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Latin American applied research

versión impresa ISSN 0327-0793

Lat. Am. appl. res. vol.39 no.3 Bahía Blanca jul. 2009

 

Input-to-output stability properties of switched perturbed nonlinear control systems

J.L. Mancilla-Aguilar and R.A. Garcia

Department of Physics and Mathematics, ITBA
jmancill@itba.edu.ar
Department of Physics and Mathematics, ITBA
ragarcia@itba.edu.ar

Abstract — In this paper we obtain the extension of results on input-to-output stability properties of switched systems to switched systems whose dynamics are described by perturbed forced differential equations and whose outputs are obtained via switched perturbed functions. We also present Lyapunov characterizations of these input-output stability properties, obtained in terms of certain conceptual output functions.

Keywords — Switched Systems. Input-To-Output Stability. Lyapunov Characterizations.

I. INTRODUCTION

In the last decade, the study of the properties of switched systems described by

(1)

with σ : [0, +∞) → Γ an arbitrary switching signal and Γ the index set, has attracted a great amount of attention, mainly motivated by the rapidly development of the area of intelligent control (see Liberzon (2003) and references therein for details). In fact, switched systems (1) enable us, for example, to model the continuous portion of a hybrid system, see Branicky (1998) and Lygeros et al. (2003).

In this context, different stability properties of system (1) were studied and characterized in terms of Lyapunov functions (see among others Liberzon and Morse, 1999, Liberzon, 2003, Mancilla-Aguilar and García, 2000, 2001).

On the other hand, although under mild regularity conditions, differential equation (1) provides, for each initial condition and each switching signal, a complete description of the time evolution of the state x and, in consequence, a tractable analysis of the closed loop system (with u(t)= k(x(t))), it lacks in robustness. Hence, in order to take into account some of the uncertainty caused by errors and disturbances that are inevitable in any realworld control problem, a more general model was considered in Mancilla-Aguilar et al. (2005): switched systems described by forced differential inclusions

(2)

In that work representations of system (2) were obtained by means of perturbed control systems described by differential equations, whose inputs are those of the original system and perturbations that evolve in compact sets.

As immediate applications of the results on the representations, Lyapunov characterizations of the input-output-tostate stability and several input-to-output stability properties were developed.

Nevertheless the model (2) does not take into account the fact that the output could be obtained by switching different output maps, and that for these maps uncertainties caused by errors and disturbances should also be taken into account.

This would lead us to consider the general model

(3)

The treatment of such a model involves subtle and lengthy technicalities, and for this reason in this paper we will focus instead on the relatively easier to study class of switched systems described by

(4)

where ω and ζ model the uncertainties and disturbances, and leave the treatment of systems (3), to be presented elsewhere.

In this work, then, we obtain necessary and sufficient conditions for different input-to-output stability properties for systems (4).

The conditions obtained are characterized by certain conceptual output maps that enable us to reduce the treatment of system (4) to that of the (considerably easier to handle) system

(5)

As immediate application of our results, we extend previous results on Lyapunov characterization of those inputto-output stability properties to systems (4).

The paper is organized as follows. In Section II we establish the basic notation and the class of switched systems we consider. In Section III we state the notions of stability that we will address. Section IV presents the main result. In Section V, we develop Lyapunov characterizations of several input-to-output stability properties. In that section we also present a result on equivalence between inputto-output properties (uOLIOS and uSIIOS, see details below) modulo a time reparametrization. Finally, in Section VI, the conclusions are given.

II. BASIC NOTATION AND DEFINITIONS

We first introduce some notation and definitions that will be used in the sequel. We use |·| to denote the Euclidean norm for any given and stands for the closed ball centered at 0 of radius r in . Given a metric space E, we denote by the set of Lebesgue measurable functions η : [0, +∞) → E that are locally essentially bounded. In case E = , we write instead of . For an input u and any t ≥ 0, we denote by ||u||t the -norm of u restricted to the interval [0,t]. As usual, by a -function we mean a function α : that is strictly increasing and continuous, and satisfies α(0) = 0, by a -function one that is in addition unbounded, and we let be the class of functions × which are of class on the first argument and decrease to zero on the second argument.

As was previously said, the switched systems with inputs and outputs we consider in this work are modeled by

(6)

where x is the state, u is the input, y is the output, σ : [0, ∞) → Γ is a switching signal, i.e. σ is a piecewise-constant continuous from the right function which takes values in the compact metric space Γ (the index set) and ω ∈ and ζ ∈ are, respectively, the perturbations in the dynamics and in the output, with Ω1 and Ω2 compact metric spaces.

We suppose that f : × × Γ × Ω1 and h : × Γ × Ω2 are continuous functions so that for each N, there exists lN ≥ 0 such that for all γ ∈ Γ, all ν ∈ Ω1 and all θ ∈ Ω2

and

for all ξ, ξ' ∈ and all µ, µ' ∈ . In addition,

In what follows we denote the set of all switching signals by .

Given an input u, a switching signal σ ∈ and a perturbation ω ∈ , we say that a locally absolutely continuous function x : where = [0,T] or [0,T) with 0 < T ≤ +∞ is a trajectory of (6) corresponding to u, σ and ω, if = f(x(t),u(t),σ(t),ω(t)) for almost all t. Observe that, due to the assumptions about f, for each ξ ∈ , each u, each σ ∈ and each ω ∈ , there always exists a unique maximal trajectory xξ,u,σ,ω corresponding to u, σ and ω that is defined on an interval [0,Tx), with Tx > 0 or Tx = +∞, which verifiesxξ,u,σ,ω(0) = ξ (a trajectory x corresponding to u, to σ ∈ and to ω ∈ is called maximal if it does not have a proper extension which is a trajectory corresponding to u, σ and ω).

III. INPUT-OUTPUT STABILITY OF SWITCHED SYSTEMS

Next, and following the work in (Sontag and Wang, 1999, 2001) we will introduce some notions on input-to-output stability properties for switched systems modeled by (6).

In order to do so, we must first introduce the following notion: we say that a system as in (6) is forward complete if every trajectory xξ,u,σ,ω corresponding to any ξ ∈ , any u, any σ ∈ and any ω ∈ is defined for all t ∈ [0, +∞).

Definition III..1 A forward complete system as in (6) is

1. uniformly input to output stable (uIOS) if there exist a -function β and a -function η such that for all ξ ∈ , all u, all σ ∈ , all ω ∈ and ζ ∈ , if y(t) = h(xξ,u,σ,ω(t),σ(t),ζ(t)),

(7)

2. uniformly output-Lagrange input to output stable (uOLIOS) if it is uIOS and there exists some -function ρ such that for all ξ ∈ , all u, all σ ∈ , all ω ∈ and all ζ ∈ , if y(t) = h(xξ,u,σ,ω(t),σ(t),ζ(t)),

(8)

3. uniformly state-independent input-to-output stable (uSIIOS ) if there exist some β ∈ and some η ∈ such that for all ξ ∈ , all u, all σ ∈ , all ω ∈ and all ζ ∈ , if y(t) = h(xξ,u,σ,ω(t),σ(t),ζ(t)),

(9)

IV. MAIN RESULT

In what follows we will establish the main result of this work, which asserts that the study of the different already defined input-to-output stability properties of system (6) can be reduced to that of a system like (5). In order to do so, we define the conceptual output functions hM : and hm : as follows

(10)
(11)

The following result enables us to establish that hM and hm are locally Lipschitz functions.

Proposition IV..1 Let η : × K, with K a compact metric space, be a continuous function so that for all N there exists lN ≥ 0 such that for all κ ∈ K

for all ξ, ξ' ∈ . Then

are well defined locally Lipschitz functions.

Proof. We prove the assertion for ηM only, since

That ηM is well defined follows from the facts that η is continuous and K is compact. Let N, ξ12, and let κ12K such that ηM1) = η(ξ11) and ηM2) = η(ξ22). Then

By symmetry

and the proposition follows.

If we consider in Proposition IV..1, η(ξ,κ) = |h(ξ,γ,θ)|, with κ = (γ,θ) ∈ Γ × Ω2 = K, the following result holds.

Corollary IV..2 hM and hm, as given by (10) and (11) respectively, are well defined locally Lipschitz functions.

Let us consider the following switched system with outputs

(12)

Then, we can state the main result of this work.

Theorem 1 The following equivalences hold:

1. System (6) is uIOS if and only if system (12) is uIOS.

2. System (6) is uOLIOS (resp. uSIIOS) if and only if system (12) is uOLIOS (resp. uSIIOS) and there exists some function α ∈ such that

(13)

Proof. We only prove 2. for the uOLIOS case, since 1. and 2. for the uSIIOS case can be proved in a similar way.

Sufficiency: suppose that (12) is uOLIOS and that there exists α ∈ such that (13) holds. Let ξ ∈ , u, σ ∈ ,ω ∈ and ζ ∈ and let z(t) = hM(xξ,u,σ,ω(t)) and y(t) = h(xξ,u,σ,ω(t),σ(t),ζ(t)).

Since system (12) is uOLIOS there exists ρ ∈ such that (8) holds with z instead of y. Then, taking (10) into account, we have for all t ≥ 0 that

Since z(0) = hM(ξ) ≤ α(hm(ξ)) ≤ α(|y(0)|), it follows that

Therefore, (8) holds with ρ* = max(ρ, ρ ◦ α) in place of ρ. Finally, the fact that system (6) is uIOS follows from (7) (with z(t) instead of y(t)) and the fact that |y(t)| ≤ z(t) for all t ≥ 0.

Necessity: suppose that system (6) is uOLIOS and let ξ ∈ , u, σ ∈ , ω ∈ and let z(t) = hM(xξ,u,σ,ω(t)) and τ ≥ 0. From the definition of hM, there exists some γM ∈ Γ and some θM ∈ Ω2 such that z(τ) = |h(xξ,u,σ,ω(τ),γMM)|. Let = xξ,u,σ,ω(τ), u* ∈ defined by u*(t) = u(τ + t), σ* ∈ , defined by σ*(t) = γM, ω* ∈ given by ω*(t) = ω(t + τ) and y*(t) = . It follows that y*(0) = . Consider now and , where, for a pair of functions θ and λ from [0, ∞) to a set A, denotes the function defined as for 0 ≤ t < τ and for t ≥ τ. Then, it is clear that . Since system (6) isuIOS and , we have that

(14)

which proves that system (12) is also uIOS.

From the facts that system (6) verifies (8), (0) = y(0), z(0) ≥ |y(0)| and ρ ∈ , we have that

which proves that (8) holds with z in place of y. We have proved that system (12) is uOLIOS.

It remains to show the existence of a function α ∈ such that hM(ξ) ≤ α ◦ hm(ξ) for all ξ ∈ .

Fix ξ ∈ . From the definitions of hM and hm there exist γM and γm in Γ and θM and θm in Ω2 such that hM(ξ) = |h(ξ,γMM)| and hm(ξ) = |h(ξ,γmm)|. Let σ, σ* ∈ defined by σ(t) = γm,σ*(t) = γM for all t ≥ 0, and ζ,ζ* ∈ given by ζ(t) = θm,ζ*(t) = θM for all t ≥ 0 and denote by 0 the zero input. Let ω ∈ be any dynamic perturbation, y(t) = h(xξ,0,σ,ω(t),σ*(t),ζ*(t)). Then y(0) = h(ξ,γmm), g : [0, +∞) → is a continuous function and g(0) = h(ξ, γMM).

For each τ > 0 let = xξ,0,σ,ω(τ) and yτ(t) = . It follows that yτ(0) = = g(τ). It is clear that if and . Then . Therefore, from the facts that system (6) verifies (8) and that the input to the system is 0, we have that

for all τ > 0. In consequence

and the claim about hM and hm follows taking α = ρ.

Remark IV..3 Theorem 1 exhibits the robustness of the stability properties involved with respect to the synchronism between the switching of the dynamics of the switched system and that of the output functions. In fact under the hypotheses of the theorem, it follows that if system (12) has any of these stability properties, the same holds for the switched system

(15)

with σ, σ' ∈ not necessarily the same.

In consequence, delays or perturbations in the switching signals will not affect the stability properties considered.

V. LYAPUNOV CHARACTERIZATIONS

We are now in position to present the Lyapunov characterizations of the input-to-output stability properties, but in order to do so we need to introduce the following:

Definition V..1 A system as in (6) is uniformly bounded input bounded state stable (uBIBS) if there exists some nondecreasing function φ such that for all ξ ∈ , all u, all σ ∈ and all ω ∈ ,

(16)

The following Lyapunov characterizations hold:

Theorem 2 Suppose the system (6) is uBIBS

1. The system is uIOS if and only if there exists a smooth function V : such that

     · for some α1, α2,

(17)

     · for some χ ∈ and β1, the following holds for all ξ ∈ , all µ ∈ , all γ ∈ Γ and all ν ∈ Ω1:

(18)

2. The system is uOLIOS if and only if there exists a smooth function V : such that

     · for some α12,

(19)

     · for some χ ∈ and some β1, (18) holds for all ξ ∈ , all µ ∈ , all γ ∈ Γ and all ν ∈ Ω1.

3. The system is uSIIOS if and only if there exists a smooth function V : such that

     · for some α12, (19) holds, and

     · there exist α3 and χ ∈ such that for all ξ ∈ , all µ ∈ , all γ ∈ Γ and all ν ∈ Ω1,

(20)

Proof Sufficiency: it follows from Lemma A.2 in (Mancilla-Aguilar et al., 2005) that the existence of a function V that verifies the conditions in 1. (respectively 2., 3.) implies that system (12) is uIOS (resp. uOLIOS, uSI-IOS). Hence, from Theorem 1, system (6) is uIOS. Taking into account that condition (19) implies that hM(ξ) ≤ = α(hm(ξ)) for all ξ ∈ and that = α ∈ , from Theorem 1 again, we conclude that system (6) has the stability properties stated in 2. or 3.

Necessity: let for any ξ ∈ ,µ ∈ and γ ∈ Γ

and consider the switched system

(21)

Since according to Filipov's lemma (see Clarke et al., 1998; Vinter, 2000) x is a maximal solution of (21) corresponding to u and σ ∈ that verifies x(0) = ξ if and only if there exists ω ∈ such that x = xξ,u,σ,ω, it follows that system (12) is uBIBs (resp. uIOS, uOLIOS, uSIIOS) if and only if system (21) is uBIBS (resp. uIOS, uOLIOS, uSIIOS).

Due to the properties of f and hM, we have that system (21) verifies the hypotheses of Theorem 8 in (Mancilla-Aguilar et al., 2005) and in consequence there exists a function V which satisfies conditions in 1. (resp. 2., 3.) if system (12) is uBIBs and uIOS, (resp. uOLIOS, uSIIOS), but with

instead of (19) in the cases of uOLIOS and uSIIOS.

Since in the case when system (6) is uOLIOS or uSIIOS, due to Theorem 1 there exists α ∈ such that

we have that

and in consequence, V verifies (19) with α2 ◦ α−1 in lieu of α2.

Remark V..2 In the uSIIOS case the uBIBS condition can be replaced by the forward completeness property as long as switching signals with possible Zeno behavior 1 are considered (see Remark A.3. in Mancilla-Aguilar et al., 2005, for details). Hence, we have the following:

Proposition V..3 Suppose that the class of admissible switching signals of system (6) includes Zeno switching signals also and that this system is forward complete. Then it is uSIIOS if and only if there exists a smooth function V : such that

  • for some α12, (19) holds; and
  • there exist α3 and χ ∈ such that for all ξ ∈ , all µ ∈ , all γ ∈ Γ and all ν ∈ Ω1, property (20) holds.

Next we present a result which shows that a system is uOLIOS if and only if it is uSIIOS modulo a time reparametrization. This result complements in a certain way that in Sontag and Wang (2001) which establishes that a non-switched system described by non-perturbed differential equations is IOS if and only if it is OLIOS modulo an output redefinition.

Theorem 3 The following statements are equivalent

  1. system (6) is uBIBS and uOLIOS
  2. there exists a locally Lipschitz function φ : → [1, +∞) such that the system
    (22)
    is uBIBS and uSIIOS.

Proof. First we prove that system (6) is uBIBS if and only if system (22) is so.

Suppose that system (6) is uBIBS. Then the trajectories of (6) verify (16) for some nondecreasing and nonnegative function φ. Let ξ ∈ , u, σ ∈ and ω ∈ and let z : [0,Tz) → be the maximal trajectory of (22) corresponding to u, σ and ω that verifies z(0) = ξ. We will show that z also verifies (16) for all t ∈ [0,Tz). In consequence, due to well-known results on ordinary differential equations, Tz = ∞, and system (22) is uBIBS.

Let τ(t) = φ(z(s))ds. Then τ : [0,Tz) → is a continuously differentiable, definite positive and strictly increasing function. Let τz = z could be +∞). Consider , and ; we note that , and are defined on [0,τz). In the case when τz < +∞ we extend them to [0, +∞) in such a way that , and belong to , and respectively.

Then, if x is the maximal trajectory of (6) which corresponds to , and and verifies x(0) = ξ, we have that x(t) = z−1(t)) for all t ∈ [0,τz). Since for all s ∈ [0,Tz), if t = τ(s),

and , we have that

and, in consequence, that system (22) is uBIBS.

We note that in the proof above, we have only used that φ is positive and locally Lipschitz. In consequence, that system (6) is UBIBS if system (22) is so, follows taking into account that

and that 1/φ(ξ) is positive and locally Lipschitz.

Suppose now that system (6) is uOLIOS and uBIBS. Then due to Theorem 2 there exists a smooth function V : , functions α12, χ ∈ and β1 such that (18) and (19) hold. Then according to Lemma A.2 in Sontag and Wang (2001), there exist two functions κ1 and κ2 of class such that

(23)

Pick any locally Lipschitz function κ : such that κ(t) ≥ κ2(t) for all t and let φ : → [1, +∞) given by φ(ξ) = κ(|ξ|)+1. Next we prove that V verifies (20) with φ(ξ)f(ξ, µ, γ, ν) instead of f(ξ, µ, γ, ν) and κ1 in lieu of α3, and therefore, according to Theorem 2, that system (22) is uSIIOS. Let ξ ∈ ,γ ∈ Γ, µ ∈ and ν ∈ Ω1 such that V(ξ) ≥ χ(|µ|). Since DV(ξ)f(ξ, µ, γ, ν) ≤ −β1(V(ξ),|ξ|), we have that

Now suppose that 2. holds. Let ρ(r) = max|ξ|≤rφ(ξ) and α(r) = ρ(r) − ρ(0) + r. It is clear that α is a -class function such that ρ(0) + α(|ξ|) ≥ φ(ξ) for all ξ ∈ . Since (22) is uSIIOS and uBIBS, due to Theorem 2 there exists a Lyapunov function V, functions α123 and χ ∈ which verify (19) and (20). Consider the -class function

Let ξ ∈ ,γ ∈ Γ, ν ∈ Ω1 and µ ∈ such that V(ξ) ≥ χ(|µ|). Since DV(ξ)φ(ξ)f(ξ, µ, γ, ν) ≤ −α3(V(ξ)), we have that

Therefore, V verifies (18) with β2 in place of β1, and system (6) is uOLIOS.

VI. CONCLUSIONS

In this paper we have extended the study of some input-tooutput stability properties to switched systems described by perturbed forced differential equations and outputs obtained via switched perturbed functions. We have shown that under suitable hypotheses the input-to-output stability behavior of these systems is equivalent to that of a switched system also described by the same perturbed forced differential equations but whose output is given by a conceptual output function. We also presented Lyapunov characterizations of these input-to-output stability properties in terms of these conceptual output functions. Finally, a result on equivalence between input-to-output properties modulo a time reparametrization was provided.


1σ : [0, +∞) → Γ is a Zeno switching signal if there exists a strictly increasing sequence of switching times {tn, n} such that t0 = 0, tnt* < +∞ as n → +∞, σ(t) is constant between switching times and σ(t)= σ(t*) for t ≥ t*.

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Received: October 18, 2007.
Accepted: February 9, 2009.
Recommended by Guest Editors D. Alonso, J. Figueroa, E. Paolini and J. Solsona.