Multiobjective H2/H∞/impulse-to-peak synthesis: Application to the control of an aerospace launcher D. Arzelier, D. Peaucelle LAAS-CNRS, 7 Avenue du Colonel Roche, 31 077 Toulouse, Cedex 4, France emails:
[email protected],
[email protected] October 3, 2003 Keywords: Impulse-to-peak performance, Multiobjective control, LMI optimization, Aerospace launcher.
1
Problem formulation
Let the LTI discrete plant Σ be given by its state-space minimal realization: xk+1 xk A B1 B zk = C1 D11 D wk yk uk C D21 0
(1)
where x ∈ Rn is the state vector, w ∈ Rmw is the disturbance vector, u ∈ Rmu is the input vector, z ∈ Rrz is the controlled output vector and y ∈ Rry is the measured output vector. The z and w vectors are partitioned woo wi2p w2
zoo z i2p z2
Σ
y
u K
Figure 1: Standard model for multiobjective control as indicated in figure 1.
z∞ w∞ z = zi2p w = wi2p z2 w2
The associated matrices are therefore consequently partitioned. D∞ B1 = B∞ Bi2p B2 D11 = Di2p∞ D2∞
D21 =
C∞ C1 = Ci2p C2
Dy∞
Dyi2p
Dy2
(2)
D∞i2p Di2p D2i2p
D∞2 Di2p2 D2
D∞u D = Di2pu D2u
(3)
The controller K is given by its minimal state-space realization: ηk+1 = AK ηk + BK yk uk = CK ηk + DK yk The closed-loop system Σ K is given by its state-space matrices: A + BDK C BCK B1 + BDK D21 Acl = Bcl = BK C AK BK D21 Ccl =
C1 + DDK C
DCK
1
Dcl = [D11 + DDK D21 ]
(4)
(5)
Problem 1 (multiobjective H2 /H∞ /i2p control problem) Find a controller K in the set of internally stabilizing controllers K such that: min
αi2p γi2p + α2 γ2
K∈K
under ||Σ∞ K||2∞ ≤ γ∞ ||Σi2p K||2i2p ≤ γi2p ||Σ2 K||22 ≤ γ2
2 2.1
(6)
LMI formulation of the impulse-to-peak synthesis problem The impulse-to-peak performance
Let the closed-loop discrete-time plant be given by its minimal realization: xk+1 = Acl xk + Bcl wk zk = Ccl xk + Dcl wk
x(0) = 0
(7)
Problem 2 (worst-case i2pk synthesis) Find a controller K in the set of internally stabilizing controllers K. min
K∈K
sup ||z||L∞ ||w ¯ i || ≤ 1 1 ≤ i ≤ mw
(8)
where z is the performance output response to the impulsive input wki = w ¯ i δk , wkl = 0 (l = i) and δk is the unit pulse. Theorem 1 If there exists a matrix P ∈ S∗+ satisfying: Acl P Acl − P < 0 Bcl Bcl
