# Stability Results for Switched Linear Systems with Constant Discrete Delays

Authors: M. de la Sen; A. Ibeas
Journal: Mathematical Problems in Engineering
 (2008)

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## 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.


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## Body

## 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
thenth linear and time-invariant dynamic
system with internal (i.e., in the state) delayed dynamics with  constant discrete (or point) delay: (2.1)x˙(t)=A0x(t)+A1x(t−h),
where x(t)∈Rn is the state vector and Ai∈Rn×n  (i=0,1) are matrices of delay-free and delayed
dynamics, respectively, for delay h∈[0,h¯],
either for some admissible delay upper-bound 0≤h¯<∞ or for all h∈[0,∞).
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 φ:[−h,0]→Rn with φ(0)=x(0)=x0.
The generalization of potential results for (2.1) for more general systems
involving q delays 0=h0<hi−1<hi
(i∈q¯:={1,2,…,q}) of the form
(2.2)x˙(t)=∑i=0qAix(t−hi) 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 R0+:={R∋t≥0} 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 C0-semigroup of infinitesimal generator A0 with the particular correction due to the
forcing term A1x(t−h).
If h=0, then (2.1) becomes x˙(t)=(A0+A1)x(t), 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(A0+A1) is a stable matrix (i.e., all its eigenvalues
are in Res<0). Then, it exists a maximum allowable delay h¯∈R+:={R∋t≥0} such that (2.1) is globally exponentially stable
for any delay h∈[0,h¯).Theorem 2.2.
Assume
thatA0 is a stable matrix. Then, (2.1) is globally
exponentially stable independent of the delay (i.e., for any delay h∈R0+ provided that ∥A1∥ is sufficiently small compared to the
stability abscissa of A0). For ℓ2  (spectral)-norms, a related  specific testable condition is
∥A1∥2<ρ0, for
any real constant
ρ0∈R+
satisfying
ρ0∈(0,|μ2(A0)|), where
μ2(A0):=(1/2)max1≤i≤ξReλi(A0+A0T)
is the matrix
measure of
A0
with respect to the ℓ2-norm, and λi(A0+A0T)(i∈ξ¯:={1,2,…,ξ}, ξ≤n) denote the distinct
eigenvalues of such a  matrix.If
the condition of Theorem2.2 is relaxed to ∥A1∥2≤ρ0, then the time-delay system is
guaranteed to be globally Lyapunov’s stable since −ρ0<0. Theorem 2.3(i) below states that stability
independent of delay implies that A is a stable matrix, related to the case of
arbitrarily large delay while (A0+A1) 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 ∥A1∥2≥|μ2(A0)|, with μ2(A0):=(1/2)max1≤i≤ξReλi(A0+A0T)<0  being the matrix measure of A0 with respect to the ℓ2-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 ℓ2-norm as that of A1 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 A0 is a stable matrix.Theorem 2.3.
The
following properties hold.(i)
Ifμ2(A0)<0
and
∥A1∥2<|μ2(A0)|,
then (2.1) is
globally exponentially stable independent of the delay size and both
A0
and (A0+A1) are stable matrices.(ii)
IfA0
is a stable matrix and
∥A1∥2≥|μ2(A0)|,
then there exists
a system (2.1) with a matrix A1* fulfilling ∥A1∥2=∥A1*∥2 (so that
μ2(A0)+∥A1*∥2≥0) such that
the system x˙(t)=A0x(t)+A1*x(t−h) is either critically stable or unstable.


## 3. Switching among distinct parameterizations

Consider
thenth linear and time-invariant dynamic  switched system with internal delayed
dynamics:
(3.1)x˙(t)=A0σ(t)(t)x(t)+A1σ(t)(t)x(t−h), where x(t)∈Rn is the state vector and
Aiσ:R0+→{Ai1,…,Aip}⊂M(Rn×n);
i=0,1; j∈p¯:={1,2,…,p} are matrix functions of piecewise constant
entries of delay-free and delayed dynamics, respectively, for some delay h∈[0,h¯] and some admissible delay upper-bound h¯∈R+,
or for h∈[0,∞),
where M(Rn×n) denotes the set of real-square matrices of n
order. In other words, Aiσ(t)∈{Ai1,…,Aip}⊂M(Rn×n);
i=0,1. The
piecewise constant real function σ:R0+→p¯⊂N is the switching
law defined by σ(t)=j∈p¯ for all t∈[tk,tk+1), and
{tk∈R0+}k∈Ns is a strictly increasing real sequence of switching instants and Ns⊂N 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 φ:[−h,0]→Rn with φ(0)=x(0)=x0.


## 3.1. Constrained Switching

Some
results of Section3, 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(A0j+A1j) are stable matrices for all j∈p¯.
Thus, there exists
a maximum allowable delay h¯∈R+:={R∋t≥0} such that (3.1) is globally exponentially stable
for any delay h∈[0,h¯) for any switching law σ:R0+→p¯∈N fulfilling σ(t)=j∈p¯ for all t∈[tk,tk+h¯+T) with T∈R+ being a minimum residence time which depends
on the set
{Ai1,…,Aip};
i=0,1; with
{tk}k∈Ns being the sequence of switching instants.Note
that Theorem3.1 holds also if
there are many infinite parameterizations with (A0j+A1j) being stable matrices for all j∈N with a switching law σ(t)=j∈N.
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
thatA0j are stable matrices for all j∈p¯.
Thus, (3.1) is globally exponentially stable independent of the delay provided
that the following hold.
(1)
∥A1j∥ for all j∈p¯ are sufficiently small compared to the maximum
stability abscissa of A0j for all j∈p¯.(2)
The switching lawσ:R0+→p¯∈N which generates the switching instants fulfils σ(t)=j∈p¯ for all t∈[tk,tk+h+T) with T∈R+ being a minimum residence (or dwelling) time
which depends on the set
{Ai1,…,Aip};
i=0,1,  with {tk}k∈Ns being the switching instants.Forℓ2 (spectral)-norms, a related  specific testable condition in Theorem 3.2 is max1≤j≤p(−ρ0j+∥A1j∥2)<0 for any real constant ρ0j∈R+ satisfying ρ0j∈(0,|μ2(A0j)|) for all j∈p¯, where μ2(A0j):=(1/2)max1≤i≤ξReλi(A0j+A0jT) for all j∈p¯ is the measure of the matrix A0j with respect to the ℓ2-norm, and λi(A0j+A0jT)
(i∈ξ¯:={1,2,…,ξ},j∈p¯) denote the distinct eigenvalues of such a  matrix.


## 3.1.1. Simulation Example 1

Consider
the linear switched system (3.1) with h=0.75 second, and three parameterizations: (3.2)A01=(−7−12−4),A02=(−5.51−2−8),A03=(−71−1.5−9),A11=(11−12),A12=(−21−11),A13=(−31−1−2). According to Theorem 3.2, all the matrices A0j are stable for j=1,2,3, and μ2(A01+∥A11∥2<−1.61<0, μ2(A02)+∥A12∥2<−2.8<0, and μ2(A03)+∥A13∥2<−3.7<0.
Also, note that the matrices A11 and A12 are not stable since it is not required in
Theorem 3.2. The initial conditions are φ1(t)=12500, φ2(t)=13500, t∈[−0.75,0].
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 T=1 second. The simulated results
are shown in Figure 2 for the switching rule of Figure 1.Figure 1
Switching rule evolution.Figure 2
State variables evolution.


## 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
Theorems3.1, 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 σ:R0+→p¯∈N, provided that the delay is sufficiently
small  if A0j+A1j are all stable matrices such that
(A0j+A1j)
and
(A0j+A1j)T
pair-wise commute for all j∈p¯.(2)
The switched
system (3.1) is globally exponentially stable for any arbitrary switching law σ:R0+→p¯∈N, provided that the delay is sufficiently small
if (A0j+A1j) are all stable matrices and pair-wise commute
for all j∈p¯.An
alternative condition for the existence of common Lyapunov functions follows by
direct calculation for arbitrarily close parameterizations of the switching
system. DefineΔAikj:=Aik−Aij (i=0,1) for any j∈p¯ and for all k(≠j)∈p¯ (ΔAijj=0 (i=0,1 for all j∈p¯))  and let ρj<0 be any real constant to the right of
the stability abscissa of (A0j+A1j) and close to it. Since (A0j+A1j) is a stable matrix, for any positive definite
square real n-matrix Q, P=∫0∞e(A0j+A1j)TτQe(A0j+A1j)τdτ is a positive definite symmetric solution
matrix to the Lyapunov equation (A0j+A1j)TP+P(A0j+A1j)=−Q for all j∈p¯.
For all k(≠j)∈p¯,(3.3)(A0k+A1k)TP+P(A0k+A1k)=−Q+(ΔA0k+ΔA1k)TP+P(ΔA0k+ΔA1k).
The
right-hand side of the above equation is guaranteed to be negative definite if
(3.4)λmin(Q)>Kjλmax(Q)ρj‖ΔA0k+ΔA1k‖2≥2λmax(P)‖ΔA0k+ΔA1k‖2 for all k(≠j)∈p¯, with Kj≥1 being such that ∥e(A0j+A1j)t∥2≤Kje−ρjt for all t∈R0+,∥⋅∥2 denoting the ℓ2 vector and corresponding (induced) matrix norm,
and λmin(M), λmax(M) denoting the minimum and maximum eigenvalues
of the real symmetric matrix M=P,Q. The last constraint is identical to
(3.5)λmin(Q)>Kjλmax(Q)ρj‖ΔA0k+ΔA1k‖2=Kjλmax(Q)ρjλmax1/2((ΔA0k+ΔA1k)T(ΔA0k+ΔA1k)) for all k(≠j)∈p¯ 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 Q=In can be made and Kj=1 in the better case. Thus, a common Lyapunov
function for zero delay exists, even if not all the  p
parameterizations pair-wise commute,  provided that
(3.6)ρj>maxi≠k∈p¯⁢λmax1/2((ΔA0k+ΔA1k)T(ΔA0k+ΔA1k))∀j∈p¯.	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 σ:R0+→p¯∈N provided that the delay is sufficiently small
if (A0j+A1j) are all stable matrices for all j∈p¯ and all the matrices in the set {A0i,A1i;i∈p¯} pair-wise commute.The
above result is direct since the commutation condition implies that of Theorem3.3. 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 σ:R0+→p¯∈N if and only if
A0j+A1j=fji(A0i+A1i)
for all j∈p¯, and any given i∈p¯ (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 p 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 xt=x(t+θ) for all θ∈[−h,0]. The subsequent notations M>0,M≥0,M<0,M≤0 mean that the real square matrix M is positive
definite, positive semidefinite, negative definite, and negative semidefinite,
respectively.  Consider the Krasovsky-Lyapunov functional candidate defined with matrices P=PT>0, S=ST>0 [1]:
(3.7)V(t,xt)=xT(t)Px(t)+∫t−htxT(τ)Sx(τ)dτ.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 σ:R0+→p¯ and each admissible function of initial
conditions if
(3.8)Qi:=[A0iTP+PA0i+SPA1iA1iTP−S]<0∀i∈p¯, 		for
some Rn×n∋P=PT>0, Rn×n∋S=ST>0. For such a system, the functional (3.7) is a
common Krasovsky-Lyapunov
functional  parameterization for its p distinct
parameterizations.
(2) Consider the general polytopic switched system(3.9)x˙(t)=A0σ(t)(t)x(t)+∑j=1qωjσ(t)Ajσ(t)(t)x(t−hj), where x(t)∈Rn is the state vector,  Aiσ:R0+→{Ai1,…,Aij}⊂M(Rn×n);
i∈q¯∪{0};
j∈p¯ are matrix functions defining the switched
dynamics and the hj(j∈q¯) are the q delays (being eventually distinct) for
each of the p distinct  parameterizations,
and ωji are either zero or unity   scalars while ωji is unity for at least one i∈q¯ for each j∈p¯. Thus, if
(3.10)Q¯i:=[A0iTP+PA0i+∑i=1qSiω1iPA1i⋯ωqiPAqiω1iA1iTP−S10⋯0⋮0⋮⋱⋮ωqiAqiTP0⋯−Sq]<0∀i∈p¯			for
some Rn×n∋P=PT>0, Rn×n∋Si=SiT>0
(for all i∈q¯),
then the switched system (3.9) is globally asymptotically stable independent of
the delays for any switching law σ:R0+→p¯ and each admissible function of initial
conditions. Also,
(3.11)V¯(t,xt)=xT(t)Px(t)+∑i=1q∫t−hitxT(τ)Six(τ)dτ,
with xt redefined by x(t)=x(t+θ) for all θ∈[−h^,0] and h^:=max1≤i≤q(hi) is a common Krasovsky-Lyapunov
functional  parameterization for its p distinct
parameterizations.The
proof of  Theorem3.6(1) follows straightforwardly by direct calculation noting
since the time-derivative of (3.7) along any trajectory solution of (3.1) is V˙(t)=x¯(t)Qσ(t)x¯(t)≤−η∥x¯(t)∥22<0 for some η∈R+, provided that x¯(t):=(xT(t),xT(t−h))T≠0, so that it is a negative quadratic form in R2n since Qi<0 for all i∈p¯.
Theorem 3.6(2) follows since V¯˙(t)=x¯(t)Q¯σ(t)x¯(t)≤−η¯∥x¯(t)∥22<0 if x¯(t):=(xT(t),xT(t−h1),…,xT(t−hp))T≠0 for some η¯∈R+, so that it is a negative quadratic form in Rqn since Q¯i<0 for all i∈p¯ from (3.10). Thus, V¯(t,xt) (3.11) is a Krasovsky-Lyapunov functional for the
switched system (3.9).Remark 3.7.
Theorem3.6(2)
applies when each of the p-switched parameterizations has a unique delay and
all of them are distinct, that is, q=p,
ωii=1,
ωij=0, hi≠hj for all j,i(≠j)∈q¯.  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 h for all the parameterizations,
that is, q=p, ωii=1, ωij=0,
h=hi, S=Si, for all  j,i(≠j)∈p¯.Remark 3.8.
Note that
Theorem3.6 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).Theorem3.6 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 h∈[0,h¯] and for any switching law σ:R0+→p¯ and each admissible function of initial
conditions if
(3.12)Q^i:=[(A0iT+A1iT)P+P(A0i+A1i)+h¯(S0+S1)h¯PA1iMih¯A1iTPMi−h¯S]<0∀i∈p¯,(3.13)Mi:=[A0i,…,A1i]∈Rn×2n∀i∈p¯,S:=diag(S0,S1)∈R2n×2nfor
some Rn×n∋P=PT>0, Rn×n∋Sj=SjT>0(j=1,2).
The real functional,
(3.14)V^(t,xt)=xT(t)Px(t)+∑j=12∫−jh−(j−1)h∫t+θtxT(τ)Sjx(τ)dτ⁢dθ,is
a common Krasovsky-Lyapunov
functional  for its p distinct
parameterizations.
(2) The switched system (3.9) is globally
asymptotically stable independent of the delays for any switching law σ:R0+→p¯ and all delays hi∈⌊0,h^i⌋; (for all
i∈q¯) if
(3.15)Q¯^i:=[(∑j=0qAjiT)P+P(∑j=0qAji)+∑i=1q∑j=0qh^iSijω1ih^1PA1iM1⋯ωqih^qPAqiMqω1ih^1M1TA1iTP−R10⋯0⋮0⋮⋱⋮ωqih^qMqTAqiTP0⋯−Rq]<0∀i∈p¯,(3.16)Mi:=[A0i,…,Aqi]∈Rn×(q+1)n,Ri:=diag(Si0,Si1,…,Siq)∈R(q+1)n×(q+1)n∀i∈p¯,for
some Rn×n∋P=PT>0, Rn×n∋Sij=SijT>0 (for all i∈q¯, for all j∈q¯∪{0}).
The real functional,
(3.17)V¯^(t,xt)=xT(t)Px(t)+∑i=1q∑j=0q∫−hi−hj−hj∫t+θtxT(τ)Sijx(τ)dτ⁢dθ,
with Rn×n∋P=PT>0, Rn×n∋Sij=SijT>0 (for all i∈p¯, for all j∈q¯∪{0}) is a common Krasovsky-Lyapunov functional for
its p distinct parameterizations with xt being  redefined
by x(t)=x(t+θ) for all θ∈[−h^,0] and h^:=max1≤i≤q(h^i).The
proof is direct since it is very close to that of Theorem3.6. V^˙(t)=x¯(t)Q^σ(t)x¯(t)≤−η^∥x¯(t)∥22<0 if x¯(t):=(xT(t),xT(t−h1),…,xT(t−hq))T≠0 for Theorem 3.9(1) and V¯^˙(t)=x¯(t)Q¯^σ(t)x¯(t)≤−η¯^∥x¯(t)∥22<0 for any nonzero x¯(t) 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
Qi<0, Q¯i<0,
Q^i<0, and Q¯^i<0 for all i∈p¯,
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			(3.18)x˙(t)=∑i=1ϑλi(t)(A0σ(t)(t)x(t)+∑j=1qωjσ(t)Ajσ(t)(t)x(t−hj)) with σ:R0+→p¯, ωij being either zero or unity scalars and at
least one ωji=1 for j∈q¯ for each i∈p¯ and λi:R0+→[0,1] subject to ∑i=1ϑλi(t)=1.
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 ϑ 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 Aji for all (j,i)∈(q¯∪{0})×p¯ 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 thatQ¯i<0 (3.10) for all i∈p¯.
Then, the switched system (3.18) is globally asymptotically stable independent of
the delays (i.e., for all hi∈[0,∞);
for all i∈p¯) irrespective of the switching law σ:R0+→p¯.
The real functional (3.11) is a Krasovsky-Lyapunov functional for the switched
polytopic system (3.18).(2)
IfQ¯^i<0,
(3.15) for all i∈p¯, then the switched system (3.18) is globally
asymptotically stable for any delays
hi∈⌊0,h^i⌋ for all i∈p¯, irrespective of the switching law σ:R0+→p¯.
The real functional (3.17) is a Krasovsky-Lyapunov functional for the switched
polytopic system (3.18).The
proof of Theorem3.11 is direct since all the real functions λi:R0+→[0,1] are nonnegative and they cannot be
simultaneously zero at any time instant. This leads to functional time derivatives V¯˙(t)=∑i=1ϑλi(t)x¯(t)Q¯σ(t)x¯(t)<0 and V¯^˙(t)=∑i=1ϑλi(t)x¯(t)Q¯^σ(t)x¯(t)<0 for all nonzero x¯(t) 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
thatQi≥0 (3.8) for all i∈p¯.
Then, the switched system (3.1) is not globally asymptotically stable independent
of the delay size for any switching law σ:R0+→p¯ but it can be still globally Lyapunov’s stable
for some switching law σ:R0+→p¯.
Those properties also hold for any particular delay. If Qi>0 for all i∈p¯, then the switched system (3.1) is instable for
any delay and any given switching law σ:R0+→p¯ for any given admissible function of initial
conditions.
(ii) Assume
thatQ¯i≥0,
(3.10) for all i∈p¯.
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 σ:R0+→p¯ but it can be still globally Lyapunov’s stable
for some switching law σ:R0+→p¯.
Those properties also hold for any particular delay. If Q¯i>0 for all i∈p¯, then the switched systems (3.9) and (3.18) are
both instable for any delay and any given switching law σ:R0+→p¯ for any given admissible function of initial
conditions.
(iii) Assume thatQi<0 for at least some i∈p¯.
Then, there are infinitely many switching laws σ:R0+→p¯ 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 Q¯i<0 for at least some i∈p¯.


## 3.2.1. Simulation Example 2

Consider
the linear switched system (3.1) with h=0.75 seconds and parameterizations (3.19)A01=(−7−12−4),A02=(−15−13−2),A03=(−521−0.5),A11=(11−12),A12=(−121−3.51),A13=(−4−2−1−0.5).  According to Theorem 3.3, all the matrices A0j+A1j are stable and pair-wise commute for j=1,2,3. The initial conditions are
given by φ1(t)=250,φ2(t)=−350,t∈[−0.75,0].
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.Figure 3
Switching law.Figure 4
State-space variables evolution.


## 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(3.20)x˙(t)=A0(t)x(t)+A1(t)x(t−h), 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
A(t) 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 valueA0(t) of the matrix function A0:R0+→Rn×n is a stable matrix satisfying Re(λi(A0(t)))≤−ρ0<0 for all t∈R0+.(2)
A0(t),A1(t) has uniformly bounded entries for all time.(3)
A˙0(t) exists and it is bounded almost everywhere in R0+ with sufficiently small upper-bound
γdA≥∥A˙0(t)∥2,
where A˙0(t) in R0+ exists.(4)
There exist positive definite realn-matrix functions P(t) and S(t) satisfying
(3.21)[S(t)+(K22ρ2γdA−1)InP(t)A1(t)A1T(t)P(t)−S(t−h)]<0∀t∈R0+.(5)
The time
interval between any two consecutive time instants at whichA˙0(t)
does not exist is sufficiently large according
to the amplitude-bound KA (in terms of ℓ2-norm) of the
corresponding bounded discontinuities of A0(t),
the absolute value of the stability abscissa
ρ,
γdA,
and P(t) and S(t).
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 A(t), 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 Theorem3.13, 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)
Theorem3.13 is an extension of
Theorem 3.6(1) if (3.1) is generalized to (3.20).(b)
In the same way, Theorem3.6(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)
Theorem3.9 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 (A0(t)+A1(t)) 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)
Theorem3.11(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 A0(t) 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 (A0(t)+A1(t)) 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
considerasynchronous 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:
(3.22)x˙(t)=A0σ0(t)(t)x(t)+A1σ1(t)(t)x(t−h), where σj:R0+→p¯ select for all time  matrices Ajσj(t)∈{Aj1,…,Ajp} for all j∈{0,1},
for all t∈R0+. Also, (3.9) extended to the more general one:
(3.23)x˙(t)=A0σ0(t)(t)x(t)+∑j=1qωjσj(t)Ajσj(t)(t)x(t−hj), where σj:R0+→p¯ select for all time matrices Ajσj(t)∈{Aj1,…,Ajp} for all j∈q¯∪{0},
for all t∈R0+.
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 σ(t) defined with the whole set of asynchronous
ones by a (q+1)-tuple of positive integers σ(t):=(σ0(t),σ0(t),…,σq(t)) which is a vector mapping on a Cartesian
product (3.24)σ:R0+⟶p¯×⋯×p¯q︸.The above to a synchronous switching law has at most pq+1 distinct parameterizations of the switched
system formed with all the combinations of matrices associated with each
particular delay with q=1 for the switched system (3.1). If the set of matrices
for some particular delay is less than p, 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 h=0.75 sec., φ1(t)=25,φ2(t)=−35,t∈[−0.75,0], A0(t)=(−1a0(t)0−1), and A1(t)=(−11/(2+t)0−2) with
(3.25)a0(t)={11+tn,n−1<t≤n;n=1,2,…,6,11+t7,t>6,
and a0(0)=1.
Note that a0(t) is a discontinuous function implying that A0(t) has impulsive behavior at integer times. The
solution of (3.20) with the above parameterization is depicted in Figure 5.Figure 5
State-variables evolution.


## 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.



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