finite-time stabilization of nonlinear dynamical systems via control vector lyapunov functions

ARTICLE IN PRESS

Journal of the Franklin Institute 345 (2008) 819–837

0016-0032/$3

doi:10.1016/j

$This rese

0240 and by�CorrespoE-mail ad

(W.M. Hadd

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Finite-time stabilization of nonlinear dynamicalsystems via control vector Lyapunov functions$

Sergey G. Nersesova,�, Wassim M. Haddadb, Qing Huib

aDepartment of Mechanical Engineering, Villanova University, 800 Lancaster Avenue, Villanova,

PA 19085-1681, USAbSchool of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA

Received 8 October 2007; accepted 25 April 2008

Abstract

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium

state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse

time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability

have been developed in the literature using Holder continuous Lyapunov functions. In this paper, we

develop a general framework for finite-time stability analysis based on vector Lyapunov functions.

Specifically, we construct a vector comparison system whose solution is finite-time stable and relate

this finite-time stability property to the stability properties of a nonlinear dynamical system using a

vector comparison principle. Furthermore, we design a universal decentralized finite-time stabilizer

for large-scale dynamical systems that is robust against full modeling uncertainty. Finally, we present

two numerical examples for finite-time stabilization involving a large-scale dynamical system and a

combustion control system.

r 2008 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Finite-time stability; Finite-time convergence; Vector comparison principle; Vector Lyapunov

functions; Non-Lipschitzian dynamics; Homogeneity

2.00 r 2008 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2008.04.015

arch was supported in part by the Air Force Office of Scientific Research under Grant FA9550-06-1-

the Naval Surface Warfare Center under Contract N65540-05-C-0028.

nding author. Tel.: +1610 519 8977; fax: +1 610 519 7312.

dresses: [email protected] (S.G. Nersesov), [email protected]

ad), [email protected] (Q. Hui).

www.elsevier.com/locate/jfranklin

dx.doi.org/10.1016/j.jfranklin.2008.04.015

mailto:[email protected]




ARTICLE IN PRESSS.G. Nersesov et al. / Journal of the Franklin Institute 345 (2008) 819–837820

1. Introduction

The notions of asymptotic and exponential stability in dynamical systems theory implyconvergence of the system trajectories to an equilibrium state over the infinite horizon. Inmany applications, however, it is desirable that a dynamical system possesses the propertythat trajectories that converge to a Lyapunov stable equilibrium state must do so in finitetime rather than merely asymptotically. Most of the existing control techniques in theliterature ensure that the closed-loop system dynamics of a controlled system are Lipschitzcontinuous, which implies uniqueness of system solutions in forward and backward times.Hence, convergence to an equilibrium state is achieved over an infinite time interval. Inorder to achieve convergence in finite time, the closed-loop system dynamics need to benon-Lipschitzian giving rise to non-uniqueness of solutions in backward time. Uniquenessof solutions in forward time, however, can be preserved in the case of finite-timeconvergence. Sufficient conditions that ensure uniqueness of solutions in forward timein the absence of Lipschitz continuity are given in [1–4]. In addition, it is shown in[5, Theorem 4.3, p. 59] that uniqueness of solutions in forward time along with continuityof the system dynamics ensure that the system solutions are continuous functions of thesystem initial conditions even when the dynamics are not Lipschitz continuous.Finite-time convergence to a Lyapunov stable equilibrium, that is, finite-time stability,

was rigorously studied in [6,7] using Holder continuous Lyapunov functions. Finite-timestabilization of second-order systems was considered in [8,9]. More recently, researchershave considered finite-time stabilization of higher-order systems [10] as well as finite-timestabilization using output feedback [11]. Design of globally strongly stabilizing continuouscontrollers for nonlinear systems using the theory of homogeneous systems was studied in[12]. Alternatively, discontinuous finite-time stabilizing feedback controllers have also beendeveloped in the literature [13–15]. However, for practical implementations, discontinuousfeedback controllers can lead to chattering due to system uncertainty or measurementnoise, and hence, may excite unmodeled high-frequency system dynamics.In this paper, we develop a general framework for finite-time stability analysis of

nonlinear dynamical systems using vector Lyapunov functions. Specifically, we construct avector comparison system that is finite-time stable and, using the vector comparisonprinciple [16–20], relate this finite-time stability property to the stability properties of thenonlinear dynamical system. We show that in the case of a scalar comparison system thisresult specializes to the result in [6]. Furthermore, we design universal finite-time stabilizingdecentralized controllers for large-scale dynamical systems based on the newly proposednotion of a control vector Lyapunov function [20]. In addition, we present necessary andsufficient conditions for continuity of such controllers. Moreover, we specialize theseresults to the case of a scalar Lyapunov function to obtain universal finite-time stabilizersfor nonlinear systems that are affine in the control. Finally, we demonstrate the utility ofthe proposed framework on two numerical examples.

2. Mathematical preliminaries

In this section, we introduce notation and definitions, and present some key resultsneeded for developing the main results of this paper. Let R denote the set of real numbers,Rþ denote the set of positive real numbers, Zþ denote the set of nonnegative integers, Rn

denote the set of n� 1 column vectors, and ð�ÞT denote transpose. For v 2 Rq we write

ARTICLE IN PRESSS.G. Nersesov et al. / Journal of the Franklin Institute 345 (2008) 819–837 821

vXX0, (respectively, vb0) to indicate that every component of v is nonnegative(respectively, positive). In this case, we say that v is nonnegative or positive, respectively.Let R

qþ and R

qþ denote the nonnegative and positive orthants of Rq, that is, if v 2 Rq, then

v 2 Rq

þ and v 2 Rqþ are equivalent, respectively, to vXX0 and vb0. Furthermore, let D

�

, D,and qD denote the interior, the closure, and the boundary of the set D � Rn, respectively.Finally, we write k � k for an arbitrary spatial vector norm in Rn, V 0ðxÞ for the Frechetderivative of V at x,Be; ðaÞ, a 2 Rn, e40, for the open ball centered at a with radius e,e 2 Rq for the ones vector of order n, that is, e9½1; . . . ; 1�T, and xðtÞ !M as t!1 todenote that xðtÞ approaches the set M, that is, for each e40 there exists T40 such thatdist ðxðtÞ;MÞoe for all t4T , where dist ðp;MÞ9infx2Mkp� xk.

The following definition introduces the notion of class W functions involvingquasimonotone increasing functions.

Definition 2.1 (Siljak [18]). A function w ¼ ½w1; . . . ;wq�T : Rq ! Rq is of class W if

wiðz0Þpwiðz

00Þ, i ¼ 1; . . . ; q, for all z0; z00 2 Rq such that z0jpz00j , z0i ¼ z00i , j ¼ 1; . . . ; q, iaj,where zi denotes the ith component of z.

If wð�Þ 2W we say that w satisfies the Kamke condition [21,22]. Note that if wðzÞ ¼Wz,where W 2 Rq�q, then the function wð�Þ is of class W if and only if W is essentially

nonnegative, that is, all the off-diagonal entries of the matrix W are nonnegative.Furthermore, note that it follows from Definition 2.1 that any scalar ðq ¼ 1Þ function wðzÞ

is of class W.Next, consider the nonlinear dynamical system given by

_xðtÞ ¼ f ðxðtÞÞ; xðt0Þ ¼ x0; t 2 Ix0, (1)

where xðtÞ 2 D � Rn, t 2 Ix0, is the system state vector, Ix0

is the maximal interval ofexistence of a solution xðtÞ of Eq. (1), D is an open set, 0 2 D, f ð0Þ ¼ 0, and f ð�Þ iscontinuous on D. A continuously differentiable function x : Ix0

! D is said to be asolution of Eq. (1) on the interval Ix0

� R if xð�Þ satisfies (1) for all t 2 Ix0. Recall that

every bounded solution to Eq. (1) can be extended on a semi-infinite time interval ½0;1Þ[23]. We assume that (1) possesses unique solutions in forward time for all initialconditions except possibly the origin in the following sense. For every x 2 Dnf0g thereexists tx40 such that, if y1 : ½0; t1Þ ! D and y2 : ½0; t2Þ ! D are two solutions of Eq. (1)with y1ð0Þ ¼ y2ð0Þ ¼ x, then txpminft1; t2g and y1ðtÞ ¼ y2ðtÞ for all t 2 ½0; txÞ. Without lossof generality, we assume that for each x, tx is chosen to be the largest such number in Rþ.In this case, we denote the trajectory or solution curve of Eq. (1) on ½0; txÞ satisfying theconsistency property sð0; xÞ ¼ x and the semi-group property sðt; sðt;xÞÞ ¼ sðtþ t; xÞ forevery x 2 D and t; t 2 ½0; txÞ by sð�; xÞ or sxð�Þ. Sufficient conditions for forward uniquenessin the absence of Lipschitz continuity can be found in [1; 2, Section 10; 3; 4, Section 1].

The next result presents the vector comparison principle [16–20] for nonlinear dynamicalsystems.

Theorem 2.1 (Nersesov and Haddad [20]). Consider the nonlinear dynamical system (1).Assume there exists a continuously differentiable vector function V : D! Q � Rq such that

V 0ðxÞf ðxÞppwðV ðxÞÞ; x 2 D, (2)

where w : Q! Rq is a continuous function, wð�Þ 2W, and

_zðtÞ ¼ wðzðtÞÞ; zðt0Þ ¼ z0; t 2 Iz0 , (3)


has a unique solution zðtÞ, t 2 Iz0 . If ½t0; t0 þ t� � Ix0\Iz0 is a compact interval and

V ðx0Þppz0, z0 2 Q, then V ðxðtÞÞppzðtÞ, t 2 ½t0; t0 þ t�.

The next definition introduces the notion of finite-time stability.

Definition 2.2 (Bhat and Bernstein [6]). Consider the nonlinear dynamical system (1). Thezero solution xðtÞ 0 to Eq. (1) is finite-time stable if there exist an open neighborhoodN � D of the origin and a function T :Nnf0g ! ð0;1Þ, called the settling-time function,such that the following statements hold:

(i)
Finite-time convergence: For every x 2Nnf0g, sxðtÞ is defined on ½0;TðxÞÞ, sxðtÞ 2
Nnf0g for all t 2 ½0;TðxÞÞ, and limt!TðxÞsðx; tÞ ¼ 0.
(ii) Lyapunov stability: For every e40 there exists d40 such that Bdð0Þ �N and for every
x 2 Bdð0Þnf0g, sðt;xÞ 2 Beð0Þ for all t 2 ½0;TðxÞÞ.

The zero solution xðtÞ 0 of Eq. (1) is globally finite-time stable if it is finite-time stablewith N ¼ D ¼ Rn.

Note that if the zero solution xðtÞ 0 to Eq. (1) is finite-time stable, then it isasymptotically stable, and hence, finite-time stability is a stronger notion than asymptoticstability.

3. Finite-time stability via vector Lyapunov functions

We start this section by considering an example of a finite-time stable system with acontinuous but non-Lipschitzian vector field.

Example 3.1 (Bhat and Bernstein [6]). Consider the scalar system

_xðtÞ ¼ �k signðxðtÞÞjxðtÞja; xð0Þ ¼ x0; tX0, (4)

where x0 2 R, signðxÞ9x=jxj, xa0, signð0Þ90, k40, and a 2 ð0; 1Þ. The right-hand side ofEq. (4) is continuous everywhere and locally Lipschitz everywhere except the origin.Hence, every initial condition in Rnf0g has a unique solution in forward time on asufficiently small time interval. The solution to Eq. (4) is obtained by direct integration andis given by

sðt;x0Þ ¼

signðx0Þ½jx0j1�a � kð1� aÞt�1=ð1�aÞ; to

1

kð1� aÞjx0j

1�a; x0a0;

0; tX1

kð1� aÞjx0j

1�a; x0a0;

0; tX0; x0 ¼ 0:

8>>>>><>>>>>:

(5)

It is clear from Eq. (5) that (i) in Definition 2.2 is satisfied with N ¼ D ¼ R and with thesettling-time function T : R! Rþ given by

Tðx0Þ ¼1

kð1� aÞjx0j

1�a; x0 2 R. (6)

Lyapunov stability follows by considering the Lyapunov function V ðxÞ ¼ x2, x 2 R. Thus,the zero solution xðtÞ 0 to Eq. (4) is globally finite-time stable.


Next, we present sufficient conditions for finite-time stability using a vector Lyapunovfunction involving a vector differential inequality.

Theorem 3.1. Consider the nonlinear dynamical system (1). Assume there exist a

continuously differentiable vector function V : D! Q \ Rqþ, where Q � Rq and 0 2 Q, and

a positive vector p 2 Rqþ such that V ð0Þ ¼ 0, the scalar function pTV ðxÞ, x 2 D, is positive

definite, and

V 0ðxÞf ðxÞppwðV ðxÞÞ; x 2 D, (7)

where w : Q! Rq is continuous, wð�Þ 2W, and wð0Þ ¼ 0. In addition, assume that the vector

comparison system

_zðtÞ ¼ wðzðtÞÞ; zð0Þ ¼ z0; t 2 Iz0 , (8)

has a unique solution in forward time zðtÞ, t 2 Iz0 , and there exist a continuously

differentiable function v : Q! R, real numbers c40 and a 2 ð0; 1Þ, and a neighborhood

M � Q of the origin such that vð�Þ is positive definite and

v0ðzÞwðzÞp� cðvðzÞÞa; z 2M. (9)

Then the zero solution xðtÞ 0 to Eq. (1) is finite-time stable. Moreover, if N is as in

Definition 2.2 and T :N! ½0;1Þ is the settling-time function, then

Tðx0Þp1

cð1� aÞðvðV ðx0ÞÞÞ

1�a; x0 2N, (10)

and Tð�Þ is continuous on N. If, in addition, D ¼ Rn, vð�Þ is radially unbounded, and (9) holds

on Rq, then the zero solution xðtÞ 0 to Eq. (1) is globally finite-time stable.

Proof. Note that pTV ðxÞpmaxi¼1;...;qfpigeTV ðxÞ, x 2 D. Hence, since pTV ðxÞ, x 2 D, is

positive definite, that is, pTV ð0Þ ¼ 0 and pTV ðxÞ40, xa0, it follows that the functioneTV ðxÞ, x 2 D, is also positive definite.

Let V �M be a bounded open set such that 0 2V and V � Q. Then qV is compactand 0eqV. Now, it follows from Weierstrass’ theorem that the continuous function vð�Þ

attains a minimum on qV and since vð�Þ is positive definite, minz2qVvðzÞ40. Let0obominz2qVvðzÞ and Db9fz 2V : vðzÞpbg. It follows from (9) that Db �M isinvariant with respect to Eq. (8). Furthermore, it follows from (9), the positive definitenessof vð�Þ, and standard Lyapunov arguments that for every e40 there exists d40 such thatBdð0Þ � Db �M and

kzðtÞk1pe; kz0k1od, (11)

where k � k1 denotes the absolute sum norm, Bdð0Þ is defined in terms of the absolute sumnorm k � k1, and t 2 Iz0 . Moreover, since the solution zðtÞ to (8) is bounded for all t 2 Iz0 ,it can be extended on the semi-infinite interval ½0;1Þ [23], and hence, zðtÞ is defined for alltX0. Furthermore, it follows from Theorem 2.1 with q ¼ 1, wðyÞ ¼ �cya, andzðtÞ ¼ sðt; vðz0ÞÞ, where a 2 ð0; 1Þ, that

vðzðtÞÞpsðt; vðz0ÞÞ; z0 2 Bdð0Þ; t 2 ½0;1Þ, (12)


where sð�; �Þ is given by Eq. (5) with k ¼ c. Now, it follows from (5), (12), and the positivedefiniteness of vð�Þ that

zðtÞ ¼ 0; tX1

cð1� aÞðvðz0ÞÞ

1�a; z0 2 Bdð0Þ, (13)

which implies finite-time convergence of the trajectories of (8) for all z0 2 Bdð0Þ. This alongwith (11) implies finite-time stability of the zero solution zðtÞ 0 to (8).Next, it follows from the continuity of V ð�Þ that there exists d140 such that kV ðx0Þk1od

for all kx0kod1, where k � k is the Euclidian norm on Rn. Now, choose z0 ¼ V ðx0Þ 2 Bdð0Þfor all kx0kod1. In this case, it follows from (7) and Theorem 2.1 that V ðxðtÞÞppzðtÞ on acompact interval ½0; tx0

�, where ½0; tx0Þ is the maximal interval of existence of the solution

xðtÞ to Eq. (1). Since zðtÞ, tX0, is bounded and eTV ð�Þ is positive definite it follows that xðtÞ,t 2 ½0; tx0

�, is bounded, and hence, xðtÞ can be extended to the semi-infinite interval ½0;1Þ.Using (13) it follows that

eTV ðxðtÞÞ ¼ eTzðtÞ ¼ 0; tX1

cð1� aÞðvðz0ÞÞ

1�a; z0 ¼ V ðx0Þ 2 Bdð0Þ. (14)

Since eTV ð�Þ is positive definite, it follows that

xðtÞ ¼ 0; tX1


1�a; kx0kod1, (15)

which implies finite-time convergence of the trajectories of Eq. (1) for all kx0kod1.Furthermore, it follows from Eq. (15) that the settling-time function satisfies

Tðx0Þp1


1�a; kx0kod1. (16)

Next, note that since eTV ðxÞ, x 2 D, is positive definite, there exist r40 and class Kfunctions [24] a, b : ½0; r� ! Rþ such that Brð0Þ � D, where Brð0Þ is defined in terms of theEuclidean norm k � k, and

aðkxkÞpeTV ðxÞpbðkxkÞ; x 2 Brð0Þ. (17)

Let e40 and choose 0oeominfe; rg. In this case, it follows from the Lyapunov stability ofthe nonlinear vector comparison system (8) that there exists m ¼ mðeÞ ¼ mðeÞ40 such that ifkz0k1om, then kzðtÞk1oaðeÞ, tX0. Now, choose z0 ¼ V ðx0ÞXX0, x0 2 D. SinceV ðxÞ;x 2 D, is continuous, eTV ðxÞ;x 2 D, is also continuous. Hence, for m ¼ mðeÞ40there exists d ¼ dðmðeÞÞ ¼ dðeÞ40 such that dominfd1; eg, and if kx0kod, theneTV ðx0Þ ¼ eTz0 ¼ kz0k1om, which implies that kzðtÞk1oaðeÞ, tX0. Now, withz0 ¼ V ðx0ÞXX0, x0 2 D, and the assumption that wð�Þ 2W it follows from (7) andTheorem 2.1 that 0ppV ðxðtÞÞppzðtÞ on any compact interval ½0; t�, and hence,eTzðtÞ ¼ kzðtÞk1, t 2 ½0; t�.Let t40 be such that xðtÞ 2 Brð0Þ, t 2 ½0; t�, for all x0 2 Bdð0Þ. Thus, using (17), ifkx0kod, then

aðkxðtÞkÞpeTV ðxðtÞÞpeTzðtÞoaðeÞ; t 2 ½0; t�, (18)

which implies kxðtÞkoeoe, t 2 ½0; t�. Now, suppose, ad absurdum, that for some x0 2 Bdð0Þthere exists t4t such that kxðtÞk ¼ e. Then, for z0 ¼ V ðx0Þ and the compact interval ½0; t� itfollows from (7) and Theorem 2.1 that V ðxðtÞÞppzðtÞ, which implies thataðeÞ ¼ aðkxðtÞkÞpeTV ðxðtÞÞpeTzðtÞoaðeÞ, leading to a contradiction. Hence, for a given


e40 there exists d ¼ dðeÞ40 such that for all x0 2 Bdð0Þ, kxðtÞkoe, tXt0, which impliesLyapunov stability of the zero solution xðtÞ 0 to Eq. (1). This, along with Eq. (15),implies finite-time stability of the zero solution xðtÞ 0 to Eq. (1) with N9Bdð0Þ. Eq. (10)implies that Tð�Þ is continuous at the origin, and hence, by Proposition 2.4 of [6],continuous on N.

Finally, ifD ¼ Rn and vð�Þ is radially unbounded, then global finite-time stability followsusing standard arguments. &

Assume the conditions of Theorem 3.1 are satisfied with q ¼ 1. In this case, there exists acontinuously differentiable, positive definite function V : D! Q \ Rþ such that (7) holds,and there exists a continuously differentiable, positive definite function v : Q! Rþ suchthat (9) holds. Since q ¼ 1 and M is a neighborhood of the origin, it follows that thereexists g40 such that ½0; g� �M. Furthermore, since vð�Þ is positive definite, there existsb40 such that v0ðzÞX0 for all z 2 ½0;b�. Next, consider the function ~vðxÞ9vðV ðxÞÞ, x 2 D,and note that ~vð�Þ is positive definite. DefineV9fx 2 D : V ðxÞpminfb; ggg. Then it followsfrom (7) and (9) that

_~vðxÞ ¼ v0ðV ðxÞÞV 0ðxÞf ðxÞ

pv0ðV ðxÞÞwðV ðxÞÞ

p� cðvðV ðxÞÞÞa

¼ � cð~vðxÞÞa; x 2V, (19)

which implies condition (4.7) in Theorem 4.2 of [6]. Thus, in the case where q ¼ 1,Theorem 3.1 specializes to Theorem 4.2 of [6].

The next result is a specialization of Theorem 3.1 to the case where the structure of thecomparison dynamics directly guarantees finite-time stability of the comparison system.That is, there is no need to require the existence of a scalar function vð�Þ such that (9) holdsin order to guarantee finite-time stability of the nonlinear dynamical system (1).

Corollary 3.1. Consider the nonlinear dynamical system (1). Assume there exist a

continuously differentiable vector function V : D! Q \ Rqþ, where Q � Rq and 0 2 Q, and

a positive vector p 2 Rqþ such that V ð0Þ ¼ 0, the scalar function pTV ðxÞ, x 2 D, is positive

definite, and

V 0ðxÞf ðxÞppW ðV ðxÞÞ½a�; x 2 D, (20)

where a 2 ð0; 1Þ, W 2 Rq�q is essentially nonnegative and Hurwitz, and

ðV ðxÞÞ½a�9½ðV 1ðxÞÞa; . . . ; ðV qðxÞÞ

a�T. Then the zero solution xðtÞ 0 to (1) is finite-time

stable. If, in addition, D ¼ Rn, then the zero solution xðtÞ 0 to (1) is globally finite-time

stable.

Proof. Consider the comparison system given by

_zðtÞ ¼W ðzðtÞÞ½a�; zð0Þ ¼ z0; tX0, (21)

where z0 2 Rqþ. Note that the right-hand side in (21) is of class W and is essentially

nonnegative and, hence, the solutions to (21) are nonnegative for all nonnegative initialconditions [25]. Since W 2 Rq�q is essentially nonnegative and Hurwitz, it follows fromTheorem 3.2 of [25] that there exist positive vectors p 2 R

qþ and r 2 R

qþ such that

0 ¼WTpþ r. (22)


Now, consider the Lyapunov function vðzÞ ¼ pTz, z 2 Rqþ. Note that vð0Þ ¼ 0, vðzÞ40,

z 2 Rqþ, za0, and vð�Þ is radially unbounded. Let b9mini¼1;...;qri, g9maxi¼1;...;qpa

i , where ri

and pi are the ith components of r 2 Rqþ and p 2 R

qþ, respectively. Then

_vðzÞ ¼ pTWz½a�

¼ � rTz½a�

p�bggXq

i¼1

zai

!

p�bg

Xq

i¼1

pai zai

!

p�bg

Xq

i¼1

pizi

!a

p�bgðvðzÞÞa

¼ � cðvðzÞÞa; z 2 Rqþ, (23)

where c9 bg. Thus, it follows from Theorem 4.2 of [6] that the comparison system (21) is

finite-time stable with the settling-time function Tðz0Þpð1=cð1� aÞÞðvðz0ÞÞ1�a, z0 2 R

qþ.

Next, it follows from Corollary 4.1 of [20] that the nonlinear dynamical system (1) isasymptotically stable with the domain of attraction N � D. Now, the result is a directconsequence of Theorem 3.1. &

Remark 3.1. It follows from Corollary 3.1 that the nonlinear dynamical system (1) has asettling-time function Tðx0Þpð1=cð1� aÞÞðvðV ðx0ÞÞÞ

1�a, x0 2N, where vðzÞ ¼ pTz, z 2 Rqþ,

and p 2 Rqþ is as in the proof of Corollary 3.1.

4. Finite-time stabilization of large-scale dynamical systems

In the recent paper [20], the notion of a control vector Lyapunov function was introducedas a generalization of the classical notion of a control Lyapunov function. Furthermore, auniversal stabilizing feedback control law was constructed based on a control vectorLyapunov function [20]. In this section, we show that this control law can be used tostabilize large-scale dynamical systems in finite time provided that the comparison systempossesses non-Lipschitzian dynamics.Specifically, consider the large-scale dynamical system composed of q interconnected

subsystems given by

_xiðtÞ ¼ f iðxðtÞÞ þ GiðxðtÞÞuiðtÞ; tXt0; i ¼ 1; . . . ; q, (24)

where f i : Rn ! Rni satisfying f ið0Þ ¼ 0 and Gi : R

n ! Rni�mi are continuous functions forall i ¼ 1; . . . ; q, and uið�Þ, i ¼ 1; . . . ; q, satisfy sufficient regularity conditions such that thenonlinear dynamical system (24) has a unique solution forward in time. Let V ¼

½V 1; . . . ;Vq�T : Rn ! R

qþ be a component decoupled continuously differentiable vector

function, that is, V ðxÞ ¼ ½V 1ðx1Þ; . . . ;VqðxqÞ�T, x 2 Rn, p 2 R

qþ be a positive vector, and


w : Rqþ ! Rq be a continuous function such that V ð0Þ ¼ 0, pTV ðxÞ, x 2 Rn, is positive

definite, and wð�Þ 2W with wð0Þ ¼ 0. Define aiðxÞ9V 0iðxiÞf iðxÞ, x 2 Rn, andbiðxÞ9GT

i ðxÞV0Ti ðxiÞ, x 2 Rn, and assume that

V 0iðxiÞf iðxÞowiðV ðxÞÞ; x 2 Ri; i ¼ 1; . . . ; q, (25)

where Ri9fx 2 Rn; xa0 : biðxÞ ¼ 0g, i ¼ 1; . . . ; q. Construct the feedback control lawfðxÞ ¼ ½fT

1 ðxÞ; . . . ;fTq ðxÞ�

T, x 2 Rn, given by

fiðxÞ ¼� c0i þ

ðaiðxÞ � wiðV ðxÞÞÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaiðxÞ � wiðV ðxÞÞÞ

2þ ðbTi ðxÞbiðxÞÞ

2q

bTi ðxÞbiðxÞ

0@

1AbiðxÞ; biðxÞa0;

0; biðxÞ ¼ 0;

8>>><>>>:

(26)

where c0i40, i ¼ 1; . . . ; q.The vector Lyapunov derivative components _V ið�Þ, i ¼ 1; . . . ; q, along the trajectories of

the closed-loop dynamical system (24), with u ¼ fðxÞ, x 2 Rn, given by Eq. (26), is given by

_ViðxiÞ ¼ V 0iðxiÞ½f iðxÞ þ GiðxÞfiðxÞ�

¼ aiðxÞ þ bTi ðxÞfiðxÞ

¼

�c0ibTi ðxÞbiðxÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaiðxÞ � wiðV ðxÞÞÞ

2þ ðbTi ðxÞbiðxÞÞ

2q

þwiðV ðxÞÞ; biðxÞa0;

aiðxÞ; biðxÞ ¼ 0;

8>>><>>>:

owiðV ðxÞÞ; x 2 Rn. (27)

It follows from Theorem 3.1 that if there exist v : Rqþ ! R, c40, and a 2 ð0; 1Þ such that

vð�Þ is positive definite and

v0ðzÞwðzÞp� cðvðzÞÞa; z 2M, (28)

where M is a neighborhood of Rqþ containing the origin, then the zero solution xðtÞ 0 to

Eq. (24) is finite-time stable with the settling time Tðx0Þpð1=cð1� aÞÞðvðV ðx0ÞÞÞ1�a,

x0 2 Rn. In this case, it follows from Theorem 5.1 of [20] that V ðxÞ, x 2 Rn, is a controlvector Lyapunov function.

Remark 4.1. If Ri ¼ ;, i ¼ 1; . . . ; q, then the function wð�Þ in Eq. (26) can be chosen to be

wðzÞ ¼Wz½a�; z 2 Rqþ, (29)

where W 2 Rq�q is essentially nonnegative and Hurwitz, a 2 ð0; 1Þ, and z½a�9½za1; . . . ; zaq�T.

In this case, condition (28) need not be verified and it follows from Corollary 3.1 that theclosed-loop system (24) and (26) with wð�Þ given by Eq. (29) is finite-time stable and, hence,the controller (26) is finite-time stabilizing controller for Eq. (24).

Since f ið�Þ and Gið�Þ are continuous and Við�Þ is continuously differentiable for alli ¼ 1; . . . ; q, it follows that aiðxÞ and biðxÞ, x 2 Rn, i ¼ 1; . . . ; q, are continuous functions,and hence, fiðxÞ given by Eq. (26) is continuous for all x 2 Rn if either biðxÞa0 or aiðxÞ �

wiðV ðxÞÞo0 for all i ¼ 1; . . . ; q. Hence, the feedback control law given by Eq. (26) iscontinuous everywhere except for the origin. The following result provides necessary and


sufficient conditions under which the feedback control law given by Eq. (26) is guaranteedto be continuous at the origin in addition to being continuous everywhere else.

Proposition 4.1 (Nersesov and Haddad [20]). The feedback control law fðxÞ given by

Eq. (26) is continuous on Rn if and only if for every e40, there exists d40 such that for all

0okxkod there exists ui 2 Rmi such that kuikoe and aiðxÞ þ bTi ðxÞuiowiðV ðxÞÞ,i ¼ 1; . . . ; q.

The following corollary addressing the case where q ¼ 1 is immediate from the abovearguments. In this case, the nonlinear dynamical system (24) specializes to

_xðtÞ ¼ f ðxðtÞÞ þ GðxðtÞÞuðtÞ; xðt0Þ ¼ x0; tXt0, (30)

where x0 2 Rn and f : Rn ! Rn satisfying f ð0Þ ¼ 0 and G : Rn ! Rn�m are continuousfunctions.

Corollary 4.1. Consider the nonlinear dynamical system (30). Assume there exists a

continuously differentiable function V : D! Rþ such that V ð�Þ is positive definite,wðV ðxÞÞ9� cðV ðxÞÞa, x 2 Rn, and

V 0ðxÞf ðxÞpwðV ðxÞÞ ¼ �cðV ðxÞÞa; x 2 R, (31)

where c40, a 2 ð0; 1Þ, R9fx 2 Rn;xa0 : V 0ðxÞGðxÞ ¼ 0g. Then the nonlinear dynamical

system (30) with the feedback controller u ¼ fðxÞ, x 2 Rn, given by

fðxÞ ¼� c0 þ

ðaðxÞ � wðV ðxÞÞÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðaðxÞ � wðV ðxÞÞÞ2 þ ðbTðxÞbðxÞÞ2

qbTðxÞbðxÞ

0@

1AbðxÞ; bðxÞa0;

0; bðxÞ ¼ 0;

8>>><>>>:

(32)

where c040, aðxÞ9V 0ðxÞf ðxÞ, x 2 Rn, and bðxÞ9GTðxÞV 0TðxÞ, x 2 Rn, is finite-time stable

with the settling time Tðx0Þpð1=cð1� aÞÞðV ðx0ÞÞ1�a, x0 2 Rn. Furthermore, V ð�Þ is a control

Lyapunov function.

Next, we show that the control law (32) ensures finite-time stability for a perturbedversion of Eq. (30) with bounded perturbations. Specifically, consider the more accuratedescription of system (30) given by the perturbed model

_xðtÞ ¼ f ðxðtÞÞ þ GðxðtÞÞuðtÞ þ gðt; xðtÞÞ; xðt0Þ ¼ x0; tXt0, (33)

where g : ½t0;1Þ � Rn ! Rn is a continuous function that captures disturbances,uncertainties, parameter variations, or modeling errors. Assume that there exists acontinuously differentiable function V : Rn ! Rþ such that the conditions of Corollary4.1 are satisfied. Then it follows from Theorem 5.2 of [6] that there exist d040, ‘40, t40,and an open neighborhood V of the origin such that for every continuous function gð�; �Þwith

d ¼ sup½t0;1Þ�R

nkgðt;xÞkod0, (34)

the solutions xðtÞ, tXt0, to the closed-loop system (33) with uðtÞ given by Eq. (32) andx0 2V are such that xðtÞ 2V, tXt0, and

kxðtÞkp‘dg; tXt, (35)

ARTICLE IN PRESS

Fig. 1. Large-scale dynamical system G.

S.G. Nersesov et al. / Journal of the Franklin Institute 345 (2008) 819–837 829

where g ¼ ð1� aÞ=a. Note that, if in Corollary 4.1, a 2 ð0; 12Þ, then g41 which makes the

bound in (35) smaller for sufficiently small d compared to the case when 0ogo1. Inaddition, if gð�; �Þ is such that

kgðt; xÞkpLkxk; ðt;xÞ 2 ½t0;1Þ � Rn, (36)

where LX0, then it follows from Theorem 5.3 of [6] that xðtÞ ¼ 0, tXt, for all x0 2V.Finally, if g : Rn ! Rn is only a function of the dynamical system state and

kgðxÞkpLkxk; x 2 Rn, (37)

where LX0, then it follows from Theorem 5.4 of [6] that the zero solution xðtÞ 0 to theclosed-loop system (33) with uðtÞ given by Eq. (32) is finite-time stable.

Next, consider the large-scale dynamical system G shown in Fig. 1 involving energyexchange between n interconnected subsystems. Let xi : ½0;1Þ ! Rþ denote the energy(and hence a nonnegative quantity) of the ith subsystem, let ui : ½0;1Þ ! R denote thecontrol input to the ith subsystem, and let sij : R

nþ ! Rþ, iaj, i, j ¼ 1; . . . ; n, denote the

instantaneous rate of energy flow from the jth subsystem to the ith subsystem.An energy balance yields the large-scale dynamical system [26]

_xðtÞ ¼ f ðxðtÞÞ þ GðxðtÞÞuðtÞ; xðt0Þ ¼ x0; tXt0, (38)

where xðtÞ ¼ ½x1ðtÞ; . . . ;xnðtÞ�T, tXt0, f iðxÞ ¼

Pnj¼1;jaifijðxÞ, where fijðxÞ9sijðxÞ � sjiðxÞ,

x 2 Rnþ, iaj, i; j ¼ 1; . . . ; q, denotes the net energy flow from the jth subsystem to the ith

subsystem, GðxÞ ¼ diag½G1ðx1Þ; . . . ;GnðxnÞ�, x 2 Rnþ, GiðxiÞ ¼ 0 if and only if xi ¼ 0 for all

i ¼ 1; . . . ; n, and uðtÞ 2 Rn, tXt0. Here, we assume that sijðxÞ ¼ 0, x 2 Rnþ, whenever

xj ¼ 0, iaj, i; j ¼ 1; . . . ; n. In this case, f ð�Þ is essentially nonnegative [25,26] (i.e., f iðxÞX0for all x 2 Rn

þ such that xi ¼ 0, i ¼ 1; . . . ; n). The above constraint implies that if theenergy of the jth subsystem of G is zero, then this subsystem cannot supply any energy toits surroundings. In addition, we assume that fijðx

0Þpfijðx00Þ, iaj, i; j ¼ 1; . . . ; n, for all

x0; x00 2 Rn such that x0i ¼ x00i and x0kpx00k, kai, where xi is the ith component of x.The above assumption implies that the more energy the surroundings of the ith subsystempossess, the more energy is gained by the ith subsystem from the energy exchange dueto subsystem interconnections. Finally, in order to ensure that the trajectories of the


closed-loop system remain in the nonnegative orthant of the state space for all nonnegativeinitial conditions, we seek a feedback control law uð�Þ that guarantees the closed-loopsystem dynamics are essentially nonnegative [25].For the dynamical system G, consider the control vector Lyapunov function candidate

V ðxÞ ¼ ½V1ðx1Þ; . . . ;V nðxnÞ�T, x 2 Rn

þ, given by

V ðxÞ ¼ ½x1; . . . ; xn�T; x 2 Rn

þ. (39)

Note that V ð0Þ ¼ 0 and eTV ðxÞ, x 2 Rnþ, is positive definite and radially unbounded.

Furthermore, consider the function

wðV ðxÞÞ ¼ �V1=21 ðx1Þ þ

Xn

j¼1;ja1

f1jðV ðxÞÞ; . . . ;�V1=2n ðxnÞ

"

þXn

j¼1;jan

fnjðV ðxÞÞ

#T; x 2 Rn

þ, (40)

and note that it follows from the above constraints that wð�Þ 2W and wð0Þ ¼ 0.Furthermore, note that

Ri9 x 2 Rnþ;xia0 : V 0iðxiÞGiðxiÞ ¼ 0

� �¼ x 2 Rn

þ; xia0 : xi ¼ 0� �

¼ ;,

and hence, condition (25) is satisfied for V ð�Þ and wð�Þ given by Eqs. (39) and (40),respectively.Next, consider the vector comparison system

_zðtÞ ¼ wðzðtÞÞ; zðt0Þ ¼ z0; tXt0, (41)

where z0 2 Rnþ and the ith component of wðzÞ is given by wiðzÞ ¼ �z

1=2i þ

Pnj¼1;jaifijðzÞ,

z 2 Rnþ. In addition, consider the Lyapunov function candidate vðzÞ ¼ eTz, z 2 Rn

þ, and

note that vð�Þ is radially unbounded, vð0Þ ¼ 0, vðzÞ40, z 2 Rnþ, za0, and

v0ðzÞwðzÞ ¼ �Xn

i¼1

z1=2i þ

Xn

i¼1

Xn

j¼1;jai

fijðzÞ

¼ �Xn

i¼1

z1=2i

p�Xn

i¼1

zi

!1=2

¼ � ðvðzÞÞ1=2; z 2 Rnþ. (42)

Thus, it follows from Theorem 3.1 with c ¼ 1, a ¼ 12, and M ¼ Rn

þ that the large-scaledynamical system (38) is finite-time stable with a settling time Tðx0Þp2ðeTx0Þ

1=2, x0 2 Rnþ,

and V ðxÞ, x 2 Rnþ, given by Eq. (39) is a control vector Lyapunov function for Eq. (38).

Finally, the feedback control law fðxÞ ¼ ½fT1 ðxÞ; . . . ;f

Tn ðxÞ�

T, where fiðxÞ, i ¼ 1; . . . ; n, isgiven by Eq. (26) with aiðxÞ ¼ V 0iðxiÞf iðxÞ ¼

Pnj¼1;jaifijðxÞ, biðxÞ ¼ GiðxiÞ, x 2 Rn

þ,and c0i40, i ¼ 1; . . . ; n, is a finite-time globally stabilizing decentralized feedbackcontroller for Eq. (38). It can be seen from the structure of the feedback control lawthat the closed-loop system dynamics are essentially nonnegative. Furthermore, since


aiðxÞ � wiðV ðxÞÞ ¼ ðV iðxiÞÞ1=2, x 2 Rn

þ, i ¼ 1; . . . ; n, this feedback controller is fullyindependent from f ðxÞ which represents the internal interconnections of the large-scalesystem dynamics, and hence, is robust against full modeling uncertainty in f ðxÞ.

5. Finite-time stabilization for large-scale homogeneous systems

In this section, we use geometric homogeneity developed in [7,27] to construct finite-timecontrollers for large-scale homogeneous systems. First, we introduce the concept ofhomogeneity in relation to a scaling operation or dilation.

Definition 5.1 (Bhat and Bernstein [7], Bacciotti and Rosier [27]). Letx9½x1; . . . ;xn�

T 2 Rn. A dilation DlðxÞ : ðl;x1; . . . ; xnÞ7!ðlr1x1; . . . ; l

rn xnÞ is a mapping thatassigns to every l40 a diffeomorphism DlðxÞ ¼ ðl

r1x1; . . ., lrn xnÞ, where ðx1; . . . ; xnÞ is a

suitable coordinate on Rn and ri40, i ¼ 1; . . . ; n, are constants. A function V : Rn ! R ishomogeneous of degree l 2 R with respect to the dilation DlðxÞ if V ðlr1x1; . . . ;lrn xnÞ ¼ llV ðx1; . . . ; xnÞ. Finally, a vector field f ðxÞ9½f 1ðxÞ; . . . ; f nðxÞ�

T : Rn ! R ishomogeneous of degree k 2 R with respect to the dilation DlðxÞ if f iðl

r1x1; . . . ;lrn xnÞ ¼ lkþri f iðx1; . . . ; xnÞ, l40, i ¼ 1; . . . ; n.

Proposition 5.1 (Bhat and Bernstein [7]). Consider the nonlinear dynamical system (1).Assume f ð�Þ is homogeneous of degree k 2 R with respect to the dilation DlðxÞ. Furthermore,assume f ð�Þ is continuous on D and x ¼ 0 is an asymptotically stable equilibrium point of

Eq. (1). If ko0, then x ¼ 0 is a finite-time stable equilibrium point of Eq. (1). Alternatively,suppose f ðxÞ ¼ g1ðxÞ þ � � � þ gqðxÞ, x 2 D, where for each i ¼ 1; . . . ; q, the vector field gið�Þ is

continuous on D, homogeneous of degree ki 2 R with respect to the dilation DlðxÞ, and

k1o � � �okq. If x ¼ 0 is a finite-time-stable equilibrium point of g1ð�Þ, then x ¼ 0 is a finite-

time-stable equilibrium point of f ð�Þ.

Remark 5.1. If in Theorem 3.1 the comparison function wð�Þ is homogeneous of degreeko0 with respect to the dilation DlðzÞ and z ¼ 0 is an asymptotically stable equilibriumpoint of Eq. (8), then the zero solution xðtÞ 0 to Eq. (1) is finite-time stable. In this case,there is no need to construct a scalar positive definite function vð�Þ such that (9) holds.

Now, consider the large-scale dynamical system G involving energy exchange between n

interconnected subsystems given by Eq. (38). Furthermore, assume that there exists aconstant k 2 R such that

fijðlr1x1; . . . ; l

rn xnÞ ¼ lriþkfijðx1; . . . ;xnÞ; i; j ¼ 1; . . . ; q; iaj, (43)

for every l40 and for given ri40, i ¼ 1; . . . ; n. Next, consider the decentralized controllergiven by

ui ¼ ciðxiÞ; i ¼ 1; . . . ; n, (44)

with ciðxiÞ satisfying

Giðlri xiÞciðl

ri xiÞ ¼ lriþlGiðxiÞciðxiÞ; i ¼ 1; . . . ; n; x 2 Rn, (45)

and

Xn

i¼1

GiðxiÞciðxiÞo0; x 2 Rn, (46)


for every l40 and for given ri40, i ¼ 1; . . . ; n, where l 2 R, GðxÞ ¼ diag½G1ðx1Þ; . . .,GnðxnÞ�, and GiðxiÞ ¼ 0 if and only if xi ¼ 0, i ¼ 1; . . . ; n. If l ¼ ko0, then it follows fromProposition 5.1 that the closed-loop system (38) with uðtÞ ¼ ½c1ðx1Þ; . . . ;cnðxnÞ�

T is globallyfinite-time stable. Alternatively, if lok and lo0, then it follows from Proposition 5.1 thatthe closed-loop system (38) with uðtÞ ¼ ½c1ðx1Þ; . . . ;cnðxnÞ�

T is finite-time stable.Note that if lok and lo0, then stability is only local [7]. In order to obtain a global

result in this case, we need to examine the control vector Lyapunov function of the large-scale homogeneous system. Specifically, for the dynamical system G given by Eq. (38),consider the control vector Lyapunov function candidate V ð�Þ given by Eq. (39).Furthermore, consider the function

wðV ðxÞÞ ¼ �s1ðV 1ðx1ÞÞ þXn

j¼1;ja1

f1jðV ðxÞÞ; . . . ;�snðVnðxnÞÞ

"

þXn

j¼1;jan

fnjðV ðxÞÞ

#T; x 2 Rn

þ, (47)

where sið�Þ satisfies siðlri xiÞ ¼ lriþlsiðxiÞ for each l40 and given ri40, i ¼ 1; . . . ; n, lo0,

xi 2 Rþ, sið0Þ ¼ 0, siðzÞ40 for za0, z 2 R, and fijð�Þ satisfies Eq. (43) with k4l andi; j ¼ 1; . . . ; n, iaj.Next, consider the comparison system given by Eq. (41) where the ith component of wðzÞ

is given by wiðzÞ ¼ �siðziÞ þPn

j¼1;jaifijðzÞ, z 2 Rnþ. Then it follows from Proposition 5.1

that (41) is finite-time stable. Furthermore, consider the Lyapunov function candidatevðzÞ ¼ eTz, z 2 Rn

þ, and note that vð�Þ is radially unbounded, vð0Þ ¼ 0, vðzÞ40, z 2 Rnþ,

za0, and

v0ðzÞwðzÞ ¼ �Xn

i¼1

siðziÞ þXn

i¼1

Xn

j¼1;jai

fijðzÞ

¼ �Xn

i¼1

siðziÞ

o0; za0; z 2 Rnþ, (48)

which implies that Eq. (41) is globally asymptotically stable. Hence, Eq. (41) is globallyasymptotically stable, and thus, the large-scale homogeneous system (38) with ui ¼ ciðxiÞ,i ¼ 1; . . . ; n, is globally finite-time stable and V ð�Þ given by Eq. (39) is a control vectorLyapunov function for Eq. (38). Finally, (26) with aiðxÞ ¼ V 0iðxiÞf iðxÞ ¼

Pnj¼1;jaifijðxÞ,

biðxÞ ¼ GiðxiÞ, x 2 Rnþ, and c0i40, i ¼ 1; . . . ; n, is a finite-time globally stabilizing

decentralized feedback controller for Eq. (38). It can be seen from the structure of thefeedback control law that the closed-loop system dynamics are essentially nonnegative.Furthermore, since aiðxÞ � wiðV ðxÞÞ ¼ siðViðxiÞÞ, x 2 Rn

þ, i ¼ 1; . . . ; n, this feedbackcontroller is fully independent from f ðxÞ which represents the internal interconnectionsof the large-scale system dynamics, and hence, is robust against full modeling uncertaintyin f ðxÞ.


6. Illustrative numerical examples

In our first example we consider the large-scale dynamical system shown in Fig. 1 withthe power balance equation (38) where sijðxÞ ¼ sijx

2j , sijX0, iaj, i; j ¼ 1; . . . ; n, and

GiðxiÞ ¼ x1=4i , i ¼ 1; . . . ; n. Note that in this case fijðx

0Þpfijðx00Þ, iaj, i; j ¼ 1; . . . ; n, for all

x0; x00 2 Rnþ such that x0i ¼ x00i and x0kpx00k, kai. Furthermore, with ui ¼ �2x

1=4i ,

i ¼ 1; . . . ; n, the conditions of Proposition 4.1 are satisfied, and hence, the feedbackcontrol law (26) is continuous on Rn

þ. For our simulation we set n ¼ 3, s12 ¼ 2, s13 ¼ 3,s21 ¼ 1:5,s23 ¼ 0:3, s31 ¼ 4:4, s32 ¼ 0:6, c01 ¼ 1, c02 ¼ 1, and c03 ¼ 0:25, with initialcondition x0 ¼ ½3; 4; 1�

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

For the next example we consider control of thermoacoustic instabilities in combustionprocesses. Engineering applications involving steam and gas turbines and jet and ramjetengines for power generation and propulsion technology involve combustion processes.Due to the inherent coupling between several intricate physical phenomena in theseprocesses involving acoustics, thermodynamics, fluid mechanics, and chemical kinetics, thedynamic behavior of combustion systems is characterized by highly complex nonlinearmodels [28–31]. The unstable dynamic coupling between heat release in combustionprocesses generated by reacting mixtures releasing chemical energy and unsteady motionsin the combustor develop acoustic pressure and velocity oscillations which can severelyimpact operating conditions and system performance. These pressure oscillations, knownas thermoacoustic instabilities, often lead to high vibration levels causing mechanicalfailures, high levels of acoustic noise, high burn rates, and even component melting. Hence,the need for active control to mitigate combustion-induced pressure instabilities is critical.

Next, we design a finite-time stabilizing controller for a two-mode, nonlinear time-averaged combustion model with nonlinearities present due to the second-order gas

0 0.5 1 1.5 20

1

2

3

0 0.5 1 1.5 20

2

4

0 0.5 1 1.5 20

1

2

Fig. 2. Controlled system states versus time.

ARTICLE IN PRESS

0 0.5 1 1.5 2-4

-2

0

0 0.5 1 1.5 2-4

-2

0

0 0.5 1 1.5 2-4

-2

0

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

S.G. Nersesov et al. / Journal of the Franklin Institute 345 (2008) 819–837834

dynamics. This model is developed in [28] and is given by

_x1ðtÞ ¼ a1x1ðtÞ þ y1x2ðtÞ � bðx1ðtÞx3ðtÞ þ x2ðtÞx4ðtÞÞ þ u1ðtÞ; x1ð0Þ ¼ x10, (49)

_x2ðtÞ ¼ �y1x1ðtÞ þ a1x2ðtÞ þ bðx2ðtÞx3ðtÞ � x1ðtÞx4ðtÞÞ þ u2ðtÞ; x2ð0Þ ¼ x20, (50)

_x3ðtÞ ¼ a2x3ðtÞ þ y2x4ðtÞ þ bðx21ðtÞ � x2

2ðtÞÞ þ u3ðtÞ; x3ð0Þ ¼ x30, (51)

_x4ðtÞ ¼ �y2x3ðtÞ þ a2x4ðtÞ þ 2bx1ðtÞx2ðtÞ þ u4ðtÞ; x4ð0Þ ¼ x40, (52)

where a1, a2 2 R represent growth/decay constants, y1, y2 2 R represent frequency shiftconstants, b ¼ ððgþ 1Þ=8gÞo1, where g denotes the ratio of specific heats, o1 is thefrequency of the fundamental mode, and ui, i ¼ 1; . . . ; 4, are control input signals. For thedata parameters a1 ¼ 5, a2 ¼ �55, y1 ¼ 4, y2 ¼ 32, g ¼ 1:4, o1 ¼ 1, and x0 ¼ ½2; 3; 1; 1�

T,the open-loop (i.e., uiðtÞ 0; i ¼ 1; . . . ; 4) dynamics (49)–(52) result in a limit cycleinstability.To stabilize this system in finite time we design a feedback control law given by Eq. (32),

where V ðxÞ ¼ 12x

Tx, x 2 R4, c ¼ 1, c0 ¼ 1, a ¼ 34. In this case, V 0ðxÞ ¼ xT, GðxÞ ¼ I4, and

hence, R ¼ fx 2 R4;xa0 : xT ¼ 0g ¼ ;. Thus, condition (31) is trivially satisfied and itfollows from Corollary 4.1 that the closed-loop system (49)–(52) with the feedback controllaw (32) is finite-time stable. Furthermore, the hypothesis of Proposition 4.1 are satisfiedfor the case where q ¼ 1, and hence, the control law (32) is continuous in R4. Specifically,with u ¼ �f ðxÞ � 2�3=4gðxÞ, where

f ðxÞ ¼

a1x1 þ y1x2 � bðx1x3 þ x2x4Þ

�y1x1 þ a1x2 þ bðx2x3 � x1x4Þ

a2x3 þ y2x4 þ bðx21 � x2

2Þ

�y2x3 þ a2x4 þ 2bx1x2

266664

377775; gðxÞ ¼

x1=31

x1=32

x1=33

x1=34

2666664

3777775, (53)

ARTICLE IN PRESS

Fig. 4. Controlled system states versus time.

S.G. Nersesov et al. / Journal of the Franklin Institute 345 (2008) 819–837 835

the inequality

aðxÞ þ bTðxÞupwðV ðxÞÞ; 0okxkod, (54)

is satisfied, where aðxÞ9V 0ðxÞf ðxÞ, bðxÞ9GTðxÞV 0TðxÞ, wðV ðxÞÞ ¼ �ðV ðxÞÞ3=4, x 2 R4, and

0odo1. To see this, note that

aðxÞ þ bTðxÞu ¼ � 2�3=4xTgðxÞ

¼ � 2�3=4X4i¼1

x4=3i

p� 2�3=4X4i¼1

x2i

!3=4

¼ � ðV ðxÞÞ3=4

¼ wðV ðxÞÞ; 0okxkodo1. (55)

In addition, since f ð�Þ and gð�Þ are continuous and f ð0Þ ¼ gð0Þ ¼ 0, it follows from Eq. (55)that for every e40, there exists 0odo1 such that for all 0okxkod there exists u 2 R4 suchthat kukoe and inequality (54) holds. Thus, the feedback control law (32) is continuous inR4. Fig. 4 shows the states of the closed-loop system versus time and Fig. 5 shows thecontrol signals versus time.

ARTICLE IN PRESS

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

0 0.2 0.4 0.6 0.8 1-40

-20

0

20

u1(

t)

0 0.2 0.4 0.6 0.8 1-5

0

5

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

u2(

t)u

3(t)

u4(

t)

Time

Fig. 5. Control signals in each control channel versus time.

S.G. Nersesov et al. / Journal of the Franklin Institute 345 (2008) 819–837836

7. Conclusion

A vector Lyapunov function framework for addressing finite-time stability of nonlineardynamical systems was developed. In addition, the newly developed notion of controlvector Lyapunov functions was used to construct decentralized finite-time stabilizingcontrollers for large-scale dynamical systems with robustness guarantees against fullmodeling uncertainty. Two numerical examples were presented to show the utility of thedeveloped framework. Finally, a family of continuous finite-time decentralized feedbackstabilizers was developed for a class of large-scale homogeneous dynamical systems byexploiting connections between finite-time stability and geometric homogeneity.

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finite-time stabilization of nonlinear dynamical systems via control vector lyapunov functions

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