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 en-velope, advanced linear parameter-varying (LPV) control methodologies are studied in this thesis with particular applications of actuator saturation control and switch-ing 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) op-timization 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 advan-tages over thrust vectoring in a high angle of attack region.
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 Dr. Paul I. Ro
To my husband Qifu Li
Biography
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-1-01119 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.
Contents
List of Tables vii
List of Figures viii
List of Notations x
Mathematical Symbols . . . x
Symbols for Aircraft Dynamics . . . xi
1 Introduction 1 1.1 Motivations and Objectives . . . 2
1.2 Background in Flight Control . . . 4
1.3 Thesis Outline . . . 6
2 Modeling of the Aircraft 9 2.1 Generalized Nonlinear Model for the Thrust-Vectored Aircraft . . . . 10
2.2 Longitudinal Model of an F-16 Aircraft with Thrust Vectoring . . . . 14
2.2.1 Aircraft Description . . . 14
2.2.2 Aerodynamic Model . . . 16
2.2.3 Propulsion System Model . . . 18
2.2.4 Equations of Motion . . . 18
2.3 LPV Modeling of F-16 Longitudinal Axis . . . 19
2.3.1 LPV Systems . . . 19
2.3.2 Development of F-16 LPV Models . . . 21
2.3.3 Verification of F-16 LPV Models . . . 23
2.4 Summary . . . 26
3 Actuator Saturation Control 29 3.1 Introduction . . . 30
3.2 Saturation Nonlinearity . . . 33
3.4 LPV Antiwindup Control . . . 41
3.4.1 Problem Statement and Preliminaries . . . 41
3.4.2 LPV Antiwindup Control Synthesis . . . 46
3.4.3 LPV Antiwindup Compensator Construction . . . 51
3.5 Applications to the F-16 Aircraft Model . . . 56
3.5.1 Problem Setup . . . 56
3.5.2 Design Results . . . 57
3.5.3 Nonlinear Simulations . . . 59
3.5.4 Comparison with Thrust Vectoring Control . . . 63
3.6 Summary . . . 67
4 Switching LPV Control 70 4.1 Introduction . . . 71
4.2 Analysis of Switched LPV Systems . . . 74
4.2.1 Hysteresis Switching . . . 75
4.2.2 Switching with Average Dwell Time . . . 78
4.3 Switching LPV Control Design . . . 82
4.3.1 Hysteresis Switching . . . 84
4.3.2 Switching with Average Dwell Time . . . 88
4.4 Convexified Switching LPV Synthesis Conditions . . . 94
4.4.1 Switching with Continuous Control State . . . 94
4.4.2 Switching via Control State Reset . . . 96
4.5 Applications to F-16 Aircraft Model . . . 100
4.5.1 Problem Setup . . . 100
4.5.2 Switching with Continuous Control State . . . 104
4.5.3 Switching via Control State Reset . . . 106
4.6 Summary . . . 108
5 Conclusion 115 5.1 Contributions . . . 115
5.2 Future Work . . . 119
List of Tables
2.1 Mass and geometric properties. . . 15
2.2 Control actuator models. . . 15
3.1 H∞ performance level vs. sector range [0, ki]. . . 58
4.1 Effect of weights on performance level. . . 105
List of Figures
2.1 Definition of axes and aerodynamic angles. . . 12
2.2 Vectored thrust. . . 13
2.3 F-16 aircraft model with and without thrust vectoring. . . 15
2.4 Aerodynamic data. . . 17
2.5 Flight equilibrium points. . . 23
2.6 Time responses for 0.1 step input of δth atV = 200 ft/s and α= 21◦. 24 2.7 Time responses for ±1◦ doublet input ofδ e atV = 200 ft/s andα = 21◦. 25 2.8 Time responses for −0.5◦ step input ofδ ptv atV = 200 ft/s andα = 21◦. 26 2.9 Time responses for 0.1 step input of δth atV = 160 ft/s and α= 35◦. 27 2.10 Time responses for 0.05 step input of δth atV = 160 ft/s and α= 35◦. 28 3.1 A control system with saturation nonlinearity. . . 34
3.2 A graphical representation of sect[a, b]. . . 35
3.3 Sector bounds on saturation nonlinearity. . . 35
3.4 Saturation block with two outputs. . . 36
3.5 Dead-zone representation of saturation nonlinearity. . . 36
3.6 Sector-bounded uncertainty. . . 37
3.7 Nonlinear saturation control diagram. . . 43
3.8 Open-loop interconnection for nominal LPV controller design. . . 58
3.9 Nonlinear ±1◦ doublet response with LTI nominal controller. . . . 60
3.10 Nonlinear ±2◦ doublet response w/o LTI antiwindup compensator. . 61
3.11 Nonlinear ±1.7◦ doublet response w/o LTI antiwindup compensator for an unstable open-loop plant. . . 62
3.12 Nonlinear doublet response with LPV antiwindup compensator. . . . 64
3.13 Thrust vectoring control scheme. . . 66
3.14 Nonlinear doublet response with thrust vectoring. . . 68
3.15 Comparison of nonlinear doublet response with thrust vectoring and with antiwindup compensator for an unstable open-loop plant. . . 69
4.2 Hysteresis switching region and switching signal σ. . . . 76 4.3 Piecewise continuous Lyapunov functions for hysteresis switching (ZN =
{1,2}). . . 77 4.4 Switching region and switching signal σ with dwell time. . . 79 4.5 Discontinuous Lyapunov functions for switching with dwell time (ZN =
{1,2}). . . 80 4.6 Flight conditions and partitioned flight envelope for two switching logics.101 4.7 Weighted open-loop interconnection for switching control of F-16 aircraft.102 4.8 Command input 1 for switching LPV control simulation. . . 106 4.9 Time history of parameters and switching signal under hysteresis
switch-ing with continuous control state for command input 1. . . 109 4.10 Time responses of actuators under hysteresis switching with continuous
control state for command input 1. . . 110 4.11 Time history of parameters and switching signal under switching
con-trol via concon-trol state reset for command input 1. . . 111 4.12 Time responses of actuators under switching control via control state
reset for command input 1. . . 112 4.13 Time history of parameters and switching signal under switching
con-trol via concon-trol state reset for command input 2. . . 113 4.14 Time responses of actuators under switching control via control state
List of Notations
Mathematical Symbols
0n×m the zero element of Rn×m
C1(U, V) set of continuously differentiable functions from U to V diag(a1, a2, . . . , an) the n byn diagonal matrix with elements a1, a2, . . ., an
on the diagonal line
Fl(·,·) linear fractional transformation
In the n-dimensional identity
Ker(M) the orthogonal complement of the matrix M
L2 space of square integrable functions MT 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 matrixM is negative definite (negative semi-definite)
R set of real numbers
R+ set of non-negative real numbers Rn set ofn-dimensional real vectors
Rm×n set of realm×n matrices
Sn×n set of symmetric matrices inRn×n
Sn×n
+ set of positive definite matrices inRn×n sect[a, b] conic sector{(q, p) : (p−aq)(p−bq)≤0}
kuk2 =
£R∞
0 uT(t)u(t)dt
¤1
2 foru∈ L 2
Symbols for Aircraft Dynamics
b wing span, m (ft)
Cl,t total rolling moment coefficient
Cm pitching moment coefficient about y body axis
Cmq pitching moment derivative with respect to pitch rate,
deg−1 or rad−1
Cm,t total pitching moment coefficient
Cn,t total yawing moment coefficient
Cx, Cz aerodynamic force coefficient along x and z body axes
Cxq, Czq aerodynamic stability derivatives with respect to pitch
rate, deg−1 or rad−1
Cx,t, Cy,t, Cz,t total force coefficients along x, y, and z body axes
¯
c wing mean aerodynamic chord, m (ft)
Fx, Fy, Fz total forces along x, y and z body axes, N (lb)
Fx,A, Fy,A, Fz,A aerodynamic forces along x,y and z body axes, N (lb)
Fx,G, Fy,G, Fz,G gravitational forces along x,y and z body axes, N (lb)
Fx,T, Fy,T, Fz,T thrust forces along x, y and z body axes, N (lb)
g gravity constant, 9.8 m/s2 (32.2 ft/s2)
h altitude, m (ft)
Ix, Iy, Iz moment of inertia about x, y and z body axes, kg-m2
(slug-ft2)
Ixz cross product of inertia with respect to x and z body
axes, kg-m2 (slug-ft2)
lT moment arm produced by thrust force, m (ft)
Mx, My, Mz total moments aboutx,yandzbody axes, kg-m (slug-ft)
Mx,A, My,A, Mz,A moments produced by aerodynamic forces about
respec-tive body axes, kg-m (slug-ft)
Mx,T, My,T, Mz,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
pE, pN 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/m2 (lb/ft2)
r yaw rate about z body axis, deg/s or rad/s S wing area, m2 (ft2)
T thrust force, N (lb)
u, v, w components of velocity along x, y and z body axes, m/s (ft/s)
xcg center-of-gravity location, fraction of ¯c
xcg,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-of-gravity locations, m (ft)
δe elevator angle, deg
δptv effective pitch thrust vectoring angle, deg
δth throttle position
δytv effective yaw thrust vectoring angle, deg
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 air-craft 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 at-mospheric turbulence and downbursts. Also, the UAVs can make near-vertical de-scents 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 contin-uous actuators.
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].
Thrust vectoring technology has been successfully demonstrated on several air-crafts 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 sug-gested 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.
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 cur-rent 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, high-fidelity 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 un-certainty 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
con-trol laws for fixed-wing aircraft, including PI (proportional-integral) concon-trol, optimal LQR/LQG (linear quadratic regulator / linear quadratic gaussian) control, H∞
con-trol, µsynthesis 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, includingH∞and µ-synthesis,
pro-vide an efficient means of developing linear controllers for aircraft. A robust H∞
controller within an inner/outer loop framework was designed for a supermaneuver-able 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 air-craft in Ref. [93] with incorporating flying qualities and accounting for structured uncertainties. In Ref. [18], a reduced-orderH∞ 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 dom-inant 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 sev-eral 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].
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 Vec-toring) [36, 82, 81]. The inner loop dynamic inversion/outer loopµ-synthesis control structure separately addresses operating envelope variations and robustness concerns. However, the dynamic inversion methods lack solid performance and robustness guar-antees.
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 allow-ing 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
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 anti-windup 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 anH∞ 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 au-topilot control system design at high angle of attack, and compared with the thrust vectoring control scheme through nonlinear simulations.
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].
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 refer-ence frame [95] can be written as follows, where the physical meanings of all symbols are referred to List of Notations.
Force Equations:
˙ u= Fx
m −qw+rv (2.1)
˙ v = Fy
m +pw−ru (2.2)
˙ w= Fz
m −pv+qu (2.3)
Moment Equations:
˙ p=
µ
1− I
2
xz
IxIz
¶−1· pq
µ
Ixz
Ix
+ Ixz(Ix−Iy) IxIz
¶
+qr
µ
Iy −Iz
Ix
− Ixz2
IxIz
¶
+Mx Ix
+ Ixz IxIz
Mz
¸
(2.4)
˙
q =prIz−Ix Iy
−(p2−r2)Ixz Iy
+ My Iy
(2.5)
˙ r =
µ
1− Ixz2
IxIz
¶−1·
pq
µ
Ix−Iy
Iz
+ Ixz2
IxIz
¶
+qr
µ
−Ixz
Iz
+ Ixz(Iy −Iz) IxIz
¶
+ Ixz IxIz
Mx+
Mz Iz ¸ (2.6) Kinematic Equations: ˙
φ =p+qsinφtanθ+rcosφtanθ (2.7) ˙
θ =qcosφ−rsinφ (2.8)
˙
Navigation Equations:
˙
pN =ucosθcosψ+v(−cosφsinψ+ sinφsinθcosψ)
+w(sinφsinψ+ cosφsinθcosψ) (2.10) ˙
pE =ucosθsinψ+v(cosφcosψ+ sinφsinθsinψ)
+w(−sinφcosψ+ cosφsinθsinψ) (2.11) ˙
h=usinθ−vsinφcosθ−wcosφcosθ (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 compo-nents 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 air-craft 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 =√u2+v2+w2 (2.13) β= sin−1
³v
V
´
(2.14) α= tan−1
³w
u
´
(2.15)
Differentiating the both sides of the above equations and substituting from equa-tions (2.1)–(2.3) yield the expressions for the new state derivatives as
˙ V = 1
m(Fxcosαcosβ+Fysinβ+Fzsinαcosβ) (2.16) ˙
β = 1
mV(−Fxcosαsinβ+Fycosβ−Fzsinαsinβ) +psinα−rcosα (2.17) ˙
α= 1 mV
µ
−Fx
sinα cosβ +Fz
cosα cosβ
¶
−pcosαtanβ+q−rsinαtanβ (2.18)
Generally, Fx, Fy, and Fz are forces composed of gravitational, aerodynamic and
propulsive forces, and Mx, My, and Mz are moments produced by aerodynamic and
body x-axis V u w body z-axis body y-axis D E v relative wind x-axis (wind) x-axis (stability) xcg
Figure 2.1: Definition of axes and aerodynamic angles.
Fx,G Fy,G Fz,G =g
−sinθ cosθsinφ cosθcosφ
(2.19)
The aerodynamic forces are modeled as
Fx,A Fy,A Fz,A =qS
Cx,t Cy,t Cz,t (2.20)
The aerodynamic moments acting on the body are similarly modeled as
Mx,A My,A Mz,A =qS
bCl,t cCm,t bCn,t (2.21)
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 (δptv) and yaw (δytv)
planes as shown in Figure 2.2. The roll thrust vector is usually generated by two nozzles or subnozzles, each vectoring δytv in opposite directions [32].
T Fx,T
Fy,T
Fz,T
body
y-axis
body
x-axis
body
z-axis
lT
ytv
G
ptv
G
xcg
xT
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
Fx,T
Fy,T
Fz,T
=T
cosδptvcosδytv
sinδytv
−sinδptvcosδytv
(2.22)
and the pitch and yaw moments are produced by the moment arm lT = xcg −xT
Mx,T
My,T
Mz,T
=
0
−lTT sinδptvcosδytv
−lTTsinδytv
(2.23)
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 longi-tudinal 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.
2.2.1
Aircraft Description
c.g. T
(a) Original F-16 model
lT
Fx,T
Fz,T T
G ptv c.g.
(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 δth 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) 20500
Moment of inertia Iy (slug-ft2) 55814
Wing area S (ft2) 300 Mean aerodynamic chord ¯c(ft) 11.32
Reference CG location xcg,ref 0.35¯c
Table 2.2: Control actuator models.
Actuator Deflection limit Rate limit Time constant Elevator δe ±25◦ 60◦/s 0.0495 s
2.2.2
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
coeffi-cientsCxq,Czq, andCmq are the functions of the angle of attackα, and the coefficients
Cx, Cz, and Cm depend on both of α and the elevator angle δe. Note that Figure
2.4(d) gives the value ofz-axis force coefficient when the elevator does not deflected, i.e. Cz(α,0). The value of Cz(α, δe) can be approximated by the following equation.
Cz(α, δe) =Cz(α,0)−0.19
δe
25 (2.24)
In the simulation, the data are interpolated linearly between the points, and ex-trapolated beyond the table boundaries. But note that the extrapolation may lead to unrealistic results. The total coefficients Cx,t, Cz,t, and Cm,t are calculated by
sum-ming the various aerodynamic contributions to a given force or moment coefficient.
Cx,t =Cx(α, δe) +
qc
2VCxq(α) (2.25)
Cz,t=Cz(α, δe) +
qc
2V Czq(α) (2.26)
Cm,t =Cm(α, δe) +Cz,t(xcg,ref−xcg) + qc
−10 0 10 20 30 40 50 −0.5 0 0.5 1 1.5 2 2.5 3
α (deg)
Cxq
(
α
)
(a) Damping coefficientCxq
−10 0 10 20 30 40 50
−0.1 −0.05 0 0.05 0.1 0.15
α (deg)
Cx
(
α
,δe
)
δe=−24°
δ e=12 ° δ e=0 ° δ e=12 ° δ e=24 °
(b)x-axis force coefficientCx
−10 0 10 20 30 40 50
−40 −35 −30 −25 −20 −15 −10 −5
α (deg)
Czq
(
α
)
(c) Damping coefficientCzq
−10 0 10 20 30 40 50
−2.5 −2 −1.5 −1 −0.5 0 0.5 1
α (deg)
Cz
(
α
,0)
(d)z-axis force coefficientCz
−10 0 10 20 30 40 50
−7.5 −7 −6.5 −6 −5.5 −5
α (deg)
Cmq
(α
)
(e) Damping coefficientCmq
−10 0 10 20 30 40 50 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
α (deg)
Cm
(
α
,δe
)
δe=−24°
δe=12°
δe=0°
δe=12°
δe=24
°
(f) Pitching moment coefficient Cm
2.2.3
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 δptv 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
Fx,T =Tcosδptv (2.28)
Fz,T =−Tsinδptv (2.29)
My,T =−lTTsinδptv (2.30)
It is obvious that the original aircraft model is corresponding to the case of δptv = 0.
2.2.4
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(Fxcosα+Fzsinα) (2.31) ˙
α= 1
mV (−Fxsinα+Fzcosα) +q (2.32) ˙
q=My Iy
(2.33) ˙
θ=q (2.34)
where the x and z axes forces and pitching moment are given as
Fx =qSCx,t−mgsinθ+T cosδptv (2.35)
Fz =qSCz,t+mgcosθ−T sinδptv (2.36)
Combining with the aerodynamic and propulsive models, the final nonlinear lon-gitudinal model of F-16 aircraft are given as follows.
˙
V = qS¯ c¯
2mV [Cxq(α) cosα+Czq(α) sinα]q−gsin(θ−α) +qS¯
m [Cz(α, δe) cosα+Cz(α, δe) sinα] + T
mcos (α+δptv) (2.38) ˙
α=
·
1 + qS¯ ¯c
2mV2 (Czq(α) cosα−Cxq(α) sinα)
¸
q+ g
V cos(θ−α) + qS¯
mV [Cz(α, δe) cosα−Cx(α, δe) sinα]− T
mV sin (α+δptv) (2.39) ˙
q= qS¯ c¯ 2IyV
[¯cCmq(α) + ∆Czq(α)]q
+qS¯ ¯c Iy
·
Cm(α, δe) + ∆
¯
cCz(α, δe)
¸
−lTT
Iy
sinδptv (2.40)
˙
θ=q (2.41)
It is noted that the altitudehis one of the factors to affect the thrust forceT, and it enters the equations of motion in an implicit way. From the navigation equation (2.12) and definition of aerodynamic angles, we have
˙
h=V cosαsinθ−V sinαcosθ (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.
2.3.1
LPV Systems
vary-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 ⊂Rs. In addition, its time
derivative is often assumed to be bounded and satisfy the constraintνk≤ρ˙k ≤ν¯k,k=
1,2, . . . , s. For notational purposes, denote V = {ν: νk ≤νk ≤ν¯k, k= 1,2, . . . , s},
where V is a given convex polytope in Rs that contains the origin. Given the sets P
and V, the parameterν-variation set is defined as
Fν
P =
©
ρ∈ C1(R
+,Rs) : ρ(t)∈ P, ρ(t)˙ ∈ V, ∀t≥0
ª
(2.43) SoFν
Pspecifies 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 Fν
P, a generalized nth-order
LPV system can be described by
" ˙ x(t) y(t) # = " A(ρ(t)) B(ρ(t)) C(ρ(t)) D(ρ(t)) # " x(t) u(t) # (2.44)
where ρ∈ Fν
P, the state x∈Rn, the output y∈Rny, and the input u∈Rnu. All of
the state-space data are continuous functions of the parameterρ, i.e.,A :Rs→Rn×n,
B :Rs →Rn×nu, C :Rs→Rny×n, and D:Rs→Rny×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 vectorx(t) can be decomposed into scheduling states x1(t)∈ FPν, and nonscheduling states x2(t) [61], i.e.,
xT(t) = £xT
1(t) xT2(t)
¤
(2.45) then the quasi-LPV system can be obtained as
˙ x1(t) ˙ x2(t)
y(t) =
A11(x1(t),Ω(t)) A12(x1(t),Ω(t)) B1(x1(t),Ω(t)) A21(x1(t),Ω(t)) A22(x1(t),Ω(t)) B2(x1(t),Ω(t)) C1(x1(t),Ω(t)) C2(x1(t),Ω(t)) D(x1(t),Ω(t))
x1(t) x2(t) u(t)
(2.46)
where the scheduling parameter vector isρT(t) =£xT
1(t) ΩT(t)
¤
2.3.2
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
˙
x=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 equi-librium points, which are parameterized by the scheduling parameter ρ and satisfy f(xe(ρ), ue(ρ)) = 0. Corresponding to a specified family of equilibrium points, the
family of the linearized plants can be written in the form of
"
δx(t)˙ δy(t)
#
=
"
∂f
∂x|xe(ρ),ue(ρ)
∂f
∂u|xe(ρ),ue(ρ)
∂g
∂x|xe(ρ),ue(ρ)
∂g
∂u|xe(ρ),ue(ρ)
# "
δx(t) δu(t)
#
(2.49)
where the deviation variables are defined in the obvious fashion.
δx=x−xe(ρ), δu=u−ue(ρ), δy =y−ye(ρ) (2.50)
The state-space data in (2.49) is actually in the LPV form since the equilibrium point (xe, ue) depends on the parameterρ. At each fixedρ, the linearization (2.49) describes
Before deriving the LPV model of the F-16 aircraft, we need to select the schedul-ing parameters. Generally, the schedulschedul-ing 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= [δth δe]T, and for
the thrust-vectored aircraft model, u= [δth δe δptv]T. Note that the throttle setting
δth indirectly affects the states through the power output from the engine.
There-fore, 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 asx= [V α q θ pow]T. The statepowis 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 δptv at equilibrium is always zero, i.e., the thrust nozzle does not deflect.
As mentioned before, the thrust-vectored model when δptv = 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 21◦ 20.8437]
[δth δe δptv] = [0.3210 −6.3657◦ 0◦]
20 25 30 35 40 45 50 150
160 170 180 190 200 210
α (deg)
V (ft/s)
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.
2.3.3
Verification of F-16 LPV Models
In this section, we will check if the F-16 LPV model captures the local nonlinear-ities 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 2.6– 2.8 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.
1. 0.1 step of the throttle setting effective from time = 1 s.
δth=
(
0.3210 0≤t <1 0.4210 t≥1
δe=
−6.3657◦ 0≤t <1 and t ≥9
−7.3657◦ 1≤t <5
−5.3657◦ 5≤t <9
3. −0.5◦ step of the thrust vector effective from time = 1 s.
δptv =
(
0 0≤t <1
−0.5 t≥1
0 2 4 6 8 10
199 200 201 202 203 204 205 206 207 208 209 210 time (s) V (ft/s) nonlinear model linear model
(a) VelocityV
0 2 4 6 8 10
20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21 21.1 time (s) α (deg) nonlinear model linear model
(b) Angle of attackα
0 2 4 6 8 10
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q (deg/s) nonlinear model linear model
(c) Pitch rateq
0 2 4 6 8 10
20 21 22 23 24 25 26 time (s) θ (deg) nonlinear model linear model
(d) Pitch angleθ
Figure 2.6: Time responses for 0.1 step input of δth atV = 200 ft/s and α= 21◦.
Note that the acceptable size of the command input is different for the different equilibrium point. To prove this, another flight condition V = 160 ft/s and α= 35◦
0 5 10 15 190
192 194 196 198 200 202 204 206 208
time (s)
V (ft/s)
nonlinear model linear model
(a) VelocityV
0 5 10 15
18 19 20 21 22 23 24
time (s)
α
(deg)
nonlinear model linear model
(b) Angle of attackα
0 5 10 15
−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5
time (s)
q (deg/s)
nonlinear model linear model
(c) Pitch rateq
0 5 10 15
16 17 18 19 20 21 22 23 24 25
time (s)
θ
(deg)
nonlinear model linear model
(d) Pitch angleθ
Figure 2.7: Time responses for ±1◦ doublet input ofδ
e atV = 200 ft/s and α= 21◦.
To save the space, only the 0.1 step input of δth is tested, i.e.,
δth=
(
0.6865 0≤t <1 0.7865 t≥1
and Figure 2.9 shows the corresponding time responses. Compared to the case at V = 200 ft/s and α = 21◦, the LPV model does not follow the nonlinear model very
0 2 4 6 8 10 188
190 192 194 196 198 200 202
time (s)
V (ft/s)
nonlinear model linear model
(a) VelocityV
0 2 4 6 8 10
20.5 21 21.5 22 22.5 23
time (s)
α
(deg)
nonlinear model linear model
(b) Angle of attackα
0 2 4 6 8 10
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1
time (s)
q (deg/s)
nonlinear model linear model
(c) Pitch rateq
0 2 4 6 8 10
20.5 21 21.5 22 22.5 23 23.5 24
time (s)
θ
(deg)
nonlinear model linear model
(d) Pitch angleθ
Figure 2.8: Time responses for −0.5◦ step input ofδ
ptv atV = 200 ft/s and α= 21◦.
2.4
Summary
0 5 10 15 150
152 154 156 158 160 162 164 166
time (s)
V (ft/s)
nonlinear model linear model
(a) VelocityV
0 5 10 15
33.8 34 34.2 34.4 34.6 34.8 35 35.2 35.4 35.6
time (s)
α
(deg)
nonlinear model linear model
(b) Angle of attackα
0 5 10 15
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
q (deg/s)
nonlinear model linear model
(c) Pitch rateq
0 5 10 15
34 35 36 37 38 39 40 41 42 43 44 45
time (s)
θ
(deg)
nonlinear model linear model
(d) Pitch angleθ
0 5 10 15 156
157 158 159 160 161 162 163
time (s)
V (ft/s)
nonlinear model linear model
(a) VelocityV
0 5 10 15
34.5 34.6 34.7 34.8 34.9 35 35.1 35.2 35.3
time (s)
α
(deg)
nonlinear model linear model
(b) Angle of attackα
0 5 10 15
−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
time (s)
q (deg/s)
nonlinear model linear model
(c) Pitch rateq
0 5 10 15
34 34.5 35 35.5 36 36.5 37 37.5 38 38.5 39
time (s)
θ
(deg)
nonlinear model linear model
(d) Pitch angleθ
Chapter 3
Actuator Saturation Control
Saturation of the conventional aerodynamic control surfaces is a major issue for the flight control in the high angle of attack region, especially at near stall and post-stall flight conditions. It is well recognized that actuator saturation degrades the performance of the flight control system and may lead to instability. Without cost and hardware concerns, unconventional actuators such as thrust vectoring are usu-ally used to compensate the conventional aerodynamic control surfaces. However, the incorporation of the additional thrust vectoring nozzle could complicate the de-sign of flight control laws in the post-stall regime. The antiwindup method provides an alternative approach to handle control saturation. Applying antiwindup control scheme to the flight control is promising because no additional actuator is needed to compensate control authority. The implementation of antiwindup controllers could be done by simply modifying the flight control software.
explicit compensator construction formula in Section 3.4.3. In Section 3.5, the LPV antiwindup control scheme is applied to the F-16 longitudinal model and compared with thrust vectoring control. This chapter concludes with a summary in Section 3.6.
3.1
Introduction
Saturation is probably the most widely encountered and most dangerous nonlin-earity in control systems. The result of saturation is a difference between the control input demanded by the controller, and the realized output of the actuator. If the constraints on the actuator amplitude and rate are ignored at the controller design stage, the closed-loop performance may severely degrade, and the system may even be driven to instability. In the example of flight control, the destabilizing effects of actuator saturation have been cited as a major contribution to catastrophic pilot in-duced oscillations [62]. Also, the failure of several aircraft has been attributed to the saturation of actuators. In the case of the F-22 crash in April 1992 [25], the control surface rate was limited. A similar occurrence caused the Gripen crash in August of 1992 [86].
As a result, the analysis and synthesis of controllers for dynamic systems subject to actuator saturation received increasing attention. A chronological bibliography on saturating actuators can be found in a special journal issue [11]. Also, there are several books [55, 40, 42] that report recent research activities on this topic. While analysis of systems including saturated actuators is relatively easy, the controller synthesis problem in the presence of input nonlinearity is a much more involved task. There are two principal approaches to saturation control design problem [83]:
1. direct nonlinear control design taking account of the nonlinearity from the be-ginning;
2. a two-step approach involving a linear (local) design followed by certain modi-fications to improve the nonlocal behavior.
this line [55]. In Ref. [56], the notion of semiglobal stabilization has been introduced, and low gain stabilizing approach is analyzed for linear systems subject to actuator saturation. The controller gain is constructed in such a way that the control input does not saturate for any a priori given arbitrarily large bounded set of initial condi-tions. The methods based on algebraic Riccati equation (ARE) were later developed for designing low-gain controllers utilizingH2 andH∞optimal control theory [97, 57].
However, low-gain controllers that avoid saturation will often result in low levels of performance, especially for the cases where the disturbance has intermediate or small amplitude, or large capacity actuators are available.
Nguyen and Jabbari also focus their research on this category [67, 68], and their methods are based on LMIs, which enable finding the control law in a convex search. Specifically, disturbance attenuation for systems with input saturation is studied in [67], and the controller is formulated as state feedback control with a special structure. The result was later extended to output feedback design for disturbance attenuation with actuator amplitude and rate saturation [68]. The controllers have a more general structure similar to those used in LPV approach, and operate close to maximum actuator capacity.
Due to the combination of various requirements, this single-step approach may become more conservative and more complicated. Comparatively, the second ap-proach has distinct advantages. For this two-step procedure, the actuator saturation is first ignored, and a linear controller is designed to meet the required performance specifications. Then an antiwindup compensator is designed to minimize the adverse influence of saturation. The main advantage of this approach is to separate the sta-bility and performance requirements from saturation control. Moreover, the designer has freedom to choose different design approaches as desired in the construction of the nominal controller.
the actual plant input is different from the output of the controller as a result of ac-tuator limitations, the controller output will not drive the plant, and the states of the controller will be wrongly updated. This effect is called controller windup [45].
Like other saturation control techniques, the antiwindup compensator design often assumes a linear time-invariant (LTI) plant, and models the saturation block as a sector-bounded nonlinearity. Then absolute stability conditions, such as Popov, circle, and scaled small-gain theorems [100, 99, 46, 43], are applied for the stability and performance analysis. Desirable design requirements for antiwindup compensation subject to actuator saturation are the closed-loop system stability, recovery of the linear design specifications in the absence of saturation (linear performance recovery), and the smooth degradation of the linear performance in the presence of saturation (graceful performance degradation).
Early results in antiwindup control [26, 5, 22, 109] often have the drawbacks of lacking rigorous stability analysis and clear exposition of performance objectives. A general framework that unifies a large class of existing antiwindup control schemes was developed in [21, 45]. This framework is useful for understanding different antiwindup control schemes and motivates the development of systematic procedures for designing antiwindup controllers that provide guaranteed stability and performance. However, the proposed antiwindup construction in terms of two matrix parameters does not lead to convex synthesis conditions in terms of LMIs. It is also true for the most of antiwindup control schemes in the last decade.
Until recently, the synthesis condition of static antiwindup controllers given in [64] is formulated as an LMI problem using an extended circle criterion. However, the static antiwindup synthesis problem is not always solvable. The study in [33] has further revealed that antiwindup control for stable open-loop LTI systems can be solved globally as an LMI problem with the order of antiwindup compensator no more than the order of the plant. By modeling the saturation as a sector-bounded nonlinearity, the authors recast the design of dynamic antiwindup compensator as a robust H∞ problem based on the extended circle criterion. Alternatively, the Popov
stability condition has also been applied to the antiwindup compensator design prob-lem [99]. However, the synthesis condition of the saturation controller is given in coupled Riccati equations, which are difficult to solve for the optimal solution.
It is noted that most of previous antiwindup compensator designs are only appli-cable to open-loop stable LTI systems, and thus limit their usefulness for practical problems. When the system is nonlinear and open-loop unstable, the control syn-thesis problem becomes very difficult to solve. Therefore, global stabilization cannot be achieved [98, 40]. However, in many control systems including flight control sys-tems, the system dynamics are inherently nonlinear and their linearizations at some operation points are strictly unstable. Our previous work [104] was built up on the research in [33] but removed the open-loop stability requirement. In this research, the result in [104] will be generalized to LPV systems. This generalization is very important because of the relevance of LPV systems to nonlinear systems.
3.2
Saturation Nonlinearity
sat(ui) =
umax
i ui > umaxi
ui umini ≤ui ≤umaxi
umin
i u < umini
(3.1)
where umax
i >0,umini <0, and on a vector u= (u1, u2, . . . , unu) by
sat(u) = (sat(u1), . . . ,sat(unu)) (3.2)
K P
d e
y u
sat
Figure 3.1: A control system with saturation nonlinearity.
The saturation function can be represented by a sector condition [43]. We use sect[a, b] to denote a conic sector in Figure 3.2, which is defined as
sect[a, b] ={(u, y) : (y−au)(y−bu)≤0} (3.3)
where a and b are the slopes of the two boundaries of the conic sector. It is obvious that the saturation nonlinearity defined in (3.1) belongs to sect[0,1] as shown in Figure 3.3(a). In some cases, tighter sector bounds on sat(ui) can be obtained if
the control input ui is bounded in magnitude [48]. Figure 3.3(b) shows the case of
sat(·)∈sect[k,1] with 0< k <1, and obviouslyui is restricted to be less than 1kumaxi .
There is another way to represent the saturation nonlinearity. As shown in Figure 3.4, if one more output from the saturation block is created and defined by
qi =ui−sat(ui) (3.4)
then the saturation block can be transformed to a dead-zone with ui as the input and
u y
y=au y=bu
Figure 3.2: A graphical representation of sect[a, b].
ui
sat(ui)
max
i u
min
i u
sect[0,1] saturation
(a) sect[0,1]
ui sat(ui)
max
i
u
min
i
u
sect[k,1] saturation
(b) sect[k,1] with 0< k <1
Figure 3.3: Sector bounds on saturation nonlinearity.
saturation nonlinearity defined in Figure 3.3(a). Similarly, the conic sector can also be used to denote the dead-zone nonlinearity. For example, the dead-zone in Figure 3.5(a) can be represented by sect[0,1]. Figure 3.5(b) gives the case of sect[0, k], where the maximal value of the control input ui is restricted to be less than
¡ 1
1−k
¢
umax
i .
The actuator saturation nonlinearity considered in this research is defined as
σ(ui) =
(
ui |ui| ≤umaxi
sgn(ui)umaxi |ui|> umaxi
(3.5)
ui sat(ui)
qi
Figure 3.4: Saturation block with two outputs.
qi
ui max
i u
min
i u
sect[0,1]
1
1
(a) sect[0,1]
qi
ui
max
i
u min
i
u
sect[0,k]
(b) sect[0,k] with 0< k <1
Figure 3.5: Dead-zone representation of saturation nonlinearity.
3.3
LPV Robust Analysis of Sector-Bounded
Un-certainties
Consider an LPV system connected with sector-bounded uncertainty
˙ x u e
=
A(ρ) B0(ρ) B1(ρ) C0(ρ) D00(ρ) D01(ρ) C1(ρ) D10(ρ) D11(ρ)
x q d
(3.6)
q = Φ(u) (3.7)
where x ∈ Rn is the state of the linear part of the system. q, u ∈ Rnu is the
input/output pair connecting to the uncertainty. d ∈ Rnd is the disturbance and
e ∈ Rne is the controlled output. We assume that each element of the uncertain
operator Φ : Rnu → Rnu is contained in a conic sector [0, k
i], i = 1,2, . . . , nu as
shown in Figure 3.6.
Φ(ui)
ui
ki
Figure 3.6: Sector-bounded uncertainty.
Denote Φ ∈ sect[0, K] with K = diag{k1, k2, . . . , knu}. For well-posedness of the
problem, it is required that each ki ≤1. The constraint on the uncertainty Φ can be
expressed using its input/output relation, that is
qTW(Ku−q)≥0 (3.8)
for any diagonal matrix W = diag{w1, w2, . . . , wnu}>0, andW may depend on the
parameter ρ for the LPV system.
Before presenting the analysis condition, we give an important description of the stability and performance properties in the sense of Lyapunov function. Given a linear plant "
˙ x e
#
=
"
A B C D
# "
x d
#
