Control vector Lyapunov functions for large-scale ... - Semantic Scholar

Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243 www.elsevier.com/locate/nahs

Control vector Lyapunov functions for large-scale impulsive dynamical systems Sergey G. Nersesov a,∗ , Wassim M. Haddad b,1 a Department of Mechanical Engineering, Villanova University, Villanova, PA 19085-1681, United States b School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, United States

Received 15 May 2006; accepted 15 May 2006

Abstract Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Vector Lyapunov functions; Hybrid comparison principle; Partial stability; Control vector Lyapunov functions; Hybrid decentralized control; Large-scale impulsive systems

1. Introduction The mathematical descriptions of many hybrid dynamical systems can be characterized by impulsive differential equations [1–6]. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist of three elements—namely, a continuous-time differential equation, which governs the motion of the dynamical system between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs; and a criterion for determining when the states of the system are to be reset. Since impulsive systems can involve impulses at variable times, they are in general time-varying ∗ 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. c 2006 Elsevier Ltd. All rights reserved. 1751-570X/$ - see front matter doi:10.1016/j.nahs.2006.10.006

224

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

systems, wherein the resetting events are both a function of time and the system’s state. In the case where the resetting events are defined by a prescribed sequence of times which are independent of the system state, the equations are known as time-dependent differential equations [1,2,4,7–9]. Alternatively, in the case where the resetting events are defined by a manifold in the state space that is independent of time, the equations are autonomous and are known as state-dependent differential equations [1,2,4,7–9]. Even though impulsive dynamical systems were first formulated by Mil’man and Myshkis [10,11], the fundamental theory of impulsive differential equations is developed in the monographs by Bainov, Lakshmikantham, Perestyuk, Samoilenko, and Simeonov [1,2,4,5,12]. These monographs develop qualitative solution properties, existence of solutions, asymptotic properties of solutions, and stability theory of impulsive dynamical systems. In the recent series of papers [8,9,13,14], stability, dissipativity, and optimality results have been developed for impulsive dynamical systems which include invariant set stability theorems, partial stability, dissipativity theory, and optimal control design. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuoustime systems [15] to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. Lyapunov methods have also been used by control system designers to obtain stabilizing feedback controllers for nonlinear systems. In particular, for smooth feedback, Lyapunov-based methods were inspired by Jurdjevic and Quinn [16] who give sufficient conditions for smooth stabilization based on the ability of constructing a Lyapunov function for the closedloop system. More recently, Artstein [17] introduced the notion of a control Lyapunov function whose existence guarantees a feedback control law which globally stabilizes a nonlinear dynamical system. In general, the feedback control law is not necessarily smooth, but can be guaranteed to be at least continuous at the origin in addition to being smooth everywhere else. Even though for certain classes of nonlinear dynamical systems a universal construction of a feedback stabilizer can be obtained using control Lyapunov functions [18,19], there does not exist a unified procedure for finding a Lyapunov function candidate that will stabilize the closed-loop system for general nonlinear systems. In an attempt to simplify the construction of Lyapunov functions for the analysis and control design of nonlinear dynamical systems, several researchers have resorted to vector Lyapunov functions as an alternative to scalar Lyapunov functions. Vector Lyapunov functions were first introduced by Bellman [20] and Matrosov [21], and further developed in [22–25], with [22–24,26–29] exploiting their utility for analyzing large-scale systems. The use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Weakening the hypothesis on the Lyapunov function enlarges the class of Lyapunov functions that can be used for analyzing system stability. In particular, each component of a vector Lyapunov function need not be positive definite with a negative or even negative-semidefinite derivative. Alternatively, the time derivative of the vector Lyapunov function need only satisfy an element-by-element inequality involving a vector field of a certain comparison system. Since in this case the stability properties of the comparison system imply the stability properties of the dynamical system, the use of vector Lyapunov theory can significantly reduce the complexity (i.e., dimensionality) of the dynamical system being analyzed. Extensions of vector Lyapunov function theory that include relaxed conditions on standard vector Lyapunov functions as well as matrix Lyapunov functions appear in [28–30]. The results of this paper closely parallel those in [15] and include a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Next, we develop stability theorems based on hybrid comparison inequalities as well as partial stability results for impulsive systems using vector Lyapunov functions. Furthermore, we extend the newly developed notion of control vector Lyapunov functions presented in [15] to impulsive dynamical systems and show that in the case of a scalar comparison system the definition of a control vector Lyapunov function collapses into a combination of the classical definition of a control Lyapunov function for continuous-time dynamical systems given in [17] and the definition of a control Lyapunov function for discrete-time dynamical systems given in [31,32]. In addition, using control vector Lyapunov functions, we present a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system with guaranteed gain and sector margins. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

225

2. Mathematical preliminaries In this section we introduce notation and definitions needed for developing stability analysis and synthesis results for nonlinear impulsive dynamical systems via vector Lyapunov functions. Let R denote the set of real numbers, Z+ denote the set of nonnegative integers, Rn denote the set of n × 1 column vectors, and (·)T denote transpose. For v ∈ Rq we write v ≥≥ 0 (respectively, v  0) to indicate that every component of v is nonnegative (respectively, q q positive). In this case, we say that v is nonnegative or positive, respectively. Let R+ and R+ denote the nonnegative q q and positive orthants of Rq , that is, if v ∈ Rq , then v ∈ R+ and v ∈ R+ are equivalent, respectively, to v ≥≥ 0 and ◦

v  0. Furthermore, let D and D denote the interior and the closure of the set D ⊂ Rn , respectively. Finally, we write k·k for an arbitrary spatial vector norm in Rn , V 0 (x) for the Fr´echet derivative of V at x, Bε (α), α ∈ Rn , ε > 0, for the open ball centered at α with radius ε, e ∈ Rq for the ones vector, that is, e , [1, . . . , 1]T , and x(t) → M as t → ∞ to denote that x(t) approaches the set M, that is, for each ε > 0 there exists T > 0 such that dist (x(t), M) < ε for all t > T , where dist ( p, M) , infx∈M k p − xk. The following definition introduces the notion of class W functions involving quasimonotone increasing functions. Definition 2.1 ([26]). A function w = [w1 , . . . , wq ]T : Rq ×V → Rq , where V ⊆ Rs , is of class Wc if for every fixed y ∈ V ⊆ Rs , wi (z 0 , y) ≤ wi (z 00 , y), i = 1, . . . , q, for all z 0 , z 00 ∈ Rq such that z 0j ≤ z 00j , z i0 = z i00 , j = 1, . . . , q, i 6= j, where z i denotes the ith component of z. If w(·, y) ∈ Wc we say that w satisfies the Kamke condition [33,34]. Note that if w(z, y) = W (y)z, where W : V → Rq×q , then the function w(·, y) is of class Wc if and only if W (y) is essentially nonnegative for all y ∈ V, that is, all the off-diagonal entries of the matrix function W (·) are nonnegative. Furthermore, note that it follows from Definition 2.1 that any scalar (q = 1) function w(z, y) is of class Wc . Finally, we introduce the notion of class Wd functions involving nondecreasing functions. Definition 2.2 ([25]). A function w = [w1 , . . . , wq ]T : Rq × V → Rq , where V ⊆ Rs , is of class Wd if for every fixed y ∈ V ⊆ Rs , w(z 0 , y) ≤≤ w(z 00 , y) for all z 0 , z 00 ∈ Rq such that z 0 ≤≤ z 00 . If w(z, y) = W (y)z, where W : V → Rq×q , then the function w(·, y) is of class Wd if and only if W (y) is nonnegative for all y ∈ V, that is, all the entries of the matrix function W (·) are nonnegative. Note that if w(·, y) ∈ Wd , then w(·, y) ∈ Wc . Next, consider the nonlinear comparison system given by z˙ (t) = w(z(t), y(t)),

z(t0 ) = z 0 ,

t ∈ Iz 0 ,

(1)

where z(t) ∈ Q ⊆ Rq , t ∈ Iz 0 , is the comparison system state vector, y : T → V ⊆ Rs is a given continuous function, Iz 0 ⊆ T ⊆ R+ is the maximal interval of existence of a solution z(t) of (1), Q is an open set, 0 ∈ Q, and w : Q × V → Rq . We assume that w(·, y(t)) is continuous in t and satisfies the Lipschitz condition kw(z 0 , y(t)) − w(z 00 , y(t))k ≤ Lkz 0 − z 00 k,

t ∈T,

(2)

for all z 0 , z 00 ∈ Bδ (z 0 ), where δ > 0 and L > 0 is a Lipschitz constant. Hence, it follows from Theorem 2.2 of [35] that there exists τ > 0 such that (1) has a unique solution over the time interval [t0 , t0 + τ ]. Theorem 2.1 ([15]). Consider the nonlinear comparison system (1). Assume that the function w : Q × V → Rq is continuous and w(·, y) is of class Wc . If there exists a continuously differentiable vector function V = [v1 , . . . , vq ]T : Iz 0 → Q such that V˙ (t)  w(V (t), y(t)),

t ∈ Iz 0 ,

(3)

then V (t0 )  z 0 , z 0 ∈ Q, implies V (t)  z(t),

t ∈ Iz 0 ,

(4)

where z(t), t ∈ Iz 0 , is the solution to (1). Next, we present a stronger version of Theorem 2.1 where the strict inequalities are replaced by soft inequalities.

226

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

Theorem 2.2 ([15]). Consider the nonlinear comparison system (1). Assume that the function w : Q × V → Rq is continuous and w(·, y) is of class Wc . Let z(t), t ∈ Iz 0 , be the solution to (1) and [t0 , t0 + τ ] ⊆ Iz 0 be a compact interval. If there exists a continuously differentiable vector function V : [t0 , t0 + τ ] → Q such that V˙ (t) ≤≤ w(V (t), y(t)),

t ∈ [t0 , t0 + τ ],

(5)

then V (t0 ) ≤≤ z 0 , z 0 ∈ Q, implies V (t) ≤≤ z(t), t ∈ [t0 , t0 + τ ]. Next, consider the nonlinear dynamical system given by x(t) ˙ = f (x(t)),

x(t0 ) = x0 ,

t ∈ Ix0 ,

(6)

Rn , t

where x(t) ∈ D ⊆ ∈ Ix0 , is the system state vector, Ix0 is the maximal interval of existence of a solution x(t) of (6), D is an open set, 0 ∈ D, and f (·) is Lipschitz continuous on D. The following result is a direct consequence of Theorem 2.2. Corollary 2.1 ([15]). Consider the nonlinear dynamical system (6). Assume there exists a continuously differentiable vector function V : D → Q ⊆ Rq such that V 0 (x) f (x) ≤≤ w(V (x), x), where w : Q × D →

Rq

x ∈ D,

(7)

is a continuous function, w(·, x) ∈ Wc , and

z˙ (t) = w(z(t), x(t)),

z(t0 ) = z 0 ,

t ∈ Iz 0 ,x0 ,

(8)

has a unique solution z(t), t ∈ Iz 0 ,x0 , where x(t), t ∈ Ix0 , is a solution to (6). If [t0 , t0 + τ ] ⊆ Ix0 ∩ Iz 0 ,x0 is a compact interval, then V (x0 ) ≤≤ z 0 , z 0 ∈ Q, implies V (x(t)) ≤≤ z(t), t ∈ [t0 , t0 + τ ]. If in (6) f : Rn → Rn is globally Lipschitz continuous, then (6) has a unique solution x(t) for all t ≥ t0 . A more restrictive sufficient condition for global existence and uniqueness of solutions to (6) is continuous differentiability of f : Rn → Rn and uniform boundedness of f 0 (x) on Rn . Note that if the solutions to (6) and (8) are globally defined for all x0 ∈ D and z 0 ∈ Q, then the result of Corollary 2.1 holds for any arbitrarily large but compact interval [t0 , t0 + τ ] ⊂ R+ . For the remainder of this paper we assume that the solutions to the systems (6) and (8) are defined for all t ≥ t0 . Continuous differentiability of f (·) and w(·, ·) provides a sufficient condition for the existence and uniqueness of solutions to (6) and (8) for all t ≥ t0 . 3. Stability of impulsive systems via vector Lyapunov functions In this section, we consider state-dependent impulsive dynamical systems [8] given by x(t) ˙ = f c (x(t)),

x(t0 ) = x0 ,

∆x(t) = f d (x(t)),

x(t) ∈ Z,

x(t) 6∈ Z,

t ∈ Ix0 ,

(9) (10)

where x(t) ∈ D ⊆ Rn , t ∈ Ix0 , is the system state vector, Ix0 is the maximal interval of existence of a solution x(t) to (9) and (10), D is an open set, 0 ∈ D, f c : D → Rn is Lipschitz continuous and satisfies f c (0) = 0, f d : D → Rn is continuous, ∆x(t) , x(t + ) − x(t), and Z ⊂ D ⊆ Rn is the resetting set. For a particular trajectory x(t), t ≥ 0, we let tk = τk (x0 ), x0 ∈ D, denote the kth instant of time at which x(t) intersects Z. To ensure the well-posedness of the resetting times we make the following assumptions [8]: 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(tk ) ∈ ∂Z ∩ Z, then the system states reset to x + (tk ) , x(tk ) + f d (x(tk )), according to the resetting law (10), which serves as the initial condition for the continuous-time dynamics (9). Assumption A1 ensures that, if a trajectory reaches the closure of Z at a point that does not belong to Z, then the trajectory must be directed away from Z, that is, a trajectory cannot enter Z through a point that belongs to the closure of Z but not to Z. Furthermore, A2 ensures that when a trajectory intersects the resetting set Z, it instantaneously exits Z. Finally, we note that if x0 ∈ Z then the system initially resets to x0+ = x0 + f d (x0 ) 6∈ Z, which serves as the initial condition for continuous-time dynamics (9). Since the resetting times are well defined and distinct, and since the solution to (9) exists and is unique, it follows that the solution of the impulsive dynamical system (9) and (10)

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

227

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 many resettings in a finite time, encounter the same resetting surface a finite or infinite number of times in zero time, and coincide 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 impulsive dynamical system over a forward time interval can be extended to infinity due to Zeno solutions, we assume that existence and uniqueness of solutions are satisfied in forward time. For details see [1,2,4]. The next result presents a generalization of the comparison principle given in Corollary 2.1 to impulsive dynamical systems. Theorem 3.1. Consider the impulsive dynamical system (9) and (10). Assume there exists a continuously differentiable vector function V : D → Q ⊆ Rq such that V 0 (x) f c (x) ≤≤ wc (V (x), x),

x 6∈ Z,

(11)

V (x + f d (x)) ≤≤ V (x) + wd (V (x), x),

x ∈ Z,

(12)

where wc : Q × D → Rq and wd : Q × Z → Rq are continuous functions, wc (·, x) ∈ Wc , wd (·, x) ∈ Wd , and the comparison impulsive dynamical system z˙ (t) = wc (z(t), x(t)),

z(t0 ) = z 0 ,

∆z(t) = wd (z(t), x(t)),

x(t) ∈ Z,

x(t) 6∈ Z,

t ∈ Iz 0 ,x0 ,

(13) (14)

has a unique solution z(t), t ∈ Iz 0 ,x0 , where x(t), t ∈ Ix0 , is a solution to (9) and (10). If [t0 , t0 + τ ] ⊆ Ix0 ∩ Iz 0 ,x0 is a compact interval, then V (x0 ) ≤≤ z 0 ,

z 0 ∈ Q,

(15)

implies V (x(t)) ≤≤ z(t),

t ∈ [t0 , t0 + τ ].

(16)

Proof. Without loss of generality, let x0 6∈ Z, x0 ∈ D. If x0 ∈ Z, then by Assumption A2, x0 + f d (x0 ) 6∈ Z serves as the initial condition for the continuous-time dynamics. If for x0 6∈ Z the solution x(t) 6∈ Z for all t ∈ [t0 , t0 + τ ], then the result follows from Corollary 2.1. Next, suppose the interval [t0 , t0 + τ ] contains the resetting times τk (x0 ) < τk+1 (x0 ), k ∈ {1, 2, . . . , m}. Consider the compact interval [t0 , τ1 (x0 )] and let V (x0 ) ≤≤ z 0 . Then it follows from (11) and Corollary 2.1 that V (x(t)) ≤≤ z(t),

t ∈ [t0 , τ1 (x0 )],

(17)

where z(t), t ∈ Iz 0 , is the solution to (13). Now, since wd (·, x) ∈ Wd it follows from (12) and (17) that V (x(τ1+ (x0 ))) ≤≤ V (x(τ1 (x0 ))) + wd (V (x(τ1 (x0 ))), x(τ1 (x0 ))) ≤≤ z(τ1 (x0 )) + wd (z(τ1 (x0 )), x(τ1 (x0 ))) =

z(τ1+ (x0 )).

(18)

Consider the compact interval [τ1+ (x0 ), τ2 (x0 )]. Since V (x(τ1+ (x0 ))) ≤≤ z(τ1+ (x0 )), it follows from (11) that V (x(t)) ≤≤ z(t),

t ∈ [τ1+ (x0 ), τ2 (x0 )].

(19)

Repeating the above arguments for t ∈ [τk+ (x0 ), τk+1 (x0 )], k = 3, . . . , m, yields (16). Finally, in the case of infinitely many resettings overS the time interval [t0 , t0 +  τ ], let limk→∞ τk (x0 ) = τ∞ (x0 ) ∈ (t0 , t0 + τ ]. In this case, ∞ [t0 , τ∞ (x0 )] = [t0 , τ1 (x0 )] ∪ [τ (x ), τ (x )] . Repeating the above arguments, the result can be shown for k+1 0 k=1 k 0 the interval [t0 , τ∞ (x0 )].  Note that if the solutions to (9), (10), (13) and (14) are globally defined for all x0 ∈ D and z 0 ∈ Q, then the result of Theorem 3.1 holds for any arbitrarily large but compact interval [t0 , t0 + τ ] ⊂ R+ . For the remainder of this paper we

228

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

assume that the solutions to the systems (9), (10), (13) and (14) are defined for all t ≥ t0 . Next, consider the cascade nonlinear impulsive dynamical system given by z˙ (t) = wc (z(t), x(t)), x(t) ˙ = f c (x(t)),

z(t0 ) = z 0 , x(t0 ) = x0 ,

x(t) 6∈ Z,

x(t) 6∈ Z,

∆z(t) = wd (z(t), x(t)), x(t) ∈ Z, ∆x(t) = f d (x(t)), x(t) ∈ Z,

t ≥ t0 ,

(20) (21) (22) (23)

where z 0 ∈ Q ⊆ Rq , x0 ∈ D ⊆ Rn , [z T (t), x T (t)]T , t ≥ t0 , is the solution to (20)–(23), wc : Q × D → Rq and wd : Q × Z → Rq are continuous, wc (·, x) ∈ Wc , wd (·, x) ∈ Wd , wc (0, 0) = 0, f c : D → Rn is Lipschitz continuous on D, f c (0) = 0, and f d : D → Rn is continuous. The following definition introduces several types of partial stability of the nonlinear state-dependent impulsive dynamical system (20)–(23). Definition 3.1 ([6]). (i) The nonlinear impulsive dynamical system (20)–(23) is Lyapunov stable with respect to z if, for every ε > 0 and x0 ∈ Rn , there exists δ = δ(ε, x0 ) > 0 such that kz 0 k < δ implies that kz(t)k < ε for all t ≥ 0. (ii) The nonlinear impulsive dynamical system (20)–(23) is Lyapunov stable with respect to z uniformly in x0 if, for every ε > 0, there exists δ = δ(ε) > 0 such that kz 0 k < δ implies that kz(t)k < ε for all t ≥ 0 and for all x0 ∈ Rn . (iii) The nonlinear impulsive dynamical system (20)–(23) is asymptotically stable with respect to z if it is Lyapunov stable with respect to z and, for every x0 ∈ Rn , there exists δ = δ(x0 ) > 0 such that kz 0 k < δ implies that limt→∞ z(t) = 0. (iv) The nonlinear impulsive dynamical system (20)–(23) is asymptotically stable with respect to z uniformly in x0 if it is Lyapunov stable with respect to z uniformly in x0 and there exists δ > 0 such that kz 0 k < δ implies that limt→∞ z(t) = 0 uniformly in z 0 and x0 for all x0 ∈ Rn . (v) The nonlinear impulsive dynamical system (20)–(23) is globally asymptotically stable with respect to z if it is Lyapunov stable with respect to z and limt→∞ z(t) = 0 for all z 0 ∈ Rq and x0 ∈ Rn . (vi) The nonlinear impulsive dynamical system (20)–(23) is globally asymptotically stable with respect to z uniformly in x0 if it is Lyapunov stable with respect to z uniformly in x0 and limt→∞ z(t) = 0 uniformly in z 0 and x0 for all z 0 ∈ Rq and x0 ∈ Rn . (vii) The nonlinear impulsive dynamical system (20)–(23) is exponentially stable with respect to z uniformly in x0 if there exist scalars α, β, δ > 0 such that kz 0 k < δ implies that kz(t)k ≤ αkz 0 ke−βt , t ≥ 0, for all x0 ∈ Rn . (viii) The nonlinear impulsive dynamical system (20)–(23) is globally exponentially stable with respect to z uniformly in x0 if there exist scalars α, β > 0 such that kz(t)k ≤ αkz 0 ke−βt , t ≥ 0, for all z 0 ∈ Rq and x0 ∈ Rn . Theorem 3.2. Consider the impulsive dynamical system (9) and (10). Assume that there exist a continuously q q differentiable vector function V : D → Q ∩ R+ and a positive vector p ∈ R+ such that V (0) = 0, the scalar T function v : D → R+ defined by v(x) , p V (x), x ∈ D, is such that v(x) > 0, x 6= 0, and V 0 (x) f c (x) ≤≤ wc (V (x), x),

x 6∈ Z,

V (x + f d (x)) ≤≤ V (x) + wd (V (x), x),

(24) x ∈ Z,

(25)

where wc : Q × D → and wd : Q × Z → are continuous, wc (·, x) ∈ Wc , wd (·, x) ∈ Wd , and wc (0, 0) = 0. Then the following statements hold: (i) If the nonlinear impulsive dynamical system (20)–(23) is Lyapunov stable with respect to z uniformly in x0 , then the zero solution x(t) ≡ 0 to (9) and (10) is Lyapunov stable. (ii) If the nonlinear impulsive dynamical system (20)–(23) is asymptotically stable with respect to z uniformly in x0 , then the zero solution x(t) ≡ 0 to (9) and (10) is asymptotically stable. (iii) If D = Rn , Q = Rq , v : Rn → R+ is radially unbounded, and the nonlinear impulsive dynamical system (20)–(23) is globally asymptotically stable with respect to z uniformly in x0 , then the zero solution x(t) ≡ 0 to (9) and (10) is globally asymptotically stable. (iv) If there exist constants ν ≥ 1, α > 0, and β > 0 such that v : D → R+ satisfies Rq

αkxkν ≤ v(x) ≤ βkxkν ,

Rq

x ∈ D,

(26)

and the nonlinear impulsive dynamical system (20)–(23) is exponentially stable with respect to z uniformly in x0 , then the zero solution x(t) ≡ 0 to (9) and (10) is exponentially stable.

229

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

(v) If D = Rn , Q = Rq , there exist constants ν ≥ 1, α > 0, and β > 0 such that v : Rn → R+ satisfies (26), and the nonlinear impulsive dynamical system (20)–(23) is globally exponentially stable with respect to z uniformly in x0 , then the zero solution x(t) ≡ 0 to (9) and (10) is globally exponentially stable. q

Proof. Assume there exist a continuously differentiable vector function V : D → Q ∩ R+ and a positive vector q p ∈ R+ such that v(x) = p T V (x), x ∈ D, is positive definite, that is, v(0) = 0 and v(x) > 0, x 6= 0. Since v(x) = p T V (x) ≤ maxi=1,...,q { pi }eT V (x), x ∈ D, where e , [1, . . . , 1]T , the function eT V (x), x ∈ D, is also positive definite. Thus, there exist r > 0 and class K functions α, β : [0, r ] → R+ such that Br (0) ⊂ D and α(kxk) ≤ eT V (x) ≤ β(kxk),

x ∈ Br (0).

(27)

(i) Let ε > 0 and choose 0 < εˆ < min{ε, r }. It follows from Lyapunov stability of the nonlinear impulsive dynamical system (20)–(23)Pwith respect to z uniformly in x0 that there exists µ = µ(ˆε ) = µ(ε) > 0 such that if q kz 0 k1 < µ, where kzk1 , i=1 |z i | and z i is the ith component of z, then kz(t)k1 < α(ˆε ), t ≥ t0 , for any x0 ∈ D. Now, choose z 0 = V (x0 ) ≥≥ 0, x0 ∈ D. Since V (x), x ∈ D, is continuous, the function eT V (x), x ∈ D, is also continuous. Hence, for µ = µ(ˆε ) > 0 there exists δ = δ(µ(ˆε )) = δ(ε) > 0 such that δ < εˆ and if kx0 k < δ, then eT V (x0 ) = eT z 0 = kz 0 k1 < µ, which implies that kz(t)k1 < α(ˆε ), t ≥ t0 . Now, with z 0 = V (x0 ) ≥≥ 0, x0 ∈ D, and the assumption that wc (·, x) ∈ Wc and wd (·, x) ∈ Wd , it follows from (24) and (25), and Theorem 3.1 that 0 ≤≤ V (x(t)) ≤≤ z(t) on any compact interval [t0 , t0 + τ ], and hence, eT z(t) = kz(t)k1 , [t0 , t0 + τ ]. Let τ > t0 be such that x(t) ∈ Br (0), t ∈ [t0 , t0 + τ ] for all x0 ∈ Bδ (0). Thus, using (27), it follows that for kx0 k < δ, α(kx(t)k) ≤ eT V (x(t)) ≤ eT z(t) < α(ˆε ),

t ∈ [t0 , t0 + τ ],

(28)

which implies kx(t)k < εˆ < ε, t ∈ [t0 , t0 + τ ]. Now, suppose, ad absurdum, that for some x0 ∈ Bδ (0) there exists tˆ > t0 + τ such that kx(tˆ)k ≥ εˆ . Then, for z 0 = V (x0 ) and the compact interval [t0 , tˆ] it follows from (24) and (25), and Theorem 3.1 that V (x(tˆ)) ≤≤ z(tˆ), which implies that α(ˆε ) ≤ α(kx(tˆ)k) ≤ eT V (x(tˆ)) ≤ eT z(tˆ) < α(ˆε ). This is a contradiction, and hence, for a given ε > 0 there exists δ = δ(ε) > 0 such that for all x0 ∈ Bδ (0), kx(t)k < ε, t ≥ t0 , which implies Lyapunov stability of the zero solution x(t) ≡ 0 to (9) and (10). (ii) It follows from (i) and the asymptotic stability of the nonlinear impulsive dynamical system (20)–(23) with respect to z uniformly in x0 that the zero solution x(t) ≡ 0 to (9) and (10) is Lyapunov stable, and there exists µ > 0 such that if kz 0 k1 < µ, then limt→∞ z(t) = 0 for any x0 ∈ D. As in (i), choose z 0 = V (x0 ) ≥≥ 0, x0 ∈ D. It follows q from Lyapunov stability of the zero solution x(t) ≡ 0 to (9) and (10), and the continuity of V : D → Q ∩ R+ that there exists δ = δ(µ) > 0 such that if kx0 k < δ, then kx(t)k < r, t ≥ t0 , and eT V (x0 ) = eT z 0 = kz 0 k1 < µ. Thus, by asymptotic stability of (20)–(23) with respect to z uniformly in x0 , for any arbitrary ε > 0 there exists T = T (ε) > t0 such that kz(t)k1 < α(ε), t ≥ T . Thus, it follows from (24) and (25), and Theorem 3.1 that 0 ≤≤ V (x(t)) ≤≤ z(t) on any compact interval [T, T + τ ], and hence, eT z(t) = kz(t)k1 , t ∈ [T, T + τ ], and, by (27), α(kx(t)k) ≤ eT V (x(t)) ≤ eT z(t) < α(ε),

t ∈ [T, T + τ ].

(29)

Now, suppose, ad absurdum, that for some x0 ∈ Bδ (0), limt→∞ x(t) 6= 0, that is, there exists a sequence {tn }∞ n=1 , with tn → ∞ as n → ∞, such that kx(tn )k ≥ εˆ , n ∈ Z+ , for some 0 < εˆ < r . Choose ε = εˆ and the interval [T, T + τ ] such that at least one tn ∈ [T, T + τ ]. Then it follows from (29) that α(ε) ≤ α(kx(tn )k) < α(ε), which is a contradiction. Hence, there exists δ > 0 such that for all x0 ∈ Bδ (0), limt→∞ x(t) = 0 which along with Lyapunov stability implies asymptotic stability of the zero solution x(t) ≡ 0 to (9) and (10). (iii) Suppose D = Rn , Q = Rq , v : Rn → R+ is radially unbounded, and the nonlinear impulsive dynamical q system (20)–(23) is globally asymptotically stable with respect to z uniformly in x0 . In this case, V : Rn → R+ satisfies (27) for all x ∈ Rn , where the functions α, β : R+ → R+ are of class K∞ [35]. Furthermore, Lyapunov q stability of the zero solution x(t) ≡ 0 to (9) and (10) follows from (i). Next, for any x0 ∈ Rn and z 0 = V (x0 ) ∈ R+ , identical arguments as in (ii) can be used to show that limt→∞ x(t) = 0, which proves global asymptotic stability of the zero solution x(t) ≡ 0 to (9) and (10). q (iv) Suppose (26) holds. Since p ∈ R+ , then ν ν ˆ αkxk ˆ ≤ eT V (x) ≤ βkxk ,

x ∈ D,

(30)

230

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

4 4 where αˆ = α/ maxi=1,...,q { pi } and βˆ = β/ mini=1,...,q { pi }. It follows from the exponential stability of the nonlinear impulsive dynamical system (20)–(23) with respect to z uniformly in x0 that there exist positive constants γ , µ, and η such that if kz 0 k1 < µ, then

kz(t)k1 ≤ γ kz 0 k1 e−η(t−t0 ) ,

t ≥ t0 ,

(31) q R+ ,

for all x0 ∈ D. Choose z 0 = V (x0 ) ≥≥ 0, x0 ∈ D. By continuity of V : D → Q ∩ there exists δ = δ(µ) > 0 such that for all x0 ∈ Bδ (0), eT V (x0 ) = eT z 0 = kz 0 k1 < µ. Furthermore, it follows from (24), (25), (30) and (31), and Theorem 3.1 that for all x0 ∈ Bδ (0) the inequality ν ˆ 0 kν e−η(t−t0 ) αkx(t)k ˆ ≤ eT V (x(t)) ≤ eT z(t) ≤ γ kz 0 k1 e−η(t−t0 ) ≤ γ βkx

(32)

holds on any compact interval [t0 , t0 + τ ]. This in turn implies that for any x0 ∈ Bδ (0), kx(t)k ≤

γ βˆ αˆ

!1 ν

η

kx0 ke− ν (t−t0 ) ,

t ∈ [t0 , t0 + τ ].

(33)

Now, suppose, ad absurdum, that for some x0 ∈ Bδ (0) there exists tˆ > t0 + τ such that kx(tˆ)k >

γ βˆ αˆ

!1 ν

η

kx0 ke− ν (tˆ−t0 ) .

(34)

 ˆ  ν1 η Then for the compact interval [t0 , tˆ], it follows from (33) that kx(tˆ)k ≤ γαˆβ kx0 ke− ν (tˆ−t0 ) , which is a contradiction. Thus, inequality (33) holds for all t ≥ t0 establishing exponential stability of the zero solution x(t) ≡ 0 to (9) and (10). (v) The proof is identical to the proof of (iv).  q

If V : D → Q ∩ R+ satisfies the conditions of Theorem 3.2 we say that V (x), x ∈ D, is a vector Lyapunov function [26]. Note that for stability analysis each component of a vector Lyapunov function need not be positive definite, nor does it need to have a negative definite time derivative along the trajectories of (9) and (10) between resettings and negative semi-definite difference across the resettings. This provides more flexibility in searching for a vector Lyapunov function as compared to a scalar Lyapunov function for addressing the stability of impulsive dynamical systems. Next, we use the vector Lyapunov stability results of Theorem 3.2 to develop partial stability analysis results for nonlinear impulsive dynamical systems [14,36]. Specifically, consider the nonlinear impulsive dynamical system (9) and (10) with partitioned dynamics2 given by x˙I (t) = f Ic (xI (t), xII (t)), x˙II (t) = f IIc (xI (t), xII (t)), ∆xI (t) = f Id (xI (t), xII (t)), ∆xII (t) = f IId (xI (t), xII (t)),

xI (t0 ) = xI0 , xII (t0 ) = xII0 , x(t) ∈ Z, x(t) ∈ Z,

x(t) 6∈ Z, x(t) 6∈ Z,

t ≥ t0 ,

(35) (36) (37) (38)

where xI (t) ∈ DI , t ≥ t0 , DI ⊆ Rn I is an open set such that 0 ∈ DI , xII (t) ∈ Rn II , t ≥ t0 , ∆xI (t) = xI (t + ) − xI (t), ∆xII (t) = xII (t + ) − xII (t), f Ic : DI × Rn II → Rn I is such that for all xII ∈ Rn II , f Ic (0, xII ) = 0 and f Ic (·, xII ) is locally Lipschitz in xI , f IIc : DI × Rn II → Rn II is such that for every xI ∈ DI , f IIc (xI , ·) is locally Lipschitz in xII , f Id : DI × Rn II → Rn I is continuous and f Id (0, xII ) = 0 for all xII ∈ Rn II , f IId : DI × Rn II → Rn II is continuous, T , x T ]T , and n + n = n. For the Z ∈ DI × Rn II , x(t) , [xIT (t), xIIT (t)]T ∈ D = DI × Rn II ⊆ Rn , t ≥ t0 , x0 , [xI0 I II II0 nonlinear impulsive dynamical system (35)–(38) the definitions of partial stability given in [14] hold. Furthermore, for a particular trajectory x(t) = (xI (t), xII (t)), t ≥ 0, we let tk (=τk (xI0 , xII0 )) denote the kth instant of time at 2 Here we use the Roman subscripts I and II as opposed to Arabic subscripts 1 and 2 for denoting the partial states of x in order not to confuse the partial states with the component states of the vector Lyapunov function.

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

231

which x(t) intersects Z and we assume that Assumptions 1 and 2 hold for x(t) = (xI (t), xII (t)), t ≥ 0. Note that T (x , x )]T , (x , x ) ∈ D × Rn II , and for the impulsive dynamical system (9) and (10), f c (xI , xII ) = [ f IcT (xI , xII ), f IIc I II I II I T T T n II f d (xI , xII ) = [ f Id (xI , xII ), f IId (xI , xII )] , (xI , xII ) ∈ DI × R . For the following result define ∆V (xI , xII ) , V (xI + f Id (xI , xII ), xII + f IId (xI , xII )) − V (xI , xII ),

(39)

for a given vector function V : DI × Rn II → Rq . Theorem 3.3. Consider the nonlinear impulsive dynamical system (35)–(38). Assume that there exist a continuously q q differentiable vector function V : DI × Rn II → Q ∩ R+ , a positive vector p ∈ R+ , and class K functions α(·) and β(·) such that the scalar function v : DI × Rn II → R+ defined by v(xI , xII ) , p T V (xI , xII ) satisfies α(kxI k) ≤ v(xI , xII ) ≤ β(kxI k),

(xI , xII ) ∈ DI × Rn II ,

(40)

and V 0 (xI , xII ) f (xI , xII ) ≤≤ wc (V (xI , xII ), xI , xII ), ∆V (xI , xII ) ≤≤ wd (V (xI , xII ), xI , xII ),

(xI , xII ) 6∈ Z,

(41)

(xI , xII ) ∈ Z,

(42)

where wc : Q × DI → and wd : Q × Z → where Z ⊂ DI wd (·, xI , xII ) ∈ Wd , and wc (0, 0, 0) = 0. Then the following statements hold: × Rn II

Rq

Rq ,

× Rn II ,

are continuous, wc (·, xI , xII ) ∈ Wc ,

(i) If the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is Lyapunov (respectively, asymptotically) stable with respect to z uniformly in (xI0 , xII0 ), then the nonlinear impulsive dynamical system (35)–(38) is Lyapunov (respectively, asymptotically) stable with respect to xI uniformly in xII0 . (ii) If DI = Rn I , Q = Rq , the functions α(·) and β(·) are class K∞ , and the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is globally asymptotically stable with respect to z uniformly in (xI0 , xII0 ), then the nonlinear impulsive dynamical system (35)–(38) is globally asymptotically stable with respect to xI uniformly in xII0 . (iii) If there exist constants ν ≥ 1, α > 0, and β > 0 such that v : DI × Rn II → R+ satisfies αkxI kν ≤ v(xI , xII ) ≤ βkxI kν ,

(xI , xII ) ∈ DI × Rn II ,

(43)

and the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is exponentially stable with respect to z uniformly in (xI0 , xII0 ), then the nonlinear impulsive dynamical system (35)–(38) is exponentially stable with respect to xI uniformly in xII0 . (iv) If DI = Rn I , Q = Rq , there exist constants ν ≥ 1, α > 0, and β > 0 such that v : Rn I × Rn II → R+ satisfies (43), and the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is globally exponentially stable with respect to z uniformly in (xI0 , xII0 ), then the nonlinear impulsive dynamical system (35)–(38) is globally exponentially stable with respect to xI uniformly in xII0 . q

Proof. Since p ∈ R+ is a positive vector it follows from (40) that α(kxI k)/ max { pi } ≤ eT V (xI , xII ) ≤ β(kxI k)/ min { pi }, i=1,...,q

i=1,...,q

(xI , xII ) ∈ DI × Rn II .

(44)

Next, let ε > 0 and note that it follows from Lyapunov stability of the nonlinear impulsive dynamical system (20), (22) and (35)–(38) Pq with respect to z uniformly in (xI0 , xII0 ) that there exists µ = µ(ε) > 0 such that if kz 0 k1 < µ, where kzk1 , i=1 |z i | and z i is the ith component of z, then kz(t)k1 < α(ε)/ maxi=1,...,q { pi }, t ≥ t0 , for any (xI0 , xII0 ) ∈ DI × Rn II . Now, choose z 0 = V (xI0 , xII0 ) ≥≥ 0, (xI0 , xII0 ) ∈ DI × Rn II . Since V (·, ·) is continuous, the function eT V (·, ·) is also continuous. Moreover, it follows from the continuity of β(·) that for µ = µ(ε) there exists δ = δ(µ(ε)) = δ(ε) > 0 such that δ < ε and if kxI0 k < δ, then β(kxI0 k)/ mini=1,...,q { pi } < µ which, by (44), implies that eT V (xI0 , xII0 ) = eT z 0 = kz 0 k1 < µ for all xII0 ∈ Rn II , and hence, kz(t)k1 < α(ε)/ maxi=1,...,q { pi }, t ≥ t0 . In addition, it follows from (41) and (42), and Theorem 3.1 that 0 ≤≤ V (xI (t), xII (t)) ≤≤ z(t) on any compact interval [t0 , t0 + τ ], and hence, eT z(t) = kzk1 , [t0 , t0 + τ ]. Thus, it follows from (44) that for all kxI0 k < δ, xII0 ∈ Rn II , and t ∈ [t0 , t0 + τ ], α(kxI (t)k)/ max { pi } ≤ eT V (xI (t), xII (t)) ≤ eT z(t) < α(ε)/ max { pi }, i=1,...,q

i=1,...,q

(45)

232

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

which implies that kxI (t)k < ε, t ∈ [t0 , t0 + τ ]. Next, suppose, ad absurdum, that for some xI0 ∈ DI with kxI0 k < δ and for some xII0 ∈ Rn II there exists ˆt > t0 + τ such that kxI (tˆ)k ≥ ε. Then, for z 0 = V (xI0 , xII0 ) and the compact interval [t0 , tˆ] it follows from Theorem 3.1 that V (xI (tˆ), xII (tˆ)) ≤≤ z(tˆ) which implies that α(ε)/ maxi=1,...,q { pi } ≤ α(kxI (tˆ)k)/ maxi=1,...,q { pi } ≤ eT V (xI (tˆ), xII (tˆ)) ≤ eT z(tˆ) < α(ε)/ maxi=1,...,q { pi }. This is a contradiction and hence, for a given ε > 0, there exists δ = δ(ε) > 0 such that for all xI0 ∈ DI with kxI0 k < δ and for all xII0 ∈ Rn II , kxI (t)k < ε, t ≥ t0 , which implies Lyapunov stability of the nonlinear impulsive dynamical system (35)–(38) with respect to xI uniformly in xII0 . The remainder of the proof involves similar arguments as above and as in the proof of parts (ii)–(v) of Theorem 3.2 and, hence, is omitted.  Remark 3.1. Note that Theorem 3.3 allows us to address stability of time-dependent nonlinear impulsive dynamical systems via vector Lyapunov functions. In particular, with xI (t) ≡ x(t), xII (t) ≡ t, n I = n, n II = 1, f Ic (xI , xII ) = f c (x(t), t), f IIc (xI (t), xII (t)) = 1, f Id (xI , xII ) = f d (x(t), t), f IId (xI (t), xII (t)) = 0, and Z = D × T , with T , {t1 , t2 , . . .}, Theorem 3.3 can be used to establish stability results for the nonlinear time-dependent impulsive dynamical system given by x(t) ˙ = f c (x(t), t),

x(t0 ) = x0 ,

∆x(t) = f d (x(t), t),

t = tk ,

t 6= tk ,

t ≥ t0 ,

(46) (47)

where x0 ∈ D ⊆ Rn . For details on the unification between partial stability of state-dependent impulsive systems and stability theory for time-dependent impulsive systems see [14]. 4. Control vector Lyapunov functions for impulsive systems In this section, we consider a feedback control problem and generalize the notion of a control vector Lyapunov function introduced in [15] to nonlinear impulsive dynamical systems. Specifically, consider the nonlinear controlled impulsive dynamical system given by x(t) ˙ = Fc (x(t), u c (t)),

x(t0 ) = x0 ,

∆x(t) = Fd (x(t), u d (t)),

x(t) ∈ Z,

x(t) 6∈ Z,

t ≥ t0 ,

(48) (49)

where x0 ∈ D, D ⊆ Rn is an open set with 0 ∈ D, u c (t) ∈ Uc ⊆ Rm c , t ≥ t0 , u d (tk ) ∈ Ud ⊆ Rm d , tk denotes the kth instant of time at which x(t) intersects Z for a particular trajectory x(t) and input (u c (·), u d (·)), Fc : D × Uc → Rn is Lipschitz continuous for all (x, u c ) ∈ D × Uc and satisfies Fc (0, 0) = 0, and Fd : D × Ud → Rn is continuous. Here, we assume that u c (·) and u d (·) are restricted to the class of admissible control inputs consisting of measurable functions such that (u c (t), u d (tk )) ∈ Uc × Ud for all t ≥ t0 and k ∈ Z[t0 ,t) , {k : t0 ≤ tk < t}, where the constraint set Uc × Ud is given with (0, 0) ∈ Uc × Ud . Furthermore, we assume that u c (·) and u d (·) satisfy sufficient regularity conditions such that the nonlinear impulsive dynamical system (48) and (49) has a unique solution forward in time. Let φc : D → Uc be such that φc (0) = 0 and let φd : Z → Ud . If (u c (t), u d (tk )) = (φc (x(t)), φd (x(tk ))), where x(t), t ≥ t0 , satisfies (48) and (49), then (u c (·), u d (·)) is called a hybrid feedback control. q

Definition 4.1. If there exist a continuously differentiable vector function V = [v1 , . . . , vq ]T : D → Q ∩ R+ , continuous functions wc = [wc1 , . . . , wcq ]T : Q × D → Rq and wd = [wd1 , . . . , wdq ]T : Q × Z → Rq , q and a positive vector p ∈ R+ such that V (0) = 0, v(x) , p T V (x), x ∈ D, is positive definite, wc (·, x) ∈ Wc , q q wd (·, x) ∈ Wd , wc (0, 0) = 0, Fc (x) , ∩i=1 Fci (x) 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, Fd (x) , ∩i=1 Fdi (x) 6= Ø, x ∈ Z, where Fci (x) , {u c ∈ Uc : vi0 (x)Fc (x, u c ) < wci (V (x), x)}, x ∈ D, x 6∈ Z, x 6= 0, i = 1, . . . , q, and Fdi (x) , {u d ∈ Ud : vi (x + Fd (x, u d )) − vi (x) ≤ wdi (V (x), x)}, x ∈ Z, i = 1, . . . , q, then the vector function q V : D → Q ∩ R+ is called a control vector Lyapunov function candidate. It follows from Definition 4.1 that if there exists a control vector Lyapunov function candidate, then there exists a hybrid feedback control law φc : D → Uc and φd : Z → Ud such that V 0 (x)Fc (x, φc (x))  wc (V (x), x),

x ∈ D,

V (x + Fd (x, φd (x))) ≤≤ V (x) + wd (V (x), x),

x 6∈ Z, x ∈ Z.

x 6= 0,

(50) (51)

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

233

Moreover, if the nonlinear impulsive dynamical system z˙ (t) = wc (z(t), x(t)),

z(t0 ) = z 0 ,

x(t) ˙ = Fc (x(t), φc (x(t))),

x(t) 6∈ Z,

x(t0 ) = x0 ,

t ≥ t0 ,

x(t) 6∈ Z,

∆z(t) = wd (z(t), x(t)), x(t) ∈ Z, ∆x(t) = Fd (x(t), φd (x(t))), x(t) ∈ Z,

(52) (53) (54) (55)

where z 0 ∈ Q and x0 ∈ D, is asymptotically stable with respect to z uniformly in x0 , then it follows from Theorem 3.2 q that the zero solution x(t) ≡ 0 to (53) and (55) is asymptotically stable. In this case, the vector function V : D → R+ given in Definition 4.1 is called a control vector Lyapunov function for impulsive dynamical system (48) and (49). This is a generalization of the notion of control vector Lyapunov functions introduced in [15] for continuous-time systems. Furthermore, if D = Rn , Q = Rq , Uc = Rm c , Ud = Rm d , v : Rn → R+ is radially unbounded, and the system (52)–(55) is globally asymptotically stable with respect to z uniformly in x0 , then the zero solution x(t) ≡ 0 to (48) and (49) is globally asymptotically stabilizable. Remark 4.1. If in Definition 4.1 wc (z, x) = wc (z), wd (z, x) = wd (z), and the zero solution z(t) ≡ 0 to z˙ (t) = wc (z(t)),

z(t0 ) = z 0 ,

∆z(t) = wd (z(t)),

x(t) ∈ Z,

t ≥ t0 ,

x(t) 6∈ Z,

(56) (57)

where z 0 ∈ Q, is asymptotically stable, then it follows from Theorem 3.2, with wc (z, x) = wc (z), wd (z, x) = wd (z), q that V : D → Q ∩ R+ is a control vector Lyapunov function. In this case, the nonlinear impulsive comparison system (56) and (57) is a time-dependent impulsive dynamical system [8]. To see this, note that the resetting times τk (x0 ), k ∈ Z+ , where x(τk (x0 )) ∈ Z and x(t) is the solution to (48) and (49), are determined by the state of (48) and (49), and hence provide a prescribed sequence of the resetting times for (56) and (57) since the dynamics of the impulsive system (48) and (49) and the comparison system (56) and (57) are decoupled. In the case where q = 1, wc (z, x) ≡ wc (z), and wd (z, x) ≡ wd (z), Definition 4.1 implies the existence of a positive-definite continuously differentiable function v : D → Q ∩ R+ and continuous functions wc : Q → R and wd : Q → R, where Q ⊆ R, such that wc (0) = 0, Fc (x) = {u c ∈ Uc : v 0 (x)Fc (x, u c ) < wc (v(x))} 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, and Fd (x) = {u d ∈ Ud : v(x + Fd (x, u d )) − v(x) ≤ wd (v(x))} 6= Ø, x ∈ Z, which implies inf v 0 (x)Fc (x, u c ) < wc (v(x)),

u c ∈Uc

x 6∈ Z,

x 6= 0,

inf [v(x + Fd (x, u d )) − v(x)] ≤ wd (v(x)),

u d ∈Ud

x ∈ Z.

(58) (59)

Now, the fact that Fc (x) 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, and Fd (x) 6= Ø, x ∈ Z, implies the existence of a hybrid feedback control law φc : D → Uc and φd : Z → Ud such that v 0 (x)Fc (x, φc (x)) < wc (v(x)), x ∈ D, x 6∈ Z, x 6= 0, and v(x + Fd (x, φd (x))) − v(x) ≤ wd (v(x)), x ∈ Z. Moreover, if v : D → R+ is a control vector Lyapunov function (with q = 1), then it follows from Remark 4.1 that the zero solution z(t) ≡ 0 to the system (56) and (57) is asymptotically stable and, since q = 1, this implies that wc (z) < 0, z ∈ Q ∩ R+ , z 6= 0. Thus, since v(·) is positive definite, (58) and (59) can be rewritten as inf v 0 (x)Fc (x, u c ) < 0,

u c ∈Uc

x 6∈ Z,

x 6= 0,

inf [v(x + Fd (x, u d )) − v(x)] ≤ wd (v(x)),

u d ∈Ud

(60) x ∈ Z,

(61)

which can be regarded as a definition of a scalar control Lyapunov function for impulsive dynamical systems. Even though such a result has never been addressed in the literature, it can be seen that this is a combination of the classical definition of a control Lyapunov function for continuous-time dynamical systems given in [17] and the definition of a control Lyapunov function for discrete-time dynamical systems given in [31,32]. Next, consider the case where the control input to (48) and (49) possesses a decentralized control architecture so that the dynamics of (48) and (49) are given by x˙i (t) = Fci (x(t), u ci (t)),

x(t) 6∈ Z,

t ≥ t0 ,

i = 1, . . . , q,

(62)

234

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

∆xi (t) = Fdi (x(t), u di (t)),

x(t) ∈ Z,

i = 1, . . . , q,

(63)

where xi (t) ∈ x(t) = u ci (t) ∈ Uci ⊆ ≥ t0 , u di (tk ) ∈ Udi ⊆ ∈ Z+ , Pq Pq tk denotes the kth resetting time for a particular trajectory of (62) and (63), i=1 n i = n, i=1 m ci = m c , and P q ni i=1 m di = m d . Note that x i (t) ∈ R , t ≥ t0 , i = 1, . . . , q, as long as x(t) ∈ D, t ≥ t0 , and the sets of control inputs are given by Uc = Uc1 × · · · × Ucq ⊆ Rm c and Ud = Ud1 × · · · × Udq ⊆ Rm d . In the case of a component decoupled control vector Lyapunov function candidate, that is, V (x) = [v1 (x1 ), . . . , vq (xq )]T , x ∈ D, it suffices to require in Definition 4.1 that, for all i = 1, . . . , q, Rn i ,

[x1T (t), . . . , xqT (t)]T ,

Rm ci , t

Fci (x) = {u c ∈ Uc : vi0 (xi )Fci (x, u ci ) < wci (V (x), x)} 6= Ø,

x ∈ D,

Rm di , k

x 6∈ Z,

x 6= 0,

(64)

Fdi (x) = {u d ∈ Ud : vi (xi + Fdi (x, u di )) − vi (x) ≤ wdi (V (x), x)} 6= Ø, x ∈ Z, (65) Tq Tq to ensure that Fc (x) = i=1 Fci (x) 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, and Fd (x) = i=1 Fdi (x) 6= Ø, x ∈ Z. Note that q for a component decoupled control vector Lyapunov function V : D → Q ∩ R+ , (64) is equivalent to inf V 0 (x)Fc (x, u c )  wc (V (x), x),

x ∈ D,

u c ∈Uc

x 6∈ Z,

x 6= 0,

(66)

and (65) implies inf [V (x + Fd (x, u d )) − V (x)] ≤≤ wd (V (x), x),

u d ∈Ud

x ∈ Z,

(67)

where the infimum in (66) and (67) is taken componentwise, that is, for each component of (66) and (67) the infimum is calculated separately. It follows from the fact that Fc (x) 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, and Fd (x) 6= Ø, x ∈ Z, that there T (x), . . . , φ T (x)]T , x ∈ D, exists a hybrid feedback control law φc : D → Uc and φd : Z → Ud such that φc (x) = [φc1 cq T T T φd (x) = [φd1 (x), . . . , φdq (x)] , x ∈ Z, where φci : D → Uci , φdi : Z → Udi , vi0 (xi )Fci (x, φci (x)) < wci (V (x), x), x ∈ D, x 6∈ Z, x 6= 0, and vi (xi + Fdi (x, φdi (x))) − vi (xi ) ≤ wdi (V (x), x), x ∈ Z, i = 1, . . . , q. Remark 4.2. If wci (V (x), x) = 0 for x ∈ D with xi = 0, then condition (64) holds for all x ∈ D such that xi 6= 0. Next, we consider the special case of a nonlinear impulsive dynamical system of the form (62) and (63) with affine control inputs given by x˙i (t) = f ci (x(t)) + G ci (x(t))u ci (t),

x(t) 6∈ Z,

∆xi (t) = f di (x(t)) + G di (x(t))u di (t),

x(t) ∈ Z,

t ≥ t0 ,

i = 1, . . . , q,

(68)

i = 1, . . . , q,

(69)

where f ci : Rn → Rni satisfying f ci (0) = 0 and G ci : Rn → Rni ×m ci are smooth functions (at least continuously differentiable mappings) for all i = 1, . . . , q, f di : Rn → Rni and G di : Rn → Rn i ×m di are continuous for all i = 1, . . . , q, u ci (t) ∈ Rm ci , t ≥ t0 , and u di (tk ) ∈ Rm di , k ∈ Z+ , for all i = 1, . . . , q. Theorem 4.1. Consider the controlled nonlinear impulsive dynamical system given by (68) and (69). If there q exist a continuously differentiable, component decoupled vector function V : Rn → R+ , continuous functions q P1ui : Rn → R1×m di , P2ui : Rn → Rm di ×m di , i = 1, . . . , q, wc = [wc1 , . . . , wcq ]T : R+ × Rn → Rq , q q wd = [wd1 , . . . , wdq ]T : R+ × Z → Rq , and a positive vector p ∈ R+ such that V (0) = 0, the scalar function T n n v : R → R+ defined by v(x) , p V (x), x ∈ R , is positive definite and radially unbounded, wc (·, x) ∈ Wc , wd (·, x) ∈ Wd , wc (0, 0) = 0, and, for all i = 1, . . . , q, vi (xi + f di (x) + G di (x)u di ) = vi (xi + f di (x)) + P1ui u di + u Tdi P2ui (x)u di , vi0 (xi ) f ci (x) < wci (V (x), x),

x ∈ Rn ,

x ∈ Ri ,

u di ∈ Rm di ,

(70) (71)

1 Ď T vi (xi + f di (x)) − vi (xi ) − P1ui (x)P2ui (x)P1ui (x) ≤ wdi (V (x), x), 4 Ď

x ∈ Z,

(72)

where Ri , {x ∈ Rn , x 6= 0 : vi0 (xi )G ci (x) = 0}, i = 1, . . . , q, and P2ui (·) is Moore–Penrose generalized inverse of q P2ui (·) [37], then V : Rn → R+ is a control vector Lyapunov function candidate. If, in addition, there exist φc : Rn → T (x), . . . , φ T (x)]T , x ∈ Rn , φ (x) = [φ T (x), . . . , φ T (x)]T , x ∈ Z, Rm c and φd : Z → Rm d such that φc (x) = [φc1 d cq d1 dq

235

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

and the system (52)–(55) is globally asymptotically stable with respect to z uniformly in x0 , then the zero solution q x(t) ≡ 0 to (53) and (55) is globally asymptotically stable and V : Rn → R+ is a control vector Lyapunov function. Proof. Note that for all i = 1, . . . , q, infm

u ci ∈R

ci

vi0 (xi )( f ci (x) +

−∞, vi0 (xi ) f ci (x),

x∈ 6 Ri , x ∈ Ri ,

< wci (V (x), x),

x ∈ Rn ,

G ci (x)u ci ) =



x 6= 0,

(73)

which implies that Fci (x) 6= Ø, x ∈ x 6= 0, i = 1, . . . , q. Next, note that it follows from a Taylor series 2 ∗ expansion about xi = xi + f di (x) that P1ui (x) = vi0 (xi∗ )G di (x) and P2ui (x) = 12 G Tdi (x) ∂ v2i |xi =xi∗ G di (x). Since Rn ,

V (·) is continuously differentiable it follows that i = 1, . . . , q. Next, with u di =

Ď T (x), − 12 P2ui (x)P1ui

∂ 2 vi ∗ | ∂ xi2 xi =xi

∂ xi

is symmetric, and hence, P2ui (·) is symmetric for all

x ∈ Z, i = 1, . . . , q, it follows from (70) and (72) that

vi (xi + f di (x) + G di (x)u di ) − vi (xi ) = vi (xi + f di (x)) − vi (xi ) + P1ui (x)u di + u Tdi P2ui (x)u di 1 Ď T = vi (xi + f di (x)) − vi (xi ) − P1ui (x)P2ui (x)P1ui (x) 4 ≤ wdi (V (x), x), x ∈ Z, i = 1, . . . , q,

(74)

which implies that Fdi (x) 6= Ø, x ∈ Z. Now, the result is a direct consequence of the definition of a control vector Lyapunov function by noting that, for component decoupled vector Lyapunov functions, (64) and (65) are equivalent to Fc (x) 6= Ø, x ∈ Rn , x 6= 0, and Fd (x) 6= Ø, x ∈ Z, respectively.  Using Theorem 4.1 we can construct an explicit feedback control law that is a function of the control vector T (x), . . . , φ T (x)]T , x ∈ Lyapunov function V (·). Specifically, consider the hybrid feedback control law φc (x) = [φc1 cq T (x), . . . , φ T (x)]T , x ∈ Z, given by Rn , and φd (x) = [φd1 dq q     (α (x) − w (V (x), x)) + (αi (x) − wci (V (x), x))2 + (βiT (x)βi (x))2  i ci    βi (x), − c0i + φci (x) = βiT (x)βi (x)    0,

βi (x) 6= 0,

(75)

βi (x) = 0,

and 1 Ď T φdi (x) = − P2ui (x)P1ui (x), 2

x ∈ Z,

(76)

where αi (x) , vi0 (xi ) f ci (x), x ∈ Rn , βi (x) , G Tci (x)vi0T (xi ), x ∈ Rn , and c0i > 0, i = 1, . . . , q. The derivative V˙ (·) along the trajectories of the dynamical system (68), with u c = φc (x), x ∈ Rn , given by (75), is given by v˙i (xi ) = vi0 (xi )( f ci (x) + G ci (x)φci (x)) = αi (x) + βiT (x)φci (x)  q  −c0i βiT (x)βi (x) − (αi (x) − wci (V (x), x))2 + (βiT (x)βi (x))2 = + wci (V (x), x),   αi (x), < wci (V (x), x),

x ∈ Rn .

βi (x) 6= 0, βi (x) = 0, (77)

In addition, using (70), the difference of V (x) at the resetting instants with u d = φd (x), x ∈ Z, given by (76), is given by ∆vi (xi ) = vi (xi + f di (x) + G di (x)φdi (x)) − vi (xi ) 1 Ď T = vi (xi + f di (x)) − vi (xi ) − P1ui (x)P2ui (x)P1ui (x) 4 ≤ wdi (V (x), x), x ∈ Z.

(78)

236

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

Thus, if the zero solution z(t) ≡ 0 to (52)–(55) is globally asymptotically stable with respect to z uniformly in x0 , then it follows from Theorem 3.2 that the zero solution x(t) ≡ 0 to (68) and (69) with u c = φc (x) = T (x), . . . , φ T (x)]T , x ∈ Rn , given by (75) and u = φ (x) = [φ T (x), . . . , φ T (x)]T , x ∈ Z, given by (76) is [φc1 d d cq d1 dq globally asymptotically stable. Remark 4.3. If in Theorem 4.1 wc (z, x) = wc (z), wd (z, x) = wd (z), and the zero solution z(t) ≡ 0 to (56) and (57) is globally asymptotically stable, then it follows from Theorem 3.2 that the hybrid feedback control law given by (75) and (76) is a globally asymptotically stabilizing controller for the nonlinear impulsive dynamical system (68) and (69). Remark 4.4. In the case where q = 1, the functions w(·, ·) and wd (·, ·) in Theorem 4.1 can be set to be identically zero, that is, wc (z, x) ≡ 0 and wd (z, x) ≡ 0. In this case, the feedback control law (75) specializes to Sontag’s universal formula [18], the feedback control law (76) specializes to the discrete-time control law given in [32], and the hybrid feedback control law (75) and (76) is a global stabilizer for the nonlinear impulsive dynamical system (68) and (69). Since f ci (·) and G ci (·) are smooth and vi (·) is continuously differentiable for all i = 1, . . . , q, it follows that αi (x) and βi (x), x ∈ Rn , i = 1, . . . , q, are continuous functions, and hence, φci (x) given by (75) is continuous for all x ∈ Rn if either βi (x) 6= 0 or αi (x) − wci (V (x), x) < 0 for all i = 1, . . . , q. Hence, the feedback control law given by (75) is continuous everywhere except for the origin. The following result provides necessary and sufficient conditions under which the feedback control law given by (75) is guaranteed to be continuous at the origin in addition to being continuous everywhere else. Proposition 4.1 ([15]). The feedback control law φc (x) given by (75) is continuous on Rn if and only if for every ε > 0, there exists δ > 0 such that for all 0 < kxk < δ there exists u ci ∈ Rm ci such that ku ci k < ε and αi (x) + βiT (x)u ci < wci (V (x), x), i = 1, . . . , q. Remark 4.5. If the conditions of Proposition 4.1 are satisfied, then the feedback control law φc (x) given by (75) is continuous on Rn . However, it is important to note that for a particular trajectory x(t), t ≥ 0, of (68) and (69), φc (x(t)) is a piecewise continuous function of time due to state resettings. 5. Stability margins and inverse optimality In this section, we construct a hybrid feedback control law that is robust to sector bounded input nonlinearities. Specifically, we consider the nonlinear impulsive dynamical system (68) and (69) with nonlinear uncertainties in the input so that the impulsive dynamics of the system are given by x˙i (t) = f ci (x(t)) + G ci (x(t))σci (u ci (t)), ∆xi (t) = f di (x(t)) + G di (x(t))σdi (u di (t)),

x(t) 6∈ Z, x(t) ∈ Z,

t ≥ t0 ,

i = 1, . . . , q,

i = 1, . . . , q,

(79) (80)

where σci (·) ∈ Φci , {σci : Rm ci → Rm ci : σci (0) = 0 and 12 u Tci u ci ≤ σciT (u ci )u ci < ∞, u ci ∈ Rm ci }, σdi (·) ∈ Φdi , {σdi : Rm di → Rm di : σdi (0) = 0 and αd u 2di j ≤ σdi j (u di )u di j < βd u 2di j , u di ∈ Rm di }, i = 1, . . . , q, σdi j (·) ∈ R and u di j ∈ R, j = 1, . . . , m di , are the jth components of σdi (·) and u di , respectively, and 0 ≤ αd < 1 < βd < ∞. In addition, we show that for the dynamical system (68) and (69) the hybrid feedback control law to be defined in Theorem 5.1 is inverse optimal in the sense that it minimizes a derived hybrid performance functional over the set of stabilizing controllers S(x0 ) , {(u c (·), u d (·)) : u c (·) and u d (·) are admissible andx(t) → 0 as t → ∞}. Theorem 5.1. Consider the nonlinear impulsive dynamical system (68) and (69) and assume that the conditions of Theorem 4.1 hold with (72) replaced by 1 T P1ui (x)(R2di (x) + P2ui (x))−1 P1ui (x) ≤ wdi (V (x), x), x ∈ Z, (81) 4 where R2di : Z → Rm di ×m di is positive definite, wc (z, x) ≡ wc (z), wd (z, x) ≡ wd (z), and with the zero solution z(t) ≡ 0 to (56) and (57) being globally asymptotically stable. Then the hybrid feedback control law (φci (·), φdi (·)) vi (xi + f di (x)) − vi (xi ) −

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

237

given by (75) and 1 T φdi (x) = − (R2di (x) + P2ui (x))−1 P1ui (x), x ∈ Z, i = 1, . . . , q, 2 minimizes the performance functional given by Z ∞X q [L 1ci (x(t)) + u Tci (t)R2ci (x(t))u ci (t)]dt J (x0 , u c (·), u d (·)) = t0

+

(82)

i=1

X

q X

[L 1di (x(tk )) + u Tdi (tk )R2di (x(tk ))u di (tk )]

(83)

k∈Z[t0 ,∞) i=1

in the sense that J (x0 , φc (x(·)), φd (x(·))) =

min

(u c (·),u d (·))∈S (x0 )

J (x0 , u c (·), u d (·)),

x 0 ∈ Rn ,

where Z[t0 ,∞) = {k : t0 ≤ tk < ∞}, L 1ci (x) , −αi (x) + γi 2(x) βiT (x)βi (x), x ∈ Rn ,   1 I , βi (x) 6= 0, R2ci (x) , 2γi (x) m c i 0, βi (x) = 0, q   (αi (x) − wci (V (x))) + (αi (x) − wci (V (x)))2 + (βiT (x)βi (x))2  > 0, βi (x) 6= 0, γi (x) , c0i + βiT (x)βi (x)   0, βi (x) = 0,

(84)

(85)

(86)

T (x)(R (x) + P (x))φ (x) − v (x + f (x)) + v (x ), x ∈ Z, and P (·) and v (·) are given in L 1di (x) , φdi i i i i i 2di 2ui di di 2ui q Theorem 4.1. Furthermore, J (x0 , φc (x(·)), φd (x(·))) = eT V (x0 ), x0 ∈ Rn , where V : Rn → R+ is a control vector Lyapunov function for the impulsive dynamical system (68) and (69). In addition, the nonlinear impulsive dynamical system (79) and (80) with the hybrid feedback control law given by (75) and (82) asymptotically   stable   is globally 1 1 for all σci (·) ∈ Φci and σdi (·) ∈ Φdi , i = 1, . . . , q, where αd = maxi=1,...,q 1+θdi , βd = mini=1,...,q 1−θ , di qγ θdi , γ di , γ di , infx∈Z σmin (R2di (x)), γ di , supx∈Z σmax (R2di (x) + P2ui (x)), i = 1, . . . , q. di

Proof. To show that the hybrid feedback control law (75) and (82) minimizes (83) in the sense of (84), define the Hamiltonian H (x, u c , u d ) = Hc (x, u c ) + Hd (x, u d ),

(87)

where Hc (x, u c ) ,

q X

[L 1ci (x) + u Tci R2ci (x)u ci + vi0 (xi )( f ci (x) + G ci (x)u ci )],

(88)

i=1

Hd (x, u d ) ,

q X

[L 1di (x) + u Tdi R2di (x)u di + vi (xi + f di (x) + G di (x)u di ) − vi (xi )],

(89)

i=1

and note that H (x, φc (·), φd (·)) = 0 and H (x, u c , u d ) ≥ 0, x ∈ Rn , u c ∈ Rm c , u d ∈ Rm d , since H (x, u c , u d ) =

q X

(u ci − φci (x))T R2ci (x)(u ci − φci (x))

i=1

+

q X

(u di − φdi (x))T (R2di (x) + P2ui (x))(u di − φdi (x)),

i=1

x ∈ Rn ,

u c ∈ Rm c ,

u d ∈ Rm d .

(90)

238

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

Thus, J (x0 , u c (·), u d (·)) =

Z

∞

" Hc (x(t), u c (t)) −

t0

q X

# vi0 (xi (t))( f ci (x(t)) + G ci (x(t))u ci (t)) dt

i=1

" +

X

Hd (x(tk ), u d (tk ))

k∈Z[t0 ,∞)

+

q X

# (vi (xi (tk ))) − vi (xi (tk ) + f di (x(tk )) + G di (x(tk ))u di (tk ))

i=1

= − lim e V (x(t)) + e V (x0 ) + T

T

t→∞

Z

∞

Hc (x(t), u c (t))dt + t0

X

Hd (x(tk ), u d (tk ))

k∈Z[t0 ,∞)

≥ eT V (x0 ) = J (x0 , φc (x(·)), φd (x(·))),

(91)

which yields (84). Next, we show that the uncertain nonlinear impulsive dynamical system (79) and (80) with the hybrid feedback control law (75) and (82) is globally asymptotically stable for all σci (·) ∈ Φci (·) and σdi (·) ∈ Φdi (·). It follows from Theorem 4.1 that the hybrid feedback control law (75) and (82) globally asymptotically stabilizes the impulsive q dynamical system (68) and (69) and the vector function V : Rn → R+ is a control vector Lyapunov function for the impulsive dynamical system (68) and (69). Note that with (86) the continuous-time feedback control law (75) can be q rewritten as φci (x) = −γi (x)βi (x), x ∈ Rn , i = 1, . . . , q. Let the control vector Lyapunov function V : Rn → R+ for (68) and (69) be a vector Lyapunov function candidate for (79) and (80). Then the vector Lyapunov derivative components along the trajectories of (79) are given by v˙i (xi ) = vi0 (xi )( f ci (x) + G ci (x)σci (φci (x))) = αi (x) + βiT (x)σci (φci (x)),

x 6∈ Z,

i = 1, . . . , q.

(92)

Note that φci (x) = 0, and hence, σci (φci (x)) = 0 whenever βi (x) = 0 for all i = 1, . . . , q. In this case, it follows from (71) that v˙i (xi ) < wci (V (x)), x 6∈ Z, βi (x) = 0, x 6= 0, i = 1, . . . , q. Next, consider the case where βi (x) 6= 0, i = 1, . . . , q. In this case, note that −c0i βiT (x)βi (x) γi (x) T βi (x)βi (x) = 2 2 q (αi (x) − wci (V (x))) − (αi (x) − wci (V (x)))2 + (βiT (x)βi (x))2

αi (x) − wci (V (x)) −

+

2

< 0,

x 6∈ Z,

βi (x) 6= 0,

(93)

for all i = 1, . . . , q. Thus, the vector Lyapunov derivative components given by (92) satisfy γi (x) T β (x)βi (x) + βiT (x)σci (φci (x)) 2 i 1 1 = wci (V (x)) + φciT (x)φci (x) − φ T (x)σci (φci (x)) 2γi (x) γi (x) ci " # φciT (x)φci (x) 1 T = wci (V (x)) + − φci (x)σci (φci (x)) γi (x) 2

v˙i (xi ) < wci (V (x)) +

≤ wci (V (x)),

x 6∈ Z, βi (x) 6= 0,

(94)

for all σci (·) ∈ Φci and i = 1, . . . , q. Next, consider the Lyapunov difference of each component of V (·) at the resetting instants for the resetting dynamics (80) with u di = φdi (x) given by (82). Note that it follows from (81) that L 1di (x) + wdi (V (x), x) ≥ 0, x ∈ Z, i = 1, . . . , q, and hence, since σdi (·) ∈ Φdi it follows that

239

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

∆vi (xi ) = vi (xi + f di (x) + G di (x)σdi (φdi (x))) − vi (xi ) ≤ vi (xi + f di (x)) − vi (xi ) + P1ui (x)σdi (φdi (x)) + σdiT (φdi (x))P2ui (x)σdi (φdi (x)) + L 1di (x) + wdi (V (x), x) = P1ui (x)σdi (φdi (x)) + σdiT (φdi (x))P2ui (x)σdi (φdi (x)) T + φdi (x)(R2di (x) + P2ui (x))φdi (x) + wdi (V (x), x) T = φdi (x)(R2di (x) + P2ui (x))φdi (x) + σdiT (φdi (x))P2ui (x)σdi (φdi (x)) T − 2φdi (x)(R2di (x) + P2ui (x))σdi (φdi (x)) + wdi (V (x), x)

= [σdi (φdi (x)) − φdi (x)]T (R2di (x) + P2ui (x))[σdi (φdi (x)) − φdi (x)] − σdiT (φdi (x))R2di (x)σdi (φdi (x)) + wdi (V (x), x) ≤ γ di [σdi (φdi (x)) − φdi (x)]T [σdi (φdi (x)) − φdi (x)] − γ di σdiT (φdi (x))σdi (φdi (x)) + wdi (V (x), x) T    1 1 = γ di (1 − θdi2 ) σdi (φdi (x)) − σdi (φdi (x)) − φdi (x) φdi (x) + wdi (V (x), x) 1 + θdi 1 − θdi     m di X 1 1 2 = γ di (1 − θdi ) σdi j (φdi (x)) − φdi j (x) σdi j (φdi (x)) − φdi j (x) 1 + θdi 1 − θdi j=1 + wdi (V (x), x) ≤ wdi (V (x), x), x ∈ Z,

i = 1, . . . , q.

(95)

Since the impulsive dynamical system (56) and (57) is globally asymptotically stable it follows from Theorem 3.2 that the nonlinear impulsive dynamical system (79) and (80) is globally asymptotically stable for all σci (·) ∈ Φci and σdi (·) ∈ Φdi , i = 1, . . . , q.  Remark 5.1. It follows from Theorem 5.1 that with the hybrid feedback stabilizing control law (75) and (82) the 1 nonlinear impulsive dynamical system (68) and (69) has a sector (and hence gain) margin (( 21 , ∞), ( 1+θ , 1 )), di 1−θdi i = 1, . . . , q, in each decentralized input channel. For details on stability margins for nonlinear dynamical systems, see [38–40]. 6. Decentralized control for large-scale impulsive dynamical systems In this section, we apply the proposed hybrid control framework to decentralized control of large-scale nonlinear impulsive dynamical systems. Specifically, we consider the large-scale dynamical system G involving energy exchange between n interconnected subsystems. Let xi : [0, ∞) → R+ denote the energy (and hence a nonnegative quantity) n of the ith subsystem, let u ci : [0, ∞) → R denote the control input to the ith subsystem, let σci j : R+ → R+ , i 6= j, i, j = 1, . . . , n, denote the instantaneous rate of energy flow from the jth subsystem to the ith subsystem between n resettings, let σdi j : R+ → R+ , i 6= j, i, j = 1, . . . , n, denote the amount of energy transferred from the jth n subsystem to the ith subsystem at the resetting instant, and let Z ⊂ R+ be a resetting set for the large-scale impulsive dynamical system G. An energy balance for each subsystem Gi , i = 1, . . . , q, yields [6,41] n X

x˙i (t) =

[σci j (x(t)) − σc ji (x(t))] + G ci (x(t))u ci (t),

x(t0 ) = x0 ,

x(t) 6∈ Z,

t ≥ t0 ,

(96)

j=1, j6=i

∆xi (t) =

n X

[σdi j (x(t)) − σd ji (x(t))] + G di (x(t))u di (t),

x(t) ∈ Z,

(97)

j=1, j6=i

or, equivalently, in vector form for the large-scale impulsive dynamical system G x(t) ˙

= f c (x(t)) + G c (x(t))u c (t),

∆x(t) = f d (x(t)) + G d (x(t))u d (t),

x(t0 ) = x0 , x(t) 6∈ Z, x(t) ∈ Z,

t ≥ t0 ,

(98) (99)

240

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

P n where x(t) = [x1 (t), . . . , xn (t)]T , t ≥ t0 , f ci (x) = nj=1, j6=i φci j (x), where φci j (x) , σci j (x) − σc ji (x), x ∈ R+ , i 6= j, i, j = 1, . . . , q, denotes the net energy flow from the jth subsystem to the ith subsystem between resettings, n G c (x) = diag[G c1 (x), . . . , G cn (x)] = diag[x1 , . . . , xn ], x ∈ R+ , G d (x) = diag[G d1 (x), . . . , G dn (x)], x ∈ Z, Pn G di : Rn → R, i = 1, . . . , n, u c (t) ∈ Rn , t ≥ t0 , u d (tk ) ∈ Rn , k ∈ Z+ , f di (x) = j=1, j6=i φdi j (x), where φdi j (x) , σdi j (x) − σd ji (x), x ∈ Z, i 6= j, i, j = 1, . . . , q, denotes the net amount of energy transferred n from the jth subsystem to the ith subsystem at the instant of resetting. Here, we assume that σci j : R+ → R+ , n i 6= j, i, j = 1, . . . , n, are locally Lipschitz continuous on R+ , σci j (0) = 0, i 6= j, i, j = 1, . . . , n, and u c = [u c1 , . . . , u cn ]T : R → Rn is such that u ci : R → R, i = 1, . . . , n, are bounded piecewise continuous n functions P of time. Furthermore, we assume that σci j (x) = 0, x ∈ R+ , whenever x j = 0, i 6= j, i, j = 1, . . . , n, n and xi + j=1, j6=i φdi j (x) ≥ 0, x ∈ Z. In this case, f c (·) is essentially nonnegative [41,42] (i.e., f ci (x) ≥ 0 for all n

n

x ∈ R+ such that xi = 0, i = 1, . . . , n) and x + f d (x), x ∈ Z ⊂ R+ , is nonnegative (i.e. xi + f di (x) ≥ 0 for all x ∈ Z, i = 1, . . . , n). The above constraints imply that if the energy of the jth subsystem of G is zero, then this subsystem cannot supply any energy to its surroundings between resettings and the ith subsystem of G cannot transfer more energy to its surroundings than it possesses at the instant of resetting. Finally, in order to ensure that the trajectories of the closed-loop system remain in the nonnegative orthant of the state space for all nonnegative initial conditions, we seek a hybrid feedback control law (u c (·), u d (·)) that guarantees the continuous-time closed-loop system dynamics (98) are essentially nonnegative and the closed-loop system states after the resettings are nonnegative [43]. For the dynamical system G, consider the control vector Lyapunov function candidate V (x) = n [v1 (x1 ), . . . , vn (xn )]T , x ∈ R+ , given by V (x) = [x1 , . . . , xn ]T ,

n

x ∈ R+ .

(100) n R+ ,

n R+ ,

is such that v(0) = 0, v(x) > 0, x 6= 0, x ∈ and v(x) → ∞ as Note that V (0) = 0 and v(x) , eT V (x), x ∈ n n kxk → ∞ with x ∈ R+ . Also, note that since vi (xi ) = xi , x ∈ R , i = 1, . . . , n, are linear functions of x, it follows that P2ui (x) ≡ 0, i = 1, . . . , n, and hence, by (76), φdi (x) ≡ 0, i = 1, . . . , n. Furthermore, consider the functions " #T n n X X wc (V (x), x) = −σc11 (v1 (x1 )) + φc1 j (x), . . . , −σcnn (vn (xn )) + φcn j (x) , j=1, j6=1

j=1, j6=n

n R+ ,

wd (V (x), x) =

x∈ " n X

(101) n X

φd1 j (x), . . . ,

j=1, j6=1

#T φdn j (x)

,

x ∈ Z,

(102)

j=1, j6=n n

where σcii : R+ → R+ , i = 1, . . . , n, are positive definite functions, and note that wc (·, x) ∈ Wc , x ∈ R+ , wc (0, 0) = 0, wd (·, ·) does not depend on V (·), and hence wd (·, x) ∈ Wd , x ∈ Z. Also note that it follows from n n Remark 4.2 that Ri , {x ∈ R+ , xi 6= 0 : Vi0 (xi )G ci (x) = 0} = {x ∈ R+ , xi 6= 0 : xi = 0} = Ø, and hence, condition (71) is satisfied for V (·) and wc (·, ·) given by (100) and (101), respectively, and condition (72) is satisfied as an equality for wd (·, ·) given by (102) and P2ui (x) ≡ 0, i = 1, . . . , n. To show that the impulsive dynamical system z˙ (t)

= wc (z(t), x(t)),

∆z(t) = wd (z(t), x(t)),

z(t0 ) = z 0 ,

x(t) 6∈ Z,

x(t) ∈ Z,

t ≥ t0 ,

(103) (104)

n R+ ,

where z(t) ∈ t ≥ t0 , x(t), t ≥ t0 , is the solution to (98) and (99), the ith component of wc (z, x) is given P n n by wci (z, x) = −σcii (z i ) + nj=1, j6=i φci j (x), z ∈ R+ , x ∈ R+ , and the ith component of wd (z, x) is given by Pn wdi (z, x) = j=1, j6=i φdi j (x), x ∈ Z, is globally asymptotically stable with respect to z uniformly in x 0 , consider n

˜ is radially unbounded, v(0) ˜ = 0, v(z) ˜ > 0, the partial Lyapunov function candidate v(z) ˜ = eT z, z ∈ R+ . Note that v(·) P P Pn n n n n n Pn ˙˜ z ∈ R+ , z 6= 0, v(z) = − i=1 σcii (z i ) + i=1 φ (x) = − σ (z ) < 0, z ∈ R , z = 6 0, x ∈ R+ , + j=1, j6=i ci j i=1 cii i Pn Pn n and ∆v(z) ˜ = i=1 j=1, j6=i φdi j (x) = 0, z ∈ R+ , x ∈ Z. Thus, it follows from Theorem 2.1 of [14] that the impulsive dynamical system (103), (104), (98) and (99) is globally asymptotically stable with respect to z uniformly

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

241

Fig. 6.1. Controlled system states versus time. n

in x0 . Hence, it follows from Theorem 4.1 that V (x), x ∈ R+ , given by (100) is a control vector Lyapunov function for the dynamical system (98) and (99). P n Next, using (75) with αi (x) = vi0 (xi ) f ci (x) = nj=1, j6=i φci j (x), βi (x) = xi , x ∈ R+ , and c0i > 0, i = 1, . . . , n, we construct a globally stabilizing hybrid decentralized feedback controller for (98) and (99) given by q    2 (x ) + x 2  σcii σ (x ) +  i cii i i    xi , xi 6= 0, − c0i + 2 φci (x) = i = 1, . . . , n, (105) x i    0, xi = 0, and φdi (x) ≡ 0,

i = 1, . . . , n.

(106)

It can be seen from the structure of the feedback control law (105) and (106) that the continuous-time closed-loop system dynamics are essentially nonnegative and the closed-loop system states after the resettings are nonnegative. n Furthermore, since αi (x) − wci (V (x), x) = σcii (vi (xi )), x ∈ R+ , i = 1, . . . , n, the continuous-time feedback controller φc (·) is fully independent from f c (x) which represents the internal interconnections of the large-scale system dynamics, and hence, is robust against full modeling uncertainty in f c (x). For the following simulation we consider (98) and (99) with σci j (x) = σci j xi x j , σcii (x) = σcii xi2 , and σdi j (x) =Pσdi j x j , where σci j ≥ 0, i 6= j, i, j = 1, . . . , n, σcii > 0, i = 1, . . . , n, and σdi j ≥ 0, i 6= j, i, j = 1, . . . , n, and 1 ≥ nj=1, j6=i σd ji , i = 1, . . . , n. Note that in this case the conditions of Proposition 4.1 are satisfied, and hence n

the continuous-time feedback control law (105) is continuous on R+ . For our simulation we set n = 3, σc11 = 0.1, σc22 = 0.2, σc33 = 0.01, σc12 = 2, σc13 = 3, σc21 = 1.5, σc23 = 0.3, σc31 = 4.4, σc32 = 0.6, σd12 = 0.75, σd13 = 0.33, σd21 = 0.2, σd23 = 0.5, σd31 = 0.66, σd32 = 0.2, c01 = 1, c02 = 1, c03 = 0.25, the resetting set Z = {x ∈ R3 : x1 + x2 − 0.75 = 0}, with initial condition x0 = [3, 4, 1]T . Fig. 6.1 shows the states of the closed-loop system versus time and Fig. 6.2 shows continuous-time control signal for each decentralized control channel as a function of time. 7. Conclusion In this paper we developed vector Lyapunov stability results for nonlinear impulsive dynamical systems. Specifically, we addressed the generalized comparison principle involving hybrid comparison dynamics that are dependent on comparison system states as well as the nonlinear impulsive dynamical system states. We presented Lyapunov stability theorem to provide sufficient conditions for several types of stability using vector Lyapunov

242

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

Fig. 6.2. Control signals in each decentralized control channel versus time.

functions. In addition, we addressed partial stability of nonlinear impulsive dynamical systems via vector Lyapunov functions. Furthermore, we generalized the novel notion of control vector Lyapunov functions to impulsive dynamical systems. Using this notion, we developed universal decentralized hybrid feedback stabilizer for a decentralized affine in control nonlinear impulsive dynamical system with robustness guarantees against full modeling and input uncertainty. Finally, we designed decentralized controllers for large-scale impulsive dynamical systems and presented a numerical example. Acknowledgments This research was supported in part by the Air Force Office of Scientific Research under Grant F49620-03-1-0178 and by the Naval Surface Warfare Center under Contract N65540-05-C-0028. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. D.D. Bainov, P.S. Simeonov, Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood, Chichester, 1989. S. Hu, V. Lakshmikantham, S. Leela, Impulsive differential systems and the pulse phenomena, J. Math. Anal. Appl. 137 (1989) 605–612. D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific, Singapore, 1995. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. W.M. Haddad, V. Chellaboina, S.G. Nersesov, Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity, and Control, Princeton University Press, Princeton, NJ, 2006. R.T. Bupp, D.S. Bernstein, V. Chellaboina, W.M. Haddad, Resetting virtual absorbers for vibration control, J. Vib. Control. 6 (2000) 61–83. W.M. Haddad, V. Chellaboina, N.A. Kablar, Nonlinear impulsive dynamical systems. Part I: Stability and dissipativity, Internat. J. Control 74 (2001) 1631–1658. W.M. Haddad, V. Chellaboina, N.A. Kablar, Nonlinear impulsive dynamical systems. Part II: Stability of feedback interconnections and optimality, Internat. J. Control 74 (2001) 1659–1677. V.D. Mil’man, A.D. Myshkis, On the stability of motion in the presence of impulses, Siberian Math. J. 1 (1960) 233–237. V.D. Mil’man, A.D. Myshkis, Approximate methods of solutions of differential equations, in: Random Impulses in Linear Dynamical Systems, Publ. House. Acad. Sci. Ukr., SSR, Kiev, 1963, pp. 64–81. D.D. Bainov, P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific & Technical, London, 1993. V. Chellaboina, S.P. Bhat, W.M. Haddad, An invariance principle for nonlinear hybrid and impulsive dynamical systems, Nonlinear Anal. TMA 53 (2003) 527–550. W. Haddad, V. Chellaboina, S. Nersesov, A unification between partial stability of state-dependent impulsive systems and stability theory for time-dependent impulsive systems, Int. J. Hybrid Syst. 2 (2) (2002) 155–168. S.G. Nersesov, W.M. Haddad, On the stability and control of nonlinear dynamical systems via vector Lyapunov functions, IEEE Trans. Automat. Control 51 (2) (2006) 203–215.

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243 [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

243

V. Jurdjevic, J.P. Quinn, Controllability and stability, J. Differential Equations 28 (1978) 381–389. Z. Artstein, Stabilization with relaxed controls, Nonlinear Anal. TMA 7 (1983) 1163–1173. E.D. Sontag, A universal construction of Artstein’s theorem on nonlinear stabilization, Systems Control Lett. 13 (1989) 117–123. J. Tsinias, Existence of control Lyapunov functions and applications to state feedback stabilizability of nonlinear systems, SIAM J. Control Optim. 29 (1991) 457–473. R. Bellman, Vector Lyapunov functions, SIAM J. Control 1 (1962) 32–34. V.M. Matrosov, Method of vector Liapunov functions of interconnected systems with distributed parameters (Survey), Avtomat. i Telemekh. 33 (1972) 63–75 (in Russian). A.N. Michel, R.K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, Academic Press, New York, NY, 1977. L.T. Gruji´c, A.A. Martynyuk, M. Ribbens-Pavella, Large Scale Systems: Stability Under Structural and Singular Perturbations, SpringerVerlag, Berlin, 1987. J. Lunze, Stability analysis of large-scale systems composed of strongly coupled similar subsystems, Automatica 25 (1989) 561–570. V. Lakshmikantham, V.M. Matrosov, S. Sivasundaram, Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems, Kluwer, Dordrecht, The Netherlands, 1991. ˇ D.D. Siljak, Large-Scale Dynamic Systems: Stability and Structure, Elsevier North-Holland, New York, NY, 1978. ˇ D.D. Siljak, Complex dynamical systems: Dimensionality, structure and uncertainty, Large Scale Syst. 4 (1983) 279–294. A.A. Martynyuk, Stability by Liapunov’s Matrix Function Method with Applications, Marcel Dekker, New York, NY, 1998. A.A. Martynyuk, Qualitative Methods in Nonlinear Dynamics. Novel Approaches to Liapunov’s Matrix Functions, Marcel Dekker, New York, NY, 2002. Z. Drici, New directions in the method of vector Lyapunov functions, J. Math. Anal. Appl. 184 (1994) 317–325. G. Amicucci, S. Monaco, D. Normand-Cyrot, Control Lyapunov stabilization of affine discrete-time systems, in: Proc. IEEE Conf. Dec. Contr., San Diego, CA, 1997, pp. 923–924. V. Chellaboina, W. Haddad, Stability margins of discrete-time nonlinear-nonquadratic optimal regulators, in: Proc. IEEE Conf. Dec. Contr., Tampa, FL, 1998, pp. 1786–1791. E. Kamke, Zur Theorie der Systeme gew¨ohnlicher Differential—Gleichungen. II, Acta Math. 58 (1931) 57–85. T. Waˇzewski, Syst`emes des e´ quations et des in´egalit´es diff´erentielles ordinaires aux deuxi´emes membres monotones et leurs applications, Ann. Soc. Poln. Mat. 23 (1950) 112–166. H.K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, 1996. V. Chellaboina, W.M. Haddad, A unification between partial stability and stability theory for time-varying systems, Control Syst. Mag. 22 (6) (2002) 66–75; 23 (2003) 101 (erratum). D.S. Bernstein, Matrix Mathematics, Princeton University Press, Princeton, NJ, 2005. R. Sepulchre, M. Jankovi´c, P. Kokotovi´c, Constructive Nonlinear Control, Springer-Verlag, New York, NY, 1997. V. Chellaboina, W.M. Haddad, Stability margins of nonlinear optimal regulators with nonquadratic performance criteria involving crossweighting terms, Systems Control Lett. 39 (2000) 71–78. W.M. Haddad, V. Chellaboina, S.G. Nersesov, On the equivalence between dissipativity and optimality of nonlinear hybrid controllers, Int. J. Hybrid Syst. 1 (2001) 51–65. W.M. Haddad, V. Chellaboina, S.G. Nersesov, Thermodynamics. A Dynamical Systems Approach, Princeton University Press, Princeton, NJ, 2005. W.M. Haddad, V. Chellaboina, Stability and dissipativity theory for nonnegative dynamical systems: A unified analysis framework for biological and physiological systems, Nonlinear Anal. RWA 6 (2005) 35–65. W.M. Haddad, V. Chellaboina, S.G. Nersesov, Hybrid nonnegative and compartmental dynamical systems, J. Math. Probab. Eng. 8 (2002) 493–515.