅ 1 (t = (1 + 2 n 2 (t n 1 (t + u(t ṅ 2 (t = n 1 (t y(t = n 2 (t (4.23 which can be rearranged as an LPV equation of the form (4.18. Next, we show that the function substitution-based LPV model is identical to the original nonlinear model. By substituting r(t = u(t + 1 x 1 (t = n 1 (t + 1 x 2 (t = n 2 (t (4.24 in (4.1. Another nonlinear equation can be obtained in the form. ṅ 1 (t = n 1 (t + u(t ṅ 2 (t = n 1 (t n 2 (t n 2 (t y(t = n 2 (t (4.25 which can be rearranged as an LPV equation of the form (

82 4.4 Mismatch Uncertainty 4.4 Mismatch Uncertainty Consider the transfer function of LPV plant models (2.46 over θ Θ (with frozen θ that is given by [1] G(s, θ = D(θ + C(θ[sI A(θ] 1 B(θ (4.26 Substituting the matrices A(,..., D( of the Jacobian-based LPV model (4.5 into (4.26, the transfer function can be determined as G(s, θ = 1 (s + 1(s + 2θ (4.27 where, according to (4.5, θ = n 2 varying from to 1. This transfer function has two poles; one pole fixes at -1, the other pole varies from -2 to. The location of the varying pole of the Jacobian-based LPV model is equal to 2θ, but the true location of the varying pole of the original nonlinear plant is not equal to 2θ whenever this nonlinear plant is not in an equilibrium condition. As y(t = n 2 (t moves closer to, the mismatch uncertainty between the Jacobian-based LPV model and the original nonlinear model becomes more significant and makes the nonlinear closed-loop system unstable as shown below. Consider the state-trajectories of the closed-loop system of the Jacobian-based LPV controller and LPV model, taken from [3] with disturbance w identically zero, is ẋ = (α 1  1 + α 2  2 x (4.28 where α 1 = 1 θ, α 1 2 = θ and  1 =  2 = As a result of Proposition 3.3.4, this closed-loop system is quadratically stable. Note that, the MATLAB Robust Control Toolbox [12] also provides a function quadstab 56

83 4.5 Pole Placement Approach to test the quadratic stability for a class of affine LPV systems. However, when including the time-varying real parametric uncertianty to both Â1 and Â2 of (4.28 in a region close to the right-half s-plane G(s, θ = Equation (4.28 becomes where ẋ = 1, δ [ 1, 1] (4.29 (s + 1(s + 2θ +.4δ ( α 1 (Â1 + δa δ + α 2 (Â2 + δa δ x (4.3 A δ =.4 As a result of quadstab, the closed-loop system (4.3 is quadratically unstable. (4.31 Having determined the reasons of the closed-loop instability for the LPV controller with the original nonlinear plant, the problem can be solved by simply increasing the conservativeness of the LPV plant model. That is, by setting a new range of the time-varying parameter to cover the uncertainty in the region close to the right-half s-plane. For example, setting θ = n 2 to vary from -1 to 1, indicates that the varying pole can vary from -2 to 2 even though, in fact, it can only vary from -2 to. Using hinfgs with the same weighting function as previously but with the new range of θ, the new LPV controller is obtained with γ = The simulation results of the nonlinear closed-loop system with the new Jacobian-based LPV controller are presented in Figure 4.7. The closed-loop instability disappears but the transient performance is degraded because of setting a more conservative range of θ. 4.5 Pole Placement Approach Pole placement with state feedback can be used to overcome the closed-loop instability problem without degrading the transient performance. In this approach, we restrict ourselves to special LPV plants of the form ẋ = A(θx + Bu A(θ = A + θ 1 A θ n A n y = Cx (4.32 where A(θ is known functions and depends affinely on time-varying parameters, θ. Furthermore, in order to apply state feedback and state observer, this LPV 57

84 4.5 Pole Placement Approach Figure 4.7: Nonlinear step response from to of new Jacobian-based LPV controller with the original nonlinear plant plant is assumed to be state controllable and observable for all possible parameters trajectories θ. Figure 4.8 shows a block diagram structure of the gain-scheduled pole placement controller. We apply state feedback u = K(θx + n. The state feedback gain, K(θ, is parameter-dependent and can be a nonlinear function of θ. Substituting u in (4.32, the state feedback closed-loop system becomes ẋ = Ã(θx + Bn Ã(θ = [A(θ BK(θ] y = Cx (4.33 where n is the new input of the state feedback closed-loop system. By determining the state feedback gain, it is possible to achieve any closed-loop eigenvalue assignment. However for the example, the states cannot be measured. Hence, a state observer is used to estimate state values. A general state observer can be constructed using observer feedback gain K e (θ which is parameter-dependent and can be a nonlinear function of θ. The state observer closed-loop system is given by Subtracting (4.34 from (4.32, we obtain ẋ e = [A(θ K e (θc]x e + Bn + K e (θcx (4.34 ẋ ẋ e = [A(θ K e (θc](x x e (4.35 To demonstrate the method, we consider the Jacobian-based LPV model (4.5 taken 58

85 4.5 Pole Placement Approach e e e e Figure 4.8: Block diagram structure of the gain-scheduled pole placement controller from Leith and Leithead [61]. We also select the state feedback closed-loop system to have a realistic closed loop characteristic, i.e. natural frequency, ω n = 1 rad/s and damping ratio, ξ =.77, in order to prevent actuator saturation. Hence, the desired characteristic equation can be written as λ 2 + (2.77 1λ = (4.36 The state feedback gain K(θ can be determined by solving [ ] λ λ {[ ] 1 1 2θ [ ] 1 [k1 ] } k 2 = λ 2 + (2θ + k 1 + 1λ + (2θ[1 + k 1 ] + k 2 = (4.37 Equating coefficients of the polynomial yields the state feedback gain as k 1 = θ, k 2 = 4θ θ + 1 (4.38 Having determined the state feedback gain K(θ, the observer feedback gain K e (θ can be determined by solving [ ] λ λ {[ ] 1 1 2θ [ ke1 k e2 ] [ 1 ] } = λ 2 + (2θ + k e2 + 1λ + (2θ + k e1 + k e2 = (4.39 The dynamics of the state observer must be faster than the system being controlled. Therefore we select the state observer to have a suitable characteristic, i.e. ω n = 4 59

86 4.5 Pole Placement Approach rad/s and ξ =.77, in order to avoid amplifying the noise of the controlled output. Then, the desired characteristic equation can be written as λ 2 + (2.77 4λ = (4.4 Equating the coefficients of the polynomial yields the observer feedback gain as k e1 = , k e2 = θ (4.41 Substituting the matrices A(, C( and K e (θ of the state observer in (4.35 yields a state-space form as ] [ṅ1 ṅ e1 = ṅ 2 ṅ e2 [ ] [ ] n1 n e1 n 2 n e2 (4.42 The state observer has two poles at 28.3 ± 28.3i. Since both poles are in the left-half s-plane, the state observer is stable. Having applied the state observer and the state feedback to the Jacobian-based LPV model, the state feedback closed-loop system can be determined by substituting the matrices A(,..., C( and K(θ in (4.33 yielding the state-space form as ] [ ] [ ] [ṅ1 2θ θ = θ 1 n1 + ṅ 2 1 2θ n 2 y = [ 1 ] [ ] n 1 n 2 [ ] 1 n (4.43 where n is the new input of state feedback closed-loop system. Substituting the matrices A(,..., C( of the state feedback closed-loop system into (4.26 gives the transfer function 1 G(s = (4.44 s s + 1 The state feedback closed-loop system has two constant poles at 7.7±7.7i. Both poles are in the left-half s-plane therefore the state feedback closed-loop system is stable. According to the state feedback closed-loop transfer function, this approach shows that the parameters variation of the special LPV plants can be cancelled. In order to obtain performance from the system, an additional linear time invariant (LTI controller can be applied as an outer loop as shown in Figure 4.9. For this particular example, an H -mixed-sensitivity controller is used. Using MATLAB Robust Control Toolbox function [12], mixsyn, with the same weighting function as previously, the following H controller is obtained with γ =.761 for the state feedback closed-loop system as ẋ k = A k x k + B k (y ref y n = C k x k (4.45 6

87 4.6 Conclusion Reference Yref x' = Ax+Bu y = Cx+Du Y feedback n LTI Controller (mixsyn u = - K(thetax + n K(thetax U Nonlinear plant Y Y U U K(zetax X Xe Y Gain -scheduling state feedback Gain -scheduling observer Figure 4.9: Pole placement controller design in MATLAB Simulink environment where.2 A k = e B k = 1.296e 16, C k = [ ] e 17 The simulation results of the nonlinear closed-loop system are presented in Figures 4.1 and The closed-loop instability problem is solved with good transient performance of the system. 4.6 Conclusion The example from Leith and Leithead [61] is very interesting. The closed-loop instability of the LPV controller with the original nonlinear model occurs because the mismatch uncertainty between the Jacobian-based LPV model and the original nonlinear model is in a region close to the right-half s-plane. In addition, for this particular example, both function substitution and state transformation methods give an LPV plant model that more accurately represents the nonlinear plant than the Jacobian linearization method. In this chapter, a design method for cancelling the parameters variation of the example of Leith and Leithead [61] by pole-placement state feedback is proposed. For the example, the approach yields reliable closed-loop stability and good closed-loop transient performance of the system because it makes the nonlinear plant appear to be an LTI plant, hence well-developed LTI tools can be applied. 61

88 4.6 Conclusion State transformation-based LPV controller Function substitution-based LPV controller Gain-scheduled pole placement controller Figure 4.1: Nonlinear step response from to with the original nonlinear plant 3 25 State transformation-based LPV controller Function substitution-based LPV controller Gain-scheduled pole placement controller Figure 4.11: Control input to the original nonlinear plant 62

89 Chapter 5 Longitudinal LPV Autopilot Design: A TP Approach This chapter describes a design of a longitudinal autopilot, i.e. speed-hold and altitude-hold, for the entire flight envelope of a Jindivik UAV using a linear parametervarying (LPV technique that is based on a gain-scheduled output feedback H control [1] for which an LPV model is required for control synthesis and analysis. Typically, LPV models are derived from original nonlinear equation models using an algebraic manipulation method (see section 3.1. However, the derived LPV models from those methods are often nonlinearly dependent on the time-varying parameters; this is especially the case for a longitudinal Jacobian-based LPV model that is nonlinearly dependent on speed and altitude. A grid LPV model is usually used to synthesize a controller in the case of nonlinear parameter dependence, however, the result of heuristic gridding technique is unreliable and the analysis result is dependent on choosing the gridding points [96]. To synthesize an LPV autopilot with a finite number of LMIs and avoid the gridding technique, the TP model transformation is employed in order to transform a longitudinal nonlinearly parameter-dependent LPV model into a TP convex polytopic model form. Therefore, based on single quadratic Lyapunov functions (Theorem 3.4.3, the LMIs need only be evaluated at all vertices as shown in (3.8 ( Jacobian-Based Longitudinal LPV Model An LPV model is required for gain-scheduled H performance analysis and synthesis. Chapters 3 and 4 show that the state transformation-based LPV model is more accurately represents the original nonlinear dynamics than the Jacobian-based LPV model that only accurately represents the nonlinear dynamics about the neighbouhood of a set of equilibrium points. However, the state transformation approach requires sufficient date in order to derive a transformation-based LPV model from a nonlinear model. In addition, the transformation-based LPV model is a complex 63

90 5.1 Jacobian-Based Longitudinal LPV Model structure model. Therefore, in this work, the Jacobian method is employed to derive a longitudinal LPV model from the standard 6-DOF equations of motion because it is applicable to a general class of nonlinear models and has simple structure model. Consider the standard 6-DOF equations of motion for a conventional fixed wing aircraft, (A.1 and (A.11, where detail of the aircraft equations of motion is briefly summarized in section A.2. Although the exact form of aerodynamic coefficients structures for a general fixed wing aircraft are not certain, typical linear model structures are suggested in [55, 58] and [77]. The linear model structures, that are used in this thesis, from [55] are ( q c C X = C X + C Xα α + C Xq + C Xδe δ e (5.1 2V ( pb ( rb C Y = C Y + C Yβ β + C Yp + C Yr + C Yδa δ a + C Yδr δ r (5.2 ( 2V 2V q c C Z = C Z + C Zα α + C Zq + C Zδe δ e (5.3 2V ( pb ( rb C l = C l + C lβ β + C lp + C lr + C lδa δ a + C lδr δ r (5.4 2V ( 2V q c C m = C m + C mα α + C mq + C mδe δ e (5.5 2V ( pb ( rb C n = C n + C nβ β + C np + C nr + C nδa δ a + C nδr δ r (5.6 2V 2V Equations (A.1 (A.11 are 6-DOF dynamics in three-dimensional space, i.e. North- East-Down axis, which can often be simplified into two motions of 3-DOF dynamics in two-dimensional space, i.e. longitudinal and lateral motions. In this chapter, we only consider the longitudinal motion. In addition, we assume an aircraft is about a wings level and constant altitude and airspeed flight condition, and, assume it can manoeuvre only in North-Down plane, hence, all of the lateral states in (A.1 (A.11 frozen and equal to zero, i.e. v = p = r = β = φ = ψ = y E =. Moreover, after substituting α = tan ( 1 w u, V = u2 + w 2, and q = 1ρV 2, we get equations 2 of longitudinal motion as u = ρs [ C X + C Xα tan ( 1 w ] u 2 + ρs [ C X + C Xα tan ( 1 w ] w 2 2m u 2m u [ ( ρs c + 4m C X q u2 + w 2 ] w q g sin θ + 2m( ρs u 2 + w 2 C δ Xδe e + T (5.7 m ẇ = ρs [ C Z + C Zα tan ( 1 w ] u 2 + ρs [ C Z + C Zα tan ( 1 w ] w 2 2m u 2m u [ ( ρs c + 4m C Z q u2 + w 2 ] + u q + g cos θ + ρs ( u 2 + w 2 C Zδe 2m δ e (5.8 q = ρs c [ C m + C mα tan ( 1 w ] u 2 + ρs c [ C m + C mα tan ( 1 w ] w 2 2I y u 2I y u + ρs c ( u2 + w 4I 2 C mq q + ρs c ( u 2 + w 2 C mδe δ e (5.9 y 2I y θ = q (5.1 ḣ = (sin θu (cos θw (

91 5.1 Jacobian-Based Longitudinal LPV Model In practice, the inertias I x, I y, and I z of an UAV can be determined using a torsional pendulum experiment that is presented in [6, 36, 54] and [13] where the inertia I xz is often neglected. In addition, the engine thrust T can be modelled and estimated using the technique that is presented in [54] and [55]. However, in this work, the thrust is modelled from data in Table 5.1, where the data are obtained by using MATLAB function, trim, to trim the Jindivik nonlinear model [41] about a wings level and constant altitude and airspeed flight condition. Based on ad hoc methods, a simple model of thrust, that fits the data in Table 5.1, is T = C T +C Tu u+c Tuh u h+c Trpm rpm+c Turpm u rpm+c Thrpm h rpm+c Tuhrpm u h rpm (5.12 where all thrust coefficients in the above equation are given in Table 5.2. Figure 5.1 shows a comparison of the simulated thrust, from Table 5.1, and the estimated thrust, from (5.12. Note that, the mathematical modelling of aerodynamic forces and moments, thrust, sensors, and actuators of the Jindivik nonlinear model [41] are given in section A.3. After (A.1 (A.11 were linearized about a wings level and constant altitude and airspeed flight condition using Jacobian linearization method, we get a longitudinal LTI model as a state-space system of the form, u X u X w X q X θ ũ X δe X δrpm ẇ Z u Z w Z q Z θ w Z δe [ ] q M u M w M q q M δe δe (5.13 δ rpm θ ḣ = 1 h u h w h θ θ + h where ũ = u u trim, w = w w trim, q = q, θ = θ θ trim, h = h h trim, δ e = δ e δ etrim, δ rpm = rpm rpm trim, and X u = ρs ( CX + C Xα α trim + C Xδe m e trim utrim ρs 2m C X α w trim + 1 ( CTu + C Tuh m trim + C Turpmrpm trim + C h Tuhrpm trimrpm trim (5.14 X w = ρs ( CX + C Xα α trim + C Xδe m e trim wtrim + ρs 2m C X α u trim (5.15 X q = w trim + ρs c 4m C X q u 2 trim + w2 trim (5.16 X θ = g cos θ trim (5.17 X δe = ρs 2m C ( X δe u 2 trim + wtrim 2 (5.18 X δrpm = 1 ( CTrpm + C Turpmu trim + C m Thrpm trim + C u Tuhrpm trimh trim (5.19 Z u = ρs ( CZ + C Zα α trim + C Zδe m e trim utrim ρs 2m C Z α w trim (5.2 Z w = ρs ( CZ + C Zα α trim + C Zδe m e trim wtrim + ρs 2m C Z α u trim (5.21 Z q = u trim + ρs c 4m C Z q u 2 trim + w2 trim (5.22 Z θ = g sin θ trim (

92 5.1 Jacobian-Based Longitudinal LPV Model Table 5.1: Simulation data of Jindivik s engine * Altitude (ft Speed (ft/s RPM Thrust (lb f 2, 337 8, , 443 9, , 548 9, , 654 1, , , , 337 8, , 432 9, , 527 9, , 622 1, , , , 337 8, , 421 9, , 56 9, , 59 1, , 675 1, , 337 9, , 411 9, , 485 9, , 559 1, , 632 1, , 337 9, , 4 9, , 464 9, , 542 9, , 59 1, * about wings level and constant altitude and airspeed flight condition Z δe = ρs 2m C ( Z δe u 2 trim + wtrim 2 (5.24 M u = ρs c ( Cm + C mα α trim + C mδe Iy e trim utrim ρs c 2Iy C m α w trim (5.25 M w = ρs c Iy ( Cm + C mα α trim + C mδe δ e trim wtrim + ρs c 2Iy C m α u trim (5.26 M q = ρs c2 4Iy C m q u 2 trim + w2 trim (5.27 M δe = ρs c 2Iy C ( m δe u 2 trim + wtrim 2 (5.28 h u = sin θ trim (5.29 h w = cos θ trim (5.3 h θ = u trim cos θ trim + w trim sin θ trim (5.31 Equations (5.14 (5.31 are the stability and control derivatives (longitudinal 66

93 5.1 Jacobian-Based Longitudinal LPV Model 1 9 Simulated Thrust Estimated Thrust Figure 5.1: A comparison between the simulated thrust and the estimated thrust Table 5.2: Jindivik s engine thrust coefficients Coefficient C T C Tu C Trpm Value Coefficient C Tuh C Turpm C Thrpm C Tuhrpm Value mode. The trim values in the above equations can be calculated by setting u = ẇ = q = θ = ḣ = and ρ = ρ [ 1 ( h ] in (5.7 (5.11. Moreover, since α is actually small for this flight condition, we assume tan α α and V u. Then, the results are α trim = ( [ S 2 u2 C Zδe C m C Z C mδe (ρ 1 ( h ] mgc mδe ( ( [ S 2 u2 C Zα C mδe C Zδe C mα ρ 1 ( h ] (5.32 w trim = u tan α trim (5.33 θ trim = α trim (5.34 (C m + C mα α trim δ etrim = (5.35 C mδe ( ( [ mgα trim S 2 u2 C X + C Xα α trim + C Xδe δ etrim ρ 1 ( h ] rpm trim = ( CTrpm + C Turpmu + C h + C Thrpm T ( uh uhrpm C T + C Tu u + C Tuh uh ( CTrpm + C Turpmu + C h + C (5.36 Thrpm T uhrpm uh From (5.14 (5.36, it can be seen that, the stability and control derivatives are 67

94 5.1 Jacobian-Based Longitudinal LPV Model Table 5.3: Stability and control derivative data (longitudinal mode of Jindivik nonlinear model * u (ft/s h (ft 9,9 1,5 11,1 9,9 1,5 11,1 9,9 1,5 11,1 α trim (rad δ etrim (rad rpm trim 9,445 9,46 9,467 9,63 9,631 9,632 9,818 9,813 9,88 X u X w X q X θ X δe X δrpm Z u Z w Z q Z θ Z δe M u M w M q M δe h u h w h θ * about wings level and constant altitude and airspeed flight condition Table 5.4: Jindivik s longitudinal aerodynamic coefficients Coefficient C X C Xα C Xq C Xδe Value Coefficient C Z C Zα C Zq C Zδe Value Coefficient C m C mα C mq C mδe Value nonlinearly dependent on only speed and altitude. With the speed and altitude are fixed, (5.13 is a longitudinal LTI model. However, as the speed and altitude vary slowly over the entire flight envelope, (5.13 becomes a longitudinal nonlinearly parameter-dependent LPV model. Moreover, these equations show that the accuracy of this LPV model depends on the accuracy of the information that provides the aerodynamic and thrust coefficients. Traditionally, the aerodynamic coefficients are often determined using wind tunnel tests by measuring the aerodynamic forces and moments introduced on the aircraft. However, the wind tunnel tests are expensive in terms of schedule and budget for UAV applications. System identification techniques are an alternative approach that can be used to estimate stability and control derivatives or aerodynamic coefficients from flight data, where the details of the method are presented in [28, 36, 53, 54, 57, 58, 74, 75] and [78]. 68

95 5.2 Longitudinal TP Convex Polytopic Model In this chapter, we use the MATLAB function, linmod, to emulate aircraft parameter identification techniques. Note that Appendix B presents the details of UAV aerodynamic model identification from a racetrack manoeuvre. Using functions trim and linmod, the stability and control derivative values, shown in Table 5.3, are obtained about one flight condition (speed = 497 ft/s and altitude = 1,5 ft. After substituting the data from Table 5.3 in (5.14 (5.36, we can determine approximate aerodynamic coefficients of the Jindivik UAV as shown in Table 5.4 where the exact aerodynamic coefficients of the Jindivik nonlinear model [41] are given in sub-section A.3.1. Knowing the aerodynamic coefficients, the system matrices of the longitudinal nonlinearly parameter-dependent LPV model (5.13 at all points over the entire parameter spaces can be determined using (5.14 (5.36. Figure 5.2 shows the determined value of X q and α trim for a calculation example. In addition, Figure 5.3 shows the variation of open-loop characteristic of this nonlinearly parameter-dependent LPV model, i.e. u(s/rpm(s, over an entire flight envelope. According to Figure 5.3, two poles of short period mode are open-loop stable with variation of the damping ratio (ξ and natural frequency (ω n from.256 to.319 and 2.42 rad/s to 7.19 rad/s respectively, where the other two poles of Phugoid mode are open-loop unstable with variation of the damping ratio and natural frequency from -.19 to and.449 rad/s to.129 rad/s respectively. Moreover, the system of (5.13 also has non-minimum phase zeros as shown in Figure Longitudinal TP Convex Polytopic Model Based on the system of (5.13, the speed and altitude are the only time-varying parameters. The entire flight envelope of the Jindivik UAV, taken from [41], is that the speed and altitude vary from ft/s to ft/s and 1, ft to 18, ft respectively. To synthesize an LPV controller for the system of (5.13 with a finite number of LMIs, the gridding technique that is presented in [39, 66, 99, 1] and [11] can be used. However, the result of heuristic gridding technique is unreliable and the analysis result is dependent on choosing the gridding points [96]. A TP model transformation is an alternative approach that can be used to obtain a finite number of LMIs for which the method transforms a given nonlinearly parameter-dependent LPV model (5.13 into a TP convex polytopic model. We applied the MATLAB Tensor Product Model Transformation Toolbox from [17] to determine the LTI vertex systems, S a, and the weighting functions, w a ( p(t, as shown in (3.26. The transformation space is defined as Ω = [337.6, 759.5] [1, 18] and let the density of the sampling grid be 5 1. In addition, the weighting type of cno convex hull is used during the transformation in order to have a tight hull representation. A tensor of size was received with the singular values in speed dimension as: 39198, , , 49.12, 1.486,.32623,.5345, and e-5. and in altitude dimension as: 39191, , 13.23,.4544,.18278, and 1.439e-5. 69

96 5.2 Longitudinal TP Convex Polytopic Model The variation of Xq with speed and altitude Xq (ft/s x Altitude (ft Speed (ft/s 8 (a X q The variation of angle of attack with speed and altitude at trim condition.1 Angle of attack at trim condition (rad Altitude (ft Speed (ft/s 7 8 (b α trim Figure 5.2: X q and α trim are nonlinearly dependent on speed and altitude 7

97 5.2 Longitudinal TP Convex Polytopic Model Pole-Zero Map Short Period Mode Real Axis (a Overview.15 Pole-Zero Map.1.5 One Non-minimum Phase Zero Phugoid Mode Real Axis (b Zoom in Figure 5.3: The open-loop characteristic of transfer function longitudinal nonlinearly parameter-dependent LPV model u(s rpm(s of the Jindivik 71

98 5.3 Gain-Scheduled H Autopilot Design This means that the longitudinal nonlinearly parameter-dependent LPV model of the Jindivik UAV can exactly be given as a convex combination of 8 6 = 48 LTI vertex systems. However, in practice, a small number of controllers is preferred for implementation in real applications, therefore, we kept only the four and three largest singular values in speed and altitude dimension respectively. The number of LTI vertex systems was reduced to 5 4 = 2. Theoretically, the maximum error in L 2 matrix norm approximation is the sum of the discarded small singular values, thus, e e 5 = However, we have compared the decomposed TP polytopic model with the original nonlinearly parameter-dependent LPV model, (5.13, over 2, test points of randomly selected parameter values, i.e. speed and altitude, in the ranges given by Ω. The maximum and mean error in the L 2 matrix norm, ɛ, was received as.3594 and respectively. Thus, the decomposed TP polytopic model can be reduced to a system of half the complexity while it is still accurate enough for real world experiments. Hence, the longitudinal TP polytopic model can be written as 5 4 ( ( ( ẋ(t = w 1,i u(t w2,j h(t A i,j x(t + B i,j u(t (5.37 i=1 j=1 where the weighting functions w n,j ( pn (t are presented in Figure 5.4. Moreover, Figure 5.5 shows w 1 ( p(t and w2 ( p(t as an example for determining wa ( p(t. Some of the LTI system matrices, S a, of this TP polytopic model are shown below A 1,1 = A 2,1 = A 3,1 = A 5,4 = B 1,1 = B 2,1 = B 3,1 = B 5,4 = (5.38 (5.39 (5.4 ( Gain-Scheduled H Autopilot Design In practice, the plant model is normally augmented with some weighting functions before we can apply the H control synthesis to compute an LPV controller. In 72

99 5.3 Gain-Scheduled H Autopilot Design (a w 1,i ( u(t (b w 2,j ( h(t Figure 5.4: The cno type convex weighting functions in one-dimensional parameter, i.e. w n,j ( pn (t, of the longitudinal TP polytopic model (a w 1 ( p(t (b w 2 ( p(t Figure 5.5: The cno type convex weighting functions in two-dimensional parameters, i.e. w a ( p(t, of the longitudinal TP polytopic model 73

100 5.3 Gain-Scheduled H Autopilot Design this chapter, we used the mixed-sensitivity criterion [2, 33, 47] W 1S W 2 KS < 1 (5.42 The objective of this mixed-sensitivity function is to shape the sensitivity function S and control sensitivity function KS with performance weighting functions W 1 and robustness weighting functions W 2 respectively. Hence, we should get a controller that is good at command following (i.e. small tracking error, good at disturbance attenuation (i.e. attenuation of the effect of disturbance on output, low sensitivity to measurement noise, with reasonably small control efforts, and that is robustly stable to additive plant perturbations. Figure 5.6 shows the weighted open-loop interconnection for synthesis where (.5s+.664 W 1 (s = s (5.43 W 2 (s = W pre-filter (s =.5s+.664 s ( 1s s s s ( 5 s+5 1 s+1 (5.44 (5.45 The purpose of W pre-filter is to make matrices B 2 and D 12 of the plant model to be parameter-independent [1], hence, the gain-scheduled output feedback H controller design method of [1] can be used. In addition, the values of weighting functions W 1 and W 2 are hand-tuned until the desired objectives of performance and robustness of the closed-loop system are achieved. Having augmented the longitudinal TP convex polytopic model, taken from (5.38 (5.41, with weighting functions W 1 and W 2, an LPV controller can be synthesized using the routine hinfgs. As a result of hinfgs, the LPV controller with γ = was obtained. Once the twenty LTI system matrix vertices of the LPV controller are obtained from the routine hinfgs, this LPV controller can be constructed ( by the combination of the system matrix vertices and weighting functions, w a p(t, in the same fashion as the TP convex polytopic model, hence, x k (t = [ ] δe (t = rpm(t 5 i=1 5 i=1 u ref u(t 4 ( ( w 1,i u(t w2,j h(t A w(t k i,j x k (t + B ki,j q(t θ(t h ref h(t u ref u(t 4 ( ( w 1,i u(t w2,j h(t C w(t k i,j x k (t + D ki,j q(t θ(t (5.46 h ref h(t j=1 j=1 To confirm that the mixed-sensitivity criterion (5.42 is achieved, the singular values 74

101 5.4 Nonlinear Simulation Results 1 2 u h δe rpm ref e ref Figure 5.6: The weighted open-loop interconnection for the longitudinal TP convex polytopic plant model of the transfer matrices S and KS are computed over all θ Θ (with frozen values of θ, and are shown in Figure 5.7. Obviously, the singular values of S and KS are shaped and bound by W 1 and W 2 respectively. In addition, Figure 5.8 shows that [W1 S, W 2 KS] T < γ. Hence, the mixed-sensitivity criterion (5.42 is satisfied. 5.4 Nonlinear Simulation Results The designed H gain-scheduling autopilot is validated with the Jindivik nonlinear model [41] in( a MATLAB Simulink simulation. Note that all twenty weighting functions, w a p(t, used in the simulation were constructed using two-dimensional look-up tables. In Figure 5.9, the transient response of the simulated vehicle for small demanded changes in speed and altitude are shown for one particalar point in the flight envelope. Similar responses for other points in the flight envelope were obtained. Figure 5.1 shows a simulated flight that cover a wide range of the flight envelope. It demonstrates that the stability and robustness properties of the closedloop system were achieved over the defined flight envelope. Note that, there is an effect on regulating an altitude when the autopilot is tracking a speed demand as well as there is an effect on regulating a speed when the autopilot is tracking an altitude demand. This is because the longitudinal TP polytopic model is a quasi-lpv model where the scheduling parameters, speed and altitude, are also states of the system. This is a common problem for quasi-lpv models when synthesizing an LPV controller using single quadratic Lyapunov function since the parameter variation rate is as fast as the system states. 75

102 5.4 Nonlinear Simulation Results 2 Singular Values -2 (S sensitivity (/W1 performance bound u h Frequency (rad/sec (a σ(s 1 5 rpm Singular Values (KS control sensitivity (/W2 robustness bound e Frequency (rad/sec (b σ(ks Figure 5.7: Singular value of S and KS over θ Θ (with frozen θ 76

103 5.4 Nonlinear Simulation Results (a W 1 S (b W 2 KS Figure 5.8: H norm of W 1 S and W 2 KS over θ Θ (with frozen θ 77

104 5.4 Nonlinear Simulation Results 1.15 x 14 Altitude demand Altitude response (a Altitude (ft 52 Speed demand Speed response (b Speed (ft/s 78

105 5.4 Nonlinear Simulation Results (c Pitch angle (rad 1 x (d Pitch rate (rad/s 79

106 5.4 Nonlinear Simulation Results 4 x (e Angle of Attack (rad (f Elevator deflection (rad 8

107 5.4 Nonlinear Simulation Results (g Engine speed (RPM Figure 5.9: The transient performance of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 56 ft/s and altitude = 1, ft 81

108 5.4 Nonlinear Simulation Results 2 x 14 Altitude demand 1.8 Altitude response (a Altitude (ft 8 75 Speed demand Speed response (b Speed (ft/s 82

109 5.4 Nonlinear Simulation Results (c Pitch angle (rad 6 x (d Pitch rate (rad/s 83

110 5.4 Nonlinear Simulation Results (e Angle of Attack (rad (f Elevator deflection (rad 84

111 5.5 Conclusion 1.2 x (g Engine speed (RPM Figure 5.1: The stability and robustness properties of the closed-loop system were achieved over the defined flight envelope 5.5 Conclusion A recently proposed technique, the tensor-product (TP model transformation [18], is applied to generate a convex polytopic representation of a longitudinal nonlinearly parameter-dependent LPV model of the Jindivik UAV. The gain-scheduled output feedback H controller design method [1] was applied to the resulting TP convex polytopic model to yield a controller that guarantees the stability, robustness and performance properties of the closed-loop system over the whole grid. The method is relatively easy to apply owing to the availabilty of good computational tools [17] and [46]. The controller was tested with a full 6-DOF simulation of the vehicle. These results show that the stability and robustness properties of the closed-loop system were achieved over the defined flight envelope. 85

112 This page intentionally contains only this sentence.

113 Chapter 6 Longitudinal LPV Autopilot Design: A PDLF Approach In general, a single quadratic Lyapunov function is more conservative than a parameterdependent Lyapunov function when the parameters are time-invariant or slowly varying [45]. Hence, using the parameter-dependent Lyapunov function can reduced the conservatism of the designed LPV controller in the previous chapter and can also solve a problem of a quasi-lpv model (that the scheduling parameter is dependent on the system state where both scheduling parameter and system state variations are at the same speed. However, to synthesize a gain-scheduled output feedback H controller for a class of affine (or polytopic LPV plant model using the parameter-dependent Lyapunov function involves solving an infinite number of LMIs for which a number of convexifying techniques exist for obtaining a finite number of LMIs. In this chapter, an alternative approach for obtaining a finite number of LMIs is proposed, by simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality form is obtained. Hence, the LMIs need only be evaluated at all vertices. A technique to construct the intermediate controller variables as an affine matrixvalued function in the polytopic coordinates of the scheduled parameter is also proposed. Computational results on a numerical example [61] using the proposed approach are compared with those from a multi-convexity approach [11] in order to demonstrate the impacts of the proposed method in the parameter-dependent Lyapunov-based stability and performance analysis. The proposed method is applied to synthesize a longitudinal autopilot for a full flight envelope of the Jindivik UAV. The resulting autopilot is tested for a bounded flight envelope with a full 6-DOF Jindivik nonlinear model and the simulation results are presented to show the effectiveness of the approach. 87

114 6.1 Stability Analysis using PDLF 6.1 Stability Analysis using PDLF The system of (3.44 is said to be parameter-dependent stable if there exists a continuously differentiable parameter-dependent Lyapunov function V (x, θ = x T P (θx whose derivative is negative along all state trajectories. This is equivalent to the existence of a P (θ = P (θ T such that P (θ >, A(θ T P (θ + P (θa(θ + P (θ <, (θ, θ Θ Φ (6.1 where the rate of variation θ is well defined at all times and satisfies θ i [ v i, v i ] and θ lies in a polytope Φ, θ Φ, Φ = [v 1, v 1 ] [v 2, v 2 ]... [v n, v n ], n is the total number of θ(t. Although an exact parameter-dependent function for a continuously differentiable parameter-dependent Lyapunov variable P (θ is still not established, a basis parameter-dependent function for the parameter-dependent Lyapunov variable is suggested in [8, 1] and [11] and is to copy the plant s parameter-dependent function. Therefore, we can constrain the basis parameter-dependent function for the parameter-dependent Lyapunov variable to vary in an affine fashion. P (θ = P + θ 1 P θ n P n = α 1 ˆP1 + α 2 ˆP2 + + α r ˆPr (6.2 where r = 2 n, α i is determined using (3.24 and (3.25 and ˆP 1 1 θ 1 θ 2... θ n 1 θ n ˆP 2 1 θ 1 θ 2... θ n 1 θ n P ˆP 3 = 1 θ 1 θ 2... θ n 1 θ P 1 n... P n ˆP r 1 θ 1 θ 2... θ n 1 θ n (6.3 Although the set of feasible solutions of inequality (6.1 is reduced by defining the parameter-dependent Lyapunov function, P (θ, as affine (as in (6.2, this is a practical approach and makes inequality (6.1 is tractable. This is a common. Differentiating (6.2 with respect to time gives P (θ = θ 1 P θ n P n = β 1 P1 + β 2 P2 + + β r Pr (6.4 where β i can be determined in a similar manner to α i using (3.24 and (3.25 and P 1 v 1 v 2... v n 1 v n P 2 v 1 v 2... v n 1 v n P P 3 = v 1 v 2... v n 1 v P 1 n... P P r v 1 v 2... v n 1 v n n (6.5 88

115 6.1 Stability Analysis using PDLF Substituting (3.51, (6.2 and (6.4 into (6.1, and recalling that r i=1 α i = 1 and r i=1 β i = 1, we get r 1 +2 P (θ >, r i=1 j=i+1 k=1 r i=1 r αi 2 β k (ÂT i ˆPi + ˆP i  i + P k k=1 ( r 1 α i α j β k 2 (ÂT i ˆP j + ˆP j  i + ÂT j ˆP i + ˆP i  j + 2 P k <, (θ, θ Θ Φ (6.6 As αi 2 β k [, 1], i, k = 1,..., r, 2α i α j β k [,.5], i = 1,..., r 1, j = i + 1,..., r, k = 1,..., r, and r r i=1 k=1 α2 i β k + 2 r 1 r r i=1 j=i+1 k=1 α iα j β k = 1, by Lemma solving the above inequality for positive definite symmetric matrices P i need only be done at all vertices. Hence we get the following proposition. Proposition The system of (3.44 is parameter-dependent stable whenever there exist a positive definite symmetric matrix P i, i = 1, 2,..., r, such that the following LMI conditions hold ˆP i > (6.7  T i  T i ˆP i + ˆP i  i + P k < (6.8 ˆP j + ˆP j  i + ÂT j ˆP i + ˆP i  j + 2 P k < (6.9 for i, k = 1,..., r and 1 i < j r Note that the numbers of LMIs for (6.7 (6.9 are r, r 2 and r 2 (r 1/2, respectively. Therefore, the total number of LMIs to be solved is r(r 2 + r + 2/2. Note also that there are other approaches that could be used to obtain a finite number of LMIs, e.g. multi-convexity [11, 45], S-procedure [4], gridding parameter space [39, 66, 1, 11], etc. Proposition (Multi-convexity approach, [11, Proposition 5.1] Assume θ =, The system of (3.44 is parameter-dependent stable whenever there exist a positive definite symmetric matrix P i, i = 1, 2,..., r, and scalars λ i, i = 1, 2,..., r, such that the following LMI conditions hold ˆP k > (6.1 λ k (6.11  T ˆP k k + ˆP k  k < λ k I (6.12  T i ˆP i + ˆP i  i + ÂT ˆP j j + ˆP j  j (ÂT i ˆP j + ˆP j  i + ÂT ˆP j i + ˆP i  j (λ i + λ j I (6.13 for k = 1,..., r and 1 i < j r Corollary Assume that θ =. If Proposition is satisfied then Proposition is satisfied. 89

116 6.2 Controller Synthesis using PDLF Proof. First, we show that if (6.12 is satisfied then (6.8 is satisfied. Let ÂT k ˆP k + ˆP k  k = M k and M k + λ k I < hence, for all nonzero vector X R p, X T M k X + λ k X T X < if and only if X T M k X < λ k X T X since λ k and X T X >. This yields X T M k X < or ÂT k ˆP k + ˆP k  k <. Next, we show that if (6.13 is satisfied then (6.9 is satisfied. Let ÂT i ˆP i + ˆP i  i +  T ˆP j j + ˆP j  j = M and ÂT i ˆP j + ˆP j  i + ÂT ˆP j i + ˆP i  j = M ij. We have X T MX X T M ij X +(λ i +λ j X T X, for all nonzero vectors X R p. From (6.12, we have X T MX + (λ i + λ j X T X <, therefore > X T MX + (λ i + λ j X T X X T M ij X. This yields X T M ij X < or ÂT i ˆP j + ˆP j  i + ÂT ˆP j i + ˆP i  j < The corollary above shows that Proposition is a subset of Proposition which is more general since ÂT ˆP k k + ˆP k  k and ÂT i ˆP j + ˆP j  i + ÂT ˆP j i + ˆP i  j can be less than a negative definite symmetric matrix rather than they are just less than a diagonal negative definite matrix λ k I and (λ i + λ j I, respectively. In addition, Proposition shows that the determination of a negative definite symmetric matrix is not necessary, hence, comparing with Proposition 6.1.2, the number of LMIs, decision variables and the computational time are reduced while the achieved performance γ level is improved. 6.2 Controller Synthesis using PDLF In the previous section, a sufficient condition to guarantee the stability property of the closed-loop system using the parameter-dependent Lyapunov function has been presented, where it is sufficient to evaluate the LMIs at all vertices. Next, we consider the problem of designing a gain-scheduled output feedback H control with guaranteed L 2 -gain performance for affine LPV systems using the parameterdependent Lyapunov function for which the proposed technique in the previous section can be directly extended to synthesizing a gain-scheduled H controller. The material in this section is draws from [8, 1, 43] and [44]. Consider a given affine LPV plant model with state-space realization of the form (3.59. The gain-scheduled output feedback H control problem using the parameterdependent Lyapunov function is to compute a dynamic LPV controller, K(θ, with state-space equations ẋ k = A k (θ, θx k + B k (θy u = C k (θx k + D k (θy (6.14 which stabilizes the closed-loop system, (3.59 and (6.14, and minimizes the closedloop quadratic H performance, γ, ensures the induced L 2 -norm of the operator mapping the disturbance signal w into the controlled signal z is bounded by γ t1 t1 z T zdt γ 2 w T wdt, t 1 (6.15 9

117 6.2 Controller Synthesis using PDLF along all possible parameter trajectories, (θ, θ Θ Φ. Note that A and A k have the same dimensions, since we restrict ourselves to the full-order case. The closed-loop system, (3.59 and (6.14, is described by the state-space equations where [ ] ẋ ẋ k [ x = A cl (θ x k [ x z = C cl (θ x k ] + B cl (θw ] + D cl (θw (6.16 [ ] [ ] A cl (θ, θ A(θ + B2 D k (θc 2 B 2 C k (θ B1 = B k (θc 2 A k (θ, θ (θ + B, B cl (θ = 2 D k (θd 21 B k (θd 21 C cl (θ = [ C 1 (θ + D 12 D k (θc 2 D 12 C k (θ ], D cl (θ = D 11 (θ + D 12 D k (θd 21 (6.17 Based on the parameter-dependent Lyapunov function, V (x, θ = x T P (θx, there is an LPV controller K(θ of the form of (6.14 that stabilizes the closed-loop system, (3.59 and (6.14, and ensures the induced L 2 -norm of the operator mapping the disturbance signal w into the controlled signal z is bounded by γ along all possible parameter trajectories if and only if there exists P (θ = P T (θ such that [45] P (θ >, d ( x T P (θx + z T z γ 2 w T w <, (θ, dt θ Θ Φ (6.18 Note that, unlike the single quadratic Lyapunov function case (sub-section 3.4.1, P (θ, A k (θ, θ, B k (θ,..., D k (θ and A cl (θ, θ, B cl (θ,..., D cl (θ do not depend affinely on the scheduled parameters θ. Inequality (6.18 leads to [45] AT cl (θ, θp (θ + P (θa cl (θ, θ + P (θ P (θb cl (θ Ccl T (θ Bcl T (θp (θ γi DT cl (θ < (6.19 C cl (θ D cl (θ γi Introducing intermediate controller variables, i.e. Â k (θ, ˆBk (θ, Ĉ k (θ, as [8, 43] A k (θ, θ ( = N 1 (θ X(θẎ (θ + N(θṀ T (θ + Âk(θ X(θ ( A(θ B 2 D k (θc 2 Y (θ ˆBk (θc 2 Y (θ X(θB 2 Ĉ k (θ M T (θ (6.2 B k (θ = N (θ( 1 ˆBk (θ X(θB 2 D k (θ (6.21 C k (θ = (Ĉk (θ D k (θc 2 Y (θ M T (θ (6.22 where N(θ = X(θ + Y 1 (θ, Ṅ(θ = Ẋ(θ Y 1 (θẏ (θy 1 (θ, M(θ = Y (θ and Ṁ(θ = Ẏ (θ. A pair of positive definite symmetric matrices ( X(θ, Y (θ is taken from the structure of the parameter-dependent Lyapunov variable, P (θ, 91

118 6.2 Controller Synthesis using PDLF which is defined as ( X(θ X(θ Y 1 (θ P (θ = ( X(θ Y 1 (θ X(θ Y 1 (θ [ [Y ] Ip X(θ 1 ( ] (θ Ip = p p X(θ Y 1 (6.23 (θ Y (θ p p [ P Ẋ(θ (θ = (θẏ (θy ] 1 (θ (θẏ (θy 1 (θ Ẋ(θ + Y 1 (θẏ (θy 1 (6.24 (θ [ ] Y (θ Y (θ P (θ 1 = ( 1X(θY Y (θ X(θ Y 1 (θ (θ [ ] [ ] 1 Y (θ Ip I p ( X(θ = Y (θ p p X(θ Y 1 (6.25 (θ p p where the positive definite symmetric matrices ( X(θ, Y (θ R p p, X(θ Y 1 (θ, and rank(x(θ Y 1 (θ p [82]. Note that, (6.2 (6.22 show that A k (θ, θ, B k (θ and C k (θ can not depend affinely on the scheduled parameters θ when the symmetric matrix X or Y is parameter-dependent. In this thesis, we propose the intermediate controller variables, i.e.  k (θ, ˆBk (θ, Ĉ k (θ and D k (θ, and ( X(θ, Y (θ to depend affinely on the parameters θ as  k (θ = Âk + θ 1  k1 + + θ n  kn = α 1 à k1 + α 2 à k2 + + α r à kr (6.26 ˆB k (θ = ˆB k + θ 1 ˆBk1 + + θ n ˆBkn = α 1 Bk1 + α 2 Bk2 + + α r Bkr (6.27 Ĉ k (θ = Ĉk + θ 1 Ĉ k1 + + θ n Ĉ kn = α 1 Ck1 + α 2 Ck2 + + α r Ckr (6.28 D k (θ = D k + θ 1 D k1 + + θ n D kn = α 1 Dk1 + α 2 Dk2 + + α r Dkr (6.29 X(θ = X + θ 1 X θ n X n = α 1 ˆX1 + α 2 ˆX2 + + α r ˆXr (6.3 Y (θ = Y + θ 1 Y θ n Y n = α 1 Ŷ 1 + α 2 Ŷ α r Ŷ r (6.31 Ẋ(θ = θ 1 X 1 + θ 2 X θ n X n = β 1 X1 + β 2 X2 + + β r Xr (6.32 Ẏ (θ = θ 1 Y 1 + θ 2 Y θ n Y n = β 1 Ỹ 1 + β 2 Ỹ β r Ỹ r (6.33 Note that X j and Ỹj, j = 1,..., r, map to X i and Y i, i = 1,..., n, respectively in a similar manner to (6.5. This proposed technique offers obvious advantages in reducing computational burden and ease of controller implementation because the intermediate controller variables can be constructed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter. In addition, an existing method for computing the intermediate controller variables, that is based on explicit controller formulas [43], is given in Algorithm C..2. Define [ ] Y (θ Ip P 1 (θ = Y (θ p p (6.34 By premultiplying the first row and postmultiplying the first column of (6.19 by 92

119 6.2 Controller Synthesis using PDLF P1 T (θ and P 1 (θ respectively and substituting (6.17 (6.24 in (6.19, we get [8] ( Ẋ(θ + X(θA(θ + ˆB k (θc 2 + (  T k (θ + A(θ + B 2D k (θc 2 Ẏ (A(θY (θ + (θ + B 2 Ĉ k (θ + ( B1 T (θx(θ + D21 T ˆB k T (θ BT 1 (θ + D21D T k T (θbt 2 C 1 (θ + D 12 D k (θc 2 C 1 (θy (θ + D 12 Ĉ k (θ γi < (6.35 D 11 (θ + D 12 D k (θd 21 γi where the notation represents a symmetric matrix block. Substituting (3.61 and (6.26 (6.33 in (6.35, we have ( X k + ˆXi  i + B r r C ki 2 + ( αi 2 β k à T k i + Âi + B 2 Dki C 2 Ỹk + (Âi Ŷ i + B 2 Cki + ( i=1 k=1 ˆB 1 T ˆXi i + D21 T B k T ˆBT i 1i + D21 T D k T i B2 T Ĉ 1i + D 12 Dki C 2 Ĉ 1i Ŷ i + D 12 Cki γi ˆD 11i + D 12 Dki D 21 γi ( X k + 1 ˆXj  2 i + B C kj 2 + ˆX i  j + B C ki 2 + ( r 1 1 r r (ÃT +2 α i α j β k 2 k j + Âi + B 2 Dkj C 2 + ÃT k i + Âj + B 2 Dki C 2 ( 1 i=1 j=i+1 k=1 ˆBT 2 1 ˆXj i + D21 T B k T j + ˆB 1 T ˆXi j + D21 T B k T i 1 2 (Ĉ1i + D D 12 C kj 2 + Ĉ1 + D D j 12 C ki 2 Ỹk Inequality (6.36 can be also rewritten as r i=1 r k=1 1 2 (Âi Ŷ j + B 2 Ckj + ÂjŶi + B 2 Cki + ( ( ˆBT 1 i + D T 21 D T k j B T 2 + ˆB T 1 j + D T 21 D T k i B T 2 1 Ŷ 2 (Ĉ1i j + D 12 Ckj + Ĉ1 j Ŷ i + D 12 Cki γi 1 2 ( ˆD11i + D 12 D kj D 21 + ˆD 11j + D 12 D ki D 21 αi 2 β k (Ψ clii + Q T ˆKT i P + P T ˆKi Q r r i=1 j=i+1 k=1 γi ( r 1 α i α j β k 2 < ( Ψ clij (

120 6.2 Controller Synthesis using PDLF + Q T ˆKT i P + P T ˆKi Q + Ψ clji + Q T ˆKT j P + P T ˆKj Q < (6.37 where Ψ cl = X k + ˆX  + (  Ỹk +  Ŷ + ( ˆB 1 T ˆX ˆBT 1 γi Ĉ 1 Ĉ 1 Ŷ ˆD11 γi Q = [ [ ] Ip C, D 21, (p+q2 q 1, B = B 2 P = [ ( BT, (p+m2 m 1, D12] T Ãki Bki, ˆKi = C ki Dki ] (6.38 (6.39 (6.4 with the subscript denote (ii, ij, or ji. By Lemma and knowing the matrix vertices ( ˆX i, Ŷi, i = 1, 2,..., r, the system matrix vertices ˆK i can be determined from (6.37, that is an LMI in ˆK i, at all vertices for which ( ˆK 1, ˆK 2,..., ˆK r have to satisfy all of r 2 (r + 1/2 LMIs. Furthermore, knowing Ãk i,..., D ki, the controller system matrices A k (θ, θ,..., D k (θ can be computed on-line in real-time using (6.2 (6.22 with instantaneous measurement values of θ and θ. However, usually, the parameter derivatives either are not available or are difficult to estimate during system operation [8]. To avoid using the measured value of θ, we can constrain either X(θ or Y (θ to depend affinely on θ. This yields Ẋ(θY (θ + Ṅ(θM T (θ = (X(θẎ (θ + N(θṙT (θ = [8], hence, (6.2 becomes A k (θ = N 1 (θ(âk (θ X(θ ( A(θ B 2 D k (θc 2 Y (θ ˆBk (θc 2 Y (θ X(θB 2 Ĉ k (θ M T (θ (6.41 By Lemma , the LMIs of (6.37 are solvable for ˆK i if and only if there exist a pair of positive definite symmetric matrices ( X(θ, Y (θ that satisfy the following LMIs: r r ( T  T αi 2 i ˆX i + β k NX ˆX i  i + X k ˆXi ˆB1i Ĉ1 T i ( ˆB I 1 T ˆXi i γi ˆDT 11i NX I i=1 k=1 Ĉ 1i ˆD11i γi 1 (ÂT r 1 r r ( T 2 i ˆX j + ÂT ˆX j i + ( + X k NX ( α i α j β k ˆBT I 2 1 ˆXj i + ˆB 1 T ˆXi j i=1 j=i+1 k=1 1 2 (Ĉ1i + Ĉ1 j ( ( γi NX 1 ˆD11i + ˆD < (6.42 I 2 11j γi r r ( T  i Ŷ i + αi 2 β k NY ŶiÂT i Ỹk Ŷ i Ĉ1 T ˆB1i i ( Ĉ I 1i Ŷ i γi NY ˆD11i I i=1 k=1 ˆB 1 T ˆDT i 11i γi 94

121 r r i=1 j=i+1 k=1 r ( T NY α i α j β k I 6.2 Controller Synthesis using PDLF 1 (Âi Ŷ 2 j + ÂjŶi + ( Ỹk 1 Ŷ 2 (Ĉ1i j + Ĉ1 j Ŷ i 1 2 ( ˆBT 1 i + ˆB 1 T j ( ( γi NY 1 ˆDT 2 11 i + ˆD < ( T I j γi r ( ˆXi I α i > (6.44 I i=1 Ŷ i where N X and N Y denote bases of the null spaces of [C 2, D 21 ] and [B T 2, D T 12], respectively. Note that, (6.44 ensures X(θ, Y (θ > and X(θ Y (θ 1. By Lemma 3.3.1, (6.42 (6.44 need only be evaluated at all vertices. Note that, (6.42 (6.44 will become (3.77 (3.79 when both X and Y are constant. Moreover, the quadratic H performance γ is determined from both ( X(θ, Y and ( X, Y (θ cases and the case that gives lowest γ is selected. Theorem There exists an LPV controller K(θ guaranteeing the closed-loop system, (3.59 and (6.14, quadratic H performance γ along all possible parameter trajectories, (θ, θ Θ Φ, if and only if the following LMI conditions hold for some positive definite symmetric matrices (X(θ, Y (θ, which further satisfy Rank(X(θ Y 1 (θ p: ( NX I ( NY I ( NX I ( NY I T T T T  T i ˆX i + ˆX i  i + X k ˆXi ˆB1i Ĉ1 T i ˆB 1 T ˆXi i γi ˆDT 11i Ĉ 1i ˆD11i γi  i Ŷ i + ŶiÂT i Ỹk Ŷ i Ĉ1 T ˆB1i i Ĉ 1i Ŷ i γi ˆD11i ˆB 1 T ˆDT i 11i γi  T i ˆX j + ˆX j  i + ÂT ˆX j i + ˆX i  j + 2 X k ˆB 1 T ˆXj i + ˆB 1 T ˆXi j ˆX i ˆB1j + ˆX j ˆB1i 2γI ˆD 11i + ˆD 11j Ĉ 1i + Ĉ1 j Ĉ T 1 i + ĈT 1 j ˆDT 11i + ˆD 11 T j 2γI  i Ŷ j + ŶjÂT i + ÂjŶi + ŶiÂT j 2Ỹk Ĉ 1i Ŷ j + Ĉ1 j Ŷ i ˆB 1 T i + ˆB 1 T j Ŷ i Ĉ1 T j + ŶjĈT 1 ˆB1i i + ˆB 1j 2γI ˆD11i + ˆD 11j ˆD 11 T i + ˆD 11 T j 2γI ( NX I ( NY I ( NX I ( NY I < (6.45 < (6.46 < (6.47 < (

122 6.3 Numerical Example ( Xi I I Y i > (6.49 for i, k = 1,..., r and 1 i < j r Note that, Theorem provides a new approach and an alternative to the multiconvexity approach [11] that is given in theorem C Numerical Example We demonstrate the effectiveness of the approach through a simple numerical example taken from [61], described in Chapter 4, and where the Jacobian-based LPV model is given as (4.5. An LPV controller is synthesized with the criterion [W1 S, W 2 KS] T < 1. The performance weighting, W1, and robustness weighting, W 2, taken from [61] are given as (4.7. The results and numerical features of the LPV synthesis technique for the case where the pair (X(θ, Y (θ are affine is presented in Table 6.1. It shows that, for this example, the number of LMIs and decision variables and the computational time are reduced. Furthermore, the achieved performance γ level is less conservative when using Theorem compared with the multi-convexity technique [11]. The LMIs are solved using the MATLAB Robust Control Toolbox function [12], mincx, on a desktop PC (Intel Core(TM2 CPU 2.13 GHz with 2 GB of RAM. 6.4 Longitudinal Affine LPV Model In this thesis, a minimum least-squares method [58] is used to convert a nonlinearly parameter-dependent LPV model, shown in (5.13, into an affine LPV model. The method minimizes the sum of squared differences between a nonlinearly parameterdependent LPV model and an affine LPV model. For example, consider X u (u, h shown in (5.14, define X u (u, h to be depended affinely on u and h as X u ( u, h = Xu + ux uu + hx uh (6.5 Then, the least-squares problem for X u (u, h is to determine the best [X u X uu X uh ] T that minimizes the sum of squared differences between X u (u, h and X u (u, h. Hence, the least-squares problem for X u (u, h can be formulated as Z = Xθ + v (6.51 where Z is an N 1 vector of values computed from (5.14, θ is a 3 1 vector of unknown parameters, X is an N 3 matrix of known data vectors or regressors, and v is an N 1 vector of equation errors as shown below 96

123 6.4 Longitudinal Affine LPV Model Table 6.1: Numerical comparisons of LPV synthesis techniques; an (X(θ, Y (θ case Condition (θ, θ [, 1] [ 5, 5] # of LMI # of decision variables CPU time (s performance γ Multi-convexity Theorem SQLF * Condition (θ, θ [, 1 4 ] [ 5, 5] # of LMI # of decision variables CPU time (s performance γ Multi-convexity Theorem SQLF * Condition (θ, θ [, 1] [ 1 6, 1 6 ] # of LMI # of decision variables CPU time (s performance γ Multi-convexity Theorem SQLF * [11, Theorem 5.3] * SQLF, [1, Theorem 5.2] 97

124 6.4 Longitudinal Affine LPV Model Z = [ X u (u 1, h 1 X u (u 1, h 2 X u (u i, h j 1 X u (u i, h j ] T 1 u 1 h 1 1 u 1 h 2 X =. 1 u i h j 1 1 u i h j θ = [ X u X uu X uh ] T, v = [ v1 v 2 v N 1 v N ] T where N = i j. The best estimator of θ minimizes the sum of squared differences; the cost function, J, is given by Differentiating (6.52 with respect to θ gives [58], J(θ = 1 2 (Z XθT (Z Xθ (6.52 J(θ θ = Z T X + θ T X T X (6.53 The necessary condition for minimizing the cost is given by J(θ/ θ = giving the least-squares solution for the unknown parameter vector θ as ˆθ = ( X T X 1 X T Z (6.54 Having converted all of the stability and control derivatives, shown in (5.14 (5.31, to be depended affinely on u and h as in a similar manner to (6.5, a nonlinearly parameter-dependent LPV model, shown in (5.13, can be converted into an affine LPV model as ẋ = (A + ua u + ha h x + (B + ub u + hb h u (6.55 where x= [u w q θ h] T, u= [δ e δ rpm ] T, (u, h [337.6, 759.5] [1, 18], and A = ( A u = ( A h = (

125 6.5 Gain-Scheduled H Autopilot Design B = B h = B u = (6.59 ( Gain-Scheduled H Autopilot Design In this approach, the mixed-sensitivity criterion (5.42 is also employed in a similar manner to section 5.3 where the performance weighting, W 1, and robustness weighting, W 2, are (.5s W 1 (s = s W 2 (s = W pre-filter (s =.5s s ( 1s s s s ( 5 s+5 1 s+1 (6.61 Note that the values of weighting functions W 1 and W 2 are hand-tuned until the desired objectives of performance and robustness of the closed-loop system are achieved. After the longitudinal affine LPV model, shown in (6.55, is augmented with the weighting functions, shown in (6.61, a pair of positive definite symmetric matrices ( X(θ, Y (θ can be determined in four cases, i.e. ( X, Y, ( X(θ, Y, ( X, Y (θ, and ( X(θ, Y (θ, using parameter-dependent Lyapunov function, Theorem 6.2.1, for which the performance measure (γ, shown in Table 6.2, can be compared. The LMIs are solved using the MATLAB Robust Control Toolbox function [12], mincx. 6.6 Nonlinear Simulation Results For an entire flight envelope of the vehicle, the mismatch uncertainties between the affine LPV model, shown in (6.55, and the nonlinearly parameter-dependent LPV model, shown in (5.13, become more significant and degrade the transient performance. Leith and Leithead [61] and Chumalee and Whidborne [3] have shown that an affine LPV model can not always accurately represent the original nonlinear plant model. In severe cases, the mismatch uncertainties can cause closed-loop instability 99

126 6.6 Nonlinear Simulation Results Table 6.2: Performance γ comparison for different cases of (X(u, h, Y (u, h Flight condition (X(u, h, Y (u, h (X(u, h, Y (X, Y (u, h (X, Y * (u, h [464.1, 548.5] [75, 125] ( u, ḣ [ 1.26, 1.26] [ 2, 5] Performance γ Flight condition (X(u, h, Y (u, h (X(u, h, Y (X, Y (u, h (X, Y * (u, h [337.6, 759.5] [1, 18] ( u, ḣ [ 1.26, 1.26] [ 2, 5] Performance γ Flight condition (X(u, h, Y (u, h (X(u, h, Y (X, Y (u, h (X, Y * (u, h [464.1, 548.5] [75, 125] ( u, ḣ [ 16, 1 6 ] [ 1 6, 1 6 ] Performance γ * SQLF, [1, Theorem 5.2] for the designed controller with the original nonlinear plant model. However, Lim and How [64] have shown that the mismatch uncertainties can be reduced until they are less significant when the scheduled parameters have a small variation. Extending the proposed method to be applicable to a general class of LPV system (whose system matrices can be nonlinearly dependent on the scheduled parameters θ with a fuller range of operating condition requires further work which is discussed in the Conclusion. Hence, the designed H gain-scheduling autopilot is validated for a bounded flight envelope, i.e. (u, h [464.1, 548.5] [75, 125], ( u, ḣ [ 1.26, 1.26] [ 2, 5], with the Jindivik nonlinear model [41] in a MATLAB Simulink simulation. In Figure 6.1, the transient responses of the simulated vehicle for small demanded changes in speed and altitude are shown for one particalar point in the flight envelope. In addition, it also shows transient response of two LPV controllers, i.e. the first LPV controller (explicit formulas where the intermediate controller variables (Âk(θ, ˆBk (θ, Ĉ k (θ and D k (θ are computed using Algorithm 3.1 [43] and the second LPV controller (proposed method where the matrix vertices of intermediate controller variables are determined from (6.37. The first LPV controller s response is non-smooth because D 21 matrix is not full-row rank, hence, cause a D 21 singular problem. According to [43, Algorithm 3.1], the intermediate controller variables will be varied smoothly following the parameter θ, if the D 12 and D 21 matrices are full-column and full-row rank [8] respectively, shown in Algorithm C..2. In addition, Table 6.3 shows computational time of the intermediate controller variables using Algorithm 3.1 [43] and the proposed method using (6.26 (6.29. One can see from Table 6.3 that our proposed method has a lower computational time. Finally, Figure 6.2 shows a simulated flight that cover a wide range of the flight envelope. It demonstrates that performance robustness was achieved over the defined flight envelope. 1

127 6.7 Conclusion Table 6.3: Computational time of  k (θ, ˆBk (θ, Ĉ k (θ and D k (θ Method CPU time (us Explicit formulas 283 Proposed method * 42 [43, Algorithm 3.1] * (6.26 ( Conclusion In this chapter, new sufficient conditions for gain-scheduled H performance analysis and synthesis for a class of affine LPV systems using parameter-dependent Lyapunov function are proposed, in Theorem Compared with the multi-convexity technique [11] (Theorem C..1 fewer LMIs and decision variables are required and the computational time is lower while the achieved performance level is improved. The analysis and synthesis conditions are represented in the form of a finite number of LMIs. A numerical example is compared with the multi-convexity technique [11] results. In addition, the intermediate controller variables, i.e.  k (θ, ˆBk (θ, Ĉ k (θ and D k (θ, can be constructed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter without the need for constraints on the D 12 and D 21 matrices. The proposed method was applied to synthesize a longitudinal LPV autopilot of the Jindivik UAV. The designed controller was tested with a full 6-DOF simulation of the vehicle and nonlinear simulation results show the effectiveness of the proposed method. The main limitation is that the proposed method is not applicable to a general class of LPV systems which can be nonlinearly dependent on the time-varying parameters θ. However, the mismatch uncertainty between affine LPV model and nonlinearly parameter-dependent LPV model can be modelled as time-varying real parametric uncertianties and can be also included in an affine LPV model using a linear fractional transformation (LFT for gain-scheduled control synthesis and analysis purpose. This approach will be presented in the next chapter. 11

128 6.7 Conclusion 4 (a Altitude (ft (b Speed (ft/s 12

129 6.7 Conclusion (c Pitch angle (deg (d Pitch rate (deg/s 13

130 6.7 Conclusion (e Angle of Attack (deg (f Elevator deflection (deg 14

131 6.7 Conclusion 4 (g Engine speed (RPM Figure 6.1: The transient performance of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 56 ft/s and altitude = 1, ft 15

132 6.7 Conclusion 4 (a Altitude (ft (b Speed (ft/s 16

133 6.7 Conclusion (c Pitch angle (deg (d Pitch rate (deg/s 17

134 6.7 Conclusion (e Angle of Attack (deg (f Elevator deflection (deg 18

135 6.7 Conclusion 4 (g Engine speed (RPM Figure 6.2: The desired performance and robustness objectives are achieved across the defined flight envelope 19

136 This page intentionally contains only this sentence.

137 Chapter 7 Robust Lateral LPV Autopilot Design A major advantage of a class of affine LPV models is that sufficient conditions for gain-scheduled output feedback H control synthesis problem can be obtained in the form of a finite number of LMIs for both the single quadratic and parameterdependent Lyapunov function cases. However, an affine LPV model can rarely accurately represent the original nonlinear model. There are always mismatch uncertainties between these two models. In this chapter, time-varying real parametric uncertainties are included in the system state-space model matrices as a linear fractional transformation (LFT form in order to guarantee closed-loop stability and improve transient performance in presence of these mismatch uncertainties. Hence, a new class of LPV models is obtained called an uncertain affine LPV model which is less conservative than the existing parameter-dependent linear fractional transformation model (LPV/LFT. New algorithms of robust stability analysis and gain-scheduled controller synthesis for this uncertain affine LPV model using single quadratic and parameter-dependent Lyapunov functions are proposed. The analysis and synthesis conditions are represented in the form of a finite number of LMIs. Moreover, a technique to construct the intermediate controller variables as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter is also proposed. To demonstrate the impacts of the proposed scheme in robustness improvement of uncertain affine LPV systems, we compare our approach with the existing uncertain LPV/LFT approach on a numerical example. Furthermore, the proposed method is applied to synthesize a lateral autopilot, i.e. heading-hold, for a bounded flight envelope of the Jindivik UAV. The simulation results on a six degree-of-freedom Jindivik nonlinear model are presented to show the effectiveness of the approach. 111

138 7.1 Jacobian-Based Lateral LPV Model 7.1 Jacobian-Based Lateral LPV Model Suppose an aircraft is assumed to be about a wings level and constant altitude and airspeed flight condition. In addition, assume it can manoeuvre only in the North-East plane, hence, all of the longitudinal states in (A.1 and (A.11 are frozen and equal to the trim values, i.e. u = u trim, w = w trim, q = q trim =, α = α trim = θ = θ trim, and h = h trim. Moreover, after substituting β = sin 1 (v/v, V = u 2 trim + v2 + w 2 trim, and q = 1 2 ρv 2 into (A.1 and (A.11, we get equations of lateral motion as [ v = ρs ( C Y + C Yβ sin 1 v 2m u 2 trim + v 2 + wtrim 2 [ (ρsb + 4m C Y p u 2 trim + v2 + wtrim 2 + w trim ]p + [ (ρsb 4m C Y r u 2 trim + v2 + wtrim 2 u trim ]r ( + ( ρs g cos θ trim sin φ + u 2 trim + v 2 + wtrim 2 2m + ρs ( u 2 trim + v 2 + wtrim 2 2m [ ṗ = ρsb ( C l + C lβ sin 1 v 2I x u 2 trim + v 2 + wtrim 2 + ρsb2 4I x ( u 2 trim + v2 + w 2 trim + ρsb 2I x ( ṙ = ρsb 2I z ] ( u 2 trim + v 2 + wtrim 2 C Yδa δ a C Yδr δ r (7.1 u 2 trim + v 2 + wtrim 2 [ ( C n + C nβ sin 1 v u 2 trim + v 2 + wtrim 2 ( + ρsb2 u 2 trim 4I + v2 + wtrim 2 z + ρsb ( u 2 trim + v 2 + wtrim 2 2I z φ = p + ψ = ] ( u 2 trim + v 2 + wtrim 2 ( C lp p + ρsb2 u 2 trim 4I + v2 + wtrim 2 C lr r x C lδa δ a + ρsb 2I x ( u 2 trim + v 2 + w 2 trim C lδr δ r (7.2 ] ( u 2 trim + v 2 + wtrim 2 ( C np p + ρsb2 u 2 trim 4I + v2 + wtrim 2 C nr r z C nδa δ a + ρsb 2I z ( u 2 trim + v 2 + w 2 trim C nδr δ r (7.3 [ ( tan θtrim cos φ ] r (7.4 [ cos φ cos θ trim ] r (7.5 After (7.1 (7.5 are linearized using the Jacobian linearization method about a wings level and constant altitude and airspeed flight condition, we get a lateral LTI 112

139 7.1 Jacobian-Based Lateral LPV Model model as a state-space system of the form, v Y v Y p Y r Y φ v ṗ ṙ φ = L v L p L r p N v N p N r r 1 φ r φ + ψ ψ r ψ Y δa L δa N δa Y δr L δr N δr [ δa δ r ] (7.6 Note that, v trim = p trim = r trim = β trim = φ trim = ψ trim = δ atrim = δ rtrim =, where, Y v = ρs [ 2m Y p = Y r = [ ρsb 4m ( u 2 trim + w2 trim C Yβ (7.7 ] ( u 2 trim + w2 trim C Yp + w trim (7.8 ] ρsb ( u 2 trim 4m + w2 trim C Yr u trim (7.9 Y φ = g cos θ trim (7.1 Y δa = ρs ( u 2 trim + wtrim 2 C Yδa (7.11 2m Y δr = ρs ( u 2 trim + wtrim 2 C Yδr (7.12 2m L v = ρsb 2I x ( u 2 trim + w2 trim L p = ρsb2 4I x ( u 2 trim + w2 trim L r = ρsb2 4I x ( u 2 trim + w2 trim L δa = ρsb 2I x ( u 2 trim + w 2 trim L δr = ρsb 2I x ( u 2 trim + w 2 trim N v = ρsb 2I z ( u 2 trim + w2 trim N p = ρsb2 4I z ( u 2 trim + w2 trim N r = ρsb2 4I z ( u 2 trim + w2 trim N δa = ρsb ( u 2 trim + wtrim 2 2I z N δr = ρsb ( u 2 trim + wtrim 2 2I z C lβ (7.13 C lp (7.14 C lr (7.15 C lδa (7.16 C lδr (7.17 C nβ (7.18 C np (7.19 C nr (7.2 C nδa (7.21 C nδr (7.22 φ r = tan θ trim (7.23 ψ r = sec θ trim (

140 7.1 Jacobian-Based Lateral LPV Model Table 7.1: Stability and control derivative data (lateral mode of Jindivik nonlinear model * u (ft/s h (ft 9,9 1,5 11,1 9,9 1,5 11,1 9,9 1,5 11,1 w trim θ trim Y v Y p Y r Y φ Y δa Y δr L v L p L r L δa L δr N v N p N r N δa N δr φ r ψ r * about wings level and constant altitude and airspeed flight condition ft/s rad Table 7.2: Jindivik s lateral aerodynamic coefficients Coefficient C Y C Yβ C Yp C Yr C Yδa C Yδr Value Coefficient C l C lβ C lp C lr C lδa C lδr Value Coefficient C n C nβ C np C nr C nδa C nδr Value Equations (7.7 (7.24 are the stability and control derivatives (lateral mode, where values of w trim and θ trim can be calculated using (5.33 and (5.34 respectively. Furthermore, the stability and control derivatives in the above equations are nonlinearly dependent on only speed and altitude. With the speed and altitude frozen, (7.6 is a lateral LTI model. However, as the speed and altitude vary over the entire flight envelope, (7.6 becomes a lateral nonlinearly parameter-dependent LPV model. Moreover, these equations show that the accuracy of this LPV model depends on the accuracy of the information that provides the aerodynamic coefficients. Using functions trim and linmod, the stability and control derivative values, shown in Table 7.1, are obtained about one flight condition (speed = 497 ft/s and altitude = 1,5 ft. After substituting the data from Table 7.1 into (7.7 (7.24, we can determine approximate aerodynamic coefficients of the UAV as shown in Table

141 7.2 Stability Analysis of Uncertain Affine LPV Systems Knowing the aerodynamic coefficients, the system matrices of the lateral nonlinearly parameter-dependent LPV model (7.6 at all points over the entire parameter spaces can be determined using (7.7 (7.24. In addition, Figure 7.1 shows the variation of open-loop characteristic of the lateral nonlinearly parameter-dependent LPV model, i.e. v(s/δ r (s, over an entire flight envelope. According to Figure 7.1, one pole of spiral mode is open-loop unstable with constant damping ratio (ξ of -1 and variation of the natural frequency (ω n from.222 rad/s to.535 rad/s, one pole of roll mode is open-loop stable with constant ξ of 1 and variation of ω n from.831 rad/s to 3.21 rad/s, and the other two poles of Dutch roll mode are open-loop stable with variation of ξ and ω n from.449 to.75 and 1.6 rad/s to 4.41 rad/s respectively. Moreover, the system of (7.6 also has non-minimum phase zeros as shown in figure Stability Analysis of Uncertain Affine LPV Systems An uncertain affine LPV system that is extended from an affine LPV system [45] is given by ẋ = A ( θ, δ x, x( = x (7.25 where δ = [ δ 1,..., δ m ] T R m is a vector of time-varying real parametric uncertainty which cannot be measured. The plant state matrix A ( θ, δ is assumed to depend affinely on both the scheduled parameters θ and parametric uncertainties δ. That is A ( θ, δ = A ( θ + δ 1 A δ1 ( θ + δ2 A δ2 ( θ + + δm A δm ( θ (7.26 where A ( θ = A + θ 1 A θ n A n A δi ( θ = Aδi + θ 1 A δi1 + + θ n A δin, i = 1,..., m (7.27 with A, A 1,..., A n are known fixed matrices. A δi, A δi1,..., A δin, i = 1,..., m, are also known fixed matrices. We assume that each uncertainty δ i is assumed to lie between known bound values δ i and δ i, δ i [ δ i, δ i ], and δ lies in a polytope, δ. Define the normalized parametric uncertainties δ i [ 1, 1], i = 1,..., m as [83] δ i = δ i T i S i, T i = δ i + δ i, S i = δ i δ i 2 2 (7.28 Substitute (7.28 into (7.26, we get A(θ, δ = A(θ, δ = A(θ + δ 1 A δ1 (θ + δ 2 A δ2 (θ + + δ m A δm (θ (

142 7.2 Stability Analysis of Uncertain Affine LPV Systems 5 Pole-Zero Map One Non-minimum Phase Zero 1 Dutch Roll Mode -1 Roll Mode Real Axis (a Overview 1 x 1-3 Pole-Zero Map Spiral Mode Real Axis (b Zoom in Figure 7.1: The open-loop characteristic of transfer function v(s δ r(s lateral LPV model 116 of the Jindivik

143 7.2 Stability Analysis of Uncertain Affine LPV Systems where ( A(θ = A + m T i A δi + θ 1 (A 1 + i=1 m T i A δi1 + + θ n (A n + i=1 m T i A δin i=1 = à + θ 1 à θ n à n A δi (θ = S i A δi + θ 1 S i A δi1 + + θ n S i A δin, i = 1,..., m (7.3 and à j = A i + m T i A δij, j =, 1,..., n i=1 Based on the LFT technique [14], the parametric uncertainties δ in (7.29 (which are unknown but bounded can be separated from the system state-space model matrices (which are known as ẋ A(θ B δ1 (θ B δ2 (θ... B δm (θ x z δ1 C δ1 (θ D δ11 (θ D δ12 (θ... D δ1m (θ w δ1 z δ2 = C δ2 (θ D δ21 (θ D δ22 (θ... D δ2m (θ w δ2... z δm C δm (θ D δm1 (θ D δm2 (θ... D δmm (θ w δm w δ1 δ 1 I s1 z δ1 w δ2. = δ2 I s2 z δ2 ( w δm δm I sm z δm where w δi, z δi R s i and B δi (θ = B δi + θ 1 B δi1 + θ 2 B δi2 + + θ n B δin C δi (θ = C δi + θ 1 C δi1 + θ 2 C δi2 + + θ n C δin D δii (θ = D δii + θ 1 D δii1 + θ 2 D δii2 + + θ n D δiin (7.32 Note that B δi (θ, C δi (θ, i = 1,..., m, in (7.31 are the factors of A δi (θ in (7.29 whereby either B δi (θ or C δi (θ depends affinely on the scheduled parameters θ. Moreover, D δii (θ is introduced in order that (7.31 is in a general state-space equation form. With notation w δ = [ ] wδ T 1 wδ T 2 wδ T T m z δ = [ ] zδ T 1 zδ T 2 zδ T T m B δ (θ = [ B δ1 (θ B δ2 (θ B δm (θ ] C δ (θ = [ Cδ T 1 (θ Cδ T 2 (θ Cδ T m (θ ] T D δ11 (θ D δ12 (θ D δ1m (θ D δ21 (θ D δ22 (θ D δ2m (θ D δ (θ =. D δm1 (θ D δm2 (θ D δmm (θ (

144 7.2 Stability Analysis of Uncertain Affine LPV Systems Equation (7.31 can be rewritten as ẋ = A(θx + B δ (θw δ z δ = C δ (θx + D δ (θw δ w δ = ˆ z δ (7.34 where w δ, z δ R s, s = s 1 + s s m, ˆ = diag( δ1 I s1, δ 2 I s2,..., δ m I sm, and ˆ 1. Figure 7.2 shows the structures of both uncertain affine LPV and uncertain LPV/LFT [9] models where θ i [ 1, 1], i = 1,..., n, is the normalized timevarying parameters and can be computed using (3.36. In general, the uncertain LPV/LFT [9] model is more conservative than the proposed uncertain affine LPV model since the scheduled parameters θ of an LPV/LFT model are in the uncertainty block in which the variations of parameters are allowed to be complex, thus it introduces conservatism when the scheduled parameters are real [1, 6, 83]. The plant state matrix A(θ in (7.34 can also be written as a convex combination of the matrix vertices as } A(θ = Co {Â1, Â2,..., Âr = α 1  1 + α 2  α r  r (7.35 where r = 2 n, α i is determined using (3.24 and (3.25 and  1 1 θ 1 θ 2... θ n 1 θ n  2 1 θ 1 θ 2... θ n 1 θ n à  3 = 1 θ 1 θ 2... θ n 1 θ à 1 n... à 1 θ 1 θ 2... θ n 1 θ n n  r (7.36 Note that B δ (θ, C δ (θ and D δ (θ in (7.34 can also be written as a convex combination of the matrix vertices in a similar manner to ( Robustness Analysis using SQLF The system of (7.34 is said to be quadratically stable if there exists a quadratic Lyapunov function V (x = x T P x whose derivative is negative, d/dt ( V (x <, along all possible parameter trajectories, θ Θ, for all ˆ with ˆ 1 < 1/γ. This is equivalent to the existence of a P = P T such that [ ] T [ ] [ ] x A P >, T (θp + P A(θ P B δ (θ x Bδ T (θp < (7.37 w δ for all nonzero x satisfying [25] w T δ w δ = (C δ (θx + D δ (θw δ T ˆ T ˆ (Cδ (θx + D δ (θw δ < 1 γ 2 (C δ(θx + D δ (θw δ T (C δ (θx + D δ (θw δ (7.38 w δ 118

145 7.2 Stability Analysis of Uncertain Affine LPV Systems w D x C z w B x A x w m w w w 2 1 m z z z z 2 1 s m m s s I I I ~ ~ ~ θ(t (a Uncertain affine LPV structure m n z z z z z z z m n s m s s s n s s I I I I I I ~ ~ ~ ~ ~ ~ m n w w w w w w w (b Uncertain LPV/LFT structure Figure 7.2: Structure comparisons of uncertain affine LPV and uncertain LPV/LFT [9] models. 119

146 7.2 Stability Analysis of Uncertain Affine LPV Systems where (7.38 can be written further as [ ] T [ ] [ ] x Cδ (θ T C δ (θ C δ (θ T D δ (θ x Dδ T (θc δ(θ γ 2 I Dδ T (θd < (7.39 δ(θ w δ Following [25, pp.62-63], by applying the S-procedure, the quadratic stability of (7.34 is equivalent to the existence of P and λ satisfying [ ] A P >, λ, T (θp + P A(θ + λc δ (θ T C δ (θ P B δ (θ + λc δ (θ T D δ (θ Bδ T (θp + λdt δ (θc δ(θ λγ 2 I + λdδ T (θd < δ(θ (7.4 Having rearranged (7.4 by using the Schur complement [44, Lemma 3.2], (7.4 becomes the well-known bounded real lemma [1] inequality A T (θp + P A(θ P B δ (θ Cδ T (θ Bδ T (θp γi DT δ (θ < (7.41 C δ (θ D δ (θ γi Note that P in (7.41 has been scaled by 1/λ. The robust stability requirement is that γ < 1. However there generally exist an infinite number of the factor matrices pairs (B δ (θ, C δ (θ in which only some factor matrices pair give γ < 1. Instead of searching for such factor matrices pair manually, by introducing a scaling matrix L 1/2, we can select any factor matrices pair for which γ will always be less than unity if the system (7.34 is quadratically stable and the factor matrices pair can be determined using a singular value decomposition where L 1/2 denotes the unique positive definite square root of L L. The set L is defined as { L = L > : L ˆ = ˆ L, } δ R s s (7.42 Therefore, (7.34 can be modified further as ẋ = A(θx + B δ (θl 1 2 ẃδ ź δ = L 1 2 Cδ (θx + L 1 2 Dδ (θl 1 2 ẃδ ẃ δ = ˆ ź δ (7.43 where ź δ = L 1 2 z δ and w δ = L 1 2 ẃ δ. With parameters from (7.43, (7.41 becomes a scaled bounded real lemma [8, 9] inequality A T (θp + P A(θ P B δ (θ Cδ T (θ Bδ T (θp γl DT δ (θ < (7.44 C δ (θ D δ (θ γl 1 Substituting (7.35 into (7.44 and rearranging using the Schur complement [44, Lemma 3.2], we get r ( ÂT α i P + P Âi + ĈT δ i LĈδ i P ˆB δi + ĈT δ i L ˆD δi i ˆB δ T i P + ˆD δ T i LĈδ i γ 2 L + ˆD δ T i L ˆD < (7.45 δi i=1 By Lemma 3.3.1, solving the above inequality for some positive definite symmetric Lyapunov and scaling matrices (P, L need only be done at all vertices. Hence we get the following proposition. 12 w δ

147 7.2 Stability Analysis of Uncertain Affine LPV Systems Proposition The system of (7.34 is quadratically stable along all possible parameter trajectories, θ Θ, for all ˆ with ˆ < 1/γ if and only if the following LMI conditions hold for some positive definite symmetric Lyapunov and scaling matrices (P, L ( ÂTi P + P Âi + ĈT δ i LĈδ i P ˆB δi + ĈT δ i L ˆD δi ˆB δ T i P + ˆD δ T i LĈδ i γ 2 L + ˆD δ T i L ˆD < (7.46 δi where i = 1,..., r Note that, the minimization of γ is achieved heuristically or by a simple grid search Robustness Analysis using PDLF With parameter-dependent Lyapunov functions, V (x, θ = x T P (θx, (7.44 becomes AT (θp (θ + P (θa(θ + P (θ P (θb δ (θ Cδ T (θ Bδ T (θp (θ γl DT δ (θ < (7.47 C δ (θ D δ (θ γl 1 Substituting (7.35, (6.2 and (6.4 into (7.47, rearranging using the Schur complement [44, Lemma 3.2], and recalling that r i=1 α i = 1 and r i=1 β i = 1, we get r r ( α 2 ÂT i β i ˆPi + ˆP i  i + P k + ĈT δ i LĈδ i ˆPiˆr δi + ĈT δ i L ˆD δi k ˆr T i=1 k=1 δ ˆPi i + ˆD δ T i LĈδ i γ 2 L + ˆD δ T i L ˆD δi ( r 1 r r ÂT + α i α j β i ˆPj + ˆP j  i + ÂT ˆP j i + ˆP i  j + 2 P k + ĈT δ i LĈδ i + ĈT δ j LĈδ j k ˆr T i=1 j=i+1 k=1 δ ˆPj i + ˆr δ T ˆPi j + ˆD δ T i LĈδ i + ˆD δ T j LĈδ j ˆP iˆr δj + ˆP jˆr δi + ĈT δ i L ˆD δi + ĈT δ j L ˆD δj 2γ 2 L + ˆD δ T i L ˆD δi + ˆD δ T j L ˆD < (7.48 δj By Lemma 3.3.1, the above inequalities are sufficiently evaluated at all vertices for which the total number of LMIs to be solved is r 2 (r + 3/2. Hence we get the following proposition. Proposition The system of (7.34 is parameter-dependent stable along all possible parameter trajectories, (θ, θ Θ Φ, for all ˆ with ˆ < 1/γ if and only if the following LMI conditions hold for some positive definite symmetric Lyapunov and scaling matrices (P, L ( ÂT i ˆPi + ˆP i  i + P k + ĈT δ i LĈδ i ˆPiˆr δi + ĈT δ i L ˆD δi ˆr δ T ˆPi i + ˆD δ T i LĈδ i γ 2 L + ˆD δ T i L ˆD < (7.49 δi ( ÂT i ˆPj + ˆP j  i + ÂT ˆP j i + ˆP i  j + 2 P k + ĈT δi LĈδ i + ĈT δj LĈδ j ˆr T δ i ˆPj + ˆr T δ j ˆPi + ˆD T δ i LĈδ i + ˆD T δ j LĈδ j ˆP iˆr δj + ˆP jˆr δi + ĈT δ i L ˆD δi + ĈT δ j L ˆD δj 2γ 2 L + ˆD T δ i L ˆD δi + ˆD T δ j L ˆD δj < (

148 7.3 Controller Synthesis for Uncertain Affine LPV Systems where i, k = 1,..., r and 1 i < j r Note that, the minimization of γ is achieved heuristically or by a simple grid search. 7.3 Controller Synthesis for Uncertain Affine LPV Systems In the previous section, a sufficient condition for robust stability that can guarantee the closed-loop system property has been presented. This is especially relevant for robustness analysis using parameter-dependent Lyapunov functions, where a finite number of LMIs need only be evaluated at all vertices. By including time-varying real parametric uncertainties δ in an affine LPV model as shown in (7.34, our stability analysis can guarantee a larger stability margin over conventional stability analysis technique. Next, we consider the problem of designing a gain-scheduled output feedback H control with guaranteed L 2 -gain performance for uncertain affine LPV systems for which the proposed technique in the previous section can be directly extended to designing a gain-scheduled H controller. The material in this section is draws from [8, 1, 43] and [44]. Consider an uncertain affine LPV model that is extended from an affine LPV model (3.59 is given by ẋ = A(θ, δx + B 1 (θ, δw + B 2 u z = C 1 (θ, δx + D 11 (θ, δw + D 12 u y = C 2 x + D 21 w (7.51 where B 1 (θ, δ, C 1 (θ, δ, and D 11 (θ, δ can be written in a similar manner to (7.26. Based on the LFT technique [14], the uncertainty δ in (7.51 can be separated from the system matrices in a similar manner to (7.31 and (7.43, giving ẋ A(θ B δ1 (θl B δm (θl 1 2 B 1 (θ B 2 x z δ1 L 1 2 C δ1 (θ L 1 2 D δδ11 (θl L 1 2 D δδ1m (θl 1 2 L 1 2 D δ11 (θ L 1 2 D δ2 w δ1. =. z δm L 1 2 C δm (θ L 1 2 D δδ1m (θl L 1 2 D δδmm (θl 1 2 L 1 2 D δ1m (θ L 1. 2 D δ2 z C 1 (θ D 1δ1 (θl D 1δm (θl 1 w δm 2 D 11 (θ D 12 w y C 2 D 2δ L D 2δ L 1 2 D 21 u w δ1 δ 1 I s1 z δ1 w δ2. = δ2 I s2 z δ2 ( w δm δm I sm z δm where L L is a scaling matrix that has to be determined, w δi, z δi R s i, δ i [ 1, 1], i = 1,..., m, the normalized parametric uncertainties which can be 122

149 7.3 Controller Synthesis for Uncertain Affine LPV Systems determined using (7.28, and [ Aδi (θ B (θ ] 1δi C 1δi (θ D 11δi (θ = [ ] Bδi (θ [Cδi D 1δi (θ (θ D (θ] δ1i (7.53 Note that either [Bδ T i (θ, D1δ T i (θ] T or [C δi (θ, D δ1i (θ] depends affinely on θ. Moreover, D δδii (θ, D δ2, and D 2δ are introduced in order that (7.52 is in a general state-space equation form. With the notation of (7.33, we can rewrite (7.52 as ẋ A(θ B δ (θl 1 2 B 1 (θ B 2 x z δ z = L 1 2 C δ (θ L 1 2 D δδ (θl 1 2 L 1 2 D δ1 (θ L 1 2 D δ2 C 1 (θ D 1δ (θl w δ 1 2 D 11 (θ D 12 w y C 2 D 2δ L 1 2 D 21 u w δ = ˆ z δ (7.54 where w δ, z δ R s, s = s 1 + s s m, ˆ = diag( δ1 I s1, δ 2 I s2,..., δ m I sm, and ˆ 1. Note that matrices B δ (θ, B 1 (θ, C δ (θ, C 1 (θ, D δδ (θ, D δ1 (θ, D 1δ (θ, and D 11 (θ can also be written as a convex combination of the matrix vertices in a similar manner to (7.35 as A(θ B δ (θ B 1 (θ B 2 C δ (θ D δδ (θ D δ1 (θ D δ2 C 1 (θ D 1δ (θ D 11 (θ D 12 C 2 D 2δ D 21 = r i=1 α i  i ˆBδi ˆB1i B 2 Ĉ δi ˆDδδi ˆDδ1i D δ2 Ĉ 1i ˆD1δi ˆD11i D 12 C 2 D 2δ D 21 ( Gain-Scheduled Controller Design using SQLF The gain-scheduled output feedback H control problem using single quadratic Lyapunov functions is to compute a dynamic LPV controller, K(θ, with state-space equations ẋ k = A k (θx k + B k (θy u = C k (θx k + D k (θy (7.56 which stabilizes the closed-loop system, (7.54 and (7.56, and minimizes the closedloop quadratic H performance, γ, so ensuring that the induced L 2 -norm of the operator mapping the disturbance signal into the controlled signal is bounded by γ t1 t1 (zδ T z δ + z T zdt γ 2 (wδ T w δ + w T wdt, t 1 (7.57 along all possible parameter trajectories, θ Θ. Note that A and A k have the same dimensions, since we restrict ourselves to the full-order case. With the notation ( Ak (θ B K(θ = k (θ r = α C k (θ D k (θ i K i (7.58 i=1 ( Aki B K i = ki, i = 1, 2,..., r (7.59 C ki D ki 123

150 7.3 Controller Synthesis for Uncertain Affine LPV Systems where r is the total number of vertices and α i is determined using (3.24 and (3.25. the closed-loop system, (7.54 and (7.56, is described by the state-space equations x cl A cl (θ B δcl (θl 1 2 B 1cl (θ x cl z δ = L 1 2 C δcl (θ L 1 2 D δδcl (θl 1 2 L 1 2 D δ1cl (θ w δ z C 1cl (θ D 1δcl (θl 1 2 D 11cl (θ w w δ = ˆ z δ (7.6 where x cl = [ ] x T x T T k and [ ] A(θ r ] [Âi A cl (θ = + BK(θC = α i  cli,  cli = + BK p p i C p p i=1 [ ] Bδ (θ r [ ] ˆBδi B δcl (θ = + BK(θD δ21 = α i ˆBδcli, ˆBδcli = + BK i D δ21 i=1 [ ] B1 (θ r [ ] ˆB1i B 1cl (θ = + BK(θD 121 = α i ˆB1cli, ˆB1cli = + BK i D 121 i=1 C δcl (θ = [ C δ (θ ] r + D δ12 K(θC = α i Ĉ δcli, Ĉ δcli = [ Ĉ δi ] + D δ12 K i C with C 1cl (θ = [ C 1 (θ ] + D 112 K(θC = D δδcl (θ = D δδ (θ + D δ12 K(θD δ21 = D δ1cl (θ = D δ1 (θ + D δ12 K(θD 121 = D 1δcl (θ = D 1δ (θ + D 112 K(θD δ21 = D 11cl (θ = D 11 (θ + D 112 K(θD 121 = B = [ ] B2, C = I p i=1 r α i Ĉ cli, Ĉ cli = [ Ĉ 1i ] + D 112 K i C i=1 r α i ˆDδδcli, ˆDδδcli = ˆD δδi + D δ12 K i D δ21 i=1 r α i ˆDδ1cli, ˆDδ1cli = ˆD δ1i + D δ12 K i D 121 i=1 r α i ˆD1δcli, ˆD1δcli = ˆD 1δi + D 112 K i D δ21 i=1 r α i ˆD11cli, ˆD11cli = ˆD 11i + D 112 K i D 121 i=1 [ ] [ ] [ ] Ip, D C 2 δ21 =, D D 121 = 2δ D 21 (7.61 D δ12 = [ D δ2 ], D112 = [ D 12 ], (7.62 Based on the single quadratic Lyapunov functions V (x = x T P x, there is an LPV controller K(θ that stabilizes the closed-loop system, (7.54 and (7.56, and ensures the L 2 -induced norm of the operator mapping the disturbance signal into the controlled signal is bounded by γ along all possible parameter trajectories if and only if there exists P = P T such that [45] P >, d ( x T P x + (zδ T z δ + z T z γ 2 (wδ T w δ + w T w <, θ Θ (7.63 dt 124

151 7.3 Controller Synthesis for Uncertain Affine LPV Systems Inequality (7.63 leads to the scaled bounded real lemma [8, 9] inequality A T cl (θp + P A cl(θ P B δcl (θ P B 1cl (θ Cδ T cl (θl C1 T cl (θ Bδ T cl (θp γl Dδδ T cl (θl D1δ T cl (θ B1 T cl (θp γi Dδ1 T cl (θl D11 T cl (θ LC δcl (θ LD δδcl (θ LD δ1cl (θ γl C 1cl (θ D 1δcl (θ D 11cl (θ γi < (7.64 Substituting (7.61 into (7.64, we get r i=1 α i  T cl i P + P Âcl i P ˆB δcli P ˆB 1cli Ĉ T δ cli L Ĉ T 1 cli ˆB T δ cli P γl ˆDT δδcli L ˆD T 1δ cli ˆB T 1 cli P γi ˆDT δ1cli L ˆD T 11 cli LĈδ cli L ˆD δδcli L ˆD δ1cli γl Ĉ 1cli ˆD1δcli ˆD11cli γi < (7.65 Inequality (7.65 can be also rewritten as [44] r i=1 α i (Ψ cli + Q T Ki T P cl + PclK T i Q < (7.66 where Ψ cli = ] T ] [ ] [ ] [Âi [Âi ˆBδi ˆB1i P + P P P p p p p [Ĉδi ] T L [Ĉ1i ] T [ ] T ˆBδi P γl ˆDT δδ i L ˆDT 1δi [ ] T ˆB1i P γi ˆDT δ1 i L ˆDT 11i L [ Ĉ δi ] L ˆD δδi L ˆD δ1i γl [Ĉ1i ] ˆD1δi ˆD11i γi Q = [ ] C, D δ21, D 121, (p+q2 (q 1 +s P cl = [ ] B T P, (p+m2 (m 1 +s, Dδ T 12, D1 T 12 (7.67 (7.68 (7.69 Having determined the quadratic Lyapunov variable P R 2p 2p and the scaling matrix L L, the system matrix vertices K i of the LPV controller K(θ for each vertex Θ i, i = 1,..., r, can be determined from (7.66 that is an LMI in K i. By Lemma 3.3.1, the LMIs (7.66 are sufficiently evaluated at all vertices. Knowing K i, the controller system matrices A k (θ,..., D k (θ can be computed on-line in real-time using (7.58 and an instantaneous measurement value of θ. By Lemma with a known scaling matrix L, LMIs (7.66 are solvable for K i if and only if there exist a pair of positive definite symmetric matrices (X, Y satisfying 125

152 7.3 Controller Synthesis for Uncertain Affine LPV Systems the following LMIs: r i=1 α i r i=1 ( NX I α i ( NY I T T  T i X + XÂi X ˆB δi X ˆB 1i Ĉ T δ i Ĉ T 1 i ˆB T δ i X γl ˆDT δδi ˆDT 1δi ˆB T 1 i X γi ˆDT δ1i ˆDT 11i Ĉ δi ˆDδδi ˆDδ1i γl 1 Ĉ 1i ˆD1δi ˆD11i γi  i Y + Y ÂT i Y ĈT δ i Y ĈT 1 i ˆBδi ˆB1i Ĉ δi Y γl 1 ˆDδδi ˆDδ1i Ĉ 1i Y γi ˆD1δi ˆD11i ˆB T δ i ˆDT δδi ˆDT 1δi γl ˆB T 1 i ˆDT δ1i ˆDT 11i γi ( NX I ( NY I < (7.7 < (7.71 ( X I > I Y (7.72 where N X and N Y denote bases of the null spaces of [C 2, D 2δ, D 21 ] and [B T 2, D T δ2, DT 12], respectively. Note that (7.72 ensures X, Y > and X Y 1. By Lemma 3.3.1, (7.7 (7.72 are sufficiently evaluated at all vertices. Although (7.7 (7.72 are not standard LMI problems, they can be solved by an iterative approach, referred to as D-K iteration [14]. Like the µ-synthesis algorithms, such a scheme is not guaranteed to converge to a global minimum [14], but may find a local minimum [9]. In spite of this drawback, the D-K iteration control design technique appears to work well on many engineering problems [14]. Theorem Given a scaling matrix L, there exists an LPV controller K(θ that guarantees the closed-loop system, (7.54 and (7.56, quadratic H performance γ along all possible parameter trajectories, θ Θ, if and only if the following LMI conditions hold for some positive definite symmetric matrices (X, Y, which further satisfy rank(x Y 1 p. ( NX I ( NY I T T  T i X + XÂi X ˆB δi X ˆB 1i Ĉ T δ i Ĉ T 1 i ˆB T δ i X γl ˆDT δδi ˆDT 1δi ˆB T 1 i X γi ˆDT δ1i ˆDT 11i Ĉ δi ˆDδδi ˆDδ1i γl 1 Ĉ 1i ˆD1δi ˆD11i γi  i Y + Y ÂT i Y ĈT δ i Y ĈT 1 i ˆBδi ˆB1i Ĉ δi Y γl 1 ˆDδδi ˆDδ1i Ĉ 1i Y γi ˆD1δi ˆD11i ˆB T δ i ˆDT δδi ˆDT 1δi γl ˆB T 1 i ˆDT δ1i ˆDT 11i γi ( NX I ( NY I < (7.73 < (

153 for i = 1,..., r. 7.3 Controller Synthesis for Uncertain Affine LPV Systems ( X I I Y > (7.75 Algorithm Computation of X,Y and L Step 1: Setting L = I, solve (7.73 (7.75 for X, Y by minimizing γ. Step 2: Knowing X,Y, and L, solve (7.66 for K i. Step 3: With K i fixed, solve (7.65 for L by minimizing γ (heuristical. Step 4: With L fixed, solve (7.73 (7.75 for new X,Y by minimizing γ. Step 5: Iterate over Steps 2 to 4 until γ is converged to some minimum value. Note that, by Lemma 3.3.1, (7.65 and (7.66 are sufficiently solved at all vertices Gain-Scheduled Controller Design using PDLF With parameter-dependent Lyapunov functions, V (x, θ = x T P (θx, (7.63 becomes d ( P (θ >, x T P (θx + (zδ T z δ + z T z γ 2 (wδ T w δ + w T w <, (θ, dt θ Θ Φ (7.76 A state-space equations of a dynamic LPV controller, K(θ in (7.56, becomes ẋ k = A k (θ, θx k + B k (θy u = C k (θx k + D k (θy (7.77 Note that, unlike the single quadratic Lyapunov functions case, P (θ, A k (θ, θ, B k (θ,..., D k (θ and A cl (θ, θ, B δcl (θ,..., D 11cl do not depend affinely on the scheduled parameters θ. Inequality (7.76 becomes A T cl (θ, θp (θ + P (θa cl (θ, θ + P (θ P (θb δcl (θ P (θb 1cl (θ Cδ T cl (θl C1 T cl (θ Bδ T cl (θp (θ γl Dδδ T cl (θl D1δ T cl (θ B1 T cl (θp (θ γi Dδ1 T cl (θl D11 T cl (θ LC δcl (θ LD δδcl (θ LD δ1cl (θ γl C 1cl (θ D 1δcl (θ D 11cl (θ γi (7.78 where [ ] A cl (θ, θ A(θ + B2 D k (θc 2 B 2 C k (θ = B k (θc 2 A k (θ, θ [ ] [ ] Bδ (θ + B B δcl (θ = 2 D k (θd 2δ B1 (θ + B, B B k (θd 1cl (θ = 2 D k (θd 21 2δ B k (θd 21 C δcl (θ = [ C δ (θ + D δ2 D k (θc 2 D δ2 C k (θ ] C 1cl (θ = [ C 1 (θ + D 12 D k (θc 2 D 12 C k (θ ] D δδcl (θ = D δδ (θ + D δ2 D k (θd 2δ, D δ1cl (θ = D δ1 (θ + D δ2 D k (θd 21 D 1δcl (θ = D 1δ (θ + D 12 D k (θd 2δ, D 11cl (θ = D 11 (θ + D 12 D k (θd 21 (7.79 < 127

154 7.3 Controller Synthesis for Uncertain Affine LPV Systems Following section 6.2, inequality (7.78 leads to Ẋ(θ + X(θA(θ + ˆB k (θc 2 + (  T k (θ + A(θ + B 2D k (θc 2 Ẏ (θ + A(θY (θ + B 2Ĉk(θ + ( Bδ T (θx(θ + DT ˆB 2δ k T (θ BT δ (θ + DT 2δ DT k (θbt 2 B1 T (θx(θ + D21 T ˆB k T (θ BT 1 (θ + D21D T k T (θbt 2 LC δ (θ + LD δ2 D k (θc 2 LC δ (θy (θ + LD δ2 Ĉ k (θ C 1 (θ + D 12 D k (θc 2 C 1 (θy (θ + D 12 Ĉ k (θ γl γi < LD δδ (θ + LD δ2 D k (θd 2δ LD δ1 (θ + LD δ2 D k (θd 21 γl D 1δ (θ + D 12 D k (θd 2δ D 11 (θ + D 12 D k (θd 21 γi (7.8 where the notation represents a symmetric matrix block. Substituting (7.55 and (6.26 (6.33 in (7.8, we have r i=1 r αi 2 β k k=1 X k + ˆX i  i + B ki C 2 + ( à T k i + Âi + B 2 Dki C 2 Ỹk + ÂiŶi + B 2 Cki + ( ˆB δ T ˆXi i + D2δ T B k T i ˆBT δi + D2δ T D k T i B2 T ˆB 1 T ˆXi i + D21 T B k T i ˆBT 1i + D21 T D k T i B2 T LĈδ i + LD δ2 Dki C 2 Ĉ 1i + D 12 Dki C 2 LĈδ i Ŷ i + LD δ2 Cki Ĉ 1i Ŷ i + D 12 Cki γl γi L ˆD δδi + LD δ2 Dki D 2δ L ˆD δ1i + LD δ2 Dki D 21 γl ˆD 1δi + D 12 Dki D 2δ ˆD11i + D 12 Dki D 21 γi r 1 +2 r i=1 j=i+1 k=1 r α i α j β k ( X k + 1 ˆXj  2 i + B C kj 2 + ˆX i  j + B C ki 2 + ( 1 (ÃT 2 k j + Âi + B 2 Dkj C 2 + ÃT k i + Âj + B 2 Dki C 2 ( 1 ˆBT 2 δ ˆXj i + D2δ T B k T j + ˆB δ T ˆXi j + D2δ T B k T ( i 1 ˆBT 2 1 ˆXj i + D21 T B k T j + ˆB 1 T ˆXi j + D21 T B k T i (Ĉδi + D D δ2 C kj 2 + Ĉδj + D D δ2 C ki 2 L (Ĉ1i + D 12 D kj C 2 + Ĉ1 j + D 12 D ki C 2 128

155 (Âi Ŷ j + B 2 Ckj + ÂjŶi + B 2 Cki + ( Ỹk ( 1 ˆBT 2 δ i + D2δ T D k T j B2 T + ˆB δ T j + D2δ T D k T i B2 T ( 1 ˆBT 2 1 i + D21 T D k T j B2 T + ˆB 1 T j + D21 T D k T i B2 T (Ĉδi Ŷ j + D δ2 Ckj + Ĉδj Ŷ i + D δ2 Cki L 2 L 2 1 Ŷ 2 (Ĉ1i j + D 12 Ckj + Ĉ1 j Ŷ i + D 12 Cki 7.3 Controller Synthesis for Uncertain Affine LPV Systems γi ( ˆDδ1i + D δ2 D kj D 21 + ˆD δ1j + D δ2 D ki D ( ˆD11i + D 12 D kj D 21 + ˆD 11j + D 12 D ki D 21 Inequality (7.81 can be also rewritten as r i=1 r k=1 αi 2 β k (Ψ clii + Q T ˆKT i P + P T ˆKi Q L 2 1 γl r γl ( ˆDδδi + D D δ2 D kj 2δ + ˆD + D D δδj δ2 D ki 2δ ( ˆD1δi + D D 2 12 D kj 2δ + ˆD + D D 1δj 12 D ki 2δ γi r i=1 j=i+1 k=1 < (7.81 ( r 1 α i α j β k 2 ( Ψ clij + Q T ˆKT i P + P T ˆKi Q + Ψ clji + Q T ˆKT j P + P T ˆKj Q < (7.82 where Ψ cl = X k + ˆX  + (  Ỹk +  Ŷ + ( ˆB δ T ˆX ˆBT δ γl ˆB 1 T ˆX ˆBT 1 γi LĈδ LĈδ Ŷ L ˆD δδ L ˆD δ1 γl Ĉ 1 Ĉ 1 Ŷ ˆD1δ ˆD11 γi Q = [ [ ] ] Ip C, D δ21, D 121, (p+q2 (q 1 +s, B = B 2 P = [ ( ] BT, (p+m2 (m 1 +s, Dδ T 12, D1 T Ãki Bki 12, ˆKi = C ki Dki (7.83 (7.84 (7.85 with the subscript denote (ii, ij, or ji. By Lemma with a known scaling matrix L and matrix vertices ( ˆX i, Ŷi, i = 1, 2,..., r, the system matrix vertices ˆK i can be determined from (7.82, that is an LMI in ˆK i, at all vertices for which ( ˆK 1, ˆK 2,..., ˆK r have to satisfy all of r 2 (r + 1/2 LMIs. Furthermore, knowing à ki,..., D ki, the controller system matrices A k (θ, θ,..., D k (θ can be computed on-line in real-time using (6.2 (6.22 with an instantaneous measurement value of θ and θ. 129

156 7.3 Controller Synthesis for Uncertain Affine LPV Systems However, the parameter derivatives either are not available or are difficult to estimate during system operation [8]. We have (6.41 as a result to avoid using the measured value of θ. By Lemma with a known scaling matrix L, the LMIs of (7.82 are solvable for ˆK i if and only if there exist a pair of positive definite symmetric matrices ( X(θ, Y (θ that satisfy the following LMIs: r i=1 k=1 Ĉ T 1 i ˆD T 1δ i ˆD T 11 i γi r ( αi 2 NX β k I ( NX I T r  T i X i + X i  i + X k X i ˆBδi X i ˆB1i Ĉδ T i ˆB δ T i X i γl ˆDT δδi ˆB 1 T i X i γi ˆDT δ1i Ĉ δi ˆDδδi ˆDδ1i γl 1 Ĉ 1i ˆD1δi ˆD11i r i=1 j=i+1 k=1 r ( NX α i α j β k I γl 1 γi 1 2( ˆDδδi + ˆD ( 1 δδj 2 ˆDδ1i + ˆD δ1j γl 1 1 2( ˆD1δi + ˆD ( 1 1δj 2 ˆD11i + ˆD 11j γi r i=1 k=1 ˆB 1i ˆD δ1i ˆD 11i γi r ( αi 2 NY β k I ( NY I T + 2 r 1 T ( NX I  i Y i + Y i  T i Ỹk Y i Ĉδ T i Y i Ĉ1 T ˆBδi i Ĉ δi Y i γl 1 ˆDδδi Ĉ 1i Y i γi ˆD1δi ˆB δ T ˆDT i δδi ˆDT 1δi γl ˆB 1 T ˆDT i δ1i ˆDT 11i r i=1 j=i+1 k=1 r ( NY α i α j β k I γl 1 γi 1 2( ˆDT + ˆD T 1 δδi δδ j 2( ˆDT + ˆD T 1δi 1δ j γl 1 2( ˆDT + ˆD T 1 δ1i δ1 j 2( ˆDT + ˆD T 11i 11 j γi r ( ˆXi I α i I i=1 Ŷ i ( NY I T 1 2(ÂT i ˆXj + ÂT ˆX j i + ( + X k 1 2( ˆBT δi ˆX j + ˆB δ T ˆXi j 1 2( ˆBT 1i ˆX j + ˆB 1 T ˆXi j 1 (Ĉδi 2 + Ĉδ j (Ĉ1i + Ĉ1 j 1 2 < ( (Âi Ŷ j + ÂjŶi + ( 1 (Ĉδi 2 Ŷ j + Ĉδ j Ŷ i 1 (Ĉ1i 2 Ŷ j + Ĉ1 j Ŷ i 1 2( ˆBT + ˆB T δi δ j 1 2( ˆBT + ˆB T 1i 1 j < (7.87 > (7.88 where N X and N Y denote bases of the null spaces of [C 2, D 2δ, D 21 ] and [B T 2, D T δ2, DT 12], respectively. Note that (7.88 ensures X(θ, Y (θ > and X(θ Y (θ 1. By 13 Ỹk

157 7.3 Controller Synthesis for Uncertain Affine LPV Systems Lemma 3.3.1, (7.86 (7.88 are sufficiently evaluated at all vertices. Also note that (7.86 (7.88 will become (7.7 (7.72 when both X and Y are constant. Moreover, the quadratic H performance γ is determined for both the ( X(θ, Y and ( X, Y (θ cases and the case that gives lowest γ is selected. Theorem Given a scaling matrix L, there exists an LPV controller K(θ guaranteeing the closed-loop system, (7.54 and (7.56, quadratic H performance γ along all possible parameter trajectories, (θ, θ Θ Φ, if and only if the following LMI conditions hold for some positive definite symmetric matrices (X(θ, Y (θ, which further satisfy rank(x(θ Y 1 (θ p: ( NX I ( NY I ( NX I T T T  T i X i + X i  i + X k X i ˆBδi X i ˆB1i Ĉ T δ i Ĉ T 1 i ˆB T δ i X i γl ˆDT δδi ˆDT 1δi ˆB T 1 i X i γi ˆDT δ1i ˆDT 11i Ĉ δi ˆDδδi ˆDδ1i γl 1 Ĉ 1i ˆD1δi ˆD11i γi  i Y i + Y i  T i Ỹk Y i Ĉ T δ i Y i Ĉ T 1 i ˆBδi ˆB1i Ĉ δi Y i γl 1 ˆDδδi ˆDδ1i Ĉ 1i Y i γi ˆD1δi ˆD11i ˆB T δ i ˆDT δδi ˆDT 1δi γl ˆB T 1 i ˆDT δ1i ˆDT 11i γi (ÂT i ˆXj + ÂT ˆX j i + ( + 2 X k ˆB δ T ˆXj i + ˆB δ T ˆXi j 2γL ˆB 1 T ˆXj i + ˆB 1 T ˆXi j Ĉ δi + Ĉδ j ˆDδδi + ˆD δδj ( NX I ( NY I < < (7.89 (7.9 ( NY I T Ĉ 1i + Ĉ1 j ˆD1δi + ˆD 1δj ( 2γI NX ˆD δ1i + ˆD < (7.91 δ1j 2γL 1 I ˆD 11i + ˆD 11j 2γI (Âi Ŷ j + ÂjŶi + ( 2Ỹk Ĉ δi Ŷ j + Ĉδ j Ŷ i 2γL 1 Ĉ 1i Ŷ j + Ĉ1 j Ŷ i ˆB δ T i + ˆB δ T ˆDT j δδi + ˆD δδ T j ˆB T 1 i + ˆB T 1 j 2γI ˆD 1δ T i + ˆD 1δ T j 2γL ˆD T 11 i + ˆD T 11 j 2γI ˆDT δ1i + ˆD T δ1 j ( NY I < (

158 7.4 Numerical Example ( ˆXi I I Ŷ i > (7.93 for i, k = 1,..., r and 1 i < j r. Algorithm Computation of X i, Y i, i = 1,..., b and L Step 1: Setting L = I, solve (7.89 (7.93 for X i, Y i by minimizing γ. Step 2: Knowing X i, Y i, and L, solve (7.82 for ˆK i. Step 3: With ˆK i fixed, solve (7.81 for L by minimizing γ (heuristical. Step 4: With L fixed, solve (7.89 (7.93 for new X i, Y i by minimizing γ. Step 5: Iterate over Steps 2 to 4 until γ is converged to some minimum value. Note that, by Lemma 3.3.1, (7.81 and (7.82 are sufficiently solved at all vertices. 7.4 Numerical Example We demonstrate the robustness improvement through the numerical example of [61], described in Chapter 4. Consider the transfer function of the Jacobian-based LPV model taken from [3] P (s, θ = 1, θ [, 1] (7.94 (s + 1(s + 2θ Having added a time-varying real parametric uncertainty δ, (7.94 becomes P δ (s, θ, δ = 1, (θ, δ [, 1] [ 1, 1] (7.95 (s + 1(s + 2θ + δ To demonstrate the impacts of the proposed scheme in robustness improvement of uncertain affine LPV systems, we compare our approach with the existing uncertain LPV/LFT approach [9, 24]. Figure 7.3 shows both uncertain affine LPV and LPV/LFT closed-loop systems. First, consider uncertain LPV/LFT approach [9, 24], we have θ = (θ 5/5 [ 1, 1], using (3.36, and δ = δ [ 1, 1]. Having augmented P (s with two weighting function presented in [61], we get P a (s with state-space realization ẋ 1 1 x 1 1 ẋ 2 ẋ 3 = 1 1 x x w θ 1 L 2 w θ +.5 w + w ẋ 4 1 x 4 δ 4 x z θ 1 z θ = L x 2 w θ 1 + L 1 2 L 1 2 w θ + L 1 2 w + L 1 2 z δ 1 w δ x 3 x [ ] u ũ [ ] u ũ

159 7.4 Numerical Example [ ] [ ] x 1 [ ] z1 1 = x 2 w θ [ ] [ ] [ ] z 2 5 x 3 + L 1 2 w θ u + w +.2 ũ w x δ 4 [ ] [ ] x 1 [ ] y 1 = x 2 w θ [ ] [ ] [ ] ỹ x 3 + L 1 2 w 1 θ 1 u + w + ũ w x δ 4 w θ θ w θ = θ z θ z θ (7.96 w δ δ z δ where L 1θ L 2θ L = L T 2 θ L 3θ L δ An LPV controller can be computed using Theorem 5.1 [9]. Unfortunately, the convexity of Theorem 5.1 is destroyed by the extra uncertainty block δ, and D- K iterations are needed to compute adequate LPV controllers. Solving LMIs in Theorem 5.1 by using a MATLAB Robust Control Toolbox function [12], mincx, we got γ =.5524, a scaling matrix, and an LPV controller as shown below L = ẋ k x k ẋ k2 ẋ k3 = x k x k y ẋ k x k ỹ u = [ ] x k1 x k2 x k3 x k y ỹ ũ = [ ] x k2 x k y ỹ x k4 ỹ = δũ x k1 (7.97 Next, consider our approach, having augmented P (s, θ with two weighting function 133

160 7.4 Numerical Example (a Uncertain affine LPV closed-loop structure I L L L L L T I L L L L L T (b Uncertain LPV/LFT closed-loop structure Figure 7.3: Structure comparisons of uncertain affine LPV and LPV/LFT [9] closedloop systems. 134

161 7.4 Numerical Example presented in [61], we get P a (s, θ with state-space realization ẋ 1 1 ẋ 2 ẋ 3 = 1 2θ.5.2 ẋ 4 1 [ z1 z δ = L 1 [ ] 2 δ 1 z 2 ] = [ 1 5 ] x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 1 x 2 x 3 x 4 + L 1 [ ] 2 δ L 1 2 [ ] + L 1 2 δ w δ L 1 2 δ w δ + δ w δ + L 1 [ ] 1 2 δ w + L 2 δ [ ] [ ] w + u.2 y = [ 1 ] x 2 x 3 + [ ] L 1 2 δ w δ + [ 1 ] w + [ ] u x 4 w δ = δz δ.5 1 w + u 4 [ ] u (7.98 Solving Algorithm with the same scaling matrix of δ, i.e. L δ =.2756, and the LMIs are solved using a MATLAB Robust Control Toolbox function [12], mincx, we got γ =.2435 and an LPV controller as shown below ẋ k = ( α 1 A k1 + α 2 A k2 xk + ( α 1 B k1 + α 2 B k2 y where α 1 = (1 θ/1, α 2 = θ/1 and A k1 = A k2 = u = ( α 1 C k1 + α 2 C k2 xk + ( α 1 D k1 + α 2 D k2 y ( B k1 = , B k 2 = C k1 = [ ] C k2 = [ ] D k1 = , D k2 =

162 7.5 Lateral Uncertain Affine LPV Model Y (units Hinfgs controller Uncertain LPV/LFT controller Proposed scheme controller Time (s Figure 7.4: Nonlinear step response from -3 to of the LPV ctroller with the original nonlinear plant. Without including the uncertainty δ and using the MATLAB Robust Control Toolbox function [12], hinfgs, an LPV controller was obtained with γ =.1211 [3]. However, the closed-loop system of the LPV controller with the original nonlinear model is unstable [3, 61]. Including the uncertainty δ, shown in (7.96 and (7.98, using Theorem 5.1 [9] and Theorem 7.3.1, two LPV controllers were obtained with γ =.5524 and.2435, respectively. As an effect of including the uncertainty δ, we get more conservatism on γ but the stability property of the closed-loop system is guaranteed with a larger stability margin. Hence, the closed-loop instability for the controller with the nonlinear plant disappears without degrading the transient performance as shown in Figure 7.4. Moreover, this numerical example shows that our approach is less conservative than the uncertain LPV/LFT approach. 7.5 Lateral Uncertain Affine LPV Model Following section 6.4, the lateral nonlinearly parameter-dependent LPV model, shown in (7.6, can be converted into an affine LPV model using a minimum least-squares method [58]. Moreover, the maximum and minimum of the leastsquares conversion errors can be determined for all possible parameter trajectories, i.e. (u, h [464.1, 548.5] [75, 125], by subtracting the converted (affine stability and control derivatives from the stability and control derivatives in (7.7 (7.24. For example, consider Y δr shown in (7.6, define Y δraffine to be depended 136

163 7.5 Lateral Uncertain Affine LPV Model affinely on u and h, in a similar manner to (6.5, as Y δraffine = Y δr + uy δru + hy δrh (7.1 Then, Y δr, Y δru, and Y δrh can be determined using a minimum least-squares method [58] as presented in section 6.4. Figure 7.5 a and b shows a nonlinear Y δr and an affine Y δr that are determined using (7.12 and (7.1, respectively. The error (mismatch uncertainty between a nonlinear Y δr and an affine Y δr can be computed as δ Yδr = Y δr Y δraffine (7.11 The calculations of δ Yδr and δ Yδr (normalize, that is computed using (7.28, are shown in Figure 7.6 a and b, respectively. Therefore, an uncertain affine Y δr can be written as where Y δr (u, h, δ = ( Y δr + T Yδr + uyδru + hy δrh + S Yδr δyδr (7.12 T Yδr = δ Y δr + δ Yδr 2 (7.13 S Yδr = δ Y δr δ Yδr 2 (7.14 Figure 7.6 a also shows a maximun δ Yδr and minimum δ Yδr errors of Y δr that are equal to.3696 and , respectively. Having normalized all of the mismatch uncertainties δ Yv, δ Yp,..., δ Nδr using (7.28, and included the normalized mismatch uncertainties in the system state-space model matrices of the converted affine LPV model, we can compute the lateral uncertain affine LPV model, in a similar manner to (7.12, as ẋ = A(u, h, δx + B(u, h, δu (7.15 where x= [v p r φ ψ] T, u= [δ a δ r ] T, (u, h [464.1, 548.5] [75, 125], i.e. bounded flight envelope, and A(u, h, δ = u h

164 7.6 Robust Gain-Scheduled H Autopilot Design + B(u, h, δ = +h.79214( δ Yv.2339( δ Yp ( δ Yr.1143( δ Yφ.1125( δ Lv.1241( δ Lp.8129( δ Lr ( δ Nv ( δ Np.1572( δ Nr u ( δ Yδr.37667( δ Lδa.12133( δ Lδr.46399( δ Nδa.143( δ Nδr Note that δ Yv, δ Yp,..., δ Nδr [ 1, 1] and the uncertainty δ in (7.15 can be separated from the system matrices in a similar manner to ( Robust Gain-Scheduled H Autopilot Design The mixed-sensitivity criterion (5.42 is also employed in a similar manner to section 5.3. Figure 7.7 shows the weighted open-loop interconnection for synthesis where [ ] T [ ] T z δ = z δyv, z δyp,, z δnδr, z = zv, z ψ, z δa, z δa [ ] T [ ] T w δ = w δyv, w δyp,, w δnδr,, w = vref, ψ ref y = [ v ref v, p, r, φ, ψ ref ψ, ] T [ ] T, u = δa, δ r δ = diag( δ Yv, δ Yp,..., δ Nδr (.6667s+.221 W 1 (s = s+.221 W 2 (s = W pre-filter (s = s+1.15 s (.6s s s s+1.15 ( 5 s+5 5 s+5 (7.16 Note that the values of weighting functions W 1 and W 2 are hand-tuned until the desired objectives of performance and robustness of the closed-loop system are achieved. After the lateral uncertain affine LPV model, shown in (7.15, is augmented with the weighting functions, shown in (7.16, a pair of positive definite symmetric matrices ( X(θ, Y (θ ( ( ( ( can be determined in four cases, i.e. X, Y, X(θ, Y, X, Y (θ, and X(θ, Y (θ, using Theorem with the same scaling 138

165 7.6 Robust Gain-Scheduled H Autopilot Design (a Nonlinear Y δr (b Affine Y δr Figure 7.5: The variation of a nonlinear Y δr and an affine Y δr with speed and altitude 139

166 7.6 Robust Gain-Scheduled H Autopilot Design (a δ Yδr (b δ Yδr (Normalize Figure 7.6: The variation of δ Yδr and δ Yδr (normalize with speed and altitude 14

167 7.7 Nonlinear Simulation Results Table 7.3: Performance γ comparison for different cases of (X(u, h, Y (u, h Flight condition (X(u, h, Y (u, h (X(u, h, Y (X, Y (u, h (X, Y * (u, h [464.1, 548.5] [75, 125] ( u, ḣ [ 1.26, 1.26] [ 2, 5] Performance γ Parameter-dependent Lyapunov functions, Theorem * Single quadratic Lyapunov function, Theorem W1 W2 G(θ Pre-filter P(θ Figure 7.7: The weighted open-loop interconnection for the lateral uncertain affine LPV plant model. matrix L = diag(79.126,.43552, 91.49, 44.6, 7.34, 3.941, , 15, 15.7, 144.9, , , 3.88, 129.2, 816 for all cases, for which the performance measure (γ, shown in Table 7.3, can be compared. The LMIs are solved using the MATLAB Robust Control Toolbox function [12], mincx. 7.7 Nonlinear Simulation Results Since, for an LPV plant model with a large parameter variation region, it is often conservative to design a single LPV controller over the entire parameter space [64, 65, 66], the designed H gain-scheduling autopilot is validated for a bounded flight envelope, i.e. (u, h [464.1, 548.5] [75, 125], ( u, ḣ [ 1.26, 1.26] [ 2, 5], with the Jindivik nonlinear model [41] in a MATLAB Simulink simulation. In Figure 7.8, the transient responses of the simulated vehicle for small demanded changes in yaw angle are shown for one particular point in the bounded flight envelope. Similar responses for other points in the flight envelope were obtained. Figure 7.9 shows 141

168 7.8 Conclusion a rate one turn simulated flight, i.e. 3 per second turn, which completes a 36 turn in 2 minutes, for one particular point in the bounded flight envelope. Again, similar responses for other points in the flight envelope were obtained. These simulation results show that desired performance and robustness objectives are achieved over the defined flight envelope. 7.8 Conclusion In this chapter, the mismatch uncertainties between a nonlinear model and an affine LPV model are handled by a new class of affine LPV systems which is called an uncertain affine LPV model. New sufficient conditions of gain-scheduled H performance analysis and synthesis, for this uncertain affine LPV model, using single quadratic or parameter dependent Lyapunov function are proposed for which the proposed scheme can guarantee robust stability and robust performance for all time-varying real parametric uncertainties which are δ < 1/γ. The analysis and synthesis conditions are represented in the form of a finite number of LMIs. A numerical example was compared with uncertain LPV/LFT approach [9] results. In addition, the intermediate controller variables, i.e. Â k (θ, ˆBk (θ, Ĉ k (θ and D k (θ, can be constructed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter without the need for constraints on the D 12 and D 21 matrices. The proposed scheme was applied to synthesized a lateral LPV autopilot of the Jindivik UAV. The designed controller was tested with a full 6-DOF simulation of the vehicle and nonlinear simulation results show the effectiveness of the proposed method. 142

169 7.8 Conclusion 5 4 Yaw angle demand Yaw angle response Time (s 2 (a ψ Time (s (b φ 143

170 7.8 Conclusion Time (s 15 (c β Time (s (d p 144

171 7.8 Conclusion Time (s.4 (e r Time (s (f a y 145

172 7.8 Conclusion Time (s 1.5 (g δ r Time (s (h δ a Figure 7.8: The transient performance of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 464 ft/s and altitude = 7,5 ft. 146

173 7.8 Conclusion 2 15 Yaw angle demand Yaw angle response 1 Yaw angle (deg Time (s (a ψ (b φ 147

174 7.8 Conclusion (c β (d p 148

175 7.8 Conclusion (e r (f a y 149

176 7.8 Conclusion (g δ r (h δ a Figure 7.9: The rate one turn of H gain-scheduling autopilot is validated with the Jindivik nonlinear dynamic model about one condition inside the flight envelope, i.e. speed = 56 ft/s and altitude = 1, ft. 15

177 Chapter 8 Conclusions In this thesis, an LPV control approach has been employed to design a robust gainscheduled flight controller for a conventional fixed-wing UAV. The work presented in this thesis was motivated by the shortcomings of the conventional gain-scheduling techniques that are both expensive and time-consuming for many UAV applications. The effectiveness of the proposed methods for designing flight controllers is verified and validated through the 6-DOF nonlinear model [41] in MATLAB Simulink environment of the Jindivik UAV. However, the proposed methods are also applicable to a general class of conventional fixed-wing aircrafts. This chapter discusses some aspects of the work, lists the main contributions and provides suggestions for future work. 8.1 Conclusions & Discussions The dynamic characteristics of both lateral and longitudinal modes of an aircraft are represented by the stability and control derivatives, shown in (A.13 and (A.14, that vary following speed and altitude, shown in (5.14 (5.36 and (7.7 (7.24. With the speed and altitude fixed, both (A.13 and (A.14 are longitudinal and lateral LTI models, respectively. Suppose an aircraft is assumed about a wings level, constant altitude and airspeed flight condition, both nonlinearly parameter-dependent longitudinal and lateral LPV models, shown in (5.13 and (7.6 respectively, can be derived from a 6-DOF nonlinear model using Jacobian linearization. Two interesting features of the derived LPV models are (i they can accurately represent nonlinear dynamic characteristics of the 6-DOF nonlinear model better than the longitudinal and lateral LTI models, shown in (A.13 (A.14 respectively, because they use the time-varying parameters θ (i.e. speed and altitude to capture the nonlinear dynamic characteristics of the original nonlinear model and (ii they are still a linear system where the system matrices are functions of speed and altitude. Note that, the speed and altitude are 151

178 8.1 Conclusions & Discussions also the state variables of the system. Comparison with the 6-DOF nonlinear models, the longitudinal and lateral LPV models are easier to prove stability using the single quadratic (Theorem or parameter-dependent Lyapunov functions (Theorem Equations (5.14 (5.36 and (7.7 (7.24 also show that the accuracy of both longitudinal and lateral LPV models, shown in (5.13 and (7.6 respectively, depend on the accuracy of the information that provides the aerodynamic and thrust coefficients. To estimate stability and control derivatives or aerodynamic coefficients of a conventional fixed-wing UAV, system identification techniques are preferred to the wind tunnel tests. However, a number of difficulties arise when system identification techniques are applied to certain UAVs, described in Appendix B. In addition, Appendix B shows that, when an UAV is flown as a racetrack manoeuvre pattern, those difficulties can be overcome. To synthesize an LPV controller based on Theorem (SQLF with a finite number of LMIs and avoiding the gridding parameter technique, the Tensor- Product (TP model transformation can be applied to transform a nonlinearly parameter-dependent LPV model, shown in (5.13, into a TP-type convex polytopic model form, shown in (5.37. The TP-based LPV controller, shown in (5.46, can be constructed as a convex combination of the vertex coordinates of the scheduled parameter. Hence there is less computational on-line complexity at the gain-scheduling level than the grid-based LPV controller but its structure is still more complex than the affine-based LPV controller. An affine LPV model, shown in (6.55, is converted from a nonlinearly parameterdependent LPV model, shown in (5.13, using the minimum least-squares method [58]. Based on an affine LPV model, an LPV controller can be synthesized with a finite number of LMIs using Theorem (SQLF. However, the affine LPV model can rarely accurately represent the original nonlinear model and, in addition, the SQLF-based LPV controller is conservative when the parameters are time-invariant or slowly varying [45]. The example from Leith and Leithead [61] is very interesting. The closed-loop instability of the LPV controller, shown in (4.11, with the original nonlinear model, shown in (4.1, occurs because the mismatch uncertainty between the Jacobian-based LPV model, shown in (4.5, and the original nonlinear model is in a region close to the right-half s-plane, described in Chapter 4. The mismatch uncertainties between a nonlinear model and an affine LPV model can be handled by a new class of affine LPV systems which is called an uncertain affine LPV model, described in Chapter 7. Synthesizing an LPV controller based on this uncertain affine LPV model, shown in (7.98, the stability property of the Leith and Leithead s example closed-loop system can be guaranteed with a larger stability margin. Hence, the closed-loop instability for the LPV controller, shown in (7.99, with the nonlinear plant disappears without degrading the transient performance as shown in Figure 7.4. Based on the lateral uncertain affine LPV model, shown in (7.15, an LPV 152

179 8.2 Main Contributions autopilot can be synthesized with a finite number of LMIs using Theorem (SQLF or (PDLF for which the resulting controller can guarantee the closed-loop system with a larger stability margin over conventional affinebased LPV controller. As shown by the simulation results of both longitudinal and lateral LPV autopilots in Chapters 5 7, the desired performance and robustness objectives are achieved across the defined flight envelope. Both aims and objectives of this thesis are achieved since the proposed schemes yield an LPV controller that can handle both uncertainties and nonlinearities of a 6-DOF nonlinear model with good command following, good disturbance attenuation, low sensitivity to measurement noise, reasonably small control efforts, and that is robustly stable to additive plant perturbations. We emphasize that the proposed schemes in this thesis can be a strong and very likely candidate for the next generation of flight control systems design for a conventional fixed-wing UAV. 8.2 Main Contributions The main contributions of this thesis are the following: The determination of aircraft aerodynamic coefficient from wind-tunnel or computational fluid dynamics data can be an expensive and time-consuming procedure. A system identification techniques, i.e. equation-error method [58], is used to obtain this information from a racetrack manoeuvre with sufficient accuracy to design a satisfactory flight control system for an UAV [36]. Knowing the aerodynamic coefficients, aircraft moments of inertias, and thrust coefficients, an LPV aircraft model can be derived from a 6-DOF nonlinear model using Jacobian linearization method. A tensor-product (TP model transformation [15] is applied to transform a longitudinal nonlinearly parameter depedent LPV model to a TP-type convex polytopic model form. A gainscheduled output feedback H controller [1] that is based on single quadratic Lyapunov functions is applied to the resulting TP convex polytopic model to yield a longitudinal LPV autopilot that guarantees the stability, robustness and performance properties of the closed-loop system [33]. New sufficient conditions for gain-scheduled H performance analysis and synthesis for a class of affine LPV systems using parameter-dependent Lyapunov function are proposed, in Theorem Compared with the multi-convexity technique [11], fewer linear matrix inequalities (LMIs and decision variables are required and the computational time is lower while the achieved performance level is improved. The analysis and synthesis conditions are represented in the form of a finite number of LMIs. In addition, the intermediate controller variables, i.e. Â k (θ, ˆBk (θ, Ĉ k (θ and D k (θ, can be constructed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameter 153

180 8.3 Further Work without the need for constraints on the D 12 and D 21 matrices. The proposed method is applied to synthesize a longitudinal LPV autopilot for a bounded flight envelope of a Jindivik UAV [34]. The mismatch uncertainties between a nonlinear model and an affine LPV model are handled by a new class of affine LPV systems which is called an uncertain affine LPV model. New sufficient conditions for gain-scheduled H performance analysis and synthesis for this uncertain affine LPV model, using single quadratic or parameter-dependent Lyapunov functions are proposed. These are shown in Theorems and respectively, for which the proposed scheme can guarantee robust stability and robust performance for all possible time-varying real parametric uncertainties that are δ < 1/γ. The analysis and synthesis conditions are represented in the form of a finite number of LMIs. A numerical example [61] is compared with uncertain LPV/LFT approach [9] results. The proposed scheme is applied to synthesize a lateral LPV autopilot for a bounded flight envelope of a Jindivik UAV [35]. 8.3 Further Work This thesis shows that an LPV controller can be synthesized as a single controller that will operate over the whole range of operating condition without having to create a scheduling scheme. Having assumed the complete measurement of the time-varying parameters θ is available for the controller to incorporate in the same LPV fashion as the plant model, the resulting LPV controller exploits all available information of θ to adjust its dynamic to the current plant dynamic on-line in realtime over the defined operating conditions. This provides smooth and automatic gain-scheduling with respect to θ. However, there are still several directions that should be further researched and developed as outlined below. In this thesis, both longitudinal and lateral LPV models are derived from the 6-DOF nonlinear model of the Jindivik UAV using Jacobian linearization. However, the Jacobian-based LPV models only accurately represent the original nonlinear dynamics about the neighborhood of a set of equilibrium points and the time-varying parameters must vary slowly. Hence, the Jacobian linearization method is not suitable to derive an LPV aircraft model under the case of high angle of attack and extremely aggressive manoeuvring flight conditions. It is very interesting to apply the state transformation or function substitution methods to derived an LPV aircraft model under such highly nonlinear flight conditions. An initial research using the function substitution method to derive a quasi-lpv model of the F-16 aircraft that can cover the aircraft non-trim region has been proposed by Shin et al. [94]. In this thesis, a small angle of attack flight condition is considered therefore there are only two scheduling parameters that are speed and altitude. However, 154

181 8.3 Further Work under the case of high angle of attack region, the angle of attack have a large variation hence it has to be included in the scheduling parameters where we believe that further improvements in the transient performance of both longitudinal and lateral LPV autopilots can be achieved if the autopilot is gain-scheduled with speed and altitude as well as the angle of attack. Then, it is also very interesting to design an LPV controller having speed, altitude and angle of attack as the scheduling parameters. Commonly, a feedforward controller is used to improve the transient performance of the closed-loop system. Hence, it is also very interesting to integrate the feedforward LPV controller into the gain-scheduled output feedback H control design framework where an initial research of this approach has been proposed by Prempain and Postlethwaite [85]. Compared with an affine LPV model, a TP convex polytopic model more accurately represents the original nonlinear model. But, the TP polytopic model can not be written as an affine combination form. Therefore, to synthesize an LPV controller based on a TP polytopic model, a finite number of LMIs can be obtained only when using the single quadratic Lyapunov function. It is also very interesting (i to extend the existing TP model transformation method [15] so that the resulting LPV model can be written in both convex and affine combination forms or (ii to research a new convexifying techniques so that a finite number of LMIs can be obtained when synthesizing an LPV controller based on the existing TP convex polytopic model and using the parameter-dependent Lyapunov function. The TP-based LPV controller has a very high complexity. Hence controller complexity reduction is an important issue for the practical implementaion of the method, and this aspect is also very interesting for further work. In this thesis, the effectiveness of the designed LPV autopilots, described in Chapters 5 7, is verified and validated only through the nonlinear simulation in MATLAB Simulink environment. It is also very interesting to test those designed LPV autopilots in the hardware-in-the-loop simulation. 155

182 This page intentionally contains only this sentence.

183 Appendix A Aircraft Nonlinear Model The full details of equations of motion for a conventional fixed-wing aircraft are well known and can be found in many textbooks, e.g. Cook [38], Klein and Morelli [58], Nelson [79], etc. The following sections give an overview of the equations of motion which is formulated as ordinary differential equations for the aircraft states with algebraic equations for the measured outputs. Before the equations of motion can be developed, it is necessary to define a suitable coordinate system and sign conventions (of the aircraft states, the control surfaces, and the measured outputs for the formulation of the equations of motion. A.1 Reference Frames & Sign Conventions There are a variety of reference frames that would be used to describe aircraft movement and orientation for different purposes such as earth axes would be considered for a navigation control, and body axes would be considered for a stability control. The relevant frames are described in the following list [58]: Earth axes: X E, Y E, Z E. This reference frame is also called the topodetic axes. Its origin is at an arbitrary point on the earth surface, with positive X E axis pointing toward north, positive Y E axis pointing east, and positive Z E axis pointing to the center of the earth as shown in Figure A.1. Earth axes are fixed with respect to the earth. Vehicle-carried earth axes: X V, Y V, Z V. Origin is at the centre of gravity of the aircraft (c.g., orientation of the axes is parallel to earth axes (see Figure A.1. The centre of gravity is the point about which the aircraft would balance if suspended by a cable. This reference frame is used to show the rotational orientation of the aircraft relative to earth axes. Body axes: X B, Y B, Z B. Origin is at the aircraft c.g., with positive X B axis pointing forward through the nose of the aircraft, positive Y B axis out the right wing, and 157

184 A.1 Reference Frames & Sign Conventions +δ r Reference Frames -δ al +δ ar +δ e + M Y W Y B +β Y S -Ø Y V Z W Z S +α -θ Z B Z V X W X V +ψ X B X S + N + L Earth axes X E (North Y E (East Z E (Down Figure A.1: Reference axes & sign conventions [58] positive Z B axis is directed through the underside. Body axes are fixed with respect to the aircraft body as shown in Figure A.1. The XZ plane is commonly a plane of symetry for the aircraft. Stability axes: X S, Y S, Z S. This reference frame is a type of body axes, used in the study of small deviations from a nominal flight condition. The orientation of stability axes is related to a reference flight condition, usually defined at the start of a manoeuvre. Its origin is at the aircraft c.g., with positive X S axis forward and aligned with the projection of the velocity vector of the aircraft c.g. through the air (also called the air-relative velocity onto the XZ plane in body axes. The positive Y S axis out the right wing, and positive Z S axis is directed through the underside (see Figure A.1. Wind axes: X W, Y W, Z W. This reference frame, also called the flight-path axes, has its origin at the aircraft c.g., with positive X W axis forward and aligned with the air-relative velocity vector, positive Y W axis out the right wing, and positive Z W axis through the underside in the XZ plane in body axes (see Figure A.1. The X W axis is always tangent to the air-relative trajectory, hence wind axes are not fixed with respect to the aircraft body. Sign conventions: For translational velocities at the aircraft c.g. along body axes, e.g. axial (u, lateral (v, and normal velocities (w or applied forces at the aircraft c.g. along body axes, e.g. axial force (X, side force (Y, normal force (Z, the 158

185 A.1 Reference Frames & Sign Conventions positive sign convention follows the positive direction of a body axis (see Figure A.1. In addition, angular velocities about the aircraft c.g. along body axes, e.g. pitch rate (p, roll rate (q, and yaw rate (r or applied moments about the aircraft c.g. along body axes, e.g. rolling moment (L, pitching moment (M, and yawing moment (N, the sign convention follows the right-hand rule. If the right-hand thumb is pointed in the positive direction of a body axis, the fingers curl in the direction of positive rotation (see Figure A.1. Control surfaces are hinged surfaces that can be rotated about a hinge line to change the applied aredynamic forces and moments on an aircraft. For control surfaces of a conventional aircraft, e.g. elevator δ e, aileron δ a, and rudder δ r, the sign convention also follow the right-hand rule. For example, the positive sign of elevator control surfaces, If the right-hand thumb is pointed in the positive direction of a Y B axis, the fingers curl in the direction of positive deflection (see Figure A.1. However, some control surfaces, i.e. the ailerons, are deflected simultaneously in an asymmetric manner, which means that the individual aileron control surfaces on each wing move in opposite directions. Following Klein and Morelli [58], a positive aileron deflection is defined as one-half the right aileron deflection minus the left aileron deflection, δ a = 1 2 (δ ar δ al (A.1 In addition, the angle of attack (α and sideslip angle (β can be defined in terms of the velocity components of air-relative velocity (V as shown in Figure A.1. The positive sign convention of α and β follow the positive sign of w and v respectively. α = arctan w u β = arcsin v V V = u 2 + v 2 + w 2 u cos α cos β v = V w sin β sin α cos β (A.2 Moreover, the relative orientation of the body axes to earth axes or to vehicle-carried earth axes are the same angles, known as Euler angles, e.g. pitch angle (θ, roll angle (φ, and yaw angle (ψ as shown in Figure A.1. The positive sign convention of θ, φ, and ψ are defined by the rotation direction of the vehicle-carried earth axes to body axes, which follows the right-hand rule. If the right-hand thumb is pointed in the positive direction of a rotating axis, the fingers curl in the positive rotation direction about the rotating axis. The sequence for rotating vehicle-carried earth axes into alignment with body axes starts with a yaw angle rotation ψ about the Z V axis. In Figure A.1, it is a positive rotation direction according to right-hand rule stated above. This followed by a pitch angle rotation θ (a negative rotation direction about the Y V axis, completed by a roll angle rotation φ (also a negative rotation direction about the X V axis. Therefore, an arbitrary three dimensional vector at earth axes or vehicle-carried earth axes, 159

186 A.2 Aircraft Equation of Motion e.g. gravity force (F G, can be transformed to body axes using a transformation matrix L BV, known as Direction Cosine Matrix (DCM, as shown below, cos θ cos ψ cos θ sin ψ sin θ L BV = sin φ sin θ cos ψ cos φ sin ψ sin φ sin θ sin ψ + cos φ cos ψ sin φ cos θ cos φ sin θ cos ψ + sin φ sin ψ cos φ sin θ sin ψ sin φ cos ψ cos φ cos θ (A.3 mg x F G = mg y mg z B = L BV mg V mg sin θ = mg sin φ cos θ mg cos φ cos θ (A.4 A.2 Aircraft Equation of Motion The general motion of an aircraft, that is assumed a rigid body with fixed mass distribution and constant mass, can then be derived from Newton s second law of motion in translational and rotationnal forms, It is noted that all the given equations in this section are taken from Klein and Morelli [58], F = m V + ω mv M = I ω + ω Iω I x I xy I xz I = I yx I y I yz I zx I zy I z (A.5 where F is the applied forces at the aircraft c.g. along body axes (i.e. X, Y, and Z, m is the mass of the aircraft, V is the translational velocities at the aircraft c.g. along body axes (i.e. u, v, and w, ω is the angular velocities about the aircraft c.g. along body axes (i.e. p, q, and r, M is the applied moments about the aircraft c.g. along body axes (i.e. L, M, and N, and I is the moment of inertia matrix of the aircraft. For a rigid body with symmetry relative to the XZ plane in body axes, the moment of inertia matrix I is symmetric and I xy = I yx = I yz = I zy =. The moment of inertia matrix I then reduces to, I x I xz I = I y I xz I z (A.6 where I x = I z = V olume x 2 dm I y = z 2 dm I xz = V olume V olume V olume y 2 dm xzdm 16

187 A.2 Aircraft Equation of Motion Thus, writing (A.5 in terms of the variables defined in this section gives the force equations: X = m ( u + qw rv Y = m ( v + ru pw Z = m (ẇ + pv qu (A.7 and moment equations: L = ṗi x ṙi xz + qr (I z I y qpi xz M = qi y + pr (I x I z + ( p 2 r 2 I xz N = ṙi z ṗi xz + pq (I y I x + qri xz (A.8 Reasonably, the forces acting on an aircraft in flight consist of aerodynamic (i.e. X aero, Y aero, and Z aero, thrust (i.e. T x, T y, and T z, and gravitational forces (i.e. mg x, mg y, and mg z. Since gravity acts through the aircraft c.g., and the gravity field is assumed uniform, there is no gravity moment acting on the aircraft. Furthermore, to simplify the equation (A.7 and (A.8, the thrust from the propulsion system is assumed to act along the X B body axis and through the c.g., thus give T y = T z =. The angular momentum due to the propulsion system is also neglected therefore there is only aerodynamic (i.e. L aero, M aero, and N aero moment acting on an aircraft. The aerodynamic forces and moments acting on the aircraft can be expressed in terms of nondimensional coefficients: X aero Y aero Z aero = qs C X C Y C Z L aero bc l M aero = qs cc m N aero bc n q = 1 2 ρv 2 (A.9 where C X, C Y, C Z, C l, C m, and C n are the aerodynamic coefficients that primarily are a function of the Mach number, Reynolds number, angle of attack, and sideslip angle; They are secondary functions of the time rate of change of angle of attack and sideslip, and the angular velocity of the aircraft. q is the dynamic pressure, V is the magnitude of the air-relative velocity (also called the airspeed, ρ is the air density, S is the wing reference area, b is the wing span, and c is the mean aerodynamic chord of the wing. Substituting the preceding expressions, including (A.4, into the dynamic equation (A.7 and (A.8 gives, 161

188 A.2 Aircraft Equation of Motion Force equations: u = rv qw + qs m C X g sin θ + T m v = pw ru + qs m C Y + g cos θ sin φ ẇ = qu pv + qs m C Z + g cos θ cos φ (A.1 Moment equations: ṗ I xz I x ṙ = qsb I x C l (I z I y I x I y qr + I xz I x qp q = qs c C m (I x I z pr + I xz ( p 2 r 2 I y I y ṙ I xz ṗ = qsb I z I z C n (I y I x I z pq I xz I z qr (A.11 where T = T x. However, there are a lot more variety of equations that are interesting and often used in flight simulation application such as by differentiating (A.2 with respect to time gives, V = 1 (u u + v v + wẇ ( V uẇ w u α = u 2 + w 2 β = (u2 + w 2 v v (u u + wẇ V 2 u 2 + w 2 (A.12 Substituting in (A.12 for u, v, and ẇ form (A.1, and for u, v, and w from (A.2, gives the wind axes force equations as, where V = qs m C D W + T cos α cos β + g (cos φ cos θ sin α cos β m + g (sin φ cos θ sin β sin θ cos α cos β qs α = mv cos β C L + q tan β (p cos α + r sin α T sin α mv cos β g + (cos φ cos θ cos α + sin θ sin α V cos β β = qs mv C Y W + sin β V + p sin α r cos α + g cos β sin φ cos θ ( V g cos α sin θ g sin α cos φ cos θ + T cos α m C L = C Z cos α + C X sin α C D = C X cos α C Z sin α C DW = C D cos β C Y sin β (A.13 C YW = C Y cos β + C D sin β (A

189 A.2 Aircraft Equation of Motion C L and C D are lift coefficient and drag coefficient in stability axes respectively, whereas they are obtained from body axes components by rotation through α and β. The positive sign convention of C L and C D are directed along the Z S and X S stability axes, respectively. Similarly, C DW and C YW are lift coefficient and side force coefficient in wind axes respectively. The positive sign convention of C DW are directed along the X W and +Y W stability axes, respectively. and C YW Furthermore, the Rotational kinematic equations relate the rate of change of the Euler angles to the body axis components of angular velocity. p = φ ψ sin θ q = θ cos φ + ψ sin φ cos θ r = ψ cos φ cos θ θ sin φ (A.15 or φ = p + tan θ (q sin φ + r cos φ θ = q cos φ r sin φ q sin φ + r cos φ ψ = cos θ (A.16 Navigation equations are most often used to calculate the position of the aircraft. Its dynamics can be determined using a transformation matrix L V B, that transforms an arbitrary three dimensional vector at body axes to earth axes or vehicle-carried earth axes: 1 L V B = L BV cos ψ cos θ cos ψ sin θ sin φ sin ψ cos φ cos ψ sin θ cos φ + sin ψ sin φ = sin ψ cos θ sin ψ sin θ sin φ + cos ψ cos φ sin ψ sin θ cos φ cos ψ sin φ sin θ cos θ sin φ cos θ cos φ (A.17 x E = u cos ψ cos θ + v (cos ψ sin θ sin φ sin ψ cos φ + w (cos ψ sin θ cos φ + sin ψ sin φ y E = u sin ψ cos θ + v (sin ψ sin θ sin φ + cos ψ cos φ + w (sin ψ sin θ cos φ cos ψ sin φ ḣ = u sin θ v cos θ sin φ w cos θ cos φ (A.18 where h is altitude (height above the ground, = Z E. Other commonly used variables, i.e. azimuth or heading angle (χ, and the flight path angle (γ, which can be computed directly from the aircraft s state. These are defined as, χ = β + ψ γ = arcsin ḣ V = θ α (A

190 A.3 Jindivik Nonlinear Mathematical Model A.3 Jindivik Nonlinear Mathematical Model The Jindivik UAV, shown in Figure 1.1, is a remotely piloted fixed wing aircraft, which has been used as an aerial target drone in Australia and the UK for 5 years. Its specifications are given in Table A.1. Based on manufacturer s (Australian Government Aircraft Factory wind tunnel data and subsequent flight trial validation, a 6-DOF nonlinear mathematical model of the UAV has been developed in the MAT- LAB Simulink environment by Fitzgerald [41], shown in Figure A.2. This section only gives a brief overview of the Jindivik nonlinear model; for further details refer to [41] and [42]. The aerodynamic model of the Jindivik nonlinear model is only valid within the following flight condition: Altitude between sea level and 2, ft. True airspeed between 18 knots and 53 knots. Bank angles up to 8. Normal acceleration in the range -2g to +8g. Maximum Mach number.86. A.3.1 Aerodynamic Force and Moment Models The aerodynamic forces (i.e. X aero, Y aero, and Z aero and moments (i.e. L aero, M aero, and N aero acting on the vehicle are computed using (A.9. Following [42], (A.9 can be written further as Axial drag force due to aerodynamics, X aero, X aero = 1 2 ρv 2 S[C Lwb sin(α C D cos(α]+ 1 2 ρv 2 S T [C LT sin(α T α R ] (A.2 where C D is the drag coefficient, C LT is the tailplane lift coefficient, C Lwb is the wing-body lift coefficient, S = 76 ft 2 is the wing area, S T = 14 ft 2 is the tailplane area, α R =.5 deg is the tailplane rigging angle, and α T is the tailplane angle of attack. α T is defined as α T = α + α R ε + q l T V + lag T (A.21 where l T = 9.49 ft is the tail moment arm that is measured from wing quarter chord to tailplane quarter chord, ε is the downwash angle, and lag T is the tailplane angle of attack lag angle due to downwash. ε is defined as ε = ε + ε α (α α 164 (A.22

191 Table A.1: The Jindivik UAV specifications Description Details fuselage length 28 ft 8.75 in wing span 18 ft 9.6 in wing area 76 ft 2 wing chord 4 ft maximum height 6 ft 9.85 in weight 18.7 slug * inertia I x 2,228 slug-ft 2* inertia I y 1,789 slug-ft 2* inertia I z 3,934 slug-ft 2* speed range 2-45 knot service ceiling 18 ft range 82 miles propulsion Bristol Siddeley Viper Mk.21 turbojet; 18 lb f * with fuel of 1 gallon A.3 Jindivik Nonlinear Mathematical Model where α is the zero lift angle of attack, ε is the downwash angle at zero angle of attack, and ε/ α is the rate of change of downwash angle with respect to angle of attack of the flexible aircraft. ε is defined as ε = ε δf =1 + ( εδf =2 ε δf =1 19 (δ f 1 (A.23 where ε and ε δf =1 δf are the downwash angles at zero angle of attack with =2 flaps deflection δ f = 1 and δ f = 2 respectively from which they are defined as ε δf =1 ε δf =2.611 =.323 βpg 2 =.99 rad (A β 8 pg where the Prandtl-Glauert factor, β pg, is defined as rad (A.24 β pg = 1 + M 2, M = V a (A.26 where M and a are the Mach number and the speed of sound, respectively. ε/ α is defined as ε α = K ε ff2 (A.27 α R 165

192 A.3 Jindivik Nonlinear Mathematical Model cg t = Data Store mass Gravity mass Euler Angles Euler Angles Inertia Sum of Forces rates Euler Parameters DCM Earth From Body Axes DCMEB Sum of Moments Dynamics (large pertubation Model Info Mon Oct :38 :49 22 Pio Fitzgerald Jindivik Mk 4A with Rudder and Thrust Vect Thu May 6 11 :9 :14 21 DCMEB Earth Velocities D N E Earth Positions qdyn Mach International Standard Atmosphere New Jindivik Mk 4A Aerodynamics Flight Control System Rolls -Royce Viper Mk 21 Turbojet Autopilot Control Panel Figure A.2: The Jindivik nonlinear model with its autopilot in MATLAB Simulink environment [41] 166

193 A.3 Jindivik Nonlinear Mathematical Model where K ff2 is the flexible factor applied to the rigid body rate of change of downwash with respect to angle of attack and it is a function of Mach number and altitude as presented in Fitzgerald [42, Figure 3-6]. The rate of change of downwash angle with respect to angle of attack of the rigid aircraft, ε, is R defined as where ε α = R ε and ε α δf =1 α δf =2 ε α δf =1 + ε α δf =2 19 ε α δf =1 (δ f 1 α (A.28 are the rate of change of downwash angle with respect to angle of attack with flaps deflection δ f = 1 and δ f = 2 respectively in which they are defined as ε α δf =1 ε α δf =2 = a 1wb ε C L (A.29 =.412 (A.3 where a 1wb is the wing-body combination lift curve slope of the flexible aircraft and ε/ C L is the rate of change of downwash angle with lift coefficient. a 1wb is defined as a 1wb = K ff1 a 1wbR (A.31 where K ff1 is the flexibility factor applied to the rigid wing-body combination lift curve slope and it is a function of Mach number and altitude as presented in Fitzgerald [42, Figure 3-5]. The wing-body combination lift curve slope of the rigid aircraft, a 1wbR, is defined as a 1wbR = a 1wbRδf =1 + ( a1wbrδf =2 a 1 wbrδf =1 19 (δ f 1 (A.32 a 1wbRδf =1 = 5.5 rad-1, a 1wbRδf =2 = (1 β pg rad -1 (A.33 ε/ C L is defined as ε C L = β 8 pg lag T is defined as lag T = α ε l T α V In addition, the lift and drag coefficients are defined as (A.34 (A.35 C L = C Lwb + S T S C L T + C Ls (A.36 C D = C Di + C DZ + C DRe + C DM + C DCL >C Lcrit + C Ds + C Duc (A.37 where C Ls is the lift coefficient increment due to the deflection of spoilers [42, pages 26-29], C Ds is the drag coefficient increment due to symmetric spoiler 167

194 A.3 Jindivik Nonlinear Mathematical Model deflection [42, pages 3-31], C Duc is the drag coefficient increment due to the extension of undercarriage [42, pages 31-32], C Di is the induced drag coefficient, C DZ is the constant Reynolds number profile drag coefficient, C DRe is the Reynolds number dependent profile drag coefficient, C DM is the drag coefficient due to Mach number, C DCL >C Lcrit is the increment of drag coefficient due to the lift coefficient, C L, being greater than the critical lift coefficient, C Lcrit. C Lwb is defined as C Lwb = a 1wb (α wb α α wb = α + α w (A.38 (A.39 where α wb is the wing-body combination angle of attack and α w = 1 is the wing setting angle. C LT is defined as C LT = a 1T α T + a 2T δ e (A.4 a 1T = (1 β pg rad -1 (A.41 a 2T = (1 β pg rad -1 (A.42 where a 1T is the tailplane lift curve slope and a 2T is the elevator effectiveness. C Di + C DZ is defined as ( [CDi + C DZ C Di +C DZ = [C Di +C DZ ] δf=1 + ] δf=2 [C Di + C ] DZ δf=1 (δf 1 19 (A.43 where [C Di +C DZ ] δf=2 and [C Di +C DZ ] δf=1 are a function of C L as presented in [42, Figure 3-7]. C DRe is defined as C DRe = log(re log(re 2 (A.44 where the Reynolds number, Re, is defined as Re = ρv c η (A.45 where c = 4 ft is the wing chord and the dynamic viscosity, η, is defined as η = T 1.5 T (A.46 where T is the outside air temperature (at altitude in kelvin. C DM is defined as ( ( 1 1 C DM = k3 + k4 1 β pg 1 β pg β pg1 (A.47 where k3 =.4, k4 = if M < M crit else k4 =.65, M crit is the aircraft Mach number in which the airflow first reaches the speed of sound, and the Prandtl-Glauert factor at the critical mach number, β pg1, is defined as β pg1 = 1 M crit M crit =.86.2C L +.24C 2 L (A.48 (A

195 A.3 Jindivik Nonlinear Mathematical Model C DCL >C Lcrit is defined as C DCL >C Lcrit = k2 ( C 2 L C 2 L crit if CL > C Lcrit else C DCL >C Lcrit = (A.5 where k2 =.1 and C Lcrit =.6 Sideforce due to aerodynamics, Y aero, Y aero = ρv SC Y (A.51 where the sideforce coefficient, C Y, is defined as C Y = C Yp p b 2 + C Y v v + C Yδr 1 2 V δ r (A.52 where b = 18.8 ft is the wingspan, C Yp =.215 is the sideforce coefficient due to roll rate, C Yv =.38 is the sideforce coefficient due to lateral velocity, and C Yδr =.1176 is the sideforce coefficient due to rudder deflection. Normal lift force due to aerodynamics, Z aero, Z aero = 1 2 ρv 2 S[C Lwb cos(α + C D cos(α] 1 2 ρv 2 S T [C LT sin(α T α R ] Rolling moment due to aerodynamics, L aero, (A.53 L aero = ρv SC l b 2 where the rolling moment coefficient, C l, is defined as (A.54 C l = ( C lr r + C lp p b 2 + C l δa V δ a + C lv V β + C lδr V δ r (A.55 where C lr, C lp =.57, C lδa, C lv, and C lδr =.6215 are the rolling moment coefficients due to yaw rate, roll rate, aileron deflection, lateral velocity, and rudder deflection respectively, and C lr =.459.6C L (A.56 ( C lδa = V 2 (A.57 Vδ 2 ar where V δar = 63 kts is the aileron reversal speed. Pitching moment due to aerodynamics, M aero, C lv = C L (A.58 M aero = 1 2 ρv 2 SC m ρv 2 S T [ (l T + (.25 h c(c LT cos(α T α R (T wl h wl C LT sin(α T α R ] (A

196 A.3 Jindivik Nonlinear Mathematical Model where c = 4 ft is the mean aerodynamic chord, T wl = ft is the height of the tailplane from the waterline, h wl = is the height of the centre of gravity from the waterline, h is the aerodynamic centre position as a percentage of the mean aerodynamic chord, and the pitching moment coefficient, C m, is defined as ( hwl c 1/4wl C m = C m1/4 + C Zwb (.25 h cg + C Xwb (A.6 c where c 1/4wl = is the height of the quarter chord from the waterline, h cg is the centre of gravity position as a percentage of the mean aerodynamic chord [42, Figure 3-2], C Xwb is the wing-body combination coefficient of axial force, C Zwb is the wing-body combination coefficient of normal force, and C m1/4 is the quarter chord pitching moment coefficient, and C Xwb = C Lwb sin α C D cos α C Zwb = C Lwb cos α C D sin α C m1/4 = C Lwb (.25 h + C m where C m is the zero lift pitching moment. h and C m are defined as h = h δf =1 + ( hδf =2 h δf =1 19 C m = C mδf =1 + ( Cmδf =2 C m δf =1 19 (A.61 (A.62 (A.63 (δ f 1 (A.64 (δ f 1 (A.65 where h and h δf =1 δf =.12 are the aerodynamic centre position as a percentage of mean aerodynamic chord with flaps deflection δ f = 1 (as presented =2 in [42, Figure 3-8] and δ f = 2 respectively, and C and C mδf =1 m = δf =2.174 are the zero lift pitching moment with flaps deflection δ f = 1 and δ f = 2 respectively. C mδf is defined as =1 C mδf =1 =.9.24 β pg Yawing moment due to aerodynamics, N aero, (A.66 b N aero = ρv SC n (A.67 2 where the yawing moment coefficient, C n, is defined as C n = ( C nr r + C np p b 2 + C n δa V δ a + C nv V β + C nδr V δ r (A.68 where C nr =.1545, C np, C nδa, C nv, and C nδr =.127 are the yawing moment coefficients due to yaw rate, roll rate, aileron deflection, lateral velocity, and rudder deflection respectively, and C np = C L (A.69 C nδa = C L (A.7 C nv = C nvδf =1 + ( Cnvδf =2 C n vδf =1 19 (δ f 1 (A.71 17

197 A.3 Jindivik Nonlinear Mathematical Model where C and C nvδf =1 n vδf are the coefficient of yawing moment due to lateral =2 velocity with flaps deflection δ f = 1 and δ f = 2 respectively, and C nvδf =1 = C3 L C L C nvδf =2 =.15.32C3 L C L (A.72 (A.73 A.3.2 Thrust Model The Bristol Siddeley Viper Mk.21 turbojet engine has been used by the Jindivik Mk 4A UAV as a propulsion system. Fitzgerald [41, 42] presented a mathematical model of this turbojet engine as shown below; thrust, T, is defined as T = T G D int ( TGnd T G = k TG P P int ratio D int = ṁv g P intratio ( (A.74 (A.75 (A.76 where T G is the gross thrust, k TG = is the gross thrust factor, P is the atmospheric pressure, T Gnd is the non dimensional gross thrust as a function of intake pressure ratio, P intratio, and non dimensional engine speed, N nd, as presented in [42, Figure 3-18], D int is the engine intake drag, and ṁ is the engine mass flow. P intratio is defined as 1 P intratio = P rec P ratio P ratio = ( 1 +.2M if M >.5, P rec = (M.5 else, P rec = k Prec ( V (A.77 (A.78 (A.79 where P rec is the engine pressure recovery, P ratio is the pressure ratio, and k Prec =.98 is the intake pressure recovery factor. N nd is defined as N nd = N N1 max T intake (A.8 where N1 is the engine speed, N1 max = 13, 8 is the engine maximum speed, and the engine intake temperature, T intake, is defined as ṁ is defined as T intake = T (1 +.2M 2 ṁ = kṁk h P intake Tintake (A.81 (A

198 A.3 Jindivik Nonlinear Mathematical Model where kṁ is the mass flow coefficient, k h is the altitude correction factor as a function of engine intake pressure, P intake, as presented in [42, Figure 3-17]. P intake is defined as P intake = P ( 1 +.2M (A kṁ is defined as N if > 1.3, kṁ = T intake ( N else, kṁ = T intake In addition, the engine idle speed, N1 idle, is defined as [ ] h N1 idle = M + ( M N1 max 2 (A.84 (A.85 A.3.3 Sensor Model The sensors model implemented in the Jindivik Simulink model [41] includes angle of attack probe, sideslip vane, accelerometers, rate gyros, attitude gyros, static and dynamic pressure sensors, Mach meter, altimeter and velocity meters. An angle of attack probe is used to measure an angle of attack of the vehicle and is modelled as α probe (s α adj (s = 1.73s + 1 α adj = α cg l pq V (A.86 (A.87 where α cg is the angle of attack at the centre of gravity that is computed using (A.2, α adj is the adjusted angle of attack due to the upwash induced by pitch rate, α probe is the angle of attack output from the probe, and the distance from the centre of gravity to the probe, l p, is defined as l p = 7.43 h cg c (A.88 The probe output, α probe, is limited in the range of to An sideslip vane is used to measure a sideslip angle of the vehicle and is modelled as β vane (s β cg (s = 1 (A.89.73s + 1 where β cg is the angle of sideslip at the centre of gravity that is computed using (A.2. Note that, the factors such as offset from the centre of gravity and local aerodynamic influences are not modelled. The output of the sideslip vane is limited to ±3. 172

199 A.3 Jindivik Nonlinear Mathematical Model Accelerometers are used to measure translational accelerations of the vehicle and are modelled as a xacc (s a xadj (s = a y acc(s a yadj (s = a z acc(s a zadj (s = s 2 + 2(.77(34.557s a xadj a xcg (q 2 + r 2 (pq ṙ (pr + q x acc a yadj = a ycg + (pq + ṙ (p 2 + r 2 (qr ṗ y acc a zadj a zcg (pr q (qr + ṗ (p 2 + q 2 z acc a xcg u + qw rv a ycg = v + ru pw a zcg ẇ + pv qu (A.9 (A.91 (A.92 where a xcg, a ycg, and a zcg are the X, Y, and Z-body axis translational accelerations at the centre of gravity respectively that are computed using (A.92. a xadj, a yadj, and a zadj are the adjusted X, Y, and Z-body axis translational accelerations respectively due to the accelerometers are offset from the centre of gravity. a xacc, a yacc, and a zacc are the X, Y, and Z-body axis translational accelerations output from the accelerometers respectively which a xacc and a yacc are limited to ±4 g and a zacc is limited to ±1 g. x acc, y acc and z acc are the X, Y, and Z-body axis distance from the centre of gravity to the accelerometers. Rate gyros are used to measure body axis roll, pitch, and yaw angular rates of the vehicle and are modelled as q gyro (s q(s p gyro (s p(s = r gyro(s r(s = = 9 2 s 2 + 2(.8(9s s 2 + 2(.89(2s (A.93 (A.94 where p, q, and r are the body axis roll, pitch, and yaw angular rates respectively that are computed using (A.11. p gyro, q gyro, and r gyro are the body axis roll, pitch, and yaw angular rate outputs from the rate gyros which p gyro is limited to ±3 while q gyro and r gyro are limited to ±1. Attitude Gyros are used to measure roll, pitch, and yaw angles of the vehicle and are modelled as φ gyro (s φ(s = θ gyro(s θ(s = ψ gyro(s ψ(s = 1.25s + 1 (A.95 where φ, θ, and ψ are the roll, pitch, and yaw angles respectively that are computed using (A.16. φ gyro, θ gyro, and ψ gyro are the roll, pitch, and yaw angle outputs from the attitude gyros which φ gyro and ψ gyro are limited to ±18 and θ gyro is limited to ±9. Static and dynamic pressure sensors are used to measure static pressure of the atmosphere and dynamic pressures of the air flow respectively. The pressure sensors are modelled as P press (s P (s = q press(s q(s 173 = 1.25s + 1 (A.96

200 A.3 Jindivik Nonlinear Mathematical Model where P and q are the static and dynamic pressures respectively that are computed following the International Standard Atmosphere [42, page 47]. P press and q press are the static and dynamic pressure output from the static and dynamic pressure sensors respectively. Mach meter is used to measure Mach number of the air speed and is modelled as M meter (s 1 = (A.97 M(s.25s + 1 where M is the Mach number which is computed following the International Standard Atmosphere [42, page 47]. M meter is the Mach number output from the Mach meter. Altimeter is used to measure altitude of the vehicle and is modelled as h meter (s 1 = (A.98 h(s.25s + 1 where h is the altitude which is computed using (A.18. h meter is the altitude output from the altimeter. Velocity meter is used to measure true air speed of the vehicle and is modelled as V meter (s 1 = (A.99 V (s.3s + 1 where V is the true air speed which is computed using (A.12. V meter is the true air speed output from the velocity meter. A.3.4 Actuator Model Actuator models of elevator, aileron, rudder, trailing edge flap, thrust vectoring paddles, and undercarriage have been implemented in the Jindivik Simulink model [41]. However, in this thesis, the trailing edge flap, thrust vectoring paddles, and undercarriage are not used in which the flap angle is seted to and the undercarriage is in the retracted position. Elevator actuator dynamics are modelled as δ eres (s δ ecmd (s = s 2 + 2(.59(3.74s (A.1 where δ eres and δ ecmd are the elevator angle command and response respectively. The rate limit is ±4 /sec and the position limit is 15 to 25. Aileron actuator dynamics are modelled as δ ares (s δ acmd (s = 75 2 s 2 + 2(.59(75s (A.11 where δ ares and δ acmd are the aileron angle command and response respectively. The rate limit is ±1 /sec and the position limit is ±4. 174

201 A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model Rudder actuator dynamics are modelled as δ rres (s δ rcmd (s = s 2 + 2(.69(72.1s (A.12 where δ rres and δ rcmd are the rudder angle command and response respectively. The rate limit is ±82 /sec and the position limit is ±35. A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model To investigate open-loop dynamic characteristics of the Jindivik nonlinear model [41] about one particular flight condition, two major lateral and longitudinal modes of the vehicle can be determined using MATLAB functions, trim and linmod. In this section, we consider the following flight configuration: (i fuel 1 gallons, (ii flap angle, (iii the undercarriage in the retracted position, and (iv wings level and constant altitude 1, ft and airspeed 56.3 ft/s straight flight condition. As a result of trim and linmod, the state-space system forms of these two modes are, (i a longitudinal LTI model: u u [ ] ẇ q = w q δe δ rpm θ 1. θ (A.13 (ii a lateral LTI model: v v [ ] ṗ ṙ = p r δa δ r φ φ (A.14 In addition, Table A.2 presents the mode characteristics of the determined longitudinal and lateral Dynamics. It can be seen that the Phugoid and spiral modes are an open-loop unstable. Moreover, Figures A.3-A.6 show open-loop dynamic responses of the vehicle due to the step inputs of elevator, engine speed, aileron, and rudder. Obviously, the determination of natural frequency ω n of the short period mode from Figures A.3 b and A.3 c, that is 3.83 (rad/s, agrees quite well with the presented data in Table A.2. Figures A.5 a, A.6 c, and A.6 e also show that the open-loop transfer functions of v(s/δ a (s, r(s/δ r (s, and ψ(s/δ r (s are non-minimum phase zero. Furthermore, the ω n = 2.6 (rad/s of the Dutch roll mode that is determined from Figures A.5 a, A.6 a, and A.6 c, agrees quite well with the presented data in Table A

202 A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model (a u(ft/s (b w(ft/s (c q(deg/s (d θ(deg (e δ e (deg Figure A.3: Aircraft open-loop dynamic responses to.1 degree elevator step 176

203 A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model (a u(ft/s Time (s (b w(ft/s (c q(deg/s (d θ(deg 1.3 x (e RP M Figure A.4: Aircraft open-loop dynamic responses to 5 RPM engine speed step 177

204 A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model s Non-Minimum Phase Zero (a v(ft/s (b p(deg/s (c r(deg/s (d φ(deg (e ψ(deg (f δ a (deg Figure A.5: Aircraft open-loop dynamic responses to.1 degree aileron step 178

205 A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model (a v(ft/s (b p(deg/s Non-Minimum Phase Zero s (c r(deg/s (d φ(deg (e ψ(deg (f δ r (deg Figure A.6: Aircraft open-loop dynamic responses to.1 degree rudder step 179

206 A.4 Open-Loop Characteristics of the Jindivik Nonlinear Model Table A.2: Mode characteristics of the determined longitudinal and lateral dynamics * Mode Pole Damping (ξ Frequency (rad/s Phugoid Mode.117,.129 1, 1.117,.129 Short Period Mode 1.16 ± 3.93i Spiral Mode Roll Mode Dutch Roll Mode.14 ± 2.61i * about wings level and constant altitude and airspeed flight condition 18

207 Appendix B UAV aerodynamic model identification from a racetrack manoeuvre The determination of aircraft stability and control derivatives from wind-tunnel and computational fluid dynamics data can be an expensive and time-consuming procedure. This appendix shows how linear system identification techniques can be used to obtain this information with sufficient accuracy to design a satisfactory flight control system for an aerial target of the Royal Thai Air Force. Just one flight was undertaken, using a racetrack manoeuvre, to provide the data for identification and validation. The system parameters were identified and a flight control system was designed. Hardware-in-the-loop simulation was used to perform initial tests on the controller and to test the control system hardware and software. A second flight was performed to test the resulting controller, and a satisfactory performance was obtained without the need to adjust the controller gains. B.1 Introduction Flight control systems are typically designed and validated using six-degree-offreedom (6-DOF dynamic models of the aircraft. Traditionally, the model parameters are determined using wind tunnel tests by measuring aerodynamic forces and moments introduced on an aircraft. Furthermore, aircraft moments of inertia are calculated and the aircraft engine model determined from experimental data. However these standard processes are both expensive and time-consuming, and may not be affordable or practicable for many UAV applications. System identification techniques provide an alternative approach that can be used to estimate stability and control derivatives or aerodynamic coefficients of both manned and unmanned aircrafts from flight data [48, 53, 57, 78]. Typical tasks of comprising the system identification process are experiment design, data compatibility analysis, 181

208 B.1 Introduction model structure determination, parameter estimation, and model validation [58, 69, 78]. An important part of the experiment design is to select input waveforms that are used in the excitation of maneuver suitable for model structure determination and parameter estimation because the shape of an input signal has a major impact on the accuracy of estimated parameters from dynamic flight measurements [69, 74, 78]. A data compatibility analysis is usually applied to the measured aircraft responses in order to remove bias and scale factor errors and reconstruct the measured response data [28, 58, 78]. Having checked the data compatibility, the model structure determination [22, 76, 71] and parameter estimation [28, 53, 57, 72] can be executed in order to determine the most appropriate form of the equations to describe the measured responses and to estimate the numerical values of the coefficients appearing in the equations, respectively. The model validation is the last step in the identification process. The identified model should have parameters that are physically reasonable values with acceptable accuracy and it should have good prediction capabilities. For these reasons, the predicted responses are usually compared with a separate set of the measured responses that is not used in the identification process. If the validation process is successful then the analysis is stopped. Otherwise, the identification process is repeated using a different equation structure or parameter estimation technique. In severe cases, the whole process may have to be performed again with a different shape of the input waveform [58, 69, 75, 78]. In practice, to obtain accurate and reliable results from system identification techniques, an aircraft is typically required to fly about a trim condition and to manoeuvre by deflecting each control surfaces deflections at a time with a suitable input waveform of appropriate amplitude [58] in order to to excite certain dynamic modes. A steady wings-level flight condition is generally most suitable. The inputs can be optimal input waveforms [69, 7, 74], input waveform [28, 73, 84], etc. Klein [58] suggests that manoeuvres about trim should not exceed angles of attack and sideslip of ±5, angular rates of ±2 /s, and translational accelerations of ±.3 g. A number of difficulties arise when system identification techniques are applied to certain UAVs. Before the automatic control system is designed, the UAV must be flown as a remotely piloted vehicle. Without pilot-eye-view video images and telemetered data, these requirements can be difficult and time-consuming to perform by a remote ground pilot using only radio control; this is especially the case for fast, large and heavy UAVs, because the remote pilot does not receive the same motion cues that they would if onboard the aircraft. Furthermore extensive test ranges must be used which is very costly. In this thesis, the aerodynamic and propulsion coefficients of a RTAF aerial target were identified from a racetrack manoeuvre. Thus the aircraft is always in visual contact with the remote pilot, and this reduces the range requirements. In addition, ground pilots are generally familiar with the racetrack pattern which is easy to perform. However, the aircraft is not always flying straight and level which is the normal condition for aircraft parameter identification methods. Moreover, due to the 182

209 B.2 Flight System Configuration Figure B.1: RTAF aerial target schedule and limited budget of this work, only one flight test for identification was undertaken. The study shows that, for this vehicle, the necessary parameters can be identified with sufficient accuracy to design a satisfactory flight control system. B.2 Flight System Configuration The RTAF aerial target is a conventional fixed wing aircraft, shown in Figure B.1. It is powered by an AR731 rotary engine from UAV Engines Ltd. The elevator and aileron deflections and throttle setting are effected by Futaba S926 servo motors. Note that the aerial target does not have a rudder. Its specifications are given in Table B.1. B.2.1 Flight Control Computer The flight computer board was developed especially for this work by the Science and Weapon System Development Center (SWSDC, RTAF, shown in Figure B.2. This board is an embedded flight computer which does not have any operating system. It is equipped with only three main chips: an Intel 8C196MH with a 16 MHz clock speed as a processor, a STMicroelectronics PSD4235G2 as a flash insystem programmable (ISP peripheral, and a Texas Instruments TL16C554 as an asynchronous communication element (ACE peripheral. The number of chips has been kept as low as possible so that the board is simple and reliable. B.2.2 Avionic Instrumentation High quality instrumentation was implemented for measuring the flight data. The sampling rate of measurement and the type of sensor output signal were the main criteria for selecting the vehicle instrumentations. As suggested by [58], the measurement sampling rate was selected as 25f max where f max represents the maximum 183

210 B.2 Flight System Configuration Figure B.2: SWSDC flight controller Table B.1: RTAF aerial target specifications Description Details Wing span (b 3.21 m Wing area (S 1.57 m 2 Wing chord ( c.547 m Stabiliser span 1.25 m Fuselage length 2.67 m Maximum height.82 m Weight (m 65 kg Speed range 4-15 knot Endurance 45 minute frequency of the rigid body modes. In general, the frequencies of the rigid-body dynamic modes are below 2 Hz, which means that the sampling rate should be at least 5 Hz. The chosen instrumentation packages are listed in Table B.6. Digital output devices were preferred because digital signals are less prone to electromagnetic interference or radio frequency interference (RFI than the analogue. In addition, the antialiasing filters did not have to be designed and ADCs implemented. However the mini air-data boom provides an analogue output. A second-order low-pass Butterworth filter [56] was used to for its anti-aliasing filters. The break frequency was chosen as 5f max as suggested in [58]. B.2.3 Radio Telemetry High quality radio frequency telemetries (RF Modem were also required in this work. The RFM96W from Pacific Crest Corporation was used for the manual piloted command uplink and was configured as MHz, 9,6 baud for RS-232 interface, and 9,6 baud for the link rate. The data packages of the piloted command were transmitted (uplink every 8 ms. The RFM96WSS (frequency hopping 184

211 B.2 Flight System Configuration spread spectrum from the same company was used for the flight data downlink and was configured as MHz, 19,2 baud for RS-232 interface and 37.5 kbps raw data for the link rate. The flight data downlink packages were transmitted, displayed and recorded every 4 msec on the ground station computer, shown in Figure B.3 which also shows an overview of the flight system configuration of the RTAF aerial target. B.2.4 Racetrack manoeuvre The flight data (racetrack manoeuvre that was used for the identification are shown in Figure B.4. Additional results can be seen in [31, 32]. Control surface deflection sensors were not used, however the control surface defections are generated by simulation from a linear model of the servo dynamics and the known pulse-width modulation (PWM signals. A technique to identify a servo linear dynamic model is presented in [54]. The Futaba S926 linear dynamic model was assumed as a first-order transfer function, 1/(τ a s + 1, where the time constant (actuator lag, τ a, was 2 ms. The measured maximum and minimum deflections of the elevator, aileron, and throttle setting were equal to ±32.4, ±19.96 and 51 and respectively. The maximum angle rate magnitude of the Futaba S926 servo is 315 /s, so the maximum rate magnitudes of the elevator, aileron, and throttle control were determined as 126 /s, /s, and 15 /s repectively because the ratio of elevator, aileron, and throttle deflections to servo output deflections were 1:2.5, 1:4, and 1:3 respectively. The altitude and velocity along the body x-axis are calculated as [1] [ ] ( SP h = u = [ T ] ( DP 2/7 5 SP (B.1 (B.2 where h is the altitude in feet, SP is the static pressure in inches of mercury (inhg, u is the velocity along the x-body axis in knots, T is the outside air temperature (at altitude in degrees Celsius, and DP is the dynamic pressure in inhg. To simplify (B.2, T is assumed to be constant and equal to a measurement temperature on the ground that was 27.6 C. Although the static pressure was not calibrated to perform altitude error corrections, the calculated altitude using the static pressure data in (B.1 agreed quite well with the global positioning system (GPS altitude as shown in Figure B.4. In addition, because of installation constraints, the mini air-data boom was installed on the wing tip of the vehicle as shown in Figure B.3. Due to upwash, such a location can affect the accuracy of the altitude and u-velocity measurements. It is a general issue for small UAVs that the air-data probes can not always be placed in aerodynamically optimal locations. 185

212 B.2 Flight System Configuration Pressure Sensors Gyro Mini Air Data Boom Pressure Tubes Vibration Absorber Angle of Attack Vane GPS Spread Spectrum Radio Modem Uplink Radio Downlink Radio Serial Interface (UARTS GARMIN GPS35 Built-in Antenna, Position Accuracy: 15 meter, Update Rate 1 sec., Tracking 12 Satellites, NMEA ASCII output Maximum Angular Rate 2 deg/s Maximum Acceleration 1 g A/D Converters Total pressure Static pressure holes Side-slip Vane Serial Interface (UARTS Engine Rotary Engine 16 bit microcontroller core Pulse Width Capture RPM Pulse 4 Mb Flash Memory (Flight Data Storage Servo Actuator Pulse Width Modulation Flight Control Computer Aileron Elevator Throttle Parachute Engine Cut 3 Deg./Sec, Torque 132 oz-in Spread Spectrum Radio Modem Uplink Antenna Downlink Antenna Downlink Flight Data Uplink Instrumentation Monitoring Display Control Flight Path & Moving Map Display Ground Control Station 35 Watt. 412 MHz UHF Radio Modem Figure B.3: Flight system configuration 186

213 B.2 Flight System Configuration Speed from dynamic pressure (knot Longitude (deg (a 2D flight path Time (sec (b u Altitude from GPS Altitude from Static Pressure Altitude (ft Pitch rate (deg/sec Time (sec (c h Time (sec (d q Angle of attack (deg Time (sec (e α Time (sec (f a z Figure B.4: Flight data (racetrack pattern for identification 187

214 B.3 Aircraft Parameter Estimation B.3 Aircraft Parameter Estimation B.3.1 Model Postulation Based on a priori knowledge about the standard 6-DOF equations of motion, the aerodynamic force and moment coefficients can be calculated from flight data as shown below [77] C X = ma x T qs C Y = ma y qs C Z = ma z qs C L = C X sin α C Z cos α C D = C X cos α C Z sin α C l = I [ x ṗ I xz (pq + ṙ + (I z I y ] qr qsb I x I x C m = I [ y q + (I x I z pr + I ] xz (p 2 r 2 qs c I y I y C n = I [ z ṙ I xz (ṗ qr + (I y I x ] pq qsb I z I z (B.3 (B.4 (B.5 (B.6 (B.7 (B.8 (B.9 (B.1 Variables a x, a y, a z, p, q, r, α, and q are measured by the instrumentation shown in Table B.6. Parameters m, S, b, and c are given in Table B.1. The inertias I x, I y, and I z are measured by a torsional pendulum experiment, i.e. the bifilar pendulum [13] and trifilar pendulum [6], using the relation I = mgt 2 R 2 4π 2 L (B.11 where I (kg-m 2 is the measured moment of inertia, T (s is the period of oscillation, R (m is the distance between cables and the distance from a cable to the center of three cables in the case of bifilar and trifilar pendulums respectively, and L (m is the cable length. In this work, the trifilar pendulum was used to measure the inertias of the aerial target and the measured values of I x, I y, and I z, were equal to kg-m 2, kg-m 2, and kg-m 2 respectively. I xz was assumed zero. The engine thrust, T (N, as a function of engine rotational speed was measured at steady engine rotational speed using a force gauge by letting the aerial target pull the force gauge only in the x-axis direction. The results of the measurement are shown in Figure B.5. A simple linear thrust model that fits the shown data was determined as T = (.716 RP M (B.12 where RPM is the measurement of engine rotational speed in revolutions per minute and lies in the range

215 B.3 Aircraft Parameter Estimation Linear fitting: y =.716*x Static Test linear fitting 2 Thrust of engine (N RPM Figure B.5: Static thrust measurement result B.3.2 Flight Data Post-Processing All of the measured aircraft response data have to be synchronized. However, as shown in Table B.6, the sampling rates for each avionic instrumentation package are different. Hence linear interpolation was used to re-sample the data at the same frequency. To remove the noise, e.g. process noise (atmospheric disturbance, engine vibration, and sensor noise, that lie outside the bandwidth of the data of interest, the received data were filtered using the standard MATLAB Filter Design Toolbox function [4], butter (i.e. Butterworth infinite impulse response digital filter, since Butterworth filters give a magnitude response that is maximally flat in the passband and is monotonic overall. In addition, they sacrifice rolloff steepness for monotonicity in the passband and stopband. In this work, the filters for V, u, φ, and θ were order 3 with a cutoff frequency of 2.35 Hz; the filters for a x, a y, a z, p, q, r, α, and β were order 3 with a cutoff frequency of 6.35 Hz, and the filters for δ a, δ e, and δ th were order 2 with a cutoff frequency of 4.75 Hz. A power spectral density of the filtered flight data shown in Figure B.6. The angle of attack, sideslip angle, and translational accelerations were corrected to the center of gravity using [27, 58] g a x a y a z α = α E + qx α V β = β E + rx β V = g u v = w a xe a ye a ze u E v E w E py α V pz β V (q 2 + r 2 (pq ṙ (pr + q + (pq + ṙ (p 2 + r 2 (qr ṗ (pr q (qr + ṗ (p 2 + q 2 r q x air + r p y air q p z air x a y a z a (B.13 (B.14 (B.15 (B

216 B.3 Aircraft Parameter Estimation where the subscript E denotes the measured value from the experiment. [x α y α z α ] T, [x β y β z β ] T, [x a y a z a ] T, and [x air y air z air ] T denote the position vectors of the angles of attack and sideslip, accelerometer, and air-data probes relative to the centre of gravity in the body axes respectively. A data compatibility analysis should be applied to the measured aircraft responses in order to verify the data accuracy because the measured response data contains bias and scale factor errors. The purpose of data compatibility analysis is to remove the bias and scale factor errors and reconstruct the measured responses [58, 78]. The measurement equation model for aircraft sensors with typical instrumentation errors is defined as [28] y m = (1 + λ y y + b y (B.17 where y m denotes the measurement of the true value of variable y, λ y is the scale factor error, and b y is the bias error. Following [58], the state-space form of the translational and rotational kinematics differential equations and the measured output equations used for data compatibility analysis is u r E b r (q E b q u g sin θ + ga xe b ax v (r E b r p E b p v ẇ ḣ = q E b q (p E b p sin θ cos θ sin φ cos θ cos φ φ 1 tan θ sin φ tan θ cos φ p E b p θ = cos φ sin φ q E b q ψ sin φ sec θ cos φ sec θ r E b r w h + g cos θ sin φ + ga ye b yx g cos θ cos φ + ga ze b zx (B.18 (B.19 V E (i = (1 + λ V u 2 (i + v 2 (i + w 2 (i + b V (B.2 [ v(i ] β E (i = (1 + λ β tan 1 + b β (B.21 u(i [ w(i ] α E (i = (1 + λ α tan 1 + b α (B.22 u(i φ E (i = (1 + λ φ φ(i + b φ (B.23 θ E (i = (1 + λ θ θ(i + b θ (B.24 ψ E (i = (1 + λ ψ ψ(i + b ψ (B.25 h E (i = (1 + λ h h(i + b h (B.26 where the subscript E again indicates measured values from the experiment, λ ( is the unknown scale factor error, and b ( is the unknown bias error. The constant unknown instrumentation error parameters in (B.18 (B.26 were estimated using a function, dcmp, of the MATLAB software package SIDPAC, which is documented in and included with [58]. The function dcmp is an output-error parameter identification technique that is based on the principle of maximum likelihood method, for further details refer to [58]. The results are shown in Table B.2. Klein [58] discusses the expected errors of a typical instrumentation package, and this was used to determine which bias and scale factor terms should be estimated. The 19

217 B.3 Aircraft Parameter Estimation 1 2 a a y p r Frequency (Hz 1 2 e a z q Frequency (Hz 1 3 th 1 2 a x u V Frequency (Hz Figure B.6: Power spectral densities 191

218 B.3 Aircraft Parameter Estimation Table B.2: Estimated instrumentation error parameters b ax (m/s ±.135 b ay (m/s ±.129 b az (m/s ±.59 b p (rad/s -.8 ±. b q (rad/s -.12 ±. b r (rad/s ±.1 b β (rad.193 ±.34 b α (rad.145 ±.5 λ φ.71 ±.3 λ θ.894 ±.25 b φ (rad -.39 ±.3 b θ (rad -.67 ±.4 bias and scale factor of yaw angle were not estimated since there was no measurement data of the yaw angle. The selection of bias and scale factor terms in function dcmp were hand-tuned until the reasonable parameters, shown in Table B.2, were obtained with the best match between the measured and reconstructed responses, shown in Figure B.7. The match between the measured and reconstructed responses for V, α, and β, shown in Figure B.7 a, were not as good as the match between the measured and reconstructed responses for φ and θ, shown in Figure B.7 b. This is because the random measurement errors of the translational accelerations and angular rates were neglected in (B.18. Moreover, Figure B.4 also shows that all of the measured responses have a significant noise; this is especially the case for the translational accelerations. Although all of the measured responses were filtered, the process noise (i.e. atmospheric turbulence typically resides in the same frequency band as the aircraft dynamics making it difficult to filter out post-flight without also filtering out the data of interest. Furthermore, the upwash also induced some errors on the u-velocity. B.3.3 Equation-error Method Although the exact forms of aerodynamic coefficients structures are not certain, typical linear model structures are suggested in [5, 58, 77]. The linear model structures that were used in this work, from [5], are C D = C D + C Dα α + C Dδe δ e (B.27 ( pb ( rb C Y = C Y + C Yβ β + C Yp + C Yr + C Yδa δ a (B.28 2V 2V ( α c ( q c C L = C L + C Lα α + C L α + C Lq + C Lδe δ e (B.29 2V 2V ( pb ( rb C l = C l + C lβ β + C lp + C lr + C lδa δ a (B.3 2V 2V 192

219 B.3 Aircraft Parameter Estimation 2 18 Measured Reconstructed Maneuver length = 3 (sec time (sec (a V, β, and α 5 Maneuver length = 3 (sec -5 Measured Reconstructed (b φ and θ Figure B.7: Data compatibility analysis time (sec 193

220 B.3 Aircraft Parameter Estimation ( α c ( q c C m = C m + C mα α + C m α + C mq + C mδe δ e (B.31 2V 2V ( pb ( rb C n = C n + C nβ β + C np + C nr + C nδa δ a (B.32 2V 2V RP M = C + C α α + C V V + C δth δ th (B.33 After substituting the aerodynamic coefficients that were calculated using (B.3 (B.1 and the measured state and control variables in (B.27 (B.33, we obtain a set of equations of the unknown aerodynamic parameters for which are solved using an equation-error method (least-squares method. For example, following [58], the least-squares problem for the lift coefficient C L is formulated using the model structure in (B.29 as Z = Xθ + v (B.34 where Z is an N 1 vector of values computed from (B.6, θ is a 5 1 vector of unknown parameters, X is an N 5 matrix of measurement data vectors or regressors, and v is an N 1 vector of equation errors as shown below Z = [ C L (1 C L (2 C L (N ] T θ = [ ] T C L C Lα C L α C Lq C Lδe c α(1 cq(1 1 α(1 δ 2V 2V e (1 c α(2 cq(2 1 α(2 δ 2V 2V e (2 X =. 1 α(n δ e (N c α(n 2V cq(n 2V v = [ v(1 v(2 v(n ] T The best estimator of θ minimizes the sum of squared differences between the dependent variable measurements Z and the model; the cost function, J, is given by J(θ = 1 2 (Z XθT (Z Xθ (B.35 Differentiating (B.35 with respect to θ gives [27], J(θ θ = ZT X + θ T X T X (B.36 The necessary condition for minimizing the cost is given by J(θ/ θ = giving the least-squares solution for the unknown parameter vector θ as ˆθ = ( X T X 1 X T Z (B.37 The estimated parameter covariance matrix from [58] is [ Cij ] = ˆσ 2 (X T X 1, i, j = 1, 2,..., n P (B.38 where n P is the dimension of the unknown parameter vector and ˆσ 2 = (Z Xˆθ T (Z Xˆθ N n P (B

221 B.3 Aircraft Parameter Estimation Table B.3: Parameter correlation coefficient matrix (coefficients of drag, lift, and pitching moment C (D,L,m C (D,L,mα C (L,m α C (L,mq C (D,L,mδe C (D,L,m C (D,L,mα C (L,m α C (L,mq C (D,L,mδe The standard error of the estimated parameters is [77] s (ˆθj = Cjj, j = 1, 2,..., n P (B.4 The correlation coefficient between two estimated parameters is [27] ρ ij = C ij Cii C jj (B.41 The correlation coefficient is a measure of the pair-wise correlation between the two parameters. A value of ρ ij.9 means that the two regressors, X i and X j, are linearly dependent and are in some way related to each other. In that case, some additional action must be taken [58]. B.3.4 Results A single flight was undertaken lasting 1,38 sec. The flight data of the period from sec was used for identification. Some of the measurements are shown in Figure B.4. The identification results are presented in Table B.7. Figure B.8 show how well of the match between the estimated and calculated aerodynamic coefficients using the flight data of period sec. The correlation coefficients of these results are presented in Tables B.3 to B.5. The results were also validated by comparing the estimated aerodynamic coefficients with the values calculated by (B.3 (B.1 using the flight data of period sec. Some results are shown in Figure B.9. Additional results can be seen in [31, 32]. The offset in the RPM plot occurs because engine nonlinear effects are likely to be significant at low engine speeds (below 4,5 RPM, the engine speed for the identification period ranged between 5,2 RPM to 6,8 RPM, but was below 4,4 RPM for the validation period. Moreover, the regressor showing the correlation between V and δ th, shown in Table B.5, shows these are linearly dependent. This could cause an inaccurate estimated parameter. Although we can not suppose the identified results are the exact aerodynamic coefficient parameters of the vehicle, these results should be reliable and accurate enough for control synthesis and analysis. 195

222 B.3 Aircraft Parameter Estimation Table B.4: Parameter correlation coefficient matrix (coefficients of sideforce, rolling, and yawing moments C (Y,l,n C (Y,l,nβ C (Y,l,np C (Y,l,nr C (Y,l,nδa C (Y,l,n C (Y,l,nβ C (Y,l,np C (Y,l,nr C (Y,l,nδa Table B.5: Parameter correlation coefficient matrix (engine speed coefficient C C α C V C δth C C α C V C δth Calculated C D Calculated C L.11 Estimated C D.6 Estimated C L (a C D (b C L.3.2 Calculated C m Estimated C m Measured RPM Estimated RPM (c C m (d RP M Figure B.8: Estimation of aerodynamic coefficients 196

223 B.3 Aircraft Parameter Estimation.3 Coefficient of drag Calculated CD Estimated CD 1.4 Coefficient of lift Calculated CL Estimated CL Non-dimensional Unit.2.15 Non-dimensional Unit Time (sec Time (sec (a C D (b C L.15.1 Coefficient of pitching moment Calculated Cm Estimated Cm Measured RPM Estimated RPM.5 46 Non-dimensional Unit RPM Time (sec (c C m Time (sec (d RP M Figure B.9: Validation of aerodynamic coefficients 197

224 B.4 Design, HIL simulation and flight test B.4 Design, HIL simulation and flight test B.4.1 PID Autopilot Design Three autopilot functions, altitude-hold, speed-hold, and GPS waypoint navigation, were designed using a proportional, integral and derivative (PID control methodology drawn from [59]. The PID autopilot were implemented as a discrete-time controller with antiwindup. The details of its structure are shown in Figure B.1. All gain values in each loop were tuned manually, based on the identified 6-DOF nonlinear model, until the autopilot s functions performances are satisfied. The angle calculation box in Figure B.1 b calculates an angle υ from a current position (Latitude, Longitude of the vehicle to the command position (destination. The purpose of pitch and roll limiters in Figures B.1 a and B.1 b is to ensure that the aerial target will manoeuvre with small amplitude of controls, i.e. elevator, aileron, and throttle. Hence, its dynamic will not be faraway from trim condition (or small-disturbance condition. B.4.2 Hardware-In-the-Loop (HIL Simulation In practice, before a real flight test, HIL simulation is often used to validate the reliability of the hardware and software of the flight control system as well as the effectiveness of the designed flight control law. The details of the HIL simulation are shown in Figure B.11. Based on the identified aerodynamic coefficients, the 6- DOF nonlinear model of the vehicle was written in MATLAB/Simulink Real-Time Workshop [2] environment with a 32-bit xpc Target system being an Intel Pentium III computer. Two PCI-CTR5 counter/timer boards from Measurement Computing Corporation were installed inside the xpc target computer and used to capture the elevator, aileron, and throttle PWM signals that were generated from the flight control computer. Five RS-232 serial port signals were required (see Figure B.11, so a PCI- ESC-1 serial board from Quatech Inc. were also installed to provide these. For the 6-DOF nonlinear Simulink model, the xpc Target Toolbox blocks PCI-CTR5 PWM and ESC-1 Quatech [3] were used to program the PCI-CTR5 boards and PCI- ESC-1 board respectively. The 6-DOF nonlinear Simulink model was automatically compiled by VisualC and downloaded to the xpc target using the MATLAB function, xpcexplr [3]. The designed PID autopilot was programmed manually using the Phyton 8C196 C language development kit. The sensor interface modules of the flight computer board were programmed using assembly language. The Intel Hex-file format was the result of compiling both assembly and C program modules. This Intel Hex-file was burned to the flash in-system programmable peripheral (PSD4235G2 on the flight computer board. 198

225 B.5 Conclusions The ground station application programs, i.e. instrumentation monitoring and moving map displays, were programmed using Visual Basic 6.. Furthermore, this HIL simulation actually can simulate flying in both manual and automatic modes. Mode selection is controlled by three switches, i.e. altitude-hold on-off, speed-hold on-off, and GPS waypoint navigation on-off, on the joystick control unit. Microsoft Flight Simulator was used for visualisation. An experienced ground pilot used the manual mode to simulate flying with the flight dynamic visualization display, shown in Figure B.11, in order to validate the identified 6-DOF nonlinear model. B.4.3 Flight Test After the HIL simulation results were deemed successful and satisfactory, a real flight test was performed. The altitude command was set to 2, ft, the speed command to 15 knot, and three GPS waypoint navigation settings of (lat: N , long: E , (lat: N , long: E1 4.8, and (lat: N , long: E Some of the flight test data are presented in Figure B.12. It can be seen that the PID autopilot successfully performed its functions without any need to adjust the controller gains. Additional results can be seen in [31, 32]. B.5 Conclusions This appendix demonstrates how system identification techniques can be used for UAV control system design and development in a cost-effective manner. An ordinary piloted manoeuvre and off-trim condition flight data (racetrack manoeuvre was studied and identified in order to estimate the aerodynamic coefficients of the RTAF aerial target. As shown by the flight test results of the PID autopilot, the identified 6-DOF non-linear model was sufficiently reliable and accurate for the design of a satisfactory control system. In this work, only two flight tests had to be undertaken. The first flight test was done to record flight data by controlling the aerial target manually for a flight duration of 12 min. The second flight test was done to validate the PID autopilot for a flight duration of 26 min. However, both flight tests were performed in one flight condition; it is envisaged that gain-scheduling will be required to cover a fuller range of flight conditions. An advanced robust gain-scheduling technique, namely linear parameter-varying control, should be employed where the details of LPV control approach are already presented in the main chapters of this thesis. 199

226 B.5 Conclusions Altitude (ft Command PI Pitch from Altitude (5 Hz.1524 Pitch (deg Command -15 to +15 (deg Pitch Limiter PID Elevator from Pitch (5 Hz PWM Width 1 to 2 usec Elevator Limiter To Elevator Servo Motor Altitude (ft Feedback (6 Hz Pitch (deg.76 Feedback.3367 (75 Hz Anti Windup If Pitch command < -15 or Pitch command > 15 then Disable integrator End if.1667 d dt 15 Anti Windup If PWM Width < 1 or PWM Width > 2 then Disable integrator End if (a Altitude-hold υ (deg Command.87 Roll (deg Command -3 to +3 (deg Roll Limiter PID Aileron frompi Roll(5 from Hz CTS (1 Hz PWM Width 1 to 2 usec Aileron Limiter To Aileron Servo Motor GPS Course Feedback (deg (1 Hz Lat, Long (deg Command Lat, Long (deg Feedback (1 Hz Angle Calculation Roll (deg.1 Feedback -.25 (75 Hz Anti Windup υ (deg Command If Roll command < -3 or Roll command > 3 then Disable integrator End if North Long, Lat Command υ (deg Course to Steer (CTS GPS Course Long(n, Lat(n -.83 d dt 15 Anti Windup If PWM Width < 1 or PWM Width > 2 then Disable integrator End if Long(n-1, Lat(n-1 (b GPS waypoint navigation Speed (knot Command PI Throttle from Speed (5 Hz PWM Width 1 to 2 usec Throttle Limiter To Throttle Servo Motor Speed (knot Feedback (6 Hz (c Speed-hold 15 Anti Windup If PWM Width < 1 or PWM Width > 2 then Disable integrator End if Figure B.1: Structure design of the autopilot 2

227 B.5 Conclusions Microsoft Flight Simulator (Flight Dynamic Visualization RS ,2 bps (5 Hz PCI-ESC-1 xpc Target, 6-DOF nonlinear dynamic model (1, Hz PCI-CTR5 RFM96W GPS, RS-232 4,8 bps (1 Hz SP, RS ,2 bps (6 Hz DP, RS-232 Flight Control Computer 19,2 bps (6 Hz Discrete-time autopilot (5Hz Vertical gyro, RS ,4 bps (75 Hz Up Link, RS-232 9,6 bps (12.5 Hz RPM Elevator PWM Aileron PWM Throttle PWM Down Link, RS ,2 bps (25 Hz Elevator Servo Aileron Servo Throttle Servo RFM96WSS Joystick Controller RFM96W Ground station Instrumentation monitoring display Flight path & moving map display RS ,2 bps (25 Hz RFM96WSS (a Block diagram PCI-ESC-1 from Quatech, 8 channel RS-232 serial port PCI-CTR5 from Measurement Computing, PWM capture RS ,2 b/s (5 Hz PCI bus PCI bus xpc Target, Real-Time HIL Simulation at 1, Hz, 6-DOF nonlinear dynamic model Elevator Servo Flight Dynamic Visualization RFM96W GPS SP DP Gyro Up Link RPM PWM (ch1 PWM (ch2 PWM (ch3 Discrete-time autopilot (5Hz Down Link Aileron Servo Throttle Servo RFM96WSS RFM96WSS RS ,2 b/s (25 Hz (b Image diagram Ground station Instrumentation monitoring display Flight path & moving map display Figure B.11: Real-time hardware-in-the-loop simulation environment 21