**Finite-Time Convergence Stability of Lur’e Discrete-Time Systems**

**Boutheina SFAIHI, Sirine FEKIH, Mohamed BENREJEB **

Université de Tunis El Manar, Ecole Nationale d’Ingénieurs de Tunis,

Laboratoire de Recherche en Automatique (LARA),

B.P. 37, 1002 Tunis, Le Belvédère, Tunisia

Boutheina.sfaihi@isetr.rnu.tn; Sirine.fekih@enit.rnu.tn; Mohamed.benrejeb@enit.rnu.tn

**Abstract: **In this paper, a constructive design methodology for Finite-Time Stability (FTS) of a class of nonlinear discrete-time systems is proposed. A dead-beat controller that can achieve *n*–finite-time stability is constructed. Furthermore, stability conditions, based on the use of Borne and Gentina practical stability criterion and matrices in the Benrejeb arrow form, are synthesized. Similarity between transient behaviors of dead-beat controlled linearized and nonlinear third order systems are shown and concluding remarks are formulated.

**Keywords:** Discrete Lur’e Postnikov systems; Finite-time stability; Dead-beat control; Vector norm; Benrejeb arrow form matrix.

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**CITE THIS PAPER AS**:

Boutheina SFAIHI, Sirine FEKIH, Mohamed BENREJEB, **Finite-Time Convergence Stability of Lur’e Discrete-Time Systems**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 25(4), pp. 401-410, 2016. https://doi.org/10.24846/v25i4y201601

**Introduction**

Finite Time Stability (FTS) was introduced in the control literature in the Sixties in [14, 35]. Since that, FTS has become one of the most fundamental and challenging problems of the nonlinear control. Finite-time stability (or short-time stability) is a much stronger requirement than classical asymptotic stability. It requires that every solution trajectory of the studied system reaches the origin in a finite time, called settling time [9]. Many approaches were proposed in the literature to solve FTS of nonlinear continuous systems. In [9], a solid development for finite-time stability theory of non-Lipschitzian systems is provided and sensitivity of finite-time-stable systems to perturbations is investigated. Some necessary and sufficient conditions for FTS of systems with the uniqueness of solutions in forward time are given in [25]. In [33], finite-time stability and finite-time boundedness sufficient conditions for systems with polynomial vector fields are provided and computational method checking developed conditions introduced.

Moreover, finite-time control and stabilization techniques have developed, in the last decades, an increasing attention in nonlinear control systems theory [17]. Considered first, in the literature of time-optimal control, one of the main advantages of the finite-time control strategy, is its ability to force a control system to reach a specified target in finite time. By consequence, various theoretical control techniques were developed for different classes of nonlinear continuous systems. In this context, feedback finite-time stabilization controllers of double integrators systems was considered in [8] and [19], perturbed double integrator systems in [32], chain of integrators systems in [27], higher-order controllable systems in [18], systems with parametric and dynamic uncertainties in [20], large-scale interconnected dynamical systems in [26], stochastic systems in [36] and nonlinear systems which can be represented by affine fuzzy system in [23].

Although the method has potential application to practical discrete-time processes, the study of feedback finite-time stabilizing controllers of discrete-time systems is quite underdeveloped and most of the results in the literature are focused on the linear case. Actually, feedback finite-time stabilizing controllers are synthesized for linear perturbed systems in [2], for uncertain linear perturbed systems in [38], for linear time-varying systems in [1] and, for linear systems with time-varying delay in [37]. In the last few years, other contributions on finite-time stabilization of nonlinear discrete-time systems have been introduced in [23], for systems which can be represented by affine fuzzy system, and in [15], for a class of lower-triangular nonlinear systems.

In this work, the finite-time stabilization problem for the class of discrete-time nonlinear Lur’e Postnikov systems [24] is considered. A procedure showing how to develop a compensator ensuring the system trajectories convergence to zero, in finite sampling time, is introduced. Transient behaviours of the controlled nonlinear and linearized systems are then discussed.

This paper is organized as follows. In Section 2, the class of the discrete-time Lur’e Postnikov–type systems is introduced and the problem formally stated. In Section 3, the existence conditions of a state feedback dead-beat controller guaranteeing *n*–FTS convergence of the *n*–order linearized discrete-time system, is provided. In Section 4, sufficient stability conditions for Lur’e system are developed via the Borne and Gentina stability criterion and the Benrejeb arrow form matrix, and the case of a third order Lur’e system discussed. Concluding remarks are provided in Section 5.

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