Volume 13, Issue 1 p. 1-16
Review Article
Free Access

Overview of recent advances in stability of linear systems with time-varying delays

Xian-Ming Zhang

Xian-Ming Zhang

School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC, 3122 Australia

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Qing-Long Han

Corresponding Author

Qing-Long Han

School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC, 3122 Australia

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Alexandre Seuret

Alexandre Seuret

LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France

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Frédéric Gouaisbaut

Frédéric Gouaisbaut

LAAS-CNRS, Université de Toulouse, CNRS, UPS, Toulouse, France

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Yong He

Yong He

School of Automation, China University of Geosciences, Wuhan, 430074 People's Republic of China

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First published: 01 January 2019
Citations: 196

Abstract

This study provides an overview and in-depth analysis of recent advances in stability of linear systems with time-varying delays. First, recent developments of a delay convex analysis approach, a reciprocally convex approach and the construction of Lyapunov–Krasovskii functionals are reviewed insightfully. Second, in-depth analysis of the Bessel–Legendre inequality and some affine integral inequalities is made, and recent stability results are also summarised, including stability criteria for three cases of a time-varying delay, where information on the bounds of the time-varying delay and its derivative is totally known, partly known and completely unknown, respectively. Third, a number of stability criteria are developed for the above three cases of the time-varying delay by employing canonical Bessel–Legendre inequalities, together with augmented Lyapunov–Krasovskii functionals. It is shown through numerical examples that these stability criteria outperform some existing results. Finally, several challenging issues are pointed out to direct the near future research.

1 Introduction

A time-delay system is also called a system with after-effect or dead-time [1]. A defining feature of a time-delay system is that its future evolution is related not only to the current state but also to the past state of the system. Time-delay systems are a particular class of infinite dimensional systems, which have complicated dynamic properties compared with delay-free systems. A large number of practical systems encountered in areas, such as engineering, physics, biology, operation research and economics, can be modelled as time-delay systems [2, 3]. Therefore, time-delay systems have attracted continuous interest of researchers in a wide range of fields in natural and social sciences, see, e.g. [49].

Stability of time-delay systems is a fundamental issue from both theoretical and practical points of view. Indeed, the presence of time-delays may be either beneficial or detrimental to stability of a practical system. Time-delays are usually regarded as a factor of system destabilisation. However, practical engineering applications reveal that, for some dynamical systems, intentional introduction of a specific time-delay may stabilise an unstable system [4]. Hence, one main concern about a time-delay system is to determine the maximum delay interval on which the system remains stable. For non-linear time-delay systems, it is quite challenging due to complicated dynamical properties. For linear time-delay systems, a lot of effort has been made on delay-dependent stability analysis in the last two decades, see, e.g. [711]. In this paper, we focus on the following linear system with a time-varying delay described by
urn:x-wiley:17518644:media:cth20001:cth20001-math-0001(1)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0002 is the system state; A and urn:x-wiley:17518644:media:cth20001:cth20001-math-0003 are real urn:x-wiley:17518644:media:cth20001:cth20001-math-0004 constant matrices; urn:x-wiley:17518644:media:cth20001:cth20001-math-0005 is the initial condition; and urn:x-wiley:17518644:media:cth20001:cth20001-math-0006 is a time-delay satisfying urn:x-wiley:17518644:media:cth20001:cth20001-math-0007. If urn:x-wiley:17518644:media:cth20001:cth20001-math-0008, the system (1) is called a linear system with an interval time-varying delay [12], which allows the system to be unstable at urn:x-wiley:17518644:media:cth20001:cth20001-math-0009. Networked control systems and event-triggered control systems can be modelled as such a system with an interval time-varying delay [1320]. For simplicity of presentation, in what follows, let urn:x-wiley:17518644:media:cth20001:cth20001-math-0010 if no specific declaration is made.

Recalling some existing results, there are two types of approaches to delay-dependent stability of linear time-delay systems: frequency domain approach and time domain approach. Frequency domain approach-based stability criteria have been long in existence [1, 9]. For some recent developments in the frequency domain, we mention an integral quadratic constraint framework [2123], which describes the stability of a system in the frequency domain in terms of an integral constraint on the Fourier transform of the input/output signals [24]. In the time domain approach, the direct Lyapunov method is a powerful tool for studying stability of linear time-delay systems [1, 9]. Specifically, there are complete Lyapunov functional methods and simple Lyapunov–Krasovskii functional methods for estimating the maximum admissible delay upper bound that the system can tolerate and still maintain stability. Complete Lyapunov functional methods can provide necessary and sufficient conditions on stability of linear systems with a constant time-delay [2528]. Simple Lyapunov-Krasovskii functional methods only provide sufficient conditions on stability of linear time-delay systems. Compared with stability criteria based on complete Lyapunov functional methods, although stability criteria based on simple Lyapunov-Krasovskii functional methods are more conservative, they can be applied easily to control synthesis and filter design of linear time-delay systems [29].

A simple Lyapunov–Krasovskii functional method for estimating the maximum delay upper bound for linear time-delay systems is based on a proper simple Lyapunov–Krasovskii functional candidate, which usually includes a double integral term as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0011(2)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0012 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0013 being either a constant or a time-varying function. A delay-dependent stability criterion can be derived based on an estimate of the time derivative of the Lyapunov–Krasovskii functional, in which such an integral term is included
urn:x-wiley:17518644:media:cth20001:cth20001-math-0014(3)
The key to the stability criterion is how to deal with the quadratic integral term urn:x-wiley:17518644:media:cth20001:cth20001-math-0015. Typically, there are several approaches reported in the literature.
  1. Model transformation approach. The model transformation approach employs the Leibniz–Newton formula urn:x-wiley:17518644:media:cth20001:cth20001-math-0016 to transform the system (1) to a system such that a cross-term urn:x-wiley:17518644:media:cth20001:cth20001-math-0017 is introduced in the derivative of the Lyapunov–Krasovskii functional. Then using the basic inequality (or called Young's inequality)

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0018(4)
    or the improved basic inequality [30], for urn:x-wiley:17518644:media:cth20001:cth20001-math-0019
    urn:x-wiley:17518644:media:cth20001:cth20001-math-0020
    or the general basic inequality [31]: if urn:x-wiley:17518644:media:cth20001:cth20001-math-0021
    urn:x-wiley:17518644:media:cth20001:cth20001-math-0022(5)
    the cross-term urn:x-wiley:17518644:media:cth20001:cth20001-math-0023 can be bounded by urn:x-wiley:17518644:media:cth20001:cth20001-math-0024, which exactly ‘offsets ’ the quadratic integral term urn:x-wiley:17518644:media:cth20001:cth20001-math-0025 in the derivative of the Lyapunov-Krasovskii functional, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0026 is a proper quadratic function on urn:x-wiley:17518644:media:cth20001:cth20001-math-0027 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0028. As a result, a delay-dependent stability criterion can be derived in terms of linear matrix inequalities. There are a number of model transformation approaches proposed in the literature, namely, ‘first-order transformation’ [32], ‘parameterised first-order transformation’ [33], ‘second-order transformation’ [34], ‘neutral transformation’ [35] and ‘descriptor model transformation’ [36, 37]. As pointed out in [38, 39], under the first-order transformation, or the parameterised first-order transformation, or the second-order transformation, the transformed system is not equivalent to the original one due to the fact that additional eigenvalues are introduced into the transformed system. Under the neutral transformation, although no explicit additional eigenvalue is introduced, some additional eigenvalue constraints for the stability of an appropriate operator should be satisfied [39]. The descriptor model transformation delivers some larger delay upper bounds since the transformed system is equivalent to the original one.

  2. Free-weighting matrix approach. Compared with model transformation approaches, a free-weighting matrix approach can provide an easier way to deal with the quadratic integral term urn:x-wiley:17518644:media:cth20001:cth20001-math-0029. By introducing some proper zero-valued terms as [40, 41]

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0030
    where urn:x-wiley:17518644:media:cth20001:cth20001-math-0031 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0032 are called free-weighting matrices, the derivative of the Lyapunov–Krasovskii functional can be expressed as urn:x-wiley:17518644:media:cth20001:cth20001-math-0033, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0034 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0035 is a certain matrix. Then a delay-dependent stability criterion is obtained. It is clear that the model transformation and the bounding of cross-terms are obviated by the free-weighting matrix approach. Moreover, simulation results have shown that free-weighting matrix approaches can produce larger delay upper bounds than model transformation approaches.

  3. Integral inequality approach. An integral inequality approach directly provides an upper bound for the quadratic integral term urn:x-wiley:17518644:media:cth20001:cth20001-math-0036 [4244]. By using the Leibniz–Newton formula, an integral inequality for urn:x-wiley:17518644:media:cth20001:cth20001-math-0037 is proposed, which reads as

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0038(6)
    where urn:x-wiley:17518644:media:cth20001:cth20001-math-0039 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0040 is a free matrix.

In [45], Jensen integral inequality is introduced for the first time in the stability of time-delay systems. In [46], Jensen integral inequality is used to derive a different upper bound for urn:x-wiley:17518644:media:cth20001:cth20001-math-0041 as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0042(7)
This inequality can be also derived from (6), where the free weighting matrix M is selected as urn:x-wiley:17518644:media:cth20001:cth20001-math-0043. The integral inequality approach also employs neither model transformations nor cross-term bounding. It is verified that, a stability criterion based on the integral inequality (6) or (7) can obtain the same delay upper bound as that based on the free-weighting matrix approach. Notice that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0044(8)
Thus, the relationship between the integral inequalities (6) and (7) can be disclosed by
urn:x-wiley:17518644:media:cth20001:cth20001-math-0045
which means that the integral inequality (7) provides a minimum upper bound for urn:x-wiley:17518644:media:cth20001:cth20001-math-0046 among the set of upper bounds given by the integral inequality (6). Therefore, the integral inequality (7) has been attractive in the stability analysis since it can derive some stability criteria without introducing any extra matrix variable, if compared with the integral inequality (6) and the free-weighting matrix approach.
In [47], a tight bound for urn:x-wiley:17518644:media:cth20001:cth20001-math-0047 is obtained as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0048(9)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0049. Based on the tight bound, a less conservative stability criterion for linear systems with time-varying delay is expected incorporating with the convex delay analysis method [48], the reciprocally convex approach [49] and a proper Lyapunov–Krasovskii functional. It is shown through a number of numerical examples that the obtained stability criterion can produce an admissible maximum upper bound closely approaching to the system analytical value [50]. Boosted by the Wirtinger-based integral inequality, increasing attention is paid to integral inequality approaches, and a great number of results have been reported in the open literature, see, e.g. [5160].

In this paper, we provide an overview and in-depth analysis of integral inequality approaches to stability of linear systems with time-varying delays. First, an insightful overview is made on convex delay analysis approaches, reciprocally convex approaches and the construction of Lyapunov–Krasovskii functionals. Second, in-depth analysis of Bessel–Legendre inequalities and some affine integral inequalities is made, and recent stability results based on these inequalities are reviewed. Specifically, the refined allowable delay sets are discussed with insightful understanding. Third, we develop a number of stability criteria based on a canonical Bessel–Legendre inequality recently reported, taking three cases of time-varying delay into account. Simulation results show that the canonical Bessel–Legendre inequality plus an augmented Lyapunov–Krasovskii functional indeed can produce a larger delay upper bound than some existing methods. Finally, some challenging issues are proposed for the near future research.

The remaining part of the paper is organised as follows. Section 2 gives an overview of recent advances in convex and reciprocally convex delay analysis approaches, as well as the construction of Lyapunov–Krasovskii functionals. Recent integral inequalities and their applications to stability of linear systems with time-varying delay are reviewed in Section 3. A canonical Bessel–Legendre inequality and its affine version, together with a proper augmented Lyapunov–Krasovskii functional is developed to derive some stability criteria for three cases of time-varying delays. Section 5 concludes this paper and proposes some challenging problems to be solved in the future research.

Notation: The notations in this paper are standard. urn:x-wiley:17518644:media:cth20001:cth20001-math-0050. urn:x-wiley:17518644:media:cth20001:cth20001-math-0051 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0052 denote a block-diagonal matrix and a block-column vector, respectively. urn:x-wiley:17518644:media:cth20001:cth20001-math-0053 stands for a polytope generated by two vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0054 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0055. urn:x-wiley:17518644:media:cth20001:cth20001-math-0056 represents the set of polynomials of degree less than N, where N is a positive integer; and the notation urn:x-wiley:17518644:media:cth20001:cth20001-math-0057 refers to binomial coefficients given by urn:x-wiley:17518644:media:cth20001:cth20001-math-0058. A symmetric term in a symmetric matrix is symbolised by a ‘ urn:x-wiley:17518644:media:cth20001:cth20001-math-0059 ’.

2 Recent advances in convex analysis approaches, reciprocally convex approaches and the construction of Lyapunov–Krasovskii functionals

2.1 Convex delay analysis approach

The convex delay analysis approach provides an effective way to handle the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0060 to ensure less conservatism of stability criteria [48, 61]. Suppose that a stability condition is given in the following form:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0061(10)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0062 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0063 are real symmetric matrices irrespective of urn:x-wiley:17518644:media:cth20001:cth20001-math-0064. By exploiting the convex property, the condition (10) is equivalent to two boundary linear matrix inequalities (LMIs) as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0065(11)
Some other conditions on stability are of the following form [62]:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0066(12)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0067 are real symmetric matrices independent of urn:x-wiley:17518644:media:cth20001:cth20001-math-0068. The difference from (10) is that urn:x-wiley:17518644:media:cth20001:cth20001-math-0069 is a matrix-valued quadratic function of the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0070. If urn:x-wiley:17518644:media:cth20001:cth20001-math-0071, then urn:x-wiley:17518644:media:cth20001:cth20001-math-0072 is convex with respect to urn:x-wiley:17518644:media:cth20001:cth20001-math-0073 on urn:x-wiley:17518644:media:cth20001:cth20001-math-0074, leading to a necessary and sufficient condition as urn:x-wiley:17518644:media:cth20001:cth20001-math-0075 [63]. However, if the constraint urn:x-wiley:17518644:media:cth20001:cth20001-math-0076 is not satisfied, the above conclusion is not necessarily true. In this situation, two sufficient conditions are established in [64] and [65], which are given, respectively, as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0077(13)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0078(14)
Clearly, these two sufficient conditions described in (13) and (14) are independent. The relationship between them needs to be further investigated. It should be mentioned that Theorem 1 in [66] is not correct. In fact, urn:x-wiley:17518644:media:cth20001:cth20001-math-0079 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0080 is not equivalent to urn:x-wiley:17518644:media:cth20001:cth20001-math-0081 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0082 unless the coefficient matrix of urn:x-wiley:17518644:media:cth20001:cth20001-math-0083 is semi-positive definite, where the symbols are defined in [66, Theorem 1].

2.2 Reciprocally convex delay analysis approach

In the stability analysis of the system (1), applying some integral inequalities usually yields a reciprocally convex combination on the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0084 as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0085(15)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0086; urn:x-wiley:17518644:media:cth20001:cth20001-math-0087 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0088 are urn:x-wiley:17518644:media:cth20001:cth20001-math-0089 definite-positive matrices, and urn:x-wiley:17518644:media:cth20001:cth20001-math-0090 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0091 are two n real vectors. A reciprocally convex inequality for urn:x-wiley:17518644:media:cth20001:cth20001-math-0092 is proposed in [49]: For any urn:x-wiley:17518644:media:cth20001:cth20001-math-0093 real matrix S satisfying
urn:x-wiley:17518644:media:cth20001:cth20001-math-0094(16)
the following inequality holds:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0095(17)
The reciprocally convex inequality (17) provides a lower bound for urn:x-wiley:17518644:media:cth20001:cth20001-math-0096, which is independent of urn:x-wiley:17518644:media:cth20001:cth20001-math-0097. Its application to stability analysis of the system (1) usually leads to a significant stability criterion in the sense of two aspects: (i) it requires less decision variables; and (ii) it can derive the same delay upper bound as the one using the free-weighting matrix approach, which is verified through some numerical examples [49]. Nevertheless, the second aspect only holds for stability conditions based on Jensen's inequality. This is indeed not the case anymore when employing for instance the Wirtinger-based integral inequality, which can be seen from [67] or the simulation results in the next section. Moreover, insight analysis of the reciprocally convex approach [49] is made in [68], which points out that the reciprocally convex inequality (17) can be interpreted as a discretised version of the Jensen's inequality.
Recently, an improved reciprocally convex inequality is proposed in [69] and developed in [70], which reads as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0098(18)
Compared with (17), the significance of (18) lies in two aspects: (i) the matrix constraint (16) is removed from (18); and (ii) the improved reciprocally convex inequality (18) provides a larger lower bound than (17) for urn:x-wiley:17518644:media:cth20001:cth20001-math-0099, see [70] in detail.
By introducing more slack matrix variables, a general reciprocally convex inequality is proposed in [71]: For any urn:x-wiley:17518644:media:cth20001:cth20001-math-0100 real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0101, urn:x-wiley:17518644:media:cth20001:cth20001-math-0102, urn:x-wiley:17518644:media:cth20001:cth20001-math-0103 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0104 such that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0105(19)
the following inequality holds:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0106(20)
The reciprocally convex inequality (20) is rather general: (i) Taking urn:x-wiley:17518644:media:cth20001:cth20001-math-0107 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0108, the inequality (20) reduces to (17); (ii) Taking urn:x-wiley:17518644:media:cth20001:cth20001-math-0109, urn:x-wiley:17518644:media:cth20001:cth20001-math-0110 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0111, the inequality (20) becomes (18). However, four slack matrix variables are introduced and two constraints are imposed on them, which lead to higher computation complexity of a stability criterion. Fortunately, following the idea in [70], an improved reciprocally convex inequality for urn:x-wiley:17518644:media:cth20001:cth20001-math-0112 is obtained in [72]: For any urn:x-wiley:17518644:media:cth20001:cth20001-math-0113 real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0114 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0115 with appropriate dimensions, one has
urn:x-wiley:17518644:media:cth20001:cth20001-math-0116(21)
Clearly, just two slack matrix variables are introduced and the constraints in (19) are also removed from (21). Combining the improved reciprocally convex inequality (21) with the convex delay analysis approach, some less conservative stability criteria can be derived, which is verified through some numerical examples, for details see [72].
If setting urn:x-wiley:17518644:media:cth20001:cth20001-math-0117 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0118, the basic inequality (4) immediately reduces to
urn:x-wiley:17518644:media:cth20001:cth20001-math-0119(22)
Based on (22), an estimate of urn:x-wiley:17518644:media:cth20001:cth20001-math-0120 is obtained in [52], which is in a different form as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0121(23)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0122 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0123 is a slack matrix variable. A detailed proof of (23) can be referred to [73]. The link between (21) and (23) is revealed if one considers the following slack variables:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0124
Several calculations allow indeed to recover (21) from (23).

In the sequel, it will be shown that the inequality (23) is a special case of the inequality obtained from the following lemma (see [74, Lemma 1]).

Lemma 1.Let urn:x-wiley:17518644:media:cth20001:cth20001-math-0125 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0126 be real column vectors with dimensions of urn:x-wiley:17518644:media:cth20001:cth20001-math-0127 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0128, respectively. For given real positive symmetric matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0129 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0130, the following inequality holds for any scalar urn:x-wiley:17518644:media:cth20001:cth20001-math-0131 and matrix urn:x-wiley:17518644:media:cth20001:cth20001-math-0132 satisfying urn:x-wiley:17518644:media:cth20001:cth20001-math-0133

urn:x-wiley:17518644:media:cth20001:cth20001-math-0134(24)

Apply (24) to each term in (15) to obtain
urn:x-wiley:17518644:media:cth20001:cth20001-math-0135(25)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0136 are two vectors with appropriate dimensions, and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0137(26)
If one sets urn:x-wiley:17518644:media:cth20001:cth20001-math-0138, urn:x-wiley:17518644:media:cth20001:cth20001-math-0139 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0140, the inequality (25) immediately reduces to (23).
From the analysis above, it is clear that the inequality (25) is the general form of those reciprocally convex inequalities (17), (18), (20), (21) and (23). Following the idea above, a general inequality for the reciprocally convex combination
urn:x-wiley:17518644:media:cth20001:cth20001-math-0141(27)
is derived, which can be referred to [75, Lemma 4].

2.3 Construction of Lyapunov–Krasovskii functionals

A proper Lyapunov–Krasovskii functional is crucial for deriving less conservative stability criteria for time-delay systems. However, it is still challenging to construct an exact Lyapunov–Krasovskii functional so that a necessary and sufficient stability condition can be derived for the system (1). In general, such a Lyapunov–Krasovskii functional is based on parameters which are solutions to partial differentiable equations, see, e.g. [9]. Hence, many researchers have turned to a simple Lyapunov–Krasovskii functional as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0142(28)
If urn:x-wiley:17518644:media:cth20001:cth20001-math-0143 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0144, it is proven that the positive-definiteness of the matrix Q can be weakened by [76]
urn:x-wiley:17518644:media:cth20001:cth20001-math-0145(29)
such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0146 is positive-definite. In order to reduce the conservatism of a stability criterion, a great number of Lyapunov–Krasovskii functionals are constructed on the basis of (28). In the following, typically we mention the several kinds of Lyapunov–Krasovskii functionals.

2.3.1 Augmented Lyapunov–Krasovskii functionals

An augmented Lyapunov functional is introduced in [77, 78]. A key feature of it is to augment some terms in (28) such that more information on the delayed states is exploited to derive a stability criterion. For example, in [77, 78], the first two terms urn:x-wiley:17518644:media:cth20001:cth20001-math-0147 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0148 in (28) are augmented, respectively, by
urn:x-wiley:17518644:media:cth20001:cth20001-math-0149
The augmented Lyapunov–Krasovskii functional can make the system state and some delayed states coupled closely, which possibly enhances the feasibility of the related LMIs in a stability criterion. Numerical examples show that such a stability criterion indeed can produce a larger upper bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0150 that the related system remains stable, see, e.g. [62, 79].

The purpose of the augmentation of a Lyapunov–Krasovskii functional is to help provide a tighter estimate on its derivative by introducing some new matrix variables as well as some new state-related vectors. It is true that the estimate of the derivative of a Lyapunov–Krasovskii functional depends mainly on the treatment with some integral terms. However, such an estimate sometimes is not enough for a less conservative stability criterion. In both [72, 80], it has been proven that the Wirtinger-based inequality can produce a tighter estimate on the derivative of a Lyapunov–Krasovskii functional than Jensen integral inequality, but both the obtained stability criteria are of the same conservatism if the Lyapunov–Krasovskii functional is not augmented. Recent research [81, 82] shows that using an augmented Lyapunov–Krasovskii functional plus the N -order Bessel–Legendre inequality indeed can yield nice stability criteria of less conservatism.

2.3.2 Lyapunov–Krasovskii functionals with multiple-integral terms

Another development on constructing a proper Lyapunov–Krasovskii functional is to introduce a triple-integral term as [83]
urn:x-wiley:17518644:media:cth20001:cth20001-math-0151
Following this idea, a quadruple-integral term is introduced to the augmented Lyapunov–Krasovskii functional, which is in a different form as [62]
urn:x-wiley:17518644:media:cth20001:cth20001-math-0152
More generally, multiple-integral terms are introduced as [51]
urn:x-wiley:17518644:media:cth20001:cth20001-math-0153
where m is a certain positive integer. Based on the augmented Lyapunov-Krasovskii functionals with multiple-integral terms, it is shown through numerical examples that the resulting delay-dependent stability conditions for the system (1) are less conservative, see, e.g. [84]. However, the introduction of multiple-integral terms gives rise to some new integral terms to be estimated in the derivative of the Lyapunov–Krasovskii functional [51, 63, 85].

2.3.3 Lyapunov–Krasovskii functionals for linear systems with interval time-varying delays

For a system with an interval time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0154 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0155, the stability can be analysed by constructing a proper Lyapunov–Krasovskii functional such as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0156(30)
Based on (30), a number of (augmented) Lyapunov-Krasovskii functionals are introduced by exploiting more delayed states such as urn:x-wiley:17518644:media:cth20001:cth20001-math-0157 and one can refer to [52, 83, 84, 8691] and the references therein.

It should be mentioned that, if both lower and upper bounds of urn:x-wiley:17518644:media:cth20001:cth20001-math-0158 are known to be constants, a novel Lyapunov–Krasovskii functional is introduced in [61], where the Lyapunov matrix P is chosen as a convex combination urn:x-wiley:17518644:media:cth20001:cth20001-math-0159 on urn:x-wiley:17518644:media:cth20001:cth20001-math-0160, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0161 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0162 are two positive-definite symmetric matrices. This idea is also applicable to the case of urn:x-wiley:17518644:media:cth20001:cth20001-math-0163.

2.3.4 Lyapunov–Krasovskii functionals based on a delay-fractioning approach

Another kind of Lyapunov–Krasovskii functionals is based on the delay-fractioning approach [92]. The key idea is to introduce fractions urn:x-wiley:17518644:media:cth20001:cth20001-math-0164 of urn:x-wiley:17518644:media:cth20001:cth20001-math-0165 so that the following Lyapunov–Krasovskii functional is constructed, where r is a positive integer
urn:x-wiley:17518644:media:cth20001:cth20001-math-0166(31)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0167. It is proven [92] that, as the integer r becomes larger, the obtained stability criterion is less conservative. The idea of delay-fractioning is extensively used to construct various Lyapunov–Krasovskii functionals, see, e.g. [8, 29, 9395].

3 Recent developments of integral inequality approaches to stability of linear systems with time-varying delays

In this section, we focus on the recent developments of integral inequality approaches to stability of the system (1). To begin with, we first give an overview of integral inequalities developed recently.

3.1 Recent integral inequalities

Employing the Wirtinger inequality provides a larger lower bound than the well-used Jensen integral inequality for a non-negative integral term [47]. Soon after, by introducing a proper auxiliary function, an auxiliary-function-based integral inequality is reported in [53]. Both of them are given in the following.

Lemma 2.For any constant matrix urn:x-wiley:17518644:media:cth20001:cth20001-math-0168, two scalars a and b with urn:x-wiley:17518644:media:cth20001:cth20001-math-0169, and a vector function urn:x-wiley:17518644:media:cth20001:cth20001-math-0170 such that the integrations below are well defined, the following inequalities hold:

  1. Wirtinger-based integral inequality:

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0171(32)

  2. Auxiliary-function-based integral inequality:

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0172(33)

where

urn:x-wiley:17518644:media:cth20001:cth20001-math-0173(34)

Clearly, the auxiliary-function-based integral inequality (33) is an improvement over the Wirtinger-based integral inequality (32). A natural inspiration from (33) is to extend the inequality to a general form, which is completed by introducing the Legendre polynomials, leading to the canonical Bessel–Legendre inequality [81, 82].

Lemma 3.Under the assumption in Lemma 2, the following inequality holds:

urn:x-wiley:17518644:media:cth20001:cth20001-math-0174(35)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0175(36)
where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0176(37)

The canonical Bessel–Legendre inequality includes the Wirtinger-based integral inequality and the auxiliary-function-based integral inequality as its special cases. The underlying idea of Bessel–Legendre inequality is to provide a generic and expandable integral inequality which is asymptotically (in the sense that N goes to infinity) not conservative, because of the Parseval's identity [96]. The proof of (35) relies on the expansion of the following non-negative quantity:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0177(38)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0178 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0179 being Legendre polynomials. The principles of using Bessel inequality together with orthogonal polynomials, as for instance Legendre polynomials, can be interpreted as the minimisation of the distance between the function urn:x-wiley:17518644:media:cth20001:cth20001-math-0180 and the set of polynomials of degree less than N denoted by urn:x-wiley:17518644:media:cth20001:cth20001-math-0181. Classical theories on Hilbert spaces guarantee that this minimisation problem has a unique solution, which is given by an orthogonal projection of the function urn:x-wiley:17518644:media:cth20001:cth20001-math-0182 over an orthogonal basis of urn:x-wiley:17518644:media:cth20001:cth20001-math-0183. This orthogonal projection is unique but can be expressed on a different polynomial basis such as the Legendre polynomial basis or a canonical orthogonal basis.
Lemma 3 gives a canonical Bessel–Legendre inequality of two different forms (35) and (36), where urn:x-wiley:17518644:media:cth20001:cth20001-math-0184 is a function of urn:x-wiley:17518644:media:cth20001:cth20001-math-0185 while urn:x-wiley:17518644:media:cth20001:cth20001-math-0186 is a function of urn:x-wiley:17518644:media:cth20001:cth20001-math-0187. It should be pointed out that the orthogonal polynomials chosen in [64, Lemma 1] can be expressed on the basis of urn:x-wiley:17518644:media:cth20001:cth20001-math-0188, where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0189(39)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0190 are Legendre polynomials. Thus, the integral inequality in [64] is equivalent to the above canonical Bessel–Legendre inequality if setting urn:x-wiley:17518644:media:cth20001:cth20001-math-0191.

Since urn:x-wiley:17518644:media:cth20001:cth20001-math-0192 in (35) depends on the Legendre polynomial, the inequality (35) is not convenient for use. In [82], a useful form of the canonical integral inequality is developed for stability analysis of time-delay systems, which is given in the following lemma.

Lemma 4.For an integer urn:x-wiley:17518644:media:cth20001:cth20001-math-0193, a real symmetric matrix urn:x-wiley:17518644:media:cth20001:cth20001-math-0194, two scalars a and b with urn:x-wiley:17518644:media:cth20001:cth20001-math-0195, and a vector-valued differentiable function urn:x-wiley:17518644:media:cth20001:cth20001-math-0196 such that the integrations below are well defined, then

urn:x-wiley:17518644:media:cth20001:cth20001-math-0197(40)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0198, and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0199(41)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0200(42)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0201(43)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0202(44)
with
urn:x-wiley:17518644:media:cth20001:cth20001-math-0203(45)

Notice that urn:x-wiley:17518644:media:cth20001:cth20001-math-0204 is a k -integral of the vector urn:x-wiley:17518644:media:cth20001:cth20001-math-0205. Thus, the integral inequality (40) discloses an explicit relationship between urn:x-wiley:17518644:media:cth20001:cth20001-math-0206 and the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0207, urn:x-wiley:17518644:media:cth20001:cth20001-math-0208 and the multiple integrals urn:x-wiley:17518644:media:cth20001:cth20001-math-0209.

The other class of integral inequalities is called affine integral inequalities, or free-matrix-based integral inequalities, where the coefficient urn:x-wiley:17518644:media:cth20001:cth20001-math-0210 appears linearly rather than in the form of its inverse. An affine version of the integral inequality (40) can be readily obtained based on the fact that the following inequality holds for any real matrix M with compatible dimensions
urn:x-wiley:17518644:media:cth20001:cth20001-math-0211
Thus, an affine canonical Bessel–Legendre inequality is given by
urn:x-wiley:17518644:media:cth20001:cth20001-math-0212(46)

The affine versions of (32) and (33) can be found in [67, 97] or [98]. As pointed out in [97] and [58], the affine version and its corresponding integral inequality provide an equivalent lower bound for the related integral term. It should be mentioned that those affine integral inequalities can be regarded as special cases of (46). For example, Lemma 1 in [98] is a special case of (46) with urn:x-wiley:17518644:media:cth20001:cth20001-math-0213.

3.2 Recent developments on stability of the system (1) using recent integral inequalities

Although the canonical Bessel-Legendre inequality in Lemma 3 provides a lower bound for the integral term as tight as possible if urn:x-wiley:17518644:media:cth20001:cth20001-math-0214, in the recent years, most researchers' interest is focused on its special cases such as urn:x-wiley:17518644:media:cth20001:cth20001-math-0215 [47, 99, 100] and urn:x-wiley:17518644:media:cth20001:cth20001-math-0216 [52, 53, 64, 101]. It is proven in [80] that a tighter bound of the integral term in the derivative of the Lyapunov–Krasovskii functional should not be responsible for deriving a less conservative stability criterion. Therefore, although the integral inequality (33) provides a tighter bound than (32), it is not a trivial thing to derive a less conservative stability criterion using the inequality (33). The main difficulty is that the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0217 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0218 in (34) are not easily handled in the stability analysis of the system (1). It is shown from [72] that the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0219 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0220 should occur in the derivative of the Lyapunov–Krasovskii functional so that a less conservative stability criterion for the system (1) can be obtained using the integral inequality (33). The recent development on this issue is briefly summarised as follows.

For simplicity of presentation, suppose that the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0221 in the system (1) belongs to one of three cases:
  • Case 1: urn:x-wiley:17518644:media:cth20001:cth20001-math-0222 is differentiable and satisfies

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0223(47)

  • Case 2: urn:x-wiley:17518644:media:cth20001:cth20001-math-0224 is differentiable and satisfies

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0225(48)

  • Case 3: urn:x-wiley:17518644:media:cth20001:cth20001-math-0226 is continuous and satisfies

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0227(49)
    where urn:x-wiley:17518644:media:cth20001:cth20001-math-0228, urn:x-wiley:17518644:media:cth20001:cth20001-math-0229 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0230 are real constants.

In what follows, we consider the three cases.

3.2.1 Case 1

Since information on the upper and lower bounds of the time-varying delay and its time-derivative is available, in order to formulate some less conservative stability criteria, an augmented Lyapunov–Krassovskii functional is introduced in [69] on the basis of the Lyapunov–Krassovskii functional in (28), where the first term urn:x-wiley:17518644:media:cth20001:cth20001-math-0231 is augmented with urn:x-wiley:17518644:media:cth20001:cth20001-math-0232, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0233, urn:x-wiley:17518644:media:cth20001:cth20001-math-0234, urn:x-wiley:17518644:media:cth20001:cth20001-math-0235, urn:x-wiley:17518644:media:cth20001:cth20001-math-0236, urn:x-wiley:17518644:media:cth20001:cth20001-math-0237. Then taking the time-derivative of the term urn:x-wiley:17518644:media:cth20001:cth20001-math-0238 yields some vectors similar to urn:x-wiley:17518644:media:cth20001:cth20001-math-0239 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0240 induced by the auxiliary-function-based integral inequality (33). Combining with the extended reciprocally convex inequality (18), some nice results are derived therein.

In [72], an augmented Lyapunov–Krasovskii functional is constructed as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0241(50)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0242 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0243. The significance of (50) lies in two aspects: (a) The quadratic term urn:x-wiley:17518644:media:cth20001:cth20001-math-0244 in (28) is merged into the first two integral terms such that the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0245, urn:x-wiley:17518644:media:cth20001:cth20001-math-0246 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0247 are closely coupled by urn:x-wiley:17518644:media:cth20001:cth20001-math-0248 on urn:x-wiley:17518644:media:cth20001:cth20001-math-0249 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0250 on urn:x-wiley:17518644:media:cth20001:cth20001-math-0251; and (b) The vectors induced by the auxiliary-function-based integral inequality (33) are included in the derivative of urn:x-wiley:17518644:media:cth20001:cth20001-math-0252. By employing the improved reciprocally convex inequality (21), a less conservative stability criterion is presented in [72].
In [81], using the Bessel–Legendre inequality (36), an N -dependent stability criterion is established by choosing the following augmented Lyapunov–Krasovskii functional
urn:x-wiley:17518644:media:cth20001:cth20001-math-0253(51)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0254,
urn:x-wiley:17518644:media:cth20001:cth20001-math-0255
Similar to [54], it is proven that the N -dependent stability criterion presented in [81] also forms a hierarchy, which means that its conservatism will be reduced if N is increased. On the other hand, an observation is that Lyapunov–Krasovskii functional urn:x-wiley:17518644:media:cth20001:cth20001-math-0256 is dependent on the Legendre polynomials urn:x-wiley:17518644:media:cth20001:cth20001-math-0257, which is inherited from the Bessel–Legedre inequality.

3.2.2 Case 2

Information on the lower bound of urn:x-wiley:17518644:media:cth20001:cth20001-math-0258 is not known. In this case, an augmented Lyapunov–Krasovskii functional is constructed in [64], in which two augmented terms are given as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0259(52)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0260 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0261. Applying the integral inequality similar to (33) yields four vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0262 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0263 as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0264
However, taking the derivative of the Lyapunov–Krasovskii functional just gives three vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0265, urn:x-wiley:17518644:media:cth20001:cth20001-math-0266 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0267. In order to ensure that the vector urn:x-wiley:17518644:media:cth20001:cth20001-math-0268 also appears in the derivative of the Lyapunov–Krasovskii functional, the following identity is used
urn:x-wiley:17518644:media:cth20001:cth20001-math-0269
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0270. By employing the free-weighting matrix approach and the quadratic convex condition (13), a stability criterion for the system (1) is obtained [64].

3.2.3 Case 3

The time-varying delay is only known to be continuous (possibly not differentiable), which implies that information on the derivative of the time-varying delay is unavailable. Thus, the above Lyapunov–Krasovskii functionals in Cases 1 and 2 can be no longer used to produce the vectors similar to urn:x-wiley:17518644:media:cth20001:cth20001-math-0271 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0272 in its time-derivative. In this case, an augmented Lyapunov–Krasovskii functional is constructed in [52], where the quadratic urn:x-wiley:17518644:media:cth20001:cth20001-math-0273 in (28) is augmented by urn:x-wiley:17518644:media:cth20001:cth20001-math-0274 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0275, urn:x-wiley:17518644:media:cth20001:cth20001-math-0276. By dividing the integral interval urn:x-wiley:17518644:media:cth20001:cth20001-math-0277 into two parts as urn:x-wiley:17518644:media:cth20001:cth20001-math-0278 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0279, those vectors induced from the integral inequality (35) appear in the derivative of the Lyapunov–Krasovskii functional, for details see [52, Theorem 1]. It should be pointed that stability of the system (1) with unknown information on the derivative of urn:x-wiley:17518644:media:cth20001:cth20001-math-0280 is also investigated in [53] using the auxiliary-function-based integral inequality (33), but those vectors induced from (33) do not exist in the derivative of the chosen Lyapunov–Krasovskii functional. Thus, one can claim that [53, Theorem 1] is of the same conservatism as that using the Wirtinger-based integral inequality (32) instead of (33).

From the above analysis, one can see that, it is still challenging to investigate the stability for the system (1) with time-varying delay based on the recent integral inequalities. When both the time-varying delay and its derivative are bounded from above and from below, most existing stability criteria are based on the Wirtinger-based or the auxiliary-function-based integral inequalities or the second-order Bessel–Legendre inequality. In the other cases where the information on the derivative of the time-varying delay is partly known or completely unknown, relatively few results on stability of the system (1) are obtained, even using the second-order Bessel–Legendre inequality.

3.3 Refinement of allowable delay sets

Recently, another development on stability of the system (1) is the refinement of allowable delay sets, see [81, 102]. To make it clear, suppose that a stability condition can be derived from the matrix inequality urn:x-wiley:17518644:media:cth20001:cth20001-math-0281, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0282 is a matrix-valued function that depends linearly on both urn:x-wiley:17518644:media:cth20001:cth20001-math-0283 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0284. In Case 1, urn:x-wiley:17518644:media:cth20001:cth20001-math-0285 satisfies urn:x-wiley:17518644:media:cth20001:cth20001-math-0286 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0287, which means that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0288(53)
The set urn:x-wiley:17518644:media:cth20001:cth20001-math-0289 is called an allowable delay set, which can be described as a polytope (see Fig. 1) with four vertices as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0290(54)
A stability condition is thus readily obtained provided that urn:x-wiley:17518644:media:cth20001:cth20001-math-0291 is satisfied at four vertices of urn:x-wiley:17518644:media:cth20001:cth20001-math-0292 [103]. However, it is pointed out [102] that such a stability condition is conservative in some situation. In fact, if urn:x-wiley:17518644:media:cth20001:cth20001-math-0293 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0294, two vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0295 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0296 contradict the fact that 0 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0297 are the minimum and maximum values of urn:x-wiley:17518644:media:cth20001:cth20001-math-0298. In other words, it is impossible for urn:x-wiley:17518644:media:cth20001:cth20001-math-0299 to arrive at these two vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0300 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0301 for any time urn:x-wiley:17518644:media:cth20001:cth20001-math-0302. For example, let [81]
urn:x-wiley:17518644:media:cth20001:cth20001-math-0303(55)
Then urn:x-wiley:17518644:media:cth20001:cth20001-math-0304 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0305. We now show that the vertex urn:x-wiley:17518644:media:cth20001:cth20001-math-0306 will be never reached. Note that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0307(56)
Setting urn:x-wiley:17518644:media:cth20001:cth20001-math-0308 gives urn:x-wiley:17518644:media:cth20001:cth20001-math-0309, urn:x-wiley:17518644:media:cth20001:cth20001-math-0310. Nevertheless, urn:x-wiley:17518644:media:cth20001:cth20001-math-0311, urn:x-wiley:17518644:media:cth20001:cth20001-math-0312. Likewise, it is not difficult to verify that the vertex urn:x-wiley:17518644:media:cth20001:cth20001-math-0313 is also not reached. Hence, in order to derive a less conservative stability condition, the allowable delay set urn:x-wiley:17518644:media:cth20001:cth20001-math-0314 should be refined so that all vertices can be reached. In [81, 102], the vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0315 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0316 are replaced with urn:x-wiley:17518644:media:cth20001:cth20001-math-0317 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0318, respectively, leading to a new allowable delay set urn:x-wiley:17518644:media:cth20001:cth20001-math-0319 as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0320(57)
Such an idea is extended to neural networks with time-varying delays [75, 82]. It has been shown through simulation that, a stability criterion based on the new delay set urn:x-wiley:17518644:media:cth20001:cth20001-math-0321 can produce much larger delay upper bounds than that based on urn:x-wiley:17518644:media:cth20001:cth20001-math-0322, especially for fast time-varying delays. However, there may exist some issue about those results aforementioned.
Details are in the caption following the image

Allowable delay sets urn:x-wiley:17518644:media:cth20001:cth20001-math-0323 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0324 in Case 1

On the one hand, the above analysis just keeps an eye on the two vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0325 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0326 in urn:x-wiley:17518644:media:cth20001:cth20001-math-0327 while no attention is paid to the other vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0328 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0329. In some situation, the vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0330 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0331 can also not be reached. Let us still consider the above delay function urn:x-wiley:17518644:media:cth20001:cth20001-math-0332 in (55). Set urn:x-wiley:17518644:media:cth20001:cth20001-math-0333. Then urn:x-wiley:17518644:media:cth20001:cth20001-math-0334, leading to urn:x-wiley:17518644:media:cth20001:cth20001-math-0335. Similarly, if setting urn:x-wiley:17518644:media:cth20001:cth20001-math-0336, then we have that urn:x-wiley:17518644:media:cth20001:cth20001-math-0337. In a word, these two vertices urn:x-wiley:17518644:media:cth20001:cth20001-math-0338 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0339 for this delay function can never be reached.

On the other hand, an important observation is that the allowable delay set urn:x-wiley:17518644:media:cth20001:cth20001-math-0340 may be not suitable for the description of the time-varying delay function. To reveal such a fact, we stick to urn:x-wiley:17518644:media:cth20001:cth20001-math-0341 in (55). After simple algebraic manipulations, it is found that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0342(58)
which means that the function urn:x-wiley:17518644:media:cth20001:cth20001-math-0343 belongs to a convex ellipsoid (denoted by urn:x-wiley:17518644:media:cth20001:cth20001-math-0344), see the shadow part in Fig. 2. Unfortunately, the set urn:x-wiley:17518644:media:cth20001:cth20001-math-0345 can not cover the convex ellipsoid urn:x-wiley:17518644:media:cth20001:cth20001-math-0346 for any finite value of urn:x-wiley:17518644:media:cth20001:cth20001-math-0347, which implies that a stability criterion based on urn:x-wiley:17518644:media:cth20001:cth20001-math-0348 cannot ensure the stability of the system (1) with urn:x-wiley:17518644:media:cth20001:cth20001-math-0349 being defined in (55). Thus, the set urn:x-wiley:17518644:media:cth20001:cth20001-math-0350 is not suitable for urn:x-wiley:17518644:media:cth20001:cth20001-math-0351 in (55).
Details are in the caption following the image

Allowable delay sets for urn:x-wiley:17518644:media:cth20001:cth20001-math-0352 in (55)

It is a good idea to refine the allowable delay set such that less conservative stability criteria can be obtained. However, based on the above analysis, one cannot claim that the set urn:x-wiley:17518644:media:cth20001:cth20001-math-0353 in (57) is a refinement of urn:x-wiley:17518644:media:cth20001:cth20001-math-0354. A set urn:x-wiley:17518644:media:cth20001:cth20001-math-0355 is called a refinement of urn:x-wiley:17518644:media:cth20001:cth20001-math-0356 only if urn:x-wiley:17518644:media:cth20001:cth20001-math-0357, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0358 denotes the real domain of urn:x-wiley:17518644:media:cth20001:cth20001-math-0359. Therefore, the refinement of urn:x-wiley:17518644:media:cth20001:cth20001-math-0360 aims at seeking a possible ‘minimum’ polytope within urn:x-wiley:17518644:media:cth20001:cth20001-math-0361 to cover the real domain urn:x-wiley:17518644:media:cth20001:cth20001-math-0362. How to do it depends on the delay function itself. For the delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0363 given in (55), one can build a polygon (such as the octagon with green dashed lines in Fig. 3) as small as possible to cover the ellipsoid, while for other different urn:x-wiley:17518644:media:cth20001:cth20001-math-0364 it may be not the case. It should be pointed out that the vertices of a refinement delay set are not necessarily reached by urn:x-wiley:17518644:media:cth20001:cth20001-math-0365 when a stability criterion is established.

Details are in the caption following the image

One refined allowable delay set for urn:x-wiley:17518644:media:cth20001:cth20001-math-0366 in (55)

4 Stability criteria based on the canonical Bessel–Legendre inequalities (40) and (46)

In this section, we develop some stability criteria using the canonical Bessel–Legendre inequalities (40) and (46), in order to show the effectiveness of canonical Bessel–Legendre inequalities, and confirm some claims made in the previous sections as well.

4.1 Stability criteria for case 1

Under case 1, the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0367 is differentiable satisfying (47). In this situation, we choose the following augmented Lyapunov–Krasovskii functional:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0368(59)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0369 are Lyapunov matrices to be determined, and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0370
with urn:x-wiley:17518644:media:cth20001:cth20001-math-0371 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0372, and for urn:x-wiley:17518644:media:cth20001:cth20001-math-0373
urn:x-wiley:17518644:media:cth20001:cth20001-math-0374(60)

The first augmented term in (59) is motivated from Lemma 4 such that the vectors in (60) induced from the integral inequality (40) appear in the derivative of the Lyapunov–Krasovskii functional urn:x-wiley:17518644:media:cth20001:cth20001-math-0375. The second and the third augmented terms are taken from [62]. It should be mentioned that the Lyapunov–Krasovskii functional urn:x-wiley:17518644:media:cth20001:cth20001-math-0376 in (59) is different from the one in (51), which is dependent on the Legendre polynomials.

4.1.1 N -dependent stability criteria

Proposition 1.For constants urn:x-wiley:17518644:media:cth20001:cth20001-math-0377, urn:x-wiley:17518644:media:cth20001:cth20001-math-0378 and a positive integer N, the system (1) subject to urn:x-wiley:17518644:media:cth20001:cth20001-math-0379 is asymptotically stable if there exist real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0380, urn:x-wiley:17518644:media:cth20001:cth20001-math-0381, urn:x-wiley:17518644:media:cth20001:cth20001-math-0382 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0383 and real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0384 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0385 with appropriate dimensions such that, for urn:x-wiley:17518644:media:cth20001:cth20001-math-0386

urn:x-wiley:17518644:media:cth20001:cth20001-math-0387(61)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0388(62)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0389 is defined in (41), and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0390(63)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0391(64)
with urn:x-wiley:17518644:media:cth20001:cth20001-math-0392 being defined in (42), urn:x-wiley:17518644:media:cth20001:cth20001-math-0393 being the i th block-row matrix such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0394, urn:x-wiley:17518644:media:cth20001:cth20001-math-0395 is a urn:x-wiley:17518644:media:cth20001:cth20001-math-0396 identify matrix, urn:x-wiley:17518644:media:cth20001:cth20001-math-0397 and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0398

Proof.First, we introduce a vector as urn:x-wiley:17518644:media:cth20001:cth20001-math-0399, urn:x-wiley:17518644:media:cth20001:cth20001-math-0400, urn:x-wiley:17518644:media:cth20001:cth20001-math-0401, urn:x-wiley:17518644:media:cth20001:cth20001-math-0402, urn:x-wiley:17518644:media:cth20001:cth20001-math-0403, urn:x-wiley:17518644:media:cth20001:cth20001-math-0404, urn:x-wiley:17518644:media:cth20001:cth20001-math-0405. It is easy to verify that urn:x-wiley:17518644:media:cth20001:cth20001-math-0406 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0407, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0408 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0409 are defined in Proposition 1. Taking the derivative of urn:x-wiley:17518644:media:cth20001:cth20001-math-0410 in (59) along with the trajectory of the system (1) yields

urn:x-wiley:17518644:media:cth20001:cth20001-math-0411(65)
Now, we estimate the integral term in (65) using the integral inequality (40). Apply the integral inequality (40) to obtain
urn:x-wiley:17518644:media:cth20001:cth20001-math-0412
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0413, urn:x-wiley:17518644:media:cth20001:cth20001-math-0414 is defined in (41), and urn:x-wiley:17518644:media:cth20001:cth20001-math-0415 are defined in Proposition 1. Employing the improved reciprocally convex inequality (21), one has
urn:x-wiley:17518644:media:cth20001:cth20001-math-0416(66)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0417 is defined in (64). Substituting (66) into (65) yields
urn:x-wiley:17518644:media:cth20001:cth20001-math-0418(67)
where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0419(68)
Note that urn:x-wiley:17518644:media:cth20001:cth20001-math-0420 is linear on urn:x-wiley:17518644:media:cth20001:cth20001-math-0421 and also on urn:x-wiley:17518644:media:cth20001:cth20001-math-0422. If the LMIs in (61) and (62) are satisfied, by the Schur complement, one has
urn:x-wiley:17518644:media:cth20001:cth20001-math-0423
Thus, from (67), there exists a scalar urn:x-wiley:17518644:media:cth20001:cth20001-math-0424 such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0425, which concludes that the system (1) is asymptotically stable for urn:x-wiley:17518644:media:cth20001:cth20001-math-0426. □

Instead of the integral inequality (40), one can also use its affine version (46) to derive another N -dependent stability criterion by slightly modifying the Lyapunov–Krasovskii functional (59), where the term urn:x-wiley:17518644:media:cth20001:cth20001-math-0427 is replaced with urn:x-wiley:17518644:media:cth20001:cth20001-math-0428. The result is stated in the following proposition.

Proposition 2.For constants urn:x-wiley:17518644:media:cth20001:cth20001-math-0429, urn:x-wiley:17518644:media:cth20001:cth20001-math-0430 and an integer urn:x-wiley:17518644:media:cth20001:cth20001-math-0431, the system (1) subject to urn:x-wiley:17518644:media:cth20001:cth20001-math-0432 is asymptotically stable if there exist real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0433, urn:x-wiley:17518644:media:cth20001:cth20001-math-0434, urn:x-wiley:17518644:media:cth20001:cth20001-math-0435 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0436 and real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0437 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0438 with appropriate dimensions such that, for urn:x-wiley:17518644:media:cth20001:cth20001-math-0439

urn:x-wiley:17518644:media:cth20001:cth20001-math-0440
where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0441(69)
and the other notations are the same as those in Proposition 1.

Remark 1.Propositions 1 and 2 deliver two N -dependent stability criteria for the system (1) subject to (47), thanks to the canonical integral inequality (40). The number of required decision variables can be calculated as urn:x-wiley:17518644:media:cth20001:cth20001-math-0442 for Proposition 1 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0443 for Proposition 2. Moreover, the positive definiteness of the matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0444 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0445 can be relaxed if one follows the line in [76] or [81].

4.1.2 Hierarchy of LMI stability criteria

In [81], it is proven that the stability criterion in terms of LMIs forms a hierarchy. In the following, it is shown that such a hierarchical characteristic is also hidden in the LMIs of Propositions 1 and 2. Based on Proposition 1, one has

Proposition 3.For the system (1) subject to (47), one has that

urn:x-wiley:17518644:media:cth20001:cth20001-math-0446(70)
where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0447(71)
with urn:x-wiley:17518644:media:cth20001:cth20001-math-0448, urn:x-wiley:17518644:media:cth20001:cth20001-math-0449 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0450 being defined in (54), (61) and (62), respectively.

Proof.Without loss generality, suppose that urn:x-wiley:17518644:media:cth20001:cth20001-math-0451 is not empty. From the definition of urn:x-wiley:17518644:media:cth20001:cth20001-math-0452, there exist real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0453, urn:x-wiley:17518644:media:cth20001:cth20001-math-0454, urn:x-wiley:17518644:media:cth20001:cth20001-math-0455 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0456 and real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0457 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0458 with appropriate dimensions such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0459 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0460, which are equivalent to

urn:x-wiley:17518644:media:cth20001:cth20001-math-0461
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0462 is defined in (68). Let
urn:x-wiley:17518644:media:cth20001:cth20001-math-0463
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0464 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0465. Then, we only need to prove that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0466(72)
In fact, denote
urn:x-wiley:17518644:media:cth20001:cth20001-math-0467
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0468 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0469 are proper real matrices and urn:x-wiley:17518644:media:cth20001:cth20001-math-0470. It follows that
urn:x-wiley:17518644:media:cth20001:cth20001-math-0471
Let urn:x-wiley:17518644:media:cth20001:cth20001-math-0472. Then
urn:x-wiley:17518644:media:cth20001:cth20001-math-0473
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0474 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0475 are some proper real matrices. Thus
urn:x-wiley:17518644:media:cth20001:cth20001-math-0476
Denote
urn:x-wiley:17518644:media:cth20001:cth20001-math-0477
After some algebraic manipulations, one has
urn:x-wiley:17518644:media:cth20001:cth20001-math-0478(73)
where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0479
Since urn:x-wiley:17518644:media:cth20001:cth20001-math-0480 is non-singular and urn:x-wiley:17518644:media:cth20001:cth20001-math-0481, there exist sufficiently small scalar urn:x-wiley:17518644:media:cth20001:cth20001-math-0482 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0483 such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0484 if urn:x-wiley:17518644:media:cth20001:cth20001-math-0485 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0486, which concludes urn:x-wiley:17518644:media:cth20001:cth20001-math-0487. □

Similar to Proposition 3, one can prove that the LMIs in Proposition 2 also form a hierarchy. For given scalars urn:x-wiley:17518644:media:cth20001:cth20001-math-0488 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0489, we denote by urn:x-wiley:17518644:media:cth20001:cth20001-math-0490 the admissible maximum upper bound of the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0491 using Proposition 1 or 2. Then from the hierarchy feature, one can draw a conclusion that urn:x-wiley:17518644:media:cth20001:cth20001-math-0492.

4.2 Stability criteria for case 2

In the case where the time-delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0493 satisfies (48), it is challenging to establish an N -dependent stability criterion using Corollary 4 since one cannot exploit the convex property to cope with the derivative of urn:x-wiley:17518644:media:cth20001:cth20001-math-0494 caused from the vectors in (60) if urn:x-wiley:17518644:media:cth20001:cth20001-math-0495 is unbounded from below. However, for urn:x-wiley:17518644:media:cth20001:cth20001-math-0496, we can derive some novel results based on the following augmented Lyapunov–Krasovskii functional as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0497(74)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0498 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0499.

Proposition 4.For constants urn:x-wiley:17518644:media:cth20001:cth20001-math-0500 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0501, the system (1) subject to (48) is asymptotically stable if there exist real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0502, urn:x-wiley:17518644:media:cth20001:cth20001-math-0503, urn:x-wiley:17518644:media:cth20001:cth20001-math-0504 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0505 and real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0506 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0507 with appropriate dimensions such that

urn:x-wiley:17518644:media:cth20001:cth20001-math-0508(75)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0509(76)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0510, urn:x-wiley:17518644:media:cth20001:cth20001-math-0511 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0512 are defined in (42) and (43) with urn:x-wiley:17518644:media:cth20001:cth20001-math-0513; and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0514(77)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0515(78)
urn:x-wiley:17518644:media:cth20001:cth20001-math-0516(79)
with urn:x-wiley:17518644:media:cth20001:cth20001-math-0517 being the i th urn:x-wiley:17518644:media:cth20001:cth20001-math-0518 row-block of the urn:x-wiley:17518644:media:cth20001:cth20001-math-0519 identity matrix, urn:x-wiley:17518644:media:cth20001:cth20001-math-0520; and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0521

Proof.Taking the time-derivative of urn:x-wiley:17518644:media:cth20001:cth20001-math-0522 in (74) yields

urn:x-wiley:17518644:media:cth20001:cth20001-math-0523(80)
Denote urn:x-wiley:17518644:media:cth20001:cth20001-math-0524, urn:x-wiley:17518644:media:cth20001:cth20001-math-0525, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0526 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0527 are defined in (60), and
urn:x-wiley:17518644:media:cth20001:cth20001-math-0528(81)
Then for any real matrix urn:x-wiley:17518644:media:cth20001:cth20001-math-0529 with appropriate dimensions, the following equation holds:
urn:x-wiley:17518644:media:cth20001:cth20001-math-0530(82)
From (48), (80) and (82), one obtains
urn:x-wiley:17518644:media:cth20001:cth20001-math-0531(83)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0532 is defined in (77). Apply the integral inequality (40) with urn:x-wiley:17518644:media:cth20001:cth20001-math-0533 to get
urn:x-wiley:17518644:media:cth20001:cth20001-math-0534
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0535 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0536. By the improved reciprocally convex inequality (21), one has
urn:x-wiley:17518644:media:cth20001:cth20001-math-0537(84)
Substituting (84) into (83) yields
urn:x-wiley:17518644:media:cth20001:cth20001-math-0538(85)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0539 defined in (79). If the LMIs in (75) and (76) are satisfied, urn:x-wiley:17518644:media:cth20001:cth20001-math-0540 holds by using the Schur complement. Thus, one can conclude that the system (1) subject to (48) is asymptotically stable. □

Similar to Proposition 2, if using the affine integral inequality (46) instead of (40), we have the following result.

Proposition 5.For constants urn:x-wiley:17518644:media:cth20001:cth20001-math-0541 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0542, the system (1) subject to (48) is asymptotically stable if there exist real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0543, urn:x-wiley:17518644:media:cth20001:cth20001-math-0544, urn:x-wiley:17518644:media:cth20001:cth20001-math-0545 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0546 and real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0547 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0548 with appropriate dimensions such that

urn:x-wiley:17518644:media:cth20001:cth20001-math-0549(86)
where
urn:x-wiley:17518644:media:cth20001:cth20001-math-0550(87)
and the other notations are defined in Proposition 4.

Remark 2.Propositions 4 and 5 provide two stability criteria for the system (1) subject to (48). Compared with [64, Theorem 1], the main difference lies in that Propositions 4 and 5 are derived based on such a condition as urn:x-wiley:17518644:media:cth20001:cth20001-math-0551 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0552, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0553 is a linear matrix-valued function on urn:x-wiley:17518644:media:cth20001:cth20001-math-0554, leading to a necessary and sufficient condition urn:x-wiley:17518644:media:cth20001:cth20001-math-0555 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0556 such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0557 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0558. This linear matrix-valued function contributes to the introduction of the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0559 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0560 in (81). However, in the proof of [64, Theorem 1], urn:x-wiley:17518644:media:cth20001:cth20001-math-0561 is a quadratic function on urn:x-wiley:17518644:media:cth20001:cth20001-math-0562. Thus, applying the quadratic convex approach in (13) only gives a sufficient condition such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0563 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0564.

Remark 3.The purpose of introducing the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0565 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0566 in (81) is to absorb urn:x-wiley:17518644:media:cth20001:cth20001-math-0567 such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0568 is linear on urn:x-wiley:17518644:media:cth20001:cth20001-math-0569; Otherwise, urn:x-wiley:17518644:media:cth20001:cth20001-math-0570 will be a triple matrix-valued polynomial function on urn:x-wiley:17518644:media:cth20001:cth20001-math-0571, which is difficult in deriving a stability criterion for the system (1) in Case 2. The number of decision variables required is urn:x-wiley:17518644:media:cth20001:cth20001-math-0572 for Proposition 4 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0573 for Proposition 5.

4.3 Stability criteria for case 3

Under case 3, the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0574 is known to be continuous, but no any information on the derivative of the time-varying delay is available in the stability analysis. In this case, the augmented Lyapunov–Krasovskii functional can be chosen as
urn:x-wiley:17518644:media:cth20001:cth20001-math-0575(88)
where urn:x-wiley:17518644:media:cth20001:cth20001-math-0576 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0577 are defined in (74). Using the integral inequality (40) and its affine version (46), similar to the proof of Propositions 4 and 5, we have the following two stability criteria.

Proposition 6.For a constant urn:x-wiley:17518644:media:cth20001:cth20001-math-0578, the system (1) subject to (49) is asymptotically stable if there exist real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0579, urn:x-wiley:17518644:media:cth20001:cth20001-math-0580, and urn:x-wiley:17518644:media:cth20001:cth20001-math-0581 and real matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0582 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0583 with appropriate dimensions such that one of the following two statements is true.

  1. The LMIs in (75) and (76) are satisfied, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0584 is replaced with urn:x-wiley:17518644:media:cth20001:cth20001-math-0585 as

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0586

  2. The LMIs in (86) are satisfied, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0587 is replaced with urn:x-wiley:17518644:media:cth20001:cth20001-math-0588 as

    urn:x-wiley:17518644:media:cth20001:cth20001-math-0589
    The other notations are defined in Proposition 4.

Proof.The proof can be completed by the following the proof of Proposition 4.□

Remark 4.In Case 3, Proposition 6 presents two stability criteria for the system (1). By using the second-order Bessel–Legendre inequality, a stability criterion for the system (1) with (49) is also reported in [52, Theorem 1]. The main difference between them lies in the chosen Lyapunov–Krasovskii functional. In Proposition 6, an augmented vector urn:x-wiley:17518644:media:cth20001:cth20001-math-0590 is introduced in urn:x-wiley:17518644:media:cth20001:cth20001-math-0591 in (88), but not in [52, Theorem 1]. As a result, taking the derivative of the augmented term yields

urn:x-wiley:17518644:media:cth20001:cth20001-math-0592
That is, the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0593 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0594, urn:x-wiley:17518644:media:cth20001:cth20001-math-0595 in (81) are coupled by Q, which enhances the feasibility of the stability conditions in Proposition 6.

Remark 5.The number of decision variables required in Proposition 6 is urn:x-wiley:17518644:media:cth20001:cth20001-math-0596 for the condition (i) and urn:x-wiley:17518644:media:cth20001:cth20001-math-0597 for the condition (ii), which are smaller than urn:x-wiley:17518644:media:cth20001:cth20001-math-0598 in [52, Theorem 2].

Remark 6.It should be pointed out that, the proposed results in this section can be easily extended to a linear system with an interval time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0599 provided that one modifies the chosen Lyapunov–Krasovskii functionals by taking the lower bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0600 into account. Because of their similarities, those results are omitted in the paper.

4.4 Illustrative examples

In this section, we compared the above stability criteria with some existing ones through two numerical examples.

Example 1.Consider the system (1), where

urn:x-wiley:17518644:media:cth20001:cth20001-math-0601(89)
The time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0602 is a differentiable function on urn:x-wiley:17518644:media:cth20001:cth20001-math-0603.

Example 1 is well used to calculate the admissible maximum upper bound (AMUB) urn:x-wiley:17518644:media:cth20001:cth20001-math-0604 for the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0605. In order to make a comparison, we consider two cases of urn:x-wiley:17518644:media:cth20001:cth20001-math-0606.

Case 1: urn:x-wiley:17518644:media:cth20001:cth20001-math-0607 satisfies (47) with urn:x-wiley:17518644:media:cth20001:cth20001-math-0608. We compared the stability criteria with some existing ones obtained for urn:x-wiley:17518644:media:cth20001:cth20001-math-0609 defined in (54). For different values of urn:x-wiley:17518644:media:cth20001:cth20001-math-0610, Table 1 lists the obtained AMUBs of urn:x-wiley:17518644:media:cth20001:cth20001-math-0611 by Seuret and Gouaisbaut [102, Theorem 1], Zhang et al. [72, Proposition 1], Zhang et al. [69, Theorem 2], Lee et al. [100, Theorem 1], Zeng et al. [67, Corollary 1], Seuret and Gouaisbaut [81, Theorem 8 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0612], the IQC approach [21], the quadratic separation approach [104] and Propositions 1 and 2 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0613 in this paper. From Table 1, one can see that

Table 1. AMUB urn:x-wiley:17518644:media:cth20001:cth20001-math-0614 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0615 (case 1 for Example 1)
Method urn:x-wiley:17518644:media:cth20001:cth20001-math-0616 0 0.1 0.5 0.8
[21] 6.117 4.714 2.280 1.608
[104] 6.117 4.794 2.682 1.957
[69] 6.165 4.714 2.608 2.375
[67] 6.059 4.788 3.055 2.615
[102] 6.0593 4.71 2.48 2.30
[100] 6.0593 4.8313 3.1487 2.7135
[72] 6.168 4.910 3.233 2.789
[81] 6.1725 5.01 3.19 2.70
Proposition 1 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0617) 6.0593 4.8344 3.1422 2.7131
Proposition 1 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0618) 6.1689 4.9192 3.1978 2.7656
Proposition 1 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0619) 6.1725 4.9203 3.2164 2.7875
Proposition 1 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0620) 6.1725 4.9246 3.2230 2.7900
Proposition 2 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0621) 6.0593 4.8377 3.1521 2.7278
Proposition 2 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0622) 6.1689 4.9217 3.2211 2.7920
Proposition 2 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0623) 6.1725 4.9239 3.2405 2.8159
Proposition 2 (urn:x-wiley:17518644:media:cth20001:cth20001-math-0624) 6.1725 4.9297 3.2527 2.8230
  • urn:x-wiley:17518644:media:cth20001:cth20001-math-0625 Results in this line are obtained from Theorem 8 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0626 in [81].
  • • Propositions 1 and 2 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0627 obtain a larger upper bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0628 than the criteria in [67, 69, 72, 100, 102], the IQC approach [21] and the quadratic separation approach [104]. Even for urn:x-wiley:17518644:media:cth20001:cth20001-math-0629 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0630, Propositions 1 and 2 outperform [69, Theorem 2], [100, Theorem 1], [67, Corollary 1], the IQC approach [21] and the quadratic separation approach [104].
  • • For urn:x-wiley:17518644:media:cth20001:cth20001-math-0631, Seuret and Gouaisbaut [81, Theorem 8 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0632] give a larger delay upper bound than Propositions 1 and 2 with urn:x-wiley:17518644:media:cth20001:cth20001-math-0633 due to that the positive definiteness of the matrices urn:x-wiley:17518644:media:cth20001:cth20001-math-0634 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0635 are relaxed. However, for urn:x-wiley:17518644:media:cth20001:cth20001-math-0636 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0637, Propositions 1 and 2 offer better results that [81, Theorem 8].
  • • For the same N, Proposition 2 delivers a larger upper bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0638 than Proposition 2 at the cost of higher computation burden, which means that a stability criterion using the affine integral inequality (46) can derive a larger upper bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0639 than that using the integral inequality (40) and the improved reciprocally convex inequality (21).

Case 2: The time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0640 satisfies (48). In order to show the effectiveness of Propositions 4 and 5, the AMUBs of urn:x-wiley:17518644:media:cth20001:cth20001-math-0641 are listed in Table 2 for different values of urn:x-wiley:17518644:media:cth20001:cth20001-math-0642. From this table, one can see that (i) Propositions 4 and 5 indeed can derive some larger upper bounds of urn:x-wiley:17518644:media:cth20001:cth20001-math-0643 than [64, Theorem 1] and [79, Theorem 1], while Propositions 4 and 5 require more decision variables than [79, Theorem 1] (urn:x-wiley:17518644:media:cth20001:cth20001-math-0644) and [64, Theorem 1] (urn:x-wiley:17518644:media:cth20001:cth20001-math-0645); and (ii) The affine integral inequality (46) can result in a larger upper bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0646 than the integral inequality (40) plus the improved reciprocally convex inequality (21).

Example 2.Consider the system (1) subject to (49), where

urn:x-wiley:17518644:media:cth20001:cth20001-math-0647(90)
The time delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0648 is continuous but not differentiable on urn:x-wiley:17518644:media:cth20001:cth20001-math-0649.

Table 2. AMUB urn:x-wiley:17518644:media:cth20001:cth20001-math-0650 for different urn:x-wiley:17518644:media:cth20001:cth20001-math-0651 (case 2 for Example 1)
Method urn:x-wiley:17518644:media:cth20001:cth20001-math-0652 0 0.1 0.5 0.8 1
[79] 6.059 4.704 2.420 2.113 2.113
[64] 6.168 4.733 2.429 2.183 2.182
Proposition 4 6.168 4.800 2.533 2.231 2.231
Proposition 5 6.168 4.800 2.558 2.269 2.263

This example is taken to illustrate the validity of Proposition 6.

For comparison, we calculate the upper bound of urn:x-wiley:17518644:media:cth20001:cth20001-math-0653 such that the system remains stable. Applying [47, Theorem 7], [53, Theorem 1], [52, Theorem 2] and Proposition 6, the obtained results and the required number of decision variables are listed in Table 3, from which one can see that Proposition 6 outperforms those methods in [47, 52, 53]. Moreover, it is clear that using the affine integral inequality (46) can yield a larger upper bound urn:x-wiley:17518644:media:cth20001:cth20001-math-0654 than that using the integral inequality (40) though more decision variables are required.

Table 3. AMUB urn:x-wiley:17518644:media:cth20001:cth20001-math-0655 for Example 2
Method urn:x-wiley:17518644:media:cth20001:cth20001-math-0656 Number of decision variables
[47] 1.59 urn:x-wiley:17518644:media:cth20001:cth20001-math-0657
[53] 1.64 urn:x-wiley:17518644:media:cth20001:cth20001-math-0658
[52] 2.39 urn:x-wiley:17518644:media:cth20001:cth20001-math-0659
Proposition 6-(i) 2.39 urn:x-wiley:17518644:media:cth20001:cth20001-math-0660
Proposition 6-(ii) 2.53 urn:x-wiley:17518644:media:cth20001:cth20001-math-0661

In summary, through two well-used numerical examples, it is shown that, the obtained stability criteria in this paper are more effective than some existing ones in deriving a larger upper bound for a linear system with a time-varying delay.

As a counterpart of integral inequalities, finite-sum inequalities for stability analysis of discrete-time systems with time-varying delays also have gained much attention. A large number of finite-sum inequalities and stability criteria have been reported in the published literature, see, [105114]. Since discrete-time systems with time-varying delays are not the focus of the paper, stability criteria based on finite-sum inequalities developed recently are not mentioned in the paper.

5 Conclusion and some challenging issues

An overview and in-depth analysis of recent advances in stability analysis of time-delay systems has been provided, including recent developments of integral inequalities, convex delay analysis approaches, reciprocally convex approaches and augmented Lyapunov–Krasovskii functionals. Then, some existing stability conditions have been reviewed by taking into consideration three cases of time-varying delay, where information on the upper and lower bounds of the delay-derivative are totally known, partly known and completely unknown. Furthermore, a number of stability criteria have been developed by employing the recent canonical Bessel–Legendre integral inequalities and an augmented Lyapunov–Krasovskii functional. When information on the lower and upper bounds of both urn:x-wiley:17518644:media:cth20001:cth20001-math-0662 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0663 is known, the obtained stability criteria have been proven to be hierarchical.

However, although there has been significant progress in stability analysis of time-delay systems, the following issues are still challenging.
  • • If the positive integer N approaches to infinity, the canonical N -order Bessel–Legendre inequality can provide an accurate estimate on the integral term. Thus, using such an integral inequality, it is possible to derive a necessary and sufficient condition on stability for linear systems with time-varying delays, which is interesting but challenging. Moreover, extending the canonical N -order Bessel–Legendre inequality to multi-dimensional systems like 2D systems with time-varying delays is also an interesting topic [115, 116].
  • • For the system (1) subject to (48) or (49), no N -dependent stability criteria are derived using the integral inequality (40) due to the vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0664 in which the scalar urn:x-wiley:17518644:media:cth20001:cth20001-math-0665 appears in the form of its inverse. Applying the integral inequality (40) to the system (1) possibly yields such a stability condition as urn:x-wiley:17518644:media:cth20001:cth20001-math-0666, where urn:x-wiley:17518644:media:cth20001:cth20001-math-0667 are real matrices irrespective of the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0668. How to obtain a necessary and sufficient feasible condition such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0669 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0670 is a significant problem.
  • • In the proof of Proposition 4, four extra vectors urn:x-wiley:17518644:media:cth20001:cth20001-math-0671 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0672 are introduced to absorb urn:x-wiley:17518644:media:cth20001:cth20001-math-0673 such that the obtained stability condition urn:x-wiley:17518644:media:cth20001:cth20001-math-0674 is dependent linearly on the time-varying delay urn:x-wiley:17518644:media:cth20001:cth20001-math-0675. If not doing so, urn:x-wiley:17518644:media:cth20001:cth20001-math-0676 will be a triple matrix-valued polynomial function on urn:x-wiley:17518644:media:cth20001:cth20001-math-0677. As a result, a necessary and sufficient condition, i.e. urn:x-wiley:17518644:media:cth20001:cth20001-math-0678 and urn:x-wiley:17518644:media:cth20001:cth20001-math-0679, can be derived such that urn:x-wiley:17518644:media:cth20001:cth20001-math-0680 for urn:x-wiley:17518644:media:cth20001:cth20001-math-0681. How to extend this technique to a general case is an interesting issue.
  • • The integral inequality (40) is established based on a sequence of orthogonal polynomials. Is it possible to formulate some integral inequality based on a sequence of non-orthogonal polynomials such that the scalars urn:x-wiley:17518644:media:cth20001:cth20001-math-0682 disappear or appears linearly? Answering this question is beneficial for deriving less conservative stability criteria for linear systems with time-varying delay, which is significant and challenging.
  • • Simulation in this paper shows that the canonical Bessel-Legendre inequality approach can yield some nice results on stability. However, how to apply it to deal with control problems of a number of practical systems, such as networked control systems [117, 118], event-triggered control systems [119121], vibration control systems [122124], formation control systems [125] and multi-agent systems [126129], deserves much effort of researchers.

6 Acknowledgment

This work are supported in part by the Australian Research Council Discovery Project under Grant DP160103567.