SLIDE 1
2
u u u Bu F ≤ ≤ =
Control allocation
u u u Bu M ≤ ≤ =
- Whic
Dynamic Control Allocation using Constrained QP Ola Hrkegrd - - PDF document
Dynamic Control Allocation using Constrained QP Ola Hrkegrd - - PDF document
Dynamic Control Allocation using Constrained QP Ola Hrkegrd Linkping University S weden How can we utilize actuator redundancy? Ac tuator limits Frequenc y division 1 Control allocation = = M Bu F Bu
SLIDE 2
SLIDE 3
3
Example: v=u1+u2
Desired total control
5 10 15 0.05 0.1 0.15 0.2 Time (s) v Virtual control v
Example: v=u1+u2
- 2
2 1
u u 2 1 min Dynamic allocation
5 10 15 0.05 0.1 0.15 0.2 Time (s) v, u Control signals v u1 u2 5 10 15 0.05 0.1 0.15 0.2 Time (s) v, u Control signals v u1 u2
SLIDE 4
4
Why dynamic allocation?
S pecify frequency range for each actuator. Improve c losed loop behaviour.
How?
( ) ( ) ( )
t u t u t u ≤ ≤
( ) ( )
t v t Bu =
( )
( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2 2 s 1 t u
T t u t u W t u t u W min − − + −
S olve
W1, W2 → frequency charac teristic s us → steady state distribution
SLIDE 5
5
Properties
Without actuator constraints: S olution:
( ) ( ) ( ) ( )
t Gv T t Fu t Eu t u
s
+ − + =
( ) ( )
t v t Bu =
( )
( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2 2 s 1 t u
T t u t u W t u t u W min − − + −
Stability
( ) ( ) ( ) ( )
t Gv T t Fu t Eu t u
s
+ − + =
Filter: If W1 is nonsingular then
( )
1 F < λ ≤
× × × ×
(asymptotically stable)
SLIDE 6
6
Steady state
( ) ( )
t v t Bu =
- =
If Bus(t)=v(t) then
( ) ( )
∞ → → t as t u t u
s
( )
( ) ( ) ( ) ( ) ( ) ( )
2 2 2 2 2 s 1 t u
T t u t u W t u t u W min − − + −
Example: v=u1+u2
= = = 10 W 2 1 W v 1 u
2 1 s 5 10 15 0.05 0.1 0.15 0.2 Time (s) v, u Control signals v u1 u2 10 10
−2
10
−1
10 Frequency (rad/sec) |Gvu| Control distribution u1 u2
SLIDE 7
7
Admire
u3 u4 u1 u2 u5 u6 u7
- 1000 m, Mac
h 0.5 Canards for HF
Minimum drag Improved nz response
× = = = 7 3 B C C C v u u u
n m l 7 1
- Design parameters
u u v Bu u min arg u
2 1 2 s
= = = = v x x x us = ⇒
- (
) ( )
10 ... 10 5 5 diag W , 2 ... 2 diag W
2 1
= =
( ) ( ) ( )
t Gv T t Fu t u + − =
SLIDE 8
8
Control distribution
10 10
2
10
−5
10
−4
10
−3
10
−2
10
−1
10 Frequency (rad/sec) Control distribution Roll Canard wings Outboard elevons Inboard elevons Rudder 10 10
2
10
−3
10
−2
10
−1
10 Frequency (rad/sec) Pitch 10 10
2
10
−4
10
−3
10
−2
10
−1
10 Frequency (rad/sec) Yaw
Simulation results
2 4 6 −50 50 100 150 200 250 Time (s) p (deg/s) Roll rate 2 4 6 −10 −5 5 10 15 20 25 30 Time (s) q (deg/s) Pitch rate 2 4 6 −3 −2 −1 1 2 3 4 Time (s) β (deg) Sideslip
SLIDE 9
9
Control surfaces
2 4 6 −15 −10 −5 5 10 Time (s) u1 (deg) Right canard 2 4 6 −15 −10 −5 5 10 Time (s) u4 (deg) Right inboard elevon
Dynamic vs static
1 2 3 1 2 3 4 5 6 7 Time (s) nz (−) Pilot load factor 1 1.1 1.2 1.3 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (s) nz (−) Non−minimum phase behavior δc for high freq. min ||δ|| δc=0