Application of Finite-Time Stability Concepts to the Control of ATM Networks

When dealing with the stability of a system, a distinction should be made between classical Lyapunov Stability and Finite-Time Stability (FTS) (or ShortTime Stability). The concept of Lyapunov Asymptotic Stability is largely known to the control community; on the other hand a system is said to be finite-time stable if, once we fix a time-interval, its state does not exceeds some bounds during this time-interval. Often asymptotic stability is enough for practical applications, but there are some cases where large values of the state are not acceptable, for instance in the presence of saturations. In these cases, we need to check that these unacceptable values are not attained by the state; for these purposes FTS could be used. Some early results on FTS can be found in [9], [12] and [8]; more recently the concept of FTS has been revisited in the light of recent results coming from Linear Matrix Inequalities (LMIs) theory, which has allowed to find less conservative conditions guaranteeing FTS and finite time stabilization of uncertain, linear continuous-time systems (see [3]). In this note we consider the problem of applying some sufficient conditions for finite time stabilization to design the control algorithm of an ATM network described via a discrete-time system. The extended abstract is organized as follows: in Section 2 we provide a sufficient condition for finite time stabilization of a discrete time system; in ∗F. Amato is with the Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Università degli Studi di Reggio Calabria, Reggio Calabria, Italy. †This author is responsible for correspondence. M. Ariola is with the Dipartimento di Informatica e Sistemistica, Università degli Studi di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy. E-mail: [email protected], tel.:+39-0817683862, fax: +39-0817683816. ‡C.T. Abdallah is with the Department of EECE, University of New Mexico, Albuquerque, NM 87131, USA. §C. Cosentino is with the Dipartimento di Informatica e Sistemistica, Università degli Studi di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy.