Elsevier

Automatica

Volume 80, June 2017, Pages 218-224
Automatica

Brief paper
Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays

https://doi.org/10.1016/j.automatica.2017.02.034Get rights and content

Abstract

A stability criterion for the exponential stability of systems with multiple pointwise and distributed delays is presented. Conditions in terms of the delay Lyapunov matrix are obtained by evaluating a Lyapunov–Krasovskii functional with prescribed derivative at a pertinent initial function that depends on the system fundamental matrix. The proof relies on properties connecting the delay Lyapunov matrix and the fundamental matrix, which are proven to be valid for both stable and unstable systems. The conditions are applied to the determination of the exact stability region for some examples.

Introduction

One of the best known results of control theory is without doubt the exponential stability criteria for linear time-invariant systems of ordinary differential equations, namely, the positive-definiteness of the Lyapunov matrix, solution of the Lyapunov equation, that defines the quadratic Lyapunov function. This paper is devoted to an extension of this result to the case of time-delay systems with multiple pointwise and distributed delays.

This is done in the Lyapunov–Krasovskii framework (Krasovskii, 1956), where the general results of the Lyapunov theory are extended to systems with delays by using, instead of functions, functionals that capture the whole state of the delay system.

This framework is widely used in the derivation of sufficient stability conditions obtained via the proposal of functionals of prescribed form, a topic that has been a fertile field of investigation in the past decades, due to the formulation of the conditions in terms of easy to test Linear Matrix Inequalities (see for example Fridman, 2014; Niculescu, 2001, and the references therein).

The converse approach, that consists of finding the form of the functional with prescribed derivative along the system trajectory was first addressed in Repin (1965). A detailed account of the results of the early contributors on the topic, Datko (1972), Huang (1989), Infante and Castelan (1978), and Louisell (2001), is given in Kharitonov (2013, p. 73).

In Kharitonov and Zhabko (2003), a class of complete type functionals that admit a quadratic lower bound if the corresponding system is exponentially stable was introduced. These functionals can be considered as a generalization of the quadratic Lyapunov functions, usually used for ordinary differential equations. They are defined by a matrix-valued function over the delay interval, which is the analogue of the Lyapunov matrix, and is the solution of a set of three properties that play the role of the Lyapunov equation. The reader is referred to Kharitonov (2013) for a comprehensive treatment of this topic, for different classes of delay systems; Some aspects of the computation of the Lyapunov matrices have been addressed in Huesca, Mondié, and Santos (2009), Jarlebring, Vanbiervliet, and Michiels (2011) and Kharitonov (2013).

Complete type functionals have been applied successfully to the robust stability analysis (Kharitonov & Zhabko, 2003), exponential estimates of solutions (Kharitonov & Hinrichsen, 2004), solution of the Bellman equation for time delay systems (Santos, Mondié, & Kharitonov, 2009), computation of the norm of the transfer matrix (Jarlebring et al., 2011), proof of the stability of predictor-based control scheme for state and input delay systems (Kharitonov, 2014).

It should be noted that these applications rely at some point on some stability assumption. It was only until recently that the direct application of complete type functionals to the stability analysis has received due attention. A criterion of the exponential stability of the scalar single delay equation has been given in Mondié (2012) and in Egorov and Mondié (2013). Some new sufficient stability conditions based on Lyapunov functionals of complete type were given in Medvedeva and Zhabko (2013). In the past few years, the complete type functionals have been used in order to derive families of necessary stability conditions for linear systems with multiple delays (Egorov and Mondié, 2014a, Egorov and Mondié, 2014b), and with distributed delays (Cuvas & Mondié, 2015).

The visible efficacy of the above mentioned necessary exponential stability conditions in the accurate determination of the stability region of a variety of examples, added to the availability of a criterion for the scalar single delay case, points naturally towards a proof of sufficiency of these conditions. A preliminary result in this direction was presented in Egorov (2014).

The main contribution of the present paper is to prove necessary and sufficient conditions for the exponential stability of linear systems with multiple pointwise and distributed delays. The proof of this criterion relies on the presentation of fundamental stability/unstability results for the Lyapunov–Krasovskii functionals we introduce, the determination of the class of initial functions that reveal the role of the delay Lyapunov matrix, and on the approximation of any initial function by an element in this class.

The organization of the paper is as follows: Section  2 is devoted to some preliminaries on the delay systems under consideration. In Section  3, well known results on Lyapunov–Krasovskii functionals with prescribed derivative are recalled, and some new ones are introduced. The choice of initial functions leading to an expression of the functional in terms of the delay Lyapunov matrix, without assumption of stability of the system, is discussed in Section  5. The main result with its proof is given in Section  6. The paper ends with illustrative examples in Section  7, followed by concluding remarks.

Notation: The notations Q>0,Q0,Q⩾̸0, mean that the symmetric matrix Q is positive definite, positive semidefinite, and not positive semidefinite, respectively. The square block matrix with ith row and jth column element Aij is written as {Aij}i,j=1r. The symbol indicates transposed terms in symmetric block matrices.

Section snippets

Preliminaries

We consider linear systems of the form ẋ(t)=j=0mAjx(thj)+H0G(θ)x(t+θ)dθ, where t0,Aj,j=0,,m, are real n×n matrices, delays are ordered as 0=h0<h1<<hm=H, and G(θ), is a real piecewise continuous matrix function valued on θ[H,0]. The initial function φ is assumed to be piecewise continuous, φH=PC([H,0],Rn), i.e., it has a finite number of discontinuity points of the first kind. The restriction of the solution x(t,φ) of system (1) to the interval [tH,t] is denoted by xt(φ):θx(t+θ,φ),θ

Lyapunov–Krasovskii framework

A functional v0(xt(φ)) with prescribed derivative along the trajectories of system (1), given by dv0(xt(φ))dt=xT(t,φ)Wx(t,φ), where W is a positive definite matrix, was introduced in Huang (1989). It has the form v0(φ)=φT(0)U(0)φ(0)+2φT(0)j=1mhj0U(θhj)Ajφ(θ)dθ+k=1mj=1mhk0φT(θ1)AkThj0U(θ1+hkθ2hj)Ajφ(θ2)dθ2dθ1+2φT(0)H0HθU(ξθ)G(ξ)dξφ(θ)dθ+2j=1mhj0H0Hθ2φT(θ1)AjTU(θ1+hjθ2+ξ)G(ξ)φ(θ2)dξdθ2dθ1+H0H0φT(θ1)Hθ1Hθ2GT(ξ1)U(θ1ξ1θ2+ξ2)G(ξ2)dξ2dξ1φ(θ2)dθ2dθ1. Each term

The pertinent choice for the initial function

It is clear that the substitution of any particular initial function into the functional v1, combined with the quadratic lower bound on the functional will provide a set of necessary stability conditions. For example, for the choice φ(θ)={μ,θ=0,0,elsewere , where μ is an arbitrary non-null vector, all the summands of v1, except the first one, vanish. Then, the symmetry of the delay Lyapunov matrix at zero along with inequality (6) implies that U(0)>0.

Having in mind the delay free case, one can

Stability criterion

The main result of this contribution, a stability criterion for systems with pointwise and distributed delays, is presented.

Theorem 5

System   (1)   is exponentially stable if and only if the Lyapunov condition holds and for every natural number r2{U(jir1H)}i,j=1r>0.Moreover, if the Lyapunov condition holds and system   (1)   is unstable, then there exists a natural number r such that{U(jir1H)}i,j=1r⩾̸0.

Illustrative examples

In this section, we illustrate how the presented necessary and sufficient conditions can be applied to determine the exact stability region in the space of parameters, including delays. At equally spaced points of a grid of the chosen space of parameters, we check the Lyapunov condition, and compute the delay Lyapunov matrix using the semianalytical method introduced in Kharitonov (2013). Next, we test the positivity condition (12) of Theorem 5 for increasing values of r, for example r=2:(U(0)U(

Conclusions

Necessary and sufficient conditions for systems with multiple pointwise and distributed delays are proved. In analogy with the delay free case, they depend exclusively on the delay Lyapunov matrix. A significant advantage of addressing the problem in the Lyapunov–Krasovskii framework is that, in addition to the determination of the exact stability region, a useful functional is obtained at each stable point.

Alexey V. Egorov graduated in 2010 and received his Ph.D. degree in 2013, both from St. Petersburg State University (SPbSU). Dr. Egorov is an Associate Professor of Department of Control Theory at the Faculty of Applied Mathematics and Control Processes of this university. His scientific interests include time-delay systems, control theory and stability analysis.

References (26)

  • Egorov, A.V. (2014). A new necessary and sufficient stability condition for linear time-delay systems. In 19th IFAC...
  • A.V. Egorov et al.

    A stability criterion for the single delay equation in terms of the Lyapunov matrix

    Vestnik St. Petersburg University Series 10

    (2013)
  • A.V. Egorov et al.

    Necessary conditions for the exponential stability of time-delay systems via the Lyapunov delay matrix

    International Journal of Robust and Nonlinear Control

    (2014)
  • Cited by (63)

    • Spectrum assignment and stabilization by static output feedback of linear difference equations with state variable delays

      2022, European Journal of Control
      Citation Excerpt :

      In [42] a novel periodic Lyapunov–Krasovskii functional is proposed for stability analysis of discrete-time linear systems with time-varying delays. The approach of Lyapunov matrix equation presented in [11,12,18,19] for continuous-time systems is used for solving the problem of exponential stability for delayed linear discrete-time systems in [22]. Some results on stability analysis for discrete time-delay systems based on new finite-sum inequalities are obtained in [40].

    View all citing articles on Scopus

    Alexey V. Egorov graduated in 2010 and received his Ph.D. degree in 2013, both from St. Petersburg State University (SPbSU). Dr. Egorov is an Associate Professor of Department of Control Theory at the Faculty of Applied Mathematics and Control Processes of this university. His scientific interests include time-delay systems, control theory and stability analysis.

    Carlos Cuvas is an electrical engineer who graduated from ITP, Hidalgo, México, in 2003. He received the M.S. degree from IPN, México City, in 2006 and the Ph.D. degree from CINVESTAV, México City, in 2015. He is currently a postdoctoral research associate at the UAEH, Hidalgo, México. His research interests include systems with delays, control systems theory and stability analysis.

    Sabine Mondié (S’96–M’99) received the B.S. degree in industrial engineering from the ITESM, Mexico City, and the M.S. and Ph.D. degrees in electrical engineering from the CINVESTAV, Mexico City and the IRCyN, Nantes, France, in 1883 and 1996, respectively. Since 1996, she has been a professor at the Department of Automatic Control at CINVESTAV, Mexico City, Mexico. Her research interests include time delay systems, the structural approach of linear systems, and their applications.

    This work was supported by Project Conacyt   180725 and by RFBR, research project No. 16-31-00436 a. The material in this paper was partially presented at the 19th World Congress of the International Federation of Automatic Control (IFAC 2014), August 24–29, 2014, Cape Town, South Africa. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen.

    View full text