Delay-Dependent Filtering for Discrete-Time ... - Semantic Scholar

Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

ThB07.2

Delay-Dependent Filtering for Discrete-Time Systems with Finite Frequency Small Gain Specifications Xiao-Ni Zhang and Guang-Hong Yang Abstract— This paper is concerned with the delay-dependent filtering problem for linear discrete-time multi-delay systems with small gain conditions in finite frequency ranges. A new multiplier method is developed to convert the resulting nonconvex filtering synthesis conditions to the ones based on linear matrix inequalities (LMIs). Thus, sufficient conditions for the existence of feasible filters are given in terms of solutions to a set of LMIs. Finally, the procedures and advantages of the proposed approach in comparison with existing ones in the entire frequency range are illustrated via a numerical example.

I. I NTRODUCTION The topic of H∞ filtering for time-delay systems is a research subject of great practical and theoretical significance, which has received considerable attention in the past decades. It is known that the delay-dependent conditions are generally less conservative than the delay-independent ones, especially when delays are small. Therefore, recently, great interest has been devoted to delay-dependent H∞ filtering problems in various settings, such as continuous-time systems [1]-[7], and discrete-time systems [8]-[11]. To mention a few for the discrete-time setting, [9] presented a criterion for the existence of an admissible H∞ filter for uncertain discretetime systems with a time-varying state-delay by introducing a new finite sum inequality. [10] revisited the problem and derived a design procedure based on a new linearization technique incorporating with a bounding technique. It should be noticed that the above-mentioned results were given mainly based on time-domain methods from the viewpoint of the entire frequency domain. However, various engineering design problems can be formalized in terms of specifications expressed by frequency domain inequalities (FDIs). In particular, each design specification is often given not for the entire frequency range but rather for a certain frequency range of relevance. Recently, many results for sysThis work was supported in part by the Funds for Creative Research Groups of China (No. 60821063), the State Key Program of National Natural Science of China (Grant No. 60534010), National 973 Program of China (Grant No. 2009CB320604), the Funds of National Science of China (Grant No. 60674021), the 111 Project (B08015) and the Funds of PhD program of MOE, China (Grant No. 20060145019). Xiao-Ni Zhang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China. She is also with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China.

[email protected]

Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China. He is also with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China. Corresponding author: [email protected]

978-1-4244-3872-3/09/$25.00 ©2009 IEEE

tems with finite frequency specifications have been reported in the literature; see [12]-[18] and the references therein. The objective of this paper is to design delay-dependent filters for discrete-time multi-delay such that the filtering error systems are asymptotic stable and satisfy small gain conditions in finite frequency ranges. To our knowledge, this problem has not been addressed in the literature. Here, we use the idea in [19] and Finsler’s Lemma to introduce additional slack multipliers into our analysis conditions, such that a kind of decoupling for the system matrices is exhibited. Then, a new multiplier method is developed to render the multi-objective problem convex. The main contribution of this paper is that sufficient conditions for the existence of feasible filters are given in terms of solutions to a set of LMIs. The paper is organized as follows. Section II gives the problem statement and some preliminaries, and Section III presents the filtering analysis and design. In Section IV, a numerical example is proposed to illustrate the design procedure and their effectiveness. Some concluding remarks are shown in Section V. The following notation is used throughout this paper. For a matrix A, its complex conjugate transpose is denoted by A∗ . The Hermitian part of a square matrix A is denoted by He(A) := A + A∗ . The symbol Hn stands for the set of n × n Hermitian matrices. I denotes the identity matrix with an appropriate dimension. For matrices Φ and P, Φ ⊗ P means the Kronecker product. σmax (G) represents the maximum singular value of G. For matrices G ∈ Cn×m and Π ∈ Hn+m , a function σ : Cn×m × Hn+m → Hm is defined by · ¸∗ · ¸ G G σ (G, Π) := Π . Im Im II. P ROBLEM STATEMENT AND PRELIMINARIES Consider the following asymptotically stable discrete-time multi-delay system described by r

x(k + 1) = Ax(k) + ∑ Ai x(k − di ) + Bϖ (k) i=1 r

z(k) = C1 x(k) + ∑ C1i x(k − di ) + D1 ϖ (k) i=1 r

y(k) = C2 x(k) + ∑ C2i x(k − di ) + D2 ϖ (k)

(1)

i=1

where x(k) ∈ Rnx is the state vector, ϖ (k) ∈ Rnϖ is the noise input, z(k) ∈ Rnz is the signal to be estimated and y(k) ∈ Rny is the measured output, respectively. The positive integers di (i = 1, 2, . . . , r) are the constant delays of the system. A, Ai ,

4420

ThB07.2 B, C1 , C2 , C1i , C2i , D1 and D2 are known constant matrices of appropriate dimensions. Here, we will design a filter of the following form: x(k ˆ + 1) = AF x(k) ˆ + BF y(k) zˆ(k) = CF x(k) ˆ + DF y(k)

(2)

where AF , BF , CF and DF are to·be determined. Defining ¸ x(k) the augmented state vector x(k) ¯ = and the estimation x(k) ˆ error z¯(k) = z(k) − zˆ(k), we can obtain the following filtering error system: r

x(k ¯ + 1) = Ax(k) ¯ + ∑ Ai x(k ¯ − di ) + Bϖ (k) i=1 r

z¯(k) = Cx(k) ¯ + ∑ Ci x(k ¯ − di ) + Dϖ (k)

(3)

i=1

where

¸ · ¸ A 0 Ai 0 , Ai := , BF C2 AF BF C2i 0 · ¸ £ ¤ B B := , C := C1 − DF C2 −CF , BF D2 £ ¤ Ci := C1i − DF C2i 0 , D := D1 − DF D2 .

III. F ILTERING ANALYSIS AND DESIGN WITH SMALL GAIN CONDITIONS IN FINITE FREQUENCY RANGES

In this section, we present sufficient conditions for the existence of a desired filter of form (2). First, the following lemma provides delay-dependent analysis conditions for the filtering error system (3) capturing · the specification ¸ · (6). ¸ 1 0 1 −1 Theorem 1: Let matrices Φ := , Ψ0 := 0 −1 −1 1 and Π ∈ Hnϖ +nz be given and define Λ by (7). Then, the filtering error system (3) is asymptotically stable and the specification (6) is captured for any delays di satisfying 0 < di ≤ d i if there exist Ps = Ps∗ > 0, X1i = X1i∗ > 0 (i = 1, . . . , r), P = P∗ , Q = Q∗ > 0, Z ji = Z ∗ji > 0 ( j = 1, 2) and X2i = X2i∗ such that · ¸∗ r A A1 . . . Ar (Φ ⊗ Ps + Ψ0 ⊗ ∑ d i Z1i ) I 0 ... 0 i=1

·

· A × I

A :=

· A A1 I 0 (4) ×

The transfer function matrix from ϖ to z¯ is denoted by r

r

G(λ ) = (C + ∑ λ −di Ci )(λ I − A − ∑ λ −di Ai )−1 B + D. i=1

i=1

×Π

(5)

Our aim in this paper is to design a filter (2) such that • The filtering error system (3) is asymptotically stable when ϖ (k) ≡ 0; • The small gain condition σmax [G(λ )] ≤ γ in a certain finite frequency range¸ of relevance characterized by, · I 0 , given Π := 0 −γ 2 I

σ (G(λ ), Π) < 0,

∀λ ∈ Λ(Φ, Ψ)

. . . Ar ... 0

· C C1 0 0

. . . Cr ... 0

 r ∑i=1 X ji  0  X j :=  .  ..

0 −X j1 .. .

0 0  −1 r − ∑i=1 d i Z ji  −1  d 1 Z j1 Z j :=  ..   .

(6)

−1

d r Z jr

(7)

and Λ := Λ if Λ is bounded and Λ := Λ ∪ {∞} if unbounded. Remark 1: By virtue of [12], the frequency variable λ = e jω , for the discrete-time setting, can be characterized by · ¸ 1 0 σ (λ , Φ) = 0, Φ := . A certain frequency range of 0 −1 relevance can be specified by an additional choice of Ψ as follows:  · ¸ 0 1   {e jω : |ω | ≤ ωl }   1 −2 cos ωl   " #  ω1 +ω2  0 ej 2 Ψ := {e jω : ω1 ≤ ω ≤ ω2 } ω +ω ω2 −ω1 −j 12 2  e −2 cos  2  · ¸   0 −1   {e jω : |ω | ≥ ωh }.  −1 2 cos ωh (8)

(9)

¸∗ r B (Φ ⊗ P + Ψ ⊗ Q + Ψ0 ⊗ ∑ d i Z2i ) 0 i=1 ¸ · B C C1 + 0 0 0

0 0 .. .

...

−X jr

.. . 0

¸ 0 0, X1i > 0(i = 1, . . . , r), P, Q > 0, Z ji > 0( j = 1, 2) and X2i , and matrices Y j , V j , Gk (k = 1, . . . , 5), U, Aˆ F , Bˆ F , CˆF and Dˆ F such that   ∆1 ∆2 ∆3 ∆4  ? ∆5 ∆6 ∆7    < 0, (12) ? ? ∆8 0  ? ? ? ∆9

4421

ThB07.2  Γ1 Γ3 Γ4

Γ2 0 Γ5

  0 Ξ1 0  + He Ξ5 Γ7 Ξ9

0 0 Γ6

r

−1

Ξ2 Ξ6 Ξ10

Ξ3 Ξ7 Ξ11

 Ξ4 Ξ8  < 0, UB (13)



r

−1

Moreover, an admissible filter is given by −1 ˆ −1 ˆ −1 ˆ ˆ AF = G−1 2 AF , BF = G2 BF ,CF = G5 CF , DF = G5 DF . (15)

Proof. By Schur complement lemma, (9) is equivalent to  −1 −1 −Ps − ∑ri=1 d i Z1i + ∑ri=1 X1i d 1 Z11 ...  −1  ? −d 1 Z11 − X11 . . .   .. .. ..  . . .   ? ? . . .   ? ? ... ? ? ... −1

d r Z1r 0 .. . −1

−d r Z1r − X1r ? ?

i=1

 ∗  Ψ12 Q − ∑ri=1 d i Z2i 0  0 0    . ..  , .. Γ4 :=  .    0 0 0 0   −1 r −P + Ψ22 Q + ∑i=1 (d i Z2i + X2i − d i Z2i )   −1   d 1 Z21   .. , Γ5 :=    .   −1   d r Z2r 0   −1 −1 d 1 Z21 ... d r Z2r   −1 −X21 − d 1 Z21 . . .  0   . . .  , Γ6 :=  .. .. ..    −1  0 . . . −X2r − d Z2r  0

...

0



  · ¸   −G1 −G2 0   Γ7 :=  ,  , Ξ1 := −G3 −G2 0    0  −γ 2 I · ¸ G A + Bˆ F C2 Aˆ F , Ξ2 := 1 G3 A + Bˆ F C2 Aˆ F · ¸ G A + Bˆ C 0 . . . G1 Ar + Bˆ F C2r 0 Ξ3 := 1 1 ˆ F 21 , G3 A1 + BF C21 0 . . . G3 Ar + Bˆ F C2r 0 ¸ · £ ¤ G B + Bˆ F D2 Ξ4 := 1 , Ξ5 := −G4 0 −G5 , G3 B + Bˆ F D2 £ ¤ Ξ6 := G4 A + G5C1 − Dˆ F C2 −CˆF , ¤ £ Ξ7 := Ξ71 . . . Ξ7r , £ ¤ Ξ7i := G4 Ai + G5C1i − Dˆ F C2i 0 , £ ¤ £ ¤ Ξ8 := G4 B + G5 D1 − Dˆ F D2 , Ξ9 := −U 0 0 , £ ¤ £ ¤ Ξ10 := UA 0 , Ξ11 := UA1 0 . . . UAr 0 . (14)

∆1 := −Ps − ∑ d i Z1i + ∑ X1i , i=1 i=1 h i −1 −1 ∆2 := d 1 Z11 . . . d r Z1r , · ∗ ∗ ¸ A Y1 +C2∗ Bˆ ∗F −Y1∗ A∗V1∗ +C2∗ Bˆ ∗F −V1∗ ∆3 := , Aˆ ∗F − G∗2 Aˆ ∗F − G∗2 · ∗ ∗ ¸ A Y2 +C2∗ Bˆ ∗F A∗V2∗ +C2∗ Bˆ ∗F ∆4 := , Aˆ ∗F Aˆ ∗F   −1 −X11 − d 1 Z11 . . . 0   .. .. .. , ∆5 :=  . . .   ? . . . −X1r − d r Z1r  ∗ ∗  ∗ ∗ ∗ B ˆ ∗F A1Y1 +C21 Bˆ F A∗1V1∗ +C21   0 0     . . .. .. ∆6 :=  ,    A∗Y ∗ +C∗ Bˆ ∗ A∗V ∗ +C∗ Bˆ ∗  r 1 r 1 2r F 2r F 0 0   ∗ ∗ ∗ ∗ ∗ ∗ ∗ B ˆ ˆ ∗F A1Y2 +C21 BF A1V2 +C21   0 0     . . .. .. ∆7 :=  ,   ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗  A Y +C Bˆ Ar V2 +C2r Bˆ F  r 2 2r F 0 0 · ¸ r ∗ −He(Y1 ) −G2 −V1 ∆8 := + ∑ d i Z1i , ? −He(G2 ) i=1 · ¸ −He(Y2 ) −G2 −V2∗ + Ps , ∆9 := ? −He(G2 ) ¤ £ Γ1 := P + ∑ri=1 d i Z2i 0 , r £ ¤ Γ2 := Ψ12 Q − ∑ d i Z2i , Γ3 := 0 I ,

0 0 .. .

∑ri=1 d i (A − I)∗ Z1i ∑ri=1 d i A∗1 Z1i .. . ∑ri=1 d i A∗r Z1i − ∑ri=1 d i Z1i ?

A∗ Ps A∗1 Ps .. .



     < 0. (16)  A∗r Ps   0  −Ps

From Lemma 4 in Appendix III, it is equivalent to  −1 −1 −Ps − ∑ri=1 d i Z1i + ∑ri=1 X1i d 1 Z11  −1  ? −d 1 Z11 − X11   .. ..  . .   ? ?   ? ? ? ? −1

d r Z1r 0 .. . −1

−d r Z1r − X1r ? ?

r

(A − I)∗W1∗ A∗1W1∗ .. .

A∗W2∗ A∗1W2∗ .. .

A∗r W1∗ r ∑i=1 d i Z1i − He(W1 )

A∗r W2∗ 0 Ps − He(W2 )

?

... ... .. . ... ... ...       < 0.     (17)

4422

ThB07.2 Moreover, (10) is equivalent to   · ¸∗ Γ1 Γ2 0 0 · ¸ J  J Γ3 0 0 0 < 0. (18) I I Γ4 Γ5 Γ6 Γ7 · ¸ A A1 . . . Ar B where J := and Γ j ( j = 1, . . . , 7) are C C1 . . . Cr D defined by (14). By virtue of Lemma 5 in Appendix III, (18) is equivalent to   Γ1 Γ2 0 0 ¤¢ ¡ £ Γ3 0 0 0  + He W3 −I J < 0. (19) Γ4 Γ5 Γ6 Γ7   G1 G2 0 G3 G2 0  [(2r+2)×nx +nϖ ]×nx . In  Set W3 :=  G4 0 G5  with U ∈ C U 0 0 order to solve the multi-objective problem subject · ¸ · to (17) ¸ Y1 W12 Y2 W12 and (19), we define W1 = ˆ and W2 = ˆ . Y1 W22 Y2 W22 From Lemma 6 ·in Appendix ¸ III, W1 and · W2 can ¸ be recharacY2 G2 Y1 G2 , respectively. terized as W1 = and W2 = V1 G2 V2 G2 Defining the new variables Aˆ F := G2 AF ,

Bˆ F := G2 BF ,

CˆF := G5CF ,

Dˆ F := G5 DF , (20)

we can obtain (12) and (13) from (17) and (19), respectively. Thus, the proof is completed. Remark 3: Theorem 2 presents sufficient conditions for the existence of admissible filters such that the filtering error systems are asymptotically stable and satisfy the small gain conditions in finite frequency ranges. In order to solve the nonconvex multi-objective filtering synthesis problem, we set that W1 and W2 have the same elements in the second column and specify the structure of W3 such that a common variable can be used, which will lead to some conservatism. IV. E XAMPLE Consider a linear discrete-time single-delay system in the form of (1) with · ¸ · ¸ · ¸ 1 0 −0.1 −0.1 0 A= , A1 = , B= , 0 0.8 −0.2 −0.2 1 £ C1 = 1 £ ¤ C2 = 1 0 ,

¤ 2 ,

C11 = 0,

filter, the method obtained by Theorem 2 (Q=0) is less conservative than those in [9] and [10]. Furthermore, Theorem 2 which achieves the filter design with a small gain condition in the high frequency range produces much less conservative results than others. TABLE I C OMPARISON OF ACHIEVED

Methods [9] [10] Theorem 2 (Q=0) Theorem 2

d=2 6.0597 6.0597 5.9701 2.8639

d=3 8.7286 8.7286 8.1293 3.5280

γ

d=4 10.5120 10.5120 9.9049 3.5285

d=5 26.1197 26.1197 24.7137 3.5345

V. C ONCLUSION In this paper, the filter design problem for linear discretetime state-delayed systems with the small gain conditions in finite frequency ranges has been addressed. Sufficient conditions for the existence of feasible filters are given in terms of solutions to a set of LMIs. Finally, a numerical example is given to demonstrate the utility of the proposed design approach. A PPENDIX I P ROOF OF T HEOREM 1 Under the condition of the theorem, we first show the asymptotic stability of system (3). To this end, we consider system (3) with ϖ (k) = 0. Now, choose a Lyapunov functional candidate as r

V (x(k)) ¯ =x¯∗ (k)Ps x(k) ¯ +∑

k−1

∑

x¯∗ ( j)X1i x( ¯ j)

i=1 j=k−di

r

−1

+∑

k−1

∑ ∑

y¯∗ (m)Z1i y(m) ¯

(21)

i=1 j=−di m=k+ j

where y(k) ¯ := x(k ¯ + 1) − x(k). ¯ Defining ∆V := V (x(k ¯ + 1)) − V (x(k)), ¯ we have r

∆V =x¯∗ (k + 1)Ps x(k ¯ + 1) − x¯∗ (k)Ps x(k) ¯ + ∑ x¯∗ (k)X1i x(k) ¯ i=1

r

r

− ∑ x¯∗ (k − di )X1i x(k ¯ − di ) + ∑ di y¯∗ (k)Z1i y(k) ¯

D1 = 0,

£ ¤ C21 = 0.5 0 ,

d=1 4.2551 4.2551 4.2255 1.1984

MINIMUM

i=1 r k−1

∑

−∑

D2 = 1.

i=1

y¯∗ ( j)Z1i x(y). ¯

(22)

i=1 j=k−di

Our objective is to design a filter (2) such that the filtering error system (3) is asymptotically stable and satisfies π σmax [G(e jω )] < γ , |ω | ≥ . 5 · ¸ 0 Ψ12 Here, Ψ can be given by Ψ := = Ψ∗12 Ψ22 · ¸ 0 −1 . The results are compared in Table 1. −1 2 cos π5 It is clearly demonstrated that, for the case with a H∞

By virtue of Jensen integral inequality, it follows that

4423

k−1

−

∑

y¯∗ ( j)Z1i x(y) ¯

j=k−di

1 ≤− di =−

Ã

k−1

∑

j=k−di

!∗ y( ¯ j)

à Z1i

k−1

∑

! y( ¯ j)

j=k−di

¤∗ £ ¤ 1£ x(k) − x(k − di ) Z1i x(k) − x(k − di ) . di

(23)

ThB07.2 Therefore, it follows from (9) that for all 0 < di ≤ d i ∆V ≤ ς ∗ φ ς < 0

(24) £ ∗ ¤ ∗ where ς := x (t) x∗ (t − d1 ) . . . x∗ (t − dr ) and φ is defined as the left side of (9). This implies that the system (3) with ϖ (k) = 0 is asymptotically stable for any delays di satisfying 0 < di ≤ d i . Next, we shall establish the performance. The specification (6) can be denoted by the FDI · ¸∗ · ¸ · ¸· ¸ ˆ D ∗ ˆ D ζ ζ C C Π 0}. (38) (29) Lemma 2: Let matrices Φ, Ψ ∈ H2 and F ∈ (27) can be denoted by C4n×[(2r+2)×n+nϖ ] be given such that Λ in (6) represents ∗ ξ Θξ < 0, ∀ξ ∈ G1 . (30) curves. Then, the set M defined by (38) is admissible and rank-one separable. Furthermore, for ∀ξ ∈ G1 , we have Γλ F ξ = 0(λ ∈ Λ) where Remark 2: Lemmas 1 and 2 basically follow from [12]. ¤ ½ £ Lemma 3 (Further generalization of the strict S-procedure): I −λ I¤ (λ ∈ Λ) £ , (31) Let admissible sets Mi ⊂ Hq (i = 1, 2, . . . , p) and Ni ∈ Hq Γλ := 0 −I (λ = ∞) · ¸ (i = 1, 2, . . . , s) be given and define G by A A1 . . . Ar B F := . (32) I 0 ... 0 0 G := {ξ ∈ Cq : ξ 6= 0, ξ ∗ Mi ξ ≥ 0∀Mi ∈ Mi , ξ ∗ Ni ξ ≥ 0}. (39) Defining Then, if and only if Mi ⊂ Hq are rank-one separable, the G2 := {ξ ∈ C(2r+2)×n+nϖ : ξ 6= 0, Γλ F ξ = 0, λ ∈ Λ}, (33) following statements are equivalent. ∗ we can see that G1 ⊂ G2 . From Lemma 1 in Appendix II, i) ξ Θξ < 0 ∀ξ ∈ G. p the set G2 can be characterized by (37) with M in (38). Then ii)s There exist Mi ∈ Mi and τi ≥ 0 such that Θ + ∑i=1 Mi + ∑i=1 τi Ni < 0 for an arbitrary Θ ∈ Hq . define

G1 :=

(2r+2)×n+nϖ

ξ ∈C

nϖ

G := {ξ ∈ C(2r+2)×n+nϖ : ξ 6= 0, ξ ∗ M ξ ≥ 0∀M ∈ M, ξ ∗ N ξ ≥ 0} (34) where M is given by (38) and · ¸ r Z2 0 N := + F ∗ (Ψ0 ⊗ ∑ d i Zi )F 0 0 i=1

(35)

A PPENDIX III Lemmas 4-6 are presented for the proof of Theorem 2. Lemma 4 Consider the system described by (3). Then the following statements are equivalent: (i)There exist symmetric matrices Ps > 0, X1i > 0(i = 1, . . . , r) and Z1i > 0 such that (16) holds. (ii) There exist symmetric matrices Ps > 0, X1i > 0(i =

4424

ThB07.2 ¸ I 0 ∗ ˜ ∗ ˜ where T := −1 . Set Ps := T Ps T , X1i := T X1i T 0 W12W22 and Z˜ 1i := T Z1i T ∗ . Pre- and post-multiplying (17) by diag{T, T, . . . , T, T, T } and its complex conjugate transpose, respectively, (17) is equivalent to (40) by A˜ F := ∗ )−1W ∗ A (W −1 )∗W ∗ and B ∗ )−1W ∗ B . Thus, ˜ F := (W12 (W12 22 F 12 22 F 22 the proof is completed. ·

1, . . . , r) and Z1i > 0, and matrices W j ( j = 1, 2) such that (17) holds. Proof. (i)⇒(ii): Choose W1 = W1∗ = ∑ri=1 d i Z1i and W2 = W2∗ = Ps . (ii)⇒(i): Assume that (17) holds. Hence, He(W1 ) − ∑ri=1 d i Z1i > 0 and He(W2 ) − Ps > 0. It implies that W j ( j = 1, 2) are both nonsingular matrices. Furthermore, we have (∑ri=1 d i Z1i − W1 )(∑ri=1 d i Z1i )−1 (∑ri=1 d i Z1i − W1 )∗ ≥ 0 and (Ps − W2 )Ps−1 (Ps − W2 )∗ ≥ 0. Therefore, r and W1 (∑i=1 d i Z1i )−1W1∗ ≥ He(W1 ) − ∑ri=1 d i Z1i W2 Ps−1W2∗ ≥ He(W2 )−Ps . Then, replace ∑ri=1 d i Z1i −He(W1 ) and Ps − He(W2 ) with −W1 (∑ri=1 d i Z1i )−1W1∗ and −W2 Ps−1W2∗ in (17), respectively. Multiplying it by diag{I, I, . . . , I, ∑ri=1 d i Z1iW1−1 , PsW2−1 } on the left and its complex conjugate transpose on the right, respectively, it is equivalent to (16). Thus, the proof is completed. Lemma 5 (Finsler’s Lemma): Let x ∈ Rn , symmetric matrix Ξ ∈ Rn×n , and Γ ∈ Rm×n such that rank(Γ) = r < n. Then the following statements are equivalent: i) x∗ Ξx < 0, for any x 6= 0 and Γx = 0. ii) ∃∆ ∈ Rn×m : Ξ + He(∆Γ) < 0. Lemma 6: Consider the system described by (3). Then the following statements are equivalent: (i)There exist symmetric matrices Ps >¸0, X1i > 0(i = 1, . . . , r) · Y j W12 and Z1i > 0, matrices W j = ˆ ( j = 1, 2) and a filter Y j W22 described by (2) such that (17) holds. (ii)There exist symmetric matrices· P˜s > 0,¸ X˜1i > 0(i = Y G2 1, . . . , r) and Z˜ 1i > 0, matrices W˜ j = j ( j = 1, 2) and V j G2 a filter described by (2) with A˜ F = AF and B˜ F = BF such that  −1 −1 d 1 Z˜ 11 ... −P˜s − ∑ri=1 d i Z˜ 1i + ∑ri=1 X˜1i  −1 ˜ ˜  ? −d 1 Z11 − X11 . . .   .. .. ..  . . .   ? ? . . .   ? ? ... ? ? ... −1 d r Z˜ 1r 0 .. . −1 −d r Z˜ 1r − X˜1r ? ?

˜ − I)∗W˜ ∗ ˜ ∗W˜ ∗ (A A 1 2 ∗ ∗ ˜ W˜ ˜ ∗W˜ ∗ A A 1 1 1 2 .. .. . . ∗ ∗ ∗ ˜ ˜ ˜ Ar W1 Ar W˜ 2∗ r ˜ ˜ 0 ∑i=1 d i Z1i − He(W1 ) ? P˜s − He(W˜ 2 )

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