H1, 1, 2

1. , 250061

E-mail: cuipeng@sdu.edu.cn

2. , 250101

: , ,

, α, H . (LMI) ,

, α H ,

H . .

: H , , ,

Robust H∞ Control for Uncertain Stochastic System with Spectral Restraint

Peng CUI1, Chenghui ZHANG1, Mei ZHANG2

1. School of Control Science and Engineering, Shandong University, Jinan, P.R.China 250061

E-mail: cuipeng@sdu.edu.cn

2. School of Information and Electrical Engineering, Shandong Jianzhu University, Jinan, P.R.China 250101

Abstract: A design procedure for state feedback controllers is derived for uncertain stochastic systems. The controller

ensures that the closed loop system is asymptotical mean square stable with stabilizing degree α, and H∞ bound is less

than some given scalars. Based on linear matrix inequalities (LMI) method and convex optimization algorithm, this paper

gets one sufficient condition, which guarantees the closed-loop is with stabilizing degree α and the H∞ bound is least.

Furthermore, the robust H∞ optimization controller design and the least H∞ bound are presented via solving one convex

optimization problem with LMI constraints.

Key Words: H∞ control, spectrum, convex optimization, LMI

1

,

, [1]-[9], W. Zhang

[6] , ,

,

, [7] W. Zhang

,

, ,

. [8]

.

H∞ . LMI ,

.

(61174141),

(JQ201219)

(BS2011DX019, BS2011SF009)

H∞ .

, , H∞. ,

.

P > 0 P ; AT : A ;

‖z‖2L2

�= E

∫∞0

zT (t)z(t)dt; ϕn : n × n ,

ϕ− : .

2

( t )

⎧⎪⎪⎪⎨⎪⎪⎪⎩

dx = [(A+�A)x+ (B +�B)u+ (B1 +�B1)v]dt

+[(C +�C)x+ (D +�D)u]dw�= (Ax+ Bu+ B1v)dt+ (Cx+ Du)dw

z(t) = C1x+D1u(1)

, x(t) ∈ Rn , u(t) ∈ Rm

, v(t) ∈ Rp , v(t) ∈L2[0,∞], z(t) ∈ Rq , w(t)

2568978-1-4673-5534-6/13/$31.00 c©2013 IEEE

Wiener ; A,B,B1, C,D,C1, D1 ,

�A,�B,�B1,�C,�D ,

:

[ �A �B �B1 �C �D]

= MF (t)[N1 N2 N3 N4 N5

] (2)

M,Ni(i = 1, · · · , 5) ,

F (t) ∈ Rl1×l2 FT (t)F (t) ≤ I , I

.

u(t) = Kx(t) (3)

(1)-(3) .

(a) v(t) ≡ 0 , (1)-(3)

, α, α > 0 .

(b) H∞ γ(γ > 0), J =

‖z(t)‖2L2− γ2‖v(t)‖2L2

< 0

1. v(t) ≡ 0 , (1)-(3)

, x0,

dx = (A+ BK)x(t)dt+ (C + DK)x(t)dw (4)

limt→+∞E[xT (t)x(t)] = 0.

1.([5]) (4) ,

X > 0

(A+BK)TX+X(A+BK)+(C+DK)TX(C+DK) < 0

(A+BK)X+X(A+BK)T+(C+DK)X(C+DK)T < 0

2.([6]) (4) Lk : ϕn →ϕn

Lk : Z ∈ ϕn → (A+ BK)Z + Z(A+ BK)T

+(C + DK)Z(C + DK)T

Lk σ(Lk) = {λ ∈ ϕ : Lk(Z) =

λZ,Z ∈ ϕn, Z = 0}.

2.([6]) (4) ,

Lk .

3. (4) ,

α > 0,

σ(Lk) ⊂ ϕ−α := {λ : Reλ < −α}

3.([7]) (4) ,

α, X :

(A+ BK + α2 I)

TX +X(A+ BK + α2 I)

+(C + DK)TX(C + DK) < 0

(A+ BK + α2 I)X +X(A+ BK + α

2 I)T

+(C + DK)X(C + DK)T < 0

. ([7]), ,

.

(4) , α,

σ(Lαk ) ⊂ ϕ−, Lα

k

dx = (A+ BK +α

2I)x(t)dt+ (C + DK)x(t)dw

, 1 . .

4. (Schur ) M = MT , R =

RT > 0, N , :

1). M +NR−1NT < 0

2).

[M N

NT −R

]< 0.

5. Y1, H1 E1, Y1 ,

FT (t)F (t) ≤ I ,

Y1 +H1F (t)E1 + ET1 F

T (t)HT1 < 0

ε > 0,

Y1 + εH1HT1 + ε−1ET

1 E1 < 0

3

(4) ,

, (1)

α ,

(3) .

1. v(t) ≡ 0 ,

X , Y , ε,

⎡⎢⎢⎣

Γ1 (CX +DY )T

CX +DY −X + εMMT

N1X +N2Y 0

N4X +N5Y 0

(N1X +N2Y )T (N4X +N5Y )T

0 0

−εI 0

0 −εI

⎤⎥⎥⎦ < 0

(5)

, Γ1 = AX+BY +XAT +Y TBT +αX+εMMT .

(1) (3),

, α, u(t) =

Y X−1x(t).

. 3 , (3),

(4) , α

X > 0,[Γ2 (C + DK)T

C + DK −X−1

]< 0 (6)

, Γ2 = (A+ BK+ α2 I)

TX+X(A+ BK+ α2 I).

diag{X−1, I}, diag{X−1, I},

Y = KX−1, X = X−1, , X > 0,

K (6) , X > 0,

Y :[Γ3 (CX + DY )T

CX + DY −X

]< 0 (7)

2013 25th Chinese Control and Decision Conference (CCDC) 2569

, Γ3 = AX + BY + XAT + Y T BT + αX .

, 5

, :

Σ =

[Γ4 (CX +DY )T

CX +DY −X

](8)

, Γ4 = AX +BY + XAT + Y TBT + αX .

(7) :

Σ+

[M

M

] [F (t)

F (t)

] [N1X +N2Y 0

N4X +N5Y 0

]

+

([M

M

] [F (t)

F (t)

] [N1X +N2Y 0

N4X +N5Y 0

])T

< 0(9)

5 , , ε > 0,

Σ+ ε

[M

M

] [MT

MT

]

+ε−1

[(N1X +N2Y )T (N4X +N5Y )T

0 0

][

N1X +N2Y 0

N4X +N5Y 0

]< 0

Schur , (5) . .

1 (3),

(1), α

, X,Y ε

,

Matlab . (1)

H∞ .

2. (3)

, α, H∞ γ

X > 0, Y ,

⎡⎢⎢⎣

Γ3 B1 (CX + DY )T (C1X +D1Y )T

� −γ2I 0 0

� � −X 0

� � � −I

⎤⎥⎥⎦ < 0

(10)

H∞ u(t) = Y X−1x(t).

. 1 , (10) , 1

(7) , 1

α. H∞γ .

, Lyapunov-Krasovskii : V (x(t)) =

xT (t)Xx(t), x0 = 0 , ∀T > 0, :

J = ‖z(t)‖2L2− γ2‖v(t)‖2L2

= E∫ T

0zT (t)z(t)dt− γ2E

∫ T

0vT (t)v(t)dt

= E∫ T

0[xT (t)(C1 +D1K)T (C1 +D1K)x

−γ2vT (t)v(t)]dt

= E∫ T

0[xT (t)(C1 +D1K)T (C1 +D1K)x

−γ2vT (t)v(t) + dV (x(t))]dt− EV (x(T ))

≤ E∫ T

0[xT (t)(C1 +D1K)T (C1 +D1K)x

−γ2vT (t)v(t) + dV (x(t))]dt

= E∫ T

0{xT (t)[(C1 +D1K)T (C1 +D1K)

+X(A+ BK) + (A+ BK)TX

+(C + DK)TX(C + DK)]x+ 2xTXB1v

−γ2vT (t)v(t)}dt= E

∫ T

0

[xT (t) vT (t)

] [ Ω XB1

BT1 X −γ2I

] [x

v

]dt

, Ω = (C1 +D1K)T (C1 +D1K) +X(A+ BK) +

(A+ BK)TX+(C+DK)TX(C+DK). J < 0,

[Ω XB1

BT1 X −γ2I

]< 0 (11)

Schur

⎡⎢⎢⎣

Γ5 XB1 (C + DK)T (C1 +D1K)T

� −γ2I 0 0

� � −X−1 0

� � � −I

⎤⎥⎥⎦ < 0

(12)

Γ5 = X(A+BK)+(A+BK)TX . ,

diag{X−1, I, I, I}, X = X−1, Y = KX−1,

(12)

⎡⎢⎢⎣

Γ6 B1 (CX + DY )T (C1X +D1Y )T

� −γ2I 0 0

� � −X 0

� � � −I

⎤⎥⎥⎦ < 0

(13)

Γ6 = AX + XAT + BY + Y T BT .

(10) , (13) . .

2 (1) H∞(a)(b) ,

, , 1 ,

(10) ,

.

3. (3),

, α, H∞ γ

X > 0, Y , ε > 0

2570 2013 25th Chinese Control and Decision Conference (CCDC)

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Ψ B1 (CX +DY )T (C1X +D1Y )T

−γ2I 0 0

−X + εMMT 0

−I

(N1X +N2Y )T (N4X +N5Y )T

NT3 0

0 0

0 0

−εI 0

−εI

⎤⎥⎥⎥⎥⎥⎥⎥⎦< 0

(14)

Ψ = AX + XAT +BY + Y TBT +αX + εMMT ,

H∞ u(t) = Y X−1x(t).

1 , .

3 γ2 = γ, (14)

B(X, Y, ε, γ) < 0 (15)

(15) X, Y, ε, γ ,

Matlab ,

(1) α H∞.

4. (3),

, α, H∞X > 0, Y, ε > 0, γ > 0,

minX>0,Y,ε>0,γ>0

γ

subject to LMI(15)(16)

H∞ u(t) = Y X−1x(t), H∞√γ.

. (1) H∞ ,

A =

[ −4 1

−1 −3

], B =

[1

1

], B1 =

[1

1

],

C =

[1 2

1 1

], D =

[2

1

], C1 =

[1 2

],

D1 = 1, M =

[0.1 0 0.1

0 1 0

], N1 =

⎡⎣ 1 0.1

0 0

0 0.1

⎤⎦ ,

N2 =

⎡⎣ 1

0

0.2

⎤⎦ , N3 =

⎡⎣ 0

1

0

⎤⎦ , N4 =

⎡⎣ 1 0.1

0 0

0 0

⎤⎦ ,

N5 =

⎡⎣ 0.1

0

0.1

⎤⎦ ,

α = 0.1( 1) , H∞u(t) =

[ −1.0000 −2.0000]x(t), H∞

9.0575e− 005.

α = 6( 2) , H∞u(t) =

[ −0.5345 −0.8928]x(t), H∞

4.4572.

, ,

, H∞ .

0 2 4 6 8 10 12 14 16 18−10

−5

0

5

10

15

20

25

time [s]

output of x1

output of x2

α = 0.1

0 2 4 6 8 10 12 14 16 18−10

−5

0

5

10

15

20

25

time [s]

output of x1

output of x2

α = 6

4

H∞ , ,

, α

. α H∞, LMI .

, H∞H∞ ,

.

2013 25th Chinese Control and Decision Conference (CCDC) 2571

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2572 2013 25th Chinese Control and Decision Conference (CCDC)