5

H∞ Control and Estimation of Retarded

Linear Discrete-Time Systems

5.1 Introduction

In this chapter we address the problems of H∞ state-feedback control andfiltering of state-delayed, discrete-time, state-multiplicative linear systems viathe input–output approach based on the stability and Bounded Real Lemma(BRL) of these systems, which are developed here. The multiplicative noiseappears in the system model in both the delayed and the non delayed statesof the system.

A major part of the stochastic H∞ control and estimation research for LTIdelayed systems, has centered around continuous-time systems. As reviewedin the Introduction, this includes the stability and control of stochastic de-layed systems of various types (i.e constant time-delay, slow and fast vary-ing delay etc.). The field of discrete-time control of delayed systems in thestochastic state-multiplicative context has not been fully investigated. Overthe last decade, the mean square exponential stability and the control and fil-tering problems of these systems were treated by several groups [131], [135]. In[131], the state-feedback control problem solution is solved for norm-boundeduncertain systems, for the restrictive case where the same multiplicative noisesequence multiplies both the states and the input of the system. The solutionin [131] is delay-dependent.

The point of view taken in the present chapter is similar to the one takenfor the solution of both the continuous-time state-feedback control and filter-ing problems in Chapter 2 (see also [55],[59]). Here, we adopt the input–outputapproach of Chapter 2, to delay-dependent solutions of the above discrete-timecounter part stochastic problems. This approach is based on the representa-tion of the system’s delay action by linear operators, with no delay, whichin turn allows one to replace the underlying system with an equivalent onewhich possesses a norm-bounded uncertainty, and therefore may be treatedby the theory of norm bounded uncertain, non-retarded systems with state-multiplicative noise [53].

E. Gershon & U. Shaked: Advanced Topics in Control & Estimation, LNCIS 439, pp. 95–113.DOI: 10.1007/978-1-4471-5070-1_5 c© Springer-Verlag London 2013

96 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

Similarly to Chapter 2, in our system we allow for a time-varying delaywhere the uncertain stochastic parameters multiply both the delayed and thenon delayed states in the state space model of the system. Based on the input–output approach, the solution of the stability issue is achieved in Section 5.3,followed by the solution of the BRL in Section 5.4. The state-feedback controlproblem is treated in Section 5.5. The filtering problem solution is achievedin Section 5.6, for a general-type filter, resulting in a single LMI that yieldsthe filter parameters. Each of the above sections is ended with a numericalexample that demonstrates the tractability of our solution method.

Notation: For convenience we bring the necessary notations for the discrete-time setting. We denote, by L2(Ω,Rn) the space of square-integrable Rn−valued functions on the probability space (Ω,F ,P), where Ω is the samplespace, F is a σ algebra of a subset of Ω called events and P is the probabilitymeasure on F . By (Fk)k∈N we denote an increasing family of σ-algebras Fk ⊂F . We also denote by l2(N ;Rn) the n-dimensional space of nonanticipativestochastic processes {fk}k∈N with respect to (Fk)k∈N where fk ∈ L2(Ω,Rn).On the latter space the following l2-norm is defined:

||{fk}||2l2 = E{∑∞

0 ||fk||2} =∑∞

0 E{||fk||2} <∞, {fk} ∈ l2(N ;Rn),

(5.1)

where || · || is the standard Euclidean norm. Throughout this chapter we referto the notation of exponential l2 stability, or internal stability, in the sense of[13] (see Definition 2.1, page 927, there).

5.2 Problem Formulation

We consider the following linear retarded system:

xk+1 = (A0 +Dνk)xk + (A1 + Fμk)xk−τ(k) +B1wk + (B2 +Gζk)uk,xl = 0, l ≤ 0,yk = C2xk +D21nk

(5.2)

with the objective vector

zk = C1xk +D12uk, (5.3)

where xk ∈ Rn is the system state vector, wk ∈ Rq is the exogenous dis-turbance signal, nk ∈ Rp is the the measurement noise signal, uk ∈ R isthe control input, yk ∈ Rm is the measured output and zk ∈ Rr is the statecombination (objective function signal) to be regulated and where the timedelay is denoted by the integer τk and it is assumed that 0 ≤ τk ≤ h, ∀k. Thevariables {ζk} {μk} and {νk} are zero-mean real scalar white-noise sequencesthat satisfy:

E{νkνj} = δkj , E{ζkζj} = δkj , E{μkμj} = δkj ,

5.2 Problem Formulation 97

E{ζkμj} = E{ζkνj} = E{μkνj} = 0, ∀k, j ≥ 0.

The matrices in (5.2a,b) and (5.3) are constant matrices of appropriate di-mensions.

We treat the following two problems:

i) H∞ state-feedback control:We consider the system of (5.2a) and (5.3) and the following performanceindex:

JEΔ= ||zk||2l2 − γ

2||wk||2l2 . (5.4)

Our objective is to find a state-feedback control law uk = Kxk that achievesJE < 0, for the worst-case of the process disturbance wk ∈ l2Fk

([0,∞);Rq)and for the prescribed scalar γ > 0.

ii) H∞ filtering:We consider the system of (5.2a,b) and (5.3), where B2 = 0, G = 0 andD12 = 0 and the following general form estimator:

xk+1 = Acxk +Bcyk, xi = 0, ∀i ≤ 0zk = Ccxk.

(5.5)

We denoteek = xk − xk, and zk = zk − zk, (5.6)

and for a given scalar parameter γ, we consider the following cost function:

JFΔ= ||zk||2l2 − γ

2[||wk||2l2 + ||nk+1||2l2 ]. (5.7)

Given γ , we seek an estimate Ccxk of C1xk over the infinite time horizon[0,∞) such that JF given by (5.7) is negative for all nonzero wk, nk wherewk ∈ l2Fk

([0,∞);Rq), nk ∈ l2Fk([0,∞];Rp).

In the robust stochastic H∞control and estimation problems treated here,we assume that the system parameters lie within the following polytope

ΩΔ=[A0A1B1B2C1C2D12D21DF G

], (5.8)

which is described by the vertices:

Ω = Co{Ω1, Ω2, ..., ΩN}, (5.9)

where

ΩiΔ=

[A

(i)0 A

(i)1 B

(i)1 B

(i)2 C

(i)1 C

(i)2 D

(i)12 D

(i)21 D

(i) F (i) G(i)]

(5.10)

and where N is the number of vertices. In other words:

Ω =

N∑i=1

Ωifi,

N∑i=1

fi = 1 , fi ≥ 0. (5.11)

98 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

5.3 Mean-Square Exponential Stability

In order to solve the above two problems, we investigate first the stability ofthe retarded discrete-time system. We introduce the following scalar operatorswhich are needed, in the sequel, for transforming the delayed system to anequivalent norm-bounded nominal system:

Δ1(gk) = gk−h, Δ2(gk) =k−1∑

j=k−h

gj . (5.12)

Denoting yk = xk+1 − xk and using the fact that Δ2(yk) = xk − xk−h, thefollowing state space description of the system is obtained:

xk+1 = (A0 +Dνk +M)xk + (A1 −M + Fμk)Δ1(xk)−MΔ2(yk) +B1wk+

(B2 +Gζk)uk, xl = 0, l ≤ 0,

yk = C2xk +D21nk,zk = C1xk +D12uk,

(5.13)

where the matrix M is a free decision variable that will be determined later.We consider then the following auxiliary system where we take B2 = 0

and G = 0:

xk+1 = (A0 +Dνk +M)xk + (A1 −M + Fμk)w1,k −Mw2,k +B1wk,(5.14)

with the feedback

w1,k = Δ1(xk), w2,k = Δ2(yk). (5.15)

We consider the system of (5.14 ) where B1 = 0 and the following Lyapunovfunction:

VkΔ= xTkQxk. (5.16)

Taking expectation with respect to vk and μk and solving for (5.14 ) we obtain:

E{Vk+1} − Vk =E{[xTk (AT

0 +MT+DT νk)+wT1,k(A

T1 −MT+FTμk)−wT

2,kMT ]Q

[(A0 +M +Dνk)xk + (A1−M+Fμk)w1,k−Mw2,k]} − xTkQxk.(5.17)

We thus arrive at the following condition for E{Vk+1} − Vk < 0.

Theorem 5.1. The exponential stability in the mean square sense of the sys-tem (5.2a) where B1 = 0, B2 = 0, and G = 0, is guaranteed if there existn × n matrices Q > 0, R1 > 0, R2 > 0, and M that satisfy the followinginequality:

5.3 Mean-Square Exponential Stability 99

ΓΔ=

⎡⎢⎢⎢⎣

Γ11 (A0 +M)TQ 0 0 h(AT0 +MT )R2 −R2h

∗ −Q Q(A1 −M) QM 0

∗ ∗ −R1 + F T (Q+ h2R2)F 0 h(AT1 −MT )R2

∗ ∗ ∗ −R2 −hMTR2

∗ ∗ ∗ ∗ −R2

⎤⎥⎥⎥⎦ < 0,

(5.18)

where Γ11 = −Q+DT (Q + h2R2)D +R1.

Proof:Define xk+1 = xk+1−Dνkxk−Fμkw1,k and yk = yk−Dνkxk−Fμkw1,k

and denoteηk = col{xk, xk+1, w1,k, w2,k, hyk}.

If (5.18) is satisfied for the appropriate Q, R1, R2, andM , then the followingholds:

θkΔ= ηTk Γηk < 0, ηk �= 0. (5.19)

Carrying out the multiplications in (5.19) we find that

θk = −xTkQxk + xTkDT (Q+ h2R2)Dxk + xTkR1xk − xTk+1Qxk+1 + ηk

−wT1,k(R1 − FT (Q +R2)F )w1,k + h2yTk R2yk − wT

2,kR2w2,k,

where ηk = 2xTk (AT0 +MT )Qxk+1 − 2wT

2MTQxk+1 + 2xTk+1Q(A1 −M)w1 =

2xTk+1Qxk+1.

It thus follows that

θk = xTk+1Qxk+1 − xT

kQxk + xTk D

T (Q+ h2R2)Dxk + [xTk R1xk − wT

1,kR1w1,k]+[h2yT

k R2yk − wT2,kR2w2,k] + wT

1,kFT (Q+ h2R2)Fw1,k.

Since

E{yTk R2yk} = E{yTkR2yk}+ E{xTkDTR2Dxk}+ E{wT1,kF

TR2Fw1,k}and

E{xTk+1Qxk+1} = E{xTk+1Qxk+1}+E{xTkDTQDTxk}+E{wT1,kF

TQFw1,k},we find that if (5.18) is satisfied then:

E{θk} = E{xTk+1Qxk+1 − xTkQxk}+ E{yTk h2R2yk − wT2,kR2w2,k}+

E{xTkR1xk − wT1,kR1w1,k} < 0.

(5.20)

Since for all sequences {rk} in Rn

‖Δ1rk‖2 ≤ ‖rk‖2, and ‖Δ2rk‖2 ≤ h2‖rk‖2,

we have that E{h2yTk R2yk−wT2,kR2w2,k}>0 and E{xTkR1xk−wT

1,kR1w1,k}>0.

Thus, (5.20) implies that E{Vk+1} − Vk = E{xTk+1Qxk+1} − xTkQxk < 0and the stability is guaranteed.

100 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

We note that inequality (5.18) is bilinear due to the terms QM and R2M.Similar bilinearity is observed in the continuous-time counterpart of the sta-bility problem (see Chapter 2, Section 2.3.1). There exist few algorithms thatmay solve bilinear matrix inequalities, however they do not always convergeto a global minimum and they may require considerable computational effort.In order to remain in the linear domain, we choose R2 = εQ where ε is apositive tuning scalar. Defining QM = QM we obtain the following result:

Corollary 5.3.1 The exponential stability in the mean square sense of thesystem (5.2a) where B1 = 0, B2 = 0, and G = 0, is guaranteed if there existn×n matrices Q > 0, R1 > 0 and QM , and a tuning scalar ε > 0 that satisfythe following inequality:

ΓΔ=

⎡⎢⎢⎢⎣

Γ11 AT0 Q+QT

M 0 0 εh[AT0 Q+QT

M ]− εhQ∗ −Q QA1 −QM QM 0∗ ∗ −R1 + (1 + εh2)F TQF 0 εh[AT

1 Q−QTM ]

∗ ∗ ∗ −εQ −hεQTM

∗ ∗ ∗ ∗ −εQ

⎤⎥⎥⎥⎦ < 0,

(5.21)

where Γ11 = −Q+DTQ[1 + εh2]D +R1.

In the uncertain case we obtain the following result:

Corollary 5.3.2 The exponential stability in the mean square sense of thesystem (5.2a) where B1 = 0, B2 = 0, and G = 0 and where the systemmatrices lie within the polytope Ω of (5.8) is guaranteed if there exist n × nmatrices Q > 0, R1 > 0 and QM , and tuning scalar ε > 0 that satisfy thefollowing set of LMIs:

Γ i Δ=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Q+R1 Γi12 0 0 Γ i

15 Di,TQh 0∗ −Q QAi

1 −QM QM 0 0 0

∗ ∗ −R1 0 εh[Ai,T1 Q−QT

M ] 0 εF i,TQ∗ ∗ ∗ −εQ −hεQT

M 0 0∗ ∗ ∗ ∗ −εQ 0 0∗ ∗ ∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,

(5.22)∀i, i = 1, 2, ...., N, where

Γ i12 = Ai,T

0 Q+QTM ,

Γ i15 = εh[Ai,T

0 Q +QTM ]− εhQ,

ε2 = 1 + εh2.

5.4 The Bounded Real Lemma 101

5.3.1 Example – Stability

We consider the system of (5.2a), with the following system matrices:

A =

[0.8 00 0.97

], D =

[0.1 00 0

], A1 =

[−0.1 0−0.1 −0.1

], F =

[0.05 00 0

],

where B1 = 0, B2 = 0, and G = 0. Applying the result of Corollary 5.3.1, weobtain a maximum delay bound of h = 7 for ε = 0.08.

5.4 The Bounded Real Lemma

In the case where wk �= 0, (5.19) implies that E{[xk+1 − B1wk]TQ[xk+1 −

B1wk]} − xTkQxk < 0. Denoting

JB = E{xTk+1Qxk+1} − xTkQxk + zTk zk − γ2wTk wk, (5.23)

where zk = C1xk, we add B1wk to the previously defined xk+1 and readilyfind that JB < 0, ∀ wk ∈ l2Fk

([0,∞);Rq) if

ηTk Γ ηk < 0, (5.24)

where ηk = col{xk, xk+1, w1,k, w2,k, hyk, wk, zk} and where

ΓΔ=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

Γ11 (A0+M)TQ 0 0 h(AT0 +MT )R2 −R2h 0 CT

1

∗ −Q Q(A1 −M) QM 0 QB1 0

∗ ∗ Γ33 0 h(AT1 −MT )R2 0 0

∗ ∗ ∗ −R2 −hMTR2 0 0∗ ∗ ∗ ∗ −R2 hR2B1 0∗ ∗ ∗ ∗ ∗ −γ2Iq 0∗ ∗ ∗ ∗ ∗ ∗ −Ir

⎤⎥⎥⎥⎥⎥⎥⎥⎦,

(5.25)

withΓ11 = −Q+DT (Q+ h2R2)D + R1,

Γ33 = −R1 + FT (Q + h2R2)F.

Note that when carrying out the multiplications in (5.24) the productxTk+1Γ2ηk is zero, where Γ2 denotes the second row block of Γ . We also notethat the matrix in the 5th row and the 6th column blocks in the latter inequal-ity stems from the fact that the expression for yk includes now the additionalterm B1wk.

Similarly to the stability result of Corollary 5.3.1, the following result isreadily obtained:

102 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

Theorem 5.2. Consider the system (5.2a) and (5.3) with B2 = 0, G = 0,and D12 = 0 . The system is exponentially stable in the mean square senseand, for a prescribed scalar γ > 0 and a given scalar tuning parameter εb > 0,the requirement of JB < 0 is achieves for all nonzero w ∈ l2Fk

([0,∞);Rq), ifthere exist n× n matrices Q > 0, R1 > 0 and a n× n matrix Qm that satisfythe following LMI:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

Γ11 AT0Q+QT

m 0 0 Γ15 0 CT1

∗ −Q QA1 −Qm Qm 0 QB1 0

∗ ∗ Γ33 0 εbh[AT1 Q−QT

m] 0 0∗ ∗ ∗ −εbQ −hεbQT

m 0 0∗ ∗ ∗ ∗ −εbQ εbhQB1 0∗ ∗ ∗ ∗ ∗ −γ2Iq 0∗ ∗ ∗ ∗ ∗ ∗ −Ir

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦< 0,

(5.26)where

Γ11 = −Q+DTQ[1 + εbh2]D +R1,

Γ33 = −R1 + (1 + εbh2)FTQF.

In the uncertain case we obtain the following result:

Corollary 5.4.1 Consider the system (5.2a) and (5.3) with B2 = 0, G = 0,and D12 = 0 and where the system matrices lie within the polytope Ω of(5.8). The system is exponentially stable in the mean square sense and, fora prescribed γ > 0 and a given tuning parameter εb > 0, the requirement ofJB < 0 is achieves for all nonzero w ∈ l2Fk

([0,∞);Rq), if there exist n × nmatrices Q > 0, R1 > 0 and a n× n matrix Qm that satisfy the following setof LMIs:

Γ i Δ=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Γ i11 Γ

i12 0 0 Γ i

15 0 Ci,T1 Γ i

18 0

∗ −Q Γ i23 Qm 0 QBi

1 0 0 0

∗ ∗ −R1 0 Γ i35 0 0 0 εF i,TQ

∗ ∗ ∗ −εbQ −hεbQTm 0 0 0 0

∗ ∗ ∗ ∗ −εbQ εbhQBi1 0 0 0

∗ ∗ ∗ ∗ ∗ −γ2Iq 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −Ir 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (5.27)

∀i, i = 1, 2, ...., N, where

5.5 State-Feedback Control 103

Γ i11 = −Q+R1,

Γ i12 = Ai,T

0 Q+QTm

Γ i15 = εbh[A

i,T0 Q+QT

m]− εbhQ,Γ i18 = εDi,TQ,

Γ i23 = QAi

1 −Qm

Γ i35 = εbh[A

i,T1 Q−QT

m]

ε2 = 1 + εbh2.

5.4.1 Example – BRL

We consider the system of (5.2a) and (5.3) with the following system matrices:

A =

[0.1 0.6−1 −0.5

], D =

[0 0.630 0

], A1 =

[0 0.10 0

], F =

[0 0.020 0

],

B1 =

[−0.2250.45

], C1 =

[−0.5 040 0

],

where B2 = 0, G = 0, and D12 = 0. Applying the result of Theorem 5.2, weobtain for a delay interval of h = 120, a near minimum attenuation level ofγ = 44.97 for εb = 10−6.

5.5 State-Feedback Control

In this section we address the problem of finding the following state-feedbackcontrol law

uk = Kxk, (5.28)

that stabilizes the system and achieves a prescribed level of attenuation. Weconsider the system of (5.2a) and (5.3) and we apply the control law of (5.28),where A0 is replaced by (A0 + B2K) and C1 is replaced by C1 +D12K. Weobtain the following inequality:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−Q+R1 + Υ11,a Υ12 0 0 Υ15 0 Υ17∗ −Q QA1 −QM QM 0 QB1 0∗ ∗ Υ33 0 εbh[A

T1Q−QT

M ] 0 0∗ ∗ ∗ −εbQ −hεbQT

M 0 0∗ ∗ ∗ ∗ −εbQ hεbQB1 0∗ ∗ ∗ ∗ ∗ −γ2Iq 0∗ ∗ ∗ ∗ ∗ ∗ −Ir

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦< 0,

(5.29)where

104 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

Υ11,a = DTQ[1 + εbh2]D +KTGTQ[1 + εbh

2]GK,

Υ12 = [A0 +B2K]TQ+QTM ,

Υ15 = εbh[(A0 +B2K)TQ+QTM ]− εbhQ,

Υ17 = [C1 +D12K]T ,

Υ33 = −R1 + (1 + εbh2)FTQF.

Multiplying the above inequality by diag{Q−1, Q−1, Q−1, Q−1, Q−1, Iq, Ir},from the left and the right and denoting , R1 = Q−1R1Q

−1, PΔ= Q−1, MP =

MP, KP = KP , we obtain the following LMI:

ΥΔ=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

−P + R1 Υ12 0 0 Υ15 0 Υ17 εPDT 0 εKTP GT

∗ −P A1P − MP MP 0 B1 0 0 0 0

∗ ∗ −R1 0 Υ35 0 0 0 εPFT 0

∗ ∗ ∗ −εbP −hεbMTP 0 0 0 0 0

∗ ∗ ∗ ∗ −εbP hεbB1 0 0 0 0

∗ ∗ ∗ ∗ ∗ −γ2Iq 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −Ir 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −P 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(5.30)

whereΥ12 = PAT

0 +KTPB

T2 +MT

P ,

Υ15 = εbh[PAT0 +KT

PBT2 +MT

P ]− εbhP,Υ17 = PCT

1 +KTPD

T12,

Υ35 = εbh[PAT1 −MT

P ],

ε2 = 1+ εbh2.

We thus arrive at the following theorem:

Theorem 5.3. Consider the system (5.2a) and (5.3). For a prescribed scalarγ > 0, and positive tuning scalar εb > 0, there exists a state-feedback gain thatachieves negative JE for all nonzero w ∈ l2Fk

([0,∞);Rq), if there exist n× nmatrices P > 0, R1 > 0, n×n matrix MP and a l×n matrix KP that satisfythe LMI of (5.30). In the latter case the state-feedback gain is given by:

K = KPP−1. (5.31)

In the uncertain case we obtain the following result:

Corollary 5.5.1 Consider the system (5.2a) and (5.3), where the system ma-trices lie within the polytope Ω of (5.8). For a prescribed scalar γ > 0, andpositive tuning scalar εb > 0, there exists a state-feedback gain that achievesnegative JE for all nonzero w ∈ l2Fk

([0,∞);Rq), if there exist n× n matrices

5.5 State-Feedback Control 105

P > 0, R1 > 0, n × n matrix MP and a l × n matrix KP that satisfy thefollowing set of LMIs:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ i11 Υ

i12 0 0 Υ i

15 0 Υ i17 Υ

i18 0 Υ i

1,10

∗ −P Ai1P −MP MP 0 Bi

1 0 0 0 0∗ ∗ −R1 0 Υ i

35 0 0 0 εPF i,T 0∗ ∗ ∗ −εbP −hεbMT

P 0 0 0 0 0∗ ∗ ∗ ∗ −εbP hεbB

i1 0 0 0 0

∗ ∗ ∗ ∗ ∗ −γ2Iq 0 0 0 0∗ ∗ ∗ ∗ ∗ ∗ −Ir 0 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ −P 0 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P 0∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(5.32)∀i, i = 1, 2, ...., N, where

Υ i11 = −P + R1,

Υ i12 = PAi,T

0 +KTPB

i,T2 +MT

P ,

Υ i15 = εbh[PA

i,T0 +KT

PBi,T2 +MT

P ]− εbhP,

Υ i17 = PCi,T

1 +KTPD

i,T12 ,

Υ i18 = εPDi,T ,

Υ i1,10 = εKT

PGi,T ,

Υ i35 = εbh[PA

i,T1 −MT

P ],

ε2 = 1 + εbh2

In the latter case, the state-feedback gain is given by (5.31).

5.5.1 Example – Robust State-Feedback

We consider the system of (5.2a) and (5.3) with the following system matrices:

A =

[0.1 0.6± a−1 −0.5

], D =

[0 0.630 0

], A1 =

[0 0.10 0

], F =

[0 0.050 0

],

B1 =

[−0.2250.45

], C1 =

[−0.5 040 0

], B2 =

[0.04± b0.05

], D12 =

[00.1

],

where G = 0. Taking a = 0, b = 0 for the nominal case and applying the resultof Theorem 5.3, we obtain for a delay bound of h = 11, a near minimumattenuation level of γ = 6.79 for εb = 0.0001. The controller gain is K =[16.099 5.804] and the closed-loop system poles are at 0.5223 0.0119. Takinga ∈ [−0.1 0.1] and b ∈ [−0.02 0.02], we obtain for the latter delay bounda near minimum attenuation level of γ = 7.68 for εb = 10−6. The controllergain is K = [13.8653 3.1584].

106 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

5.6 Delayed Filtering

In this section we address the filtering problem of the delayed state-multiplicative noisy system. We consider the system of (5.2a,b) and (5.3)with B2 = 0, G = 0, D12 = 0 and the general type filter of (5.5). Denoting

ξTkΔ= [xTk xTk ], w

Tk

Δ= [wT

k nTk ] we obtain the following augmented system:

ξk+1 = A0ξk + Bwk + A1ξk−τ(k) + Dξkνk + F ξk−τ(k)μk, ξl = 0, l ≤ 0,

zk = Cξk,(5.33)

where

A0 =

[A0 0

BcC2 Ac

], B =

[B1 0

0 BcD21

], A1 =

[A1 0

0 0

], D =

[D 0

0 0

],

F =

[F 0

0 0

], C = [C1 − Cc].

(5.34)

Using the BRL result of Section 5.4 we obtain the following inequality condi-tion:

ΥΔ=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 0 0 Υ15 0 CT Υ18∗ −Q QA1 − QM QM 0 QB 0 0

∗ ∗ Υ33 0 εfh[AT1 Q− QT

M ] 0 0 0

∗ ∗ ∗ −εf Q −hεfQTM 0 0 0

∗ ∗ ∗ ∗ −εfQ hεf QB 0 0∗ ∗ ∗ ∗ ∗ −γ2Iq+p 0 0∗ ∗ ∗ ∗ ∗ ∗ −Ir 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −Q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(5.35)where

Υ11 = −Q+ R1,

Υ12 = AT0 Q+ QT

M ,

Υ15 = εfh[AT0 Q+ QT

M ]− εfhQ,Υ18 = DT Q

√1 + εfh2,

Υ33 = −R1 + (1 + εfh2)FT QF .

Defining P = Q−1, denoting the following partitions

5.6 Delayed Filtering 107

P =

[X MT

M T

],

Q =

[Y NT

N W

],

J =

[X−1 Y

0 N

],

we multiply (5.35) by J = diag{PJ, PJ, P J, P J, PJ, I, I, PJ} from the rightand by JT , from the left. We obtain, denoting Rp = JT P R1PJ,

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Υ11 Υ12 0 0 Υ15

∗ −JT P J Υ23 JT P QM P J 0

∗ ∗ −Rp 0 Υ35

∗ ∗ ∗ −εfJT P J Υ45

∗ ∗ ∗ ∗ −εfJT PJ

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

0 JT P CT εJT P DTJ 0

JT B 0 0 0

0 0 0 εJT P FTJ

0 0 0 0

hεbJT B 0 0 0

−γ2Iq+p 0 0 0

∗ −Ir 0 0

∗ ∗ −JT P J 0

∗ ∗ ∗ −JT P J

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (5.36)

where

108 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

Υ11 = −JT P J + Rp,

Υ12 = JT P AT0 J + JT P QT

M P J,

Υ15 = εfh[JT P AT

0 J + JT P QTM P J ]− εfhJT P J,

Υ23 = JT A1P J − JT P QM P J,

Υ35 = εfh[JT P AT

1 J − JT P QTM P J ],

Υ45 = −hεfJT P QTM P J,

ε2 = 1 + εfh2.

DenotingX = X−1, PM = JT P QT

M P J,

we obtain:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−[X XX Y

]+ Rp Ψ12 0 0 Ψ15

∗ −[X XX Y

]Ψ23 PM 0

∗ ∗ −Rp 0 Ψ35

∗ ∗ ∗ −εf[X XX Y

]Ψ45

∗ ∗ ∗ ∗ −εf[X XX Y

]

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

0 JT P CT εJT P DTJ 0

JT B 0 0 0

0 0 0 εJT P FTJ

0 0 0 0

hεbJT B 0 0 0

−γ2Iq+p 0 0 0

∗ −I 0 0

∗ ∗ −[X XX Y

]0

∗ ∗ ∗ −[X XX Y

]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (5.37)

where

5.6 Delayed Filtering 109

Ψ12 = JT P AT0 J + PM ,

Ψ15 = εfh[JT P AT

0 J + PM ]− εfh[X XX Y

],

Ψ23 = JT A1P J − PM ,

Ψ35 = εfh[JT P AT

1 J − PM ],

Ψ45 = −hεf PM ,

ε2 = 1 + εfh2.

Carrying out the various multiplications and denoting K0 = NTAcMX, U =NTBc and Z = CcMX, we obtain the following result:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−[X XX Y

]+ Rp Ψ12 0 0 Ψ15

∗ −[X XX Y

]Ψ23 PM 0

∗ ∗ −Rp 0 Ψ35

∗ ∗ ∗ −εf

[X XX Y

]−hεf PM

∗ ∗ ∗ ∗ −εf

[X XX Y

]

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

0

[CT

1 − ZT

CT1

] [DT X DTY

DT X DTY

]0

[XB1 0

Y B1 UD21

]0 0 0

0 0 0

[F T X F TY

F T X F TY

]

0 0 0 0

hεf

[XB1 0

Y B1 UD21

]0 0 0

−γ2Iq+p 0 0 0

∗ −I 0 0

∗ ∗ −[X XX Y

]0

∗ ∗ ∗ −[X XX Y

]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0, (5.38)

110 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

where

Ψ12 =

[AT

0 X AT0 Y + CT

2 UT +KT

0

AT0 X AT

0 Y + CT2 U

T

]+ PM ,

Ψ15 = εfh

[AT

0 X AT0 Y + CT

2 UT +KT

0

AT0 X AT

0 Y + CT2 U

T

]+ εfhPM − εfh

[X XX Y

],

Ψ23 =

[XA1 XA1

Y A1 Y A1

]− PM ,

Ψ35 = εfh

[AT

1 X AT1 Y

AT1 X AT

1 Y

]− εfhPM ,

ε2 = 1 + εfh2.

We thus arrive at the following theorem:

Theorem 5.4. Consider the system of (5.2a,b) and (5.3) with B2 = 0 andG = 0, D12 = 0. For a prescribed scalar γ > 0 and a positive tuning scalar εf ,there exists a filter of the structure (5.5) that achieves JF < 0, where JF isgiven in (5.7), for all nonzero w ∈ l2([0,∞);Rq), n ∈ l2([0,∞);Rp), if thereexist n× n matrices X > 0, Y > 0, 2n× 2n matrix Rp > 0, n× n matrices

K0 and U, 2n×2n matrix PM and a n× l matrix Z, that satisfy (5.38). In thelatter case the filter parameters can be extracted using the following equations:

Ac = N−TK0XM−1, Bc = N−TU, Cc = ZXM−1. (5.39)

5.6 Delayed Filtering 111

In the uncertain case we obtain the following set of LMIs:

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−[X XX Y

]+ Rp Ψ i

12 0 0 Ψ i15

∗ −[X XX Y

]Ψ i23 PM 0

∗ ∗ −Rp 0 Ψ i35

∗ ∗ ∗ −εf[X XX Y

]−hεf PM

∗ ∗ ∗ ∗ −εf[X XX Y

]

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

0

[Ci,T

1 − ZT

Ci,T1

] [Di,T X Di,TY

Di,T X Di,TY

]0

[XBi

1 0

Y Bi1 UD

i21

]0 0 0

0 0 0

[F i,T X F i,TY

F i,T X F i,TY

]

0 0 0 0

hεb

[XBi

1 0

Y Bi1 UD

i21

]0 0 0

−γ2Iq+p 0 0 0

∗ −I 0 0

∗ ∗ −[X XX Y

]0

∗ ∗ ∗ −[X XX Y

]

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

< 0,

(5.40)∀i, i = 1, 2, ...., N, where

112 5 H∞ Control and Estimation of Retarded Linear Discrete-Time Systems

Ψ i12 =

[Ai,T

0 X Ai,T0 Y + Ci,T

2 UT +KT0

Ai,T0 X Ai,T

0 Y + Ci,T2 UT

]+ PM ,

Ψ i15 = εfh

[Ai,T

0 X Ai,T0 Y + Ci,T

2 UT +KT0

Ai,T0 X Ai,T

0 Y + Ci,T2 UT

]+ εfhPM − εfh

[X XX Y

],

Ψ i23 =

[XAi

1 XAi1

Y Ai1 Y A

i1

]− PM ,

Ψ i35 = εfh

[Ai,T

1 X Ai,T1 Y

Ai,T1 X Ai,T

1 Y

]− εfhPM ,

ε2 = 1 + εfh2.

Noting that XY − MTN = I, the filter matrix parameters Ac, Bc, andCc can be readily found, without any loss of generality, by a singular valuedecomposition of I −XY.

Corollary 5.6.1 Consider the system of (5.2a,b) and (5.3) with B2 = 0 andG = 0, D12 = 0, where the system matrices lie within the polytope Ω of (5.8).For a prescribed scalar γ > 0 and a positive tuning scalar εf , there exists afilter of the structure (5.5) that achieves JF < 0, where JF is given in (5.7),for all nonzero w ∈ l2([0,∞);Rq), n ∈ l2([0,∞);Rp), if there exist matricesX > 0, Y > 0, K0, U, Rp, PM and Z, as in Theorem 5.4, that satisfy (5.40).In the latter case the filter parameters can be extracted using (5.39).

5.6.1 Example – Filtering

We consider the system of (5.2a,b) and (5.3) with the following system ma-trices:

A =

[0.1 0.6−1 −0.5

], D =

[0 0.0630 0

], A1 =

[0 0.10 0

], F =

[0 0.010 0

],

B1 =

[−0.2250.45

], C1 =

[−0.5 040 0

], C2 =

[0 1

], D21 =

[0.01

],

where B2 = 0, D12 = 0, and G = 0. Applying the result of Theorem 5.4,we obtain for a delay bound of h = 12, a near minimum attenuation level ofγ = 4.28 for εf = 0.001. The filter matrix parameters are:

Ac =

[5.496 −9.30016.406 −29.483

], Bc =

[−9.3442.363

], Cc =

[2.1765 9.5492

0 0

].

5.7 Conclusions 113

5.7 Conclusions

In this chapter the theory of linear H∞ state-feedback control and filtering ofstate-multiplicative noisy systems is developed for discrete-time delayed sys-tems, where the stochastic uncertainties are encountered in both the delayedand the non delayed states in the state space model of the system. The delay isassumed to be unknown and time-varying where only the bound on its size isgiven. Delay dependent analysis and synthesis methods are developed whichare based on the input–output approach, in accordance with the approachtaken in Chapter 2 for the solution of the continuous-time state-feedback andfiltering problems. This approach transforms the delayed system to a nonre-tarded system with norm-bounded operators.

Sufficient conditions are thus derived for the stability of the system andthe existence of a solution to the corresponding BRL. Based on the BRLderivation, the state-feedback control and filtering problems are formulatedand solved.

An inherent overdesign is admitted to our solution due to the use of thebounded operators which enable us to transform the retarded system to annorm-bounded one. Some additional overdesign is also admitted in our solu-tion due to the special structure imposed on R2. The numerical examples atthe end of each section demonstrate the efficiency of the proposed results.