Dynamic Control Allocation using Constrained QP Ola Hrkegrd - PDF document

Dynamic Control Allocation using Constrained QP Ola Härkegård Linköping University S weden How can we utilize actuator redundancy? � Ac tuator limits � Frequenc y division 1

Control allocation = = M Bu F Bu ≤ ≤ ≤ ≤ u u u u u u Whic h u should we pick? Whic h u should we pick? � System overview v u Feedback Control S ystem r law allocation dynamics x � Direc t CA = v Bu � Daisy chaining ≤ ≤ u u u � Linear prog. � Quadratic prog. 2

Example: v=u 1 +u 2 � Desired total control Virtual control 0.2 0.15 0.1 v 0.05 v 0 0 5 10 15 Time (s) Example: v=u 1 +u 2     1 0 u     1 min � Dynamic allocation     � 0 2 u     2 2 Control signals Control signals 0.2 0.2 0.15 0.15 v, u v, u 0.1 0.1 0.05 0.05 v v u 1 u 1 u 2 u 2 0 0 0 5 10 15 0 5 10 15 Time (s) Time (s) 3

Why dynamic allocation? � S pecify frequency range for each actuator. � Improve c losed loop behaviour. How? � S olve ( ( ) ( ) ) ( ( ) ( ) ) 2 2 − + − − min W u t u t W u t u t T ( ) 1 s 2 2 2 u t ( ) ( ) = Bu t v t ( ) ( ) ( ) ≤ ≤ u t u t u t � W 1 , W 2 → frequency charac teristic s � u s → steady state distribution 4

Properties � Without actuator constraints: ( ( ) ( ) ) ( ( ) ( ) ) 2 2 − + − − min W u t u t W u t u t T ( ) 1 s 2 2 2 u t ( ) ( ) = Bu t v t ( ) ( ) ( ) ( ) = + − + � S olution: u t Eu t Fu t T Gv t s Stability ( ) ( ) ( ) ( ) = + − + � Filter: u t Eu t Fu t T Gv t s � If W 1 is nonsingular then ( ) ≤ λ < 0 F 1 × × × × (asymptotically stable) 5

Steady state ( ( ) ( ) ) ( ( ) ( ) ) 2 2 − + − − min W u t u t W u t u t T ( ) 1 s 2 2 2 u t � � � � � � � ( ) ( ) = = 0 Bu t v t � If Bu s (t)=v(t) then ( ) ( ) → → ∞ u t u t as t s Example: v=u 1 +u 2       0 1 0 0 0 =   =   =   u v W W       s 1 2 1 0 2 0 10       Control signals Control distribution 0 0.2 10 0.15 |G vu | v, u −1 0.1 10 0.05 v u 1 u 1 u 2 u 2 −2 0 10 0 5 10 15 0 10 Time (s) Frequency (rad/sec) 6

u 3 u 1 u 4 u 7 Admire u 5 u 2 u 6       u C       1 l = = = ×       u � v C B 3 7 � m       u C       7 n � 1000 m, Mac h 0.5 � Canards for HF � Minimum drag � Improved n z response Design parameters   0 0 0 =   u arg min u s � 2   0 0 0 ⇒ = = u s v Bu v   x x x   = = u u 0   1 2  �  ( ) ( ) = = W diag 2 ... 2 , W diag 5 5 10 ... 10 � 1 2 ( ) ( ) ( ) = − + u t Fu t T Gv t 7

Control distribution Roll Pitch Yaw 0 0 0 10 10 10 −1 10 −1 10 Control distribution −1 10 −2 10 −2 10 −3 10 −2 10 Canard wings −3 10 −4 10 Outboard elevons Inboard elevons Rudder −3 −5 −4 10 10 10 0 2 0 2 0 2 10 10 10 10 10 10 Frequency (rad/sec) Frequency (rad/sec) Frequency (rad/sec) Simulation results Roll rate Pitch rate Sideslip 250 30 4 25 3 200 20 2 150 15 p (deg/s) q (deg/s) β (deg) 1 100 10 0 5 50 −1 0 0 −2 −5 −50 −10 −3 0 2 4 6 0 2 4 6 0 2 4 6 Time (s) Time (s) Time (s) 8

Control surfaces Right canard Right inboard elevon 10 10 5 5 0 0 u 1 (deg) u 4 (deg) −5 −5 −10 −10 −15 −15 0 2 4 6 0 2 4 6 Time (s) Time (s) Dynamic vs static Pilot load factor Non−minimum phase behavior 7 1.8 δ c for high freq. min || δ || 6 1.6 δ c =0 5 1.4 4 n z (−) n z (−) 1.2 3 1 2 0.8 1 0 0.6 0 1 2 3 1 1.1 1.2 1.3 Time (s) Time (s) 9