Stability analysis of a certain class of time-varying hybrid dynamical systems

J. L. Mancilla Aguilar¹ and R. A. García²

Depto. de Matemática, Fac. de Ingeniería, Universidad de Buenos Aires.
Paseo Colón 850, (1063)Buenos Aires, Argentina.
¹ jmancil@fi.uba.ar
² rgarcia@fi.uba.ar

Abstract ¾ In this work we study the exponential stability of a class of hybrid dynamical systems that comprises the sampled-data systems consisting of the interconnection of a time-varying nonlinear continuous-time plant and a time-varying nonlinear discrete-time controller, assuming that the sampling periods are not necessarily constant. For this purpose we develop an Indirect Lyapunov Method of analysis, and show that under adequate hypotheses the exponential stability of the hybrid dynamical system is equivalent to the exponential stability of its linearization.

Keywords ¾ Sampled-Data Systems. Lyapunov Stability. Hybrid Systems. Discrete-Time Systems.

I. INTRODUCTION

Sampled-data control systems consisting of the interconnection of a continuous-time nonlinear plant (described by a system of non autonomous first order ordinary differential equations) and a nonlinear digital controller (described by a time-varying system of first order difference equations) is a hybrid system, in the sense that some of its variables evolve smoothly in continuous time while the others change only in a discrete set of time instants. The coexistence of these two different time scales makes it hard to analyze the stability properties of this kind of systems.

Stability analysis of sampled-data systems in the whole time scale was primarily studied for linear systems (Francis and Georgiou, 1988; Iglesias, 1994). The nonlinear case was addressed in the recent papers (Hou et al., 1997; Mancilla-Aguilar et al., 2000; Hu and Michel, 2000a, 2000b), in which qualitative properties of sampled-data control systems, where the plant and the controller are time-invariant, were obtained.

In (Mancilla Aguilar et al., 2000), in particular, we analyzed the stability of a class of hybrid dynamical systems, those described by time-invariant hybrid equations, that contains as particular cases the interconnected system consisting of a continuous time plant and a digital controller and that of a continuous time plant and a certain class of hybrid controllers presented in (Rui et al., 1997). In this work we consider the class of hybrid systems described by time-varying hybrid equations with non regular sampling times, and extend some results of (Mancilla Aguilar et al., 2000) to the class of hybrid systems described by this type of equations. To be more precise, we obtain necessary and sufficient conditions for the exponential stability of the system in terms of its linearization, developing in this way an Indirect Lyapunov Method for this kind of dynamical systems. Our results can be straightforwardly applied to the study of the exponential stability of sampled-data systems whose plant and controller are nonlinear and time-varying, as these systems are a particular class of those under study. They also justify the method of design of nonlinear digital local ex­ponential stabilizers based on the discretized model of the linearization of the original nonlinear control system (a more complete treatement of these topics can be found in (Mancilla Aguilar, 2001)). Two reasons motivated the treatment of non-regular sampling. The first is the uncertainty that could appear in the frequency of the sampler oscillator, that in certain applications should be of importance. The second is that our results can also be applied to the stability analysis of certain classes of switched systems whose switching times are not necessarily regularly spaced.

The paper is organized as follows. In section II we establish some notation and state the main result of the paper. In section III we present some results about the exponential stability of a perturbed discrete-time time-varying linear system. In section IV we use these results to obtain stability properties of perturbed hybrid linear systems, that enable us to develop the Indirect Lyapunov Method already mentioned, and prove the main result of the paper. Finally, in section V we present some conclusions.

II. NOTATION AND MAIN RESULT

First, we introduce some notation that will be used in this work.

Let ₀, and ⁺ the sets of non-negative integer, real and non-negative real numbers respectively. We consider the real q-space ^q as a normed space, with norm . We denote by W = ⁿ ´ ^m, its elements (x, z) = w, and we consider in W the norm ||w|| = |x| + |z|. B_p Ì W is the closed ball of radius r centered at the origin. Given r > 0 a function j: [0, r] ® ⁺ is of class , if it is continuous, strictly increasing and j(0) = 0.

We denote with P := {t_k, k Î ₀} Ì ⁺ the set of sampling points and assume that 0 - t₀ < t₁ < ..., sup_k(t_k+1 - t_k) < ¥ and lim_{k® ¥} t_k = ¥; ||P|| = sup_k(t_k+1 - t_k) is the norm of P.

In this paper we will study qualitative properties of the hybrid system S_H described by the time-varying hybrid equations

(1)

As was mentioned in the Introduction, this type of systems includes, as a particular case, the sampled-data system consisting of a plant described by the set of time-varying equations

(2)

where the state x(t) Î ⁿ, the control u(t) Î ^l, the output y(t) Î ^p and t Î ⁺, and a digital controller described by the difference equations

(3)

with states z(t_k) Î ^m, inputs w(t_k) Î ^p, and outputs v{t_k) Î ^l, interconnected via sampling and zero-order hold (ZOH), i. e. w(t_k) = y(t_k) and u(t) = v(t_k), t_k < t < t_k+1 k Î ₀.

In order to assure the existence and uniqueness of the solutions of (1) (see (Mancilla Aguilar et al., 2000; Mancilla Aguilar, 2001) for the definition of solution of (1)) we assume that the function f₁ : ⁺ ´ ⁿ ´ W ® ⁿ verifies:

H1 For each t Î ⁺ and (x, z) Î W, f₁(t,×,x,z) is continuous in ⁿ, and for each (x,x,z) Î ⁿ ´ W, f₁(×,x,x,z) is Lebesgue measurable in ⁺.

H2 For each compact set K Ì ⁿ and each (x, z) Î W there exist a constant L^* ³ 0 such that .

We suppose in addition that the origin is an equilibrium for (1), i. e. f₁(t, 0, 0, 0) = 0 and f₂(k, 0, 0) = 0 , that f₁(t, ×, ×, ×) is differentiable at (0,0,0) for all and that f₂(k, ×, ×) is differentiate at (0,0) for all . Let

	(4)
	(5)

Then, associated with S_H, we have the linear hybrid system S_LH described by the linearization of (1) at the origin:

Let us recall the definition of local exponential stability of system (1): we say that the origin is an exponentially stable equilibrium of system (1) if there exist positive numbers h, m and r such that for any w = (x,z) Î B_r, and any t_k Î P, , for all t ³ t_k, where f(t,t_k,w) is the maximally defined solution of (1), starting at w in time

The following theorem, which is the main result of this work, establishes stability properties of the origin for S_H based on its stability properties for S_LH. In other words, we will develop an Indirect Lyapunov Method for dynamic systems described by equations of the type (1).

Theorem II..1 Suppose that A(t), B₁₁(t) and B₁₂(t) in (4) are bounded and that the nonlinear terms resulting from the linearization of (1) and verify

H3

uniformly in

and

H4

uniformly in

Then, the origin is an exponentially stable equilibrium for S_H if and only if it is an exponentially stable equilibrium for S_LH.

Remark II..1 Hypothesis H3 and H4 (without the term t_k+1 - t_k in the denominator) are standard ones of the First Method of Lyapunov in the stability the­ory of Ordinary Differential and Difference Equations respectively. The term t_k+1 - t_k must be included in H4 because we do not consider that d_k = t_k+1 - t_k is bounded from below by a positive constant. In the case that inf d_k = 0, Theorem II..1 is not valid if that term is removed from H4.

III. ON THE EXPONENTIAL STABILITY OF DISCRETE-TIME SYSTEMS

In order to prove Theorem II..1 we need some results about the exponential stability of the discrete-time system described by the equation

(6)

where H_k : W_k ® W and W_k Í W for all .

We denote with the solution in time t_k of (6) with the initial condition , and assume that there exists r > 0 such that B_r Ì W_k and that H_k(0) = 0 for all . It follows from the last assumption that the origin is an equilibrium of (6). The following result, that can be easily proved employing standard techniques of Lyapunov Stability theory, gives a sufficient condition for the exponential stability of the origin of (6) in terms of Lyapunov functions.

Proposition III..1 Suppose there exists a scalar function such that

1.

2. ,

with 0 < r' £ r and, c₁, c₂ and c₃ positive constants. Then the origin is an exponentially stable (ES) equilibrium of (6), i. e., there exist positive numbers r^*, m and h such that , .

It is possible, when the discrete-time system is linear, to obtain propositions that are converse to Proposition III..1, i. e. it is possible to obtain converse Lyapunov theorems. Next, we consider the discrete-time linear system

w(t_k+1) = A_kw(t_k) (7)

with A_k matrices of proper dimensions. Then, W_k = W and 0 will always be an equilibrium for (7).

In addition, , where is the transition matrix, that verifies

Remark III..1 Due to the linearity of with respect to w₀, the following facts are easily established:

1) If the origin is an exponentially stable equilibrium, it is a global one, i. e. there exist positive constants h and m such that for all k ³ k₀ and all w₀ Î ⁿ.

2)   Due to this last inequality, verifies for all k ³ k₀.

The following converse Lyapunov theorem will be useful in the sequel.

Theorem III..1 Suppose the origin is an ES equilibrium for (7). Then, there exists a scalar function V : ₀ ´ W ® ⁺ that verifies, for all and all w, w' Î W:

a);

b);

c),

with c₁, ..., c₄ positive constants.

Proof Consider the semidefinite positive function defined by

(8)

with P(k) the positive definite matrix given by

(9)

In order to prove that this matrix is well defined we recall that as the origin is exponentially stable, then (see Remark III1); hence

since the series above is uniformly bounded. In consequence, . In addition, from (8) and the bound we easily obtain

for all w, w' Î W.

On the other hand, as for all j > k ³ 0 and all w Î W, it follows that

Consider now

This function is well defined since, due to the exponential stability of the origin, for all k ³ 0. In addition, as F(t_k,t_k) = I, then and as , we have .

Let us fix k ³ 0; it follows that for a given w Î W there exists j^* = j^*(w) ³ k such that

As a consequence, for w, w' Î W,

with h as above. Then, by symmetry, , and we obtain

Recalling once again that for all j > k ³ 0 and all w Î W, we deduce that

Finally, consider V(k,w) = V₁(k,w) + V₂(k,w). It is easy to see that this function verifies the thesis with c₁ = c₃ = 1 and c₂ = c₄ = M + h².

The previous results enable us to study the qualitative properties of perturbations of system (7), described by an equation of the form

w(t_k+1) = A_kw(t_k) + F_k(w(t_k)) (10)

where F_k : W_k ® W il is the perturbation term, and W_k are subsets of W such that there exists r > 0 with B_r Ì W _k for all . Next we will address the case of vanishing perturbations (when F_k(0) = 0 for all ) and will establish robust stability results in the sense of Lyapunov of the trivial solution of (10).

Proposition III..2 Consider the discrete-time system described by equation (10) and assume that

(11)

uniformly in . Then, if the origin is an ES equilibrium of (7), it is also an ES equilibrium of (10).

Proof Due to the exponential stability of the origin with respect to (7) for every k ³ k₀ and certain positive numbers h, m (see Remark III..1); it follows that for all . Consider now the scalar function given by Theorem III..1 and pick e < 1 that verifies

Then, due (11) there exists d = d(e) > 0 such that

for all and all w Î B_d. Pick now such an w; then

Then, the function V satisfies the hypotheses of Proposition III..1 with d and instead of r' and c₃ respectively. It follows that the origin is exponentially stable for the perturbed system (10).

Remark III..2 Examples can be exhibited which show that Proposition III..2 fails if condition (11) is replaced by the following weaker one:

(12)

uniformly in .

Nevertheless the converse of Proposition III..2 holds under this hypothesis.

Proposition III..3 Consider the discrete-time system described by equation (10) and assume that it verifies (12). Then, if the origin is an ES equilibrium of (10), it is also an ES equilibrium of (7).

IV. STABILITY OF PERTURBED HYBRID LINEAR SYSTEMS

In this section we apply the results of the previous section to the analysis of the stability properties of the origin of W = ⁿ ´ ^m for the perturbed hybrid linear system S_HLP described by the equations

(13)

where A(×), B₁₁(×) and B₁₂(×) are bounded Lebesgue measurable matrix functions in ⁺, and B₂₁(×) and B₂₂(×) are matrix functions defined in ₀. We assume that the perturbation term g₁ verifies H1 and H2. We will also assume that the perturbations are vanishing, i. e. g₁(t,0,0,0) = 0 and g₂(k,0,0) = 0 for all t Î ⁺ and respectively. Using (13) we compute the discretized perturbed hybrid linear system S_DHLP

(14)

where

and F(t, s) is the transition matrix corresponding to the matrix equation , X(s) = I. Let w(t_k) = (x(t_k)^T, z(t_k)^T)^T; then (14) can be written

(15)
where
and

Let us also consider the unperturbed (i. e. with g₁ º 0 and g₂ º 0) hybrid linear system S_HLU associated with

and its discretization, the discretized unperturbed hybrid linear system S_DULS which coincides with the unperturbed discrete hybrid linear system associated with S_DHLP

Now we may state the main result of this section.

Theorem IV..1 Suppose that g₁ and g₂ in (13) verify in addition H3 and H4, respectively.

Then, the following properties are equivalent:

1. The origin is an ES equilibrium for S_HLP

2. The origin is an ES equilibrium for S_HLU

3. The origin is an ES equilibrium for S_DHLP

4. The origin is an ES equilibrium for S_DHLU

Proof 1. Þ 3. and 2. Þ 4. hold trivially and since 4. Þ 2. is a particular case of 3. Þ 1., then it suffices to prove 3. Þ 1. and 4. Û 3.

First we show that 3. Þ 1. In order to prove it we need a technical lemma. Let w = (x,z) Î W; we denote with f(t,t_k,w) the solution of (13) with initial conditions (t_k,w) and with [t_k,T(k,w)) its maximal interval of definition. We have the following lemma.

Lemma IV..l Asumme that the hypotheses of Theorem IV..1 hold. Then there exist positive constants c and r^* such that for all and all , .

Proof Since g₁ satisfies H3, there exist r > 0 and of class such that |g₁(t,x,x,z)| £ r(|x| + |x| + |z|)(|x| + |x| + |z|) for all and all (x,x,z) Î ⁿ ´ W: |x| + |x| + |z| £ r.

Take m: 0 < m < r and with

and z > 0 such that , and consider . Then, for all with .

Let ; then, since . In consequence and it follows from the continuity of x(×) that with t^** £ t^*.

Consider now , then and in consequence, for all ,

where the last inequality is obtained applying Gronwall's Lemma. It follows that t^** - t^*; (if t^** < t^*, then due to the continuity of x(×), and which is a contradiction). In consequence, for all t Î [t_k,t^*), and all , . Finally we prove that t^* = t_k+1; be this not the case t^* = T(k,w). But f(t,t_k,w) Î B_c for all and, due to standard theorems about ordinary differential equations, t^* < T(k,w) which is a contradiction, and the lemma follows.

Now we prove 3. Þ 1. Suppose then that 0 is an ES equilibrium of S_DHLP and let the solution of (14) with initial conditions . Then there exist positive numbers h^*, m^* and k^* such that ,

Let r^* and c as in Lemma IV..1 and ; hence ,

Consider for w ÎB_k and t such that for . Then, by Lemma IV..1,

and the origin is an ES equilibrium for S_HLP.

Next we prove the statement 3. Û 4. Let with r^* as in Lemma IV..1; first we prove that 5i(fc,£,z) in (14) verifies uniformly in . If x(t) is the solution of (13) with initial conditions (t_k,w), then according to Lemma IV..1, |x(t)| £ d(|x| + |z|) and (|x(t)| + |x| + |z|) < m < r for all t Î [t,t_k+1), for all . Then, from (13) - (14), we obtain

since, with z as in the proof of Lemma IV..1. If we define as , is a function of class and it follows, by the inequalities above, that , and in consequence, uniformly in .

Then, due to the assumption about g₂(k,x,z), F_k(w) in (15) verifies uniformly in . Hence, according to Propositions III..2 and III..3, the origin is ES for S_DHLP if and only if it is ES for S_DHLU.

Remark IV..l Part 1. Û 3. of Theorem IV1 extends the results obtained by Iglesias (1994) for time-varying linear systems in two ways: we consider time-varying sampling periods instead of constant ones and we con­sider time-varying nonlinear equations.

In addition, some sufficient conditions for the uniform asymptotic stability reported in the literature may be readily obtained from this theorem. In fact, Theorems 1 of (Hou et al., 1997), Theorem 1 of (Rui et al., 1997) and Theorem 2.1 of (Hu and Michel, 2000b) are corollaries of Theorem IV1, since the conditions under which their results hold imply the exponential stability of S_DHLU.

Proof of Theorem II..1 A(t), B₁₁(t) and B₁₂(t) in (4) are bounded (by hypothesis), and are also Lebesgue measurable since f₁ in (1) verifies H1 and H2; it follows that g₁ and g₂ also verify H1 and H2. In addition, since (1) can be written in the form (13), the hypotheses of Theorem IV..1 hold for S_H and S_LH instead of S_HLP and S_HLU respectively. In consequence, the theorem follows from Theorem IV..1.

V. CONCLUSIONS

In this work we presented results about the exponential stability of the class of hybrid systems described by time-varying hybrid equations, which comprises the class of sampled-data systems consisting of the interconnection of a time-varying nonlinear continuous time plant and a time-varying nonlinear discrete-time controller, assuming that the sampling periods are not necessarily constant. For this purpose we developed an Indirect Lyapunov Method of analysis, and showed that under adequate hypotheses the exponential stability of the hybrid dynamical system is equivalent to the exponential stability of its linearization.

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