1. No category

Control Allocation
What, why, and how?
Ola Härkegård
Aircraft maneuvering
T he pilot controls
• Pitch
• R oll
• Yaw
• (S peed)
33 DOF
DOF
1
T raditional configuration
Modern configuration
Canard wings
R udder
L eading edge flaps
T railing edge flaps
2
Control allocation
How
Howdo
dowe
wedis
distribute
tributethe
thecontrol
control action
action
among
amongaaredundant
redundantsset
etof
of actuators
actuators??
Modular control des ign
• Aircraft dynamics :
~
~
x& = f (x, u) ≈ f (x, m(x, u)) = f (x, v)
dim ≈10
dim 3
1.
1. Des
Design
ignv=k(x,r)
v=k(x,r) for
for clos
closed
edloop
loop
performance.
performance.
2. S olve m(x,u) ≈ Bu=v for u.
3
Controller overview
Prefilter
r
S tate
feedback
v
Control
alloc.
u
x
Why is modularity good?
• Not all control des igns methods handle
redundancy.
• S eparate control allocation s implifies
actuator cons traint handling.
• If an actuator fails , only control
reallocation is needed.
4
Practical cons iderations
... while s olving Bu=v:
• u is cons trained in
pos ition and in rate.
u≤u≤ u
• Minimum-phas e
res pons e.
• Want to minimize
– drag
• T he actuators have
– radar s ignature
limited bandwidth.
– s tructural load
• Actuators s hould not
• We are in a hurry!
counteract
(50-100 Hz)
eachother.
S olutions
Bu = v
u≤u≤ u
• Optimization bas ed approaches
• Direct control allocation
• Dais y chaining
5
Dais y chaining
1. Us e elevators
2. Us e T VC for
until they s aturate.
additional control.
Direct control allocation
v=Bu
v
u
6
Optimization bas ed CA
min f(u)
u
Bu = v
u≤u≤ u
• How do we choos e f(u)?
• Can we s olve the problem in real time?
Ps eudo-invers e
T he optimal s olution to
min u 2
u
Bu = v
is given by u = B T (BB T ) v = B† v
−1
E xtens ion:
min W(u − up )
u
2
7
Half-time s ummary
S o far, s tatic CA: u(t) = h(v(t))
S ame relative control dis tribution
regardles s of s ituation:
•maneuvering (trans ient)
•trimmed flight (s teady s tate)
Dynamic control allocation
• E xplicit filtering:
v
LP
u1
HP
u2
How can we impos e
Bu = v
u≤u≤ u
?
Incorporate
Incorporatefiltering
filteringinto
intoan
an
optimization
optimization framework.
framework.
8
Main idea
min W1(u(t) − us (t)) 2 + W2 (u(t) − u(t − 1)) 2
2
2
u(t)
= W(u(t) − u0 (t)) 2 + K
2
Bu = v
u≤u≤ u
• S tability?
• Control dis tribution?
T he non-s aturated cas e
min W1(u(t) − us (t)) 2 + W2 (u(t) − u(t − 1)) 2
2
2
u(t)
Bu = v
is s olved by
u(t) = Eus (t) + Fu(t − 1) + Gv(t)
9
Is v→u s table?
T hm: If W1 is non-s ingular then all
eigenvalues of F s atis fy
0 ≤ λ (F ) < 1
• As ymptotically s table.
• Not os cillatory.
S teady s tate dis tribution?
T hm: If us
s atis fies
then
Bus = v
lim u(t) = us
t→∞
us can be computed from
min W(us − up )
us
Bus = v
10
Des ign example
• Mach 0.5, 1000 m
• Pitching only
– canards
– elevons
– T VC
• T rimmed flight: elevons only
• Canards : high frequencies
• T VC: midrange frequencies
Parameters
• Dynamics : v = Bu = [8 .0 − 20 .2 − 0 .87 ] u
• Des ign variables :
0



us = − 1/ 20 .2 v W1 =


0
1 0 0 
1 0 0
0 1 0  W = 0 10 0 
2




0 0 0 .1
0 0 2 
11
F requency dis tribution
100
Control effort distribution
—Canards
10-1
—E levons
—T VC
10-2
10-3
10-4
10-2
10-1
100
101
102
Frequency (rad/sec)
Aircraft res pons e
25
5
q
r
20
u [deg]
q [deg/s]
15
10
0
5
—Canards
—E levons
0
—T VC
-5
0
2
4
time [s]
6
8
-5
0
2
4
time [s]
6
8
12
Computing the s olution
min W(u − u0 ) 2
u
u ∈ argmin
Bu W
= av(Bu − v) 2
u≤u≤ u
Can this problem be s olved in real-time?
Not according to the litterature.
Problem s pecific info
• S imple inequality bounds .
• F rom t-1:
–u
– active cons traints
• Convergence in one s ample not
neces s ary.
TTrim
rimexis
existing
ting
methods
methods!!
13
S ummary
• Dynamic control allocation new concept.
a
• Need for efficient s olvers .
• New field ⇒ lot’s to do!
14