Nonlinear Fault-Tolerant Guidance and Control for Damaged Aircraft. Gong Xin Xu

Transcription

1 Nonlinear Fault-Tolerant Guidance and Control for Damaged Aircraft by Gong Xin Xu A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto Copyright c 211 by Gong Xin Xu

2 Abstract Nonlinear Fault-Tolerant Guidance and Control for Damaged Aircraft Gong Xin Xu Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto 211 This research work presents a fault-tolerant flight guidance and control framework to deal with damaged aircraft. Damaged scenarios include the loss of thrust, actuator malfunction and airframe damage. The developed framework objective is to ensure that damaged aircraft can be stabilized and controlled at all times. The guidance system is responsible for providing the airspeed, vertical and horizontal flight path angle commands while considering aircraft dynamics. The control system, designed by the nonlinear state-dependent Riccati equation (SDRE) control method, is used to track the guidance commands and to stabilize the damaged aircraft. The versatility of SDRE allows it to passively adapt to the aircraft parameter variations due to damage. A novel nonlinear adaptive control law is proposed to improve the controller performance. The new control law demonstrated improved tracking ability. The framework is implemented on the nonlinear Boeing 747 and NASA Generic Transport Model (GTM) to investigate the simulation results. ii

3 Acknowledgements I would like to express my deepest gratitude to my thesis supervisor Professor Hugh Liu for giving me the opportunity to work on the fault-tolerant flight control topic, and for his continuous guidance and support throughout the research. His encouragement and advice led me to the right path and are greatly appreciated. I would also like to thank the other members of my research committee, Professor Peter Grant and Professor Christopher Dameren for their valuable feedback and comments. My heartfelt appreciation also goes to the friends and colleagues at the Flight Systems and Control (FSC) group at UTIAS, Chen, Connie, Difu, Everett, Jason, Keith, Sohrab and others. They made my life at FSC an enjoyable and memorable experience. I would also like to extend my deepest gratitude to my family for their unconditional love and support. iii

4 Contents 1 Introduction Aircraft Flight Control System Design Research Motivation Literature Review Research Objectives & Contribution Thesis Organization Aircraft Dynamics Introduction Aircraft Modeling Nonlinear Equations of Motion Nonlinear Aerodynamic Coefficients Boeing 747-1/ NASA GTM Damaged Aircraft Modeling Trim Analysis Summary Fault-tolerant Flight Guidance and Control Problem Introduction Problem Formulation Remarks iv

5 3.4 Aircraft Guidance Law Design Guidance Law Design Summary State-Dependent Riccati Equation Control Method SDRE Control Method Stability and Optimality Analysis Stability Analysis Optimality Analysis The Art and Capabilities of SDRE Simulation Studies Loss-of-thrust example - UAV example Damaged Aircraft - B Summary Adaptive State-Dependent Riccati Equation Control Method Adaptive Control Method Stability Studies Stability Analysis Simulation Studies Baseline Control Adaptive Law Design Simulations Summary Conclusions and Future Work 78 A Derivations 8 B State-dependent coefficients 83 v

6 Bibliography 87 List of Tables 2.1 Failure Modes Trimmed States Trimmed Controls Straight and Level Flight Trim Results Design Parameters GTM Trim Results List of Figures 1.1 Military and Civil Aircraft Mechanical and FBW Systems Aircraft Reference Frames Boeing NASA GTM Simulink Environment GTM Damage Case Example Trim Routine Procedure B747 Trimmed Airspeed B747 Trimmed Roll Angle B747 Trimmed Angle of Attack B747 Trimmed Pitch Angle vi

7 2.1 B747 Trimmed Sideslip Angle B747 Trimmed Yaw Angle B747 Trimmed Roll Rate B747 Distance B747 Trimmed Pitch Rate B747 Lateral Distance B747 Trimmed Yaw Rate B747 Altitude Proposed Guidance and Control Framework Guidance Law Illustration SDRE Design Flowchart Undamaged UAV: Airspeed Time History Undamaged UAV: Flight Path Angle Time History Undamaged UAV: Angle of Attack Time History Undamaged UAV: Pitch Rate Time History Undamaged UAV: Throttle Control Time History Undamaged UAV: Elevator Time History Undamaged UAV: Tracking Distance Error Time History Undamaged UAV: Altitude Time History Damaged UAV: Airspeed Time History Damaged UAV: Flight Path Angle Time History Damaged UAV: Angle of Attack Time History Damaged UAV: Pitch Rate Time History Damaged UAV: Throttle Control Time History Damaged UAV: Elevator Time History Damaged UAV: Pitch Angle Time History vii

8 4.17 Damaged UAV: Altitude Time History Loss-of-thrust Case II Simulation Results B747 Actuator Damage: Roll Rate Time History B747 Actuator Damage: Pitch Rate Time History B747 Actuator Damage: Yaw Rate Time History B747 Actuator Damage: Elevator Time History B747 Actuator Damage: Aileron Time History B747 Actuator Damage: Rudder Time History B747: Flight Path Loop Damaged B747: Flight Path Angle Time History Damaged B747: Elevator Time History Damaged B747: Airspeed Time History Damaged B747: Throttle Control Time History Damaged B747: Trajectory Time History Adaptive framework GTM Trim: angle of attack GTM Trim: flight path angle GTM stabilizing: angle of attack GTM stabilizing: pitch rate GTM stabilizing: elevator deflection GTM tracking: angle of attack GTM tracking: elevator deflection GTM tracking: elevator deflection comparison perfect vs damage GTM tracking: angle of attack comparison GTM tracking: elevator deflection comparison viii

9 Chapter 1 Introduction 1.1 Aircraft Flight Control System Design The aircraft flight control system is a vital component, among other critical aircraft systems, to ensure flight performance and safety. It was successfully introduced by the Wright brothers in 192. The original design features the three-axis control, with coupled roll and yaw control to alleviate the adverse yaw effects [43]. Such a design paved the foundation of the modern aircraft flight control system, which has flourished with revolutionary changes. By the 195s, analog flight control computers emerged to allow artificial modification of the aircraft handling qualities in addition to the basic autopilot stabilization tasks [16]. The Canadian Avro CF-15 Arrow interceptor (Fig. 1.1a) equipped with an analog flight control computer demonstrated impressive performance capabilities. Subsequently, digital fly-by-wire (FBW) technology was introduced to replace the analog flight control computers. In 1972, the technology was flown by an F-8 Crusader (Fig. 1.1b) in flight experiments conducted by NASA. In the civil aviation field, Airbus A32 was the first commercial airliner utilized the FBW control system on main control surfaces in

10 Chapter 1. Introduction 2 (a) Avro CF-15 Arrow (b) F-8 (c) Airbus A32 Fig. 1.1: Military and civil aircraft. Sources: (a) DND (b) NASA (c) airliners.net Concurrently, the conventional mechanical flight control system (Fig. 1.2a), as still seen in small aircraft nowadays, also gradually evolved to the mechanical-hydraulic systems, which include a hydraulic system to generate actuators forces to move the control surfaces. Although the hydro-mechanical system makes pilots flying the aircraft less demanding and allows large forces on the control surfaces, it adds additional complexity and weight to the already highly complex mechanical system. The FBW flight control system (Fig. 1.2b) became the solution to replace the previous two systems. Because the digital computers are used to receive and send signals, it allows for easier control implementation, thus better handling qualities. The digital FBW technology improved the flight reliability, maneuverability as well as safety while providing drastic cost reduction. (a) Hawk aircraft mechanical system (b) Dgital FBW flight control system Fig. 1.2: Flight control systems [18]

11 Chapter 1. Introduction Research Motivation The advent of digital FBW system brings a new chapter to the flight control system design. The computer-based system allows better handling, greater aircraft maneuverability and agility, weight reduction and so on. These benefits come with the most stringent safety requirement. For example, an FBW system must have the same level of safety and integrity as the simple mechanical system. It means that the probability of a failure occurring in the FBW system which would result in catastrophic consequences to the aircraft must be less than 1 in 1 9 per flight hour [14]. The intense focus on the safety level and the need to improve the system integrity require the FBW aircraft to be fitted with a back up system. This system has the ability to generate fault-tolerant control commands that take advantages of the system redundancy in terms of controls, sensors, and computing. The control effector redundancy provides a unique opportunity for the back up control system to reconfigure itself to mitigate and compensate for the failure of the aircraft with the objective of increasing survivability. In addition, recent civil aviation safety data show that about 16% of the accidents that happened in between 1992 and 27 belongs to the category of Loss of Control Inflight (LOS-I), which is caused by pilot error, technical malfunctions, or unusual upsets due to external disturbances [16]. For example, in the late 197s an American Airlines DC-1 crashed in Chicago (Flight 191, May 25, 1979) due to engine seperation. The pilot only had 15s to react before the plane crashed. The subsequent investigations showed the accident could have been avoided [36]. More recently, an El Al cargo Flight 1862 (October 4, 1992) fatal crash was also demonstated in simulation to be avoidable [31]. These catastrophic crashes underscore the need to have intelligent, fault-tolerant flight control (FTFC) systems. NASA commenced the Integrated Resilient Aircraft Control (IRAC) project of NASA Aviation Safety Program (AvSP) to investigate and research advanced flight controls that can be implemented to ensure safety in the presence of unforeseen, adverse conditions. Similarly, European GARTEUR Flight Mechanics Action

12 Chapter 1. Introduction 4 Group FM-AG(16) on fault-tolerant flight control has also focused on the topic. In addition to the FBW system, the FTFC system is only activated when the fault is positively detected and diagnosed on the aircraft. The FTFC system is specifically designed to stabilize and control the aircraft without exacerbating the situation into a more serious situation while ensuring post-damage performance with acceptable degradation so that the survivability and safety are greatly enhanced. 1.3 Literature Review The fault-tolerant flight control system design are reviewed in this section. In general, fault-tolerant control systems can be categorized into passive and active systems. Passive controllers are designed based on a pre-determined set of requirements. They are fixed and are robust against a class of presumed faults [17]. Controllers need neither a fault detection, isolation and diagnosis scheme nor controller reconfiguration, but only limited fault-tolerant capabilities are achievable. On the other hand, as the name suggests, the active controllers respond to the system malfunction by actively reconfiguring control laws to ensure stability and performance requirements are met. Sometimes, certain degrees of performance degradation have to be accepted due to severe damage. The overall design objectives of fault-tolerant controllers are to meet the system transient and steady-state performance requirements not only under normal operating conditions, but also in fault situations. To achieve these objectives, a large number of controllers have been designed and implemented on both linear and nonlinear systems. The rest of this section presents some common control methods designed for fault-tolerant flight control, covering both the passive and active systems. Marcos and Balas [32] presented a quasi-linear parameter varying (LPV) model for LPV control synthesis that guarantees stability and robustness of the closed loop system. LPV was chosen to model the damaged aircraft because it is represented by its varying parameter description. Three different linearization approaches were used in the

13 Chapter 1. Introduction 5 paper, namely Jacobian linearization, state transformation and function substitution. Shin [49] presented an LPV model based on fault parameters. The fault parameters were also scheduling parameters in this case and were estimated on-line by using a two-stage adaptive Kalman filter. Aligned with the work of Shin, Shin and Gregory [48] presented another LPV formulation that involved function substitution. Unlike the previous work, aerodynamic coefficients were fitted from the plots and the uncertainties of which were included in the model. Maciejowski and Jones [31] applied Model Predictive Control (MPC) to the El Al flight 1862 model based on the assumption that a perfect fault detection and isolation model was available. Kale [26] and Miksch [35] also explored MPC as the control system. However, the main disadvantage in their formulations [26, 31, 35] was that the cost function has two different horizons, predictive and control horizons. By doing so, however the closed loop stability was not guaranteed and additional complexity is introduced. To overcome the stability problem, Almeida and Leissling [2] presented a new formulation using MPC in which fault-tolerant MPC with infinite prediction horizon approach was studied. Since MPC is applied over a long horizon, it would be only possible to implement such a method to dynamically slow systems, and hence not applicable to fault-tolerant flight control systems. Very recently, Fabio [15] provided a possible alternative to solve the problem by reducing the horizon to a shorter one. The Sliding mode control (SMC) method was also studied extensively by [3, 4, 47]. In the first paper [3], Alwi et al made an assumption that there was no FDI available. Unlike the MPC approach [31], the presented SMC did not require exact knowledge of the post-damage aircraft. With the popularity of control allocation in the field of FTFC systems, Shin et al [47] used the idea of control allocation, but an adaptive sliding mode control method was implemented. Most of the research related to SMC are lack of a detailed analysis in stability. As a result, in the paper by Alwi et al [4], the control allocation and SMC were combined and a rigorous design procedure was presented.

14 Chapter 1. Introduction 6 Adaptive control is another popular method being widely used in the flight control system [28, 4, 46]. A direct adaptive reconfigurable flight control method was proposed by Kim et al [28] to deal with the nuisance of having a system identification process in the indirect adaptive control. System identification was often used to accompany the indirect adaptive control to handle the model mismatch and parameter uncertainty issues. The timescale separation principle was applied in a model following scheme to control both inner loop state and the outer loop state of the flight system simultaneously. Adaptive control is often used in conjunction with a neural control scheme, particularly in nonlinear dynamic systems. In the paper by Napolitano [4], an integrated sensor and actuator failure detection, identification and accommodation within an FTFC system was studied. Such an integration is able to detect failures and pass the information to the controller with the goal of minimizing the false alarm rate and incorrect failure identification. Lombaerts et al [3] presented the use of nonlinear dynamic inversion (NDI) technique in which a real-time identified physical model of the damaged aircraft was included to avoid NDIs sensitivity to modeling error. An Iterated Extended Kalman Filter (IEKF) was used to estimate the aircraft states. The usage of which can potentially increase the computational cost and lead to real-time implementation difficulties. DI and MPC were combined together as a control method in [25] to tackle the FTFC problem. The combination is intuitive since the DI provides a linearized model that MPC can work on. As a result, a reconfigurable, nonlinear controller was designed. The above literature review covers some of the prominent control methods used in the field of fault-tolerant flight control system. However, few of the existing methods provide a systematic and efficient design approach to deal with the problem. Necessary features such as the flexibility and versatility, which these control methods lack, make the design process difficult, especially when transferring from one model to another one. It is also important to recognize the inherent nonlinear nature of aircraft dynamics. Nonlinear control methods may be better to handle the fault-tolerant tasks. As a result, a nonlinear

15 Chapter 1. Introduction 7 control method that possesses the systematic and efficient design approach is proposed in this research to deal with damaged aircraft. Fault-tolerant flight control systems are often complemented by a robust guidance system to achieve safe landing objective. For example, Menon et al. [33] implemented a robust guidance algorithm for impaired aircraft based on a point mass nonlinear aircraft model. The guidance algorithm was formulated with the finite interval differential game. The guidance commands then were inverse transformed into the roll, pitch and yaw attitude commands. Chawla et al [6] studied a partial integrated guidance and control system based on the nonlinear dynamic inversion to perform obstacle avoidance of UAVs. The collision cone concept was used in the derivation to transform the problem into a sequential target interception problem. The guidance algorithm was then derived under the frame of a collision cone. Other guidance algorithms based on optimization methods, such as mixed integer linear programming or model predictive control techniques are not suitable in our applications due to their heavy requirement of computational resources. In this thesis, a robust, feedback based guidance algorithm is implemented for damaged aircraft. The guidance algorithm takes into consideration damaged aircraft dynamics to adjust its commands in a feedback fashion. It also needs little modification to the existing control system architectures, unlike the above mentioned ones which require guidance command transformations. 1.4 Research Objectives & Contribution This research focuses on the design of nonlinear fault-tolerant flight control laws to ensure flight safety in the presence of adverse conditions. Additionally, it investigates the integration of guidance laws and control laws in the context of damaged aircraft to guarantee fast response, and safe landing. The benefits of the proposed framework over the existing ones are also explored. Although there have been many control laws designed for the purpose of mitigat-

16 Chapter 1. Introduction 8 ing faults and recovering the flight performance, few have demonstrated the ability to integrate with guidance laws to ensure safe landing. In addition, most of the exiting controllers, as mentioned in the previous section, are categorized as linear controllers, which require extensive gain scheduling to cover a wide flight envelope. This work aims to provide a sophisticated, yet designer-friendly solution to achieve the objective of safe landing of the damaged aircraft while taking the advantage of nonlinear control. The benefits of integration are critical in the damaged case. The smooth integration can support the post-damage planning, guidance, and control in a unified manner, which not only saves precious time, but also increases the survivability and safety. This work contributes to the research and development of the fault-tolerant flight control system mainly in the following areas: To develop a nonlinear fault-tolerant flight control system to handle damaged aircraft; To implement the guidance and control laws to expedite post-damage recovery and ensure safe landing; To investigate and verify the proposed framework performance by comparing with the existing control method. 1.5 Thesis Organization The thesis is organized in the following manner. Chapter 2 presents both healthy and damaged aircraft dynamics and modeling. The fault-tolerant flight guidance and control problem is introduced in Chapter 3, which covers the problem formulation and design framework. Chapter 4 focuses on the aircraft guidance law design. In Chapter 5, nonlinear fault-tolerant control methods are discussed. A novel nonlinear adaptive control law is derived in Chapter 6. Simulation studies are performed in that chapter to demonstrate its promising results. Finally, the concluding remarks and possible future works are offered in Chapter 7.

17 Chapter 2 Aircraft Dynamics This chapter presents the aircraft dynamics. In Section 2.2, the general nonlinear equations of motion (EOM) are derived and formulated. Section 2.3 introduces the nonlinear aerodynamic coefficients that are included in the models. Section 2.4 deals with the damaged aircraft modeling and Section 2.5 presents the optimization based trim routine used to seek the steady state level flight condition. 2.1 Introduction In this chapter, the general six degrees of freedom (6DoF) nonlinear EOM are introduced. Two aircraft models, Boeing 747-1/2 and NASA Generic Transport Model (GTM), are used throughout the thesis as simulation test beds. Both models are covered in details in Section 2.3. The Damaged aircraft modeling is studied in Section 2.4. Several damage scenarios as well as their possible outcomes are reviewed in that section. 2.2 Aircraft Modeling Before diving into the derivation of the equations of motion, it is important to establish the frames of reference. Throughout the thesis, the following right-handed and orthogonal reference frames are used: the earth-fixed inertial reference frame, F E ; the vehicle carried local earth reference frame, F O whose origin is fixed at the centre of gravity of the vehicle 9

18 Chapter 2. Aircraft Dynamics 1 and is assumed to have the same orientation as F E ; the wind-axes reference frame, F W, obtained by three successive rotations of horizontal flight path angle χ, vertical flight path angle γ, and bank angle µ from F O ; the stability-axes frame, F S, obtained from a rotation from F W by a rotation of β; the body-fixed frame, F B, obtained by rotations of yaw angle ψ, pitch angle θ, and roll angle φ from F O. These frames are shown in Fig Fig. 2.1: Aircraft Reference Frames Transformation from one frame to another is done by using the rotational matrices. For example, rotational matrices for F B to F S and F B to F W are defined in Eq.(2.1) and Eq.(2.2), respectively. cosα sinα F SB = 1 (2.1) sinα cosα cosα cosβ sinβ sinα sinβ F W B = cosα sinβ cosβ sinα sinβ (2.2) sinα cosα In order to derive the equations of motion, a number of assumptions must be made: The aircraft is a rigid body; The earth is flat and non-rotating; The aircraft mass properties are constant, any mass variation is negligible; The aircraft has a plan of symmetry, which is the X B Z B plane. It implies that moment of inertia I yz and I xy are equal to zero. This assumption is valid for

19 Chapter 2. Aircraft Dynamics 11 undamaged aircraft. When aircraft suffer from asymmetric damage, the assumption does not apply any more; Nonlinear Equations of Motion The aircraft equations of motion can be derived from Newton s Second Law. Mathematically, Newton s Second Law can be expressed as the following in the inertial frame F = d dt (mv t) E (2.3) M = d dt (H) E (2.4) where F is the sum of all external forces; m is the aircraft mass; M represents the sum of all external moments about the centre of the mass; H is the angular momentum about the centre of mass. The above equations can be written in the body-fixed frame, F B as F = d dt (mv t) B + ω mv t (2.5) M = d dt (H) B + ω H (2.6) where ω is the total angular velocity of the aircraft with respect to the Earth. The vector terms in Eq.(2.5) and Eq.(2.6) can be expressed as V t = ui + vj + wk (2.7) ω = pi + qj + rk (2.8) H = Iω (2.9) where I is defined as I x I xz I = I y I xz I z (2.1)

20 Chapter 2. Aircraft Dynamics 12 Substituting Eq.(2.7)-Eq.(2.9) into Eq.(2.5) and Eq.(2.6) and expanding terms yields, F x = m( u + qw rv) (2.11) F y = m( v + ru pw) (2.12) F z = m(ẇ + pv qu) (2.13) M x = ṗi x ṙi xz + qr(i z I y ) pqi xz (2.14) M y = qi y + pq(i x I z ) + (p 2 r 2 )I xz (2.15) M z = ṙi z ṗi xz + pq(i y I x ) + qri xz (2.16) where the external forces are the aerodynamic forces, thrust forces and gravity forces and the external moments include the aerodynamic moments and the engine moments. F x = qsc xb + F T x mgsinθ (2.17) F y = qsc yb + F T y + mgcosθsinφ (2.18) F z = qsc zb + F T z + mgcosθcosφ (2.19) M x = qsbc lb + M engx (2.2) M y = qs cc mb + M engy (2.21) M z = qsbc nb + M engz (2.22) where q = 1 2 ρv 2 t is the dynamic pressure. The equations presented above are collected together and rearranged into a set of twelve first order, aircraft equations of motion. Force equations: u = rv qw mgsinθ + 1 m ( qsc xb + F T x ) (2.23) v = ru + pw + gsinφcosθ + 1 m ( qsc yb + F T y ) (2.24) ẇ = qu pv + gcosφcosθ + 1 m ( qsc yb + F T y ) (2.25)

21 Chapter 2. Aircraft Dynamics 13 Kinematic equations: φ 1 sinφ tanθ cosφ tanθ p θ = cosφ sinφ q ψ r Moment equations: ṗ I xx I xz q = I yy ṙ I xz I zz 1 sinφ cosθ cosφ cosθ M x + (I yy I zz )qr + I xz pq M y + (I zz I xx )pr + I xz (r 2 p 2 ) M z + (I xx I yy )pq I xz qr (2.26) (2.27) Navigation equations: ẋ e cosθ cosψ sinφ sinθ cosψ cosφ sinψ cosφ sinθ cosψ + sinφ sinψ u ẏ e = cosθ sinψ sinφ sinθ sinψ + cosφ cosψ cosφ sinθ sinψ sinφ cosψ v ḣ e sinθ sinφ cosθ cosφ cosθ w (2.28) where (u, v, w) are the velocity components; (φ, θ, ψ) are the Euler angles, roll, pitch and yaw angle; (p, q, r) are the roll, pitch and yaw rate; (x e, y e, h e ) are the inertial positions. For the fault-tolerant flight control design, it is more sensible to introduce the airspeed, angle of attack and slideslip angle as state variables to replace u, v, w in the force equations. The main reasons are: first of all, Some of aerodynamic derivatives obtained from wind tunnel or flight tests, are tabulated based on α, β. As a result, it is easier to use these variables as state instead of converting from other variables. Thus, greater accuracy may be preserved. Secondly, when aircraft suffer from abnormalities in flight, their behavior can be difficult to predict. For instance, the upper limit of the pitch rate q may reach as high as.2rad/s, similar to the case of agile aircraft and aircraft can fly at a high airspeed (e.g. Vt = 6m/s). It means the term qu in eq.(2.25) may become as large as 12g s. However, in reality the upper limit of the normal acceleration can be only a few g s. Hence greater accelerations are introduced into equations because of the high rotation rates which the body-axes experience. This means much less favorable computer

22 Chapter 2. Aircraft Dynamics 14 scaling and hence much poorer solution accuracy for a given computer precision if the simulation is based on u, v and w instead of V t, α, and β [45]. The following equations are used to replace the force equations. Their derivations are included in the Appendix A. α = 1 mv t cosβ ( F xsinα + F z cosα + mv t ( pcosαsinβ + qcosβ rsinαsinβ)) (2.29) β = 1 mv t ( F x cosαsinβ + F y cosβ F z sinαsinβ mv t ( psinα + rcosα)) (2.3) V t = 1 m (F xcosαcosβ + F y sinβ + F z cosβsinα) (2.31) Thus, the 6DOF state vector is [ ] T x = V t α β φ θ ψ p q r x e y e h e (2.32) 2.3 Nonlinear Aerodynamic Coefficients Boeing 747-1/2 As mentioned earlier, both B747 and GTM models are considered in the thesis. In this section, the B747 nonlinear aerodynamic coefficients are presented. The NASA GTM ones are briefly reviewed later in this section. The Boeing 747-1/2 (Fig. 2.2) is an inter-continental wide-body transport with four turbofan jet engines designed to operate from international airports. It exhibits a wide array of characteristics (leading and trailing edge flaps, spoilers, variety of control surfaces, four fan jet engines...) which make it the perfect representative for any of the commercial airplanes flying today. The physical properties and aerodynamic data used in this thesis are obtained from NASA technical reports [22, 23]. The aerodynamic coefficients are based on a number of stability derivatives, which are defined in the stability frame of reference. Since the EOM are in F B, the aerodynamic coefficients must be in F B

23 Chapter 2. Aircraft Dynamics 15 as well. Thus, the following relationships are employed to accomplish the transformation. C Xb = C D cosα + C L sinα (2.33) C Zb = C D sinα C L cosα (2.34) C mb = C m (2.35) C Y b = C Y (2.36) C lb = C l cosα C n sinα (2.37) C nb = C l sinα + C n cosα (2.38) Fig. 2.2: Boeing 747, Source: Airliners.net The complete expressions of the coefficients can be found in the technical reports. However, in order to facilitate the fault-tolerant flight controller investigation and design, the model complexity will be reduced by eliminating some stability derivatives that contribute little to the overall aerodynamic coefficients. The following simplified aerodynamic coefficient equations are used: C L = C Lbasic + dc L dq q s c 2V t + ( dc L dδ EI δ EI + dc L dδ EO δ EO ) (2.39) C D = KC Dbasic + (1 K)C Dmach + C Dsideslip (2.4) C Y = dc Y dβ β + dc Y dp C l = dc l dβ β + dc l dp C m = C mbasic + dc m.25 dq C n = dc n dβ β + dc n dp p s b + C Yrudders 2V t (2.41) p s b + dc l r s b + C linbd 2V t dr 2V ailerons + C lrudders t (2.42) q s c 2V t + ( dc m.25 dδ EI δ EI + dc m.25 p s b + dc n 2V t dr δ EO ) (2.43) dδ EO r s b + C ninbd 2V ailerons + C nrudders (2.44) t

24 Chapter 2. Aircraft Dynamics 16 In the cases of lift and pitching moment coefficients, the contributing factors include the basic lift and pitching moment coefficients, the dynamic stability derivatives dc L dq and dcm dq as well as the contributions from the both inboard and outboard elevators, respectively. It is also assumed in this case that the centre of gravity coincides with the aerodynamic centre at the quarter chord location. The drag coefficient, C D, is mainly dictated by the basic drag coefficient, drag coefficient due to Mach number as well as the sideslip angle. K is an aircraft specific constant. C Y is determined from the contribution of β, p, and rudders. Similarly, C l and C n depend on β, p, r, inboard ailerons, and rudders. The stability derivatives in Eq.(2.39)- Eq.(2.44) are then put into the look-up tables (LUT) in Matlab for easy access during the simulation. Thus, the nonlinear B747 model is obtained using the reduced aerodynamic coefficients and the previous derived nonlinear EOM NASA GTM The NASA GTM model is provided in a Simulink package (Fig. 2.3). It includes the comprehensive aircraft information in terms of aerodynamic look-up tables as well as Simulink blocks. The difference between the previous mentioned EOM and the one implemented in the GTM model is that the aerodynamic forces are calculated based on the center of pressure (CP) instead of the aerodynamic center (ac). It is believed that because of the extensive wind tunnel data, such practices becomes possible. As a result, the moment equation requires modification to accommodate the change. In addition, the CG location is no longer assumed at the ac as in the case of B747. The moment equation implemented in the model is the following: M = M aero + M eng + (CP CG) F aero (2.45)

25 Chapter 2. Aircraft Dynamics 17 Fig. 2.3: NASA GTM Simulink Environment 2.4 Damaged Aircraft Modeling Aircraft damage can range from single component failure/malfunction to severe airframe and engine damage. Different failure situations pose different levels of severity and threat to the flight safety. In the case of sensor failure, the original system could be recovered as long as the correct information is available elsewhere, either from physically redundant sensors or from observers or estimators based on analytical redundancy. Actuator failures are more involved than the sensor case. After the actuator failure occurs, if the original performance is still desired, the remaindering actuators have to operate beyond their design capabilities. This means actuator saturation and further system performance degradation. Thus, in the case of actuator failures the system should accept graceful degradation in performance. The airframe structural damage can be the most difficult to deal with. Not only does it compromise the aircraft integrity, but also alters the aircraft original flight envelope. Thus, great effort must be put into the structural damage scenarios. Additionally, it is important to have a close representation of the fault when

26 Chapter 2. Aircraft Dynamics 18 designing the fault-tolerant control. Table 2.1 lists a number of common fault scenarios and their respective effects. In this work, the primary focus is on actuator failures and airframe structural damage. In the case of the B747, loss of actuator effectiveness will be considered. Let u be the actuator vector of the control design, [ ] T u = u 1 u 2 u i (2.46) where i = 1 s (max number of actuators). Let Λ be the control effective matrix to model the actuator faults. Λ is a diagonal matrix with positive elements. Λ 1... Λ 2... Λ = Λ i (2.47) The actuator fault model is ũ = Λu (2.48) where < Λ i 1. When Λ i = 1, it means no faults occurred in the i th actuator. If Λ i < 1, it implies the faults has impaired the i th actuator s function.

27 Chapter 2. Aircraft Dynamics 19 Table 2.1: Failure modes [3] Failure Mode Control loss on actuators Structural loss on control surface Engine(s) out Effect Surface stuck at last position Control effectiveness reduced minor change in aerodynamics Asymmetric thrust, increased drag due to β Large change in possible operating region Severe structural damage significant change in aerodynamics, mass and moments of inertia Fig. 2.4: GTM Damage Case [2], Note the included model support string in the grids. The NASA GTM model includes six damage models, ranging from rudder off to left horizontal stabilizer off. Each damage scenario provides a unique design challenge. However, this work concentrates on the damage scenario six (Fig. 2.4), which is the loss of entire left horizontal stabilizer and the left elevator. The damage reduces the longitudinal stability as well as the pitch control power due to the elimination of the left elevator. The control asymmetry generates an undesired rolling moment which needs to be compensated for with some roll control. In addition, the aircraft is no longer symmetric along the fuselage centerline, which means off-diagonal inertias are non-zero and the CG location is shifted forward, down and to the right of the fuselage centerline. The GTM simulink model is based on a 5.5% scaled down aircraft model. For the sake of simlicity, the simulink model is used instead of the scaled up aircraft model. The damage

28 Chapter 2. Aircraft Dynamics 2 involves the following variables change: W =.59lbs (2.49) x =.553in (2.5) y = +.88in (2.51) z =.32in (2.52) I xx =.918sl ft 2 (2.53) I yy =.27315sl ft 2 (2.54) I zz =.2849sl ft 2 (2.55) I xz =.1559sl ft 2 (2.56) I xy = +.437sl ft 2 (2.57) I yz = +.265sl ft 2 (2.58) where W is the change in the aircplane weight; x, y and z are the C.G. shift; The rest describes the change in the moment of inertia. 2.5 Trim Analysis Aircraft trim analysis is an important procedure to evaluate the aircraft behavior. As a part of the analysis, the trim routine is used to find an equilibrium point of the aircraft under a given set of constraints. Generally, an aircraft in-flight can be trimmed in several conditions: steady-state level flight, steady state climbing/descending or constant turning. The steady-state level flight is particularly interesting in this case. The steadystate flight condition means that the time derivatives of the state variables are zero. A steady-state point is often used as an initial point of a simulation. Thus, it is important to find a set of control inputs and state values corresponded to an equilibrium point of the system. As a result, the objective of the trim routine is to solve the aircraft nonlinear equations of motion which are first order differential equations, to obtain state and control vectors that ensure the time derivatives of state variables are zero.

29 Chapter 2. Aircraft Dynamics 21 A simplex optimization problem is formulated based on the cost function Eq.(2.59) adopted from [5], aircraft dynamics and the steady-state level flight constraint, Eq.(2.6) to obtain trimmed state and control vectors. J = V 2 t + 1( α 2 + β 2 ) + 1(ṗ 2 + q 2 + ṙ 2 ) (2.59) γ = (2.6) where γ = means the flight path angle must be zero during the steady-state level flight. Fig. 2.5: Trim routine procedure The objective of the optimization is to minimize the cost function by varying the control input variables. In the case of B747, the state vector is [ ] T x = V t α β φ θ ψ p q r x e y e h e and the control vector is [ ] T u = δ th δ e δ a δ r, where δ th is the thrust setting from -1; δ e is the elevator deflection; δ a is the aileron deflection; δ r is the rudder deflection. Fig.2.5 illustrates the trim routine procedure. The numerical trim analysis is performed on the Boeing 747 model. The steady-state flight condition is based on the aircraft altitude h e and airspeed V t, which are specified at the beginning of the trim routine. In this case, h e = 3m and V t = 15m/s are specified for the aircraft descent and approach scenario. Table 2.2 and 2.3 list the trimmed state and control vectors. Fig show the 3 seconds open-loop time history.

30 Chapter 2. Aircraft Dynamics 22 Table 2.2: Trimmed States State Value Derivative State Value Derivative V t 15m/s -5.31e-35 m/s 2 p rad/s 4.58e-18 rad/s 2 α 9.36e-2 rad rad/s q rad/s rad/s 2 β rad 1.79e-19 rad/s r rad/s rad/s 2 φ rad rad/s x e m 15 m/s θ 9.36e-2 rad rad/s y e m -2.96e-16 m/s ψ rad rad/s z e 3m m/s Table 2.3: Trimmed Controls Control Value Control Value δ th 1.9e-1 δ e -8.54e-1deg δ a 9.36e-2deg δ r deg V t (m/s) 15 φ (deg) Fig. 2.6: airspeed Fig. 2.7: roll angle

31 Chapter 2. Aircraft Dynamics α (deg) 5.35 θ (deg) Fig. 2.8: Angle of attack Fig. 2.9: pitch angle β (deg) ψ (deg) Fig. 2.1: Sideslip angle Fig. 2.11: yaw angle 1 5 p (deg/s).5.5 x e (m) Fig. 2.12: Roll rate Fig. 2.13: x e 1 x q (deg/s) 4 y e (m) Fig. 2.14: pitch rate Fig. 2.15: y e

32 Chapter 2. Aircraft Dynamics r (deg/s) h e (m) Fig. 2.16: Yaw rate Fig. 2.17: Altitude 2.6 Summary This section covered the fundamental aircraft dynamics. The nonlinear equations of motion were derived in this chapter. Two aircraft models were established to be used as test beds for the fault-tolerant control design in the later sections. Furthermore, the damaged aircraft dynamics were also presented. The trim routine was implemented to find the steady state conditions for the simulation scenarios.

33 Chapter 3 Fault-tolerant Flight Guidance and Control Problem 3.1 Introduction In this chapter, the fault-tolerant flight guidance and control problem is identified and formulated and the guidance law design is derived. The objective is to provide a feasible framework that is capable of handling aircraft that have suffered from damage so that stabilization and safe landing are achieved. The framework comprises the guidance and control loops. Each loop has its own design approach and objective, but overall acts in an integrated, continuous fashion. 3.2 Problem Formulation In this section, a detailed description of the fault-tolerant flight guidance and control problem is introduced. The proposed framework that provides a feasible and viable solution is also presented in the section. When the aircraft encounters damage in flight, the conventional control design may not be adequate and robust enough to handle the situation. Eventually, the aircraft may become uncontrollable and unstable. Most of the time, human pilot intervention is required to prevent the situation from deteriorating to 25

34 Chapter 3. Fault-tolerant Flight Guidance and Control Problem 26 the worst. However, human error is becoming a major contributing factor to aviation accidents. Thus, it is desired to implement the advanced control system that is capable of actively providing intelligent and effective actuator control in such situations as well as to backup the conventional flight control in normal flight conditions. In addition, damaged aircraft can behave drastically different from the original aircraft specifications. The flight envelope can be altered as well. Since the aircraft dynamics are intrinsically nonlinear, linear control methods sometimes are not adequate to handle the complex dynamics. The nonlinear robust controller on the other hand is able to tolerant significant aircraft parameter variation. For sudden, large scale behavior changes, nonlinear controller is far superior than the linear controller which may not be able to control the plant at all. The nonlinear controller does not require extensive and timeconsuming gain scheduling due to the large number of design points. The unpredictable nature of damaged scenarios can also increase the level of complexity of gain scheduling in the linear controller. Thus, the nonlinear controller is more suitable in the fault-tolerant flight control. Furthermore, the ultimate goal for any damaged aircraft is to land safely. A robust guidance law is a pre-requisite for the control system. The robust guidance law should take the altered aerodynamics and performance change into consideration when generating guidance commands. In addition, from a practical point of view, it is beneficial to have the robust guidance design fitted into the existing flight control design system so that little system modification is necessary. The proposed fault-tolerant flight guidance and control framework addresses the issues mentioned above and provides a sophisticated system to deal with damaged aircraft. Fig. 3.1 depicts the proposed system. The control part of the system adopts the conventional three loop flight control system design with a separate speed controller. Alternatively, an integrated speed controller can be augmented with the inner loop. Since the guidance loop generates the airspeed

35 Chapter 3. Fault-tolerant Flight Guidance and Control Problem 27 tracking command, it is more sensible to have a separate speed control loop to track the signal. In addition, the inner loop has fast states, such as the roll rate, pitch rate, and yaw rate, and the airspeed dynamics is not as fast as the inner loop state; as a result, the tracking performance may not be as accurate as expected. The inner loop consists of the [ p; q ; r ] T state vector. The output is the control vector [δ e ; δ a ; δ r ] T. The inner loop is responsible for stabilizing the aircraft as well as tracking the commands from the outer loop. The outer loop is designed with [α ;β; µ ] T, where µ is the bank angle. The flight path angle loop is responsible for keeping track of guidance commands, which are [γ ; χ ] T, the vertical flight path angle and the heading angle. The guidance loop also provides the speed command which feeds directly to the speed controller, whose output is the throttle setting δ th. The input to the guidance loop is based on the trajectory information defined in the trajectory loop. State feedback is required for all of the loops. Fig. 3.1: Proposed fault-tolerant flight guidance and control system The control system design is based on the nonlinear state-dependent Riccati equation (SDRE) control method. In each loop, the SDRE controller is designed to track the commands generated by the previous loop. State feedback is required in each loop to provide the information for SDRE. The guidance commands are generated by the zero effort miss concept guidance law [41] that transforms the traditional trajectory tracking problem into the aircraft-target intercepting problem. The integration between the guidance and control systems is done in a harmonious fashion. Little modification is required to accommodate the guidance commands into the traditional three loop control architecture.

36 Chapter 3. Fault-tolerant Flight Guidance and Control Problem Remarks The proposed fault-tolerant flight guidance and control framework is the backbone of the work. Not only does it provide the guideline to the guidance and control systems design, but also illustrates a sophisticated and advanced system. In the following chapters, the actual system design work are carried out in details. The flight simulation results are also included. 3.4 Aircraft Guidance Law Design In this section, the aircraft guidance law design is presented. The guidance law is used in the framework to provide the flight path angle, the heading angle as well as the airspeed commands. Additionally, the guidance law can be readily fit into the existing control system architecture so that little modification is required during the system integration. In the following sections, the detailed design is based on the work of No et al. [41]. The guidance system is based on the concept of zero effort miss, which is a common notion in the missile guidance community and has been used in a number of proportional navigation guidance laws [56]. Essentially, the aircraft under the guidance law commands tries to intercept the reference trajectory on which an ideal imaginary aircraft flies. The guidance law navigates the aircraft to follow the imaginary aircraft as close as possible. By doing so, the trajectory tracking objective is achieved with minimum error. Thus, the traditional trajectory tracking problem is reformulated into an aircraft-target intercept problem. 3.5 Guidance Law Design The core of the guidance law is the zero effort miss concept. Since the guidance law is based on the aircraft-target interception problem, the zero effort vector is defined in such a scenario. Let there be an ideal, imaginary target aircraft flying on the trajectory. The

37 Chapter 3. Fault-tolerant Flight Guidance and Control Problem 29 real aircraft tries to intercept the imaginary target. Denote (d, v) and (d, v ) as the position and velocity vectors for the aircraft and target, respectively. These vectors can be expanded into the components in the reference inertia frame (e x, e y, e z ) as d = d x e x + d y e y + d z e z (3.1) v = v x e x + v y e y + v z e z (3.2) d = d xe x + d ye y + d ze z (3.3) v = vxe x + vye y + vze z (3.4) Assume both the aircraft and target maintain their speed and direction, the distance vector between them as shown in Fig. 3.2, at some time in future t f can be expressed as d tgo = (d d) + (v v)t go (3.5) = T x e x + T y e y + T z e z (3.6) where t go is the time-to-go until the future time t f, t go = t f t (3.7) (T x, T y, T z ) are the components of the zero effort miss vector is the fixed frame. T x = d x d x + (vx v x )t go (3.8) T y = d y d y + (vy v y )t go (3.9) T z = d z d z + (vz v z )t go (3.1) The vector in Eq.(3.5) is often referred to as the zero effort miss vector. Fig. 3.2: Zero effort miss vector

38 Chapter 3. Fault-tolerant Flight Guidance and Control Problem 3 The actual guidance commands are derived through the use of a Lyapunov-like function. V = 1 2 d t go d tgo (3.11) = 1 2 (T 2 x + T 2 y + T 2 z ) (3.12) Taking time derivative of Eq.(3.12), dv dt = T x( v x v x )t go + T y ( v y v y )t go + T z ( v z v z )t go (3.13) The velocity vector can then be expressed in terms of the flight path angle γ, and the heading angle χ. v = vcosγcosχe x + vcosγsinχe y vsinγe z (3.14) Assuming the coordinate turn is achieved, so the sideslip angle β =, then the heading angle can be approximated by the yaw angle ψ, χ ψ (3.15) Substituting the above two equations into the Eq.(3.13), dv dt = T x( v x vcosψcosγ + v ψsinψcosγ + v γcosψsinγ)t go + T y ( v y vsinψcosγ v ψcosψcosγ + v γsinψsinγ)t go + T z ( v z + vsinγ + v γcosγ)t go (3.16) Following No et al [41], Eq.(3.13) can be transformed from the fixed frame (e x, e y, e z ) to a control frame (e v, e ψ, e γ )which includes the airspeed, the flight angle and the heading angle, where e v is the unit direction vector along the velocity; e γ is a unit vector perpendicular to e v and is positive in the direction of the increasing longitudinal flight path angle; e ψ is along the direction of the increasing yaw angle and follows the right hand rule. As a result, dv dt = ( v v v)t v t go + ( v ψ v ψcosγ)t ψ t go + ( v γ + v γ)t γ t go (3.17)

39 Chapter 3. Fault-tolerant Flight Guidance and Control Problem 31 where ( v v, v ψ, v γ) denote the target acceleration vector in the control frame. (T v, T γ, T ψ ) are the components in the control frame, T v = T x cosψcosγ + T y sinψcosγ T z sinγ (3.18) T ψ = T x sinψ + T y cosψ (3.19) T γ = T x cosψsinγ + T y sinψsinγ + T z cosγ (3.2) To ensure the Lyapunov stability theorem can be employed, Eq.(3.13) must be negative definiteness, which is achieved by dv dt = 2NV (3.21) where N is a positive constant. One of the advantages of the guidance law is that the aircraft dynamics are taken into consideration. However, the entire dynamics are too complex to include. First order approximations are used to describe the control channels for the airspeed v, the flight path angle γ, and the heading angle ψ. Speed control loop: v = 1 τ v (v c v) (3.22) Flight path angle loop: γ = 1 τ γ (γ c γ) (3.23) Heading angle loop: ψ = 1 τ ψ (ψ c ψ) (3.24) where τ v, τ γ and τ ψ are the time constants of each control loop. v c, γ c and ψ c are the input commands to the control loops. Finally, Eq.(3.21) becomes, dv dt = ( v v v)mt v t go + ( v ψ v ψcosγ)t ψ t go + ( v γ + v γ)t ψ t go (3.25) = 2N V (3.26) = NT 2 v NT 2 ψ NT 2 γ (3.27)

40 Chapter 3. Fault-tolerant Flight Guidance and Control Problem 32 As suggested by No et al. [41], the natural selection of guidance commands to satisfy Eq.(3.27) appear to be, v c = v + N t go τ v T v + τ v v v (3.28) ψ c = ψ + N t go τ ψ vcosγ T ψ + τ ψ vcosγ v ψ (3.29) γ c = γ N t go τ γ v T γ τ γ v v γ (3.3) The set of guidance commands provide the airspeed, the flight path angle and the heading angle to intercept the target. By enforcing a small miss distance error, in other words keeping zero effort miss vector d tgo small, for short t go, the aircraft follows the imaginary target and stays on the desired trajectory with a small error. As expected, the guidance laws Eq.(3.28), (3.29), and (3.3) are feedback based commands. For the impaired aircraft case, the ideal aircraft must consider the impaired aircraft performance degradation. For example, vx = v cosγ (3.31) vz = v sinγ (3.32) where v and γ are the feedback values of the impaired aircraft. 3.6 Summary In this chapter, The fault-tolerant flight guidance and control problem was formulated. The guidance law design was introduced. It is based on the zero effort miss concept that has been used in a number of proportional guidance law designs. The design transformed the traditional guidance law into an aircraft-target interception problem. By intercepting the target aircraft, the real aircraft stays on the desired trajectory with a small error. The guidance law has several advantageous features. The guidance commands are based on the feedback as well as the aircraft dynamics. The design parameters are similar to the control gains, requiring proper tuning.

41 Chapter 4 State-Dependent Riccati Equation Control Method In this chapter, the control method implemented in the fault-tolerant flight guidance and control framework is discussed in detail. The state-dependent Riccati equation (SDRE) control method is reviewed first. The background mathematical preliminaries, control problem formulation and design technique are also presented. The SDRE is a unique control method among nonlinear control methods. It embraces the advantages of linear controller design techniques while applying to nonlinear system dynamics. In the end, simulation results are included and discussed. The nonlinear controller design is intrinsically more difficult than the linear controller design. It requires rigorous and sophisticated mathematical background to ensure proper formulation and analysis are performed. Despite these difficult obstacles, research on the topic of nonlinear control method has flourished and made noticeable advances in recent years [24, 27]. However, there are still challenging questions awaiting to be answered in the field. The lack of connection between the theoretic work and the practical implementation prevents many modern nonlinear control methods from being applied. In addition, stability, performance and robustness continued to be the issues that nonlinear control methods struggle to address satisfactorily. 33

42 Chapter 4. State-Dependent Riccati Equation Control Method 34 The SDRE control method appearers to be a very practical nonlinear control method for the systematic design of nonlinear controllers. It has become very popular within the control community over the last decade, providing an extremely effective algorithm for synthesizing nonlinear feedback controls by allowing nonlinearities in the system states, while additionally offering great design flexibility through design metrics [9]. The control method was originally introduced by Pearson [44] in the 197s and later refined by Wernli and Cook [52]. In recent years, Cloutier, D Souza and Mracek [11, 12, 38] independently studied the control method. The SDRE method provides a straightforward and efficient computational algorithm to solve difficult nonlinear problems, which are often complicated by nonaffine-in-control, control, or state constraints. The backbone of the method is state parameterization. It allows the nonlinear dynamics expressed by differential equations to be parametrized into the product of a matrix-valued function and the state vector while preserving the original system nonlinearities. In the end, a linear-like structure is obtained in state space form. The coefficients are state-dependent and non-unique. The control method has been successfully implemented in a variety of practical applications across disciplines. Specifically, Mracek and Cloutier [37] applied the SDRE method to a full envelope missile longitudinal autopilot. Cimen [8] proposed an approximate SDRE nonlinear tracking method which was used to design a supertanker s autopilot. Gao [21] implemented the SDRE control method in a re-entry tracking problem for a reusable launch vehicle (RLV). Bogdanov [5] flight tested the SDRE controller on board a small unmanned helicopter. Flight tests were flown to evaluate the accuracy of tracking under SDRE control. These works demonstrate that the SDRE control method is a capable nonlinear control method and has great potential in the practical implementations. The control method solves an algebraic Riccati equation (ARE) to construct the suboptimal control law. The interesting fact is that because of the state-dependent nature of the coefficients, the ARE is solved at each step with varying coefficients. It means the feedback control gain varies at each step as well. This is certainly a desirable feature

43 Chapter 4. State-Dependent Riccati Equation Control Method 35 of SDRE in the fault-tolerant flight control design. The control law can actively modify itself in response to the aircraft parameter changes. In addition, extra design freedom is available through the non-uniqueness of state-dependent coefficients. In the following sections, the SDRE nonlinear control method is first reviewed in Sec. 4.1 with the control problem formulation. The section also covers the state-dependent coefficient parameterization or extended linearization. The SDRE stability and optimality analysis are offered in Sec The SDRE design techniques are presented in Sec Simulation studies are followed in Sec SDRE Control Method Consider the general autonomous, affine-in-control, nonlinear system dynamics in the form of, ẋ(t) = f(x) + B(x)u(t) x() = x (4.1) where state vector x R n and control vector u R m ; f : R n R n and B : R n R n m with B, x. The nonlinear regulator problem is formed as the following. Minimize the infinitehorizon performance index, J = 1 2 (x T Q(x)x + u T R(x)u)dt (4.2) with respect to the state vector x and the control vector u subject to the nonlinear system dynamics Eq.(4.1). The state and control weighting matrices Q(x), R(x) are state-dependent, such that Q(x) is positive semi-definite and R(x) is positive definite for all x. Additionally, Q(x) can be expressed as Q(x) = C(x) T C(x). In order to proceed with the SDRE control law, the state-dependent coefficients (SDCs) must be introduced. SDCs are obtained through a procedure known as extended linearization [19], apparent linearization [52], or SDC parameterization [12]. It

44 Chapter 4. State-Dependent Riccati Equation Control Method 36 is a procedure to bring the nonlinear dynamics into a linear-like structure expressed by SDCs in addition to the state and control vectors. It is important to assume, Assumption 1. f(x) is continuously differentiable with respect to x for all x. Assumption 2. Without the loss of generality, the origin x = is an equilibrium point of the system with u =. It implies f() = and B(). so that the existence of a global SDC parameterization of f(x) is guaranteed [51]. As a result, the nonlinear differential equations, Eq.(4.1) can be expressed as, ẋ = A(x)x + B(x)u(t) x() = x (4.3) f(x) = A(x)x (4.4) where A(x) and B(x) are the state-dependent coefficients. The following definitions are associated with the SDCs. Definition 1. A(x) is a controllable parameterization of the nonlinear system if the pair {A(x), B(x)} is controllable for all x Definition 2. A(x) is a stabilizable parameterization of the nonlinear system if the pair {A(x), B(x)} is stabilizable for all x Definition 3. A(x) is Hurwitz if all the eigenvalues of A(x) are in the open left plane (negative real parts) for all x In addition to the assumptions mentioned above, the following assumption must also be met, Assumption 3. A( ), B( ), Q( ), and R( ) are C 1 (R n ) matrix-valued functions Assumption 4. The pair {A(x), B(x)} and {A(x), Q 1/2 (x)} are pointwise stabilizable and detectable SDC parameterizations of the nonlinear system 4.1 for all x, respectively.

45 Chapter 4. State-Dependent Riccati Equation Control Method 37 The SDRE control design is similar to the Linear Quadratic Regulator (LQR) control method. In the case of SDRE, the state-dependent Riccati equation is solved at each step to construct the control law. The state feedback controller shares the similar form with LQR. u(x) = R 1 (x)b T (x)p(x)x (4.5) where P(x) is the unique, symmetric, positive definite solution to the state-dependent Riccati equation, P(x)A(x) + A T (x)p(x) P(x)B(x)R 1 (x)b T (x)p(x) + Q(x) = (4.6) The closed loop dynamics become: ẋ = [A(x) B(x)R 1 (x)b T (x)p(x)]x (4.7) The nonlinear state feedback gain is, K(x) = R 1 (x)b T (x)p(x) (4.8) Fig. 4.1: SDRE design flowchart Clearly, the control gain is dependent on the state vector x. It also varies every time the SDRE is solved. The direct benefits of SDRE method is its simplicity and effectiveness. There is no attempt to solve the Hamilton-Jacobi-Bellman equation. When the coefficients and weighting matrices are constant, the SDRE problem becomes the well-known LQR problem.