**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.

**>>Full text**

**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.

**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.

**REFERENCES**

- AMATO, F., M. ARIOLA, C. COSENTINO,
**Finite-time control of discrete-time linear systems: Analysis and design conditions**, Automatica, 46(5), 2010, pp. 919 – 924. - AMATO, F., M. ARIOLA,
**Finite-time control of discrete-time linear systems**, IEEE Transactions on Automatic Control, 50(5), 2005, pp. 724 – 729. - AMRI, I., D. SOUDANI, M. BENREJEB,
**Delay dependent robust exponential stability criterion for perturbed and uncertain neutral systems with time varying delays**, Studies in Informatics and Control, 19(2), 2010, pp. 135–144. - BENREJEB, M., P. BORNE,
**On an algebraic stability criterion for nonlinear processes. Interpretation in the frequency domain**, in: Proceedings of the Measurement and Control International Symposium (MECO’78), Athens, 1978. - BENREJEB, M., P. BORNE, F. LAURENT,
**On an application of the arrow form representation in the processes analysis**(in frensh), RAIRO Automatique, 16(2), 1982, pp. 133–146. - BENREJEB, M., A. SAKLY, K. Ben OTHMAN, P. BORNE,
**Choice of conjunctive operator of TSK fuzzy systems and stability domain study**, Mathematics and Computers in Simulation, 76(5-6), 2008, pp. 410 – 421. - BENREJEB, M.,
**Stability study of two level hierarchical nonlinear systems**, Plenary lecture in: The 12th IFAC Large Scale Systems Symposium: Theory and Applications (IFAC LSS 2010), Lille, 2010. - BHAT, S. P., D. S. BERNSTEIN,
**Continuous finite-time stabilization of the translational and rotational double integrators**, IEEE Transactions on Automatic Control, 43(5), 1998, pp. 678–682. - BHAT, S. P., D. S. BERNSTEIN,
**Finite-time stability of continuous autonomous systems,**SIAM Journal on Control and Optimization, 38(3), 2000, pp. 751–766.

- BORNE, P.,
**Analyse des systèmes asservis échantillonnés à paramètres périodiques**(Doctorat Thesis), Faculté de Sciences de l’Université de Lille, 1970. - BORNE, P., J. C. GENTINA, F. LAURENT,
**Stability study of large scale nonlinear discrete systems by use vector norms**, in: Proceedings of the IFAC Symposium on Large Scale Systems: Theory and Applications (IFAC LSS), Udine, 1976. - BORNE, P., J. P. RICHARD, N. E. RADHY, in: A. J. Fossard and D. Normand-Cyrot (Eds),
**Stability, stabilization, regulation using vector norms**, Chapman and Hall, London, 1996, pp. 45–90. - CASAVOLA, A., E. MOSCA, P. LAMPARIELLO,
**Robust ripple-free deadbeat control design**, International Journal of Control, 72(6), 1999, pp. 564–573. - DORATO, P.,
**Short time stability in linear time-varying systems**, in: Proceedings of the IRE International Convention Record, New York, 1961. - DU, H., C. QIAN, M. T. FRYE, S. LI,
**Global finite-time stabilisation using bounded feedback for a class of non-linear systems**, IET Control Theory & Applications, 6(14), 2012, 2326 – 2336. - EMAMI, A., G. F. FRANKLIN,
**Dead-beat control and tracking of discrete-time systems**, IEEE Transactions on Automatic Control, 27(1), 1982, pp. 176–181. - HADDAD, W. M., A. L’AFFLITTO,
**Finite-time stabilization and optimal feedback control,**IEEE Transactions on Automatic Control, 61(4), 2016, pp. 1069–1074 - HONG, Y.,
**Finite-time stabilization and stabilizability of a class of controllable systems**, Systems & Control Letters, 46(2), 2002, pp. 231–236. - HONG, Y., J. HUANG, Y. XU,
**On an output feedback finite-time stabilization problem**, IEEE Transactions on Automatic Control, 46(2),2001, pp. 305–309. - Hong, Y., Z. P. Jiang,
**Finite-Time stabilization of nonlinear systems with parametric and dynamic uncertainties**, IEEE Transactions on Automatic Control, 51(12), 2006, pp. 1950–1956. - KHALIL, H. K.,
**Nonlinear systems**, third ed., Prentice Hall, New Jersey, 2002. - KOTELYANSKI, D. M.,
**Some properties of matrices with positive elements**, Matematicheskii, 31(73), 1961, pp. 961–979. - LIU, H., X. ZHAO, H. ZHANG,
**New approaches to finite-time stability and stabilization for nonlinear systems**, Neurocomputing, 138(22), 2014, pp. 218-228. - LUR’E A. I., V. N. POSTNIKOV,
**On the theory of stability of control systems**, Applied Mathematics and Mechanics, 8(3), 1944, 246–248. - MOULAY, E., W. PERRUQUETTI,
**Finite time stability and stabilization of a class of continuous systems**, Journal of Mathematical Analysis and Applications, 323(2), 2006, pp. 1430–1443. - NERSESOV, S. G., W. M. HADDAD, Q. HUI,
**Finite-time stabilization of nonlinear dynamical systems via control vector Lyapunov functions**, Journal of the Franklin Institute, 345(7), 2008, pp. 819–837. - POLYAKOV, A., D. EFIMOV, W. PERRUQUETTI,
**Finite-time stabilization using implicit Lyapunov function technique**, in: Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems (IFAC Nolcos 2013), Toulouse, 2013. - POPOV, V. M.,
**Absolute stability of nonlinear systems of automatic control**, Automatikai Telemekhanika, 22(8), 1961, pp. 961–979. - SERAJI, H,
**Deadbeat control of discrete-time systems using output feedback**, International Journal of Control, 21(2), 1975, pp. 213 – 223. - SFAIHI, B., M. BENREJEB,
**On stability analysis of nonlinear discrete singularly perturbed T-S fuzzy models,**International Journal of Dynamics and Control, 1(1), 2013, pp. 20 – 31. - STREJC, V.,
**State space approach to linear computer control**, in: Proceedings of the 7th IFAC/IFIP/IMACS Conference on Digital Computer Applications to Process Control, Vienna, 1985. - SU, Y., C. ZHENG,
**Robust finite-time output feedback control of perturbed double integrator**, Automatica, 60, 2015,pp. 86–91. - TABATABAEIPOUR, S. M., M. BLANKE,
**Compositional finite-time stability analysis of nonlinear systems**, in: Proceedings of the 2014 American Control Conference (ACC 2014), Portland, 2014. - WANG, H., Y. TIAN, C. VASSEUR,
**Non-Affine Nonlinear Systems Adaptive Optimal Trajectory Tracking Controller Design and Application**, Studies in Informatics and Control, 24(1), 2015, pp. 05–12. - WEISS, L., E. F. INFANTE,
**Finite-time stability under perturbing forces and on product spaces**, IEEE Transactions on Automatic Control, 12(1), 1967, pp. 341 – 350. - ZHA, W., J. ZHAI, S. FEI, Y. WANG,
**Finite-time stabilization for a class of stochastic nonlinear systems via output feedback**, ISA Transactions, 53(3), 2014, pp. 709–716. - ZHANG, ZH., ZE. Zhang , H. ZHANG, B. ZHENG, H.R. KARIM,
**Finite-time stability analysis and stabilization for linear discrete-time system with time-varying delay**, Journal of the Franklin Institute, 351(6), 2014, pp. 3457–3476. - ZHU, L., Y. SHEN, C. LI,
**Finite-time control of discrete-time systems with time-varying exogenous disturbance**, Communications in Nonlinear Science and Numerical Simulation, 14(2), 2009, pp. 361–370.