Flight Control Design Using Backstepping Ola Hrkegrd, T orkel Glad - PDF document

Flight Control Design Using Backstepping Ola Härkegård, T orkel Glad Linköping University Background Can you design a single c ontroller that will give stability and performance � throughout the entire flight regime? � Avoid tedious gain-scheduling. � Previous work: feedback linearization. T ry backstepping! T ry backstepping! ���� 1

Controller overview δ r u Pre- S tate Control filter feedback alloc. x Control objectives β α p s p = ref β = α = α p ref 0 s s 2

Angle of attack dynamics L α − M L ( ) mg α = − + q � α mV V α δ M ( , q , ) = q � J δ Aerodynamic efforts 2 0.5 1 0 C m C L 0 -1 -0.5 -40 -40 -20 -20 0 0 40 40 20 20 0 0 δ (deg) δ (deg) α (deg) α (deg) 3

Ideas for control design � Linearize the airc raft dynamic s for a set of flight cases ( gain-scheduling ). � Cancel the nonlinear system behaviour using feedback ( feedback linearization ). � Refrain from canc elling “ harmless” nonlinearities ( backstepping ). 2-stage control design α − L ( ) mg ( ) α = − + = α + q f q � mV � Dynamic s: α δ M ( , q , ) = = q u � J ( ) = α 1. Design . u k , q 2. Allocate control surfac es. 4

Backstepping ( ) α = α + � First c onsider f q � ( ) α f α α ref � Virtual c ontrol law: ( ) ( ) = − α − α − α q f k des ref 1 ref Backstepping, contd. � Create Lyapunov function ( ) ( ) = α − α + − 2 V F q q ref des � Demand dV = − < W 0 dt and solve for u. 5

Resulting control law � Backstepping ( ) ( ( ) ) = − α − α − + α u k k q f 1 ref 2 ref � Feedback linearization ( ) ( ( ) ) ( ( ) ) ′ = − α − α − + α + α u k k f q f 1 ref 2 Step responses ref = ref = α α 60 deg 20 deg 30 60 α [deg] α [deg] 20 40 20 10 0 0 0 2 4 6 8 0 2 4 6 8 200 100 q [deg/s] q [deg/s] 0 0 -100 -200 0 2 4 6 8 0 2 4 6 8 500 1000 u [deg/s 2 ] u [deg/s 2 ] 0 0 -500 -1000 0 2 4 6 8 0 2 4 6 8 time [s] time [s] 6

Step responses ref = ref = α α 20 deg 60 deg 30 80 60 20 α (deg) α (deg) 40 10 20 0 0 0 2 4 6 8 0 2 4 6 8 500 1000 2 ) 2 ) u (deg/s u (deg/s 0 0 -500 -1000 0 2 4 6 8 0 2 4 6 8 time (s) time (s) Control law properties ( ) α δ M , q , ( ) ( ( ) ) = − α − α − + α k k q f 1 ref 2 ref J � Global stability. d α . � Does not involve dL/ � Inverse optimal ⇒ gain margin. 7

Full controller α =f 1 ( α , β , p s )+q �  ( ) ⇒ = α α β ref  u k , , q , , p 1 1 s q  =u 1 � =f 2 ( β , α )+r s β �  ( ) ⇒ = β  u k , r 2 2 s r  =u 2 � s ( ) } ⇒ = ref − p =u 3 � u K p p s 3 s s ( ( ) ) = − δ − ω × ω 1 u SJ M J Simulations 20 200 p s (deg/s) 100 α (deg) 10 0 0 -100 0 5 10 0 5 10 5 20 0 β (deg) δ (deg) 0 -20 Aileron Elevator Rudder -5 -40 0 5 10 0 5 10 time (s) 8