Nonlinear multivariable flight control Ola Hrkegrd Linkpings - PDF document

Nonlinear multivariable flight control Ola Härkegård Linköpings Tekniska Högskola Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Background Nonlinear Multivariable Different flight cases 6 DOF � � High angle-of-attack Control surface redundancy � � Rigid body dynamics Unconventional control surfaces � � Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 1

High angle of attack Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Unconventional control surfaces vs. Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 2

Outline � Aircraft � Backstepping � Control allocation Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Outline � Aircraft Why fly-by-wire? Why fly-by-wire? � � Control objectives Control objectives � � Actuators Actuators � Backstepping � � � Control allocation Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 3

Why fly-by-wire? control stick surf. Control system sensors visual info, cockpit displays, etc. Stabilize aircraft � Handling qualities � Advanced control surfaces � Autopilot functionality � Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Control objectives � Longitudinal control n z q α V � Pitch rate � Load factor � Angle of attack Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 4

Control objectives � Lateral control β V p � Sideslip angle � Roll rate Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Actuators (TVC) Canards Elevons Rudder Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 5

System overview � Inner control loop: r u α , β , p Flight Flight control sys. control sys. x � Modular control design: M u M Control Control Control Control Aircraft Aircraft Actuators Actuators laws laws allocation allocation dynamics dynamics Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Outline � Aircraft � Backstepping What is backstepping? What is backstepping? � � Why use it? Why use it? � � Research at LiTH Research at LiTH � Control allocation � � Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 6

What is backstepping? � Constructive nonlinear control design method. � Model structure: ( ) x f x , x = � Same requirement as 1 1 1 2 ( ) in feedback linearization x f x , x , x = � 2 2 1 2 3 � ( ) x = f x , x , x , , x , u � � n n 1 2 3 n Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Why use backstepping? � Can benefit from “useful” nonlinearities May require less � control effort � modeling information → robustness � Can achieve GAS when feedback linearization fails Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 7

Design procedure ( ) x = f x , x � 1 1 1 2 ( ) x = f x , x , x � 2 2 1 2 3 � ( ) x f x , x , x , , x , u = � � n n 1 2 3 n 2 V = x decreases if 1 1 ( ) x x des x = 2 2 1 Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Step backwards ( ) x f x , x = � 1 1 1 2 ( ) x = f x , x , x � 2 2 1 2 3 � ( ) x f x , x , x , , x , u = � � n n 1 2 3 n ( ) 2 V V x x des = + − decreases if 2 1 2 2 ( ) x x des x , x = 3 3 1 2 Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 8

Step backwards ( ) x = f x , x � 1 1 1 2 ( ) x = f x , x , x � 2 2 1 2 3 � ( ) x = f x , x , x , , x , u � � n n 1 2 3 n ( ) 2 V V x x des = + − decreases if n n − 1 n n ( ) u k x , x , x , , x = � 1 2 3 n Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Research at LiTH � Are there useful nonlinearities in the aircraft dynamics? � Well, at least harmless. � Can backstepping be applied to multivariable flight control? � Yes, applicable to general rigid body dynamics. Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 9

Sideslip regulation r Sideforce Y( β ) β V 4 2 x 10 1.5 1 ( ( ) ) r � Y F sin β = β − β − 1 T mV 0.5 Y (N) 0 N 1 -0.5 ( ) r N , r , u = β � -1 J -1.5 z -2 -20 -10 0 10 20 beta (deg) F T Linear Linear � � Backstepping N = – k 1 β – k 2 r Independent of Y( β ) Independent of Y( β ) � � Inverse optimal Inverse optimal � � Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Outline � Aircraft � Backstepping � Control allocation What is control allocation? What is control allocation? � � Why use it? Why use it? � � Research at LiTH Research at LiTH � � Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 10

What is control allocation? How should the total control effort be How should the total control effort be distributed among the actuators? distributed among the actuators? Control design: • Determine desired total control effort • Distribute the control effort among the actuators Control Control Control Control System System Actuators Actuators laws laws allocation allocation dynamics dynamics Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Applications Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 11

Why use control allocation? � Easy to reconfigure � Cheap way to handle actuator limits � ”Poor man’s MPC” � Necessary for certain control design methods � Feedback linearization (NDI) � Backstepping � Modularity Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Example: Hardover Max deflection after 1 s Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 12

Mathematical formulation u v Actuators Actuators u = true control signal � v = virtual control signal (total control effort) � Model: v = g(u) � Linearization: v = Bu � u ≤ u ≤ u Constraints:  min max ( ) ( ) � u t ≤ u ≤ u t  u ≤ u ≤ u � � �  min max Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Example x = − v  � x = − 2 u − u ⇔ Dynamics: �  1 2 v = 2 u + u  1 2 0 ≤ u ≤ 1 Constraints: 1 0 ≤ u ≤ 2 2 v = 3 v = 4 v = 5 u 2 Control law: v = x 2 Allocation problem: 2 u u v + = 1 2 u 1 1 Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 13

Optimization based control allocation Minimize cost function. 2 u u v + = u 0 = 1 2 d 0 u 1 ≤ ≤ W = I 1 u Bu v = 0 ≤ u ≤ 2 W 1 = 2 v u ≤ u ≤ u v = 3 . 5 u Ω 1   2 u =   ( ) 2   • Ω = arg min W Bu − v 1 . 5 2   v u ≤ u ≤ u • ( ) 2 • u = arg min W u − u u d u ∈ Ω u • u 1 d 1 Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Research at LiTH � Can standard QP methods be used for control allocation in real time? � Yes. � How can filtering be included in the allocation? � Also penalize changes in the control signal. � How is control allocation related to LQ control? � Equivalent without constraints. Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 14

Dynamic control allocation � What? Bu v = G(s) v u Constraints: G(s) u ≤ u ≤ u � Why? � Actuator dynamics � ”Practical aspects” Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Dynamic control allocation � How? Also penalize changes in the control signal. ( ) ( ( ) ( ) ) 2 2 min W u t W u t u t T + − − ( ) ( ) ( ) 1 2 ⇒ u t Fu t T Gv t ( ) 2 = − + 2 u t Bu = v Stable linear filter Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 15

Example: Flight control 100 —Canards Control effort distribution —Elevons 10-1 —TVC 10-2 10-3 10-4 -2 10-1 0 1 102 10 10 10 Frequency (rad/sec) Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Control allocation vs LQ control v r LQ CA u r LQ u LQ CA LQ x x B u = B B v x = Ax + B u x = Ax + B v � � u v v Bu = ∞ u ∫ min x T Q x u T R u d t + 1 1 ∞ 0 v ∫ v = − L x min x T Q x + v T R v d t 2 2 2 u = − L x 0 1 u L v min Wu då Bu = v = 3 2 u u = − L L x Q 1 R , 3 2 1 Q , R , W 2 2 Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 16

Example δ δ � Admire (FOI) c re � Mach 0.22, 3000 m � x = ( α β p q r) δ r δ le � Approximations: � Ignore actuator dynamics � View control surfaces as moment generators x Ax B v � Model (for control): = + � v v B = δ angular acc. control surfaces Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 Simulation results LQ LQ+CA with constraints Ola Härkegård Nonlinear multivariable flight control Lund 2003-11-13 17