Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities
Introduction
It is well-known that time-delays are frequently present in various physical, industrial and engineering systems. The delays may cause poor performance or even instability of systems, therefore much attention has been devoted to obtain tractable stability criteria for systems with time delay during the past few decades (see e.g., the monographs [1], [2], some recent papers [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27] and the references therein).
Let us consider the following system with time-varying delay: where is the state, and are given matrices, the time-varying delay τ is supposed to be a differentiable function satisfying conditions with known bounds and function φ gives the initial condition.
Several approaches have been elaborated and successfully applied for the stability analysis of time delay systems (see the references above for excellent overviews). A major issue in this respect is to develop delay-dependent stability conditions, which provide as wide delay bounding interval guaranteeing stability as possible. Many important results on asymptotic stability of time-delay systems have been established using the Lyapunov–Krasovskii functional (LKF) approach.
In order to enhance the effectiveness, more and more involved Lyapunov–Krasovskii functionals have been introduced during the past decades involving single/multiple integral terms with respect to quadratic functions. On the other hand, much effort has been devoted to derive more and more tight inequalities (Jensen’s inequality and different forms of Wirtinger’s inequality [1], [2], [4], [5], [22], [28], etc.) for the estimation of single, double and multiple quadratic integral terms in the derivative of the LKF. Lately, the so-called Bessel–Legendre inequality has been developed in [12] to deal with single integral terms of quadratic functions, and a wide range of multiple integral inequalities has been derived in [9], [10], [11]. Very recently, general integral inequalities that encompass all the above-mentioned inequalities are parallel presented in [16], [17] based on orthogonal polynomials in different Euclidean spaces with integral inner products.
Utilizing these results, sufficient stability conditions for linear time delay systems partly with constant, partly with time varying delays are derived in the form of linear matrix inequalities (LMIs). We note that a great part of recent works investigates time-delay systems with bounded delays (i.e., when ) and with different conditions on the delay derivative.
The aim of the present work is to investigate the stability problem of linear systems with interval time-varying delays (i.e., when τ >0) having known bounds of the delay derivative via the application of the inequalities proposed in [17].
It has to be emphasized that the time-variance of the delay causes some new problems compared to the case of constant delay: (1) new terms and different augmented variables have to be considered in the LKF for efficiency; (2) the presence of time-varying delays implies the non-convexity of the obtained matrix inequalities with respect to the length of the intervals, which necessitates special handling.
The contribution of the paper can be summarized as follows:
- 1.
A parameterized family of LKFs is introduced involving multiple integral terms and augmented state variables that are chosen in accordance with the applied estimations. The possible choice of the parameters gives great flexibility to find a compromise between the computational complexity and the reduction of conservatism.
- 2.
Relaxed sufficient LMI stability conditions for systems with interval time-varying and bounded time-varying delays are derived.
- 3.
The derivation is based on the multiple integral inequalities published in [17].
- 4.
It will be shown that the proposed condition yields less conservative results with substantially less number of decision variables than several others known from the literature.
The paper is organized as follows. Some preliminary results will be recalled in Section 2. The main result will be derived in Section 3, which starts by introducing a new Lyapunov–Krasovskii functional in Section 3.1. Next, the analysis problem will be investigated in Section 3.2. In Section 4, some numerical examples illustrate the effectiveness of the results. Finally, the conclusion will be drawn.
Standard notations are used. As usual, P > 0 ( ≥ 0) denotes the positive (semi-)definiteness of P. Symbols () denote the set of symmetric (and positive definite) matrices of size n × n. For any symbol He(A) is defined as For the sake of brevity, asterisks replace the blocks in hypermatrices that are inferred readily by symmetry. If and/or then vector and matrix are considered to be empty. The Euclidean vector norm in is ‖·‖. We also denote .
Section snippets
Preliminaries
This paper aims to derive sufficient stability conditions in LMI form based on some multiple integral inequalities recently published in [17]. For convenience of the readers, we evoke these estimations in the following lemmas.
Lemma 1 Let let E be a Euclidean space with the scalar product ⟨.,.⟩, and let form an orthogonal system. If ν ≥ 0 is a given integer, then for any the following inequality holds
where and the scalar product is[17]
Lyapunov–Krasovskii functional
The stability of system (1) can be analysed via different Lyapunov–Krasovskii functionals (LKFs). In order to define the LKF, let mi, Mij be nonnegative integers chosen according to Table 1.
Let and let the extended state variable be introduced as where if if
Numerical examples
In this section, we apply the proposed method to examples that have been used in the literature to compare the results. Consider system (1) with coefficient matrices listed in Table 2, where denote the analytical delay bounds of constant delays. The computations have been performed by using YALMIP [31] together with MATLAB.
Case τ >0. First, we consider Example 1, which is one of the most frequently used numerical examples, it is considered in numerous papers, to mention but a few, in
Conclusions
This paper has been devoted to the stability problem for continuous-time delay systems with interval time-varying delays having known bounds on the delay derivatives. A parameterized family of Lyapunov–Krasovskii functionals involving multiple integral terms has been introduced, and novel multiple integral inequalities proposed by Gyurkovics and Takács [17] have been utilized to derive sufficient stability conditions in the form of LMIs. As a corollary, sufficient stability condition has been
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2022, Journal of the Franklin InstituteCitation Excerpt :One is to construct suitable L–K functionals, for example, delay-partition-based L–K ones [14], multiple-integral based ones [15–17], delay-product-type ones [18], refined-function-based ones [19,20] and inequality-based ones [21]. The other is to develop and improve advanced techniques to tightly bound the time derivatives of L–K functionals, e.g., reciprocally convex combination lemmas [22–25] and various types of integral inequalities (i.e., Jensen, Wirtinger-based, Bessel–Legendre, auxiliary-function-based and free-matrix-based ones) [26–34]. To sum up, Corollary 1 achieves the lest conservatism with the largest number of decision variables involved.
New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay
2020, Applied Mathematics and ComputationFurther refinements in stability conditions for time-varying delay systems
2020, Applied Mathematics and ComputationCitation Excerpt :The present paper is based on the latter methodology. In the literature, when considering the Lyapunov-Krasovskii method combined with LMI-based approaches, several strategies have been proposed to reduce the conservatism, such as: discretization/partition delay method [1,20,21], new functional choices [22–28], free-weighting matrices [29–33], and improved quadratic integral inequalities [34–39]. This last approach is regarded as a convenient manner to relax the stability criteria due to the low complexity of the resulting LMIs when compared to the discretization methods.
State estimation strategy for continuous-time systems with time-varying delay via a novel L-K functional
2020, Control Strategy for Time-Delay Systems: Part II: Engineering ApplicationsA linear matrix inequality-based extended dissipativity criteria for linear systems with additive time-varying delays
2019, IFAC Journal of Systems and ControlCitation Excerpt :However, the construction of an appropriate Lyapunov functional still remains a more challenging task which plays a vital role in the reduction of conservatism. In the recent trends, it can be perceived that the inclusion of some multiple integral terms based on multiple integral inequalities may lead to some fair stability results for dynamical systems (Gyurkovics, Szabó-Varga, & Kiss, 2017; Qian, Cong, Li, & Fei, 2012; Wang, Li, Zhang, & Fei, 2017). In view of this, in this paper, the triple and quadruple integral terms have been included in the Lyapunov–Krasovskii functional which is augmented with state vectors.
Stability analysis of systems with time-varying delay via novel augmented Lyapunov–Krasovskii functionals and an improved integral inequality
2019, Applied Mathematics and ComputationCitation Excerpt :In order to improve the coupling relationships of the system states, the augmented LKF, which plays a great part in the stability analysis, was proposed [16]. After that, several improved LKFs were suggested based on the idea of containing more delay and system information and considering more cross-term relationships, such as the matrix-refined-function based LKF [11,17], the delay-product-type LKF [18–20], the multiple integral LKF [21–24], and the delay-partitioning LKF [25,26]. However, no matter how complicated those types of LKFs are, the augmented terms are shown as components of them.