Finite-time stabilization of nonlinear impulsive dynamical ... · Finite-time stabilization of nonlinear impulsive dynamical systems ... events are defined by a prescribed sequence

Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845www.elsevier.com/locate/nahs

Finite-time stabilization of nonlinear impulsive dynamical systemsI

Sergey G. Nersesova,∗, Wassim M. Haddadb,1

a Department of Mechanical Engineering, Villanova University, Villanova, PA 19085-1681, United Statesb School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, United States

Received 11 June 2007; accepted 11 December 2007

Abstract

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finitetime. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time,and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to resetthe system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics.In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar andvector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems thatare robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scaleimpulsive dynamical systems.c© 2008 Elsevier Ltd. All rights reserved.

Keywords: Finite-time stability; Finite-time convergence; Impulsive systems; Zeno solutions; Non-Lipschitzian dynamics; Finite-time stabilization;Control Lyapunov functions

1. Introduction

The mathematical descriptions of many hybrid dynamical systems can be characterized by impulsive differentialequations [1–6]. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist ofthree elements—namely, a continuous-time differential equation, which governs the motion of the dynamicalsystem between impulsive or resetting events; a difference equation, which governs the way the system states areinstantaneously changed when a resetting event occurs; and a criterion for determining when the states of the systemare to be reset. Since impulsive systems can involve impulses at variable times, they are in general time-varyingsystems, wherein the resetting events are both a function of time and the system’s state. In the case where the resettingevents are defined by a prescribed sequence of times which are independent of the system state, the equations areknown as time-dependent differential equations [1,2,4,7–9]. Alternatively, in the case where the resetting events are

I This research was supported in part by the Air Force Office of Scientific Research under Grant F49620-03-1-0178 and by the Naval SurfaceWarfare Centre under Contract N65540-05-C-0028.

∗ Corresponding author. Tel.: +1 (610) 519 8977; fax: +1 (610) 519 7312.E-mail addresses: [email protected] (S.G. Nersesov), [email protected] (W.M. Haddad).

1 Tel.: +1 (404) 894 1078; fax: +1 (404) 894 2760.

1751-570X/$ - see front matter c© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.nahs.2007.12.001

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845 833

defined by a manifold in the state space that is independent of time, the equations are autonomous and are known asstate-dependent differential equations [1,2,4,7–9].

Finite-time stability implies Lyapunov stability and convergence of system trajectories to an equilibrium state infinite-time, and hence, is a stronger notion than asymptotic stability. For continuous-time dynamical systems, finite-time stability implies non-Lipschitzian dynamics [10,11] giving rise to nonuniqueness of solutions in reverse time.Uniqueness of solutions in forward time, however, can be preserved in the case of finite-time convergence. Sufficientconditions that ensure uniqueness of solutions in forward time in the absence of Lipschitz continuity are given in[12–15]. Finite-time convergence to a Lyapunov stable equilibrium for continuous-time systems, that is, finite-timestability, was rigorously studied in [11,16] using Holder continuous Lyapunov functions.

Finite-time stability of impulsive dynamical systems has not been studied in the literature. For impulsive dynamicalsystems, it may be possible to reset the system states to an equilibrium state, in which case finite-time convergenceof the system trajectories can be achieved without the requirement of non-Lipschitzian dynamics. In addition, due tosystem resettings, impulsive dynamical systems may exhibit nonuniqueness of solutions in reverse time even whenthe continuous-time dynamics are Lipschitz continuous.

In this paper, we develop sufficient conditions for finite-time stability of nonlinear impulsive dynamical systems.Furthermore, we present stability results using vector Lyapunov functions wherein finite-time stability of the impulsivesystem is guaranteed via finite-time stability of a hybrid vector comparison system. We use these results to constructhybrid finite-time stabilizing controllers for impulsive dynamical systems. In addition, we construct decentralizedfinite-time stabilizers for large-scale impulsive dynamical systems. Finally, we present a numerical example to showthe utility of the proposed framework.

2. Mathematical preliminaries

In this section, we introduce notation and definitions needed for developing finite-time stability analysis andsynthesis results for nonlinear impulsive dynamical systems. Let R denote the set of real numbers, Z+ denote theset of nonnegative integers, Rn denote the set of n ×1 column vectors, and (·)T denote transpose. For v ∈ Rq we writev ≥≥ 0 (respectively, v � 0) to indicate that every component of v is nonnegative (respectively, positive). In this case,we say that v is nonnegative or positive, respectively. Let Rq

+ and Rq+ denote the nonnegative and positive orthants of

Rq , that is, if v ∈ Rq , then v ∈ Rq+ and v ∈ Rq

+ are equivalent, respectively, to v ≥≥ 0 and v � 0. Furthermore,

let◦

D, D, and ∂D denote the interior, the closure, and the boundary of the set D ⊂ Rn , respectively. Finally, we write‖ · ‖ for an arbitrary spatial vector norm in Rn , V ′(x) for the Frechet derivative of V at x , Bε(α), α ∈ Rn , ε > 0, forthe open ball centered at α with radius ε, e ∈ Rq for the ones vector given by e , [1, . . . , 1]

T, AĎ for Moore–Penrosegeneralized inverse of A ∈ Rn×m [17], and x(t) → M as t → ∞ to denote that x(t) approaches the set M, that is,for every ε > 0 there exists T > 0 such that dist (x(t),M) < ε for all t > T , where dist (p,M) , infx∈M ‖p − x‖.

The following definition introduces the notion of classWc functions involving quasimonotone increasing functions.

Definition 2.1 ([18]). A function w = [w1, . . . , wq ]T

: Rq→ Rq is of class Wc if wi (z′) ≤ wi (z′′), i = 1, . . . , q,

for all z′, z′′∈ Rq such that z′

j ≤ z′′

j , z′

i = z′′

i , j = 1, . . . , q, i 6= j , where zi denotes the i th component of z.

If w(·) ∈Wc, then we say that w satisfies the Kamke condition [19,20]. Note that if w(z) = W z, where W ∈ Rq×q ,then the function w(·) is of classWc if and only if W is essentially nonnegative, that is, all the off-diagonal entries ofthe matrix W are nonnegative. Furthermore, note that it follows from Definition 2.1 that any scalar (q = 1) functionw(z) is of classWc.

Finally, we introduce the notion of classWd functions involving nondecreasing functions.

Definition 2.2 ([21]). A function w = [w1, . . . , wq ]T

: Rq→ Rq is of class Wd if w(z′) ≤≤ w(z′′) for all

z′, z′′∈ Rq such that z′

≤≤ z′′.

If w(z) = W z, where W ∈ Rq×q , then the function w(·) is of classWd if and only if W is nonnegative, that is, allthe entries of the matrix W are nonnegative. Note that if w(·) ∈ Wd, then w(·) ∈ Wc, however, the converse is notnecessarily true.

834 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845

3. Finite-time stability of impulsive dynamical systems

Consider the nonlinear state-dependent impulsive dynamical system G [6] given by

x(t) = fc(x(t)), x(0) = x0, x(t) 6∈ Z, t ∈ Ix0 , (1)

∆x(t) = fd(x(t)), x(t) ∈ Z. (2)

where x(t) ∈ D ⊆ Rn , t ∈ Ix0 , is the system state vector, Ix0 is the maximal interval of existence of a solutionx(t) to (1) and (2), D is an open set, 0 ∈ D, fc : D → Rn is continuous and satisfies fc(0) = 0, fd : D → Rn iscontinuous, ∆x(t) , x(t+) − x(t), x(t+) , x(t) + fd(x(t)) = limε→0 x(t + ε), x(t) ∈ Z , and Z ⊂ D ⊆ Rn isthe resetting set. A function x : Ix0 → D is a solution to the impulsive dynamical system (1) and (2) on the intervalIx0 ⊆ R with initial condition x(0) = x0 if x(·) is left-continuous and x(t) satisfies (1) and (2) for all t ∈ Ix0 . For aparticular trajectory x(t), t ≥ 0, we let tk = τk(x0), x0 ∈ D, denote the kth instant of time at which x(t) intersectsZ and we let x+

k , x(τ+

k (x0)) , x(τk(x0)) + fd(x(τk(x0))) denote the state of (1) and (2) after the kth resetting. Toensure well-posedness of the resetting times we make the following assumptions [6]:

A1. If x(t) ∈ Z \ Z , then there exists ε > 0 such that, for all 0 < δ < ε, x(t + δ) 6∈ Z .A2. If x ∈ Z , then x + fd(x) 6∈ Z .

Assumption A1 ensures that if a trajectory reaches the closure of Z at a point that does not belong to Z , then thetrajectory must be directed away fromZ; that is, a trajectory cannot enterZ through a point that belongs to the closureof Z but not to Z . Furthermore, A2 ensures that when a trajectory intersects the resetting set Z , it instantaneouslyexits Z . It was shown in [6] that assumptions A1 and A2 ensure that the resetting times are well defined and distinct.Furthermore, note that if x0 ∈ Z , then the system initially resets to x+

0 = x0 + fd(x0) 6∈ Z , which serves as the initialcondition for continuous-time dynamics (1). Finally, we note that if the origin is the equilibrium point of (1) and (2),then it follows from A2 that 0 6∈ Z . For details, see [6].

We assume that (1) possesses unique solutions in forward time for all initial conditions in D except possiblythe origin in the following sense. For every x ∈ D \ {0} there exists τx > 0 such that, if y1 : [0, τ1) → D andy2 : [0, τ2) → D are two solutions of (1) with y1(0) = y2(0) = x , then τx ≤ min{τ1, τ2} and y1(t) = y2(t) for allt ∈ [0, τx ). Without loss of generality, we assume that for each x ∈ D, τx is chosen to be the largest such number inR+. Sufficient conditions for forward uniqueness of solutions to continuous-time dynamical systems in the absenceof Lipschitz continuity of the system dynamics can be found in [12], [13, Sect. 10], [14], and [15, Sect. 1].

Since the resetting times are well defined and distinct, and since the solution to (1) exists and is unique, it followsthat the solution of the impulsive dynamical system (1) and (2) also exists and is unique over a forward time interval.However, it is important to note that the analysis of impulsive dynamical systems can be quite involved. In particular,such systems can exhibit Zenoness and beating, as well as confluence, wherein solutions exhibit infinitely manyresettings in a finite time, encounter the same resetting surface a finite or infinite number of times in zero time, andcoincide after a certain point in time. In this paper we allow for the possibility of confluence and Zeno solutions;however, A2 precludes the possibility of beating. Furthermore, since not every bounded solution of an impulsivedynamical system over a forward time interval can be extended to infinity due to Zeno solutions, we assume thatexistence and uniqueness of solutions are satisfied in forward time. For details, see [1,2,4]. Finally, we denote thetrajectory or solution curve of (1) and (2) satisfying x(0) = x by s(·, x) or sx (·).

The following definition introduces the notion of finite-time stability for impulsive dynamical systems.

Definition 3.1. Consider the nonlinear impulsive dynamical system G given by (1) and (2). The zero solution x(t) ≡ 0to (1) and (2) is finite-time stable if there exist an open neighbourhood N ⊆ D of the origin and a functionT : N \ {0} → (0, ∞), called the settling-time function, such that the following statements hold:

(i) Finite-time convergence. For every x ∈ N \{0}, sx (t) is defined on [0, T (x)), sx (t) ∈ N \{0} for all t ∈ [0, T (x)),and limt→T (x) s(x, t) = 0.

(ii) Lyapunov stability. For every ε > 0 there exists δ > 0 such that Bδ(0) ⊂ N and for every x ∈ Bδ(0) \ {0},s(t, x) ∈ Bε(0) for all t ∈ [0, T (x)).

The zero solution x(t) ≡ 0 to (1) and (2) is globally finite-time stable if it is finite-time stable with N = D = Rn .

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845 835

Note that if the zero solution x(t) ≡ 0 to (1) and (2) is finite-time stable, then it is asymptotically stable, andhence, finite-time stability is a stronger notion than asymptotic stability. In the case of impulsive dynamical systemsit may be possible to reset the states to the origin, and hence, s(t, x0) = 0, t > τk(x0) = T (x0). The following resultprovides sufficient conditions for finite-time stability of impulsive systems using a Lyapunov function involving ascalar differential inequality.

Theorem 3.1. Consider the nonlinear impulsive dynamical system G given by (1) and (2). Assume there exists acontinuously differentiable function V : D → R+ satisfying V (0) = 0, V (x) > 0, x ∈ D, x 6= 0, and

V ′(x) fc(x) ≤ −c(V (x))α, x 6∈ Z, (3)

V (x + fd(x)) ≤ V (x), x ∈ Z, (4)

where c > 0 and α ∈ (0, 1). Then the zero solution x(t) ≡ 0 to (1) and (2) is finite-time stable. If, in addition,D = Rn

and V (·) is radially unbounded, then the zero solution x(t) ≡ 0 to (1) and (2) is globally finite-time stable.

Proof. Note that it follows from Theorem 2.1 of [6] that the zero solution to (1) and (2) is asymptotically stable. Thus,it remains to be shown that for all initial conditions in some neighborhoodN ⊆ D of the origin the trajectories of (1)and (2) converge to the origin in finite time. Since the system (1) and (2) is asymptotically stable, it follows that thereexists δ > 0 such that for all x0 ∈ Bδ(0) ⊂ D the trajectory s(t, x0) → 0 as t → ∞. Next, we separately consider thecases when the trajectories of (1) and (2) have a finite and infinite number of resettings.

Assume that for some x0 ∈ Bδ(0) the trajectory s(t, x0), t ≥ 0, exhibits a finite number of resettings withresetting times τk(x0), k = 1, . . . , m. If s(τ+

m (x0), x0) = 0, then since fc(0) = 0 it follows that s(t, x0) = 0,t ≥ τm(x0), which implies that s(t, x0), t ≥ t0, converges to the origin in finite time with a settling-time functionT (x0) = τm(x0). Alternatively, if s(τ+

m (x0), x0) 6= 0, then for all t > τm(x0) the continuous time dynamics are activeand it follows from (3) and Theorem 4.2 of [11] that the trajectory s(t, s(τ+

m (x0), x0)), t ≥ 0, converges to the originin finite time given by 1

c(1−α)[V (s(τ+

m (x0), x0))]1−α . In this case, the settling-time function for s(t, x0), t ≥ 0, is

T (x0) ≤ τm(x0) +1

c(1−α)[V (s(τ+

m (x0), x0))]1−α .

Alternatively, assume that for some x0 ∈ Bδ(0) the trajectory s(t, x0), t ≥ 0, exhibits an infinite number ofresettings with the resetting times τk(x0), k = 0, 1, . . . , where τ0(x0) , 0. Let x+

k , s(τ+

k (x0), x0), k = 0, 1, . . . ,

where x+

0 , x0, and note that since (1) and (2) is asymptotically stable it follows that τ1(x+

k ) → 0 as k → ∞.It was shown in [11] that with (3), the continuous-time dynamics are finite-time stable for the case when Z = Ø.Furthermore, note that τ1(x+

k ) < Tc(x+

k ), k = 0, 1, . . . , since (1) and (2) exhibit an infinite number of resettings,where Tc(·) is the settling-time function when Z = Ø. Moreover, as shown in [11],

V (s(t, y)) ≤ [(V (y))1−α− c(1 − α)t]

11−α , t ∈ [0, Tc(y)), y ∈ Bδ(0) (5)

and hence, since τ1(x0) < Tc(x0), it follows that

V (x1) ≤ [(V (x0))1−α

− c(1 − α)τ1(x0)]1

1−α . (6)

Thus, since V (x + fd(x)) ≤ V (x), x ∈ Z , it follows from (6) that

τ1(x+

1 ) < Tc(x+

1 )

≤1

c(1 − α)(V (x+

1 ))1−α

≤1

c(1 − α)(V (x1))

1−α

≤1

c(1 − α)[(V (x0))

1−α− c(1 − α)τ1(x0)]. (7)

Similarly, it follows from (5) that, for y = x+

2 ,

τ1(x+

2 ) < Tc(x+

2 )

≤1

c(1 − α)(V (x+

2 ))1−α

836 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845

≤1

c(1 − α)(V (x2))

1−α

≤1

c(1 − α)[(V (x+

1 ))1−α− c(1 − α)τ1(x+

1 )]

≤1

c(1 − α)[(V (x1))

1−α− c(1 − α)τ1(x+

1 )]

≤1

c(1 − α)[(V (x0))

1−α− c(1 − α)τ1(x0) − c(1 − α)τ1(x+

1 )]. (8)

Recursively repeating this procedure for k = 3, 4, . . . , it follows that, with τ1(x+

0 ) = τ1(x0),

τ1(x+

k ) <1

c(1 − α)[(V (x0))

1−α− c(1 − α)

k−1∑i=0

τ1(x+

i )]. (9)

Next, let k → ∞ and note that since x(t) ≡ 0 is asymptotically stable, it follows that limk→∞ τ1(x+

k ) = 0. Hence,

0 = limk→∞

τ1(x+

k ) <1

c(1 − α)[(V (x0))

1−α− c(1 − α)

∞∑i=0

τ1(x+

i )]. (10)

Thus,

T (x0) =

∞∑i=0

τ1(x+

i ) <1

c(1 − α)(V (x0))

1−α < ∞, (11)

which implies that the trajectory s(·, x0) is Zeno [6] and converges to the origin in finite time with an infinite numberof resettings, that is, s(t, x0) → 0 as t → T (x0).

Finally, suppose, ad absurdum, that s(t ′, x0) 6= 0 for some t ′ > T (x0), x0 ∈ Bδ(0). Then, since V (·) is positivedefinite, V (s(t ′, x0)) = β > 0. Furthermore, since s(t, x0) → 0 as t → T (x0), there exists t ′′ < T (x0) such thatV (s(t ′′, x0)) < β. Now, since V (s(t, x0)) is a decreasing function of time, it follows that for t ′′ < T (x0) < t ′,

β = V (s(t ′, x0)) < V (s(t ′′, x0)) < β, (12)

which leads to a contradiction. Hence, s(t, x0) = 0, t ≥ T (x0), x0 ∈ Bδ(0), which implies convergence in finite timewith N , Bδ(0). This completes the proof of finite-time stability.

Finally, if D = Rn and V (·) is radially unbounded, then global finite-time stability follows using standardarguments. See, for instance, [6]. �

Remark 3.1. Conditions (3) and (4) are only sufficient conditions for guaranteeing finite-time stability of impulsivedynamical systems. Alternatively, finite-time stability can also be achieved by imposing additional conditions on thediscrete-time dynamics. For example, if for some x0 ∈ D, x(tk) ∈ Z ∩ {x ∈ D : x − fd(x) = 0}, then the trajectoryx(·) resets to the origin and, since fc(0) = 0, finite-time convergence is achieved. A convergent Zeno solution is yetanother example of finite-time convergence to the equilibrium point.

The next theorem generalizes Theorem 3.1 to the case of vector Lyapunov functions involving a vector differentialinequality.

Theorem 3.2. Consider the nonlinear impulsive dynamical system given by (1) and (2). Assume there exist acontinuously differentiable vector function V : D → Q ∩ Rq

+, where Q ⊂ Rq and 0 ∈ Q, continuous functionswc : Q → Rq and wd : Q → Rq , and a positive vector p ∈ Rq

+ such that V (0) = 0, wc(·) ∈ Wc, wd(·) ∈ Wd,wc(0) = 0, wd(0) = 0, the scalar function pTV (x), x ∈ D, is positive definite, and

V ′(x) fc(x) ≤≤ wc(V (x)), x 6∈ Z, (13)

V (x + fd(x)) ≤≤ V (x) + wd(V (x)), x ∈ Z. (14)

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845 837

In addition, assume that the hybrid vector comparison system

z(t) = wc(z(t)), z(0) = 0, x(t) 6∈ Z, (15)

∆z(t) = wd(z(t)), x(t) ∈ Z, (16)

has a unique solution z(t), t ≥ 0, in forward time, and there exist a continuously differentiable function v : Q → R,real numbers c > 0 and α ∈ (0, 1), and a neighbourhood M ⊆ Q of the origin such that v(·) is positive definite and

v′(z)wc(z) ≤ −c(v(z))α, z ∈M, (17)

v(z + wd(z)) ≤ v(z), z ∈M. (18)

Then the zero solution x(t) ≡ 0 to (1) and (2) is finite-time stable. If, in addition,D = Rn , v(·) is radially unbounded,and (17) and (18) hold on Rq , then the zero solution x(t) ≡ 0 to (1) and (2) is globally finite-time stable.

Proof. First, note that since the dynamics of the impulsive dynamical system (1) and (2), and the impulsivecomparison system (15) and (16) are decoupled, the comparison system (15) and (16) is a time-dependent impulsivedynamical system [6]; that is, the resetting times for the trajectories of (15) and (16) are predetermined by thetrajectories of (1) and (2). Hence, the stability analysis for the comparison system (15) and (16) involves a time-dependent impulsive dynamical system [6]. Since v(·) is positive definite, there exist r > 0 and classK functions [22]α, β : [0, r ] → R+ such that Br (0) ⊂M and

α(‖z‖) ≤ v(z) ≤ β(‖z‖), z ∈ Br (0), (19)

and hence,

v′(z)wc(z) ≤ −c(v(z))α ≤ −c(α(‖z‖))α, z ∈ Br (0). (20)

Thus, it follows from (18)–(20), and Theorem 2.6 of [6] that the zero solution z(t) ≡ 0 to the impulsive comparisonsystem (15) and (16) is uniformly asymptotically stable. Furthermore, using identical arguments as in the proofof Theorem 3.1, it can be shown that the trajectories of (15) and (16) converge to the origin in finite time for allz0 ∈ Br (0), and hence, the zero solution z(t) ≡ 0 to (15) and (16) is finite-time stable. Moreover, it follows from (13)and (14), the asymptotic stability of the comparison system (15) and (16), and Theorem 2.11 of [6] that the nonlinearimpulsive dynamical system G given by (1) and (2) is asymptotically stable. Hence, it remains to be shown that thereexists a neighbourhood N ⊆ D of the origin such that the trajectories of (1) and (2) converge to the origin in finitetime for all x0 ∈ N .

Since V (·) is continuous there exists δ > 0 such that V (x0) ∈ Br (0) for all x0 ∈ Bδ(0). Let x0 ∈ Bδ(0) andz0 = V (x0) ∈ Br (0). It follows from (13) and (14), and the hybrid comparison principle [6, Theorem 2.10] that

0 ≤≤ V (x(t)) ≤≤ z(t), t ∈ [0, τ ], (21)

where x(t), t ≥ t0, is the solution to (1) and (2) with the initial condition x0 ∈ Bδ(0), z(t), t ≥ t0, is the solutionto (15) and (16) with the initial condition z0 = V (x0) ∈ Br (0), and [0, τ ] is any arbitrarily large compact time interval.Now, forming pT (21) yields

0 ≤ pTV (x(t)) ≤ pTz(t), t ∈ [0, τ ]. (22)

Since z(·) converges to the origin in finite time and pTV (·) is positive definite it follows that x(t), t ≥ t0, converges tothe origin in finite time. Hence, the nonlinear impulsive dynamical system G given by (1) and (2) is finite-time stablewith N , Bδ(0).

Finally, if D = Rn , v(·) is radially unbounded, and (17) and (18) hold on Rq , then global finite-time stabilityfollows using standard arguments. �

The next result is a specialization of Theorem 3.2 to the case where the structure of the comparison dynamicsdirectly guarantees finite-time stability of the impulsive comparison system. That is, there is no need to require theexistence of a scalar function v(·) such that (17) and (18) hold in order to guarantee finite-time stability of the nonlinearimpulsive dynamical system (1) and (2).

838 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845

Corollary 3.1. Consider the nonlinear impulsive dynamical system given by (1) and (2). Assume there exist acontinuously differentiable vector function V : D → Q ∩ Rq

+, where Q ⊂ Rq and 0 ∈ Q, and a positive vectorp ∈ Rq

+ such that V (0) = 0, the scalar function pTV (x), x ∈ D, is positive definite, and

V ′(x) f (x) ≤≤ W (V (x))[α], x 6∈ Z, (23)

V (x + fd(x)) ≤≤ V (x), x ∈ Z, (24)

where α ∈ (0, 1), W ∈ Rq×q is essentially nonnegative and Hurwitz, and (V (x))[α] , [(V1(x))α, . . . , (Vq(x))α]T.

Then the zero solution x(t) ≡ 0 to (1) and (2) is finite-time stable. If, in addition, D = Rn , then the zero solutionx(t) ≡ 0 to (1) and (2) is globally finite-time stable.

Proof. Consider the impulsive comparison system given by

z(t) = W (z(t))[α], z(0) = z0, x(t) 6∈ Z, (25)

∆z(t) = 0, x(t) ∈ Z, (26)

where z0 ∈ Rq+. Note that the right-hand side of (25) is of class Wc and is essentially nonnegative and, hence, the

solutions to (25) and (26) are nonnegative for all nonnegative initial conditions [23]. Since W ∈ Rq×q is essentiallynonnegative and Hurwitz, it follows from Theorem 3.2 of [24] that there exist positive vectors p ∈ Rq

+ and r ∈ Rq+

such that

0 = W T p + r. (27)

Now, consider the Lyapunov function candidate v(z) = pTz, z ∈ Rq+. Note that v(0) = 0, v(z) > 0, z ∈ Rq

+,z 6= 0, and v(·) is radially unbounded. Let β , mini=1,...,q ri and γ , maxi=1,...,q pα

i , where ri and pi are the i thcomponents of r ∈ Rq

+ and p ∈ Rq+, respectively. Then

v(z) = pTW z[α]

= −rTz[α]

≤ −β

γγ

(q∑

i=1

zαi

)

≤ −β

γ

(q∑

i=1

pαi zα

i

)

≤ −β

γ

(q∑

i=1

pi zi

)α

≤ −β

γ(v(z))α

= −c(v(z))α, z ∈ Rq+, (28)

and

∆v(z) = 0, z ∈ Rq+, (29)

where c , βγ

. Thus, it follows from Theorem 3.1 that the impulsive comparison system (25) and (26) is finite-timestable. In addition, it follows from Theorem 3.2 of [25] that the nonlinear impulsive dynamical system (1) and (2) isasymptotically stable with the domain of attraction N ⊂ D. Now, the result is a direct consequence of Theorem 3.2.

�

4. Finite-time stabilization of impulsive dynamical systems

In this section, we design hybrid finite-time stabilizing controllers for nonlinear affine in the control impulsivedynamical systems. In addition, for large-scale impulsive dynamical systems we design decentralized hybrid finite-time stabilizers predicated on a control vector Lyapunov function [26]. Consider the controlled nonlinear impulsive

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845 839

dynamical system given by

x(t) = fc(x(t)) + Gc(x(t))uc(t), x(t) 6∈ Z, t ≥ t0, (30)

∆x(t) = fd(x(t)) + Gd(x(t))ud(t), x(t) ∈ Z, (31)

where fc : Rn→ Rn satisfying fc(0) = 0 and Gc : Rn

→ Rn×mc are continuous functions, fd : Rn→ Rn and

Gd : Rn→ Rn×md are continuous, uc(t) ∈ Rmc , t ≥ t0, and ud(tk) ∈ Rmd , k ∈ Z+.

Theorem 4.1. Consider the controlled nonlinear impulsive dynamical system given by (30) and (31). If there exista continuously differentiable function V : Rn

→ R+ and continuous functions P1u : Rn→ R1×md and

P2u : Rn→ Rmd×md such that V (·) is positive definite and

V (x + fd(x) + Gd(x)ud) = V (x + fd(x)) + P1u(x)ud + uTd P2u(x)ud, x ∈ Rn, ud ∈ Rmd , (32)

V ′(x) fc(x) ≤ −c(V (x))α, x ∈ R, (33)

V (x + fd(x)) − V (x) −14

P1u(x)PĎ2u(x)PT

1u(x) ≤ 0, x ∈ Z, (34)

where c > 0, α ∈ (0, 1), and R , {x ∈ Rn, x 6∈ Z : V ′(x)Gc(x) = 0}, then the nonlinear impulsive dynamicalsystem (30) and (31) with the hybrid feedback control law (uc, ud) = (φc(·), φd(·)) given by

φc(x) =

−

(c0 +

(α(x) − wc(V (x))) +

√(α(x) − wc(V (x)))2 + (βT(x)β(x))2

βT(x)β(x)

)β(x),

β(x) 6= 0, x 6∈ Z,

0, β(x) = 0, x 6∈ Z,

(35)

and

φd(x) = −12

PĎ2u(x)PT

1u(x), x ∈ Z, (36)

where α(x) , V ′(x) fc(x), x ∈ Rn , β(x) , GTc (x)V ′T(x), x ∈ Rn , wc(V (x)) , −c(V (x))α , x ∈ Rn , and c0 > 0, is

finite-time stable.

Proof. Note that between resettings the time derivative of V (·) along the trajectories of (30), with uc = φc(x), x ∈ Rn ,given by (35), is given by

V (x) = V ′(x)( fc(x) + Gc(x)φc(x))

= α(x) + βT(x)φc(x)

=

{−c0β

T(x)β(x) −

√(α(x) − wc(V (x)))2 + (βT(x)β(x))2 + wc(V (x)), β(x) 6= 0,

α(x), β(x) = 0,

≤ wc(V (x)), x 6∈ Z. (37)

In addition, using (32) and (34), the difference of V (·) at the resetting instants, with ud = φd(x), x ∈ Z , given by (36),is given by

∆V (x) = V (x + fd(x) + Gd(x)φd(x)) − V (x)

= V (x + fd(x)) − V (x) + P1u(x)φd(x) + φd(x)PĎ2u(x)φd(x)

= V (x + fd(x)) − V (x) −14

P1u(x)PĎ2u(x)PT

1u(x)

≤ 0, x ∈ Z. (38)

Hence, it follows from Theorem 3.1 that the zero solution x(t) ≡ 0 to (30) and (31) with uc = φc(x), x 6∈ Z , givenby (35) and ud = φd(x), x ∈ Z , given by (36), is finite-time stable, which proves the result. �

840 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845

In the next result we develop decentralized finite-time stabilizing hybrid controllers for a large-scale impulsivedynamical system composed of q interconnected subsystems given by

xi (t) = fci (x(t)) + Gci (x(t))uci (t), x(t) 6∈ Z, i = 1, . . . , q, (39)

∆xi (t) = fdi (x(t)) + Gdi (x(t))udi (t), x(t) ∈ Z, i = 1, . . . , q, (40)

where t ≥ 0, fci : Rn→ Rni satisfying fci (0) = 0 and Gci : Rn

→ Rni ×mci are continuous functions for alli = 1, . . . , q, fdi : Rn

→ Rni and Gdi : Rn→ Rni ×mdi are continuous for all i = 1, . . . , q, uci (t) ∈ Rmci , t ≥ 0,

and udi (tk) ∈ Rmdi , k ∈ Z+, for all i = 1, . . . , q.

Theorem 4.2. Consider the controlled nonlinear impulsive dynamical system given by (39) and (40). Assume thereexist a continuously differentiable, component decoupled vector function V = [V1(x1), . . . , Vq(xq)]T

: Rn→ Rq

+,

continuous functions P1ui : Rn→ R1×mdi , P2ui : Rn

→ Rmdi ×mdi , i = 1, . . . , q, wc = [wc1, . . . , wcq ]T

: Rq+ → Rq ,

wd = [wd1, . . . , wdq ]T

: Rq+ → Rq , and a positive vector p ∈ Rq

+ such that V (0) = 0, the scalar function pTV (x),x ∈ Rn , is positive definite, wc(·) ∈Wc, wd(·) ∈Wd, wc(0) = 0, wd(0) = 0, and, for all i = 1, . . . , q,

Vi (xi + fdi (x) + Gdi (x)udi ) = Vi (xi + fdi (x)) + P1ui (x)udi + uTdi P2ui (x)udi , x ∈ Rn, udi ∈ Rmdi , (41)

V ′

i (xi ) fci (x) ≤ wci (V (x)), x ∈ Ri , (42)

Vi (xi + fdi (x)) − Vi (xi ) −14

P1ui (x)PĎ2ui (x)PT

1ui (x) ≤ wdi (V (x)), x ∈ Z, (43)

where Ri , {x ∈ Rn, x 6∈ Z : V ′

i (xi )Gci (x) = 0}, i = 1, . . . , q. In addition, assume there exist a positive definite

function v : Rq+ → R+, real numbers c > 0 and α ∈ (0, 1), and a neighbourhood M ⊆ Rq of the origin such that

v′(z)wc(z) ≤ −c(v(z))α, z ∈M ∩ Rq+, (44)

v(z + wd(z)) ≤ v(z), z ∈M ∩ Rq+. (45)

Then the nonlinear impulsive dynamical system (39) and (40) is finite-time stable with the hybrid feedback controllaw uc = φc(x) = [φT

c1(x), . . . , φTcq(x)]T, x 6∈ Z , and ud = φd(x) = [φT

d1(x), . . . , φTdq(x)]T, x ∈ Z , where for

i = 1, . . . , q,

φci (x) =

−

c0i +

(αi (x) − wci (V (x))) +

√(αi (x) − wci (V (x)))2 + (βT

i (x)βi (x))2

βTi (x)βi (x)

βi (x),

βi (x) 6= 0,

0, βi (x) = 0,

(46)

for x 6∈ Z , and

φdi (x) = −12

PĎ2ui (x)PT

1ui (x), x ∈ Z, (47)

where αi (x) , V ′

i (xi ) fci (x), x ∈ Rn , βi (x) , GciT(x)V ′T

i (xi ), x ∈ Rn , and c0i > 0, i = 1, . . . , q.

Proof. Using identical arguments as in the proof of Theorem 4.1 it can be shown that for the closed-loop system (39),(40), (46) and (47) the time derivative and the difference of the vector function V (·) between resettings and resettinginstants satisfy, respectively,

V (x) ≤≤ wc(V (x)), x 6∈ Z, (48)

V (x + fd(x) + Gd(x)φd(x)) ≤≤ V (x) + wd(V (x)), x ∈ Z, (49)

where Gd(x) , block-diag [Gd1(x), . . . , Gdq(x)], x ∈ Z . The result now follows immediately from (44) and (45),and Theorem 3.2. �

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845 841

Remark 4.1. If, in Theorem 4.2,Ri = Ø, i = 1, . . . , q, and (43) is satisfied with wd(z) ≡ 0, then the function wc(·)

in (46) can be chosen to be

wc(z) = W z[α], z ∈ Rq+, (50)

where W ∈ Rq×q is essentially nonnegative and asymptotically stable, α ∈ (0, 1), and z[α] , [zα1 , . . . , zα

q ]T. In this

case, conditions (44) and (45) need not be verified and it follows from Corollary 3.1 that the close-loop system (39),(40), (46) and (47) with wc(·) given by (50) is finite-time stable and, hence, the hybrid controller (46) and (47) is afinite-time stabilizing controller for (39) and (40).

Since fci (·) and Gci (·) are continuous and Vi (·) is continuously differentiable for all i = 1, . . . , q, it follows thatαi (x) and βi (x), x ∈ Rn , i = 1, . . . , q , are continuous functions, and hence, φci (x) given by (46) is continuous forall x ∈ Rn if either βi (x) 6= 0 or αi (x) − wci (V (x)) < 0 for all i = 1, . . . , q. Hence, the feedback control lawgiven by (46) is continuous everywhere except for the origin. The following result provides necessary and sufficientconditions under which the feedback control law given by (46) is guaranteed to be continuous at the origin in additionto being continuous everywhere else.

Proposition 4.1 ([26]). The feedback control law φc(x) given by (46) is continuous on Rn if and only if for everyε > 0, there exists δ > 0 such that for all 0 < ‖x‖ < δ there exists uci ∈ Rmci such that ‖uci‖ < ε andαi (x) + βT

i (x)uci < wci (V (x)), i = 1, . . . , q.

Remark 4.2. Identical necessary and sufficient conditions apply in the case where q = 1 to ensure continuity of thefeedback control law given by (35).

Remark 4.3. If the conditions of Proposition 4.1 are satisfied, then the feedback control law φc(x) given by (46) iscontinuous on Rn . However, it is important to note that for a particular trajectory x(t), t ≥ 0, of (39) and (40), φc(x(t))is left-continuous on [0, ∞) and is continuous everywhere on [0, ∞) except on an unbounded closed discrete set oftimes when the resettings occur for x(t), t ≥ 0.

5. Finite-time stabilizing control for large-scale impulsive dynamical systems

In this section, we apply the proposed hybrid control framework to decentralized control of large-scale nonlinearimpulsive dynamical systems. Specifically, we consider the large-scale dynamical system G shown in Fig. 5.1involving energy exchange between n interconnected subsystems. Let xi : [0, ∞) → R+ denote the energy (andhence a nonnegative quantity) of the i th subsystem, let uci : [0, ∞) → R denote the control input to the i th subsystem,let σci j : Rn

+ → R+, i 6= j , i, j = 1, . . . , n, denote the instantaneous rate of energy flow from the j th subsystemto the i th subsystem between resettings, let σdi j : Rn

+ → R+, i 6= j , i, j = 1, . . . , n, denote the amount of energytransferred from the j th subsystem to the i th subsystem at the resetting instant, and let Z ⊂ Rn

+ be a resetting set forthe large-scale impulsive dynamical system G.

An energy balance for each subsystem Gi , i = 1, . . . q, yields [27,6]

xi (t) =

n∑j=1, j 6=i

[σci j (x(t)) − σc j i (x(t))] + Gci (xi (t))uci (t), x(t0) = x0, x(t) 6∈ Z, t ≥ t0, (51)

∆xi (t) =

n∑j=1, j 6=i

[σdi j (x(t)) − σd j i (x(t))] + Gdi (x(t))udi (t), x(t) ∈ Z, (52)

or, equivalently, in vector form

x(t) = fc(x(t)) + Gc(x(t))uc(t), x(t0) = x0, x(t) 6∈ Z, t ≥ t0, (53)

∆x(t) = fd(x(t)) + Gd(x(t))ud(t), x(t) ∈ Z, (54)

where x(t) = [x1(t), . . . , xn(t)]T, t ≥ t0, fci (x) =∑n

j=1, j 6=i φci j (x), where φci j (x) , σci j (x) − σc j i (x), x ∈ Rn+,

i 6= j , i, j = 1, . . . , q , denotes the net energy flow from the j th subsystem to the i th subsystem between resettings,

842 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845

Fig. 5.1. Large-scale dynamical system G.

Fig. 5.2. Controlled system states versus time.

Gc(x) = diag[Gc1(x1), . . . , Gcn(xn)], x ∈ Rn+, Gci : R → R, i = 1, . . . , n, is such that Gci (xi ) = 0 if and

only if xi = 0 for all i = 1, . . . , n, Gd(x) = diag[Gd1(x), . . . , Gdn(x)], x ∈ Z , Gdi : Rn→ R, i = 1, . . . , n,

uc(t) ∈ Rn, t ≥ t0, ud(tk) ∈ Rn , k ∈ Z+, and fdi (x) =∑n

j=1, j 6=i φdi j (x), where φdi j (x) , σdi j (x)−σd j i (x), x ∈ Z ,i 6= j , i, j = 1, . . . , q , denotes the net amount of energy transferred from the j th subsystem to the i th subsystem atthe instant of resetting. Here, we assume that σci j (0) = 0, i 6= j , i, j = 1, . . . , n, and uc = [uc1, . . . , ucn]

T: R → Rn

is such that uci : R → R, i = 1, . . . , n, are bounded piecewise continuous functions of time. Furthermore, we assumethat σci j (x) = 0, x ∈ Rn

+, whenever x j = 0, i 6= j , i, j = 1, . . . , n, and xi +∑n

j=1, j 6=i φdi j (x) ≥ 0, x ∈ Z . In this

case, fc(·) is essentially nonnegative [24,27] (i.e., fci (x) ≥ 0 for all x ∈ Rn+ such that xi = 0, i = 1, . . . , n) and

x + fd(x), x ∈ Z ⊂ Rn+, is nonnegative (i.e., xi + fdi (x) ≥ 0 for all x ∈ Z , i = 1, . . . , n).

The above constraints imply that if the energy of the j th subsystem of G is zero, then this subsystem cannot supplyany energy to its surroundings between resettings and the i th subsystem of G cannot transfer more energy to itssurroundings than it already possesses at the instant of resetting. Finally, in order to ensure that the trajectories of theclosed-loop system remain in the nonnegative orthant of the state space for all nonnegative initial conditions, we seeka hybrid feedback control law (uc(·), ud(·)) that guarantees the continuous-time closed-loop system dynamics (53)are essentially nonnegative and the closed-loop system states after resettings are nonnegative [6,23].

For the dynamical system G, consider the Lyapunov function candidate V (x) = eTx , x ∈ Rn+. Note that V (0) = 0

and V (x) > 0, x 6= 0, x ∈ Rn+. Also, note that since V (x) = eTx , x ∈ Rn

+, is a linear function of x , it follows from (32)

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845 843

Fig. 5.3. Control signals in each control channel versus time.

that P1u(x) = eTGd(x), x ∈ Rn+, and P2u(x) ≡ 0, and hence, by (36), φd(x) ≡ 0. Define wc(V (x)) , −(V (x))

12 ,

x ∈ Rn+, and note that R , {x ∈ Rn

+, x 6∈ Z : V ′(x)Gc(x) = 0} = {x ∈ Rn+, x 6∈ Z : x = 0} = {0}, since 0 6∈ Z ,

and hence, condition (33) is satisfied. In addition, note that since V (·) is linear in x , condition (32) is trivially satisfiedand inequality (34) is satisfied as an equality.

Next, with α(x) = V ′(x) fc(x) =∑n

i=1∑n

j=1, j 6=i φci j (x) = 0, β(x) = [Gc1(x1), . . . , Gcn(xn)]T, x ∈ Rn+, and

c0 > 0, we construct a finite-time stabilizing hybrid feedback controller for (53) and (54) given by

φc(x) =

−

c0 +(eTx)1/2

+

√(eTx) +

(βT(x)β(x)

)2βT(x)β(x)

β(x), x 6= 0,

0, x = 0,

(55)

and

φd(x) ≡ 0. (56)

It can be seen from the structure of the feedback control law (55) and (56) that the continuous-time closed-loop systemdynamics are essentially nonnegative and the closed-loop system states after resettings are nonnegative. Furthermore,since α(x) − wc(V (x)) = (V (x))1/2, x ∈ Rn

+, the continuous-time feedback controller φc(·) is fully independentfrom fc(x) which represents the internal interconnections of the large-scale system dynamics, and hence, is robustagainst full modeling uncertainty in fc(x). Finally, it follows from Theorem 4.1 that the closed-loop system (53)–(56)is finite-time stable.

For the following simulation we consider (53) and (54) with σci j (x) = σci j xi x j and σdi j (x) = σdi j x j , and

Gci (xi ) = x1/4i , where σci j ≥ 0, i 6= j , i, j = 1, . . . , n, σdi j ≥ 0, i 6= j , i, j = 1, . . . , n, and 1 ≥

∑nj=1, j 6=i σd j i ,

i = 1, . . . , n. To show that the conditions of Proposition 4.1 are satisfied for the case when q = 1, let uc =

[−x1/41 , . . . ,−x1/4

n ]T and note that

α(x) + β(x)uc = −

n∑i=1

x1/2i

< −

(n∑

i=1

xi

)1/2

= −(V (x))1/2

= wc(V (x)), x ∈ Rn+, x 6= 0. (57)

Thus, for every ε > 0 there exists δ > 0, such that for all 0 < ‖x‖ < δ there exists uc ∈ Rmc such that‖uc‖ < ε and α(x) + βT(x)uc < wc(V (x)), and hence, the continuous-time feedback control law (55) is continuouson Rn

+. For our simulation we set n = 2, σc12 = 2, σc21 = 1.5, σd12 = 0.25, σd21 = 0.33, c0 = 1, and

844 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 2 (2008) 832–845

Fig. 5.4. Phase portrait of the controlled system.

Z = {x ∈ R2, x 6= 0 : x2 −23 x1 = 0}, with initial condition x0 = [4, 6]

T. Fig. 5.2 shows the states of the closed-loopsystem versus time and Fig. 5.3 shows continuous-time control signal as a function of time. Fig. 5.4 shows the phaseportrait of the closed-loop system where the dashed line denotes the resetting set Z .

6. Conclusion

The notion of finite-time stability was extended to impulsive dynamical systems using scalar and vector Lyapunovfunctions. These results were used to develop finite-time stabilizing controllers for nonlinear impulsive systems usingscalar and vector control Lyapunov functions. A hybrid decentralized feedback stabilizer was also developed fora class of large-scale impulsive dynamical systems with robustness guarantees against full modelling uncertainty.Finally, it is important to note that finite-time stability and stabilization can also be achieved by imposing additionalconditions on the discrete-time dynamics. In this case, deadbeat controllers [28] can be used to achieve finite-timestabilization for impulsive dynamical systems.

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