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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 2, FEBRUARY 2006 203

On the Stability and Control of Nonlinear DynamicalSystems via Vector Lyapunov Functions

Sergey G. Nersesov, Member, IEEE, and Wassim M. Haddad, Senior Member, IEEE

Abstract—Vector Lyapunov theory has been developed toweaken the hypothesis of standard Lyapunov theory in orderto enlarge the class of Lyapunov functions that can be used foranalyzing system stability. In this paper, we extend the theory ofvector Lyapunov functions by constructing a generalized compar-ison system whose vector field can be a function of the comparisonsystem states as well as the nonlinear dynamical system states.Furthermore, we present a generalized convergence result which,in the case of a scalar comparison system, specializes to theclassical Krasovskii–LaSalle invariant set theorem. In addition,we introduce the notion of a control vector Lyapunov functionas a generalization of control Lyapunov functions, and show thatasymptotic stabilizability of a nonlinear dynamical system isequivalent to the existence of a control vector Lyapunov function.Moreover, using control vector Lyapunov functions, we constructa universal decentralized feedback control law for a decentralizednonlinear dynamical system that possesses guaranteed gain andsector margins in each decentralized input channel. Furthermore,we establish connections between the recently developed notion ofvector dissipativity and optimality of the proposed decentralizedfeedback control law. Finally, the proposed control framework isused to construct decentralized controllers for large-scale non-linear systems with robustness guarantees against full modelinguncertainty.

Index Terms—Comparison principle, control vector Lyapunovfunctions, decentralized control, gain and sector margins, invari-ance principle, inverse optimality, large-scale systems, partial sta-bility, vector Lyapunov functions.

I. INTRODUCTION

ONE OF THE MOST basic issues in system theory is thestability of dynamical systems. The most complete con-

tribution to the stability analysis of nonlinear dynamical sys-tems is due to Lyapunov [1]. Lyapunov’s results, along withthe Krasovskii–LaSalle invariance principle [2]–[4], provide apowerful framework for analyzing the stability of nonlinear dy-namical systems. Lyapunov methods have also been used bycontrol system designers to obtain stabilizing feedback con-trollers for nonlinear systems. In particular, for smooth feed-back, Lyapunov-based methods were inspired by Jurdjevic andQuinn [5] who give sufficient conditions for smooth stabiliza-tion based on the ability of constructing a Lyapunov function forthe closed-loop system. More recently, Artstein [6] introduced

Manuscript received May 21, 2004; revised October 30, 2005. Recommendedby Associate Editor J. M. Berg. This work was supported in part by the Air ForceOffice of Scientific Research under Grant F49620-03-1-0178.

S. G. Nersesov is with the Department of Mechanical Engineering, VillanovaUniversity, Villanova, PA 19085 USA (e-mail: [email protected]).

W. M. Haddad is with the School of Aerospace Engineering, GeorgiaInstitute of Technology, Atlanta, GA 30332 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2005.863496

the notion of a control Lyapunov function whose existence guar-antees a feedback control law which globally stabilizes a non-linear dynamical system. In general, the feedback control lawis not necessarily smooth, but can be guaranteed to be at leastcontinuous at the origin in addition to being smooth everywhereelse. Even though for certain classes of nonlinear dynamical sys-tems a universal construction of a feedback stabilizer can be ob-tained using control Lyapunov functions [7], [8], there does notexist a unified procedure for finding a Lyapunov function candi-date that will stabilize the closed-loop system for general non-linear systems.

In an attempt to simplify the construction of Lyapunov func-tions for the analysis and control design of nonlinear dynam-ical systems, several researchers have resorted to vector Lya-punov functions as an alternative to scalar Lyapunov functions.Vector Lyapunov functions were first introduced by Bellman [9]and Matrosov [10], and further developed in [11]–[14], with[11]–[13] and [15]–[18] exploiting their utility for analyzinglarge-scale systems. The use of vector Lyapunov functions indynamical system theory offers a very flexible framework sinceeach component of the vector Lyapunov function can satisfyless rigid requirements as compared to a single scalar Lyapunovfunction. Weakening the hypothesis on the Lyapunov functionenlarges the class of Lyapunov functions that can be used foranalyzing system stability. In particular, each component of avector Lyapunov function need not be positive definite with anegative or even negative–semidefinite derivative. Alternatively,the time derivative of the vector Lyapunov function need onlysatisfy an element-by-element inequality involving a vector fieldof a certain comparison system. Since in this case the stabilityproperties of the comparison system imply the stability proper-ties of the dynamical system, the use of vector Lyapunov theorycan significantly reduce the complexity (i.e., dimensionality) ofthe dynamical system being analyzed. Extensions of vector Lya-punov function theory that include relaxed conditions on stan-dard vector Lyapunov functions as well as matrix Lyapunovfunctions appear in [17]–[19].

In this paper, we extend the theory of vector Lyapunov func-tions in several directions. Specifically, we construct a general-ized comparison system whose vector field can be a functionof the comparison system states as well as the nonlinear dy-namical system states. Next, using partial stability notions [20],[21] for the comparison system we provide sufficient condi-tions for stability of the nonlinear dynamical system. In addi-tion, we present a convergence result reminiscent to the invari-ance principle that allows us to weaken the hypothesis on thecomparison system while guaranteeing asymptotic stability ofthe nonlinear dynamical system via vector Lyapunov functions.

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204 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 2, FEBRUARY 2006

Furthermore, we introduce the notion of a control vector Lya-punov function as a generalization of control Lyapunov func-tions and show that asymptotic stabilizability of a nonlinear dy-namical system is equivalent to the existence of a control vectorLyapunov function. In addition, using control vector Lyapunovfunctions, we present a universal decentralized feedback sta-bilizer for a decentralized affine in the control nonlinear dy-namical system with guaranteed gain and sector margins. Fur-thermore, we establish connections between vector dissipativitynotions [22] and inverse optimality of decentralized nonlinearregulators. These results are then used to develop decentralizedcontrollers for large-scale dynamical systems with robustnessguarantees against full modeling and input uncertainty. Finally,it is important to stress that the main purpose of this paper isnot to present a procedure for constructing a generalized com-parison system or constructing vector Lyapunov functions, butrather to develop a new and novel analysis and control designframework for nonlinear systems based on control vector Lya-punov functions. As such, a key contribution of the paper isa control design framework for nonlinear dynamical systemspredicated on existing vector Lyapunov methods.

II. MATHEMATICAL PRELIMINARIES

In this section, we introduce notation and definitions neededfor developing stability analysis and synthesis results for non-linear dynamical systems via vector Lyapunov functions. Letdenote the set of real numbers, denote the set of nonnega-tive integers, denote the set of 1 column vectors, anddenote transpose. For we write (respectively,

) to indicate that every component of is nonnegative (re-spectively, positive). In this case, we say that is nonnegativeor positive, respectively. Let and denote the nonnegativeand positive orthants of , that is, if , then and

are equivalent, respectively, to and . Fur-

thermore, let and denote the interior and the closure of theset , respectively. Finally, we write for an arbitraryspatial vector norm in , for the Fréchet derivative ofat , , , , for the open ball centered at withradius , for the ones vector, that is, ,and as to denote that approaches theset , that is, for each there exists such that dist

for all , where dist .

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

Definition 2.1 [15]: A function, where , is of class if for every fixed

, , , for allsuch that , , , , wheredenotes the th component of .

If we say that satisfies the Kamke condition[23], [24]. Note that if , where

, then the function is of class if and only ifis essentially nonnegative for all , that is, all the off-diagonal entries of the matrix function are nonnegative.Furthermore, note that it follows from Definition 2.1 that anyscalar ( ) function is of class .

Finally, we introduce the notion of class functions in-volving nondecreasing functions.

Definition 2.2 [14]: A function, where , is of class if for every fixed

, for all suchthat .

Note that if , then .

III. GENERALIZED DIFFERENTIAL INEQUALITIES

In this section, we develop a generalized comparison prin-ciple involving differential inequalities, wherein the underlyingcomparison system is partially dependent on the state of a dy-namical system. Specifically, we consider the nonlinear com-parison system given by

(1)

where , , is the comparison systemstate vector, is a given continuous function,

is the maximal interval of existence of a solutionof (1), is an open set, , and .

We assume that is continuous in and satisfies theLipschitz condition

(2)

for all , where and is a Lipschitzconstant. Hence, it follows from [25, Th. 2.2] that there exists

such that (1) has a unique solution over the time interval.

Theorem 3.1: Consider the nonlinear comparison system (1).Assume that the function is continuous and

is of class . If there exists a continuously differentiablevector function such that

(3)

then , , implies

(4)

where , , is the solution to (1).Proof: Since , , is continuous it follows that

for sufficiently small :

(5)

Now, suppose, ad absurdum, inequality (4) does not hold on theentire interval . Then there exists such that

, , and for at least one

(6)

and

(7)

Since , it follows from (3), (6), and (7) that

(8)

NERSESOV AND HADDAD: ON THE STABILITY AND CONTROL OF NONLINEAR DYNAMICAL SYSTEMS 205

which, along with (6), implies that for sufficiently small, , . This contradicts the fact that

, , and establishes (4).Next, we present a stronger version of Theorem 3.1 where the

strict inequalities are replaced by soft inequalities.Theorem 3.2: Consider the nonlinear comparison system (1).

Assume that the function is continuous andis of class . Let , , be the solution to (1)

and be a compact interval. If there exists acontinuously differentiable vector functionsuch that

(9)

then , , implies ,.

Proof: Consider the family of comparison systems givenby

(10)

where , , and , and let the solution to(10) be denoted by , . Now,it follows from [26, p. 17, Th. 3] that there exists a compactinterval such that ,

, is defined for all sufficiently large . Moreover, itfollows from Theorem 3.1 that

(11)

for all sufficiently large . Since the functions, , , are continuous in

, decreasing in , and bounded from below, it follows that thesequence of functions converges uniformlyon the compact interval as , that is, thereexists a continuous function such that

(12)

uniformly on . Hence, it follows from (11) and (12)that

(13)

Next, note that it follows from (10) that

(14)

for all , which implies that and, sinceand are continuous,

as uniformly on . Hence, takingthe limit as on both sides of (14) yields

(15)

which implies that is the solution to (1) on the interval. Hence, by uniqueness of solutions of (1) we ob-

tain that , . This along with (13) provesthe result.

Next, consider the nonlinear dynamical system given by

(16)

where , , is the system state vector,is the maximal interval of existence of a solution of (16),

is an open set, , and is Lipschitz continuous on .The following result is a direct consequence of Theorem 3.2.

Corollary 3.1: Consider the nonlinear dynamical system(16). Assume there exists a continuously differentiable vectorfunction such that

(17)

where is a continuous function, ,and

(18)

has a unique solution , , where , , isa solution to (16). If is a compactinterval, then , , implies ,

.Proof: For any given , the solution , ,

to (16) is a well defined function of time. Hence, define, , and note that (17) implies

(19)

Moreover, if is a compact in-terval, then it follows from Theorem 3.2, withand , that ,

, which establishes the result.If in (16), is globally Lipschitz continuous, then

(16) has a unique solution for all . A more restrictivesufficient condition for global existence and uniqueness of solu-tions to (16) is continuous differentiability of anduniform boundedness of on . Note that if the solutionsto (16) and (18) are globally defined for all and ,then the result of Corollary 3.1 holds for any arbitrarily large butcompact interval . For the remainder of thispaper, we assume that the solutions to the systems (16) and (18)are defined for all . Continuous differentiability ofand provides a sufficient condition for the existence anduniqueness of solutions to (16) and (18) for all .


IV. STABILITY THEORY VIA VECTOR LYAPUNOV FUNCTIONS

In this section, we develop a generalized vector Lyapunovfunction framework for the stability analysis of nonlinear dy-namical systems using the generalized comparison principle de-veloped in Section III. Specifically, consider the cascade non-linear dynamical system given by

(20)

(21)

where , , , , isthe solution to (20) and (21), is continuous,

, , is Lipschitz continuouson , and . The following definition involving thenotion of partial stability is needed for the next result.

Definition 4.1 [21]: The nonlinear dynamical systemgiven by (20) and (21) is Lyapunov stable with respect to if,for every and , there existssuch that implies that for all . isLyapunov stable with respect to uniformly in if, for every

, there exists such that implies thatfor all and for all . is asymptotically

stable with respect to if it is Lyapunov stable with respect toand, for every , there exists such that

implies that . is asymptoticallystable with respect to uniformly in if it is Lyapunov stablewith respect to uniformly in and there exists suchthat implies that for all .is exponentially stable with respect to uniformly in if thereexist positive scalars , , and such that impliesthat , , for all . Finally,

is globally asymptotically (respectively, exponentially) stablewith respect to uniformly in if the previous two statementshold for all and .

Theorem 4.1: Consider the nonlinear dynamical system(16). Assume that there exist a continuously differentiablevector function and a positive vectorsuch that , the scalar function definedby , , is such that , , and

(22)

where is continuous, , and. Then, the following statements hold.

i) If the nonlinear dynamical system (20), (21) is Lya-punov stable with respect to uniformly in , thenthe zero solution to (16) is Lyapunov stable.

ii) If the nonlinear dynamical system (20), (21) is asymp-totically stable with respect to uniformly in , thenthe zero solution to (16) is asymptoticallystable.

iii) If , , is radiallyunbounded, and the nonlinear dynamical system (20),(21) is globally asymptotically stable with respect to

uniformly in , then the zero solution to(16) is globally asymptotically stable.

iv) If there exist constants , , and suchthat satisfies

(23)

and the nonlinear dynamical system (20), (21) is ex-ponentially stable with respect to uniformly in ,then the zero solution to (16) is exponen-tially stable.

v) If , , there exist constants ,, and such that satisfies (23), and

the nonlinear dynamical system (20), (21) is globallyexponentially stable with respect to uniformly in ,then the zero solution to (16) is globallyexponentially stable.

Proof: Assume there exist a continuously differentiablevector function and a positive vector

such that , , is positive definite,that is, and , . Note that since

, , the func-tion , , is also positive definite. Thus, there exist

and class functions [27] such thatand

(24)

i) Let and choose . It fol-lows from Lyapunov stability of the nonlinear dynam-ical system (20), (21) with respect to uniformly in

that there exists such thatif , where denotes the absolute sumnorm, then , , for any .Now, choose , . Since ,

, is continuous, the function , ,is also continuous. Hence, for there ex-ists such that , andif , then ,which implies that , . Now, with

, , and the assumption that, , it follows from (22) and Corol-

lary 3.1 that on any compactinterval , and hence, ,

. Let be such that ,, for all . Thus, using (24),

if , then

(25)which implies , . Now,suppose, ad absurdum, that for some thereexists such that . Then, for

and the compact interval it followsfrom (22) and Corollary 3.1 that ,which implies that

. This is a contradiction, and hence, fora given there exists such that forall , , , which impliesLyapunov stability of the zero solution to(16).


ii) It follows from i) and the asymptotic stability of thenonlinear dynamical system (20), (21) with respectto uniformly in that the zero solution to (16) isLyapunov stable and there exists such that if

, then for any . Asin i), choose , . It followsfrom Lyapunov stability of the zero solution to (16)and the continuity of that there exists

such that if , then ,, and . Thus,

by asymptotic stability of (20), (21) with respect touniformly in , for any arbitrary there

exists such that ,. Thus, it follows from (22) and Corollary

3.1 that on any compactinterval , and hence, ,

, and, by (24)

(26)Now, suppose, ad absurdum, that for some

, , that is, there ex-ists a sequence , with as ,such that , , for some .Choose and the interval such thatat least one . Then it follows from(26) that , which is acontradiction. Hence, there exists such that forall , which, along withLyapunov stability, implies asymptotic stability of thezero solution to (16).

iii) Suppose , , is a ra-dially unbounded function, and the nonlinear dynam-ical system (20), (21) is globally asymptotically stablewith respect to uniformly in . In this case, for

the inequality (24) holds for all ,where the functions are of class[27]. Furthermore, Lyapunov stability of the zero so-lution to (16) follows from i). Next, for any

and , identical argumentsas in ii) can be used to show that ,which proves global asymptotic stability of the zero so-lution to (16).

iv) Suppose (23) holds. Since , then

(27)

where and. It follows from the

exponential stability of the nonlinear dynamicalsystem (20), (21) with respect to uniformly inthat there exist positive constants , , and such thatif , then

(28)

for all . Choose , .By continuity of , there exists

such that for all ,

. Furthermore, it follows from (22),(27), (28), and Corollary 3.1 that, for all ,the inequality

(29)

holds on any compact interval . This in turnimplies that, for any

(30)Now, suppose, ad absurdum, that for some

there exists such that

. Then for

the compact interval , it follows from (30) that

, whichis a contradiction. Thus, inequality (30) holds for all

establishing exponential stability of the zerosolution to (16).

v) The proof is identical to the proof of iv).

If satisfies the conditions of Theorem 4.1we say that , , is a vector Lyapunov function [15].Note that for stability analysis each component of a vector Lya-punov function need not be positive definite with a negative def-inite or negative-semidefinite time derivative along the trajecto-ries of (20), (21). This provides more flexibility in searching fora vector Lyapunov function as compared to a scalar Lyapunovfunction for addressing the stability of nonlinear dynamical sys-tems. It is important to note here that comparison systems withvector fields dependent on the states of both the system dy-namics and the comparison system have been addressed in theliterature [12], [28], [29], with [12] providing stability analysisusing partial stability notions. However, a key difference be-tween our formulation and the results given in [12] is in thedefinitions of partial stability used to analyze the stability ofthe generalized comparison system. Specifically, the partial sta-bility definitions used in [12] (see Definitions 2 and 3 on pages161 and 162) require that the entire initial system state of thegeneralized comparison system lie in a neighborhood of theorigin, whereas in our definition of partial stability the initialsystem state corresponding to (21) can be arbitrary. This weakerassumption leads to stronger results. Furthermore, in Theorem4.1 each component of the vector Lyapunov function is depen-dent on the entire state of the dynamical system, while in The-orem 29 of [12] (see p. 210) the vector Lyapunov function iscomponent-decoupled, that is, ,

, . In addition, in Theorem 4.1 we onlyrequire that the scalar function , , bepositive definite, while in [12, Th. 29] each component of avector Lyapunov function , , is assumed to be apositive–definite function of its argument.

Remark 4.1: Sufficient conditions for partial stability of thenonlinear dynamical system (20), (21) are given in [21]. Specif-ically, [21, Th. 1] establishes partial stability of (20), (21) in


terms of a scalar Lyapunov function that is dependent on boththe states and . Alternatively, [21, Cor. 1] provides partialstability of (20), (21) in terms of a scalar Lyapunov functionthat is only dependent on the comparison system state whichcan simplify the stability analysis. In this case, the expanded di-mension of the system (20), (21) does not introduce additionalcomplexity for the partial stability analysis of the generalizedcomparison system. As in standard vector Lyapunov theory, thisensures a reduced dimension for the analysis of the comparisonsystem while addressing a more general class of nonlinear sys-tems. This point is further illustrated in Section VII.

The following corollary to Theorem 4.1 is immediate and cor-responds to the standard vector Lyapunov theorem addressed inthe literature [15].

Corollary 4.1: Consider the nonlinear dynamical system(16). Assume that there exist a continuously differentiablevector function and a positive vectorsuch that , the scalar function definedby , , is such that , , and

(31)

where is continuous, , and .Then, the stability properties of the zero solution to

(32)

where , imply the corresponding stability properties ofthe zero solution to (16).

Proof: The proof is a direct consequence of Theorem 4.1with .

Next, we present a convergence result via vector Lyapunovfunctions that allows us to establish asymptotic stability of thenonlinear dynamical system (16) using weaker conditions thanthose assumed in Theorem 4.1.

Theorem 4.2: Consider the nonlinear dynamical system(16), assume that there exist a continuously differentiablevector function and apositive vector such that , the scalar function

defined by , , is such that, , and

(33)

where is continuous, , and, such that the nonlinear dynamical system (20),

(21) is Lyapunov stable with respect to uniformly in . Let, .

Then there exists such that asfor all . Moreover, if contains no

trajectory other than the trivial trajectory, then the zero solutionto (16) is asymptotically stable.

Proof: Since the nonlinear dynamical system (20), (21) isLyapunov stable with respect to uniformly in , it followsthat there exists such that if , then the partialsystem trajectories , , of (20), (21) are bounded forall . Furthermore, since , , is continuous, itfollows that there exists such thatfor all . In addition, it follows from Theorem 4.1

that the zero solution to (16) is Lyapunov stable, and

hence, for a given such that there existssuch that if , then ,

, where , , is the solution to (16). Chooseand define . Then for all

and , it follows that , ,and , , is bounded.

Next, consider the function

(34)

It follows from (33) that

(35)

which implies that , , is a nonin-creasing function of time, and hence, ,

, exists. Moreover,for all since , ,

, is continuous. Now suppose, ad ab-surdum, that for some initial condition ,

, . Since thefunction , , , is continuous onthe compact set , it follows that , , isbounded and, hence, ,

. Now, it follows from (33) and Corollary 3.1that , , for . Notethat since it follows that , , is bounded.Furthermore, since it follows that

(36)

for all . Since , , is bounded and ,, , is continuous, it follows that there exists

such that ,

, . This is a contradiction and, hence,, , exists and is finite for

every . Thus, for every and , it fol-lows that

(37)

and, hence, for all ,

(38)


exists and is finite.Next, since is Lipschitz continuous on and

for all and it follows that

(39)

where is the Lipschitz constant on . Thus, it follows from(39) that for any there existssuch that , , which showsthat , , is uniformly continuous. Next, since isuniformly continuous and , ,

, is continuous, it follows that, , is uniformly continuous at

every . Hence, it follows from Barbalat’s Lemma [25, p.192] that asfor all and . Repeating the previous anal-ysis for all , it follows thatfor all . Finally, if contains no trajectory other thanthe trivial trajectory, then and, hence, as

for all , which proves asymptotic stability ofthe zero solution to (16).

Remark 4.2: Note that since . Fur-thermore, recall that for every bounded solution , ,to (16) with initial condition , the positive limit set

of (16) is a nonempty, compact, invariant, and connectedset with as . If and ,then it can be shown that the Lyapunov derivative vanisheson the positive limit set , , so that .Moreover, since is a positively invariant set with respectto (16), it follows that for all , the trajectory of (16) con-verges to the largest invariant set contained in . In this case,Theorem 4.2 specializes to the classical Krasovskii–LaSalle in-variant set theorem [4].

Remark 4.3: If for some ,and , , , then .In this case, it follows from Theorem 4.2 that the zero solution

to (16) is asymptotically stable. Note that even thoughfor the time derivative , , is negativedefinite, the function , , can be nonnegative defi-nite, in contrast to classical Lyapunov stability theory, to ensureasymptotic stability of (16).

Finally, we give a generalization of the converse Lyapunovtheorem that establishes the existence of a vector Lyapunovfunction for an asymptotically stable nonlinear dynamicalsystem. This result is used in the next section to establishthe equivalence between asymptotic stabilizability and theexistence of a control vector Lyapunov function.

Theorem 4.3: Consider the nonlinear dynamical system (16).Let and , and assume thatis continuously differentiable and the zero solutionto (16) is asymptotically stable. Then there exist a continuouslydifferentiable componentwise positive definite vector function

and a continuous function

such that , ,, , , and the zero

solution to

(40)

where , is asymptotically stable.Proof: Since the zero solution to (16) is asymp-

totically stable it follows from [25, Th. 3.14] that there exist acontinuously differentiable positive definite function

and class functions [27] , , and such that

(41)

(42)

Furthermore, it follows from (41) and (42) that

(43)

where “ ” denotes the composition operator andis the inverse function of , and

hence, and are class functions.Next, define such that

, , . Then, it followsthat and , ,where is such that

, . Note thatand . To show that the zero solution to(40) is asymptotically stable, consider the Lyapunov functioncandidate , . Note that , ,

, , and ,, . Thus, the zero solution to (40) is

asymptotically stable which completes the proof.

V. CONTROL VECTOR LYAPUNOV FUNCTIONS

In this section, we consider a feedback control problem andintroduce the notion of a control vector Lyapunov function as ageneralization of control Lyapunov functions. Specifically, con-sider the nonlinear controlled dynamical system given by

(44)

where , is an open set with ,, , is the control input, and

is Lipschitz continuous for all and satisfies. We assume that the control input in (44) is re-

stricted to the class of admissible controls consisting of measur-able functions such that for all , where theconstraint set is given with . Furthermore, we assumethat satisfies sufficient regularity conditions such that thenonlinear dynamical system (44) has a unique solution forwardin time. A measurable mapping satisfyingis called a control law. Furthermore, if , where

is a control law and , , satisfies (44), then iscalled a feedback control law.

Definition 5.1: If there exist a continuously differentiablevector function , acontinuous function ,and a positive vector such that ,


, , is positive definite, ,, , , , where

, ,, , then the vector function

is called a control vector Lyapunov function candidate.It follows from Definition 5.1 that if there exists a control

vector Lyapunov function candidate, then there exists a feed-back control law such that

(45)

Moreover, if the nonlinear dynamical system

(46)

(47)

where and , is asymptotically stable with respectto uniformly in , then it follows from Theorem 4.1 that thezero solution to (47) is asymptotically stable. In thiscase, the vector function given in Definition5.1 is called a control vector Lyapunov function. Furthermore,if , , , is radiallyunbounded, and the system (46), (47) is globally asymptoticallystable with respect to uniformly in , then the zero solution

to (44) is globally asymptotically stabilizable.Remark 5.1: If in Definition 5.1 and the

zero solution to

(48)

where , is asymptotically stable, then it follows fromCorollary 4.1 that is a control vector Lyapunovfunction.

Conversely, suppose that there exists a stabilizing feedbackcontrol law such that the zero solutionto (47) is asymptotically stable. Then it follows from Theorem4.3 that there exist a continuously differentiable vector function

, a continuous function, and a positive vector such

that , the scalar function defined by, , is positive definite, ,

, and , , .Thus, , , . Moreover, since, by Theorem4.3, the zero solution to (48) is asymptotically stable, itfollows from Remark 5.1 that is a control vectorLyapunov function. Hence, a given nonlinear dynamical systemof the form (44) is feedback asymptotically stabilizable if andonly if there exists a control vector Lyapunov function.

In the case where and , Definition 5.1implies the existence of a positive-definite continuously differ-entiable function and a continuous func-tion , where , such that and

, ,, which is equivalent to

(49)

Now, (49) implies the existence of a feedback control lawsuch that , , .

Moreover, if is a control vector Lyapunov function(with ), then it follows from Remark 5.1 that the zerosolution to the system (48) is asymptotically stableand, since , this implies that , ,

. Thus, since is positive definite, (49) can be rewrittenas

(50)

which is equivalent to the standard definition of a control Lya-punov function [6].

Next, consider the case where the control input to (44) pos-sesses a decentralized control architecture so that the dynamicsof (44) are given by

(51)

where , ,, , , and . Note that

, , , as long as , ,and the set of control inputs is given by .In the case of a component decoupled control vector Lyapunovfunction candidate, that is, ,

, it suffices to require in Definition 5.1 that

(52)

to ensure that , , .Note that for a component decoupled control vector Lyapunovfunction , (52) holds if and only if

(53)

where the infimum in (53) is taken componentwise, that is, foreach component of (53) the infimum is calculated separately. Itfollows from (53) that there exists a feedback control law

such that , , where, and , ,

, .Remark 5.2: If for with ,

then condition (52) holds for all such that .Next, we consider the special case of a nonlinear dynamical

system of the form (51) with affine control inputs given by

(54)

where satisfying andare smooth functions (at least continuously differen-

tiable mappings) for all , and , ,.

Theorem 5.1: Consider the controlled nonlinear dynamicalsystem given by (54). If there exist a continuously differentiable,component decoupled vector function , a contin-uous function , and apositive vector such that , the scalar function

defined by , , is positive


definite and radially unbounded, , , and

(55)where ,

, then is a control vector Lyapunov func-tion candidate. If, in addition, there exists suchthat , , and the system(46), (47) is globally asymptotically stable with respect to uni-formly in , then the zero solution to (47) is globallyasymptotically stable and is a control vectorLyapunov function.

Proof: Note that for all ,

(56)

which implies (53). Now, the proof is a direct consequence ofthe definition of a control vector Lyapunov function by notingthe equivalence between (52) and (53) for component decoupledvector Lyapunov functions.

Using Theorem 5.1 we can construct an explicit feedbackcontrol law that is a function of the control vector Lyapunovfunction . Specifically, consider the feedback control law

, , given by (57), as shownat the bottom of the page, where , ,

, , and , . Thederivative along the trajectories of the dynamical system(54), with , , given by (57), is given by (58),as shown at the bottom of the page.

Thus, if the zero solution to (46), (47) is globallyasymptotically stable with respect to uniformly in , then itfollows from Theorem 4.1 that the zero solution to(54) with , , given by(57) is globally asymptotically stable.

Remark 5.3: If in Theorem 5.1 and the zerosolution to (48) is globally asymptotically stable, thenit follows from Corollary 4.1 that the feedback control law givenby (57) is a globally asymptotically stabilizing controller for thenonlinear dynamical system (54).

Remark 5.4: In the case where , the functionin Theorem 5.1 can be set to be identically zero, that is,

. In this case, the feedback control law (57) special-izes to Sontag’s universal formula [7] and is a global stabilizerfor (54).

Since and are smooth and is continuouslydifferentiable for all , it follows that and

, , , are continuous functions, andhence, given by (57) is continuous for all if either

or for all .Hence, the feedback control law given by (57) is continuouseverywhere except for the origin. The following result providesnecessary and sufficient conditions under which the feedbackcontrol law given by (57) is guaranteed to be continuous at theorigin in addition to being continuous everywhere else.

Proposition 5.1: The feedback control law given by(57) is continuous on if and only if for every , thereexists such that for all there existssuch that and ,

.Proof: First note that since , , is a non-

negative function and , it follows from a Taylor seriesexpansion about that , , and hence,

. To show necessity assume that the feedback controllaw given by (57) is continuous on , that is, is contin-uous on for all . Then for every , thereexists such that for all and, by(57), , . Thus,necessity follows with , .

To show sufficiency, assume that for , there existssuch that for all there exists such that

and , . Inthis case, since it follows from the Cauchy-Schwartzinequality that , .Furthermore, since , , is continuously differ-entiable and , , is continuous it follows thatthere exists such that for all , ,

. Hence, for all , where, it follows that and

, . Furthermore, if , then, and if , then it follows from (57) that

(59)

(57)

(58)


and

(60)

Hence, it follows that for every there existssuch that, for all , , which

implies that , , is continuous at the origin, andhence, is continuous at the origin.

VI. STABILITY MARGINS, INVERSE OPTIMALITY, AND VECTOR

DISSIPATIVITY

In this section, we show that the feedback control law givenby (57) is robust to sector bounded input nonlinearities. Specif-ically, we consider the nonlinear dynamical system (54) withnonlinear uncertainties in the input so that the dynamics of thesystem are given by

(61)where

. In addition, we show that for the dynam-ical system (54) the feedback control law given by (57) isinverse optimal in the sense that it minimizes a derived per-formance functional over the set of stabilizing controllers

.Theorem 6.1: Consider the nonlinear dynamical system

(61) and assume that the conditions of Theorem 5.1 hold with, and with the zero solution to (48)

being globally asymptotically stable. Then with the feedbackcontrol law given by (57) the nonlinear dynamical system (61) isglobally asymptotically stable for all , .Moreover, for the dynamical system (54) the feedback controllaw (57) minimizes the performance functional given by

(62)

in the sense that

(63)

where

(64)

and is given by (65) for all . See (65), asshown at the bottom of the page. Finally,

, , where is a control vectorLyapunov function for the dynamical system (54).

Proof: It follows from Theorem 5.1 that the feedback con-trol law (57) globally asymptotically stabilizes the dynamicalsystem (54) and the vector function is a controlvector Lyapunov function for the dynamical system (54). Notethat with (65) the feedback control law (57) can be rewrittenas , , . Let the con-trol vector Lyapunov function for (54) be avector Lyapunov function candidate for (61). Then, the vectorLyapunov derivative components are given by

(66)

Note that and, hence, wheneverfor all . In this case, it follows from

(55) that , , , ,. Next, consider the case where , .

In this case, note that

(67)

for all . Thus, the vector Lyapunov derivative com-ponets given by (66) satisfy

(68)

(65)


for all and . Since the dynamical system(48) is globally asymptotically stable it follows from Corollary4.1 that the nonlinear dynamical system (61) is globally asymp-totically stable for all , .

To show that the feedback control law (57) minimizes (62) inthe sense of (63), define the Hamiltonian

(69)and note that and , ,

, since ,, . Thus

(70)

which yields (63).Remark 6.1: It follows from Theorem 6.1 that with the

feedback stabilizing control law (57) the nonlinear dynamicalsystem (54) has a sector (and hence gain) marginin each decentralized input channel. For details on stabilitymargins for nonlinear dynamical systems, see [30] and [31].

Finally, note that Theorem 6.1 implies that

(71)for all and . Thus, if

, , ,, then ,

, . In this case, the vector Lyapunovderivative components for the dynamical system (54) with theoutput , where , ,

, satisfy

(72)

Inequality (72) imples that (54) is exponentially vector dis-sipative with respect to the vector supply rate

, where, , and with the control vector Lyapunov

function being a vector storage function. Fordetails regarding vector dissipativity, see [22] and [32].

Fig. 1. Large-scale dynamical system G.

VII. DECENTRALIZED CONTROL FOR LARGE-SCALE

NONLINEAR DYNAMICAL SYSTEMS

In this section, we apply the proposed control framework todecentralized control of large-scale nonlinear dynamical sys-tems [15]. Specifically, we consider the large-scale dynamicalsystem shown in Fig. 1 involving energy exchange between

interconnected subsystems. Let denote theenergy (and hence a nonnegative quantity) of the th subsystem,let denote the control input to the th sub-system, and let , , , denotethe instantaneous rate of energy flow from the th subsystem tothe th subsystem.

An energy balance yields the large-scale dynamical system[33]

(73)where , ,

, where ,, , , denotes the net en-

ergy flow from the th subsystem to the th subsystem,,

, and , . Here, we assume that, , , are locally Lipschitz

continuous on , , , , andis such that ,

, are bounded piecewise continuous functionsof time. Furthermore, we assume that , ,whenever , , . In this case,is essentially nonnegative [33], [34] (i.e., for all

such that , ). The previousconstraint implies that if the energy of the th subsystem of

is zero, then this subsystem cannot supply any energy to itssurroundings. Finally, in order to ensure that the trajectories ofthe closed-loop system remain in the nonnegative orthant ofthe state space for all nonnegative initial conditions, we seeka feedback control law that guarantees the closed-loopsystem dynamics are essentially nonnegative [34].

For the dynamical system , consider the control vector Lya-punov function candidate ,

, given by

(74)


Note that and , , is posi-tive definite and radially unbounded. Furhtermore, consider thefunction

...

(75)

where , , are positive definitefunctions, and note that , , and .Also, note that it follows from Remark 5.2 that

and, hence, condition (55) is satisfied for and givenby (74) and (75), respectively. To show that the dynamicalsystem

(76)

where , , , , is the solution to(73), and the th component of is given by

, , , is globallyasymptotically stable with respect to uniformly in , considerthe partial Lyapunov function candidate , .Note that is radially unbounded, , ,

, , and

, . Thus, it follows from [21, Cor. 1] that the dy-namical system (76), (73) is globally asymptotically stable withrespect to uniformly in . Hence, it follows from Theorem5.1 that , , given by (74) is a control vector Lya-punov function for the dynamical system (73).

Next, using (57) with, , , and ,

, we construct a globally stabilizing decentralizedfeedback controller for (73). It can be seen from the structureof the feedback control law that the closed-loop systemdynamics are essentially nonnegative. Furthermore, since

, , ,this feedback controller is fully independent from whichrepresents the internal interconnections of the large-scalesystem dynamics, and hence, is robust against full modelinguncertainty in . Moreover, it follows from Theorem 6.1and Remark 6.1 that the dynamical system (73) with thefeedback stabilizing control law (57) has a sector (and hencegain) margin in each decentralized input channel,and hence, additionally guarantees robustness to multiplicativeinput uncertainty. Finally, the feedback controller minimizesthe derived cost functional given by (62).

Fig. 2. Controlled system states versus time.

Fig. 3. Control signals in each decentralized control channel versus time.

For the following simulation we consider (73) withand , where , ,, and , . Note that in this case the

conditions of Proposition 5.1 are satisfied, and hence, the feed-back control law (57) is continuous on . For our simulationwe set , , , , ,

, , , , , ,, and , with initial condition .

Fig. 2 shows the states of the closed-loop system versus timeand Fig. 3 shows control signal for each decentralized controlchannel as a function of time.

VIII. CONCLUSION

A generalized vector Lyapunov function framework for ad-dressing stability of nonlinear dynamical systems was devel-oped. In addition, a convergence result which specializes to theKrasovskii–LaSalle invariant set theorem for the case of a scalarcomparison system was also presented. Moreover, the notion ofa control vector Lyapunov function was introduced as a gener-alization of control Lyapunov functions, and its existence wasshown to be equivalent to asymptotic stabilizability of a non-linear dynamical system. Finally, the proposed control frame-work was used to construct decentralized controllers for large-


scale nonlinear dynamical systems with robustness guaranteesagainst full modeling uncertainty and multiplicative input un-certainty.

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Sergey G. Nersesov (S’99–M’05) received the B.S.and M.S. degrees in aerospace engineering fromthe Moscow Institute of Physics and Technology,Zhukovsky, Russia, in 1997 and 1999, respectively,with specialization in dynamics and control ofaerospace vehicles. He received the M.S. degreein applied mathematics and the Ph.D. degree inaerospace engineering, both from the GeorgiaInstitute of Technology, Atlanta, in 2003 and 2005,respectively.

From 1998 to 1999, he served as a Researcher inthe Dynamics and Control Systems Division of the Central Aero-HydrodynamicInstitute (TsAGI), Zhukovsky, Russia. Since 2005, he has been an AssistantProfessor in the Department of Mechanical Engineering at Villanova Univer-sity, Villanova, PA. His research interests include nonlinear robust and adap-tive control, nonlinear dynamical system theory, large-scale systems, hierar-chical nonlinear switching control, hybrid and impulsive control for nonlinearsystems, system thermodynamics, thermodynamic modeling of mechanical andaerospace systems, and nonlinear analysis and control of biological and physi-ological systems. He is a coauthor of the books Thermodynamics: A DynamicalSystems Approach (Princeton Univ. Press, 2005) and Impulsive and Hybrid Dy-namical Systems: Stability, Dissipativity, and Control (Princeton Univ. Press,2006).

Wassim M. Haddad (S’87–M’87–SM’01) receivedthe B.S., M.S., and Ph.D. degrees in mechanical en-gineering from Florida Institute of Technology, Mel-bourne, in 1983, 1984, and 1987, respectively, withspecialization in dynamical systems and control.

From 1987 to 1994, he served as a Consultant forthe Structural Controls Group of the GovernmentAerospace Systems Division, Harris Corporation,Melbourne, FL. In 1988, he joined the faculty of theMechanical and Aerospace Engineering Departmentat Florida Institute of Technology, where he founded

and developed the Systems and Control Option within the graduate program.Since 1994, he has been a member of the Faculty in the School of AerospaceEngineering, the Georgia Institute of Technology, Atlanta, where he holdsthe rank of Professor. His research contributions in linear and nonlineardynamical systems and control are documented in over 450 archival journaland conference publications. His recent research is concentrated on nonlinearrobust and adaptive control, nonlinear dynamical system theory, large-scalesystems, hierarchical nonlinear switching control, analysis and control of non-linear impulsive and hybrid systems, system thermodynamics, thermodynamicmodeling of mechanical and aerospace systems, network systems, nonlinearanalysis and control for biological and physiological systems, and activecontrol for clinical pharmacology. He is a coauthor of the books HierarchicalNonlinear Switching Control Design with Applications to Propulsion Systems(Springer-Verlag, 2000), Thermodynamics: A Dynamical Systems Approach(Princeton Univ. Press, 2005), and Impulsive and Hybrid Dynamical Systems:Stability, Dissipativity, and Control (Princeton Univ. Press, 2006).

Dr. Haddad is a National Science Foundation Presidential Faculty Fellow anda member of the Academy of Nonlinear Sciences.

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