# Stability Analysis of Linear Discrete-Time Systems with Interval Delay: A Delay-Partitioning Approach

Authors: Priyanka Kokil; Haranath Kar; V. Krishna Rao Kandanvli
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

This paper considers the problem of asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By using a delay partitioning-based Lyapunov functional, a new criterion for the asymptotic stability of such systems is proposed in terms of linear matrix inequalities (LMIs). The proposed stability condition depends on both the size of delay and partition size. The presented approach is compared with previously reported approaches.


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## Body

## 1. Introduction

Mathematical models with time delays are frequently encountered in various physical, industrial, and engineering systems due to measurement and computational delays, transmission and transport lags. During the last two decades, there has emerged a considerable interest in the system theoretic problems of time-delay systems. In many applications, time delays must be taken into account in a realistic system design, for instance, chemical processes, thermal processes, echo cancellation, local loop equalization, multipath propagation in mobile communication, array signal processing, congestion analysis and control in high-speed networks, neural networks and long transmission line in pneumatic systems [1–6]. The presence of time delays may result in instability of the designed systems. An excellent survey on the stability results of time-delay systems has been presented in [7]. According to dependence of delay, the available stability criteria are generally classified into two types: delay-dependent criteria and delay-independent criteria. Delay-dependent stability criteria generally lead to less conservative results than delay-independent ones, especially if the size of time delay is small [8–12].Many publications relating to the delay-dependent stability analysis of continuous time-delay systems have appeared (see, e.g., [4, 12–21] and the references cited therein). In contrast, little effort has been made for studying the problem of stability of discrete time-delay systems. Utilizing Lyapunov functional method, several delay-dependent stability criteria for discrete-time systems have been reported in the literature [2, 22–27]. By employing novel techniques to estimate the forward difference of Lyapunov functional, stability criteria for linear discrete-time systems with interval-like time-varying delays have been proposed in [26]. The criteria proposed in [26] are less conservative with smaller numerical complexity than [22, 28–31]. The delay partitioning technique has been efficiently utilized in [32–36] to the stability analysis of systems with time-varying delays. Recently, by introducing free-weighting matrices and adopting the concept of delay partitioning, improved stability criteria have been established in [37]. The criteria reported in [37] are not only dependent on the delay but also dependent on the partitioning size. Though the approach in [37] provides less conservative results than [22, 29], it would lead to heavier computational burden and more complicated synthesis procedure.Motivated by these developments, this paper studies the problem of stability analysis of linear discrete-time system with interval-like time-varying delay in the state. The paper is organized as follows. Section2 introduces model description and preliminaries. By utilizing the delay partitioning idea of [37], a new linear-matrix-inequality- (LMI-) based criterion for the asymptotic stability of discrete-time state-delayed systems is proposed in Section 3. The proposed criterion depends on the size of delay as well as partition size. An example highlighting the usefulness of the presented criterion is given in Section 4.


## 2. Model Description and Preliminaries

The following notations are used throughout the paper:Rp×q: set of p×q real matrices,Rp: set of p×1 real vectors,0: null matrix or null vector of appropriate dimension; the orders are specified in subscripts as the need arises,Ip: p×p identity matrix,BT: transpose of the matrix (or vector) B,B>0:B is positive-definite symmetric matrix,B<0:B is negative-definite symmetric matrix.In this paper, we consider a linear, autonomous, multivariable discrete-time system with interval-like time-varying delay in the state. Specifically, the system under consideration is represented by the difference equation:(2.1a)x(k+1)=Ax(k)+A1x(k-d(k)),(2.1b)x(k)=ϕ(k),k=-h2,-h2+1,…,0,where x(k)∈Rn is the system state vector, AandA1 are constant matrices with appropriate dimensions, and d(k) is a positive integer representing interval-like time-varying delay satisfying (2.2)1≤h1≤d(k)≤h2,
where h1 and h2 are known positive integers representing the lower and upper delay bounds, respectively, and ϕ(k) is an initial value at time k. Let the lower bound of the delay h1 be divided into m number of partitions such that (2.3)h1=τm,
where τ is an integer representing partition size.Definition 2.1.
The equilibrium statexe=0 of the system (2.1a) and (2.1b)–(2.3) is asymptotically stable if, for any ε>0, there exists δ>0 such that if ∥ϕ(k)∥<δ, k=-h2,-h2+1,…,0, then ∥x(k)∥<ε, for every k≥0 and limk→∞x(k)=0.Lemma 2.2 (see [30]).
For any positive definite matrixW∈Rn×n, two positive integers r and r0 satisfying r≥r0≥1, and vector function x(i)∈Rn, one has
(2.4)(∑i=r0rx(i))TW(∑i=r0rx(i))≤(r-r0+1)∑i=r0rxT(i)Wx(i).


## 3. Main Result

In this section, an LMI-based criterion for the asymptotic stability of system (2.1a) and (2.1b)–(2.3) is established. The main result may be stated as follows.Theorem 3.1.
For given positive integersτ, m, and h2, the system described by (2.1a) and (2.1b)–(2.3) is asymptotically stable if there exist real matrices P=PT>0, Qi=QiT>0  (i=1,2,3), Zi=ZiT>0  (i=1,2), X,Y and W such that
(3.1)[-Z2YTXYΨWXTWT-Z2]<0,
where
(3.2)Ψ=Λ1TΦ1Λ1+Λ2TΦ2Λ2+Λ3TΦ3Λ3-Λ4TQ2Λ4+Λ1TΦ4Λ3+Λ3TΦ4TΛ1-Λ5TZ1Λ5+Λ1TZ1Λ5+Λ5TZ1Λ1+[0Y-Y+W-W]+[0Y-Y+W-W]T,whereΛ1=[In0n×(m+2)n],Λ2=[Imn0mn×3n0mn×nImn0mn×2n],Λ3=[0n×(m+1)nIn0n],Λ4=[0n×(m+2)nIn],Λ5=[0nIn0n×(m+1)n],Φ1=ATPA-P+Q2+(h2-τm+1)Q3-Z1+(A-In)T(τ2Z1+(h2-τm)2Z2)(A-In),Φ2=[Q100-Q1],Φ3=A1T(P+τ2Z1+(h2-τm)2Z2)A1-Q3,Φ4=ATPA1+(A-In)T(τ2Z1+(h2-τm)2Z2)A1.Proof.
Choose a Lyapunov functional candidate as(3.3)V(k)=xT(k)Px(k)+∑i=k-τk-1ΥT(i)Q1Υ(i)+∑i=k-h2k-1xT(i)Q2x(i)+∑j=-h2-τm∑i=k+jk-1xT(i)Q3x(i)+∑j=-τ-1∑i=k+jk-1τΔxT(i)Z1Δx(i)+∑j=-h2-τm-1∑i=k+jk-1(h2-τm)ΔxT(i)Z2Δx(i),
where
(3.4)Υ(i)=[xT(i)xT(i-τ)⋯xT(i-(m-1)τ)]T,(3.5)Δx(i)=x(i+1)-x(i).
Taking the forward difference of (3.3) along the solution of (2.1a) and (2.1b), we have
(3.6)ΔV(k)=V(k+1)-V(k)=[Ax(k)+A1x(k-d(k))]TP[Ax(k)+A1x(k-d(k))]-xT(k)Px(k)+ΥT(k)Q1Υ(k)-ΥT(k-τ)Q1Υ(k-τ)+xT(k)Q2x(k)-xT(k-h2)Q2x(k-h2)+(h2-τm+1)xT(k)Q3x(k)-∑i=k-h2k-τmxT(i)Q3x(i)+ΔxT(k)(τ2Z1+(h2-τm)2Z2)Δx(k)-∑i=k-τk-1τΔxT(i)Z1Δx(i)-∑i=k-h2k-τm-1(h2-τm)ΔxT(i)Z2Δx(i).
Using Lemma 2.2, we obtain
(3.7)-∑i=k-τk-1τΔxT(i)Z1Δx(i)≤-(x(k)-x(k-τ))TZ1(x(k)-x(k-τ)).
Note that
(3.8)-∑i=k-h2k-τmxT(i)Q3x(i)≤-xT(k-d(k))Q3x(k-d(k)).
Now, we have the following relations:
(3.9)0=2ζT(k)Y[x(k-τm)-x(k-d(k))-∑i=k-d(k)k-τm-1Δx(i)],0=2ζT(k)W[x(k-d(k))-x(k-h2)-∑i=k-h2k-d(k)-1Δx(i)],
where Y and W are constant matrices of appropriate dimensions and
(3.10)ζ(k)=[ΥT(k)xT(k-τm)xT(k-d(k))xT(k-h2)]T.
It follows from (2.1a) and (3.5) that
(3.11)Δx(k)=(A-In)x(k)+A1x(k-d(k)).
Employing (3.6)–(3.11), we have the following inequality:
(3.12)ΔV(k)≤ζT(k)Ψζ(k)-2ζT(k)Y∑i=k-d(k)k-τm-1Δx(i)-2ζT(k)W∑i=k-h2k-d(k)-1Δx(i)-∑i=k-d(k)k-τm-1(h2-τm)ΔxT(i)Z2Δx(i)-∑i=k-h2k-d(k)-1(h2-τm)ΔxT(i)Z2Δx(i),
where Ψ is given by (3.2). Equation (3.12) can also be rearranged as
(3.13)ΔV(k)≤1(h2-τm)∑i=k-d(k)k-τm-1[ζ(k)-(h2-τm)Δx(i)]T[ΨYYT-Z2][ζ(k)-(h2-τm)Δx(i)]+1(h2-τm)∑i=k-h2k-d(k)-1[ζ(k)-(h2-τm)Δx(i)]T[ΨWWT-Z2][ζ(k)-(h2-τm)Δx(i)].
Using [17, Lemma 4.1], it can be shown that there exists a matrix X of appropriate dimensions satisfying (3.1) if and only if
(3.14)[ΨYYT-Z2]<0,[ΨWWT-Z2]<0,
which together with (3.13) implies ΔV(k)<0 for all nonzero ζ(k). This completes the proof.Remark 3.2.
A comparison of the number of the decision variables involved in several recent stability results is summarized in Table1. It may be observed that the size of complexity in [22, 26, 31] is only related to state dimension n, whereas the complexity of [37] and Theorem 3.1 depends on both n and m.
The total number of scalar decision variables of Theorem3.1 is D1=(n/2)[n(m2+4m+19)+(m+5)], and the total row size of the LMIs is L1=2n(m+5). The numerical complexity of Theorem 3.1 is proportional to L1D13 [16]. In [37], the total number of scalar decision variables of Theorem 2 is D2=(n/2)[n(3m2+18m+41)+(3m+11)], the total row size of the LMIs is L2=n(7m+31), and the numerical complexity is proportional to L2D23. Therefore, Theorem 3.1 has much smaller numerical complexity than Theorem 2 of [37].Table 1
Comparison of the number of decision variables involved in various methods.
MethodsNumber of the decision variablesTheorem 1 in [22](23n2+5n)/2Theorem 3 in [22](67n2+9n)/2Theorem 1 in [31]9n2+3nTheorem 1 in [23]13n2+5nProposition 1 in [26]8n2+3nTheorem 2 in [37](n/2)[n(3m2+18m+41)+(3m+11)]Theorem3.1(n/2)[n(m2+4m+19)+(m+5)]Remark 3.3.
For a givenh1, the allowable maximum value of h2 for guaranteeing the asymptotic stability of system (2.1a) and (2.1b)–(2.3) can be obtained by iteratively solving the LMI (3.1).Remark 3.4.
Withm=1, Y=[Y1TY2T0n×2n]T, and W=[W1TW2T0n×2n]T, Theorem 3.1 reduces to an equivalent form of [26, Proposition 1]. Thus, as compared to [26, Proposition 1], Theorem 3.1 provides additional degrees of freedom in the selection of m, Y, and W which would result in an enhanced stability region in the parameter space.Remark 3.5.
Using similar steps as in the proof of [37, Proposition 8], it is easy to establish that the conservatism of the stability result obtained via Theorem 3.1 is nonincreasing as the number of partitions increases.
It may be noted that, in the derivation of Theorem3.1, the lower bound of the delay is assumed to be h1=τm≥1. For the situation where h1=0, we have the following result.Theorem 3.6.
The system (2.1a) and (2.1b) with 0≤d(k)≤h2 is asymptotically stable if there exist real matrices P=PT>0, Qi=QiT>0(i=1,2), Z=ZT>0, X, Ỹ, and W̃ such that
(3.15)[-ZỸTXỸΨ̃W̃XTW̃T-Z]<0,
where
(3.16)Ψ̃=Λ̃1TΦ̃1Λ̃1+Λ̃1TΦ̃2Λ̃2+Λ̃2TΦ̃2TΛ̃1+Λ̃2TΦ̃3Λ̃2-Λ̃3TQ1Λ̃3+[Ỹ-Ỹ+W̃-W̃]+[Ỹ-Ỹ+W̃-W̃]T,whereΛ̃1=[In0n×2n],Λ̃2=[0nIn0n],Λ̃3=[0n×2nIn],Φ̃1=ATPA-P+h22(A-In)TZ(A-In)+Q1+(h2+1)Q2,Φ̃2=ATPA1+h22(A-In)TZA1,Φ̃3=A1T(P+h22Z)A1-Q2.Proof.
Choosing the Lyapunov functional as(3.17)V(k)=xT(k)Px(k)+∑i=k-h2k-1xT(i)Q1x(i)+∑j=-h20∑i=k+jk-1xT(i)Q2x(i)+∑j=-h2-1∑i=k+jk-1h2ΔxT(i)ZΔx(i)
and employing the steps of the proof of Theorem 3.1 discussed earlier, one can easily arrive at the above theorem. The details of the proof of Theorem 3.6 are, therefore, omitted.Remark 3.7.
WithL̃1=9n  rows and D̃1=9n2+2n scalar decision variables, LMI (3.15) in Theorem 3.6 has the numerical complexity proportional to L̃1D̃13 [16], while the condition in [37, Proposition 10] has the numerical complexity proportional to L̃2D̃23, where L̃2=21n and D̃2=(n/2)(25n+7). So, Theorem 3.6 has smaller numerical complexity than [37, Proposition 10].


## 4. A Numerical Example

To demonstrate the applicability of the presented results and compare them with previous results, we now consider a specific example of system (2.1a), (2.1b) with (4.1)A=[0.800.050.9],A1=[-0.10-0.2-0.1]
and 0≤d(k)≤h2. This example has been considered in [22, 29, 37]. Using Matlab LMI toolbox [38, 39], it is found from Theorem 3.6 that the present system is stable for 0≤d(k)≤17. In this case, the number of decision variables involved in Theorem 3.6 is 40. On the other hand, to obtain the upper bound h2=17, [37, Proposition 10] requires 57 decision variables. This demonstrates the numerical efficiency of the proposed method.Next, consider the system described by (2.1a), (2.1b)–(2.3), and (4.1). For different values of h1, the admissible upper bound h2 is listed in Table 2. From Table 2, it is clear that Theorem 3.1 can provide a larger upper bound h2 than the previously reported stability results [22, 23, 26, 29]. It may also be observed that, for the example under consideration, Theorem 3.1 leads to upper bound h2 which is identical to that arrived at via Theorem 2 in [37]. However, as discussed in Remark 3.2, Theorem 3.1 has much smaller numerical complexity than Theorem 2 in [37].Table 2
Admissible upper bound ofh2 for various h1.
Methodh1=4h1=12Theorem 1 in [29]813Theorem 1 in [22]1316Theorem 1 in [23]1721Proposition 1 in [26]1721Theorem 2 in [37]17(m=2,τ=2)21(m=1,τ=12)18(m=4,τ=1)22(m=2,τ=6)Theorem3.117(m=2,τ=2)21(m=1,τ=12)18(m=4,τ=1)22(m=2,τ=6)


## 5. Conclusion

In this paper, the problem concerning the asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state has been considered. Using the concept of delay partitioning, an LMI-based criterion for the asymptotic stability of such systems has been established. The criterion depends on the size of delay as well as partition size. The presented approach may imply asymptotic stability for a broader class of time-varying state-delayed systems, as compared to previous approaches [22, 23, 26, 29]. The presented criterion is numerically less complex than [37]. The stability results discussed in this paper can easily be extended to delayed discrete-time systems with parameter uncertainties.



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