Stability of Non-Neutral and Neutral Dynamic Switched Systems Subject to Internal Delays

This study deals with the quadratic stability and li ear state-feedback and output-feedback stabilization of switched delayed linear dynamic sy stems with, in general, a finite number of non commensurate constant internal point delays. The re sults are obtained based on Lyapunov’s stability analysis via appropriate Krasovsky-Lyapunov’s funct ionals and the related stability study is performed to obtain both delay independent and delay dependen t results. It is proved that the stabilizing switch ng rule is arbitrary if all the switched subsystems ar e quadratically stable and that it exists a (in gen eral, non-unique) stabilizing switching law when the syst em is polytopic, stable at some interior point of t he polytope but with non-necessarily stable parameteri zations at the vertices defining the subsystems.


INTRODUCTION
Switching systems are hybrid dynamical systems composed of subsystems with their own parameterizations subject to a rule orchestrating the switching law between the various subsystems. In the last years, there has been increasing interest in stability and stabilization for switched dynamic systems [1][2][3][4][5][6][7][8][9][10] where appropriate switching laws decide through time which subsystem parameterizes the system so that stability is guaranteed. In particular, switching rules are applied [8][9][10] among several estimation schemes of a given linear plant which are then used to obtain different time-updated parameterizations of the adaptive controller. The switching law orchestrating the various estimators to obtain the active one which parameterizes the adaptive controller is interpreted as a higher hierarchical decision level of the whole adaptive system while the basic adaptation scheme is the lowest decision level. The switching law is designed so that the identification error is minimized in real time while the closed-loop system is guaranteed to be stable. A key motivation for studying switched systems is that many practical systems are inherently multi-model in the sense that several dynamic subsystems describe their whole behavior depending on multiple environmental factors [5] . On the other hand, time-delay systems offer an increasing interest since many real -life examples are subject to delays, like, for instance, population growth models, signal transmission, tele-operation problems, wear/peace models and actuator monitored processes with noisy sensors etc., [11] . Delays may be classified as point delays or distributed delays according to their nature and as external (i.e. in the inputs or outputs) and internals (i.e., in the state) according to the signals they influence. Point delays may be commensurate if each delay is an integer multiple of a base delay or, more generally, incommensurate if they are arbitrary real numbers [11][12][13][14] . The presence of internal delays leads to a large complexity in the resulting system's dynamics since the whole dynamical system becomes infinite-dimensional. This fact increases, in addition, the difficulty in the study of basic properties, like for instance controllability, observability, stability and stabilization and robustness, compared to the delay-free case since the transfer functions consist of transcendent numerator and denominator quasi-polynomials  . By those reasons, the design of exact or approximate poleplacement controllers towards the achievement of a finite or infinite closed-loop spectrum becomes of increased difficulty related to the delay-free case [9][10][11][12][13][14][15][16][17][18] . Neutral delay systems, which are those where the delayed time-derivative influences the system's dynamics [11,28,29] , present even a higher analysis and design difficulty. A great effort has been devoted to the investigation of the behavior of time-delay systems in sliding mode and in the use of such a property for synthesizing appropriate controllers [23][24][25][26][27][28][29][30][31][32][36][37][38] including applications to vibrations in heat exchanger tubes and aircraft dynamics [35,36] . An important point is that different types of delays appear in a natural way when modeling discrete systems and some classes of hybrid systems [39][40][41] . The objective of this study is to investigate the stability and stabilization properties of linear switched time-delay dynamic systems subject to, in general, multiple incommensurate known internal point delays.

Notation:
* The sets R (Z), R + (Z + ) and R 0 + (Z 0 + ) denote, respectively, the sets of real (integer) numbers, positive real (integer) members and non negative real (integer) numbers. * It is said that a complex matrix is strictly Hurwitzian, or a stability matrix, if all its eigenvalues have negative real parts. * An unforced linear system with r finite incommensurate internal point delays h i of state associate systems without delays, namely: Which describes the above so-called current delay-free system time-delay system which is called the nominal delay-free system which describes the above time-delay system when A i = 0 , or when h i → ∞ ; i = 1 , r .
Both systems have to be stable in order that the delay system is a stable independent of the delays. The system is said to be g.a.s. if it is globally asymptotically Lyapunov stable and g.u.a.s. If it is globally uniformly asymptotically stable: * The l 2 -norm of a matrix (or vector ) M is denoted as ( )

STABILITY AND STABILIZATION WITH ARBITRARY SWITCHING LAW
Asymptotic stability independent of and dependent on the delays: Consider the time-varying switched linear dynamic system: where, ( ) x t ∈ n R , ( ) m u t ∈ R , ( ) p y t ∈ R are the nstate, m-input and p-output, respectively, The following result is concerned with arbitrary switching laws while generalizing previous results in [12] to multiple point delays.  ii. The switched unforced system with quadratic stability for any arbitrary switching law σ (t) N ∈ , 0 t ≥ and any delays , if there exist ε + ∈ R and real matrices > , j, k r ∀ ∈ , such that the following set of matrix constraints holds:   1 1 Note that if Theorem 1 (i) holds then Theorem 1 (ii) holds with 0 Then, the stability depends on the delays may be checked from (7.b) to establish a range of maximum allowable delays.

Asymptotic stabilization independent of and dependent on the delays:
The results of Section 2.1 may be applied to forced stabilizable systems if a stabilizing regulation control law is applied. The discussion is limited to state and output linear feedback. The first definitions are first given.
such that the closed-loop system: is g.u.a.s. With quadratic stability. The following result whose proof is omitted holds.

Theorem 2:
The following items hold: i. Assume that the switched unforced system j r ∈ , i N ∈ is sufficiently small.
ii. Assume that the switched unforced system a.s. for arbitrary switching law Definitions 1 may be generalized in a natural way for linear state and output-stabilizability via linear regulation delay-dependent control laws [20][21][22][23]32,33] . For that purpose, consider the following control laws: N i∈ specifying the controller gains. Define n x n real matrices: The following technical result, concerned with the choice of the controller gains corresponding to the delayed dynamics if the control laws (9) or (10), so that the closed-loop delayed dynamics is annihilated or nearly annihilated. ≥ then for any given real matrix * i j Â , it exists a unique matrix: such that gives the minimum value of ( ) ≥ then for any given real matrix gives exact matching

ASYMPTOTIC STABILITY AND STABILIZATION WITH A SWITCHING LAW AMONG THE VERTICES OF A POLYTOPIC SYSTEM
The main result of this section follows below. An interpretation of Theorem 3 is as follows. The conditions of Theorem 3 (i) imply that if a polytopic system: all nonzero x (t) and all 0 t ≥ ( [2,[4][5][6]8] for a delay-free system). It is now interesting to investigate quadratic stability of a switched system composed of two subsystems with an (uncertain) polytopic-type parameterization which are not necessarily stable. Assume that the switched unforced system is defined for all 0 t ≥ by one of the two subsequent subsystems: Assume that i Σ (i=1,2) are uncertain polytopic systems defined by: r j∈ defining the delay-free and delayed dynamics at the vertices. For simplicity of exposition and mathematical proofs, it is assumed in the sequel without loss of generality that the number of extreme points 2 N i = ; i= 1, 2. In order to make the subsequent discussion nontrivial, the following assumption is made. there does not exist real square symmetric n-matrices 0 P i > ( i=1,2) such that: The matrix inequality (17)   . Note that if Assumption 1 holds then Theorem 1 cannot be applied because of the instability or critical stability at the vertices. The following result, whose proof is omitted, is related to the stabilization of (15) via switching. ii.

ASYMPTOTIC STABILITY OF A CLASS OF UNFORCED NEUTRAL SYSTEMS
A standard class of unforced neutral systems involving a single point delay is now focused on [11,28] . The extensions in the cases of multiple point delays and regulating inputs are direct by using direct extensions with the tools of this class of systems. They are omitted by the sake of simplicity. Consider the neutral system: where, the function of initial conditions is any absolutely continuous function The following result is related to the stability of (20) via switching:   [43] .
Now, the polytopic structures of Theorems 4-5 are extended for the given class of switching laws which conditionally stabilize the switched system under convexity-type constraints. The extensions of all the results in this section to the case of forced systems are directly obtained by using linear state/output feedback laws under either controllability/observability or stabilizability/detectability assumptions of the appropriate parameterizations of the subsystems. Some stability results for the neutral system of this section based on the properties of its subsystems are summarized in the following result.    hold as well by the same reasons that those pointed out related to Theorem 4. The extension of Theorem 6 (ii) to general parameterizations defined by (25) with 3 N i ≥ ; i=1,2 is direct by using more constraint (26) by involving the corresponding necessary i N 2 constraints in the same way as Theorem 4 is extendable to this situation. The extension is omitted for the sake of simplicity.

CONCLUSION
This study has been devoted to investigate the stability and stabilization properties of linear switched time-delay dynamic systems being subject to, in general, multiple incommensurate known internal point delays. Firstly, the uniform asymptotic quadratic Lyapunov stability (both independent of and dependent on the delays) for unforced systems has been investigated under arbitrary switching laws in nonpolytopic systems parameterized by a finite set of stable subsystems. The results have been extended to prove the existence of stabilizing switching laws in polytopic systems under testable convexity-type conditions for the vertices. Further stability results have been derived from forced systems for linear state/output feedback control laws under certain controllability and observability/stabilizability and detectability conditions. The stability results have been also extended to a class of switched polytopic systems and two switched systems consisting of a set of polytopic subsystems, subject to mutual switchings through time, which fulfill a convexity-type condition by each combination of sets of vertices, one corresponding to each polytope.
Numerical simulated examples have corroborated some of the obtained results.