Short Communication
Novel delay-partitioning approaches to stability analysis for uncertain Lur’e systems with time-varying delays

https://doi.org/10.1016/j.jfranklin.2021.02.030Get rights and content

Abstract

This work deals with the problem of absolute stability analysis for a class of uncertain Lur’e systems with time-varying delays. Novel delay-partitioning approaches are presented, which are dividing the variation interval of the delay into three subintervals. Some new augment Lyapunov–Krasovskii functionals (LKFs) are defined on each of the obtained subintervals which can efficiently make use of the information of the delay and relate to the reciprocally convex combination technique and the Wirtinger-based integral inequality method. Several improved delay-dependent criteria are derived in terms of the linear matrix inequalities (LMIs). The merit of the proposed criteria lies in their less conservativeness and lower numerical complexity than relative literature. Two numerical examples are included to illustrate the effectiveness and the improvement of the proposed method.

Introduction

A Lur’e system is a class of nonlinear systems which nonlinear element satisfies certain sector constraints and is originally from a pilot robot [1]. Many systems such as Chua’s circuits, Goodwin models, Swarm models, n-scroll attractors and hyperchaotic attractors can be represented as Lur’e-type systems [2]. It is well known, time delay frequently occurs in various systems, such as nuclear reactors, chemical engineering systems, biological systems and population dynamics models, and is often the source of instability [3], [4], [5]. Thus stability and performance analysis for the delayed Lur’e systems have been extensively studied in [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], and references there in. Depending upon whether or not the stability criteria for such systems contains time-delay information, the obtained stability criteria can be divided into two kinds: delay-independent [6], [7] and delay-dependent ones. Since delay-dependent criteria take advantage of the information on the size of time-delay, many effective methods have been developed to derive less conservative delay-dependent stability criteria for delayed Lur’e systems [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].

It should be noted that LKF method is regarded as a powerful tool to deal with the stability analysis for delayed Lur’e systems and the maximum allowable delay bounds (MADBs) such that the Lur’e system remains absolute stability for any time-delay less than the MADBs has become a key index for judging the conservatism of stability criteria. In order to reduce the conservatism of stability conditions, various kinds of methods such as free weight matrix method [8], [9], Jensen integral inequality approach [10], [11], delay-decomposition approaches with augmented LKF approach [12], delay-partitioning approaches [13], [14], the triple integral LKF and the combined convex technique [15], free-matrix-based inequality method in combination with the convex combination technique [16], were employed to obtain the delay-dependent stability criteria. In addition, absolute stability of Lur’e control systems with multiple time delays and nonlinearities were studied in [17]. The robustly absolute stability for neutral-type Lur’e systems with mixed time-varying delay were discussed in [18], [19], [20]. Guarantee absolute stability for multiple delay general Lur’e control systems with multiple nonlinearities and time-varying delay feedback controllers for master-slave synchronization of Lur’e systems were designed in [21] and [22], [23], respectively. Very recently, Liu et al. [24] introduce a LKF related to the second-order Bessel-Legendre inequality, the time-derivative of the LKF was estimated by a linear function on the time-varying delay, some new absolute stability criteria were derived. However, when using the second-order Bessel–Legendre inequality [25] to deal with the bound of the non-negative integral term, the extra vectors such as 1τ(t)tτ(t)tx(s)ds, 1τ2(t)tτ(t)t(ts)x(s)ds, 1(dd(t))tdtd(t)x(s)ds and 1(dd(t))2tdtd(t)(td(t)s)x(s)ds have to introduce, which do not occur in the derivative of the LKFs and involve larger decision variables and higher computational complexity than using the Jensen integral inequality [26], the Wirtinger-based integral inequality [27] and reciprocally convex combination method [28], where x(t), d(t) and d represent the state vector, time-varying delay of x(t), and the upper bound of d(t), respectively.

It is also worth noting that the delay-partitioning method is an effective one to reduce the a criterion’s conservatism [29], [30], [31]. The main idea of the delay-partitioning method is dividing the variation interval of the time delay into some equidistant subintervals. Obviously, the interval segmentation is finer, the range of the subintervals is smaller, the more information of the delay can be obtained. However, the term tditϱT(s)Qϱ(s)ds is inevitably introduced in LKFs, where ϱT(s)=[xT(s),xT(sd1),,xT(sdN01)]T,di=dN0 and N0 is the delay-partitioning number. One can easily see that the derived conditions become complicated for the number of decision variables involved grows sharply as the delay-partitioning number N0 increases.

Motivated by the mentioned above, the aim of this work is to revisit the robust stability analysis for uncertain Lur’e system with time-varying delays. The main contribution of the work is as follows.

  • (1)

    Novel delay-partitioning approaches umerare proposed. Different from the delay-partitioning or delay-decomposing approaches used in [12], [13], [14], [29], [30], [31], [32], the time delay interval [0,d] is firstly divided into three subintervals [0,d1],[d1,d2] and [d2,d], where d1=dN,d2=(N1)dN, N2,NN and N no longer represents the number of the delay partitioning.

  • (2)

    Some new LKFs are introduced. In this work, two novel integral terms td1tϱ1T(s)Qϱ1(s)ds and td2tϱ2T(s)Qϱ2(s)ds are included in LKFs, where ϱ1T(s)=[xT(s),xT(sd2)], ϱ2T(s)=[xT(s),xT(sd1)]. The merit of the idea lies that the relationship of the vectors such as x(t),x(td1),x(td2) and x(td) are fully established by the above two integral terms and their derivatives.

  • (3)

    The obtained criteria are with less conservativeness and lower numerical complexity than relative literature. Firstly, the proposed delay-partitioning approaches can effectively reduce conservatism of the obtained criteria, but the computation amount of do not grow while N increasing. Then, to avoid involving the extra vectors which do not occur in the derivative of the LKFs by using the second-order Bessel–Legendre inequality to estimate the bounds of the non-negative integral terms, reciprocally convex combination technique and the Wirtinger-based integral inequality are utilized. The stability criteria are presented in terms of LMIs. Finally, two well-known numerical examples are given to demonstrate the effectiveness and less conservatism over the existing results.

Notation: In this paper, Rn denotes n-dimensional Euclidean space and Rn×m is the set of all n×m real matrices, N stands for natural number set. For symmetric matrices X and Y, the notation X>Y(respectively, XY) means that the matrix XY is positive definite(respectively, nonnegative). The subscript T denotes the transpose of the matrix. In denotes the identity matrix.

Section snippets

Problem statements and preliminaries

Consider a class of uncertain Lur’e systems with time-varying delays and sector-bound nonlinearities described as follows{x˙(t)=[A+ΔA]x(t)+[Ad+ΔAd]x(td(t))+[D+ΔD]w(t),z(t)=Mx(t)+Nx(td(t)),w(t)=ϕ(t,z(t)),x(t)=ϕ(t),t[d,0]where x(t)Rn is the state vector, w(t)Rm is the system input and z(t)Rm is the system output. A,Ad,D,M,N are constant matrices with appropriate dimensions. ΔA,ΔAd,ΔD are unknown matrices that represent the time-varying parameter uncertainties and are assumed to be of[ΔA,ΔA

Robust stability results

The objective of this paper is to formulate the robust stability conditions of system (1). First, the system (1) subject to Eqs. (2), (3), (4), (5) is inferred as follows:{x˙(t)=Ax(t)+Adx(td(t))+Dw(t)+Gp(t),p(t)=F(t)q(t),q(t)=H1x(t)+H2x(td(t))+H3w(t),z(t)=Mx(t)+Nx(td(t)),w(t)=ϕ(t,z(t)),x(t)=ϕ(t),t[d,0]

Note d1=dN,(N2,NN),d2=(N1)dN. Then, we divide the time interval [0,d] into three subintervals [0,d1],[d1,d2] and [d2,d]. In the following, we will propose some criteria for three casesC1:0

Absolute stability results

In this section, we consider the absolute stability for the nominal Lur’e system (1) subject to Eqs. (2), (3), (4), (5) with F(t)=0, i.e.,{x˙(t)=Ax(t)+Adx(td(t))+Dw(t),z(t)=Mx(t)+Nx(td(t)),w(t)=ϕ(t,z(t)),x(t)=ϕ(t),t[d,0]

Based on Theorems 1–3, one can easily obtain the following

Corollary 1

For given scalars N3,NN,d>0,|μ|0, the system (30) subject to the conditions (4) and (5) is absolutely stable in sector [K1,K2] if there exist n×n symmetric positive definite matrices S,R1,R2,R3, 2n×2n symmetric

Illustrative example

In this section, we will use two numerical examples to show the effectiveness and benefits of our results.

Example 1

Consider the following uncertain Lur systems (1) subject to Eqs. (2), (3), (4), (5) with the parametersA=[2000.9],Ad=[1011],G=[0.1000.1]D=[0.20.3],H1=H2=I,H3=[0,0]T,M=[0.3,0.1];N=[0.1,0.2],K1=0.2,K2=0.5.

The purpose of this example is to show the effectiveness of the proposed method and less conservatism than relative references. We compute the MADBs of d such that the Lur’e system (1)

Conclusions

This paper has discussed the absolute stability problem for uncertain Lur’e system with time-varying delays. Based on novel delay-partitioning approaches, several improved stability criteria have been derived by constructing some appropriate LKFs on the subintervals to relate to the Wirtinger-based integral inequality method and reciprocally convex technique. The obtained stability criteria are less conservatism and lower computational complexity than relative ones. Finally, two examples have

CRediT authorship contribution statement

Liang-Dong Guo: Conceptualization, Methodology, Software, Resources, Writing - original draft, Writing - review & editing, Formal analysis, Supervision, Project administration, Funding acquisition. Sheng-Juan Huang: Conceptualization, Methodology, Investigation, Software, Writing - original draft, Writing - review & editing. Li-Bing Wu: Conceptualization, Methodology, Formal analysis, Resources, Project administration.

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

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    The work was supported by the Nature Science Foundation (No.61503058 and 61773013).

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