Strong delay-independent stability analysis of neutral delay systems with commensurate delays☆
Introduction
From a practical point of view, many systems can be modeled by delay-differential equations of neutral type, such as lossless transmission line [13], partial element equivalent circuits [1], [2], controlled constrained manipulators [21], complex dynamical networks [18], etc. From a theoretical point of view, the stability analysis problem of such a system is very complex due to the presence of exponential type transcendental terms in the system characteristic equation [27]. Thus, the stability analysis of neutral delay systems has attracted considerable attention during the past decades [5], [7], [8], [23], [33]. In this paper, we consider a class of neutral delay systems in the form ofwhere are given constant matrices of size , and is the basic time delay.
The stability analysis of neutral delay systems is often divided into two categories: delay-independent stability and delay-dependent stability, depending respectively on whether or not the stability criteria contains information of delays [11]. Since the information of delays is often hard to be obtained in practice, delay-independent stability has been extensively investigated, for example [10], [19]. Delay-independent stability often involves two different notions: strong delay-independent stability and weak delay-independent stability. The strong delay-independent stability, although being a special case of the weak delay-independent stability, is sufficiently general from the point that it is usually more robust against perturbations in the system matrices [3]. In this paper, we focuses on strong delay-independent stability.
Delay-independent stability has been studied in the literature based on time-domain techniques [29], [31], [32]. Approaches by quadratic Lyapunov-Krasovskii functionals are intensively used, which lead to conditions expressed by linear matrix inequalities [16], [20]. These stability criteria are always easy to check, but only sufficient. In order to derive nonconservative stability conditions, frequency domain methodologies have been used to study the delay-independent stability of time delay systems [28], [35]. For example, for retarded systems with a single delay, necessary and sufficient stability criteria were proposed based on frequency discretization technique in [17], and a linear matrix inequality was established by using the Kalman-Yakubovich-Popov lemma in [9], [25]. The method in [25] was also extended to solve the strong delay-independent stability problem of system (1), which results in a stability criterion [26]. However, such a condition appears as a nonlinear function of the coefficients of system (1), and thus seems difficult to be used for robust stability analysis. In [4], Bliman established a set of linear matrix inequalities for testing the strong delay-independent stability of system (1) with . Meanwhile, the proposed method was directly applied to the case of by transforming system (1) into a higher dimensional neutral system with a single delay. However, this treatment will lead to a dramatic increase of computational burden, which is undesirable in practice.
Motivated by the aforementioned discussion, in this paper we revisit the strong delay-independent stability analysis problem of neutral delay systems with commensurate delays. The main contributions of this paper can be summarized as follows:
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A set of linear matrix inequalities is developed for assessing the strong delay-independent stability of the neutral delay system (1). More precisely, we give a family of linear matrix inequalities of increasing size, and each of them is sufficient for the delay-independent stability of system (1); reciprocally, the strong delay-independent stability of system (1) implies that the linear matrix inequalities are solvable beyond a certain size.
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Different from the method in [4], in which system (1) is first transformed into a higher dimensional neutral system with a single delay, our proposed method directly deals with the original system (1). As a consequence, the established condition in this paper possesses a lower computational burden when compared with the existing results [4], [26].
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A time-domain interpretation of the proposed stability criterion is given in terms of a quadratic integral Lyapunov functional.
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Based on the fact that the proposed condition involves matrices that are linear functions of the coefficients of the neutral delay systems, the robust stability problem is solved for neutral delay systems with norm bounded uncertainty.
The rest of this paper is organized as follows. Section 2 provides some preliminary results. In Section 3.1, the strong delay-independent stability condition is presented. A time-domain interpretation and a comparison of different approaches are given in Section 3.2 and Section 3.3, respectively. The robust strong delay-independent stability analysis problem is discussed in Section 4. Algorithms and examples are presented in Section 5. Section 6 concludes the paper. Notation 1 Matrices and are the identity matrix, the zero matrix and the zero matrix, respectively. We use , , and to denote sets of complex, real numbers, positive integers and symmetric real matrices of size , respectively. Let denote the closed right half plane of the complex plane and denote the unit disk on the complex plane {}. For an arbitrary matrix we use , and to denote the transpose, conjugate-transpose and rank of respectively. For a square matrix , and are the spectral radius, the spectral abscissa and the spectral norm of . The standard notation means that is a symmetric positive definite (semi-definite) matrix. For easy reading, we list the abbreviations to be used in this paper in Table 1.
Section snippets
Preliminaries
To facilitate reading and understanding, we gather some necessary notions and technical lemmas in this section. Let’s first introduce two notions of delay-independent stability for the neutral time delay system (1). Define
The first notion is about the weak delay-independent stability, which can be stated as follows. Definition 1 The neutral delay system (1) is said to be weakly delay-independent stable if andwhere and [22]
Strong delay-independent stability analysis
In this section, we first present a new family of LMIs which constitutes a necessary and sufficient condition for the strong delay-independent stability of the neutral delay system (1). Then we give a time-domain interpretation of the proposed LMI condition and a comparison with some existing results.
To introduce the main result, for any , we define matricesand
Robust strong delay-independent stability analysis
In this section, we consider the perturbed systemwhere , are the same as in (1) andwhere , are known matrices, and denotes the norm bounded uncertainty that satisfies the inequalityIn what follows, we investigate the strong delay-independent stability of perturbed neutral delay system (15). For notation simplicity, we define
Algorithms
In this subsection, we are ready to give algorithms to show how to use Theorem 1 to find the exact region of some parameter in the system matrices , . Now, Algorithm 1 is given to show how to find the maximum value of (denoted as ) by using Theorem 1 in the case of . Similar algorithms can be applied to find the for and for . The flowchart of Algorithm 1 is shown in Fig. 2.
Examples
Example 1 Consider the case of with the following matrices (borrowed from
Conclusion
This paper has studied the strong delay-independent stability analysis problem of neutral delay systems with commensurate delays. A new family of LMIs was derived to assess strong delay-independent stability. A time-domain interpretation of the proposed LMI condition was given. Based on the proposed LMI condition, the robust strong delay-independent stability analysis problem of neutral delay systems with norm bounded uncertainty was solved. Compared the proposed results with some existing
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank Professor Bin Zhou for his valuable suggestions.
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This work was supported in part by the National Science Foundation of China (61903102, 61773387), and the Fundamental Research Funds for the Central Universities.