discrete time mixed h2 / h control · introduction continuous time mixed h2/h∞ control problem:...
TRANSCRIPT
Discrete time mixed H2/H∞ control
Yang Xu
Department of Automatic Control
Lund University
May 25, 2016
Introduction
Continuous time mixed H2/H∞ control problem:
◮ Zhou, Kemin, et al. ”Mixed H2 and H∞ performanceobjectives. I. Robust performance analysis.” AutomaticControl, IEEE Transactions on 39.8 (1994): 1564-1574.
◮ Doyle, John, et al. ”Mixed H2 and H∞ performanceobjectives. II. Optimal control.” Automatic Control, IEEETransactions on 39.8 (1994): 1575-1587.
◮ ...
Discrete time mixed H2/H∞ control problem:
◮ Muradore, Riccardo, and Giorgio Picci. ”Mixed H2/H∞
control: the discrete-time case.” Systems & control letters54.1 (2005): 1-13.
Some notations
◮ w is deterministic disturbance, w0 is stochastic disturbance
◮ ∂D∆= {z : |z |= 1}
◮ For stochastic processes x and y , x ⊥ y meansE (x − x)(y − y)T = 0, where x = Ex , y = Ey .
Mixed H2/H∞ control problemLet G (z) be a discrete-time system with realization
σx = Ax + B0w0 + B1w + B2u
z0 = C0x + D00w0 + D01w + D02u
z = C1x + D10w0 + D11w + D12u
y = C2x + D20w0 + D21w
which satisfies the following assumptions:A1. (A,B2) is stabilizable and (C2,A) is detectable,A2. D02 has full column rank and D20 has full row rank,
A3.
[
A− zI B2
C0 D02
]
has full column rank ∀z ∈ ∂D,
A4.
[
A− zI B0
C2 D20
]
has full row rank ∀z ∈ ∂D,
A5. w ∈ P is a bounded power signal and w0 ∼ GW (0, I ) is anormalized Gaussian white noise. Moreover, w0(k) ⊥ w(j), ∀k ≥ j ,A6. the initial condition x(0) ∼ G (x0,R0) is independent of w0,∀k ≥ 0.
Mixed H2/H∞ controlThe mixed H2/H∞ problem: find u∗ and w∗ such that
J0(w∗, u∗) ≥ J0(w , u∗), J2(w
∗, u∗) ≤ J2(w∗, u)
is solvable, if and only if, the coupled algebraic Riccati equations
(A+ B2F2)TX∞
(
I − γ−2B1BT
1 X∞
)
−1
(A+ B2F2)
− X∞ + (C1 + D12F2)T (C1 + D12F2) = 0
(
AC +(
B1 − B2R−102 D02D01
)
H∞
)TX2
(
AC
+(
B1 − B2R−102 D02D01
)
H∞
)
− X2
+ (C0 + D21H∞)T R02 (C0 + D21H∞) = 0
(
AF +(
B1 − B1DT
20R−120 D21
)
H∞
)
Y2
(
I
+ CT
2 R−120 C2Y2
)
−1 (
AF +(
B1 − B1DT
20R−120 D21
)
H∞
)T
− Y2 + B0R20BT
0 = 0
Mixed H2/H∞ controlIn the three Ricaati equations,
F2 =−(
DT
02D02 + BT
2 X2B2
)
−1
(
BT
2 X2 (A+ B1H∞) + DT
02 (C0 + D01H∞))
L2 =−(
(A+ B1H∞)Y2 (C2 + D21H∞)T + B0DT
20
)
(
D20DT
20 + (C2 + D21H∞)Y2 (C2 +D21H∞)T)
−1
AC =A− B2R−102 DT
02C0
AF =A− B0DT
02R−120 C2
R02 =DT
02D02
R02 =I −D02R−102 D02T
R20 =D20DT
20
R20 =I −DT
20R−120 D20
Mixed H2/H∞ control
The three Ricaati equations have stabilizaing solutions X∞ ≥ 0,X2 ≥ 0 and Y2 ≥ 0.The optimal controller u∗ = K (z)y has a realization given by
σxK = (A+ B1H∞ + B2F2 +2 C2 + L2D21H∞) xK − L2y
u∗ = F2xK
Moreover the optimal value of the performance functionalJ∗2 (w
∗, u∗,w0) is
J∗2 = tr(
BT
0 X2B0 + DT
00D00
)
+tr((
BT
2 X2B2 + DT
02D02
)
F2Y2FT
2
)
A numerical example
A discrete-time system
σx = Ax + B0w0 + B1w + B2u = 1.05x + w0 + w − 0.55u
z0 = C0x + D00w0 + D01w + D02u = 0.94x + 1.36u
z = C1x + D10w0 + D11w + D12u = −0.54x + 0.57u
y = C2x + D20w0 + D21w = −0.59x + 0.52w0 + w
Use MATLAB to solve nonlinear equations. The optimal mixedH2/H∞ controller is
σxK = (A+ B1H∞ + B2F2 +2 C2 + L2D21H∞) xK − L2y
= 0.9376xK + 0.1406y
u∗ = F2xK = 0.3457xK
The optimal H2 cost is J∗2 = 3.42.
Future work
◮ Compare the mixed H2/H∞ performance with pure H2 andH∞ controllers
◮ LMI, Nash game (MIMO)
◮ Minimize H∞ cost with constraint of H2 cost.