Dissipative filtering for linear discrete-time systems via LMI

Chun-Jie Li1, Hai-Yi An2 and Yi-Fu Feng3

1. School of Information Science and Engineering, Bohai University, Liaoning Jinzhou 121003, China.E-mail: aaa840430@163.com

2. Panjin senior middle school, Liaoning Panjin, 124000, ChinaE-mail: aaa840430@163.com

3. College of Information Science and Engineering, Northeastern University, Liaoning Shenyang, 110004, ChinaE-mail: yf19692004@163.com

Abstract: The problem of the dissipative filtering for linear discrete systems is dealt with. A sufficient condition forexistence of a dissipative filter such that the error filtering system is asymptotically stable and guarantees dissipativeperformance, are derived in the sense of Lyapunov asymptotic stability and are formulated in the format of linear matrixinequalities (LMIs). Numerical example is provided to demonstrate the feasibility of the proposed condition and filtersdesign procedure.

Key Words: Linear Discrete-time Systems, Dissipative Filter, Lyapunov, LMI

1 INTRODUCTION

Estimation has been attracting considerable attention inthe past decades. Several approaches have been proposedto solve the estimation problem[1][2]. The H∞ filtering forsystems with a polytopic type of parameter uncertainty hadbeen addressed in Palhares and Peres[1]. The recent workGao et al.[2] provided complete results on the induced L2

and generalized H2 filtering problem for a class of nonlin-ear discrete-time systems.

On the other hand, since the notion of dissipative systemswas introduced by Willems in 1972[3], it has played animportant role in systems and control problems. Dissipa-tiveness has been extensively applied in the analysis of sta-bility , the controller designing for nonlinear systems andadaptive systems etc. , see, e.g., [4], [5], and the referencetherein. Many existing basic tools, such as bounded reallemma, passitivity lemma and circle criterion, are shownto be special cases under the framework of dissipativeness.Literature on dissipative systems theory can be found on[4], [5], [6], [7] and references therein. In [4], the problemof quadratic dissipative control for linear discrete-time sys-tems is discussed. In [5], the conditions for robust strictlydissipative control for linear discrete time-delay systemsare given, but doesn’t consider the cases when the con-troller’s parameters vary and the system is nonlinear. [6]has developed a robust quadratic dissipative control tech-nique for a class of uncertain linear systems which are sub-jected to matched nonlinear perturbation, norm-boundeduncertainty and dissipative uncertainty. In [7], the prob-lem of reliable dissipative control for a class of stochastichybrid systems is dealt with.

In this paper, the dissipative filter for linear discrete-time systems is studied. The sufficient conditions for ex-

This work is supported by Scientific Research Program for The Edu-cation Department of Liaoning Province (2008017).

istence of a dissipative filter such that the error filteringsystem is asymptotically stable and guarantees strictly dis-sipative performance, is derived in the sense of Lyapunovasymptotic stability and are formulated in the format of lin-ear matrix inequalities (LMIs).

2 PROBLEM FORMULATION

Consider a nonlinear discrete-time system described bythe following:

x(k + 1) = Ax(k) + Bw(k)y(k) = Cx(k) + Dw(k)z(k) = Lx(k)

(1)

where x(k) ∈ Rn is the state, w(k) ∈ Rr is the distur-bance input belongs to l2[0,∞), z(k) ∈ Rq is the regulatedoutput and y(k) ∈ Rp is the measured output, respectively.The system matrices A, B, C, D and L are known constantmatrices of appropriate dimensions. We begin our analysisby making the following definitions.Definition 1.[6][8][9] A discrete-time system is called dis-sipative if

T∑k=0

J(z(k), w(k)) + β(x0) ≥ 0,∀w(k),∀T ≥ 0 (2)

for some finite function β. Furthermore, a discrete-timesystem is called strictly dissipative if for some sufficientlysmall scalar α > 0

T∑k=0

J(z(k), w(k)) + β(x0) ≥ αN∑

k=0

wT (k)w(k),∀w(k),

∀T ≥ 0.(3)

Also the J(z(k), w(k)) function is known as the supplyrate (or power function).

3866978-1-4244-2723-9/09/$25.00 c© 2009 IEEE

To ensure the achievement of filter design objective, thefollowing basic assumption is also assumed to be valid.Assumption 1. A is stable.Remark 1. This is required in order to get a stable filteringerror dynamics. If this assumption is not satisfied, then astabilizing output feedback controller using reliable sensormeasurement is required.

In this paper, we are interested in designing a full-orderfilter as follow

x(k + 1) = Af x(k) + Bfy(k)z(k) = Cf x(k)

(4)

where x(k) ∈ Rn is the filter state, Af , Bf and Cf are thefilter parameters to be designed.

Then, applying the filter (4) to the system (1), we obtainthe filtering error system

ξ(k + 1) = Aξ(k) + Bw(k)z(k) = Cξ(k)

(5)

where ξT (k) =[

xT (t) xT (t)], z(k) = z(k) − z(k)

is the estimation error, and A =[

A 0BfC Af

], B =[

BBfD

], C =

[L −Cf

].

The transfer function matrix of the filtering error system(5) from w(k) to e(k) is given by

G(z) = C(zI − A)−1B. (6)

Our objective is to design the dissipative filter (4) suchthat the filtering error system (5) satisfies the following re-quirements 1 and 2 simultaneously.

1. Under the exogenous disturbance w(k) = 0, the fil-tering error system (5) is asymptotically stable;

2. Under the zero-initial condition, the filtering error sys-tem (5) is strictly dissipative.

3 STABLE AND DISSIPATIVE ANALYSIS

In order to analyze the stabilization and dissipativeness ofthe filtering error system (5), we are necessary to introducefollowing concepts and definitions.

The energy supply function J associated with the filteringerror system (5) is defined by

J(z, w, T ) = < z, Qz >T +2 < z, Sw >T + < w,Rw >T ,∀T ≥ 0

(7)

where < z, w >T =T∑

k=0

z(k)T w(k), Q, S and R are real

matrices of appropriate dimensions with Q and R symmet-ric. Then, we have the following definitions.Definition 2. The filtering error system (5) with initial stateξ(0) = x0 is said to be (Q, S, R)-dissipative if for somereal function β(·) which only depends on the initial condi-tion ξ(0) = x0, such that

J(z, w, T ) + β(x0) ≥ 0, ∀T ≥ 0, w(k) ∈ l2[0, T ). (8)

Definition 3. The filtering error system (5) with initial stateξ(0) = x0 is said to be strictly (Q, S, R)-dissipative if forany T ≥ 0, some real function β(·) and some sufficientlysmall scalar α > 0, the following condition is satisfied

J(z, w, T ) + β(x0) > α < w, w >T (9)

Furthermore, making β(x0) = 0 under zero initial statex0 = 0, for scalar α is sufficiently small, then the filteringerror system (5) is strictly (Q, S, R)-dissipative, if

J(z, w, T ) > 0 (10)

Remark 2. Dissipativeness is a kind of critical condition,which the system maybe lose of dissipative performancefor a little perturbation of parameters, while strictly dissipa-tive possesses robust performance. Therefore, we only con-sider the strictly dissipative performance in this paper. Thetheory of dissipative systems generalize the system theory,including the bounded real (small gain) theorem, passivitytheorem, circle criterion, and sector bounded nonlinearity,and so on. To see this, a few special cases fall out immedi-ately be setting the (Q, S, R) parameters. For example,

1) H∞ performance: R = γ2, γ > 0, S = 0, and Q =−I .

2) Mixed performance: R = θγ2I, S = (1 − θ)I, θ ∈[0, 1], Q = −θI.

3) Sector bounded performance: R = − 12 (KT

1 K2 +KT

2 K1), S = 12 (K1 + K2)T , and Q = −I, for some

constant matrices K1, K2.

Without loss of generality, we shall make the followingassumption.Assumption 2. Matrix Q satisfies the following condition

Q1/2− ∗ Q

1/2− = Q− = −Q ≥ 0 (11)

Remark 3. It can be observed that Assumption 2 holds forall the cases in Remark 2.

The following theorem provides a solution to the strictly(Q, S, R)-dissipative fuzzy output feedback control withmultiplicative gain variations.Theorem 1. Let Q, S, R be given matrices with Q and Rsymmetric. Consider the system (1) satisfying Assumption1. Then the filtering error system (5) is asymptotically sta-ble and strictly (Q, S, R)-dissipative if there exist matricesP = PT > 0 and matrices Af , Bf and Cf satisfying thefollowing LMI:

⎡⎢⎢⎣

−P ∗ ∗ ∗−ST C −R ∗ ∗

PA PB −P ∗Q

1/2− C 0 0 −I

⎤⎥⎥⎦ < 0 (12)

Remark 4. Obviously, (12) reduces to the inequality (12)in [10] when Q = −I, S = 0 and R = γ2I , i.e., the strict(Q, S, R)-dissipatively reduces to an H∞ performance re-quirement.

2009 Chinese Control and Decision Conference (CCDC 2009) 3867

3.1 Stable AnalysisDefine the Lyapunov function as V (k) = ξ(k)T Pξ(k),

where P is a symmetric positive matrix, then we have

ΔV (k) = V (k + 1) − V (k)

= ξT (k + 1)Pξ(k + 1) − ξT (k)Pξ(k)

= (Aξ(k) + Bw(k))T P (Aξ(k) + Bw(k)) − ξT (k)Pξ(k)

= η(k)T (Θ −[

−P 00 0

])η(k),

(13)where η(k)T =

[ξT (k) wT (k)

], Θ =[

AT

BT

]P

[AT

BT

]T

.

Under zero exogenous input w(k) = 0, the equation (13)can be simplified as follow:

ΔV (k) = η(k)T {AT PA − P}η(k) (14)

According to Schur complement theorem, inequality (12)imply the following inequality hold

AT PA − P < 0 (15)

namely ΔV (k) < 0, thus if inequality (12) holds, the fil-tering error system (5) is asymptotically stable.

3.2 Dissipative Analysis

From the definition of energy supply function, we have

−J(z, w, T )= − < z, Qz >T −2 < z, Sw >T − < w, Rw >T

= −T∑

k=0

(Cξ(k))T QCξ(k) + 2(Cξ(k))T Sw(k)

+ w(k)T Rw(k))

=T∑

k=0

(−η(k)T

[CT QC CT SST C R

]η(k)

)

(16)And with zero initial state xcl(0) = 0, V (0) = 0, such

that

−J(z, w, T )

= −J(z, w, T ) +T∑

k=0

ΔV (k) + V (0) − V (T + 1)

≤T∑

k=0

ηT (k)Φη(k)

(17)

where Φ =[

CT Q−CT − P −CT S−ST C −R

]+

[AT

BT

]P

[AT

BT

]T

.

According to Schur complement theorem, inequality(12)is equivalent to Φ < 0, that is −J(z, w, T ) < 0. There-fore, the filtering error system (5) is strictly (Q, S, R)-dissipative if inequality (12)hold. This completes the proof.

It is obvious that (12) is not LMI due to the product ofthe variables P with the filtering error system matrices Aand B, respectively. Moreover, the construct product be-tween the Lyapunov matrix and the filtering error systemmatrices introduces conservativeness. For the sake of over-coming the above disadvantage, we design the filter with

the following method. Firstly, we give out an equivalentcondition to Theorem 1.Theorem 2. Let Q, S, R be given matrices with Q and Rsymmetric. Consider the system (1) satisfying Assumption1.Then the filtering error system (5) is asymptotically sta-ble and strictly (Q, S, R)-dissipative if there exist matricesP = PT > 0 and matrices G, Af , Bf and Cf satisfyingthe following LMI:

⎡⎢⎢⎣

−P ∗ ∗ ∗−ST C −R ∗ ∗GT A GT B P − G − GT ∗

Q1/2− C 0 0 −I

⎤⎥⎥⎦ < 0 (18)

Proof. To proof necessity is a very simple task, since itsuffices to choose G = GT = P > 0 so that we recover(12). The proof of sufficiency is not so obvious. Noticethat the regularity of G is implied by the diagonal blocks of(18) since GT + G > P > 0 so that G is nonsingular and(P − G)T P−1(P − G) ≥ 0 always holds. Therefore, theinequality GT P−1G ≥ G + GT − P enables to concludethat

⎡⎢⎢⎣

−P ∗ ∗ ∗−ST C −R ∗ ∗GT A GT B −GT P−1G ∗

Q1/2− C 0 0 −I

⎤⎥⎥⎦ < 0 (19)

which recovers (12) if multiplied on the right by Γf =diag{I, I, G−1P, I} and on the left by ΓT

f . This completesthe proof.

The inequality of (12) is equivalent to (18). The maindifference between them is that the system matrices of theinequality of (12) are coupling with the Lyapunov matrix,while the system matrices of the inequality of (18) are de-coupling with the Lyapunov matrix and are multiplicativewith a freedom matrix. The conservatism of Theorem 1and 2 are similar for certain systems, while the Theorem 2can reduce the conservatism when deals with the uncertainsystems.

4 DISSIPATIVE FILTER VIA LMI AP-PROACH

In this section, we will provide a solution to the strictly(Q, S, R)-dissipative filter problem in terms of LMI tech-nique. The following theorem gives the existence of a dis-sipative filter such that the filtering error system (5) is as-ymptotically stable and strictly (Q, S, R)-dissipative.Theorem 3. Let Q, S, R be given matrices with Q and Rsymmetric, then the filtering error system (5) is asymptoti-cally stable and strictly (Q, S, R)-dissipative, if there existsome matrices X, Y, N , Af , Bf , Cf , such that

⎡⎢⎢⎢⎢⎢⎢⎣

−P1 ∗ ∗ ∗ ∗ ∗−P2 −P3 ∗ ∗ ∗ ∗Π1 −ST L −R ∗ ∗ ∗Y A Y A Y B Ω1 ∗ ∗Π2 Π3 Π4 Ω2 Ω3 ∗Π5 Q

1/2− L 0 0 0 −I

⎤⎥⎥⎥⎥⎥⎥⎦

< 0, (20)

3868 2009 Chinese Control and Decision Conference (CCDC 2009)

where Π1 = −ST L + ST Cf , Π2 = XT A + BfC + Af ,

Π3 = XT A + BfC, Π4 = XT B + BfD, Π5 = Q1/2− L−

Q1/2− Cf , Ω1 = P1 − Y − Y T , Ω2 = P2 − N − XT − Y ,

Ω3 = P3 − XT − X .Moreover, if there exist solutions of this inequality, the

reliable filter can be given by

Af = N−1Af , BF = N−1Bf , Cf = Cf . (21)

Proof. If the inequality of (18) in Theorem 2 is hold, thenthe filtering error system (5) is asymptotically stable andstrictly (Q, S, R)-dissipative.

We need look for a suitable change of variables whichis able to linearize the synthesis problem. The change-of-variables technique stated below is based on the resultScherer et al.[11]. Let us first partition G and is inverseG−1 as

G =[

X ?U ?

], G−1 =

[Y −1 ?V ?

](22)

where X and Y have appropriate dimensions and ′?′ denoteblocks in these matrices with no importance in the sequel.Furthermore, X and Y are invertible(see 1997, Scherer[11]for detail). There is no loss of generality in assuming thatU and V are nonsingular[21,22].

Construct the following matrices

J1 =[

Y −1 IV 0

], J2 =

[I X0 U

](23)

From GG−1 = I , we infer

GJ1 = J2 (24)

JT1 GT J1 = JT

2 J1 =[

Y −1 IUT V + XT Y −1 XT

](25)

Applying the congruence transformations to (18) bydiag{J1, I, J1, I}, we have

⎡⎢⎢⎣

−JT1 PJ1 ∗ ∗ ∗

−ST CJ1 −R ∗ ∗JT

1 GT AJ1 JT1 GT B Υ ∗

Q1/2− CJ1 0 0 −I

⎤⎥⎥⎦ < 0, (26)

where Υ = JT1 (P − G − GT )J1.

Let JT1 PJ1 =

[Y −T P1Y

−1 Y −T PT2

P2Y−1 P3

]> 0, and ap-

plying the parameters of the error filter system (5) to (26),we can obtain⎡⎢⎢⎢⎢⎢⎢⎣

−Y −T P1Y−1 ∗ ∗ ∗ ∗ ∗

−P2Y−1 −P3 ∗ ∗ ∗ ∗

Ξ1 −ST L −R ∗ ∗ ∗AY −1 A B Ξ5 ∗ ∗

Ξ2 Ξ3 Ξ4 Ξ6 Ξ7 ∗Ξ8 Q

1/2− L 0 0 0 −I

⎤⎥⎥⎥⎥⎥⎥⎦

< 0,

(27)where Ξ1 = −ST LY −1 + ST CfV, Ξ2 = XT AY −1 +UT BfCY −1 + UT AfV, Ξ3 = XT A + UT BfC, Ξ4 =XT B +UT BfD, Ξ5 = Y −T P1Y

−1−Y −1−Y −T , Ξ6 =

P2Y−1 −UT V −XT Y −1 − I , Ξ7 = P3 −XT −X, Ξ8 =

Q1/2− LY −1 − Q

1/2− CfV .

Similarly, applying the congruence transformationsto (27) by diag{Y, I, I, Y, I, I}, furthermore, let N =UT V Y , we obtain⎡

⎢⎢⎢⎢⎢⎢⎣

−P1 ∗ ∗ ∗ ∗ ∗−P2 −P3 ∗ ∗ ∗ ∗Λ1 −ST L −R ∗ ∗ ∗Y A Y A Y B Ω1 ∗ ∗Λ2 Λ3 Λ4 Ω2 Ω3 ∗Λ5 Q

1/2− L 0 0 0 −I

⎤⎥⎥⎥⎥⎥⎥⎦

< 0. (28)

where Λ1 = −ST L+ST CfV Y,Λ2 = XT A+UT BfC +UT AfV Y,Λ3 = XT A + UT BfC,Λ4 = XT B +UT BfD,Λ5 = Q

1/2− L − Q

1/2− CfV Y and Ω1, Ω2 and Ω3

are defined in (20).Then, we define Af = UT AfV Y , Bf = UT Bf ,

Cf = CfV Y , the filter transfer function can be describedas following

G(z) = Cf (zI − Af )−1Bf

= CfY −1V −1(zI − U−T AfY −1V −1)−1U−T Bf

= Cf (zUT V Y − Af )−1Bf

= Cf (zN − Af )−1Bf

= Cf (zI − N−1Af )−1N−1Bf

(29)hence, we can get (20) and (21) immediately. This com-pletes the proof.Remark 5. It is noted that the conditions in Theorem 3 areLMI conditions. Therefore, the dissipative problem can besolved by using convex optimization algorithms. And thedesigned filter’s parameters can be obtained by (21).

5 AN ILLUSTRATIVE EXAMPLE

In this section, we aim to demonstrate the effectivenessand applicability of the proposed method, and an exampleis given to illustrate that the dissipative control approachunifies the H∞ filter, the sector bounded filter and otherkinds of filter.

Consider a discrete-time system described by

A =

⎡⎣ 0.2 −0.8 0.2

0.4 0.3 −0.60.2 −0.4 −0.6

⎤⎦ , B =

⎡⎣ 0.2

0.40.3

⎤⎦

C =[ −0.3 0.1 −0.4

], D = 0.1

L =[

0.4 0.1 0.8].

By solving the feasibility problem in Theorem 3, we shalldesign three filters by choosing different values of Q, S andR, as special cases of the general dissipative controller.a) Q = −1, S = 0 and R = 1, i.e. the H∞ filter case. Andwe can obtain the filter parameters as follows

Af =

⎡⎣ 0.1229 −0.4809 0.0147

0.4378 0.2039 −0.33270.4342 −0.3168 −0.2704

⎤⎦ ,

Bf =

⎡⎣ −0.1363

0.38290.7390

⎤⎦ ,

Cf =[

0.3972 0.0794 0.7697].

2009 Chinese Control and Decision Conference (CCDC 2009) 3869

b) Q = −1, S = 0.75 and R = 2.5, i.e. sector boundedcase with K1 = −1.0 and K2 = 2.5[4]. The filter parame-ters are

Af =

⎡⎣ 0.0948 −0.4613 0.0205

0.4128 0.1853 −0.26940.4018 −0.3073 −0.2576

⎤⎦ ,

Bf =

⎡⎣ −0.1779

0.40700.7349

⎤⎦ ,

Cf =[

0.3157 0.0955 0.8664].

c) Q = −1, S = −0.3 and R = 0.89, i.e. the generaldissipative case. The filter parameters are

Af =

⎡⎣ 0.0961 −0.4840 −0.0154

0.4061 0.2086 −0.37050.4038 −0.3189 −0.2998

⎤⎦ ,

Bf =

⎡⎣ −0.2194

0.27860.6371

⎤⎦ ,

Cf =[

0.4850 0.0935 0.6755].

From these numerical studies, it is clear that our dissi-pative filter formulation is very general and flexible. Bychoosing different values of Q, S and R, the dissipativefilter can include various types of filter as special cases.Furthermore, the filter also guarantee the error filteringsystem (5) is asymptotically stable and strictly (Q, S, R)-dissipative.

In order to show the effectiveness of our method moreclearly, a simulation is also reformed. In the following sim-ulation, let the system initial state be x0 =

[0 0 0

]and the filter initial state be x0 =

[0 0 0

]. In addi-

tion, we consider the H∞ filter case. And we assume thedisturbance input w(k) as following:

w(k) =

⎧⎨⎩

0.5cos(0.2t) 5 ≤ k ≤ 10−0.5sin(0.1t) 25 ≤ k ≤ 300 otherwise

. (30)

which can be showed in Fig. 1. Fig.1 also shows the es-timation error e(k) response of the filters designed by theproposed methods. Obviously, the designed filter guaran-tees that the error filter system (5) is asymptotically stable.

6 CONCLUSION

We have addressed the dissipative filter problem for lin-ear discrete-time systems. Based on the matrix inequalitytechnology, this paper gives the sufficient conditions for theexistence of the dissipative filter and the algorithm simul-taneously. The proposed approach can be further extendedto deal with the robust dissipative filter design for uncer-tain discrete-time systems and uncertain delay discrete-time systems.

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−0.2

0

0.2

0.4

0 5 10 15 20 25 30 35 40−0.1

−0.05

0

0.05

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w(k)

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3870 2009 Chinese Control and Decision Conference (CCDC 2009)