Automatica 46 (2010) 919–924

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Automatica

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Brief paper

Finite-time control of discrete-time linear systems: Analysis anddesign conditionsI

Francesco Amato a, Marco Ariola b,∗, Carlo Cosentino aa School of Computer and Biomedical Engineering, Università degli Studi Magna Græcia di Catanzaro, Viale Europa, Campus di Germaneto, 88100 Catanzaro, Italyb Dipartimento per le Tecnologie, Università degli Studi di Napoli Parthenope, Centro Direzionale, Isola C4, 80143, Napoli, Italy

a r t i c l e i n f o

Article history:Received 22 April 2009Received in revised form2 November 2009Accepted 2 February 2010Available online 5 March 2010

Keywords:Discrete-time systemsLinear systemsFinite-time stabilityLMIsOutput feedback

a b s t r a c t

In this paper we deal with some finite-time control problems for discrete-time, time-varying linearsystems. First we provide necessary and sufficient conditions for finite-time stability; these conditionsrequire either the computation of the state transition matrix of the system or the solution of a certaindifference Lyapunov inequality. Then we address the design problem. The proposed conditions allow usto find output feedback controllers which stabilize the closed loop system in the finite-time sense; allthese conditions can be expressed in terms of LMIs and therefore are numerically tractable, as shown inthe example included in the paper.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

When dealing with the stability of a system, a distinctionshould bemade between classical Lyapunov stability and finite-timestability (FTS) (or short-time stability). The concept of Lyapunovasymptotic stability is largely known to the control community;conversely a system is said to be finite-time stable if, once we fix atime-interval, its state does not exceed some bounds during thistime-interval. Often asymptotic stability is enough for practicalapplications, but there are some cases where large values of thestate are not acceptable, for instance in the presence of saturations.In these cases, we need to check that these unacceptable values arenot attained by the state; for these purposes FTS could be used.Most of the results in the literature are focused on Lyapunov

stability. Some early results on FTS can be found in Dorato (1961)and Weiss and Infante (1967); more recently the concept of FTShas been revisited in the light of recent results coming fromlinear matrix inequalities (LMIs) theory (Boyd, Ghaoui, Feron, &Balakrishnan, 1994), which has enabled us to find less conservativeconditions guaranteeing FTS and finite time stabilization of

I Thematerial in this paper was partially presented at 16th IFACWorld Congress,July 4–8, 2005, Prague (CZ). This paper was recommended for publication in revisedform by Associate Editor Tong Zhou under the direction of Editor Roberto Tempo.∗ Corresponding author. Tel.: +39 081 5476719; fax: +39 081 5476777.E-mail address: ariola@uniparthenope.it (M. Ariola).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.02.008

uncertain, linear continuous-time systems (see Amato and Ariola(2001) and Amato, Ariola, and Cosentino (2006)). An approachbased on Lyapunov differential matrix equations for finite-timestabilization via state feedback is proposed in Garcia, Tarbouriech,and Bernussou (2009), whereas the use of Lyapunov differentialinequalities in this context had been introduced in Amato, Ariola,Cosentino, Abdallah, and Dorato (2003). The discrete-time caseis dealt with in the paper Amato and Ariola (2005), wheresufficient conditions for finite-time stabilization via state feedback,expressed in terms of LMIs, are given. More recently, othercontributions on FTS have been given in Ambrosino, Calabrese,Cosentino, and Tommasi (2009) and Zhao, Sun, and Liu (2008) forthe case of impulsive systems.In this paper, which is the extended version of the conference

paper Amato, Ariola, Carbone, and Cosentino (2005), we considertime-varying discrete-time systems. Our main analysis theoremguarantees FTS if and only if either a certain inequality involvingthe state transition matrix is satisfied, or a symmetric matrixfunction solving a certain Lyapunov difference inequality exists.The condition involving the state transition matrix cannot be usedas the starting point to solve the synthesis problem. Therefore, inview of the design problem, we focus on the condition involvingthe Lyapunov inequality. However this condition can becomecomputationally hard to apply, since it requires to study thefeasibility of N difference inequalities, if [1,N] is the time intervalin which FTS is studied. For this reason a sufficient conditionfor FTS which requires to check the feasibility of only one

920 F. Amato et al. / Automatica 46 (2010) 919–924

inequality is used to address the problem of designing outputfeedback controllers guaranteeing some finite-time performance.This sufficient condition for FTS is also compared with the onederived in Amato and Ariola (2005), showing that the new oneis much less conservative. In this paper we focus on the outputfeedback design; the state feedback design has been dealt with inAmato et al. (2005).The paper is organized as follows: in Section 2 the definition

of finite-time stability is recalled and specialized to the discrete-time case, and the problem we want to solve is formally stated.In Section 3 necessary and sufficient conditions for finite-timestability are provided. In Section 4 we address the FTS designproblems, namely some sufficient conditions for the existence ofoutput feedback controllers guaranteeing finite-time stabilizationof the closed loop system are provided. In Section 5, it is shownhow the proposed design conditions can be expressed in terms ofLMIs and, therefore, can be efficiently dealt with, and a numericalexample is provided. Some conclusions are drawn in Section 6.

2. Problem statement and preliminaries

In this paper we consider the following discrete-time time-varying linear system

x(k+ 1) = A(k)x(k)+ B(k)u(k) (1a)y(k) = C(k)x(k) (1b)

where A(k), B(k) and C(k) take value in Rn×n, Rn×m and Rp×n,respectively. The general idea of finite-time stability concerns theboundedness of the state of the system over a finite time intervalfor some given initial conditions; this concept can be formalizedthrough the following definition, which is an extension to discrete-time systems of the one given in Dorato (1961).

Definition 1 (Finite-time stability). The time-varying discrete-timelinear system

x(k+ 1) = A(k)x(k) k ∈ N0 (2)

is said to be finite-time stable with respect to (ε, R,N), where R isa positive definite matrix, and N ∈ N+, if

xT (0)Rx(0) ≤ 1⇒ xT (k)Rx(k) < ε2 ∀k ∈ {1, . . . ,N} .

Remark 2. Lyapunov asymptotic stability (LAS) and FTS areindependent concepts: a system which is FTS may be not LAS;conversely a LAS system could be not FTS if, during the transients,its state exceeds the prescribed bounds.

Remark 3. In this paper, the sets we consider in the FTS definitionare ellipsoids. An alternative choice is to consider polyhedral sets,as done for instance in Amato, Ambrosino, Ariola, and Calabrese(2008) and Garcia et al. (2009).

With respect to system (1), we consider the following dynamicoutput feedback controller:

xc(k+ 1) = AK (k)xc(k)+ BK (k)y(k) (3a)u(k) = CK (k)xc(k)+ DK (k)y(k), (3b)

where the controller state vector xc(k) has the same dimension ofx(k).The main goal of the paper is to find some sufficient conditions

which guarantee the existence of an output feedback controllerwhich finite-time stabilizes the overall closed loop system, asstated in the following problem.

Problem 4. Let us denote by RK the weight of the controller state.Then, given system (1), find an output feedback controller (3) suchthat the corresponding closed-loop system is finite-time stablewith respect to (ε, blockdiag(R, RK ),N).

Note that, for the sake of simplicity, we assume that theweighting matrix does not contain cross-coupling terms betweenthe system state and the controller state. Moreover the definitionof ε in Problem 4 must take into account the augmented state ofthe closed loop system.

3. Analysis conditions for FTS

The following theorem is fundamental for the subsequentdevelopment.

Theorem 5 (Nec. and Suff. Conditions for FTS). The followingstatements are equivalent:(i) system (2) is FTS with respect to (ε, R,N).(ii) ΦT (k, 0)RΦ(k, 0) < ε2R for all k ∈ {1, . . . ,N}, where Φ(·, ·)denotes the state transition matrix of system (2).

(iii) For each k ∈ {1, . . . ,N} there exists a symmetric matrix-valuedfunction Pk(·) : h ∈ {0, 1, . . . , k} 7→ Pk(h) ∈ Rn×n such that

AT (h)Pk(h+ 1)A(h)− Pk(h) < 0 h ∈ {0, 1, . . . , k− 1} (4a)Pk(k) ≥ R (4b)

Pk(0) < ε2R. (4c)

Moreover, each one of the above conditions is implied by the following.(iv) There exists a symmetric matrix-valued function P(·) : k ∈ {0,

1, . . . ,N} 7→ P(k) ∈ Rn×n such that

AT (k)P(k+ 1)A(k)− P(k) < 0, k ∈ {0, 1, . . . ,N − 1} (5a)P(k) ≥ R, k ∈ {1, . . . ,N} (5b)

P(0) < ε2R. (5c)

Proof. Proof that (i) is equivalent to (ii). First we prove that (ii)implies (i); let k ∈ {1, . . . ,N} and x(0) such that xT (0)Rx(0) ≤ 1.We have

x(k) = Φ(k, 0)x(0)

then

xT (k)Rx(k) = xT (0)Φ(k, 0)kRΦ(k, 0)x(0)< ε2xT (0)Rx(0)≤ ε2, for all k ∈ {1, . . . ,N}.

Conversely, assume by contradiction that system (2) is FTS andthat for some k ∈ {1, . . . ,N}, x ∈ Rn

xTΦT (k, 0)RΦ(k, 0)x ≥ ε2xTRx. (6)

Now let x(0) = λx, such that

xT (0)Rx(0) = λ2xTRx = 1; (7)

moreover let x(·) the state evolution of system (2) starting fromx(0).From (6) and (7), we have

xT (k)Rx(k) = xT (0)ΦT (k, 0)RΦ(k, 0)x(0)≥ ε2xT (0)Rx(0) = ε2.

Therefore, we have found an initial condition x(0) satisfyingxT (0)Rx(0) = 1 such that, for some k, xT (k)Rx(k) ≥ ε2. Thiscontradicts the hypothesis that the system is FTS.Proof that (i) is equivalent to (iii). First we prove that (iii) implies (i).Let k ∈ {1, . . . ,N} and assume there exists a symmetric matrix-valued Pk(·) such that (4) hold. By (4a) we have

xT (h+ 1)Pk(h+ 1)x(h+ 1)− xT (h)Pk(h)x(h)

= xT (h)(AT (h)Pk(h+ 1)A(h)− Pk(h)

)x(h) < 0. (8)

F. Amato et al. / Automatica 46 (2010) 919–924 921

By summing (8) between 0 and k, we have that

xT (k)Pk(k)x(k)− xT (0)Pk(0)x(0) < 0. (9)

From the last equation and by using (4b) and (4c), it follows that

xT (k)Rx(k) < ε2xT (0)Rx(0). (10)

Therefore if xT (0)Rx(0) ≤ 1 we have that xT (k)Rx(k) < ε2 for allk ∈ {1, . . . ,N} and then the proof follows.Now let us assume that system (1) is FTS. Then by continuity

arguments it follows that there exists a sufficiently small γ suchthat, letting z = γ x,

xT (0)Rx(0) ≤ 1⇒ xT (k)Rx(k)+ ‖z‖22 < ε2 (11)

where ‖z‖22 =∑kh=0 z

T (h)z(h).Now let Pk(·) : h ∈ {0, 1, . . . , k} 7→ Pk(h) ∈ Rn×n defined as

follows:

AT (h)Pk(h+ 1)A(h) = Pk(h)− γ 2I, Pk(k) = R (12)

and assume, by contradiction, that there exists x ∈ Rn such that

xTPk(0)x ≥ ε2xTRx.

Again let x(0) = λx such that xT (0)Rx(0) = 1 and x(k) the stateevolution starting from x(0). By (12) it follows that

xT (k)Pk(k)x(k)− xT (0)Pk(0)x(0)+ ‖z‖22 = 0

from which it follows that

xT (k)Rx(k)+ ‖z‖22 ≥ ε2xT (0)Rx(0) = ε2,

which contradicts (11).Therefore we have proven that, for any k ∈ {1, . . . ,N}, there

exists Pk(·) : h ∈ {0, 1, . . . , k} 7→ Pk(h) ∈ Rn×n such that

AT (h)Pk(h+ 1)A(h)− Pk(h) = −γ 2I < 0Pk(k) ≥ RPk(0) < ε2R.

Proof that (iv) implies (iii). It is straightforward to check that amatrix function P satisfying conditions (5) also satisfies conditions(4). �

The following are some remarks about the use of the resultscontained in Theorem 5.

Remark 6. Condition (ii) is very useful to test the FTS of a givensystem. However it cannot be used for design purposes. In thesameway condition (iii) is not useful from a practical point of view,since it requires to study the feasibility of N difference Lyapunovinequalities, if [1,N] is the time interval of interest. Converselythe sufficient condition (iv) requires to check only one differenceinequality.

Remark 7 (Comparison with earlier results). In this remark, wediscuss the conservativeness of the sufficient condition (iv)comparing it with the necessary and sufficient condition (ii).Moreover, we show that the sufficient condition proposed inthis paper is much less conservative than the sufficient conditionproposed in the paper Amato and Ariola (2005) (see Corollary 1).It is important to point out that the comparison is made only forthe analysis conditions, since the approach proposed in this paperis the only one that can be used for design purposes in the generaloutput feedback case.To carry out this comparison we have randomly generated

10,000 time-invariant third-order discrete-time linear systems.For each sample we have computed the minimum ε such that thegiven system is FTSwrt (ε, R,N)withN = 5 and R = I . Asmeasureof the conservativeness, we have evaluated the following quantity

err% = 100(εsuff − εtrue)/εtrue, (13)

where εtrue denotes the exact value of ε computed using condition(ii) and εsuff its estimated value obtained applying condition (iv) orthe condition proposed in the paper Amato and Ariola (2005). Inthe former case (13) evaluates to 1%, whereas in the latter case itevaluates to 20%. These results show that

• condition (iv) is much less conservative than the sufficientcondition proposed in Amato and Ariola (2005);• the sufficient condition (iv) gives in fact results that are veryclose to the results given by the necessary and sufficientcondition (ii).

4. Controller design

Let us consider the problem of finite-time stabilization viaoutput feedback. First we need the following technical lemma.

Lemma 8 (Gahinet (1996)). Given symmetric matrices S ∈ Rn×n andQ ∈ Rn×n, the following statements are equivalent.

(i) There exist a symmetricmatrix T ∈ Rn×n andmatricesM ∈ Rn×n,U ∈ Rn×n such that

P :=(S MMT T

)> 0, P−1 =

(Q UUT Z

). (14)

(ii) (Q II S

)> 0. (15)

Theorem 9 (FTS via output feedback). Problem 4 is solvable if thereexist positive definite matrix-valued functions Q (·), S(·), an invertiblematrix U(·), matrix-valued functions AK (·), BK (·), CK (·) and DK (·)such that(Θa(k) Θb(k)ΘTb (k) Θa(k+ 1)

)< 0, k ∈ {0, 1, . . . ,N − 1} (16a)

Q (k) Ψ12(k) Ψ13(k) Ψ14(k)Ψ T12(k) Ψ22(k) 0 0Ψ T13(k) 0 I 0Ψ T14(k) 0 0 I

≥ 0,k ∈ {1, . . . ,N} (16b)(Q (0) II S(0)

)< ε2

(∆a Q (0)RRQ (0) R

), (16c)

where

Θa(k) = −(Q (k) II S(k)

)(17a)

Θb(k) =(Q (k)AT (k)+ CTK (k)B

T (k) ATK (k)AT (k)+ CT (k)DTK (k)B

T (k) AT (k)S(k+ 1)+ CT (k)BTK (k)

)(17b)

Ψ12(k) = I − Q (k)R (17c)

Ψ13(k) = Q (k)R1/2 (17d)

Ψ14(k) = U(k)R1/2K (17e)

Ψ22(k) = S(k)− R (17f)

∆a = Q (0)RQ (0)+ U(0)RKUT (0). (17g)

Proof. The connection between system (1) and controller (3) reads(x(k+ 1)xc(k+ 1)

)= ACL(k)

(x(k)xc(k)

), (18)

922 F. Amato et al. / Automatica 46 (2010) 919–924

where

ACL(k) :=(A(k)+ B(k)DK (k)C(k) B(k)CK (k)

BK (k)C(k) AK (k)

).

From (iv) in Theorem 5 it follows that Problem 4 is solvable if thereexist a positive definite matrix function P(·) and matrices AK (·),BK (·), CK (·) and DK (·) such that(

−P(k) ATCL(k)P(k+ 1)P(k+ 1)ACL(k) −P(k+ 1)

)< 0

k ∈ {0, 1, . . . ,N − 1} (19a)P(k) ≥ blockdiag(R, RK ), k ∈ {1, 2, . . . ,N} (19b)

P(0) < ε2 blockdiag(R, RK ). (19c)

Now, according to Gahinet (1996), let us define

P(k) =(S(k) M(k)MT (k) ?

),

P−1(k) =(Q (k) U(k)UT (k) ?

),

where ? denotes a ‘don’t care’ block, and

Π1(k) =(Q (k) IUT (k) 0

), Π2(k) =

(I S(k)0 MT (k)

).

Note that by definition

S(k)Q (k)+M(k)UT (k) = I (20a)

Q (k)S(k)+ U(k)MT (k) = I (20b)P(k)Π1(k) = Π2(k). (20c)

By pre- and post-multiplying inequality (19a) by blockdiag(Π T1 (k),Π

T1 (k+1)) and blockdiag(Π1(k),Π1(k+1)) respectively,

pre- and post-multiplying (19b) and (19c) by Π T1 (k) and Π1(k)respectively, taking into account (20) and Lemma 8 the prooffollows once we let

BK (k) = M(k+ 1)BK (k)+ S(k+ 1)B(k)DK (k) (21a)

CK (k) = CK (k)UT (k)+ DK (k)C(k)Q (k) (21b)

AK (k) = M(k+ 1)AK (k)UT (k)+ S(k+ 1)B(k)CK (k)UT (k)+M(k+ 1)BK (k)C(k)Q (k)+ S(k+ 1) (A(k)+ B(k)DK (k)C(k))Q (k). (21c)

Note that (16a) implies that, at each time instant k,(Q (k) II S(k)

)> 0 (22)

which, according to Lemma 8, guarantees the reconstruction ofP(k) starting from the knowledge of S(k), Q (k) and U(k). �

Remark 10 (Controller design). Assume now that the hypothesesof Theorem 9 are satisfied; in order to design the controller thefollowing steps have to be carried out.

(i) FindQ (·), S(·),U(·), AK (·), BK (·), CK (·) andDK (·) such that (16)are satisfied.

(ii) Find matrix function M(·) such that M(k) = (I − S(k)Q (k))U−T (k).

(iii) Obtain AK (·), BK (·), CK (·) and DK (·) by inverting (21). Thecontroller matrices read

BK (k) = M−1(k+ 1)BK (k)−M−1(k+ 1)S(k+ 1)B(k)DK (k)

CK (k) = CK (k)U−T (k)− DK (k)C(k)Q (k)U−T (k)

M1 M2

kF

s1 s2

Fig. 1. The mechanical system considered in Example 12.

AK (k) = M−1(k+ 1)AK (k)U−T (k)−M−1(k+ 1)S(k+ 1)B(k)CK (k)− BK (k)C(k)Q (k)U−T (k)−M−1(k+ 1)S(k+ 1)× (A(k)+ B(k)DK (k)C(k))Q (k)U−T (k).

The procedure for the design of the output feedback controllerexpressed in terms of solution of a set of LMIs will be presented innext section.

5. Numerical implementation of the main results

In this section we will discuss the numerical implementationof Theorem 9 regarding the output feedback design. We will showthat it is possible to express the conditions in terms of a feasibilityproblem of a set of LMIs; in this way the solution, if it exists, can befound using one of the many packages which are available to solvesuch problems.Conditions (16a) and (16b) are already expressed as linear

difference matrix inequalities. Conversely, the initial condition(16c) exhibits a quadratic term in the optimization variables Q (0)and U(0). At the price of some conservativism, it is possible tosubstitute condition (16c) with the following inequality:(Q (0) II S(0)

)< ε2

(∆a Q (0)RRQ (0) R,

)(23)

where

∆a = Q (0)R1/2 + R1/2Q (0)+ UT (0)R1/2K + R

1/2K U(0)− 2I.

Indeed condition (23) implies condition (16c) since the followinginequality holds(R1/2Q (0)− I

)T (R1/2Q (0)− I

)+

(R1/2K U(0)− I

)T (R1/2K U(0)− I

)≥ 0.

In this way, also (16c) can be transformed into LMIs.Eventually, in order to compute M(·), the matrix-valued

function U(·) has to be invertible. To this end we can enforce thecondition U(k) > 0 for all k ∈ {0, 1, . . . ,N}. Indeed the followingresult is proven in Appendix A.

Lemma 11. There exists a positive definite matrix-valued functionU(k) defined for k ∈ {0, 1, . . . ,N}, satisfying (16b) and (23) if andonly if there exists a nonsingularmatrix-valued function U(k), definedfor k ∈ {0, 1, . . . ,N}, satisfying the same inequalities.

Therefore we end up with a set of LMIs. More specifically,once (ε, blockdiag(R, RK ),N) have been fixed, we need to checkthe feasibility of a set of (3N + 2) LMIs expressed in terms of(7N + 3) optimization variables: the 4N matrices A(k), B(k), C(k),D(k) and the 3(N + 1) matrices Q (0), . . . ,Q (N), S(0), . . . , S(N),U(0), . . . ,U(N).

Example 12. Let us consider the mechanical system shown inFig. 1, consisting of twomasses linked by a spring. Themasses slideon a plane without any friction. Letting x =

(s1 s1 s2 s2

)T ,u = F , y1 = s1, y2 = s2 and supposing M1 = M2 = 1 kg,

F. Amato et al. / Automatica 46 (2010) 919–924 923

State norm for four different initial conditions

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1 2 3 4 5 6 7 8 9

k

0 10

Fig. 2. Closed-loop state norm for four different initial conditions.

k = 1 N/m, the equations of the system are

x = Acx+ Bcu, (24a)y = Cx, (24b)

where

Ac =

0 1 0 0−1 0 1 00 0 0 11 0 −1 0

, Bc =

0100

,C =

(1 0 0 00 0 1 0

).

The discrete time version of system (24) obtained using the ZOHmethod is given by

x(k+ 1) = Adx(k)+ Bdu(k), (25a)y(k) = Cx(k), (25b)

where

Ad = exp(AcT ), Bd =∫ T

0exp(Acσ) dσBc .

The sampling time was chosen T = 0.2 s, which is about 1/20-thof the period of system’s oscillating modes. Our goal is to solveProblem 4 with ε = 1.3, N = 10, R = RK = I . The system underinvestigation is not asymptotically stable because of the absence ofany friction.Using the robust control toolbox (Packard et al., 1984), it is

possible to prove that this problem is feasible and an outputfeedback controller can be designed. In Fig. 2 four sampleevolutions from different initial conditions are presented, showingthat the designed controller guarantees the FTS of the closed loopsystem. These initial conditions have been obtained from a randomsampling of 500,000 samples.

6. Conclusions

In this paperwe have dealt with the finite-time control of lineartime-varying systems. Starting from some conditions guaranteeingfinite-time stability, we have provided sufficient conditions forthe solution of the output feedback problem. The proposed designconditions are expressed in terms of linear difference matrixinequalities.

Appendix. Proof of Lemma 11

Proof. Necessity is obvious.

Now assume that there exists a nonsingular U(k), k ∈ {0,1, . . . ,N} that satisfies (16b) and (23). This implies thatQ (k) Ψ12(k) Ψ13(k) 0Ψ T12(k) Ψ22(k) 0 0Ψ T13(k) 0 I 00 0 0 I

≥ 0,(Q (0) II S(0)

)< ε2

(Q (0)R1/2 + R1/2Q (0)− 2I Q (0)R

RQ (0) R

).

By continuity arguments, it is simple to recognize that thereexists a positive definite matrix U (set for example U = γ I , with γsufficiently small) such thatQ (k) Ψ12(k) Ψ13(k) UR1/2KΨ T12(k) Ψ22(k) 0 0

Ψ T13(k) 0 I 0R1/2K U 0 0 I

≥ 0,(Q (0) II S(0)

)< ε2

(Q (0)R1/2 + R1/2Q (0)+ UR1/2K + R

1/2K U − 2I Q (0)R

RQ (0) R

).

This completes the proof. �

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Francesco Amato was born in Naples on February 2,1965. He received the Laurea and the Ph.D. degree both inElectronic Engineering from University of Naples in 1990and 1994 respectively. From 2001 to 2003 he has beenFull Professor of Automatic Control at the University ofReggio Calabria. In 2003 he moved to the University ofCatanzaro, where he is currently the Dean of the Schoolof Computer and Biomedical Engineering, the Coordinatorof the Doctorate School in Biomedical and ComputerEngineering and the Director of the BiomechatronicsLaboratory. The scientific activity of Francesco Amato

has developed in the fields of systems and control theory with applications tothe aerospace and biomedical fields. He has published about 170 papers anda monography with Springer Verlag entitled Robust Control of Linear Systemssubject to Uncertain Time-Varying Parameters. More specifically, his main research

924 F. Amato et al. / Automatica 46 (2010) 919–924

interests concern analysis and control of uncertain systems, finite-time stabilityof linear systems, and, more recently, stability analysis of nonlinear quadraticsystems.

Marco Ariola was born in Naples, Italy, in 1971. Hereceived the Laurea degree in electronic engineering andthe Research Doctorate degree in electronic engineeringand computer science from the University of NaplesFederico II, Naples, Italy, in 1995 and 2000, respectively.From 1996 to 2005 he was with the Department ofComputer and Systems Engineering of the University ofNaples Federico II. Currently, he is anAssociate Professor ofAutomatic Control at the University of Naples Parthenopein the Technology Department. From September 1998to February 1999, he was a Visiting Scholar with the

Department of Electrical & Computer Engineering of University of New Mexico,Albuquerque, NM, USA. His research interests include statistical control, robustcontrol, control of nuclear fusion devices, control of aerospace systems. He has

published more than 130 journal papers, conference papers, articles in books andencyclopedias. He is co-author of the book Magnetic Control of Tokamak Plasmaspublished by Springer-Verlag in 2008.

Carlo Cosentino was born in Naples on November 18th,1978. He received the Laurea degree (M.Sc.) in ComputerEngineering and the Ph.D. degree in Computer andAutomation Engineering, both from Federico II Universityof Naples, in 2001 and 2005, respectively. From 2006he is Assistant Professor of Systems and Control Theoryat Magna Graecia University of Catanzaro and memberof the board of the Doctorate School in Biomedicaland Computer Engineering. He has published morethan 50 articles in international peer-reviewed journals,conference proceedings and edited books. His research

interests include finite-time stability of linear systems, stability of nonlinearquadratic systems and applications of systems and control theory to the field ofsystems biology.