1 The application of different Lyapunov-like functionals and some aggregate norm approximations of the delayed states for finite-time stability analysis of linear discrete time-delay systems Sreten B. Stojanovic 1,* , Dragutin Lj. Debeljkovic 2 1 University of Nis, Faculty of Technology, Department of Mathematical and Engineering Sciences, Leskovac, Serbia, sstojanovic@tf.ni.ac.rs 2 University of Belgrade, Faculty of Mechanical Engineering, Department of Control Engineering, Belgrade, Serbia, ddebeljkovic@mas.bg.ac.rs Abstract – In this paper, the finite-time stability (FTS) problem for a class of linear discrete time- delay systems is studied. Two classes of Lyapunov-like (LL) functionals are applied and some sufficient conditions are given in the form of the linear matrix inequalities (LMIs). First class of functional represents the classical LL functional for systems with time delay, while the second ones contains some additional power or exponential terms. When using the classical functional, the six aggregate norm approximations (ANA) of the delayed states are considered in order to obtain less conservative results. The ANA is necessary in order to establish a relation between the Lyapunov functional and its difference. The functionals with the power or exponential terms do not require the ANA for derivation of stability criteria. A numerical example is employed to verify the efficiency of the theoretical results. It was shown that the obtained results are less conservative than some existing ones in the literature. Key words: discrete time-delay systems; finite-time stability; Lyapunov-like functional; linear matrix inequalities; aggregate norm approximation of the delayed states 1. Introduction The last few decades, the concept of asymptotic stability is dominant in the field of the theory of the system stability. This concept considers the dynamic behavior of the system in the infinite time interval. However, in many practical applications, there are some cases where it is required to the states of the system do not exceed a certain limits. Moreover, it is required that this condition is fulfilled at a given finite-time interval. For example, in a chemical process, the state variables (such as temperature, humidity, pressure, and so on) are expected to be controlled within certain bounds for fixed time interval. In these cases, finite-time stability (FTS) could be used. A system is said to be finite-time stable if, once we fix a time interval and give a bound on the initial condition, the system state does not exceed a certain domain during this time interval. The FTS concept has appeared in middle of the last century [1-3] where only the class of regular continuous systems is discussed. In most previous results, the stability criteria are quite conservative and practically useless for numerical calculation. Recently, this concept has been extended to the class of regular discrete systems and revisited in the light of LMIs, which has allowed finding less conservative criteria. The consequence of this is the appearance many useful results; see, for instance [4-8] for continuous and [9-14] for discrete-time systems. However, in all these works, the time delay is considered. The stability of time-delay systems has been widely investigated in the last three decades. Practical examples of time-delay systems include chemical engineering, communications and biological systems. It has been shown that the existence of delay may lead to instability, oscillation and poor performance of control systems. Therefore, considerable attention has been devoted to the problem of stability of time delays systems and various categories of asymptotic and exponential stability and stabilization of time-delay systems have been studied: delay-independent, delay-dependent and robust. During the time, the class of time-delay systems becomes interesting to study the finite-time stability * Corresponding author. E-mail addresses: sstojanovic@tf.ni.ac.rs