On the Global Asymptotic Stability of

Volume 2008, Article ID 231710,31pages doi:10.1155/2008/231710

Research Article

On the Global Asymptotic Stability of

Switched Linear Time-Varying Systems with Constant Point Delays

M. de la Sen¹ and A. Ibeas²

1Department of Electricity and Electronics, Faculty of Science and Technology, University of Basque, Campus of Leioa (Bizkaia), Aptdo. 644, 48080 Bilbao, Spain

2Department of Telecommunication and Systems Engineering, Engineering School,

Autonomous University of Barcelona, Cerdanyola del Vall´es, 08193 Bellaterra, Barcelona, Spain

Correspondence should be addressed to M. de la Sen,manuel.delasen@ehu.es Received 22 July 2008; Accepted 25 September 2008

Recommended by Antonia Vecchio

This paper investigates the asymptotic stability of switched linear time-varying systems with constant point delays under not very stringent conditions on the matrix functions of parameters.

Such conditions are their boundedness, the existence of bounded time derivatives almost everywhere, and small amplitudes of the appearing Dirac impulses where such derivatives do not exist. It is also assumed that the system matrix for zero delay is stable with some prescribed stability abscissa for all time in order to obtain sufficiency-type conditions of asymptotic stability dependent on the delay sizes. Alternatively, it is assumed that the auxiliary system matrix defined for all the delayed system matrices being zero is stable with prescribed stability abscissa for all time to obtain results for global asymptotic stability independent of the delays. A particular subset of the switching instants is the so-called set of reset instants where switching leads to the parameterization to reset to a value within a prescribed set.

Copyrightq2008 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.

1. Introduction

Time-delay systems are receiving important attention in the last years. The reason is that they offer a very significant modeling tool for dynamic systems since a wide variety of physical systems possess delays either in the stateinternal delaysor in the input or outputexternal 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 forthsee, e.g.,1–9. Time- delay models are useful for modeling both linear systems see, e.g.,1–4,10and certain nonlinear physical systems,see, e.g.,4,7–9,11. 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,2–4,6–13either 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 14–

17, and characterization of minimal state-space realizations 18. Another research field of recent growing interest is the investigation in switched systems including their stability and stabilization properties. A general insight in this problem is given in19–21. Switched systems consist of a number of different parameterizationsor distinct active systemssubject to a certain switching rule which chooses one of them being active during a certain time. The problem is relevant in applications since the corresponding models are useful to describe changing operating points or to synthesize different controllers which can adjust to operate on a given plant according to situations of changing parameters, dynamics, and so forth.

Specific problems related to switched systems are the following.

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

bThe system parameters are changing so that the operation points change. Thus, a switched model which adjusts to several operation points may be useful19–21.

c 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, can 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 objective, or the best tracking objective or the best mixed identification and tracking objectives. Such strategies can improve the switched system performance compared to the use of a single estimator/controller pair5,22.

This paper is devoted to the investigation of the global asymptotic stability properties of switched systems subject to internal constant point delays, while the matrices defining the delay-free and delayed dynamics are allowed to be time varying while fulfilling some standard additional regularity conditions like boundedness, eventual time differentiability, and being subject to sufficiently slow growing rates23. The various obtained asymptotic stability results are either dependent on or independent of the delay size and they are obtained by proving the existence of “ad-hoc” Krasovsky-Lyapunov functionals. It is assumed that either the current system matrix or that describing the system under null delay is stability matrices for results independent of and dependent on the delays sizes, respectively. This idea relies on the well-known fact that both of those matrices have to be stable for any linear time-invariant configuration in order that the corresponding time-delay system may be asymptotically stable, 1, 4, 10, provided that a minimum residence time at each configuration is respected before the next switching to another configuration. The formalism is derived by assuming two classes of mutually excluding switching instants. The so-called reset-free switching instants are defined as those where some parametrical function is subject to a finite jump equivalently, a Dirac impulse at its time derivativewhich is not constrained to a finite set. The so-called reset switching instants are defined as those registering bounded jumps to values within some prescribed set of resetting parameterizations. The distinction between reset-free and reset switching instants

is irrelevant for stability analysis since in both cases at least one parameter is subject to a bounded jump, or equivalently, to a Dirac impulse in its time derivative. Impulsive systems are of growing interest in a number of applications related, for instance, to very large forces applied during very small intervals of times, population dynamics, chemostat models, pest, and epidemic models, and so forthsee, e.g.,24–28and references therein. However, it may be relevant in practical situations to distinguish a switch to prescribed time-invariant parameterizationssee the above situationsa–cfrom an undriven switching action. The paper is organized as follows. Section 2 is devoted to obtain asymptotic stability results dependent on the delay sizes.Section 3gives some extension for global asymptotic stability independent of the delays. Numerical examples are presented inSection 4, where switching through time in between distinct parameterizations is discussed. Finally, conclusions end the paper. Some mathematical derivations concerned with the results of Sections2 and 3 are derived inAppendix A.

2. Asymptotic stability dependent on the delays

Consider the nth order linear time-varying dynamic system with q internal in general, incommensurateknown point delays:

xt ˙ ^q

j0

Ajtx t−hj

2.1

for any given bounded piecewise absolutely continuous functionϕ :−h,0 → Rⁿof initial conditions, whereh : max_1≤j≤qhj with h₀ 0, for some delaysh_j ∈ 0,h_j, of finite or infinite maximum allowable delays sizeshj ∈ R₀ , for allj ∈ q : {1,2, . . . , q}, where R0

is the nonnegative real axis R₀ : R ∪ {0} {0 ≤ z ∈ R}; andA_j : R₀ → R^n×n, for all j∈q∪ {0}. The following assumptions are made.

2.1. Assumptions on the time-delay dynamic system2.1

One or more of the following assumptions are used to derive the various stability results obtained in this paper.

Assumption 2.1. All the entries of the matrix functions Aj : R₀ → R^n×n are piecewise continuous and uniformly bounded for allj ∈q∪ {0}.

Assumption 2.2. All the eigenvaluesλiq

j0Ajt of the matrix functionq

j0Ajt satisfy Reλiq

j0Ajt≤ −ρ0 <0 for allt∈R₀ , for alli∈σσ≤nfor someρ0∈R :{0< z∈R}, that is,q

j0A_jtis a stability matrix for allt∈R₀ .

Assumptions 2.3. The matrix functions Aj : R₀ → R^n×n are almost everywhere time differentiable with essentially bounded time derivative for allj∈q∪ {0}possessing eventual isolated bounded discontinuities, then ess sup_t∈R0 q

j0A˙_jt ≤ γ < ∞with γ being a -norm dependent nonnegative real constant and, furthermore,

_{t T}

t

A˙_jτdτ ≤μ_jT α_j≤μT α ∀j∈q∪ {0} 2.2 for someα_j, μ_j, α, μ∈R for allt∈R₀ , and some fixedT ∈R₀ independent oft.

At time instantst, where the time-derivative of some entry ofAjtdoes not exist for anyj ∈q∪{0}, the time derivative is defined in a distributional Dirac sense as ˙A_jt Γtδ0 what equivalently means the presence of a discontinuity attinA_jtdefined as

Aj

t Aj

t⁻

ε→lim0

_ε

−εΓjτδt−τdτ Ajt Γjt. 2.3

Assumption 2.1is relevant for existence and uniqueness of the solution of2.1. The differential system 2.1 has a unique state-trajectory solution for t ∈ R for any given piecewise absolutely continuous functionϕ:−h,0 → Rⁿof initial conditions. This follows from Picard-Lindel ¨off existence and uniqueness theorem. Assumption 2.2 establishes that q

j0A_jt is a stability matrix for all time what is known to be a necessary condition for the global asymptotic stability of the system2.1for a set of prescribed maximum delays in the time-invariant case. It is well known that even if A_j ≡ 0 for all j ∈ q, then the resulting linear time-varying delay-free system cannot be proved to be stable without some additional assumptions, like for instance, Assumptions2.3. The latest assumption is related to the smallness of the time-derivative of the delay-free system matrix everywhere it exists or generating sufficiently small bounded discontinuities inq

j0A_jtat times, where it is impulsive. An alternative assumption to Assumptions2.3which avoids the assumption of almost everywhere existence of a bounded ˙A_jt, for all j ∈ q∪ {0} see second part of Assumption 2.1might be stated in terms of sufficiently smallness ofΔAjtforΔAjt: A_jt−A^∗_j for allt∈R₀ for allj ∈q∪ {0}for some constant stability matrixq

j0A^∗_j whose eigenvalues satisfy Reλ_iq

j0A^∗_j ≤ −ρ0 < 0. Such an alternative assumption guarantees also the global existence and uniqueness of the state-trajectory solution of2.1and it allows obtaining very close stability results to those being obtainable from the given assumptions.

For global asymptotic stability dependent of the delay sizes on the first delay interval, the stability of the values taken by the matrix function q

j0Ajt is required within some real interval of infinite measure. Such an interval possesses a connected component being of infinite measure which is a necessary condition for global asymptotic stability for zero delaysseeTheorem 2.12i .

2.2. Switching function, switching sequence, and basic assumptions on the switching matrix function

Assumption 2.1admits bounded discontinuities in the entries ofAjtforj ∈q∪ {0}. At such timesA_jt denote right values of the matrix function whileA_jt⁻ is simply denoted by Ajt. A set ofpresetting systems of2.1is defined by the linear time-invariant systems:

˙ z_jt

q i0

A_ij

z_j t−h_j

2.4

for some given A_jiR^n×n for all j ∈ p for all i ∈ q∪ {0} for some given p ∈ N. Those parameterizations are used to reset the system 2.1 at certain reset instants defined later on.Assumption 2.2extends in a natural fashion to include the resetting systems as follows.

Assumption 2.4. All the eigenvaluesλkq

i0Aijsatisfy Reλkq

i0Aij≤ −ρ0 <0 for allt∈R₀ for allk ∈ σ_j0 σj0 ≤ nfor allj ∈ p; that is, q

i0A_ij are constant stability matrices with prescribed stability abscissa−ρ₀ <0 for allj∈p.

The following definitions are then used.

Definition 2.5. The switching matrix function is a mappingσ : R₀ → {Ajt −A_jt,∀j ∈ q∪ {0}} ⊂R^{n×q 1n}from the nonnegative real axis to the set of realn×q 1nmatrices.

The trivial switching matrix function is that being identically zero so that no switch occurs. If some switch occurs then the switching matrix function is nonzero. The switching matrix function is colloquially referred to in the following as the switching law.

Definition 2.6 switching instant. t ∈ R₀ is a switching instant ifAjt /Ajt for some j∈q∪ {0}.

The set of switching instants generated by the switching lawσ is denoted by STσ.

Two kinds of switching instants, respectively, reset instants and reset-free switching instants defined in Definitions2.7and2.8are considered.

Definition 2.7reset instant. t ∈ R₀ is a reset instant generated by the switching law σif t∈STσandAit Aijfor somei∈q∪ {0}and somej ∈p, provided thatAit Aik/Aij

for somek/j∈p.

The set of reset instants generated by the switching law σ is denoted by STrσ.

Note that STrσ ⊂ STσ from Definitions 2.6 and 2.7. Note also that the whole system parameterization is driven to some of the prefixed resetting systems2.4when a reset instant happens. Note that at reset instants, ˙A_it Aij−A_itδ0for somei∈q∪ {0},j∈p.

Definition 2.8reset-free switching instant. t∈R₀ is a switching reset-free instant generated by the switching lawσift∈STσandA_ij/A_it /A_itfor somei∈q∪ {0}, for allj ∈p.

The set of reset-free instants is denoted by t ∈ STrfσ. Note that at reset-free switching instants some of the switched system parameters suffer an undriven bounded discontinuity. If all the parameters jump to a parameterization2.4at the same time, then the corresponding instant is considered a reset time instant. Note that STrfσ ⊂ STσ, STσ STrσ∪STrfσ, and STrσ∩STrfσ ∅from Definitions2.6–2.8, that is, the whole set of switching instants is the disjoint union of the sets of reset and reset-free switching instants.

Definition 2.9. The partial switching sequence STσ, t, the partial switching sequence STrσ, t, and the reset-free partial switching sequence STrfσ, t, generated by the switching law σ : R₀ → {Ajt−A_jt⁻, ∀j ∈ q∪ {0}} ⊂ R^{n×q 1n} up till any time t ∈ R₀ , are defined, respectively, by STσ, t:{ti ∈STσ:t_i < t}, STrσ, t:{ti∈ST_rσ:t_i < t}, and STrfσ, t:{ti∈STrfσ:ti< t}.

Remark 2.10. An interpretation of Assumptions2.3is that the following conditions hold for any given-matrix norm for some nonnegative norm dependent real constantsμ_j,α_j,μ, and αfor allj∈q∪ {0}:

ess sup

t∈R₀

A˙jt≤μj≤max ess sup

t∈R₀

A˙jt:j∈q∪ {0}

≤μ,

τ∈STσ∩t,t T

A˙_jτ

≤

τ∈STσ∩t,t T

Xτδ0 ≤α_j

≤max

τ∈STσ∩t,t T

A˙_jτ

:j∈q∪ {0}

≤α,

2.5

whereδ0is a Dirac impulse att 0. Note that Assumptions2.3imply|q

j0A_jt τ −

q

j0A_jt| ≤ μT νt τfor allτ ∈ 0, T, with the functionν : R₀ → R₀ satisfying νt τ ≤α₀j α_1jt τ ≤αwithα_1jt τ 0 if ˙A_jt τexists withA˙_jt τ ≤α_j or αj−α0j ≥α1jt τ≥ Δjt τif ˙Ajt τ Δjt τδ0, that is, at least one of its entries is impulsive.

Note thatAssumption 2.1 implies that switching does not happen arbitrarily fast neither to reset parameters nor to reset-free ones . The subsequent result is direct.

Assertions 2.11. The following properties are true irrespective of the switching function:

it /∈STσ⇔σt σt 0_{n×q 1n}i.e., a zeron×q 1n-matrix;

iit∈STσ⇔σt −σt/0_{n×q 1n};

iiit /∈STσ⇔STσ, t STσ, t;

ivt∈STσ⇔STσ, t /STσ, t.

Proof. iσt σt 0_{n×q 1n} ⇒t /∈STrσ∧t /∈STrfσ ⇔ t /∈STσ, t /∈ST ⇔ σt 0 since switching is not arbitrarily fast,

∧ σ

t

−σt 0_{n×q 1n}

⇐⇒σ t

σt 0_{n×q 1n}. 2.6

Propertyihas been proven. Propertyiiis the contrapositive logic proposition to Property i, and thus equivalent, since switching is not arbitrarily fast.

Propertiesiii-ivare also contrapositive logic propositions, then equivalent since

STσ, t:

t_i∈STσ:t_i< t

⇐⇒ST σ, t

:

t_i∈STσ:t_i< t

t_i∈STσ:t_i≤t

⇐⇒STσ, t

⎧⎨

⎩

STσ, t ift/∈STσ, /STσ, t if t∈ STσ

2.7

since switching cannot happen arbitrarily fast. Propertiesiii-ivhave been proven.

The subsequent global stability result is proven in Appendix A by guaranteeing that the Krasovsky-Lyapunov functional candidate below is indeed a Krasovsky-Lyapunov functional:

Vt, xt x^TtPtxt q

i1

q j0

_−hj

−hi−hj

_t

t θx^TτSijxτdτ dθ. 2.8

Theorem 2.12. The following properties hold.

iAssume the following.

i.aThe matrix functionsA_jt,for allj∈q∪ {0}are subject toAssumption 2.1.

i.bThe switching lawσis such that

Qt:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

H1 h1PtA1tMt · · · hqPtAqtMt h1M^TtA^T₁tPt −R1 0· · · 0

... 0

...

. .. ...

hqM^TtA^T_qtPt 0 · · · −Rq

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0,

∀t∈R₀ , 2.9

whereH1 denotesq

j0A^T_jtPt Ptq

j0A_jt Pt ˙ q i1

q

j0h_iS_ij, for some time-differentiable real symmetric positive definite matrix functionP : R₀ → R^n×nand some real symmetric positive definite matricesS_ij ∈ R^n×n ∀i ∈ q, ∀j ∈ q∪ {0}, where

Mt:

A₀t, . . . , Aqt

∈R^{n×q 1n}, R_i:diag

S_i0, S_i1, . . . , S_iq

∈R^{q 1n×q 1n}; ∀i∈q, ∀t∈R₀ . 2.10 Thus, the system2.1is globally asymptotically Lyapunov’s stable for all delaysh_i∈ 0,hi, for all i ∈ q. A necessary condition isq

j0A^T_jtPt Ptq

j0Ajt P˙t < 0, for allt ∈ R₀ what implies that q

j0Ajt is a stability matrix of prescribed stability abscissa on R₀ except eventually on a real subinterval of finite measure of R₀ .

iiAssume the following

ii.aAit Aij, for alli ∈ q∪ {0}, for allt ∈ R₀ for some j ∈ p (eventually being dependent on t) satisfyingAssumption 2.4.

ii.bThe switching law σ is such that STrfσ ∅ (i.e., it generates reset switching instants only) with ST_rσ being arbitrary, namely, the set of reset times is either any arbitrary strictly increasing sequence of nonnegative real values (i.e., the resetting switching never ends) or any finite set of strictly ordered nonnegative real numbers with a finite maximal (i.e., the resetting switching process ends in finite time).

ii.c

Q^∗_i:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣ _q

j0

A^T_ji

P^∗ P^∗ _q

j0

A_{ji q}

i1

q j0

h_iS^∗_ij h₁P^∗A_1iM₁^∗ · · · h_qP^∗A_qiM^∗_q h₁M^∗₁^TA^T_1iP^∗ −R^∗₁ 0· · · 0

... 0

...

. .. ... h_qM^∗_q^TA^T_qiP^∗ 0 · · · −R^∗_q

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

<0;

∀i∈p 2.11

for some R^n×nP^∗P^∗T >0, R^n×nS^∗_ijS^∗_ij^T>0∀i∈q, ∀j∈q∪ {0}, where Mi:

A0i, . . . , Aqi

∈R^{n×q 1n},

R^∗_i :diag

S^∗_i0, S^∗_i1, . . . ,

∈R^{q 1n×q 1n}; ∀i∈q

2.12

Thus, the switched system 2.1, obtained from switches among resetting systems 2.4, is globally asymptotically Lyapunov’s stable and also globally exponentially stable for all delays h_i ∈ 0,h_i, for alli ∈ q. If 2.9 is replaced with Q^∗_i ≤

−2εI_{q 1n} < 0, for alli ∈ q, and some ε ∈ R then the state trajectory decays exponentially with rate −ε < 0.

iiiThere is a sufficiently smallh:max_i∈qh_isuch that Property (i) holds for anyh_i∈0,h_i, for alli∈qprovided that all the delay-free resetting systems2.4z˙_jt q

i0A_ijzjt fulfilAssumption 2.4, that is, they are globally exponentially stable.

It is of interest to discuss particular cases easy to test, guaranteeingTheorem 2.12i. 2.3. Sufficiency type asymptotic stability conditions obtained for

constant symmetric matrices P andS_ij

Assume real constant symmetric matricesPt P andSij, for alli ∈ q ,for allj ∈ q∪ {0}, for allt ∈ R₀ so that

Qt

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

H2 h1P A1tMt · · · hqP AqtMt h1M^TtA^T₁tP −R1 0· · · 0

... 0

...

. .. ... h_qM^TtA^T_qtP 0 · · · −Rq

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0, 2.13

whereH2 denotesq

j0A^T_jtP Pq

j0A_jt q i1

q

j0h_iS_ij. In this case, the Krasovsky- Lyapunov functional used in the proof of Theorem 2.12i holds defined with constant

matrices for all time irrespective of being a switching-free instant or any switching instant independently of its nature: reset time or reset-free switching instant. A practical test for 2.13to hold follows. ConsiderA^∗_i i ∈ q∪ {0} such that the time invariant system2.1 defined withAit → A^∗_i is globally asymptotically Lyapunov’s stable and define a stability realn-matrixA^∗:q

i0A^∗_i. DecomposeQt Q^∗ Qt, where

Q^∗:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

A^∗TP P A^∗ q

i1

q j0

h_iS_ij h₁P A^∗₁M^∗ · · · h_qP A^∗_qM^∗ h1M^∗TA^∗₁^TP −R^∗₁ 0· · · 0

... 0

...

. .. ... h_qM^∗TA^∗_q^TP 0 · · · −R^∗_q

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ ,

Qt :

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

A^TtP PAt h₁PΔ1t · · · h_qPΔqt h₁Δ^T₁tP 0 0· · · 0

... 0

...

. .. ... h_qΔ^T_qtP 0 · · · 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ ,

2.14

where

M^∗:diag

A^∗₀, A^∗₁, . . . , A^∗_q

, R^∗_i :diag

S^∗_i0, S^∗_i1, . . . , S^∗_iq

, Mt :Mt−M^∗, Ait:Ait−A^∗, At :

_q

j0

Ajit

−A^∗, Δit: AitM^∗ A^∗_iMt AitM^↔ t

.

2.15

Ift ∈ STσ, then

δQt :Q t

−Qt

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

H3 h1P Δ1

t

−Δ1t

· · · hqP Δ1

t

−Δ1t h1 Δ1

t

−Δ1t_T

P 0 0· · · 0

... 0

...

. .. ...

hq Δ1

t

−Δ1t_T

P 0 · · · 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦ ,

δQt

2≤2λmaxP

A^T t

−A^Tt

2

q i1

hiΔi

t

−Δit

2

2λmaxP

q i0

A^T_i t

−A^T_it

2

q i1

hiΔi

t

−Δit

2

, 2.16

whereH3 denotes A^Tt −A^TtP PAt −At, what leads to δQt

2≤2λ_maxP

q 1a hq

a _q

j0

A^∗_j2 q 1A^∗_i2 q 1a a

q 12a a a

,

2.17

where a : max_i∈q∪{0}Ai2 and a : sup_t∈STσmax_i∈q∪{0}Ait −A_i2 see A.9 in Appendix A. Direct results fromTheorem 2.12which follow from2.13to2.17are given below.

Corollary 2.13. Consider in 2.9 replacements with constant real matricesQt → Q^∗ Q^∗T, Pt → P P^T > 0,Ait → A^∗_i,At → A^∗ : q

i0A^∗_i; for alli ∈ q ,for allj ∈q∪ {0}, for allt ∈ R₀ such that A^∗ is a stability matrix. Then, Theorem 2.12(i) holds if Q^∗ < 0 for any switching lawσsuch that

1λmin−Q^∗ −λmaxQ^∗>Qt ₂:λ^1/2_maxQ^TtQt, for all t ∈ R₀ \STσ, 2a : max_i∈q∪{0}Ai2anda : sup_t∈STσmax_i∈q∪{0}Ait −A_i2are sufficiently

small such that

λmin

−Q^∗

− Qt

2

>2λ_maxP

q 1a hq

a _q

j0

A^∗_j

2 q 1max

i∈q∪{0}A^∗_i

2 q 1a a

q 12a a a

, ∀t∈STσ.

2.18

Corollary 2.14. Consider in 2.9 replacements with constant real matricesQt → Q^∗_j Q_j^∗T, Pt → P P^T > 0,A_ijt → A^∗_ij,At → A^∗_j : q

i0A^∗_ij for allj ∈ p,for allk ∈ q∪ {0}, for allt ∈ R0 such that eachA^∗_j ∀j ∈pis a stability matrix withA^∗_ij; for alli∈q, for allj ∈p being the parameterizations defining the resetting systems2.4. Assume that the system2.1is one of the resetting systems2.4att0. Then,Theorem 2.12(i) holds with a common Krasovsky-Lyapunov function for all those resetting systems ifQ^∗_j <0 ∀j∈pfor any switching lawσsuch that

1λmin−Q^∗ −λmaxQ^∗>Qt ₂:λ^1/2_maxQ^TtQt, for all t∈R₀ \STσ, 2aj:max_i∈q∪{0}Aji₂anda_j :sup_t∈STσmax_i∈q∪{0}Ait −Aji₂are sufficiently

small such that

λmin

−Q^∗

− Qt

2

>2λ_maxP

q 1a_j hq

a_{j q}

i0

Aji₂ q 1max

i∈q∪{0}Aji₂

q 1

a_j a_j

q 1

2a_j a_j a_j

2.19

t ∈STfrσ, provided that at time max t < t:t ∈STrσ, the system2.1coincides with at thej ∈ prsetting system2.4.

The proof ofCorollary 2.14is close to that ofCorollary 2.13fromA.9inAppendix A with the replacements a → aj, a → a_j for all j ∈ p. If 2.19 is rewritten with the replacementsaj → a : max_j∈paj, a_j → a : max_j∈pa_j then the reformulated weaker Corollary 2.14is valid for allt∈STfrσirrespective of the preceding reset switching. A result which guarantees Corollary 2.13, and then Theorem 2.12i, is now obtained by replacing the1,1block matrix ofQ^∗ by a Lyapunov matrix equality as follows. Consider a realn- matrixQ^∗₀ Q^∗T₀ > 0 such thatλminQ^∗₀ > λmaxq

i1

q

j0hiSijandP : _∞

0 e^A∗TτQ₀^∗e^A∗τdτ satisfying the Lyapunov equationA^∗TP P A^∗ −Q^∗₀ < 0 as its unique solution. Note that λmaxP ≤ K^∗λmaxQ^∗₀/2ρ^∗for someK^∗ ∈R , where−ρ^∗ < 0 is the stability abscissa ofA^∗ withe^A∗Tt ≤√

K^∗e^−ρ∗tfor allt∈R. Define the decompositionQt Q^∗ Qt, where

Q^∗: Block Diag

−Q^∗₀ q

i1

q j0

h_iS_ij,−R^∗₁, . . . ,−R^∗_q

,

Qt :

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

A^TtP PA t h₁P

A^∗₁M^∗ Δ1t

· · · h_qP

A^∗_qM^∗ Δqt h₁

A^∗₁M^∗ Δ1tT

P 0 0· · · 0

... 0

...

. .. ...

h_q

A^∗_qM^∗ ΔqtT

P 0 · · · 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦

⇒Qt

2≤

K^∗λ_max Q₀^∗ ρ^∗

A^Tt

2

q i1

h_iA^∗_iM^∗ Δit

2

.

2.20

Thus, the subsequent result follows fromCorollary 2.13and2.20.

Corollary 2.15. Consider the matrices of Corollary 2.13 with A^∗ being a stability matrix with stability abscissa−ρ^∗ < 0 which satisfies the Lyapunov equationA^∗TP P A^∗ −Q₀^∗ < 0. Then, Theorem 2.12(i) holds ifQ^∗<0 for any switching lawσsuch that

λ_min

−Q^∗

−λmax

Q^∗

> K^∗λmax

Q^∗₀ ρ^∗

A^Tt

2

q i1

hiA^∗_iM^∗ Δit

2

,

∀t∈R₀ , ∀t∈STσ.

2.21

2.4. Sufficiency type asymptotic stability conditions obtained for time-varying symmetric matricesPt,S_ijt S_ij

The following result, which is proven inAppendix A, holds.

Theorem 2.16. Under Assumptions2.1–2.3, the following properties hold.

iThe switched system2.1is globally asymptotically Lyapunov’s for any delaysh_i ∈0,h_i for all i ∈ qfor someh:max_i∈qhiand any switching lawσsuch that

athe switching instants are arbitrary;

bmaxess supA˙_jt:t∈R₀ , j ∈pis sufficiently small compared to the absolute value of the prescribed stability abscissa ofq

j0A_jt;

cthe support testing matrix of distributional derivativesΓAdjtof the same matrices are semidefinite negative for all time instants, where the conventional derivatives do not exist (i.e., ˙A_jt ΓAdjttδ0).

iiThe switched system2.1is globally exponentially stable for any delayshi∈0,hifor all i∈qfor someh:max_i∈qh_isuch that

amaxess supA˙jt:t∈R₀ , j ∈pis sufficiently small compared to the absolute value of the prescribed stability abscissa ofq

j0A_jt;

bmaxΓAdjtt: ˙Ajt ΓAdjttδ0, ∀t∈STσ, j ∈pis sufficiently small compared to the timeintervals in between any two consecutive switching instants.

Furthermore, if Assumptions2.1–2.4hold, then

iiithe switched system2.1is globally exponentially stable for any delaysh_i∈0,h_ifor all i∈q, for someh:max_i∈qhisuch that

amaxess supA˙jt:t∈R₀ , j ∈pis sufficiently small compared to the absolute value of the prescribed stability abscissa ofq

j0Ajt;

bthe switching lawσis such that

amaxΓAdjt: ˙A_jt ΓAdjtδ0,∀t∈ST_rfσ, j ∈pis sufficiently small compared to the lengths of time intervals between any two consecutive switching instants;

bit exists a common Krasovsky-Lyapunov functional Vt, xt defined with constant matricesP P^T >0 andSijS^T_ij>0, for alli, j∈q∪{0}×qfor all the time-invariant resetting systems2.4and some of the subsequent conditions hold for allt ∈ STrσ under the resetting action Pt P; for alli, j ∈ q∪ {0}×q:

b.1Vt , x_t≤Vt, xtwhich is guaranteed, in particular, ifPt P ≤Pt, b.2the tradeoff(a) is respected between sufficiently small norms of the matrices of distributional derivatives and the length|t−t|, at anyt∈STrσ, if any, where the condition (b.1) is not satisfied, wheretmaxτ ∈R₀ : STσ τ < t.

The characterization of the “sufficient smallness” of the involved magnitudes in Theorem 2.16 is given explicitly in its proof. The proof considers that when some entry time derivative of the involved matrices does not exist, it equivalently exists a distributional derivative at this time instant which is equivalent to the existence of a bounded jump-type discontinuity in its integral, so that the corresponding time instant is in fact a switching instant. The sufficiently large time intervals required in between any two consecutive switching times compared with the amplitudes of the amplitudein terms of norm errors among consecutive parameterizations are related to the need for a minimum residence time at each parameterization for the case when those ones do not possess a common Krasovsky- Lyapunov functional.

3. Asymptotic stability independent of the delays

Some results concerning sufficiency type properties of global asymptotic stability indepen- dent of the delays, that is, for any hi ∈ R₀ , for alli ∈ q of the switched system2.1are obtained under very close guidelines as those involved in the results on stability dependent of the delays given inSection 2. The Krasovsky-Lyapunov functional candidate ofSection 2 andAppendix Ais modified as follows:

Vt, xt x^TtPtxt ^q

i1

_t

t−hi

x^TτSiτxτdτ 3.1

whose time derivative along the state-trajectory solution of2.1is

V˙t, xt x^Tt

A^TtPt PtAt ^q

i1

Sit P˙t

xt

2x^Tt q

i1

P Aitx t−hi

− q

i1

x^T t−hi

Si

t−hi

x t−hi

x^TtQtxt<0

3.2

for all nonzerox^Tt x^Tt, x^Tt−h1, . . . , x^Tt−hqif

Qt:

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

A^TtPt PtAt q

i1

Sit Pt˙ P A1t · · · P Aqt

A^T₁tP −S1

t−h1

0· · · 0

... ... ...

A^T_qtP 0 −Sq

t−hq

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0. 3.3

Assumption 2.1ofSection 2remains unchanged while Assumptions2.2and2.4ofSection 2 are modified under similar justifications as follows.

Assumption 3.1. All the eigenvalues λ_iA0t of the matrix function A₀t satisfy ReλiA0t≤ −ρ00<0; for all t∈R₀ , for alli∈σ σ≤nfor someρ00 ∈R :{0< z∈R};

that is,A₀tis a stability matrix, for allt∈R₀ .

Assumptions 3.2. A_j : R₀ → R^n×n are almost everywhere time-differentiable with essentially bounded time derivative, for allj ∈q∪ {0}possessing eventual isolated bounded discontinuities, then ess sup_t∈R

0 A˙₀t ≤ γ₀ < ∞ with γ₀ being a -norm dependent nonnegative real constant and, furthermore,_{t T}

t A˙0τd τ ≤ μ0T α0 for someα0, μ0 ∈ R , for allt ∈ R₀ , and some fixedT ∈ R₀ independent oft. If the time derivative does not exist then it is defined in the distributional sense as in Assumptions2.3.

Assumption 3.3for the resetting systems. All the eigenvaluesλkA0jsatisfy ReλkAj0 ≤

−ρ00<0; for allt∈R₀ , for allk∈σ0jσ0j≤n, for allj∈p; that is,A0jare constant stability matrices with prescribed stability abscissa.

A parallel result toTheorem 2.12i-iiis the following.

Theorem 3.4. The subsequent properties hold.

iAssume that

i.athe matrix functionsAjt, for allj ∈ q ∪ {0}are subject toAssumption 2.1;

i.bthe switching lawσis such thatQt < 0,3.3, for allt ∈ R₀

for some time-differentiable real symmetric positive definite matrix function P : R₀ → R^n×nand some real symmetric positive definite matrix functionsS_i : R₀ → R^n×n∀i∈ q. Thus, the system2.1is globally asymptotically Lyapunov’s stable independent of the delays (i.e., for all delayshi∈0,∞, for alli ∈ q). A necessary condition isA^T₀tPt PtA0t P˙t < 0, for all t ∈ R₀ what implies that A₀tis a stability matrix of prescribed stability abscissa on R₀ except eventually on a real subinterval of finite measure of R₀ .

iiAssume that

ii.aAjt Aji, for allj ∈ q∪ {0}, for allt ∈ R₀ for some i ∈ p(eventually being dependent ont) satisfyingAssumption 3.3;

ii.bthe switching lawσis such that STrfσ ∅(i.e., it generates reset switching instants only) with ST_rσbeing arbitrary, namely, the set of reset times is either any arbitrary strictly increasing sequence of nonnegative real values (i.e., the resetting switching never ends) or any finite set of strictly ordered nonnegative real numbers with a finite maximal (i.e., the resetting switching ends in finite time);

ii.c

Q^∗_i :

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎣

A^T_0iP^∗ P^∗A_0i q

i1

S^∗_i P^∗A_1i · · · P^∗A_qi A^T_1iP^∗ −S^∗₁ 0· · · 0

... 0

...

. .. ... A^T_qiP^∗ 0 · · · −S^∗_q

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎦

<0; ∀i∈p 3.4

for some R^n×n P^∗ P^∗T > 0, R^n×n S^∗_i S^∗_i^T > 0∀i ∈ q. Thus, the switched system 2.1, obtained from switches among resetting systems 2.4, is globally asymptotically Lyapunov’s stable and also globally exponentially stable independent of the delays for alli ∈ q. If 3.4is replaced withQ^∗_i ≤ −2εI_{q 1n}<0, for alli ∈ q, and someε ∈ R then the state trajectory decays exponentially with rate−ε < 0.

Parallel results to Corollaries 2.13–2.15 are direct from Theorem 3.4 with the replacementsA₀t → A^∗₀a constant stability matrix,A_0jt → A^∗_0j, for all j ∈ pa set of constant stability matrices with prescribed stability abscissa for the resetting configurations.

Also, the subsequent result for global asymptotic stability independent of the delays, which is close toTheorem 2.16, follows by replacing Assumptions2.2–2.4by Assumptions3.1–3.3.

Theorem 3.5. Under Assumptions2.1and3.1–3.2, the following properties hold.

iThe switched system2.1is globally asymptotically Lyapunov’s stable independent of the delays, that is, for any delayshi∈0,∞, for alli ∈ pand any switching lawσsuch that

athe switching instants are arbitrary;

bmaxess supA˙0t:t∈R₀ is sufficiently small compared to the absolute value of the prescribed stability abscissa ofA0t;

cthe support testing matrix of distributional derivativesΓAd0tof the same matrices are semidefinite negative for all time instants, where the conventional derivatives do not exist (i.e., if ˙A₀t ΓAd0ttδ0) .

iiThe switched system2.1is globally exponentially stable independent of the delays if amaxess supA˙₀t:t∈R₀ is sufficiently small compared to the absolute value

of the prescribed stability abscissa ofA0t;

bmaxΓAd0tt : ˙A0t ΓAd0ttδ0, ∀t ∈ STσ is sufficiently small compared to the time intervals in between any two consecutive switching instants.