Lyapounov Stability for Impulsive Dynamical Systems

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CDC02-REG0943

Lyapounov Stability for Impulsive Dynamical Systems

Geraldo N. Silva

¹

, and Fernando L. Pereira

²

1

Dept. Computer Science and Statistics, Universidade Estadual Paulista C. P. 136, S. J. Rio Preto-SP, Brasil

gsilva@dcce.ibilce.unesp.br

2

Institute for Systems and Robotics-Porto Faculdade de Engenharia da Universidade do Porto

Rua Dr. Roberto Frias, 4200-465 Porto, Portugal flp@fe.up.pt

Abstract

This article addresses the problem of stability of impulsive control systems whose dynamics are given by measure driven differential inclusions. One important feature concerns the adopted solution which allows the consideration of systems whose singular dynamics do not satisfy the so-called Frobenius condition. After ex- tending the conventional notion of control Lyapounov pair for impulsive systems, some stability conditions of the Lyapounov type are given. After sketching the proof of this result, some conclusions are drawn.

Keywords: Control Lyapounov pair of functions, Sta- bility, Impulsive control, Measure driven differential inclusion.

1 Introduction

In this article, we address stability conditions for class of impulsive control problems, for which the dynamics are defined by a differential inclusion driven by a vector valued control measure. A concept of proper solution is presented which provides a meaning to the stabilization problem in the context defined by the considered assumptions characterizing the data of the problem.

The addressed class of problems arise in a variety of application areas such as finance, impact vibro-

1Research supported by CNPq and FAPESP of Brasil

2Research supported by INVOTAN and Funda¸c˜ao da Ciˆencia e Tecnologia project Cordyal

mechanics, [9], to the management of renewable resources, [10, 5, 20], passing by aerospace navigation, [23], (to name just a few), for which the solution should be found within the set of control processes with trajectories of bounded variation. Naturally, this fueled the relatively recent rapid development of a, by now, con- siderable body of theory for this class of systems (see, for example, [1, 7, 8, 16, 18, 19, 21, 25, 26, 27, 30, 31, 32, 35, 40, 41, 43], and references therein) and supporting control strategies computation schemes [17, 3, 4, 22].

Furthermore, the pervasive availability of computational capabilities associated to technological evolution led to the emergence of control systems paradigms, generically designated by Hybrid Systems, whose evolution is fueled by the combination of (continuum) time driven and by event dynamics, [42, 6, 15, 28]. Al- beit in a modeling context that differs from the one adopted in this article, [2] draws the attention for the fact that the impulsive control framework is suitable to capture important features of hybrid systems. There- fore, besides the motivation naturally associated with the above mentioned range of applications addressed by impulsive control, its relevance in dealing with systems of hybrid nature (in the sense defined above) depends on the type of results and algorithms that can be devel- oped in this framework. This is an additional incentive to pursue the research issues addressed in this article.

There is a vast amount of literature addressing stability problems for conventional control systems, i.e., when dynamic control systems such as ˙x=f(x, u),x(0) =x₀ or, its differential inclusion counterpart, ˙x∈F(x) are considered. A lot of effort have been spent in provid-

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ing conditions which are necessary and or sufficient for the system to satisfy various types of stability. An important role has been played by Control Lyapounov Function (CLF) (see [13] for a definition),V :<ⁿ→ <, which in the control or game paradigms allows the syn- thesis of stabilizing feedback control laws, i.e.,u=k(x) by the so-called Lyapounov design method¹.

Clearly, a number of questions of utmost importance which have been the subject of very active research, (see [14, 38, 37, 39, 11], to name just a few references):

a) Existence of regular Lyapounov functions, b) existence of continuous feedback functions, c) solution concept for ordinary differential equations with discontinuous r.h.s. (due to discontinuous feedback law) and d) robustness to errors, being measurement errors of particular importance for discontinuous feedback.

Brockett condition [33] states that a dynamic system

˙

x=f(x, u) is stabilizable by continuous feedback only if, for every neighborhood N of 0, f(N,Ω) contains the origin. A famous counterexample is the nonholo- nomic integrator which even is globally asymptotically controllable (GAC). In [12] it is shown that a GAC system possesses a, possibly discontinuous, stabilizable feedback controller being the adopted solution concept, x:= lim_N_→∞x^N where x^N satisfies

˙

x^N(t) =f(x^N(t), k(x^N_i )) ∀t∈[t_i, t_i+1), with x^N_i = x^N(t_i), and π_N ={t₀, t₁, ..., t_N} is a par- tition of the time interval defined so that max_i|t_i− t_i−1| →0 asN → ∞, [13].

It is shown in [36] that a GAC system always has a continuous Lyapounov function satisfying

inf_u∈ΩDV(x;f(x, u))≤ −W(x)<0 ∀x6= 0 whereDV(x;f) represents the Dini derivative of V at x in the direction f (for definitions, [13]). In [11], it is shown that a locally Lipschitz Lyapounov function always exists that “practically stabilizes the system”, i.e., provides a feedback driving the system to an arbi- trarily small neighborhood of the equilibrium point in finite time and a converse Lyapounov theorem is shown.

In [29], it is shown that a global locally Lipschitz Lya- pounov function always exists for a GAC system.

Robustness to measurement errors is ensured for suf- ficiently smooth Lyapounov functions. In the nonsmooth case, robustness can only be guaranteed if dynamic feedback is considered, [37].

The construction of a Lyapounov function remains an important issue in control design which has been ad-

1It involves two stages: a) Find a Control Lyapounov Func- tion and b) Select a function u = k(x) satisfying ∇V(x)· f(x, k(x))≤ −W(x),∀x∈ <ⁿ.

dressed when either there is a clear physical interpreta- tion, or the specific structure of the problem can be ex- ploited. In this later category, backstepping and sliding mode are two of the most popular classes of methods.

In spite of the extensive stability literature for conventional control systems and the vast effort in the control of measure driven dynamic systems, little is known in what concerns stability results for this kind of systems.

In [24], the stability problem for impulsive systems is addressed in the context of robustness of the solution with respect to perturbations in the control measure.

Characterizations of necessary and sufficient stability conditions are derived after the introduction of an ap- propriate notion of closure of the set of trajectories.

In the next section, a result on robustness of the solution to the measure differential inclusion adopted in this article, proved in [34], is presented.

In a modeling context substantially distinct of the one considered in this article, [2] provides stability conditions for dynamic systems with discontinuous trajectories in terms of inequalities involving a certain function, designated by Lyapounov function, which is shown to be a contingent solution to Hamilton-Jacobi variational inequalities.

This paper is organized as follows. In Section 2, the problem addressed in this article is formulated in detail.

This includes the presentation of the solution concept, its robustness properties and a discussion on stability issues. Then, in section 3, the extension to the measure driven differential inclusion context of the concept of Control Lyapounov function is presented and used to state stability conditions. In the ensuing section, an outline of the main result is sketched and some conclusions are drawn in the last section.

2 Solution Concept of Measure Differential Inclusion

In this paper, we will derive stability conditions of the Lyapounov type for impulsive dynamic control systems whose dynamics are given by differential inclusions of the form:

(D) dx(t)∈F(t, x(t))dt+G(t, x(t))µ(dt) t∈[0,∞) x(0) =x₀.

Here, F : [0,∞)×Rⁿ ,→ P(Rⁿ), and G : [0,∞)× Rⁿ ,→ P(R^n×q are given set-valued functions and µ∈ C^∗([0,∞);K), the set in the dual space of continuous functions from [0,∞) toR^q with values in K. The set Kis the positive pointed convex cone inR^q.

Now, we consider the following solution concept

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adopted to (D) in [27].

Definition 1. xis a solution to (D), then it evolves due to the contribution of an absolutely continuous component x_ac and a singular component x_s, i.e., x(t) = x_ac(t) +x_s(t)∀t∈[0,1] with,∀t∈[0,∞),





x(0) = x₀

˙

x_ac(t) ∈ F(t, x(t)) +G(t, x(t))·w_ac(t) L-a.e.

x_s(t) = R

[0,t]g_c(τ)w_c(τ)d¯µ_sc(τ) +R

[0,t]g_a(τ)d¯µ_sa(τ) . Here, ¯µ is the total variation measure associated with the vector-valued measureµ, µ_sc,µ_sa andµ_ac are, respectively, the singular continuous, the singular atomic, and the absolutely continuous components ofµ,w_acis the time derivative of µ_ac, w_sc is the Radon-Nicodym derivative ofµ_scwith respect to its total variation,g_c(·) is a ¯µ_sc measurable selection of G(t, x(t)) and g_a(·) is a ¯µ_sa measurable selection of a certain multifunction

G(t, x(t˜ ⁻);µ({t})) : [0,∞)× <ⁿ×K ,→ P(<ⁿ) specifying the reachable set of the singular dynamics at (t, z) when the control measure has an atomic measure of “weight”αat timet.

In order to define this multifunction, the concept of graph completion of a time reparametrized function is required (see [27]).

Definition 2. A family ofgraph completionsassociated with a measureµ∈C^∗(0,1;K) is the set of pairs (θ, γ) : [0,∞) ,→ <⁺× P(K) where θ : [0,∞) → < is the

“inverse” of ¯η(i.e.,θ(s) =t,∀s∈η(t)) and the function¯ γ: ¯η(t)→ <^q takes values

γ(s) :=

½ M(θ(s)) if ¯µ({t}) = 0 M(t⁻) +R_s

η(t⁻)v(σ)dσ if ¯µ({t})>0, wherev(·)∈V^tdefined by

{v: ¯η(t)→ <^q|v(s)∈K,∀s∈η(t),¯ Z

¯

η(t)v(s)ds=µ({t})}

and ¯µ(dt) =P_q

i=1µ_i(dt). Here,

• MR (·) = col(M1(·), ..., Mq(·)), with M_i(t) =

[0,t]µ_i(ds),∀t >0, andM_i(0) = 0;

• η(t) = [η(t¯ ⁻), η(t)] if ¯µ({t}) > 0, and ¯η(t) = {η(t)}, otherwise; and

• η(t) :=t+P_q

i=1M_i(t).

Now, we are in position of defining the set-valued function ˜G(t, z;α) which is given by











{G(t, z)w(t)} if|α|= 0

{[ξ(η(t))−ξ(η(t⁻))]

|α| : ˙ξ(s)∈G(t, ξ(s)) ˙γ(s) ¯η(t)-a.e., ξ(0) =z andγ(η(t))−γ(η(t⁻))) =α}

if|α|>0,

where |α| = P_q

i=1αⁱ, w(·) is the Radon-Nicodym derivative ofµw.r.t. ¯µ, (ξ, γ) are inAC([0,1];<ⁿ×<^q), and the pair (θ, γ) is a graph completionwith ˙θ(s)≡0 on ¯η(t).

The following result, proved in [34], reveals the robustness of the solution defined as above.

Proposition 1 Consider multifunctions F and G with domain [0,+∞)×<ⁿ, taking as values compact subsets of<ⁿ and<^n×q respectively, and satisfying:

(i) F(t,·) andG(·,·) have closed graphs,

(ii) F is Lebesgue×Borel measurable andGis Borel measurable, and

(iii) F(t, x) andG(t, x) are convex valued for all (t, x).

Consider T > 0 and take a sequence {xⁱ₀} in <ⁿ and a sequence {µ_i} in C^∗([0, T];K), and elements x₀ ∈ <ⁿ and µ ∈ C^∗([0, T];K) such that, as i → ∞, xⁱ₀ → x₀ and µ_i →^∗ µ. Take also a sequence {xi} inBV⁺([0, T];<ⁿ) such that x_i(·) is a robust solution to (D) with µ_i in place ofµ. Consider the following inclusion

y(s)∈˙ F(θ(s), y(s)) ˙θ(s) +G(θ(s), y(s)) ˙γ(s) (1) almost everywhere in [0, T].

For eachi, assume the existence ofβ(t)∈L¹andc >0 such thatF(t, x_i(t))⊂β(t)Ba.e. andG(t, x_i(t))⊂cB for allt.

Then, there exist: (a) a sequence of processes (y_i, θ_i, γ_i), solution to (1) withy_i(0) =xⁱ₀, (b) (y, θ, γ), solution to (1), withy(0) =x₀, and (c) a solutionxto (D), such that

x_i(t) =y_i(η_i(t))∀t∈(0,1] andx(t) =y(η(t))∀t∈(0,1].

Along a subsequence, we havedx_i →^∗dxand x_i(t)→ x(t) for allt∈¡

[0, T]\Mµ¢

∪{0, T}(whereMµdenotes the atoms ofµ) andy_i→y strongly inC([0,1];<ⁿ).

We notice that this solution concept adheres to important classes of engineering problems involving the coordinated control of multiple dynamic systems. Ba- sically, the idea is to control a dynamical system with several viable configurations in such a way that a given performance criterion is maximized during the execu- tion of the given activities. Although transitions between different configurations (which, in practice, are not instantaneous but, ideally, might be considered as such) correspond to non productive phases, the way they evolve might affect the overall performance of the

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system and, therefore, it is of interest to incorporate their management in the global optimization problem.

This can be regarded as choosing the best path between the jump endpoints.

3 Stability Conditions

In analogous way to Lyapounov stability theory for conventional control systems (see, for example [13]), we will consider x^∗ to be an equilibrium point for the impulsive dynamic system (D).

Definition 3. Functions (V, W) form a Lyapounov pair for (D) at x^∗ if they satisfy the following properties:

1) Positive definiteness: V ≥ 0, W ≥ 0 with W(x) = 0 iffx=x^∗.

2) The set {x ∈ <ⁿ|V(x) ≤ r} is compact for all r≥0.

3) For allz∈ <ⁿ, for allξ∈∂_pV(z) we have Min{< ξ, ν >:ν∈F(t, z)v₀+G(t, z)v,

(v₀, v)∈V¯} ≤ −W(z).

Here, ¯V :={(v₀, v) :v₀≥0, v∈K,P_q

i=0v_i = 1} and

∂_PV(z) denotes the set of proximal gradients of V at z.

The third item requires some explanation. Two distinct situations may arise in the minimization process:

The control measure is either absolutely continuous or singular with respect to the Lebesgue measure. In the first case, we havev₀>0 andz =x(t), and condition 3) coincides with the conventional one. However, in the second situation, v₀ = 0 and this condition has to considered along the graph completion of the trajectory between the jump endpoints, that is,z= ˜x(s) for s∈η(t).¯

SinceV(x(t⁺))−V(x(t⁻)) =R

¯

η(t)< ξ(s), ζ(s)w(s)> ds for some selections

• w(s) ofK, withR

¯

η(t)w(s)ds=µ({t}),

• ξ(s) of∂_PV(˜x(s)), and

• ζ(s) of G(t,x(s)), with ˜˜ x satisfying ˜x(η(t⁻)) = x(t⁻) and ˙˜x(s) =ζ(s)w(s).

A somewhat weaker expression of the decreasing condition in terms of the original parametrization is given by

−w¯ ≥ Min{Min{V(x(t⁺)−V(x(t⁻)), < ξ, ν >}: ν ∈F(t, z)v₀+G(t, z)v,(v₀, v)∈V ,¯

x(t⁺)∈ RG(η(t);η(t⁻), x(t⁻))}.

where ¯w= Max{R

¯

η(t)W(˜x(s))ds, W(x(t))}.

Here, ¯V is as above andRG(s_f;s_i, z) denotes the reachable set at the parameter²values_f of the singular dynamics when the trajectory path was passing through zat the parameter values_i.

We first set up our hypotheses on the set-valued maps F andG, which will remain in force for the rest of this work. Furthermore, in order to simplify the arguments, we will assume, from now on, thatF and depend on the state variable only.

(h1) For every x∈ <ⁿ, F(x) andG(x) are nonempty compact convex sets;

(h2) F andGare upper semicontinuous;

(h3) there exist constants a and b such that for all x∈ <ⁿ,

v ∈ F(x) =⇒ kvk ≤akxk+b; (2) v ∈ G(x) =⇒ kvk ≤akxk+b. (3) We recall thatF is upper semicontinuous on x if, given any² >0, there existsδ >0 such that

ky−xk< δ=⇒F(y)⊂F(x) +²B

whereB stands for the unit ball of<ⁿ. The condition (h3) of the hypotheses is known as the linear growth condition and is a natural generalization of the case whereF is a function (single valued).

Now, we are prepared to state the main result of this paper.

Theorem 1 Letx¯∈ <ⁿ and0∈F˜(¯x). Suppose that 0∈ {G(x)v/ : (0, v)∈V¯} (4) and that there exist a pair of Lyapounov functions V and W for the equilibrium point ¯x which are lower semi-continuous. Then for any x₀ ∈ <ⁿ there exists a process(x, µ)of(F,G)in[0,∞)such thatx(0) =x₀ andx(t)→x, when¯ t→ ∞.

4 Outline of the Proof

Proof: Let ¯x∈ <ⁿ and (V, W) be a Lyapounov pair for ¯x. It follows from ([13], Theorem 4.5.5) that for any x₀ ∈ <ⁿ there exists a standard trajectory y(·) of ˜F having y(0) = x₀ such that y(s) → x¯ where s

2according to the above parametrization, this parameter is the total variation of the control measure plus the current value of time.

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converges to infinity. Now, it follows by construction of the trajectory y as the minimal selection of ˜F and the definition of ˜F itself that

˙

y(s)∈ {f(s)v₀+ ¯G(s)v:v∈V¯},

for some f(s)∈F(y(s)) and ¯G(s)∈G(y(s)) a.e. s∈ [0,∞).

By a “standard” measurable selection theorem, there exists a measurable function (v₀, v₁, . . . , v_q)(s)∈V¯ a.e.

in [0,∞), such that

˙

y(s) =f(s)v₀(s) + Xq i=1

g_i(s)v_i(s). (5) We observe that, although the measurable selection theorem, as it is known, can only be applied on bounded intervals, we can consider a sequence of its application in the intervals [i, i+ 1] for i = 0,1, . . . , getting, by the principle of induction, the result on the whole unbounded interval [0,∞) =∪^∞_i=0[i, i+ 1].

Note that because of (4) the function v₀ : [0,∞) → [0,1] does not vanish in unbounded intervals.

We now sketch the construction of the process (x, µ) such thatx∈BV⁺ is the solution of differential inclusion (D), corresponding toµthat satisfies the required properties of the theorem.

We start by constructing the control measureµ. Define θ(s) :=

Z _s

0 v₀(σ)dσ, η(t) := sup{s:θ(s)≤t}; (6)

µ(A) :=

Z

θ⁻¹(A)v(s)ds ∀A∈ B([0,∞)), (7) wherev(s) := (v₁(s), . . . , v_q(s)).

As functionv₀(s) is nonnegative and does not vanish in unbounded intervals, function θ defined in (6) is non- decreasing and its image is the whole interval [0,∞).

This implies that µ, defined by (7), is a regular Borel measure with domain in the Borel σ−algebra of the interval [0,∞).

It follows easily that η is an increasing function. Let {t_i} be a sequence of points in [0,∞) where the in- tervalsθ⁻¹(t_i) are non-degenerated. That means that each t_i is an atom of the measure µ, by construction (see, (6) e (7)). We want to show that there exists ˜f(t) ∈ F(x(t)) a.e. and ˜g(t) ∈ G(x(t˜ ⁻);µ({t}))

|µ| −a.e.such that x(t) =x₀+

Z _t

0

f˜(σ)dσ+ Z

[0,t]

˜

g(σ)|µ|(dσ)

for allt∈[0,∞). Set f˜(t) := f(η(t)); and

˜ g(t) :=

½G(η(t))¯ ·v(η(t)) ∀t∈[0,∞)\ ∪^∞_i=1{t_i}

η(t)−η(t−)1

R

¯

η(t)G(s)¯ ·v(s)ds ∀t∈ ∪^∞_i=1{t_i}.

Note that x(θ(s)) =y(s) em [0,∞)\ ∪i{θ⁻¹(t_i)} and θ(s) = 0 in˙ ∪i{θ⁻¹(t_i)}. Then we have

0 =

Z

[0,∞)χ_F(y(s))( ˜f(θ(s))) ˙θ(s)ds

= Z

[0,∞)

χ_F_(x(t)))( ˜f(t))dt.

This implies that ˜f(t) ∈F(x(t)) a.e. with respect to the Lebesgue measure.

It follows readily from the definition of ˜gthat

˜

g(t)∈G(x(t˜ ⁻);µ({t})) ∀t∈ ∪i{ti}. (8) We also have

0 = Z

[0,∞)\∪i{θ⁻¹(ti)}χ_G(x(θ(s)))( ¯G(θ(s)))·v(s)ds

= Z

[0,∞)\∪i{(ti)}χ_G(x(t)))( ¯G(η(t)))µ(dt).

Hence ¯G(η(t))∈G(x(t)) a.e. in [0,∞) with respect to the measureµ.

Since ˜g(t) := ¯G(η(t))·v(η(t)) ∀t ∈[0,∞)\ ∪i{(ti)}, it follows from the construction of µ that µ(dt) = v(η(t))|µ|(dt), i.e., w(t) = v(η(t)) is the Radon- Nicodym derivative ofµwith respect to the total variation measure|µ|. Thus, we have

˜

g(t)∈G(x(t˜ ⁻);µ({t})), |µ|−a.e. em [0,∞)\∪_i{(t_i)}, which together with (8) implies that

˜

g(t)∈G(x(t˜ ⁻);µ({t})) |µ| −a.e. em [0,∞).

Taket >0. Then

x(t)−x₀ = y(η(t))−x₀

= Z

[0,η(t)]f(s)v₀(s)ds+ Z

[0,η(t)]

G(s)v(s)ds¯

= Z

[0,η(t)]

f˜(θ(s))v₀(s)ds+ Z

[0,η(t)]\∪iθ⁻¹(ti)

G(s)v(s)ds¯ +

Z

[0,η(t)]∩(∪iθ⁻¹(ti))

G(s)v(s)ds¯

= Z

[0,t]

f(τ˜ )dτ + Z

[0,t]˜g(τ)|µ|(dτ).

The last inequality above follows from change of vari- ables. This confirms that (x, µ) is indeed an admissible process of (D).

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It remains to show thatx(t)↑ ∞whent↑ ∞. Observe that ifttends to infinity thenη(t), defined by (6), tends also to infinity. Sincex(t) =y(η(t)) andy(s)→x¯when s↑ ∞it follows thatx(t)→x.¯

5 Conclusions

In this work, we explain how to extend the notions of an equilibrium point and the global asymptotic stability of such a point from standard differential inclusions to measure differential inclusions. We also introduce an extension of the control Lyapounov pair of functions from standard control systems to impulsive systems. We prove that the existence of an extended pair of control Lyapounov pair of functions for an equilibrium point implies that this point is globally asymptotically stable. The proof of the sufficient conditions for asymptotic stability provided here is a consequence of a similar result in the conventional case. The main difficulty in the proof arises in the construction of a regular measure on the unbounded interval [0,∞), which gives rise to the trajectory of the impulsive differential inclusion.

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