�A preliminary version of this paper was presented at the 14th IFACWorld Congress. This paper was recommended for publication inrevised form by Associate Editor Shinji Hara under the direction ofEditor Paul Van den Hof.

*Corresponding author. Tel.: #39-081-768-3513; fax: #39-081-768-3186.E-mail addresses: framato@unina.it (F. Amato), ariola@unina.it

(M. Ariola), peter@eece.unm.edu (P. Dorato).

Automatica 37 (2001) 1459}1463

Technical Communique

Finite-time control of linear systems subject to parametricuncertainties and disturbances�

Francesco Amato��*, Marco Ariola�, Peter Dorato��Dipartimento di Informatica e Sistemistica, Universita% degli Studi di Napoli Federico II, Via Claudio 21, 80125, Napoli, Italy

�EECE Department, The University of New Mexico, Albuquerque, NM 87131, USA

Received 17 November 1998; revised 4 January 2001; received in "nal form 9 March 2001

Abstract

In this paper we consider "nite-time control problems for linear systems subject to time-varying parametric uncertainties and toexogenous constant disturbances. The main result provided is a su$cient condition for robust "nite-time stabilization via statefeedback. It can be applied to problems with both non-zero initial conditions and unknown constant disturbances. This condition isthen reduced to a feasibility problem involving linear matrix inequalities (LMIs). A detailed example is presented to illustrate theproposed methodology. � 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Uncertain linear systems; Parametric uncertainties; Finite-time stability; Disturbance-rejection

1. Introduction

Much work has been done on the robust control oflinear systems over the past twenty years. See, for in-stance, the recent texts of Bhattacharyya, Chapellat, andKeel (1995) and Zhou and Doyle (1998). Most of theresults in this "eld relate to stability and performancecriteria de"ned over an in"nite time interval. In manypractical applications, however, the main concern is thebehavior of the system over a "xed "nite time interval. Inthis sense it appears reasonable to de"ne as stable a sys-tem whose state, given some initial conditions, remainswithin prescribed bounds in the "xed time interval, andas unstable a system which does not. Many are thepractical problems in which this kind of stability, called"nite-time stability (FTS) or short-time stability (seeDorato, 1961; Weiss & Infante, 1967) plays an importantrole: for instance the problem of not exceeding some

given bounds for the state trajectories, when some satura-tion elements are present in the control loop; or theproblem of controlling the trajectory of a space vehiclefrom an initial point to a "nal point in a prescribed timeinterval.In Section 2 we recall the de"nition of FTS given in

Dorato (1961) and we extend it introducing the conceptof "nite-time boundedness (FTB) for the state of a sys-tem, when not only given initial conditions but alsoexternal constant disturbances are considered.The result provided in this paper (Section 3) is a su$-

cient condition for state feedback "nite-time stabilizationof a linear system subject to parametric uncertainties andto constant disturbances; this condition requires the solu-tion of an linear matrix inequalities (LMI) problem. A de-tailed example is provided to illustrate the di!erent casesconsidered.

2. Problem statement

In this paper we consider the following linear systemsubject to time-varying parametric uncertainties and toexogenous disturbances:

x� (t)"A(p)x(t)#B(p)u(t)#G(p)w, (1)

where A(p)3����, B(p)3���� and G(p)3����.

0005-1098/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 0 8 7 - 5

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We assume the following:

(A1) The parameter vector function p( ) )"(p�( ) )

p�( ) )2p

�( ) ))� is any Lebesgue measurable func-

tion p( ) ) : [0,¹]PR, where

R"[p� �,p�

�]�[p

� �,p�

�]�2�[p

� �,p�

�];

we denote the vertices of R by p���, i"1,2, 2�.

(A2) The matrix-valued functionsA( ) ), B( ) ) andG( ) ) aregiven by multia$ne matrix-valued functions; forinstance A(p)"�

�� �2� ��A

���2� ��p���

2p���, where the

indexes i�,2, i

�3�0,1�.

(A3) The exogenous disturbance w is constant and satis-"es the constraint

w�w)d, d*0. (2)

Concerning system (1), we consider the following statefeedback controller:

u"Kx, (3)

where K3����.The aim of this paper is to "nd some su$cient condi-

tions which guarantee that the state of the closed loopsystem given by the interconnection of (1) with (3) isbounded over a xnite-time interval. The general idea ofFTS concerns the boundedness of the state of a systemover a "nite time interval for given initial conditions; thisconcept can be formalized through the following de"ni-tion, which is an extension of the one given in Dorato(1961).

De5nition 1 (F¹S). The time-varying linear system

x� (t)"A(t)x(t), t3[0,¹],

is said to be "nite time stable (FTS) with respect to(c

�, c

�,¹,R), with c

�'c

�and R'0 if

x�(0)Rx(0))c�Nx�(t)Rx(t)(c

�, ∀t3[0,¹].

Remark 2 (F¹S and asymptotic stability). It is worthnoting that asymptotic stability and FTS are independentconcepts: a system which is FTS may not be asymp-totically stable, while a asymptotically stable system maynot be FTS.

The idea of state boundedness, on the other hand, ismore general, and concerns the behavior of the state inthe presence of both given initial conditions and externaldisturbances.

De5nition 3 (F¹B). The time-varying linear system

x� (t)"A(t)x(t)#G(t)w, t3[0,¹],

subject to an exogenous disturbance w satisfying (2), issaid to be FTB with respect to (c

�, c

�,¹,R, d), with

c�'c

�and R'0 if

x�(0)Rx(0))c�Nx�(t)Rx(t)(c

�,

∀t3[0,¹], ∀w: w�w)d.

It is easy to see that, given our De"nition 3 of FTB,FTS can be recovered as a particular case by lettingd"0.

Remark 4 (F¹B and reachable sets). Since the concept ofFTB is, in some way, related to the concept of reachablesets, it is important to clarify the di!erences between thetwo ideas. Reachable sets are de"ned as the set of statesthat a dynamical system attains given some boundedinputs and starting from some given initial conditions(see Grantham, 1980, 1981). On the other hand, accord-ing to De"nition 3, FTB explores if, given a bound on thestate variables and a set of admissible initial states, thestate remains con"ned within the prescribed bound whenboth non-zero initial conditions and external constantdisturbances are considered. An important di!erence be-tween the two approaches is that in the reachable setanalysis the assumption of system asymptotic stability isexploited (see for example Boyd, El Ghaoui, Feron,& Balakrishnan, 1994, p. 83; Gayek, 1991), while the FTBanalysis condition provided in this paper allows to estab-lish FTB for t3[0,¹] of the system even if it is notasymptotically stable (see also Remark 2 and the examplein Section 4).

On the basis of the above considerations the aim ofthis paper is the solution of the following xnite-boundedproblem.

Finite Bounded Problem (FB). Given system (1) and(c

�, c

�,¹,R, d), xnd a state feedback controller in the form

(3) such that the closed-loop system given by the intercon-nection of (1) with (3) is FTB with respect to (c

�, c

�,¹,R, d)

for all admissible vector valued functions p( ) ) : [0,¹]PR.

Remark 5. In some cases it is of interest to minimize theinterval ¹ or the trajectory bound c

�. These problems

can be solved by simple binary search algorithms oncethe basic FB Problem is solved.

Concerning the disturbance free system

x� "A(p)x#B(p)u, (4)

we shall consider the analogous of the problem statedabove, regarding the problem of the FTS of system (4)with respect to (c

�, c

�,¹,R). This problem will be in-

dicated as the FTS problem.

1460 F. Amato et al. / Automatica 37 (2001) 1459}1463

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3. Main result

The following lemma states a su$cient condition forthe FTB of a certain, linear time-invariant system in theform

x� (t)"Ax(t)#Gw, (5)

which is fundamental to prove the main results in whatfollows.

Lemma 6. System (5) is FTB with respect to (c�, c

�,¹,R, d)

if, letting QI�"R������Q

�R������, there exist a positive

scalar � and two symmetric positive dexnite matricesQ

�3���� and Q

�3���� such that

�AQI

�#QI

�A�!�QI

�GQ

�Q

�G� !�Q

��(0, (6a)

c�

����

(Q�)#

d

����

(Q�)(

c�e���

����

(Q�), (6b)

where ����

( ) ) and ����

( ) ) indicate the maximum and min-imum eigenvalue of the argument, respectively.

Proof. Let <(x,w)"x�QI ���

x#w�Q���

w and denote, asusual, by <Q the derivative of < along the solution ofsystem (5). Suppose that the condition

<Q (x(t),w)(�<(x(t),w), (7)

holds for all t3[0,¹] and all w with w�w)d. We will"rst demonstrate that conditions (7) and (6b) imply thatsystem (5) is FTB with respect to (c

�, c

�,¹,R, d). Then, to

conclude the proof, we will show that condition (7) isequivalent to (6a).Our "rst claim is that conditions (7) and (6b) imply the

FTB of system (5) with respect to (c�, c

�,¹,R, d). Divid-

ing both sides of (7) by <(x,w), and integrating from 0 tot, with t3(0,¹], we obtain

log<(x(t),w)

<(x(0),w)(�t. (8)

Now, introducing

P"�Q��

�0

0 Q����, M"�

R��� 0

0 I��, z"�

x

w�,where I

�denotes the identitymatrix of order l, it is easy to

show that from (8) it follows that:

z�(t)MPMz(t)(z�(0)MPMz(0)e��. (9)

Now we have

z�(t)MPMz(t)"x�(t)R���Q���

R���x(t)#w�(t)Q���

w(t)

*����

(Q���)x�(t)Rx(t), (10a)

z�(0)MPMz(0)e��

"(x�(0)R���Q���

R���x(0)#w�(0)Q���

w(0))e��

)(����

(Q���

)x�(0)Rx(0)#����

(Q���)w�(0)w(0))e��

)(����

(Q���

)c�#�

���(Q��

�)d)e��. (10b)

Putting together (9) and (10) we have

x�(t)Rx(t)((�

���(Q��

�)c

�#�

���(Q��

�)d)e��

����

(Q���

)

"����

(Q�)e���

c�

����

(Q�)#

d

����

(Q�)�. (11)

From (11) it readily follows that (6b) implies, for allt3[0,¹], x�(t)Rx(t)(c

�; from this last consideration

our "rst claim follows.Nowwe shall prove that condition (6a) is equivalent to

(7). Assume that there exist an �'0 and two symmetricmatrices Q

�'0 and Q

�'0 such that inequality (6a) is

satis"ed. Pre and post-multiplying (6a) by

�QI ��

�0

0 Q����,

we obtain the equivalent condition

�QI ��

�A#A�QI ��

�!�QI ��

�QI ��

�G

G�QI ���

!�Q����(0,

which is, in turn, equivalent to (7). Therefore the prooffollows. �

The key idea in Lemma 6 is the use of an exponentialtime weighting e����� of the state with a negative expo-nent (to see this multiply both members of (9) by e���),while in the Lyapunov stability context one uses ex-ponential time weighting with a positive exponent. Thisallows to relax the classical conditions for asymptoticstability and to establish FTB of the system under con-sideration even if it is not asymptotically stable.Now let us go back to our original FB Problem.

A su$cient condition for the solution of this problem isgiven by the following theorem.

Theorem 7. Given system (1), the FB Problem is solvable if,letting QI

�"R������Q

�R������, there exist a positive scalar

�, two symmetric positive dexnite matrices Q�3���� and

Q�3���� and a matrix ¸3���� such that inequality (6b)

and

�A(p

���)QI

�#QI

�A�(p

���)#B(p

���)¸#¸�B�(p

���)!�QI

�Q

�G�(p

���)

G(p���)Q

�!�Q

��(0, i"1,2,2� (12)

hold. In this case a controller which solves the FB Problemis given by K"¸QI ��

�.

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Proof. By virtue of Lemma 6 we have that the FB Prob-lem admits a solution if there exist �'0, Q

�'0, i"1,2

such that inequality (6b) and

�(A(p)#B(p)K)QI

�#QI

�(A(p)#B(p)K)�!�QI

�Q

�G�(p)

G(p)Q�

!�Q��(0, ∀p3R (13)

hold. Since the left hand side in (13) depends on para-meters as the ratio of multia$ne polynomials, we canapply the main result in Garofalo, Celentano, andGlielmo (1993) which allows us to state that (13), byletting ¸"KQI

�(Geromel, Peres, and Bernussou, 1991),

is equivalent to (12). �

Remark 8. It is worth noting that we assumed a multiaf-"ne structure for the system matrices (see AssumptionA2). For this structure it is possible to state the verticesresult as in Theorem 7. If the system matrices do notsatisfy Assumption A2 (for example when they depend onparameters as multivariate polynomials), the procedureproposed in Amato, Garofalo, Glielmo, and Pironti(1995) allows one, at the price of some conservatism, toreplace the system matrices with suitable matricesdepending multia$nely on parameters as in Assum-ption A2.

The following Corollary of Theorem 7 provides a su$-cient condition for the FTS of the system

x� "A(p)x#B(p)u,

by means of a state feedback controller in the form (3).

Corollary 9. Given system (4) the FTS Problem is solvableif, letting QI "R������QR������, there exist a nonnegativescalar �, a symmetric positive dexnite matrix Q3���� anda matrix ¸3���� such that

A(p���)QI #QI A�(p

���)#B(p

���)¸#¸�B�(p

���)!�QI (0,

i"1,2,2�, (14a)

cond(Q)(c�c�

e���, (14b)

where cond(Q)"����

(Q)/����

(Q) denotes the conditionnumber of Q. In this case a controller which solves the FTSProblem is K"¸QI ��.

Remark 10. In Corollary 9 we allow � to be zero; thiswas not permitted in Theorem 7. It can be easily seen thatif we can guarantee that the conditions in Corollary 9are satis"ed for �"0, then the closed-loop systemx� "(A(p)#B(p)K)x is also quadratically stable (Corless,1993) and therefore it is uniformly asymptotically stable

on the in"nite interval [0,#R) for all admissible vec-tor-valued functions p( ) ) : [0,#R)PR. In Theorem 7,instead, if �"0, we cannot guarantee the negative de"-niteness of (12).

It is easy to check that condition (6b) is guaranteed byimposing the conditions

��I(Q

�(I, (15a)

��I(Q

�(�

I, (15b)

c�/�

�#d/�

�(c

�e���, (15c)

for some positive numbers ��, �

�and �

. Inequality (15c)

can be converted to an LMI using Schur complements. Itis equivalent to

�c�e��� �c

��d

�c�

��

0

�d 0 ���'0. (16)

From a computational point of view, it is important tonotice that, once we have "xed a value for �, the feasibil-ity of the conditions stated in Theorem 7 can be turnedinto the following LMIs based feasibility problem (Boyd,El Ghaoui, Feron, and Balakrishnan, 1994).

LMI Feasibility Problem (from Theorem 7). Given sys-tem (1) and (c

�, c

�,¹,R, d), let QI

�"R������Q

�R������, xx

an �'0 and xnd matrices ¸, Q�, Q

�, and positive scalars

��, �

�, �

satisfying the LMIs (12), (15a), (15b) and (16). If

the problem is feasible, the static controller K"¸QI ���

solves the FB Problem.

In a similar way, we can derive the following feasibilityproblem from the conditions stated in Corollary 9.

LMI Feasibility Problem (from Corollary 9). Givensystem (4) and (c

�, c

�,¹,R), let QI "R������QR������, xx

an �*0 and xnd matrices ¸, Q satisfying the LMIs (14a)and

c�

c�

e��I(Q(I.

If the problem is feasible, the static controller K"¸QI ��

solves the FTS Problem.

4. Numerical example

Let us consider system (1) with

A(p)"A#A

�p, B(p)"B

#B

�p,

G(p)"G#G

�p

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with

A"�

0 1

!2 !1�, A�"�

0 0

1 1�, B"�

0

1�,

B�"�

1

0�, G"�

��0�, G

�"�

0

���,

where p3[!10,10], c�"1, c

�"10, d"�

�and R"I,

where I is the identity matrix of appropriate dimensions.Given these values for c

�, c

�, d and R, we solved both the

analysis and the synthesis problem with the aid of theLMI toolbox (Gahinet, Nemirovski, Laub, and Chilali,1995) by solving the feasibility problem described in theprevious section.

� Analysis (d"0, K"0). Using the conditions of Co-rollary 9, we found that the system is FTS with respectto (c

�, c

�,¹) for a maximum¹

���"0.1 s, obtained for

�"22. Note that for p'1 the system is not asymp-totically stable. Therefore this is an example of a sys-tem which is FTS but not asymptotically stable (seeRemarks 2, 4).

� Synthesis (dO0, KO0). We solved the two problemsFTS and FB, obtaining the following results:(1) FTS-Problem: we obtained ¹

���"1 s, for

�"0.6, with the controller K"(!0.66 !1.85);(2) FB-Problem: in this case we obtained ¹

���"

0.42 s for �"1.1 with the controller K"

(!0.66 !2.54).As expected, the presence of the disturbance causesa reduction of the value of ¹

���.

5. Conclusions

In this paper we have considered the "nite-time con-trol problem for a linear system subject to uncertainparameters and to unknown constant disturbances. Firstof all we have extended the de"nition of FTS (Dorato,1961) to the de"nition of FTB to take into account alsothe presence of external disturbances. Then we haveprovided a su$cient condition guaranteeing FTB viastate feedback. This condition has been turned into anoptimization problem involving LMIs. Some numerical

examples have been included to illustrate the proposedalgorithms.

Acknowledgements

The author would like to thank the anonymous re-viewer �1 for suggesting an improvement to the condi-tion stated in Theorem 7 and the anonymous reviewer�3for pointing out some previous works on reachable sets.

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