ABSTRACT. Bei, Lu. Linear Parameter-Varying Control of an F-16 Aircraft at High Angle of Attack. (Under the direction of Dr. Fen Wu).

Transcription

1 ABSTRACT Bei, Lu. Linear Parameter-Varying Control of an F-16 Aircraft at High Angle of Attack. (Under the direction of Dr. Fen Wu). To improve the aircraft capability at high angle of attack and expand the flight envelope, advanced linear parameter-varying (LPV) control methodologies are studied in this thesis with particular applications of actuator saturation control and switching control. A standard two-step LPV antiwindup control scheme and a systematic switching LPV control approach are derived, and the advantages of LPV control techniques are demonstrated through nonlinear simulations of an F-16 longitudinal autopilot control system. The aerodynamic surface saturation is one of the major issues of flight control in the high angle of attack region. The incorporated unconventional actuators such as thrust vectoring can provide additional control power, but may have a potentially significant pay-off. The proposed LPV antiwindup control scheme is advantageous from the implementation standpoint because it can be thought of as an augmented control algorithm to the existing control system. Moreover, the synthesis condition for an antiwindup compensator is formulated as a linear matrix inequality (LMI) optimization problem and can be solved efficiently. By treating the input saturation as a sector bounded nonlinearity with a tight sector bound, the synthesized antiwindup compensator can stabilize the open-loop exponentially unstable systems. The LPV antiwindup control scheme is applied to the nonlinear F-16 longitudinal model, and compared with the thrust vectoring control approach. The simulation results show that the LPV antiwindup compensator improves the flight quality, and offers advantages over thrust vectoring in a high angle of attack region. For a thrust vectoring augmented aircraft, the actuator sets may be different at low and high angles of attack. Also due to different control objectives, a single controller may not exist over a wide angle of attack region. The proposed switching LPV control approach based on multiple parameter-dependent Lyapunov functions provides a flexible design method with improved performance. A family of LPV controllers are designed, each suitable for a specific angle of attack region. They are

2 switched according to the trajectory of angle of attack so that the closed-loop system remains stable and its performance is optimized. Two switching logics, hysteresis switching and switching with average dwell time, are examined. The control synthesis conditions for both switching logics are formulated as generally non-convex matrix optimization problems. To make the switching LPV control approach more applicable, two convexified methods are given according to the state of the controller is reset or not at switching time. The thrust vectoring augmented F-16 longitudinal model with different control objectives at low and high angles of attack is used to validate the results of the switching control scheme.

3 Linear Parameter-Varying Control of An F-16 Aircraft at High Angle of Attack by Bei Lu A dissertation submitted to the Graduate Faculty of North Carolina State University in partial satisfaction of the requirements for the Degree of Doctor of Philosophy Department of Mechanical and Aerospace Engineering Raleigh 2004 Approved By: Dr. Mo-Yuen Chow Dr. Ashok Gopalarathnam Dr. Fen Wu Chair of Advisory Committee Dr. Paul I. Ro

4 ii To my husband Qifu Li with all my love.

5 iii Biography Bei Lu was born on May , in Shanghai, China. She received her BS and MS degrees in Power and Mechanical Engineering from Shanghai Jiaotong University, China, in 1996 and 1999, respectively. She enrolled in graduate school at NCSU in the spring of 2002.

6 iv Acknowledgements I would first like to extend my gratification to Dr. Fen Wu, my adviser. Thank you for your financial support, technical guidance and consistent encouragement throughout the last three years. It has been an honor and a pleasure to work with you. I would also like to thank Dr. Paul I. Ro, Dr. Mo-Yuen Chow, and Dr. Ashok Gopalarathnam, for providing constructive comments on my work and presentation. I acknowledge the NASA Langley Research Center for the financial support of this research under Grant No. NAG through Dr. Fen Wu and Dr. Ashok Gopalarathnam (Technical Monitor: Dr. SungWan Kim). A special thank you goes to my parents. Thank you for your everlasting and unconditional love and support. Finally, I would like to thank my husband, Qifu Li, for being my greatest supporter. You inspired me to believe that I could accomplish anything.

7 v Contents List of Tables List of Figures vii viii List of Notations x Mathematical Symbols x Symbols for Aircraft Dynamics xi 1 Introduction Motivations and Objectives Background in Flight Control Thesis Outline Modeling of the Aircraft Generalized Nonlinear Model for the Thrust-Vectored Aircraft Longitudinal Model of an F-16 Aircraft with Thrust Vectoring Aircraft Description Aerodynamic Model Propulsion System Model Equations of Motion LPV Modeling of F-16 Longitudinal Axis LPV Systems Development of F-16 LPV Models Verification of F-16 LPV Models Summary Actuator Saturation Control Introduction Saturation Nonlinearity LPV Robust Analysis of Sector-Bounded Uncertainties

8 vi 3.4 LPV Antiwindup Control Problem Statement and Preliminaries LPV Antiwindup Control Synthesis LPV Antiwindup Compensator Construction Applications to the F-16 Aircraft Model Problem Setup Design Results Nonlinear Simulations Comparison with Thrust Vectoring Control Summary Switching LPV Control Introduction Analysis of Switched LPV Systems Hysteresis Switching Switching with Average Dwell Time Switching LPV Control Design Hysteresis Switching Switching with Average Dwell Time Convexified Switching LPV Synthesis Conditions Switching with Continuous Control State Switching via Control State Reset Applications to F-16 Aircraft Model Problem Setup Switching with Continuous Control State Switching via Control State Reset Summary Conclusion Contributions Future Work Bibliography 120

9 vii List of Tables 2.1 Mass and geometric properties Control actuator models H performance level vs. sector range [0, k i ] Effect of weights on performance level Effect of constants λ 0 and µ on average dwell time switching

10 viii List of Figures 2.1 Definition of axes and aerodynamic angles Vectored thrust F-16 aircraft model with and without thrust vectoring Aerodynamic data Flight equilibrium points Time responses for 0.1 step input of δ th at V = 200 ft/s and α = Time responses for ±1 doublet input of δ e at V = 200 ft/s and α = Time responses for 0.5 step input of δ ptv at V = 200 ft/s and α = Time responses for 0.1 step input of δ th at V = 160 ft/s and α = Time responses for 0.05 step input of δ th at V = 160 ft/s and α = A control system with saturation nonlinearity A graphical representation of sect[a, b] Sector bounds on saturation nonlinearity Saturation block with two outputs Dead-zone representation of saturation nonlinearity Sector-bounded uncertainty Nonlinear saturation control diagram Open-loop interconnection for nominal LPV controller design Nonlinear ±1 doublet response with LTI nominal controller Nonlinear ±2 doublet response w/o LTI antiwindup compensator Nonlinear ±1.7 doublet response w/o LTI antiwindup compensator for an unstable open-loop plant Nonlinear doublet response with LPV antiwindup compensator Thrust vectoring control scheme Nonlinear doublet response with thrust vectoring Comparison of nonlinear doublet response with thrust vectoring and with antiwindup compensator for an unstable open-loop plant A switching signal in the case of two parameter subsets

11 4.2 Hysteresis switching region and switching signal σ Piecewise continuous Lyapunov functions for hysteresis switching (Z N = {1, 2}) Switching region and switching signal σ with dwell time Discontinuous Lyapunov functions for switching with dwell time (Z N = {1, 2}) Flight conditions and partitioned flight envelope for two switching logics Weighted open-loop interconnection for switching control of F-16 aircraft Command input 1 for switching LPV control simulation Time history of parameters and switching signal under hysteresis switching with continuous control state for command input Time responses of actuators under hysteresis switching with continuous control state for command input Time history of parameters and switching signal under switching control via control state reset for command input Time responses of actuators under switching control via control state reset for command input Time history of parameters and switching signal under switching control via control state reset for command input Time responses of actuators under switching control via control state reset for command input ix

12 x List of Notations Mathematical Symbols 0 n m the zero element of R n m C 1 (U, V ) set of continuously differentiable functions from U to V diag(a 1, a 2,..., a n ) the n by n diagonal matrix with elements a 1, a 2,..., a n on the diagonal line F l (, ) linear fractional transformation I n the n-dimensional identity Ker(M) the orthogonal complement of the matrix M L 2 space of square integrable functions M T the transpose of the matrix M M 1 the inverse of the invertible matrix M M > 0 (M 0) the matrix M is positive definite (positive semi-definite) M < 0 (M 0) the matrix M is negative definite (negative semi-definite) R set of real numbers R + set of non-negative real numbers R n set of n-dimensional real vectors R m n set of real m n matrices S n n set of symmetric matrices in R n n S n n + set of positive definite matrices in R n n sect[a, b] conic sector {(q, p) : (p aq)(p bq) 0} u 2 = [ u T (t)u(t)dt ] 1 2 for u L 0 2 x = (x T x) 1 2 for vector x

13 xi Symbols for Aircraft Dynamics b wing span, m (ft) C l,t total rolling moment coefficient C m pitching moment coefficient about y body axis C mq pitching moment derivative with respect to pitch rate, deg 1 or rad 1 C m,t total pitching moment coefficient C n,t total yawing moment coefficient C x, C z aerodynamic force coefficient along x and z body axes C xq, C zq aerodynamic stability derivatives with respect to pitch rate, deg 1 or rad 1 C x,t, C y,t, C z,t total force coefficients along x, y, and z body axes c wing mean aerodynamic chord, m (ft) F x, F y, F z total forces along x, y and z body axes, N (lb) F x,a, F y,a, F z,a aerodynamic forces along x, y and z body axes, N (lb) F x,g, F y,g, F z,g gravitational forces along x, y and z body axes, N (lb) F x,t, F y,t, F z,t thrust forces along x, y and z body axes, N (lb) g gravity constant, 9.8 m/s 2 (32.2 ft/s 2 ) h altitude, m (ft) I x, I y, I z moment of inertia about x, y and z body axes, kg-m 2 (slug-ft 2 ) I xz cross product of inertia with respect to x and z body axes, kg-m 2 (slug-ft 2 ) l T moment arm produced by thrust force, m (ft) M x, M y, M z total moments about x, y and z body axes, kg-m (slug-ft) M x,a, M y,a, M z,a moments produced by aerodynamic forces about respective body axes, kg-m (slug-ft) M x,t, M y,t, M z,t moments produced by thrust forces about respective body axes, kg-m (slug-ft) m mass, kg (slug) p roll rate about x body axis, deg/s or rad/s p E, p N geographic coordinates align east and north, m (ft) pow actual power level q pitch rate about y body axis, deg/s or rad/s q dynamic pressure, N/m 2 (lb/ft 2 ) r yaw rate about z body axis, deg/s or rad/s S wing area, m 2 (ft 2 ) T thrust force, N (lb) u, v, w components of velocity along x, y and z body axes, m/s (ft/s) V true airspeed, m/s (ft/s)

14 xii x cg center-of-gravity location, fraction of c x cg,ref reference center-of-gravity location for aerodynamic data α angle of attack, rad or deg β angle of sideslip, rad or deg distance between the reference and the actual center-ofgravity locations, m (ft) δ e elevator angle, deg δ ptv effective pitch thrust vectoring angle, deg δ th throttle position δ ytv effective yaw thrust vectoring angle, deg γ flight-path angle, deg φ, ψ, θ Euler angles, deg

15 1 Chapter 1 Introduction Operation in the high angle of attack region, especially near and post-stall regimes, is essential for air-superiority of next generation fighter aircraft and uninhabited aerial vehicles (UAVs). With enhanced maneuverability in post-stall flight, the fighter aircraft can make emergency evasive maneuvers to avoid collisions, make emergency landings in unprepared short fields, make controlled steep landing approaches into fields with tall obstructions, and avoid loss of control when encountering severe atmospheric turbulence and downbursts. Also, the UAVs can make near-vertical descents for both earth and extra-terrestrial applications. As a result, the study of flight control at high angle of attack has received growing attention in recent years. However, the potential of flight at high angle of attack presents many challenges to the control designers. Briefly speaking, a flight control design is desired to address the following problems in the high angle of attack region: 1. Some or all of the actuation devices may be temporarily saturated. 2. There may be a combination of both discrete (on-off or bang-bang) and continuous actuators. 3. There is a high degree of both parametric and dynamic uncertainty in aerodynamics of aircraft model.

16 2 It is expected that this research will provide systematic and optimized control design strategies, which can expand the flight envelope and enhance the maneuverability of modern aircraft. 1.1 Motivations and Objectives Angle of attack is a term used in aerodynamics to describe the angle between the wing s chord and the direction of the relative wind, effectively the direction in which the aircraft is currently moving. Normally, increasing the angle of attack between the wing and the airflow causes the lift generated by the wing to increase. This remains true up to the stall point, where lift starts to decrease because of airflow separation. In most cases, because the wing is no longer producing enough lift as the stall is reached, the aircraft will start to descend and the nose will pitch down. A more dangerous stall is one where the nose rises, pushing the wing deeper into the stalled state and potentially leading to an unrecoverable deep stall. For fighter aircraft, pilots would like to fly at extreme angles of attack during maneuvers to facilitate rapid turning and pointing against at an adversary. However, entering high angle of attack region, especially post-stall regime, often leads to loss of control, and results in loss of the aircraft, pilot or both. In the 1960s, some early T-tailed airliners with swept-back wings have ended up crashing with high rates of descent because of deep stall problems. More recently in the early 1990s, some canard-configured homebuilt aircraft have also got stuck in a stable, non-rotating deep stall condition because of after-cg conditions [66]. One of the contributing factors to cause accidents at high angle of attack is that the conventional aerodynamic control surfaces could not produce enough control power due to actuator saturation. It is well recognized that actuator saturation degrades the performance of the flight control system and may even lead to instability. One suggestion is to incorporate unconventional actuators such as thrust vectoring for aircraft maneuvering at and beyond the stall angle of attack. Thrust vectoring provides the capability to turn the jet exhaust, and this allows the aircraft to create lifting forces with its motors similar to the forces created by aerodynamic surfaces.

17 3 Thrust vectoring technology has been successfully demonstrated on several aircrafts to provide tactical maneuvering advantages in the slow speed, high angle of attack flight regime [90]. For example, the F-15 STOL (short takeoff and landing) used pitch-only vectoring for enhanced agility throughout the flight envelope, and was able to make very short landing when combined with thrust reversing. The YF-22 also used pitch-only thrust vectoring to provide enhanced pitch maneuvering. The F-18 HARV (high angle of attack research vehicle) and the X-31 both used pitch and yaw thrust vectoring paddles to explore maneuvering at angles of attack up to 70 degrees and performed a tactical utility evaluations. Those programs showed that the thrust vectoring can provide additional control power at high angle of attack, and prevent the aircraft from loss of control due to aerodynamic surface saturation. However, the thrust vectoring technology may have a potentially significant pay-off in a number of critical areas, including vehicle complexity, maintenance, and total cost of ownership. Without hardware and cost concerns, thrust vectoring technology is usually suggested to compensate the conventional aerodynamic control surfaces. One usual way to generate the thrust vectoring command is to group the conventional aerodynamic surfaces and thrust vectoring nozzle into a single control surface, called generalized control. The control law provides the amount of generalized control, which is further distributed among the actuators according to a control allocation function [18, 17]. In general, the thrust vectoring is activated only when the conventional aerodynamic control surfaces are insufficient, i.e. they are unable to generate the necessary forces and moments required for commanded maneuvers. Therefore, the thrust vectoring is actually a discontinuous control input, and normally inactive in the low angle of attack region. However, this so-called daisy chain control allocation method is only applicable to the case where the dimension of control inputs is very low. It is difficult to define the generalized control and distribute the actuator commands for high dimensional control inputs. Up to now, it has not been clearly addressed on how to develop the control laws with guaranteed stability and performance by considering the continuous aerodynamic force and the discontinuous thrust force in a unified framework.

18 4 The objectives of this research are to address the above problems of flight control at high angle of attack with specific application to the longitudinal dynamics of an F-16 aircraft. First, we hope to develop a software-based actuator saturation control method. That is to say, no additional actuators such as thrust vectoring nozzles are needed to compensate control authority when the conventional aerodynamic surfaces are saturated. The second goal is to provide a systematic control design scheme for the case where the thrust vectoring is incorporated. Thus, the actuator sets are different in low and high angle of attack regions, and the thrust vectoring nozzles are on and off alternatively according to the trajectory of angle of attack. We would also like to make the control design scheme applicable when the control objectives in low and high angle of attack regions are different. It is a very possible case in practical implementation. For instance, fast and accurate response is desired at low angle of attack, while stability concern is more important at high angle of attack. In order to design effective controllers, it is necessary to get reliable aerodynamic modeling, especially at high angle of attack, even post-stall regime. However, the current state of the art does not allow accurate aerodynamic modeling in the high angle of attack region. The linearized aerodynamic models do not reliably predict many of the well-known nonlinear characteristics at high angle of attack, such as wing rock, roll reversal, and yaw departure [20]. Even the computationally-expensive, highfidelity computational fluid dynamic (CFD) technique is not robust enough at angles of attack well beyond stall. Therefore, the uncertainty associated with aerodynamic modeling presents the challenge in designing flight control systems for those regimes. In this research, the aerodynamic model used is assumed to be reliable, and the uncertainty in the aerodynamic results can be handled by the robust control techniques, which is beyond this topic, and will be studied in the future work. 1.2 Background in Flight Control In recent years, the research area of flight control is very active, and there are a number of control design techniques being used by industry to design flight con-

19 5 trol laws for fixed-wing aircraft, including PI (proportional-integral) control, optimal LQR/LQG (linear quadratic regulator / linear quadratic gaussian) control, H control, µ synthesis robust control, dynamic inversion, adaptive control, neural network, and LPV (linear parameter-varying) control [6]. The trend indicates that advanced multivariable control techniques are now the standard for designing flight control laws for advanced aircraft. Modern robust multivariable design methods, including H and µ-synthesis, provide an efficient means of developing linear controllers for aircraft. A robust H controller within an inner/outer loop framework was designed for a supermaneuverable aircraft at a single flight condition, and a Herbst-like maneuver was done to demonstrate the robust performance [23]. µ-synthesis was applied to the same aircraft in Ref. [93] with incorporating flying qualities and accounting for structured uncertainties. In Ref. [18], a reduced-order H controller was designed for inner loop equalization, and structured singular value synthesis was used to design outer loop implicit model-following controllers. These techniques have proven to be particularly effective for systems whose dominant dynamics can be accurately captured by a single, linear time-invariant (LTI) model. For an aircraft maneuvering in a wide flight envelope, its dynamic behavior varies significantly during operation. Moreover, the high angle of attack region is very nonlinear compared to the low angle of attack region. Purely linear controllers are not able to effectively control supermaneuverable aircraft in a wide flight envelope. Traditionally, several point controllers are designed throughout the operation region, and gain scheduling is incorporated when implementation. However, designing several point controllers is a time-consuming and tedious process, and also there are no stability and performance guarantees for interpolation between the point controllers [94]. Some nonlinear techniques have been explored to eliminate the problems with gain scheduling. For example, dynamic inversion methods avoid the scheduling problem by using nonlinear feedback to cancel the dynamics of the aircraft. Snell et al. compared the performance of a dynamic inversion system with one designed using conventional gain scheduling [91]. Adams et al. designed dynamic inversion/µ-synthesis

20 6 inner/outer loop control for a thrust vectored F-18 [1]. Reigelsperger, Hammett, and Banda also applied the dynamic inversion within an inner/outer loop structure to the both longitudinal and lateral control of an F-16/MATV (Multi-Axis Thrust Vectoring) [36, 82, 81]. The inner loop dynamic inversion/outer loop µ-synthesis control structure separately addresses operating envelope variations and robustness concerns. However, the dynamic inversion methods lack solid performance and robustness guarantees. This research is based on LPV control theory, which is a systematic gain-scheduling technique with stability and performance guarantees. LPV control methodology is an extension of H control theory for LPV systems. It explicitly takes into account the relationship between real-time parameter variations and performance. This enables controllers to be designed for whole ranges of operating conditions with theoretical guarantees of performance and robustness throughout the region. The LPV control approach is based on first generating a set of linear matrix inequalities (LMIs) over the parameter set, and then constructing an LPV controller from the solutions of the LMIs. When the solutions of the LMIs exist over the parameter set, the obtained LPV controller guarantees the stability of the closed-loop system and achieves a certain level of performance. LPV methodologies vary in their conservatism and their complexity. Approaches based on linear fractional transformations (LFT) [70, 4] or single quadratic Lyapunov functions [10] tend to be easier to implement. But they are conservative due to allowing the parameters to vary arbitrarily fast. Use of parameter-dependent Lyapunov functions [9, 107] can reduce the conservatism, because the parameter variation rates are constrained. The LPV control method based on parameter-dependent Lyapunov functions will be advantageous for nonlinear aircraft control to establish stability and performance properties under variable operating conditions. 1.3 Thesis Outline The detailed outline of this thesis is as follows:

21 7 Chapter 2 describes the modeling of the aircraft. A general form of coupled nonlinear flight dynamic equations of motion with thrust vectoring terms is first presented. Then, the longitudinal dynamics of the F-16 aircraft is derived, which is based on the NASA Langley wind tunnel tests on a scaled F-16 aircraft [69] and augmented with a two-dimensional thrust vectoring model. The nonlinear model is further transformed to an LPV model using Jacobian linearizations around a group of equilibrium points. Chapter 3 presents a saturation control scheme for LPV systems from an antiwindup control perspective. The proposed control approach is advantageous from the implementation standpoint because it can be thought of as an augmented control algorithm to the existing control system. First, a general robust stability analysis condition applicable to sector-bounded uncertainty is given as an H problem using an LMI mechanism. By treating the saturation as a sector-bounded nonlinearity, the synthesis conditions for the antiwindup compensator is then formulated as an LMI optimization problem and can be solved efficiently. With synthesis condition established, the construction procedure for the LPV antiwindup compensator is also derived. The LPV antiwindup compensator is applied to the F-16 longitudinal autopilot control system design at high angle of attack, and compared with the thrust vectoring control scheme through nonlinear simulations. Chapter 4 presents a systematic switching LPV control design method based on multiple parameter-dependent Lyapunov functions. Two switching logics, hysteresis switching and switching with average dwell time, are considered. For each switching logic, the analysis condition that guarantees stability and performance of switched LPV systems is first given. Then, the synthesis condition is derived and formulated as matrix optimization problem, which is generally non-convex. Two convexified methods are proposed to make the switching LPV control more applicable. The proposed switching LPV control scheme is applied to the F-16 longitudinal model. Different actuator sets and control objectives are assumed in the low and high angle of attack regions. With the proposed switching LPV control scheme, two LPV controllers are designed, each suitable for a specific angle of attack region. Also, the stability of the closed-loop system is guaranteed when switching between two controllers according to the evolution of angle of attack.

22 8 Finally, Chapter 5 contains a summary of the main results, as well as provides a remark on the future work.

23 9 Chapter 2 Modeling of the Aircraft Modeling is an important process in the design of aircraft control systems. This chapter describes two stages of the aircraft modeling. First, a mathematical model of the F-16 aircraft is built up, which is based on the laws of physics and incorporates the experimental data. Then the nonlinear model is transformed to an LPV model, and this is a prerequisite to apply LPV control synthesis to the nonlinear aircraft control problem. The LPV modeling has become a key issue in the LPV control design [85]. Section 2.1 presents a general form of coupled nonlinear flight dynamic equations of motion with thrust vectoring terms. In Section 2.2, the decoupled nonlinear equations of motion in longitudinal axis are derived, which are based on the NASA Langley wind tunnel tests on a scaled F-16 aircraft and augmented with a simple thrust vectoring model. In Section 2.3, an LPV model of the F-16 aircraft is developed using Jacobian linearizations around equilibrium points. Section 2.4 summarizes the results and provides discussions.

24 2.1 Generalized Nonlinear Model for the Thrust- Vectored Aircraft Assuming an aircraft as a rigid body with constant mass and inertia, the nonlinear six degree-of-freedom (DOF) equations of motion with respect to a body-fixed reference frame [95] can be written as follows, where the physical meanings of all symbols are referred to List of Notations. Force Equations: Moment Equations: u = F x m v = F y m ẇ = F z m ( ) 1 [ ( ṗ = 1 I2 xz Ixz pq + I ) xz (I x I y ) I x I z I x I x I z ( ) Iy I z +qr I2 xz + M x I x I x I z I x 10 qw + rv (2.1) + pw ru (2.2) pv + qu (2.3) + I ] xz M z I x I z (2.4) q = pr I z I x (p 2 r 2 ) I xz + M y (2.5) I y I y I y ( ) 1 [ ( ) ṙ = 1 I2 xz Ix I y pq + I2 xz I x I z I z I x I ( z +qr I xz + I ) xz (I y I z ) + I xz M x + M ] z (2.6) I z I x I z I x I z I z Kinematic Equations: φ = p + q sin φ tan θ + r cos φ tan θ (2.7) θ = q cos φ r sin φ (2.8) ψ = q sin φ sec θ + r cos φ sec θ (2.9)

25 11 Navigation Equations: ṗ N = u cos θ cos ψ + v ( cos φ sin ψ + sin φ sin θ cos ψ) + w (sin φ sin ψ + cos φ sin θ cos ψ) (2.10) ṗ E = u cos θ sin ψ + v (cos φ cos ψ + sin φ sin θ sin ψ) + w ( sin φ cos ψ + cos φ sin θ sin ψ) (2.11) ḣ = u sin θ v sin φ cos θ w cos φ cos θ (2.12) Note that the equations (2.1) (2.3) are expressed in terms of velocity components in the aircraft body-fixed system. Since the aerodynamic force and moment components in the above equations depend on the aircraft angles and the true airspeed, it is better to have the velocity equations in terms of stability axes or wind axes variables V, α, and β [95]. The replacement of the state variables is convenient for linearizing the equations of motion and studying the dynamic behavior. Figure 2.1 shows an aircraft with the relative wind on its left side, and with the conventional right-handed (forward, starboard, and down) set of body-fixed axes, where V = u 2 + v 2 + w 2 (2.13) ( v ) β = sin 1 (2.14) ( V w ) α = tan 1 (2.15) u Differentiating the both sides of the above equations and substituting from equations (2.1) (2.3) yield the expressions for the new state derivatives as V = 1 m (F x cos α cos β + F y sin β + F z sin α cos β) (2.16) β = 1 mv ( F x cos α sin β + F y cos β F z sin α sin β) + p sin α r cos α (2.17) α = 1 ( ) sin α F x mv cos β + F cos α z p cos α tan β + q r sin α tan β (2.18) cos β Generally, F x, F y, and F z are forces composed of gravitational, aerodynamic and propulsive forces, and M x, M y, and M z are moments produced by aerodynamic and propulsive forces. The gravitational forces are modeled as

26 12 body y-axis x cg body x-axis u w v x-axis (stability) relative wind body z-axis V x-axis (wind) Figure 2.1: Definition of axes and aerodynamic angles. F x,g F y,g F z,g The aerodynamic forces are modeled as sin θ = g cos θ sin φ (2.19) cos θ cos φ F x,a F y,a = qs C x,t C y,t (2.20) F z,a C z,t The aerodynamic moments acting on the body are similarly modeled as M x,a M y,a = qs bc l,t cc m,t (2.21) M z,a bc n,t For a high performance tactical aircraft, the conventional aerodynamic surfaces may be insufficient for control as the angle of attack α increases, especially in the deep post-stall regime. When the propulsion system ignites, vectoring the thrust can

27 13 provide additional control power and allow maneuvering in post-stall flight envelope [91, 31, 32, 52, 102]. A mathematical phenomenology is developed in Ref. [31] to reassess aircraft equations of motion due to introducing thrust vectoring capability into fighter aircraft design methodologies. The thrust vectoring forces and moments are modeled using a constant thrust that is deflected by the actuator. It is assumed that the actuator can deflect the thrust vector only in the pitch (δ ptv ) and yaw (δ ytv ) planes as shown in Figure 2.2. The roll thrust vector is usually generated by two nozzles or subnozzles, each vectoring δ ytv in opposite directions [32]. body y-axis l T F x,t x cg x T ptv body x-axis body z-axis F z,t F y,t ytv T Figure 2.2: Vectored thrust. For simplicity, we assume that no roll control from the thrust vector, i.e. the thrust nozzles deflect symmetrically. Furthermore, the thrust-vectored engine is assumed to align with x-axis. The resulting thrust force components along the body-axes are F x,t cos δ ptv cos δ ytv F y,t = T sin δ ytv (2.22) sin δ ptv cos δ ytv F z,t and the pitch and yaw moments are produced by the moment arm l T times the preceding yaw and pitch forces, respectively. = x cg x T

28 M x,t M y,t M z,t 14 0 = l T T sin δ ptv cos δ ytv (2.23) l T T sin δ ytv A more complicated model of thrust vector is available in Ref. [13], but will not be considered in this research. 2.2 Longitudinal Model of an F-16 Aircraft with Thrust Vectoring Most aircraft spend most of their flying time in a wings-level steady-state flight condition. In this case, the roll angle φ is zero. If the sideslip angle β is negligible, and also the roll and yaw rates (p and r) are small, we can obtain the decoupled equations for pure longitudinal motion [95]. The system to be controlled is the longitudinal F-16 aircraft model based on NASA Langley Research Center (LaRC) wind tunnel tests [69]. The model is a collection of modules specifying the aircraft mass and geometric properties, the aircraft actuator command inputs, the equations of motion, the atmospheric model, the aerodynamics, and the propulsion system. In this research, the F-16 model is incorporated with the thrust vectoring, and only the longitudinal model is considered. For more information, the reader can refer to Ref. [95], which describes both the longitudinal and lateral-directional models in further detail Aircraft Description The F-16 is powered by an after-burning turbofan jet engine, which produces a thrust force in the x-axis, as shown in Figure 2.3(a). In this research, the F-16 aircraft is augmented with a simple thrust vectoring model as shown in Figure 2.3(b), which is similar to that in Ref. [17].

29 15 l T c.g. T c.g. F z,t Fx,T ptv T (a) Original F-16 model (b) F-16 model augmented with thrust vectoring Figure 2.3: F-16 aircraft model with and without thrust vectoring. The mass and geometric properties used in the simulation are listed in Table 2.1. The primary actuators for pitch control consist of the elevator and the thrust vector. Table 2.2 gives the position limit, rate limit, and time constant of the actuators. The dynamics of the elevator and thrust vectoring actuator are modeled as first-order lag filters. The throttle δ th is also an input to the aircraft, but it primarily controls the aircraft trajectory other than attitude [17]. Table 2.1: Mass and geometric properties. Parameter Symbol Value Weight W (lb) Moment of inertia I y (slug-ft 2 ) Wing area S (ft 2 ) 300 Mean aerodynamic chord c (ft) Reference CG location x cg,ref 0.35 c Table 2.2: Control actuator models. Actuator Deflection limit Rate limit Time constant Elevator δ e ±25 60 /s s Thrust vector δ ptv ±17 60 /s 0.07 s

30 Aerodynamic Model The aerodynamic data are derived from low-speed static and dynamic wind tunnel tests conducted with subscaled models of the F-16 aircraft [69]. The data are provided in tabular form, and cover a very wide range of angle of attack ( 20 α 90 ), and of sideslip angle ( 30 β 30 ). However, the present state of the art does not allow accurate dynamic modeling in the high angle of attack region, especially in the post-stall region. Therefore, we use the approximate data in Ref. [95], which reduce the range of the data to 10 α 45. Moreover, the size of the data is reduced from 50 lookup tables down to 10. Based on the approximation, there are only 4 lookup tables involved for the longitudinal mode. Figure 2.4 shows the aerodynamic data when β = 0, where the damping coefficients C xq, C zq, and C mq are the functions of the angle of attack α, and the coefficients C x, C z, and C m depend on both of α and the elevator angle δ e. Note that Figure 2.4(d) gives the value of z-axis force coefficient when the elevator does not deflected, i.e. C z (α, 0). The value of C z (α, δ e ) can be approximated by the following equation. C z (α, δ e ) = C z (α, 0) 0.19 δ e 25 (2.24) In the simulation, the data are interpolated linearly between the points, and extrapolated beyond the table boundaries. But note that the extrapolation may lead to unrealistic results. The total coefficients C x,t, C z,t, and C m,t are calculated by summing the various aerodynamic contributions to a given force or moment coefficient. C x,t = C x (α, δ e ) + qc 2V C xq(α) (2.25) C z,t = C z (α, δ e ) + qc 2V C zq(α) (2.26) C m,t = C m (α, δ e ) + C z,t (x cg,ref x cg ) + qc 2V C mq(α) (2.27)

31 δ e = 24 δ e =12 δ e =0 δ e =12 δ e =24 C xq (α) C x (α,δ e ) α (deg) α (deg) 5 (a) Damping coefficient C xq 1 (b) x-axis force coefficient C x C zq (α) C z (α,0) α (deg) α (deg) 5 (c) Damping coefficient C zq 0.25 (d) z-axis force coefficient C z C mq (α) α (deg) C m (α,δ e ) δ e = 24 δ e =12 δ e =0 δ e =12 δ e = α (deg) (e) Damping coefficient C mq (f) Pitching moment coefficient C m Figure 2.4: Aerodynamic data.

32 Propulsion System Model The after-burning turbofan engine model consists of nonlinear thrust tables, a first-order dynamical model, and a throttle command shaping function [95]. The thrust produced by the engine is a function of altitude, Mach number, and throttle setting. As mentioned before, a simple thrust vectoring model is added in this research to provide additional longitudinal axis control power. Denote the thrust vector angle by δ ptv as shown in Figure 2.3(b). The thrust components along the x, z axes and the pitching moment due to thrust vector are given by F x,t = T cos δ ptv (2.28) F z,t = T sin δ ptv (2.29) M y,t = l T T sin δ ptv (2.30) It is obvious that the original aircraft model is corresponding to the case of δ ptv = Equations of Motion The longitudinal nonlinear equations of motion decoupled from (2.4) (2.9) and (2.16) (2.18) are given as follows. V = 1 m (F x cos α + F z sin α) (2.31) α = 1 mv ( F x sin α + F z cos α) + q (2.32) q = M y I y (2.33) θ =q (2.34) where the x and z axes forces and pitching moment are given as F x =qsc x,t mg sin θ + T cos δ ptv (2.35) F z =qsc z,t + mg cos θ T sin δ ptv (2.36) M y =qscc m,t l T T sin δ ptv (2.37)

33 19 Combining with the aerodynamic and propulsive models, the final nonlinear longitudinal model of F-16 aircraft are given as follows. V = qs c 2mV [C xq(α) cos α + C zq (α) sin α] q g sin(θ α) + qs m [C z(α, δ e ) cos α + C z (α, δ e ) sin α] + T m cos (α + δ ptv) (2.38) [ α = 1 + qs c ] 2mV (C zq(α) cos α C 2 xq (α) sin α) q + g cos(θ α) V + qs mv [C z(α, δ e ) cos α C x (α, δ e ) sin α] T mv sin (α + δ ptv) (2.39) q = qs c 2I y V [ cc mq(α) + C zq (α)] q + qs c [ C m (α, δ e ) + c ] I C z(α, δ e ) l T T sin δ ptv (2.40) y I y θ = q (2.41) It is noted that the altitude h is one of the factors to affect the thrust force T, and it enters the equations of motion in an implicit way. From the navigation equation (2.12) and definition of aerodynamic angles, we have ḣ = V cos α sin θ V sin α cos θ (2.42) 2.3 LPV Modeling of F-16 Longitudinal Axis LPV control theory is a systematic gain-scheduling design technique [10, 70, 4, 107, 103], which has been used to design controllers for dynamical systems over a wide parameter envelope. These include high performance aircraft as representative as F- 14 [7], F-16 [87, 94], F-18 [8], missiles [96, 105], and other fighter aircrafts [92, 34]. Before applying the LPV control synthesis, it is required to transform the nonlinear model of the system to an LPV model LPV Systems The class of finite dimensional linear systems whose state-space entries depend continuously on a time-varying parameter vector ρ(t) is called linear parameter vary-

34 20 ing systems. It is assumed that the vector-valued parameter ρ evolves continuously over time and its range is limited to a compact subset P R s. In addition, its time derivative is often assumed to be bounded and satisfy the constraint ν k ρ k ν k, k = 1, 2,..., s. For notational purposes, denote V = {ν : ν k ν k ν k, k = 1, 2,..., s}, where V is a given convex polytope in R s that contains the origin. Given the sets P and V, the parameter ν-variation set is defined as F ν P = { ρ C 1 (R +, R s ) : ρ(t) P, ρ(t) V, t 0 } (2.43) So FP ν specifies the set of all allowable parameter trajectories. Note that the trajectory of the parameter is assumed to be unknown in priori, but can be measured in real time. Using the definition of the parameter ν-variation set FP ν, a generalized nth-order LPV system can be described by [ẋ(t) ] [ ] [ ] A(ρ(t)) B(ρ(t)) x(t) = y(t) C(ρ(t)) D(ρ(t)) u(t) (2.44) where ρ F ν P, the state x Rn, the output y R ny, and the input u R nu. All of the state-space data are continuous functions of the parameter ρ, i.e., A : R s R n n, B : R s R n nu, C : R s R ny n, and D : R s R ny nu. Given an LPV system as defined in (2.44), if a scheduling variable ρ(t) is also a state of the system, then this particular class of systems are called as quasi-lpv systems. More specifically, if the state vector x(t) can be decomposed into scheduling states x 1 (t) F ν P, and nonscheduling states x 2(t) [61], i.e., x T (t) = [ x T 1 (t) x T 2 (t) ] (2.45) then the quasi-lpv system can be obtained as ẋ 1 (t) A 11 (x 1 (t), Ω(t)) A 12 (x 1 (t), Ω(t)) B 1 (x 1 (t), Ω(t)) x 1 (t) ẋ 2 (t) = A 21 (x 1 (t), Ω(t)) A 22 (x 1 (t), Ω(t)) B 2 (x 1 (t), Ω(t)) x 2 (t) (2.46) y(t) C 1 (x 1 (t), Ω(t)) C 2 (x 1 (t), Ω(t)) D(x 1 (t), Ω(t)) u(t) where the scheduling parameter vector is ρ T (t) = [ x T 1 (t) Ω T (t) ], and Ω(t) are exogenous scheduling variables.

35 Development of F-16 LPV Models The LPV model can be considered as a group of local descriptions of nonlinear dynamics. There are several approaches used to obtain reliable LPV models [84, 87, 61]. The Jacobian linearization approach is the most widespread methodology and the theoretical development is very straightforward. The second approach is called as state transformation because the quasi-lpv model is obtained through exact transformations of the nonlinear states. Both of the methods are better known, but they are restrictive in terms of operational envelope [61]. Another approach called as function substitution [87, 61] has appeared recently, and the theory still has several open questions to be addressed. In this research, we use the Jacobian linearization approach to derive the LPV model of the F-16 aircraft. Given a nonlinear system ẋ = f(x, u) (2.47) y = g(x, u) (2.48) the Jacobian linearization approach can be used to create an LPV system based on the first-order Taylor series expansion of the nonlinear model. A family of linear plants are obtained by linearizing the nonlinear system with respect to a set of equilibrium points, which are parameterized by the scheduling parameter ρ and satisfy f(x e (ρ), u e (ρ)) = 0. Corresponding to a specified family of equilibrium points, the family of the linearized plants can be written in the form of [ ] [ δẋ(t) f = f x x e(ρ),u e(ρ) ] [ ] u x e(ρ),u e(ρ) δx(t) δy(t) δu(t) g x x e (ρ),u e (ρ) g u x e (ρ),u e (ρ) where the deviation variables are defined in the obvious fashion. (2.49) δx = x x e (ρ), δu = u u e (ρ), δy = y y e (ρ) (2.50) The state-space data in (2.49) is actually in the LPV form since the equilibrium point (x e, u e ) depends on the parameter ρ. At each fixed ρ, the linearization (2.49) describes the local behavior of the nonlinear dynamics around the corresponding equilibrium point.

36 22 Before deriving the LPV model of the F-16 aircraft, we need to select the scheduling parameters. Generally, the scheduling parameters for an aircraft are combination of altitude h, velocity V, angle of attack α, and/or variables that register changes in those parameters such as Mach number M or dynamic pressure q [61]. Given the nonlinear longitudinal model of F-16 aircraft in (2.38) (2.41), we define the state vector x = [V α q θ] T. In this research, the angle of attack α and the velocity V are selected as the scheduling parameters, i.e., ρ = [α V ] T. It is obvious that the resulting LPV model is actually a quasi-lpv model. For the original aircraft model, the available control inputs u = [δ th δ e ] T, and for the thrust-vectored aircraft model, u = [δ th δ e δ ptv ] T. Note that the throttle setting δ th indirectly affects the states through the power output from the engine. Therefore, the actual power level pow is also considered as a state variable in longitudinal dynamics. The expression of the time derivative of pow can be obtained from the detailed dynamic model of the engine in the NASA report. Then the final state vector is defined as x = [V α q θ pow] T. The state pow is dimensionless, and ranges from 0 to 100. Next, it is necessary to find the wings-level equilibrium solutions at several flight conditions in the design envelope. The flight envelope of interest in this research covers aircraft speeds between 160 ft/s and 200 ft/s and angles of attack between 20 and 45. The point at which the equilibrium solution is needed to find is marked by a symbol in Figure 2.5. For the modified F-16 aircraft model, it is assumed that the value of δ ptv at equilibrium is always zero, i.e., the thrust nozzle does not deflect. As mentioned before, the thrust-vectored model when δ ptv = 0 is just as same as the original aircraft model. Therefore, we can calculate the equilibrium solution based on the original F-16 aircraft model. For example, the equilibrium solution at V = 200 ft/s and α = 21 is [V α q θ pow] = [200ft/s 21 0 /s ] [δ th δ e δ ptv ] = [ ] It is easy to verify the wings-level equilibrium solution that q e = 0 and θ e = α [61].

37 V (ft/s) α (deg) Figure 2.5: Flight equilibrium points. The local linear models are then obtained by linearizing the nonlinear equations of motion at those equilibrium points using Jacobian linearization method. This group of linearized models consist of the LPV representation of the nonlinear F-16 longitudinal dynamics within the chosen flight envelope Verification of F-16 LPV Models In this section, we will check if the F-16 LPV model captures the local nonlinearities of the system. Again, we take the flight condition V = 200 ft/s and α = 21 as an example. Three command inputs are defined as follows. It is known that the LPV model is a group of local description of the nonlinear dynamics. Therefore, all of the commands are given as small perturbations from the equilibrium. Figures show the time responses of those three command inputs for the nonlinear model and the LPV model linearized at the equilibrium point. It is observed that the LPV model follow the nonlinear model quite closely step of the throttle setting effective from time = 1 s. { t < 1 δ th = t 1 2. ±1 doublet of the elevator angle applied from time = 1 s until time = 9 s.