Mathematical Problems in Engineering

Volume 2018, Article ID 8492735, 14 pages

https://doi.org/10.1155/2018/8492735

## Complex Scaling Approach for Stability Analysis and Stabilization of Multiple Time-Delayed Systems

Department of Automatic Control Engineering, School of Energy and Electrical Engineering, Hohai University, Nanjing 211100, China

Correspondence should be addressed to Jun Zhou; nc.ude.uhh@arustak

Received 7 January 2018; Revised 12 February 2018; Accepted 27 February 2018; Published 12 April 2018

Academic Editor: Filippo Cacace

Copyright © 2018 Jun Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on complex scaling, the paper copes with stability analysis and stabilization of linear time-invariant continuous-time systems with multiple time delays in state, control input, and measured output, under state and/or output feedback. More specifically, the paper establishes stability criteria for exponential/asymptotical stability with Hurwitz complex scaling being applied to the related characteristic polynomials. The stability conditions are necessary and sufficient, delay-dependent, and independent of feedback structures and open-loop poles distribution. The criteria can be implemented graphically with locus plotting or numerically without it; moreover, no prior frequency sweeping is involved, and the contour and locus encirclement orientations can be self-defined. Exploiting the complex scaling approach and embracing its technical merits, it is considered to design static state feedback control for robustly stabilizing time-delayed systems. A small-gain interpretation for the suggested stabilization is also elaborated. Examples are included to illustrate the main results.

#### 1. Introduction

This study is motivated by the fact that many frequency-domain stability criteria are developed and exploited as parametric and graphical tools for stability analysis and stabilization in classes of time-delayed systems, subject to various structural, analytical or technical assumptions, and constraints. Due to intuitive conception, simple geometry, experimental availability, and frequency-domain methods are extended to stability analysis and stabilization in distributed [1, 2], periodically time-varying [3], and time-delayed ones with or without uncertainties [4–9]. Some stability criteria are given in Nyquist-like form, which involve open-loop unstable poles and frequency-domain plots in general and thus are not useful for time-delayed systems with time-varying factors [10–12]. However, when time-delayed systems are dealt with [13–17], poles distribution is not available numerically and frequency responses need to be examined with prior sweeping. Stability criteria for time-delayed systems are created also via LMI [18, 19]. Complex-valued neural networks with time delays are attached in [20–22]. As a matter of fact, when time delay occurs in dynamics and/or their derivatives, the characteristic polynomials are quasi-polynomials in one variable and its exponential powers [4, 23] that are of infinite degree. Although finitely many poles can be computed by truncating infinite series of exponentials or other numerical algorithms, approximate poles are not meaningful for analytically exploiting Nyquist-type criteria, let alone unavoidable algorithm convergence [24] and numerical pathology.

Lately, a series of studies [25–28] are developed in various systems based on complex scaling, which enables us to cope with structure and stability issues by means of frequency-domain properties of the concerned systems without invoking pole distribution and specific frequency responses. More precisely, the study [25] talks about stability of sampled-data systems; the work [26] deals with that of continuous-time fractal-order systems; the study [27] considers the Lur’e-type feedback nonlinear systems. The paper [28] generalizes the complex scaling tool for investigating structural features in LTI systems, say controllability, observability, and so on. Tentative discussion via complex scaling for stability in open-loop time-delayed systems was reported in [29]. In this study, we carefully work through the technical details and provide rigorous proofs for internal stability criteria for LTI feedback systems with multiple time delays.

More specifically, to surmount multiple time-delay dependence, infinite-dimensionality, polynomial cancellation, and other complex analysis issues in quasi characteristic polynomials of time-delayed continuous systems in feedback configuration, the complex scaling technique is coined by using auxiliary Hurwitz polynomial to modify the return difference relationships such that the modified ones reflect exactly and completely asymptotical/exponential stability while analytical properties are retained for applying the argument principle [30]. The suggested stability criteria produce necessary and sufficient conditions for asymptotical and exponential stability (or simply internal stability), which involve neither open-loop unstable poles that are hard to know, nor prespecified contour and locus encirclement orientations. The stability criteria are implementable graphically with locus plotting or merely by numerical complex argument integration. Also using the complex scaling approach, we further contrive a new technique for stabilizing time-delayed systems with static state feedback. The stabilization results in asymptotical/exponential stability internally as well as -stability externally that are robust with respect to time delay.

*Outlines*. Section 2 collects preliminaries to LTI time-delayed continuous systems of retarded type, their feedback configuration, and return difference relationships. Section 3 states the complex scaling return difference relationships and proves the generalized criteria. Stabilization is explained in Section 4. Illustrations are sketched in Section 5, and Section 6 gives our conclusions.

*Notations 1. * and are the sets of real and complex numbers, respectively, while . is the identity matrix. means the complex conjugate transpose of a matrix . means the determinant of a square matrix . A polynomial is Hurwitz polynomial if all its roots have negative real parts, and means its degree. denotes the functions space such that means that , where is the Euclidean norm of a vector or the induced matrix norm as appropriately.

#### 2. Preliminaries to Time-Delayed Systems

##### 2.1. Linear Time-Delayed Continuous-Time Systems of Retarded Types

By linear time-delayed continuous-time system of retarded type, we mean a system in form ofwhere , , and are the state, control input, and measured output, respectively. is an integer, and , , , and , respectively, are constant matrices. In (1), the time delays are assumed to satisfyThat is, , , and are short notations for the corresponding multiple time delays.

The transfer function matrix of system (1) isConventionally, is called the characteristic function, and is the characteristic equation, whose zeros are the characteristic roots. Since is quasi when , there are infinitely many characteristic roots. Due to infinite-dimensionality, it is considerably difficult to deal with stability investigation directly in terms of numerically computing the characteristic roots.

We collect properties about the characteristic roots in the following remark, which pave the way for stability analysis and stabilization.

*Remark 2. *(i) is holomorphic; therefore, all characteristic roots are isolated; (ii) there are only finitely many characteristic roots in any vertical strip of the complex plane, where ; (iii) for any with , only finitely many characteristic roots exist to the right-hand side of the vertical line , while infinitely many ones are located to the left-hand side of [24, p. 9, Corollary 1.9]; (iv) the characteristic roots are continuous with respect to delays and coefficients [24, p. 10, Theorem 1.14]; (v) in any bounded rectangular area , , where and , there are no infinite sequences of characteristic roots that converge to points in . To see this, we notice that is entire over . Hence, if there is an infinite sequence in which are mutually distinctive, for each , and with , then the uniformity theorem tells that over .

Asymptotical and exponential stabilities in (1) are summarized by Definition 3. Proposition 4 gives necessary and sufficient conditions for exponential stability, in terms of the characteristic roots and in light of [24].

*Definition 3. *Let be the null solution of (1) with respect to the initial condition , where is defined in (2).(i)The null solution of (1) is asymptotically stable, or simply is asymptotically stable, if, for any , there exist and (resp., ) with over such that as .(ii)The null solution of (1) is exponentially stable, or simply is exponentially stable, if there exist constants and such that over for all (resp., ).

Together with Remark 2, asymptotical and exponential stabilities are equivalent whenever LTI time-delayed systems of retarded type are concerned. Hence, only exponential stability conditions are claimed in Proposition 4 explicitly by restating Proposition 1.6 [24].

Proposition 4. *The null solution of (1) is exponentially stable if and only if there is such that for each , where means a root of the characteristic equation .*

##### 2.2. Output Feedback Configuration and Return Difference Relationship

By the notation , we denote an output feedback connection with being the time-delayed system (1) and being an LTI controller, which has the state-space equationwhere , , , and are constant matrices. Apparently, the transfer function matrix of is

On the one hand, the open-loop characteristic function is denoted by and satisfiesOn the other hand, the state-space equations for the closed-loop system are given bywhere . The closed-loop characteristic function is given byBy (6) and (8), if is well-posed (i.e., is nonsingular), both the open-loop system, denoted by , and the closed-loop system are retarded type. Hence, Definition 3 and Proposition 4 apply to as well.

To attack stability in , we need to address how to compute . This relates us to the return difference relationship given below. Since the derivation algebras are standard and trivial, we omit the details:

Although the return difference relationship (9) is also in form as what we know in LTI systems, we must be cautious in connecting (9) with the argument principle so as to create stability criteria for the concerned time-delayed systems. Note the following:(i)Exponentials in delays are involved. Open-loop unstable poles in (i.e., the zeros of ), if any, cannot be calculated simply. has infinitely many zeros; it is almost meaningless to numerically compute unstable poles, if any.(ii)Since , is a quasi-polynomial; is meromorphic in the sense that all its entries are meromorphic. Hence, the argument principle does apply to (9), whenever their right-hand sides vanish nowhere on a contour.(iii)Reducible factors may exist between and ; they may stem from the following: (1) input and/or output decoupling zeros in and/or , due to uncontrollable and/or unobservable modes [31]; (2) input and/or output decoupling zeros in caused by zero/pole cancellations between and , when they are cascaded in ; and (3) algebraic cancellations, which are poles in that are not affected by feedback or poles that are assigned to locations of other poles of [32]. Removing all reducible factors, if any, not all closed-loop characteristic roots are reflected by (9). Consequently, asymptotical/exponential stability of cannot be examined accordingly. Since and are quasi-polynomials, their coprimeness actually cannot be examined.

#### 3. Stability Analysis of Time-Delayed Systems

##### 3.1. Contour, Assumptions, and Problem Formulation

Now we describe a Cauchy contour for utilizing the argument principle of complex analysis in light of [30, 33]. In Figure 1, the contour is a simply closed curve marked by the dashed line, whose vertical portion is overlapped actually with the imaginary axis. Let us denote by the interior circumscribed by to its right. Thus, is an open set and . In the paper, the entry-belonging is equivalent to saying or the point is on the imaginary axis; namely, . In the cases involving -exponential stability, we need to use the -shifted Cauchy contour that is denoted by with being a constant. The exact meaning of the relation is with or .