Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7354654, 25 pages

https://doi.org/10.1155/2017/7354654

## Recent Progress in Stability and Stabilization of Systems with Time-Delays

Systems Engineering Department, KFUPM, P.O. Box 5067, Dhahran 31261, Saudi Arabia

Correspondence should be addressed to Magdi S. Mahmoud

Received 8 November 2016; Accepted 7 March 2017; Published 13 June 2017

Academic Editor: Sabri Arik

Copyright © 2017 Magdi S. Mahmoud. 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

This paper overviews the research investigations pertaining to stability and stabilization of control systems with time-delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay-independent and delay-dependent stability results for time-delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay-dependent stability results are also discussed.

#### 1. Introduction

The occurrence of time-delay phenomenon appears to present many real-world systems and engineering applications. This takes place in either the state, the control input side, or the measurements side. It turns out that delays are strongly involved in challenging areas of communication and information technologies including stabilization of networked controlled systems and high-speed communication networks. In many cases, time-delay is a source of instability. However, for some systems, the presence of delay can have a stabilizing effect. The stability analysis and robust control of time-delay systems (TDS) are, therefore, of theoretical and practical importance.

On the other hand, time-delay systems (TDS) are also termed systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations [1]. As opposed to ordinary differential equations (ODE), TDS belong to the class of functional differential equations (FDE) which are infinite dimension [2, 3]. A wide variety of dynamical systems can be modeled as time-delay systems [4]. Loosely speaking, time-delay is usually a source of poor performance and instability of a control system. Alternatively, in some few cases, the presence of time-delay is helpful for the stabilization of some systems. Therefore, stability analysis of time-delay systems is of both practical and theoretical importance [5–9].

A great deal of the basic results is reported in [10–16]. Broadly speaking, stability conditions for time-delay systems can be broadly classified into two categories. One is delay-independent stability conditions and the other is delay-dependent stability conditions. Much attention was paid to the study of delay-dependent stability conditions as they yield less conservative results. Recently much work was presented in [17–30] covering alternative issues pertaining to stability and stabilization of dynamical systems with time-delays.

The primary objective of this paper is to(i)familiarize wider readers with TDS,(ii)provide a systematic treatment of modern ideas and techniques for researchers.The paper bridges the huge gap from some basic classical results to recent developments on Lyapunov-based analysis and design with applications to the attractive topics of network-based control and interconnected time-delay control systems. Essentially, it provides an overview on the progress of stability and stabilization of time-delay systems (TDS). Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. For simplicity in exposition, the discussions are limited to linear or linearizable systems. Some methods and techniques used to derive stability conditions for time-delay systems are reviewed. Several future research directions on this topic are also discussed.

*Notations*. Let denote the -dimensional Euclidean space equipped with the norm . We use , , , and to denote, respectively, the transpose, the inverse, the minimum eigenvalue, and the maximum eigenvalue of any square matrix and stands for a symmetrical and positive- (negative-) definite matrix . stands for unit matrix with appropriate dimension. . denotes the first difference of . We let denote the set of nonnegative real numbers; denotes the Banach space of continuous functions , and for , the associated norm is . We let .

Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. In symmetric block matrices, we use the symbol to represent a term that is induced by symmetry. Sometimes, the arguments of a function will be omitted when no confusion can arise.

The following facts are provided in [6].

*Fact 1. *Let , and be real constant matrices of compatible dimensions and let be a real matrix function satisfying Then for any satisfying , the following matrix inequality holds:

*Fact 2. *For any real matrices , , and with appropriate dimensions and , it follows that

Lemma 1 (Finsler’s lemma, [31]). *Let , , and such that rank. The following statements are equivalent:*(i)*.*(ii)*.*(iii)*.*(iv)

*2. Overview*

*There are many applications where time-delay phenomena appear quite naturally. This includes, but not limited to, the following:(A)Automotive: combustion model (ignition delay); electromechanical brakes (actuator delay).(B)Heat exchanger: distributed delay due to conduction in a tube.(C)Hydraulic networks: the transport phenomenon of water which is modeled as a varying time-delay.(D)Electrical networks.(E)Intelligent building: time-delay due to wireless transmission of sensor data.(F)Marine robotics: transport delay due to sonar measurement of depth.(G)Population dynamics: predator-prey model based on Volterra model with predator and prey populations ( is the time-life of prey):(H)Manufacturing process: the metal cutting process on a lathe which can be described as The study of this model is critical in understanding the regenerative chattering phenomenon.(I)Epidemics: understanding the dynamics of biological processes and epidemics which is a challenge for health workers engaged in managing treatment strategies. The underlying mechanisms can be revealed by considering epidemics and diseases as dynamical processes, for which the hematology dynamics can be modeled by which formulates the circulating cell populations in one compartment, where represents the circulating cell population, is the cell-loss rate, and the monotone function (describing a feedback mechanism) denotes the flux of cells from the previous compartment. The delay represents the average length of time required to go through the compartment.(J)Glucose-insulin model: letting and represent the levels of plasma glycemia and insulinemia; then where(i) is rate of glucose uptake by tissues (insulin-dependent) per pM of plasma insulin concentration,(ii) is net balance between hepatic glucose output and insulin-independent zero-order glucose tissue uptake (mainly by the brain),(iii) is apparent distribution volume for glucose,(iv) is apparent first-order disappearance rate constant for insulin,(v) is maximal rate of second-phase insulin release,(vi) is apparent distribution volume for insulin,(vii) is apparent delay with which the pancreas varies secondary insulin release in response to varying plasma glucose concentrations,(viii) is nonlinear function that models the Insulin Delivery Rate.(K)Neutral delay systems: arising, for instance, in the analysis of the coupling between transmission lines and population dynamics: evolution of forests. The model is based on a refinement of the delay-free logistic (or Pearl-Verhulst equation) where effects as soil depletion and erosion have been introduced where is the population, is the intrinsic growth rate, and is the environmental carrying capacity.*

*3. Models and Solutions*

*A general model of TDS can be expressed aswhere*

*3.1. Retarded Systems*

*It is quite natural to consider, as state-space, the set of continuous functions mapping the interval , with the topology of uniform convergence. The initial condition must be prescribed as . Observe that or may involve bounded jumps at some discontinuity instants. The nature of the solution (and of its initial value) then distinguishes FDE from ODE.*

*Definition 2 (see [3]). *A function is said to be a solution on of the retarded functional differential equation (RDE)if there are and such that , , and satisfies (10) for . For given ; , we say that is a solution of (10) with initial value at or simply a solution through if there is such that is a solution of (10) on and .

Supposing that is open and , then a function , , is referred to as a backward continuation of the solution through if and for any and is a solution of (10) on through .

*The interested reader is referred to [3] for further useful discussions.*

*3.2. Neutral Systems*

*Neutral systems also are delay systems but involve the same highest derivation order for some components of at both time and past time(s) , which implies an increased mathematical complexity. Neutral systems are represented byorwhere is a regular operator with deviating argument in time, as, for instance, with constant matrixIt is significant to observe that the solutions of retarded systems have their differentiability degree smoothed with increasing time, but this is no longer true for neutral systems due to the implied difference-equation involving ; the trajectory may replicate any irregularity of the initial condition , even if and satisfy many smoothness properties.*

*3.3. Models for Linear Time-Invariant Systems*

*In the linear, time-invariant case (LTI), the corresponding general time-delay model iswhere(i) and is constant instantaneous matrix;(ii)constant matrices represent discrete-delay phenomena;(iii)the sum of integrals corresponds to distributed delay effects, weighted by over the time intervals ;(iv)matrices account for the neutral part;(v)matrices and are input matrices;(vi)in brief, .Note that (15), , represents the output description, with discrete and distributed delayed parts as well. The special case of (14)-(15)has been investigated extensively in the literature.*

*4. Notion of Stability*

*As a starting point, we recall the following stability notion for time-delay system (3).*

*Definition 3. *If, for any and any , there exists a such that implies for all , then the trivial solution of time-delay system (3) is stable.

*The following properties are readily recognized.(i)If the trivial solution of time-delay system (3) is stable and if can be chosen independently of , then the trivial solution of time-delay system (3) is uniformly stable.(ii)If the trivial solution of time-delay system (3) is stable and if, for any and any , there exists such that implies , then the trivial solution of time-delay system (3) is asymptotically stable.(iii)If the trivial solution of time-delay system (3) is uniformly stable and there exists , such that implies for and , then the trivial solution of time-delay system (3) is uniformly asymptotically stable.(iv)If the trivial solution of time-delay system (3) is (uniformly) asymptotically stable and if can be arbitrarily large but finite number, then the trivial solution of time-delay system (3) is globally (uniformly) asymptotically stable.*

*5. Fundamental Stability Theorems*

*In the study of stability analysis of time-delay systems, the methods of Lyapunov functions and Lyapunov-Krasovskii functionals play important roles. There are two Lyapunov methods are often used:(A) Lyapunov-Krasovskii functional (LKF) method,(B)Lyapunov-Razumikhin function (LRF) method.It is significant to observe that LKF method deals with functionals which essentially have scalar values whereas Lyapunov-Razumikhin function (LRF) method involves only functions rather than functionals.*

*In this section, these two methods are reviewed; see [6] for details.*

*Consider the following time-delay system described bywhere (i); ,(ii) is continuous and is Lipschitz in ,(iii).In the sequel, we let be the solution of (17) at time with initial condition . Let be a bounded subset of and let be a bounded subset of .*

*A statement of Lyapunov-Krasovskii stability method is provided by the following theorem.*

*Theorem 4. Suppose that maps into and are continuous, nondecreasing functions with and and , for . If there exists a continuous functional such that (1),(2),where then the trivial solution of time-delay system (3) is uniformly stable. If , for , then the trivial solution of time-delay system (3) is uniformly asymptotically stable. Additionally, if , then the trivial solution of time-delay system (3) is globally uniformly asymptotically stable.*

*In some cases, the LKF involving terms depending on the state derivatives are quite effective in the derivation of the stability conditions. This will in turn requires the modification of the conditions in Theorem 4. See [8] for details.*

*A statement of Lyapunov-Razumikhin stability method is provided by the following theorem.*

*Theorem 5. Suppose that maps into and are continuous, nondecreasing functions with and and , for , and is strictly increasing. If there exists a continuous functional such that (1),(2), if where then the trivial solution of time-delay system (3) is uniformly stable. If , for , there exists a continuous nondecreasing function , for , and the foregoing condition is strengthened to , if then the trivial solution of time-delay system (3) is uniformly asymptotically stable. Additionally, if , then the trivial solution of time-delay system (3) is globally uniformly asymptotically stable.*

*The following Halanay result [7] also plays an important role in the stability analysis of time-delay systems.*

*Theorem 6. Suppose that constant scalars and satisfy , and is a nonnegative continuous function on satisfyingThen, for , one has where is the unique solution to the following equation: *

*Remark 7. *Theorems 4 through 6 can be used to derive stability conditions for the case when the delay is time-varying, which is continuous but not necessarily differentiable.

*Remark 8. *In the sequel, stability conditions for time-delay systems can be broadly classified into two types: (1)Delay-independent stability (DIS) conditions which do not include information about the delay. Generally speaking, DIS conditions are simpler to apply.(2)Delay-dependent stability (DDS) conditions which involve information on the size and pattern of the delay. DDS conditions are less conservative especially in the case when the time-delay is small.

*In the sequel, this paper focuses on the delay-dependent stability problem and the objective is twofold:(A)to develop delay-dependent conditions to provide a maximal allowable delay as large as possible,(B)to develop delay-dependent conditions by using as few as possible decision variables while keeping the same maximal allowable delay.*

*Alternatives approaches were proposed in the literature to obtain DDS conditions, among which the linear matrix inequality (LMI) approach is the most popular. The LMI approach has played a significant role due to the fact that family linear matrix inequalities can be readily converted into a convex optimization problem. The latter can be handled efficiently by resorting to recently developed numerical algorithms for solving LMIs [31]. Additional reason that makes LMI conditions appealing is their frequent readiness to solve the corresponding synthesis problems once the stability (or other performance) conditions are established, especially when state feedback is employed.*

*6. Stability Results for Linear Delay Systems*

*For the sake of simplicity, the following linear system with a single discrete delay is considered:where is the state vector, and are system matrices with appropriate dimensions, and is the time-delay factors. There are several classes of time-delay patterns considered in the literature as follows: Class A: constant delay, Class B: unknown-but-bounded delay, Class C: bounded time-varying delay, Class D: bounded time-varying delay with bounded derivative, *

*6.1. Constant Delay*

*When the time-delay is constant, the system described by (30) can be rewritten asNatural extensions of the quadratic Lyapunov functions can be particularly used to study in the framework of LTI delay systems (30) and the functionalOne obtains sufficient conditions by the following theorem.*

*Theorem 9. The time-delay system (30) is asymptotically stable for any if there exist matrices and verifyingor equivalently the LMI*

*It is significant to observe in the delay-free case, , that (33) provides the link with the Lyapunov equation for ODE. Nevertheless, in the delayed case , this sufficient condition is far from being necessary. From here, many generalizations were proposed, involving different alternative terms:*

*The following points are noteworthy:(1)Loosely speaking, the terms ; are used for the delay-independent stability of discrete delays.(2)The term is meant for distributed delays or discrete-delay dependent stability. On considering system (30) along with standard manipulation leads, with for ; for , to the following delay-dependent LMI condition: (3)Although the terms and appear, in a general form, in necessary and sufficient schemes (see [10–12]), the general computation of the time-varying matrices is excessively burden. To avoid such computational limitations, a discretization scheme incorporating piecewise-constant functions was introduced in [15, 16].*

*6.2. Time-Varying Delay*

*In what follows, we will review the LMI techniques in deriving DDS results for the single-delay case. Extension to the multiple-delay case is a straightforward task. We consider the class of time-delay systems (class B) in which the delay factor is continuous but bounded.Similar to (31), we consider the LKF of the formSince the time-varying delay may not be differentiable, we introduce the following equalities for any matrices , , and with appropriate dimensions:The following theorem summarized the main result.*

*Theorem 10. The time-delay system (37) is asymptotically stable if there exist matrices , and such that*

*Consider the time-delay system*

*According to the Lyapunov-Razumikhin stability method Theorem 5, the following stability condition can be obtained.*

*Theorem 11. The time-delay system (41) is asymptotically stable if there exist matrix and a scalar such that*

*On choosing the LKF (31), a delay-independent stability condition can be derived in the following form.*

*Theorem 12. The time-delay system (41) is asymptotically stable if there exist matrices and such that*

*Remark 13. *It should be noted that Theorem 12 is independent of the time-delay and therefore is very conservative especially when the time-delay is small. When the delay is constant, , it follows from the Schur complements that (40) is equivalent to In turn this implies that which is a necessary and sufficient condition for the stability of system (41) with zero delay.

*In the literature, the following Lyapunov functional is often used to derive delay-dependent results.It was first introduced in [32, 33]. Using the free-weighting [34], the following DDS condition can be derived based on the LKF (47).*

*Theorem 14. The time-delay system (41) is asymptotically stable if there exist matrices , , , and , and any matrices and of appropriate dimensions such thatwhere *

*6.3. Augmented Lyapunov Functional*

*Recalling that the first term in most LKFs is which involves the current state only and does not reflect the delayed state. Hence, an augmented Lyapunov functional was proposed in [35] for system described by (30).*

*Remark 15. *Compared with the Lyapunov functional (47), the augmented Lyapunov functional can lead to less conservative results. Additionally, it is also applicable for systems with time-varying delay, which can be seen in [36] and references therein.

*6.4. Triple Integral Lyapunov Functional*

*On examining the LKFs (31) and (50), it can be seen that the Lyapunov functional often contains integral terms: single and double in order to bring the effect of time-delays.*

*A natural question which arose is whether introducing triple integral terms in the Lyapunov functional would yield improvement in the stability behavior. This question is addressed [37, 38] by extending the LKFs (50)-(51) and incorporating a triple integral term to yield the form*

*Remark 16. *It is reported in [37, 38] by simulation results that the Lyapunov functional containing triple integral terms is quite effective in reduction of the conservatism of the stability conditions.

*6.5. Newton-Leibniz Formula*

*An alternative route can be pursued by using the Newton-Leibniz formulaand recalling (30) to yield*

*Remark 17. *It should be clear that the asymptotic stability of the time-delay system in (54) implies that of system (30).

*Following [6], we proceed to study the DDS of system (54) using the following LKF candidate:Define *

*The main stability result is established by the following theorem.*

*Theorem 18. The time-delay system (54) is asymptotically stable for any delay satisfying if there exist matrices , , and such that*

*Remark 19. *The technique by using the Newton-Leibniz formula to transform the time-delay system to appropriate for DDS analysis is quite useful. However, still a different route of writing (54) would beHowever, all the transformed time-delay systems by using the Newton-Leibniz formula introduce additional dynamics which may cause conservatism as the delay-dependent conditions derived based on the transformed systems.

*6.6. Bounding Techniques*

*In studying delay-dependent stability for time-delay systems, it is desirable to find methods that yield stability conditions with reduced conservatism. A wide class of early methods rely on generating improved bounds on some weighted cross products arising in the analysis of the delay-dependent stability problem. This class of methods is obtained by using the well-known algebraic inequalitywhere the vectors and matrix . An integral bounding inequality is as follows.*

*Lemma 20 (see [39]). Assume that and are given for . Then, for any and any matrix , one has*

* which when applied to time-delay systems of the type (30), it yields the following.*

*Theorem 21. The time-delay system (30) is asymptotically stable for any delay satisfying if there exist matrices , and such that*

*An improved version of Lemma 20 is expressed by the following.*

*Lemma 22 (see [40]). Assume that and and are given for . Then, for any and any matrix , one haswhere*

*By considering the following LKF,Applying Lemma 22, we obtain the following delay-dependent stability theorem.*

*Theorem 23. The time-delay system (30) is asymptotically stable for any delay satisfying if there exist matrices , and such that*

*On the other hand, deploying Lemma 1 together with Lemma 22, a different delay-dependent stability criterion is provided by the following theorem.*

*Theorem 24 (see [41]). The time-delay system (30) is asymptotically stable for any delay satisfying if there exist matrices and such that the following LMIs hold:*

*Remark 25. *The inequality in Lemma 22 is more general than both inequalities (59) and (60) and for this reason, it was extensively used in dealing with various issues related to time-delay systems to obtain delay-dependent results.

*Now, we present another important inequality, which is also effective in the derivation of DDS conditions.*

*Lemma 26 (see [42]). For any constant matrix , scalars , and vector function such that the integrations in the following are well-defined, then *

*Using Lemma 26 and selecting the LKF*

*we obtain the following stability result.*

*Theorem 27 (see [6]). The time-delay system (30) is asymptotically stable for any delay satisfying if there exist matrices , and such that*

*Alternatively, selecting the LKF*

*we obtain the following stability result.*

*Theorem 28 (see [6]). The time-delay system (37) is asymptotically stable for all continuous delay satisfying if there exist matrices and such that*

*A useful result is summarized by the following lemma.*

*Lemma 29 (the integral inequality [43]). For any constant matrix , scalar , and vector function such that the following integration is well-defined, then it holds that*

*Lemma 29 is frequently called the “integral inequality” and it is derived from Jensen’s inequality [44].*

*Remark 30. *It is significant to observe that Theorem 27 establishes that the time-delay system (30) is asymptotically stable for any delay satisfying when the LMI (69) attains a feasible solution, which implies that, for satisfying , the time-delay system (30) is asymptotically stable as well. Then, introducing the half delay into the time-delay system (30) will take more information on the system and thus may tend to reduce the conservatism in Theorem 27. For further elaboration on this argument, see [45].

*6.7. Discrete-Time Systems*

*Less attention has been paid to discrete-time systems with a time-delay because a linear discrete-time system with a constant time-delay can be transformed into a delay-free system by means of a state-augmentation approach. However this approach is not suitable for systems with either unknown or time-varying delays. For a small time-varying delays, the descriptor model transformation approach was employed [46].*

*Consider a class of discrete-time systems with state-delay is represented bywhere for , is the state and and are constant matrices. The delay factor is unknown-but-bounded in the form where the scalars and represent the lower and upper bounds, respectively, and denotes the number of samples within the delay interval.*

*Remark 31. *By setting in (73), it is readily seen that is a necessary condition for stability of system (73). From all studies on discrete-time-delay systems, it is assumed that this is always the case.

*Remark 32. *The class of systems (73) represents a nominally linear model which emerges in many areas dealing with the applications functional difference equations or delay-difference equations. These applications include cold rolling mills, decision-making processes, and manufacturing systems.

*Related results for a class of discrete-time systems with time-varying delays can be found in [47] where delay-dependent stability and stabilization conditions were derived. It should be stressed that although we consider only the case of single time-delay, extension to multiple time-delay systems can be easily attained using an augmentation procedure.*

*Intuitively if we associate with system (73) a positive-definite Lyapunov-Krasovskii functional and we find that its first difference is negative-definite along the solutions of (73), then the origin of system (73) is globally asymptotically stable. Formally, we present the following theorem for discrete-time systems of the type (73).*

*Theorem 33. The equilibrium of the discrete-time systemis globally asymptotically stable if there is a function such that (i) is a positive-definite function, decrescent, and radially unbounded,(ii) is negative-definite along the solutions of system (73).*

*For arbitrary value of , denote We haveIt is obvious that system (73) is globally asymptotically stable if and only if system (75) is globally asymptotically stable. For system (75), we definewhere and . It is easy to see that , decrescent and radially unbounded, and hence system (75) is globally asymptotically stable.*

*By selecting the Lyapunov-Krasovskii functional and invoking the Lyapunov-Krasovskii theorem, the following stability condition can be derived.*

*Theorem 34. The discrete-delay system (73) is asymptotically stable if there exist matrices and such that*

*We stress that LMI (80) is virtually delay-independent since it is satisfied no matter the size of delay is.*

*Next, sufficient delay-dependent LMI-based stability conditions are given. The approach used here does not introduce any dynamics and leads to a product separation between the matrices of the system and those from the Lyapunov-Krasovskii functional. The following theorem provides some LMI conditions depending on the values and .*

*Theorem 35. Given the delay sample number , system (73) subject to (74) is delay-dependent asymptotically stable if one of the following equivalent conditions is satisfied:(A) There exist matrices and such that(B) There exist matrices , , , , and such thatwhere In this case, the Lyapunov-Krasovskii functional (LKF)is such that*

*The result of Theorem 35 was developed in [47–49].*

*Next, we consider the following discrete-time piecewise linear systems with infinite distributed delays [50]:where is the state and denotes a partition of the state-space into a number of closed polyhedral subspaces, is the index set of subspaces, and is the control input. Matrices are constant matrices with appropriate dimensions corresponding to the th local model of the systems. When the state of the system transits from one region to another at the time , the dynamics is governed by the local model of the former one. is the convergence constants that satisfy the following condition: Distributed time-delays have been widely recognized and intensively studied for continuous-time systems [51]. However, the corresponding results for discrete-time systems have been very few due mainly to the difficulty in formulating the distributed delays in a discrete-time domain. The distributed delay term can be regarded as the discretization of the infinite integral form for the continuous-time system. The following result is recalled [51].*

*Lemma 36. Let , , and , are constants. If the series concerned is convergent, then one has*

*Introduce the following Lyapunov-Krasovskii functional candidate:By setting and invoking Theorem 34, the following result is obtained.*

*Theorem 37. Consider the piecewise linear system (86) with . If there exist matrices and such that the following linear matrix inequalities hold for :*

*We emphasize that Theorem 37 was established in [50].*

*7. Model Transformations*

*7. Model Transformations*

*It must be recalled that the prototype system (30), the independent of delay (IOD) stability demands matrix to be Hurwitz which, coherently, can be found in condition (33). On the other hand, the criteria ensuring delay-dependent stability for require the matrix to be Hurwitz as evident in condition (36). On this basis, several results concerning delay-dependent stability were derived, from the formula Consider the change of variablesThis will transform the multiple-delay system with possibly into the system having augmented delay Model (94) guarantees that the unstable nondelayed part in system (30) is absorbed in the stable part . Indeed such decomposition can be conveniently handled using LMI tools. It is shown in [5, 52] that the foregoing system can be written in the three following forms: It turns out that each of the above formulations can be studied by using specific Lyapunov-Krasovskii functionals (34) leading to the three different Riccati equations [5]:*