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Mathematical Problems in Engineering

Volume 2008 (2008), Article ID 543145, 28 pages

http://dx.doi.org/10.1155/2008/543145

## Stability Results for Switched Linear Systems with Constant Discrete Delays

^{1}Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, 48080 Bilbao, Spain^{2}Department of Telecommunication and Systems Engineering, Engineering School, Autonomous University of Barcelona, Cerdanyola del Vallés, Bellaterra, 08193 Barcelona, Spain

Received 11 June 2008; Revised 27 August 2008; Accepted 11 November 2008

Academic Editor: Tamas Kalmar-Nagy

Copyright © 2008 M. de la Sen and A. Ibeas. 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 investigates the stability properties of switched systems possessing several parameterizations (or configurations) while being subject to internal constant point delays. Some of the stability results are formulated based on Gronwall's lemma for global exponential stability, and they are either dependent on or independent of the delay size but they depend on the switching law through the requirement of a minimum residence time. Another set of results concerned with the weaker property of global asymptotic stability is also obtained as being independent of the switching law, but still either dependent on or independent of the delay size, since they are based on the existence of a common Krasovsky-Lyapunov functional for all the above-mentioned configurations. Extensions to a class of polytopic systems and to a class of regular time-varying systems are also discussed.

#### 1. Introduction

Time-delay systems are gaining important attention in the last years. The reason is that they offer a very significant modelling tool for dynamic systems since a wide variety of physical systems possess delays either in the state (internal delays) or in the input or output (external delays). Examples of time-delay systems are war/peace models, biological systems, like, for instance, the sunflower equation, Minorsky’s effect in tank ships, transmission systems, teleoperated systems, some kinds of neural networks, and so forth (see, e.g., [1–12]). Time-delay models are useful for modelling both linear systems (see, e.g., [1–4, 13]) and certain nonlinear physical systems (see, e.g., [4, 7–9, 14]). A subject of major interest in time-delay systems, as it is in other areas of control theory, is the investigation of the stability as well as the closed-loop stabilization of unstable systems, [2–4, 6–9, 13–16] either with delay-free controllers or by using delayed controllers. Dynamic systems subject to internal delays are infinite—dimensional by nature so that they have infinitely many characteristic zeros. Therefore, the differential equations describing their dynamics are functional rather than ordinary. Recent research on time-delay systems is devoted to numerical stability tests, to stochastic time-delay systems, diffusive time-delayed systems, medical and biological applications, [17–20], and characterization of minimal state-space realizations, [21]. Another research field of recent growing interest is the investigation in switched systems including their stability and stabilization properties. A general insight to this problem is given in [22–24]. Switched systems are related to the fact that a system possesses several distinct parameterizations and commutes in between them through time according a certain switching rule. The problem is relevant in applications since the corresponding models are useful to describe changing operating points or relevant to synthesize different controllers which can adjust to operate on a given plant according to situation of changing parameters, dynamics and so forth. Specific related problems are the following.

(1) The nominal order of the dynamics changes according to the frequency content of the control signal since fast modes are excited with fast input where they are not excited under slow controls. This circumstance can imply the need to use different controllers through time.

(2) The systems parameters are changing so that the operation point changes. Thus, a switched model being adjusted to several operation points may be useful, [22–24].

(3) The adaptation transient has a bad performance due to a poor estimates initialization due to very imprecise knowledge of the true parameters. In this case, a multiparameterized adaptive controller, whose parameterization varies through time governed by a parallel multiestimation scheme, might improve the whole system performance. For this purpose, the parallel multiestimation scheme selects trough time, via a judicious supervision rule, the particular estimator associated with either the best identification, tracking or mixed identification/tracking objectives. Such strategies can improve the switched system performance compared to the use of a single estimator/controller pair [5, 25]. The asymptotic stability of switched system has been investigated exhaustively along the last decade (see, e.g., [22–24] and references there in). However, parallel general results for switched time-delay systems are not abundant in the literature. Stability results of time-delay switched systems have been obtained recently for the case of one single delay by decomposition of the dynamics into a sum of a linear ordinary differential equation and a linear delay differential equation, [12]. Related results have also been obtained by adapting switching rules for ordinary differential equations to those describing time-delay systems, [11]. Some further recent related research investigates time-varying systems under point delays. In that context, switches are considered to produce an impulsive time derivative of the matrix of dynamics, [26]. This paper is devoted to the investigation of the stability properties of switched systems subject to internal constant point delays. A first package of results is concerned with the global exponential stability of switched systems either independent of or dependent on the delay sizes based on the use of Gronwall’s lemma, [27]. Exponential stability independent of the delay size is proved under the restriction of small dynamics (characterized in terms of norms) of the delayed dynamics provided that the current delay free matrix of dynamics is stable. Exponential stability dependent on the delay size is proved under the restriction of sufficiently small delays provided that the delay-free system is exponentially stable. Note that it is well known that both of them have to be stable for any linear time-invariant configuration in order that the corresponding time-delay system may be asymptotically stable, [1, 4, 13]. It has to be pointed out that Gronwall’s lemma has been chosen as elementary analysis tool for exponential stability of switched systems since Lyapunov techniques are direct only for discussing the weaker property of asymptotic stability in the case of time-delay systems since either the Lyapunov functional, its time-derivative, or both depend on the state-squared norm on a time interval depending on the delay size rather than on each current time instant. A minimum residence time at each particular configuration, which depends on the parameterization, is required in order to guarantee exponential stability for this first set of stability results. A second package of weaker stability results concerning asymptotic stability is also discussed based on the use of “ad hoc” Krasovsky-Lyapunov’s functionals, [1]. Contrarily to the first package of results, those of the second one do not require the maintenance of a minimum residence time between any two consecutive switching instants since it is assumed that all the parameterizations possess a common Krasovsky-Lyapunov functional. Basically, four types of systems are included in different sections of this paper, namely: (a) all the configurations are time invariant with identical delays, (b) the various distinct parametrical configurations can eventually posses distinct delays, (c) the system has a polytopic structure with the vertices being associated with limit configurations, and (d) the various system parameters vary continuously through time under the restriction that the dynamics possesses time derivative almost everywhere and there are finite parametrical jumps of sufficiently small sizes at isolated time instants. In this case, asymptotic stability is guaranteed if the jumps occur at sufficiently large intervals compared to available bounds depending on other parameters of the dynamic system establishing worst-case situations for the stability degree.

#### 2. Stability results based on the use of Gronwall’s lemma and matrix measures

Consider the th linear and time-invariant dynamic system with internal (i.e., in the state) delayed dynamics with constant discrete (or point) delay: where is the state vector and () are matrices of delay-free and delayed dynamics, respectively, for delay , either for some admissible delay upper-bound or for all . The first characterization will lead to stability dependent on the delay size results while the second one will lead to results about stability independent of the delay size. The initial condition is defined by any piece-wise absolutely continuous function with . The generalization of potential results for (2.1) for more general systems involving delays () of the form is direct, and some related results will be discussed in Section 3. Thus, the attention is focused on the behavior and properties of (2.1). The solution of (2.1) over is unique for each given such a function of initial conditions from the well-known Picard-Lindeloff uniqueness theorem [27]. One expression of such a solution may be calculated via the superposition principle for linear systems with the unforced solution calculated as associated with the -semigroup of infinitesimal generator with the particular correction due to the forcing term . If then (2.1) becomes so that (2.1) is delay free. Norm-upper bounds of the solution of (2.1) through time can be computed by adapting the well-known Gronwall’s lemma [27] to calculate a norm upper-bound for the solution of (2.1). Note thatGronwall’s lemma is a useful tool to prove exponential stability either under small delays or for the case of small norms of the matrices of delayed dynamics provided that the delay-free system counterpart or the system without delay dynamics matrices, respectively, are stable. Exponential stability is a stronger property than asymptotic stability which is the one usually proved for time-delay systems with Lyapunov stability tools. The following results are proved in Appendix A and extend their weaker previous statement for asymptotic stability (see, e.g., [1]). They are of interest for the subsequent study related to results on exponential stability under switching among distinct parameterizations of time-delay systems with point delays.

Theorem 2.1. *
Assume
that is a stable matrix (i.e., all its eigenvalues
are in ). Then, it exists a maximum allowable delay such that (2.1) is globally exponentially stable
for any delay .
*

Theorem 2.2. *
Assume
that is a stable matrix. Then, (2.1) is globally
exponentially stable independent of the delay (i.e., for any delay provided that is sufficiently small compared to the
stability abscissa of ). For (spectral)-norms, a related specific testable condition is
, for
any real constant
satisfying
, where
is the matrix
measure of
with respect to the -norm, and , denote the distinct
eigenvalues of such a matrix.
*

If the condition of Theorem 2.2 is relaxed to then the time-delay system is guaranteed to be globally Lyapunov’s stable since Theorem 2.3(i) below states that stability independent of delay implies that is a stable matrix, related to the case of arbitrarily large delay while is also a stable matrix which is related to the case of zero delay [1]. Theorem 2.3(ii) is to some extent the converse of Theorem 2.2. It is proved that if with being the matrix measure of with respect to the -norm, then some dynamic system of the same structure as (2.1), eventually (2.1) itself, and some matrix of delayed dynamics of the same -norm as that of becomes at least critically stable, or even instable, for small delay size. As a result, it cannot be globally exponentially stable even if the delay-free system matrix is a stable matrix.

Theorem 2.3. *
The
following properties hold.
*(i)*
If
and
then (2.1) is
globally exponentially stable independent of the delay size and both
and are stable matrices.
*(ii)*
If
is a stable matrix and
then there exists
a system (2.1) with a matrix fulfilling (so that
) such that
the system is either critically stable or unstable. *

#### 3. Switching among distinct parameterizations

Consider
the th linear and time-invariant dynamic *switched system* with internal delayed
dynamics:
where is the state vector and
;
; are matrix functions of piecewise constant
entries of delay-free and delayed dynamics, respectively, for some delay and some admissible delay upper-bound ,
or for ,
where denotes the set of real-square matrices of
order. In other words, ;
. The
piecewise constant real function is the *switching
law* defined by for all and
is a strictly increasing real sequence of *switching instants* and is the *switching indicator* of either finite cardinal (i.e., the switching
process stops in finite time) or infinity cardinal (i.e., the switching process
never ends). The initial condition function is defined by any given piece-wise
absolutely continuous function with .

#### 3.1. Constrained Switching

Some results of Section 3, obtained for a single parameterization, are directly extendable to the case of switched systems for constrained switching between different parameterization of the same structure as that of (2.1). The switching law is required to respect a minimum residence (or dwelling) time between any two consecutive switches. The following result, proved in Appendix B, extends Theorem 2.1 for sufficiently small delay.

Theorem 3.1. *
Assume
that are stable matrices for all .
Thus, there exists
a maximum allowable delay such that (3.1) is globally exponentially stable
for any delay for any switching law fulfilling for all with being a minimum residence time which depends
on the set
;
; with
being the sequence of switching instants. *

Note
that Theorem 3.1 holds also *if
there are many infinite parameterizations* with being stable matrices for all with a switching law .
Theorem 2.2 is now extended to the switched system (3.1) possessing internal delay
of arbitrary size. Its proof is given in Appendix B. It is stated that global
exponential stability holds if all the parameterizations are stable and an
appropriate minimum residence time in between any two consecutive distinct
parameterizations is respected. A minimum size of the residence time required
for stability is calculated in the proof.

Theorem 3.2. *Assume
that are stable matrices for all .
Thus, (3.1) is globally exponentially stable independent of the delay provided
that the following hold.
*(1)* for all are sufficiently small compared to the maximum
stability abscissa of for all *(2)*
The switching law which generates the switching instants fulfils for all with being a minimum residence (or dwelling) time
which depends on the set
;
, with being the switching instants. *

For (spectral)-norms, a related specific testable condition in Theorem 3.2 is for any real constant satisfying for all where for all is the measure of the matrix with respect to the -norm, and () denote the distinct eigenvalues of such a matrix.

#### 3.1.1. Simulation Example 1

Consider the linear switched system (3.1) with second, and three parameterizations: According to Theorem 3.2, all the matrices are stable for , and , and . Also, note that the matrices and are not stable since it is not required in Theorem 3.2. The initial conditions are , , . They are taken of large amplitudes to visualize the exponential behavior of the solution under eight switches of this exponentially stable system with high stability degree. The residence time is given by second. The simulated results are shown in Figure 2 for the switching rule of Figure 1.

#### 3.2. Arbitrary Switching

It is well known for the case of time-delay time-invariant systems that if all the parameterizations possess a common Lyapunov function, then the switching law can be arbitrary (i.e., without requiring a minimum residence time in contrast with Theorems 3.1, 3.2) while keeping global asymptotic stability. It has been also proved that the various parameterizations possess a common Lyapunov function independently of their stability abscissas if and only if all the matrices of dynamics pair-wise commute [2]. A stronger sufficient condition for existence of a common Lyapunov function which does not require pair-wise commutation is that the norm deviations among the various parameterizations be sufficiently small related to any of the stability abscissas. A generalization for the switched time-delay system (3.1) under sufficiently small delay is given below. Its proof is provided in Appendix B under the assumption that a sum of any number of matrices parameterizing the switched system (3.1) is a stable matrix and pair-wise commutes with its transpose. The global exponential stability of the switched system (3.1) for an arbitrary switching law is also guaranteed under the less restrictive assumption that all the above matrices pair-wise commute while being stable matrices.

Theorem 3.3. *
The following properties hold.
*(1)*
The switched
system (3.1) is globally exponentially stable for any arbitrary switching law provided that the delay is sufficiently
small if are all stable matrices such that
and
pair-wise commute for all .
*(2)* The switched
system (3.1) is globally exponentially stable for any arbitrary switching law provided that the delay is sufficiently small
if are all stable matrices and pair-wise commute
for all . *

An alternative condition for the existence of common Lyapunov functions follows by direct calculation for arbitrarily close parameterizations of the switching system. Define () for any and for all ( ( for all )) and let be any real constant to the right of the stability abscissa of and close to it. Since is a stable matrix, for any positive definite square real -matrix , is a positive definite symmetric solution matrix to the Lyapunov equation for all . For all The right-hand side of the above equation is guaranteed to be negative definite if for all with being such that for all denoting the vector and corresponding (induced) matrix norm, and , denoting the minimum and maximum eigenvalues of the real symmetric matrix . The last constraint is identical to for all which guarantees the existence of (at least) a common Lyapunov function for all the parameterizations of the switched system. More than one common Lyapunov function may exist if all the parameterizations are close to each other in terms of small norm deviations. The above constraint implies a physical one on the stability abscissas of the various parameterizations. Note that the choice can be made and in the better case. Thus, a common Lyapunov function for zero delay exists, even if not all the parameterizations pair-wise commute, provided that Two simple consequences of Theorem 3.3(i) follow as specific results.

Corollary 3.4. *
The
switched system (3.1) is globally exponentially stable for any arbitrary
switching law provided that the delay is sufficiently small
if are all stable matrices for all and all the matrices in the set pair-wise commute.
*

The above result is direct since the commutation condition implies that of Theorem 3.3. The subsequent results refer to the feature that any two matrices of product compatible orders commute if and only if any one of them is a matrix function of the other, [2].

Corollary 3.5. *
The
switched system (3.1) is globally exponentially stable for any arbitrary
switching law if and only if
for all and any given (i.e., all the matrices are function matrices
on any particular one in the set) provided that the delay is sufficiently small. *

Estimations for the maximum sizes of the delay which guarantee global exponential stability are given in the proofs of the results. It has been proved formerly that two matrices of the same order commute if and only if any of them is a matrix function of the other [2, 24]. For any arbitrary switching law, the switching instants may be arbitrarily close and the switching from each current parameterization can occur to any other in the set of parameterizations. Subsequently, sufficient-type conditions for exponential stability independent of the delay for arbitrary switching are discussed by using Krasovsky Lyapunov functionals [1] for stability testing. Denote that for all The subsequent notations mean that the real square matrix is positive definite, positive semidefinite, negative definite, and negative semidefinite, respectively. Consider the Krasovsky-Lyapunov functional candidate defined with matrices , [1]:

Theorem 3.6. *
The
following properties hold.**
(1) The switched system (3.1) is globally
asymptotically stable independent of the delay size for any switching law and each admissible function of initial
conditions if
** for
some , For such a system, the functional (3.7) is a
common Krasovsky-Lyapunov
functional parameterization for its distinct
parameterizations.**(2) Consider the general polytopic switched system
** where is the state vector, ;
;
are matrix functions defining the switched
dynamics and the are the delays (being eventually distinct) for
each of the distinct parameterizations,
and are either zero or unity scalars while is unity for at least one for each . Thus, if
** for
some ,
(for all ),
then the switched system (3.9) is globally asymptotically stable independent of
the delays for any switching law and each admissible function of initial
conditions. Also,
**
with redefined by for all and is a common Krasovsky-Lyapunov
functional parameterization for its distinct
parameterizations.
*

The proof of Theorem 3.6(1) follows straightforwardly by direct calculation noting since the time-derivative of (3.7) along any trajectory solution of (3.1) is for some provided that so that it is a negative quadratic form in since for all . Theorem 3.6(2) follows since if for some so that it is a negative quadratic form in since for all from (3.10). Thus, (3.11) is a Krasovsky-Lyapunov functional for the switched system (3.9).

*Remark 3.7. *
Theorem 3.6(2)
applies when each of the -switched parameterizations has a unique delay and
all of them are distinct, that is, ,
,
, for all . This situation means
that switching implies modifications of the delays and that of the matrices of
dynamics. Another particular case of Theorem 3.6(ii) is that dealt with in
Theorem 3.6(1), that is, there is a single delay for all the parameterizations,
that is, , , ,
, , for all .

*Remark 3.8. *
Note that
Theorem 3.6 does not involve necessary stability conditions. The reason is that
in the delay-free case of linear and time-invariant systems, the solvability
Lyapunov matrix equation, and then the existence of its associate Lyapunov
function is a necessary and sufficient condition for global asymptotic
stability (which also implies and is implied for exponential stability) but the
property does not extend to functional equations and Krasovsky-Lyapunov
functionals like (3.7) or (3.11).

Theorem 3.6 extends for stability dependent on the delay size as follows.

Theorem 3.9. *
The
following properties hold.**(1)
The switched system (3.1) is
globally asymptotically stable for any delay and for any switching law and each admissible function of initial
conditions if
**for
some , .
The real functional,
**is
a common Krasovsky-Lyapunov
functional for its distinct
parameterizations. **
(2) The switched system (3.9) is globally
asymptotically stable independent of the delays for any switching law and all delays ; (for all
) if
**for
some , (for all , for all ).
The real functional,
**
with , (for all , for all ) is a common Krasovsky-Lyapunov functional for
its distinct parameterizations with being redefined
by for all and .*

The proof is direct since it is very close to that of Theorem 3.6. if for Theorem 3.9(1) and for any nonzero for Theorem 3.9(2).

*Remark 3.10. *
Note that if the conditions (3.8), (3.10), (3.12), and
(3.15) of Theorems 3.6(1), (2) and 3.9(1), (2) are relaxed to
, ,
and for all ,
respectively, then the corresponding switched system (3.1) or
(3.9) is guaranteed to be globally Lyapunov’s
stable.

A generalization of interest of the polytopic switched system (3.9) is with , being either zero or unity scalars and at least one for for each and subject to . The results of Theorems 3.6(1), (2) and 3.9(1), (2) correspond to switches through time among the various parameterizations of (3.18) at any of the vertices of the polytope of parameters all of them being stable [3, 4]. The switched system (3.18) is more general since any parameterization built with the convex hull of the sets of matrices for all is admissible. The following result extends directly Theorems 3.6(2) and 3.9(2) to the switched polytopic system (3.18).

Theorem 3.11. *
The following properties hold.
*(1)*
Assume that (3.10) for all .
Then, the switched system (3.18) is globally asymptotically stable independent of
the delays (i.e., for all ;
for all ) irrespective of the switching law .
The real functional (3.11) is a Krasovsky-Lyapunov functional for the switched
polytopic system (3.18).
*(2)*
If ,
(3.15) for all then the switched system (3.18) is globally
asymptotically stable for any delays
for all , irrespective of the switching law .
The real functional (3.17) is a Krasovsky-Lyapunov functional for the switched
polytopic system (3.18).*

The proof of Theorem 3.11 is direct since all the real functions are nonnegative and they cannot be simultaneously zero at any time instant. This leads to functional time derivatives and for all nonzero along any nonzero state trajectory solution, respectively. The subsequent direct refusal of the conditions of Theorem 3.11, supported by “ad-hoc” Lyapunov’s instability theorems, leads to situations where asymptotic stability of the switched system is impossible.

Theorem 3.12. *
The
following properties hold.**(i) Assume
that (3.8) for all .
Then, the switched system (3.1) is not globally asymptotically stable independent
of the delay size for any switching law but it can be still globally Lyapunov’s stable
for some switching law .
Those properties also hold for any particular delay. If for all then the switched system (3.1) is instable for
any delay and any given switching law for any given admissible function of initial
conditions.
**(ii) Assume
that ,
(3.10) for all .
Then, neither the switched system (3.9) nor any parameterization of the polytopic
switched system (3.18) are globally asymptotically stable independent of the
delay size for any switching law but it can be still globally Lyapunov’s stable
for some switching law .
Those properties also hold for any particular delay. If for all then the switched systems (3.9) and (3.18) are
both instable for any delay and any given switching law for any given admissible function of initial
conditions.
**
(iii) Assume that for at least some .
Then, there are infinitely many switching laws for which the switched system (3.1) is globally
asymptotically stable and infinitely many for which it is globally Lyapunov’s
stable. The same property holds for the switched system (3.9) and infinitely many
(but not all) parameterizations of the polytopic switched system (3.18) provided
that for at least some .
*

#### 3.2.1. Simulation Example 2

Consider the linear switched system (3.1) with seconds and parameterizations According to Theorem 3.3, all the matrices are stable and pair-wise commute for . The initial conditions are given by . The arbitrary switching is defined by the switching law of Figure 3 where the residence time between consecutive switching time instants ranges from a small to a larger one. Figure 4 shows that the state variables tend asymptotically to zero.

#### 3.3. A More General Time-Varying Switched System

In the following, global asymptotic stability independent of the delay is stated for the linear time-varying system which is a generalization of (3.1) since, in general, parametrical switches are admitted at sufficiently large separated time instants. It is assumed that the matrix is a stable matrix for all time which is time differentiable almost everywhere, namely, except at the time instants where its parameters have bounded discontinuities at some of its entries. Those time instants may be considered as switching instants for the system parameterization. The proof is given in Appendix B.

Theorem 3.13. *
Consider (3.1) under the following assumptions.
*(1)*
Each value of the matrix function is a stable matrix satisfying for all .
*(2)* has uniformly bounded entries for all time.
*(3)* exists and it is bounded almost everywhere in with sufficiently small upper-bound
where in exists.
*(4)*
There exist positive definite real -matrix functions and satisfying
*(5)*
The time
interval between any two consecutive time instants at which
does not exist is sufficiently large according
to the amplitude-bound (in terms of -norm) of the
corresponding bounded discontinuities of ,
the absolute value of the stability abscissa
,
,
and and .
**
Then,
(3.20) is globally asymptotically stable independent of the delay. *

However, (3.20) is a special case of switched system, the switches being of arbitrary but sufficiently small size at sufficiently large separated time instants. The dynamics are time-varying and one of the matrices is almost everywhere time differentiable with bounded norm time derivative. An alternative characterization which does not require time differentiability of the matrix entries can be developed directly for the case that the matrix function , which is not required to be either continuous or time differentiable, is locally deviated in norm for all time with respect to a comparison stability constant matrix with sufficiently large absolute stability abscissa with respect to the constant matrix. Stability results based on Gronwall’s lemma are direct even if the matrix entries have sufficiently small bounded jumps, related to the absolute stability abscissa of the constant comparison matrix, through time.

*Remark 3.14. *
In the
light of Theorem 3.13, Theorems 3.6–3.11 can be extended
for the case of time-varying systems possessing also eventual jumps in the
entries of the matrix functions defining their dynamics. The extensions apply
to more general systems under the structures of (3.1), (3.9), and (3.18) but where
the parameters are uniformly bounded for all time and almost everywhere continuous
and almost everywhere time-differentiable functions. A common property is that
the stability is not independent of the switching law consisting of the rule
for selecting the time instants where parametrical jumps appear. The conditions
are a “mutatis-mutandis” modification of those of Theorem 3.13 by using the
extended quadratic form of the time derivatives of the corresponding borrowed Krasovsky-Lyapunov
candidates and extended directly from those in Theorems 3.6–3.11. Therefore,
such conditions are not made explicit. In particular, note that the following
hold.(a)Theorem 3.13 is an extension of
Theorem 3.6(1) if (3.1) is generalized to (3.20).(b)In the same way, Theorem 3.6(2) may be generalized for asymptotic stability
independent of the delay if the polytopic system (3.9) is generalized to one
possessing a similar structure based on the linear time-varying system (3.20).(c)Theorem 3.9 is extendable with
the ideas of Theorem 3.13 for global asymptotic stability dependent on the delay
size. Also, Theorem 3.9(1) is extendable for the extended system (3.20) from (3.1)
for stability dependent on the delay provided that is a bounded stable matrix with a prescribed
stability abscissa which is almost everywhere time differentiable with bounded
derivative and which possess jumps of sufficiently small amplitudes at
sufficiently separated time instants according to their sizes and the remaining
system parameters. Theorem 3.9(2) may be extended for the generalization of (3.9)
to (3.20).(d)
Theorem 3.11(1) is extendable for
stability independent of the delay directly to the generalization of the
polytopic system (3.18) to a similar polytopic form with being a matrix function under Theorem 3.13. However,
Theorem 3.11(2) is extendable for stability dependent on the delay size to such
an extended polytopic system provided that is bounded and stable with prescribed abscissa
and almost every where time-differentiable entries with eventual jumps at
sufficiently large time-intervals.

*Remark 3.15. *
Note that
the various systems dealt with through this manuscript might be reformulated to
consider *asynchronous switching laws* in the sense that the matrices associated with each delay may belong to its own
set of matrices each governed by an independent switching law. Consider, for
instance, (3.1) extended to the more general one:
where select for all time matrices for all ,
for all . Also, (3.9) extended to the more general one:
where select for all time matrices for all ,
for all .
A close extension is valid for the polytopic switched system (3.20). The various
given stability results either for arbitrary switching laws or those requiring a minimum residence time extend
directly to these more general systems if one considers the combined natural synchronous switching law defined with the whole set of asynchronous
ones by a -tuple of positive integers which is a vector mapping on a Cartesian
product The above to a synchronous switching law has at most distinct parameterizations of the switched
system formed with all the combinations of matrices associated with each
particular delay with for the switched system (3.1). If the set of matrices
for some particular delay is less than , the total number of distinct
parameterizations decreases accordingly.

#### 3.3.1. Simulation Example 3

Consider now the time-varying system given by (3.20) with sec., , and with and . Note that is a discontinuous function implying that has impulsive behavior at integer times. The solution of (3.20) with the above parameterization is depicted in Figure 5.

#### 4. Concluding remarks

This paper has discussed the global exponential stability and the global asymptotic stability of standard classes of switched linear system subject to internal point time delays. The considered classes of systems have been those involving a set of time-invariant parameterizations, those having a polytopic structure, and those being time varying for all time with potential parametrical switches at certain times. The properties of global exponential and global asymptotic stability dependent on and independent of the delay size have been investigated for arbitrary switching laws and for switching laws subject to minimum residence time among any two consecutive switches. As in the delay-free case, stability under arbitrary switching requires commutation conditions on the system matrices of the various parameterizations. Otherwise, minimum residence times which depend on the parameterizations are required to keep the stability of the switched system.

#### Appendices

#### A. Proofs of the results of Section 2

##### A.1. Proof of Theorem 2.1

It holds that (2.1) may be equivalently rewritten as where the standard Landau’s “small ” and “ big ” notations are used; that is, as provided that with . It is obvious that for all and some sufficiently small exists where is the th identity matrix. Thus, from (A.1). The unique solution of (A.2) over by any piece-wise absolutely continuous function with , which is the same as that of (2.1) for all some sufficiently small , is so that where is the largest time instant of maximum state norm within . Since is a stable matrix, there exist real constants (being norm-dependent) and such that . Then, one gets from (A.4) via Gronwall’s lemma [27] It follows from (A.6) that there exists a sufficiently small fulfilling such that exponentially as for all since as . Thus, and exponentially as .

##### A.2. Proof of Theorem 2.2

The solution of (2.1) is Since is a stable matrix, there exist real constants (being norm-dependent) and such that . Then, if then one gets from (A.7) with the application of Gronwall’s lemma for : If then exponentially as for all since as . Thus, and exponentially as . It turns out that for and any real constant or, equivalently, for , and any (dependent on ) real constant , for any given sufficiently large finite , it follows from (A.9) that since . If -(spectral) norms are used for vectors and matrices, then the following choices of parameters are made in (A.10): where with , denotes the matrix measure with respect to the -norm of the square matrix ; and () and denote, respectively, the distinct eigenvalues and the maximum singular value of such a matrix. Note that since is a stable matrix. The inequality is strict if the maximum eigenvalue of is multiple. The matrix measure with respect to the -norm of the matrix is identical for all similar matrices to which follows directly from the definition of such a matrix measure. Since the stability properties of linear time-invariant systems are independent of the chosen state-space representation, it follows directly from (A.10) that (2.1) is globally exponentially stable independent of the delay if for some real constant .

##### A.3. Proof of Theorem 2.3

Note that . From the definition of the matrix measure with respect to the -norm, all the eigenvalues of and also those of are in so that is a stable matrix. Since is a stable matrix, Theorem 2.2 applies so that (2.1) is globally exponentially stable independent of the delay since . Since the system is globally exponentially stable independent of the delay, it is globally exponentially stable for zero delay so that is a stable matrix. The first part of the result has been proved. The second part is proved by finding by construction a system which is not exponentially stable of delayed dynamics being of identical -norm to that of as follows. Construct with some to be specified later so that . Since is a stable matrix, is, by construction, an antistable matrix (i.e., with all its eigenvalues in ). From the definitions of spectral norms and associate matrix measures and their properties, one has and then so that for , since As a result, is not a stable matrix and the system is not globally exponentially stable independent of the delay size while being either critically stable or unstable for zero delay.

#### B. Proofs related to Section 3

#### B.1. Proof of Theorem 3.1

Note that for all for pairs since the matrices are all stable matrices for all with and . After defining so that , (A.6) in the proof of Theorem 2.1 is now replaced on the interval between two consecutive switching instants with with if and being a sufficiently large minimum residence time provided that the delay for sufficiently small Assume that the switching law satisfies for all , for all and that (A.1) holds for any with being a proper subset of of finite cardinal . Assume that for all adopting the convention From (A.1), taking initial conditions at and final ones at , it follows that for all since the minimum residence time in between consecutive switches is respected for all switches. Thus, by complete induction is exponentially fast and all the sequence is uniformly bounded provided that has infinite cardinal. Otherwise, and are kept without modification from (A.6) for with initial conditions at finite if . Thus, the proof still follows from (A.9) with for all and .

#### B.2. Proof of Theorem 3.2

Note that for all for pairs since the matrices are all stable matrices for all ; with and . After defining so that (A.9) in the proof of Theorem 2.2 is now modified on the time interval in between two consecutive switching instants , as follows after using Gronwall’s lemma for the real function : if and being a sufficiently large minimum residence time satisfying: and being identical for any switching instant provided that for any prefixed . An alternative weaker condition than (A.2) is to switch with subject to a switching-dependent minimum residence time: provided that for any prefixed and any provided that . Both constraints lead to if switching never stops. Under a maximum finite switching instant and then as with exponential convergence rate. It switching never stops, then with exponential convergence rate. Thus, global exponential stability is guaranteed by any switching law respecting a minimum residence time in between any two consecutive switching instants. If -norms are used, then the particular stability result follows directly from the above general proof and the last part of the proof of Theorem 2.2. The proof is omitted.

#### B.3. Proof of Theorem 3.3

(i) Take with being the switching instants generated by the switching law and define the extended indicator of switching instants with and interswitching periods for all . Thus, the switched system (3.1) for zero delay becomes where for all is a real diagonal matrix with real positive eigenvalues less than unity since are all stable matrices. The real nonsingular matrix defining the state transformation is orthogonal since are all symmetric and identical for any (i.e., independent on the current ) and for all , since ; for all pair-wise commuting with its transpose implies that pair-wise commute; for all , for all , [28]. Thus, is real diagonal with eigenvalues less than unity for all . During the interswitching periods,