The problem of global asymptotic stability of a class of uncertain discrete-time systems in the presence of saturation nonlinearities and interval-like time-varying delay in the state is considered. The uncertainties associated with the system parameters are assumed to be deterministic and normbounded. The objective of the paper is to propose stability criteria having considerably smaller numerical complexity. Two new delay-dependent stability criteria are derived by estimating the forward difference of the Lyapunov functional using the concept of reciprocal convexity and method of scale inequality, respectively. The presented criteria are compared with a previously reported criterion. A numerical example is provided to illustrate the effectiveness of the presented criteria. 1. Introduction During the implementation of fixed-point state-space discrete-time systems using computer or digital hardware, one encounters finite wordlength nonlinearities such as quantization and overflow. Such nonlinearities may lead to instability in the designed system [1, 2]. Saturation overflow nonlinearity is one of the well-known nonlinear phenomena studied in the real world [3]. The stability analysis of discrete-time systems with state saturation is considered to be an important subject of system theoretic study [1–19]. Physical systems may suffer from parameter uncertainties that arise due to modeling errors, variations in system parameters, or some ignored factors. The existence of parameter uncertainties may result in instability of the designed system [20]. In the modeling of physical systems, time delays are often introduced due to finite capabilities of information processing and data transmission among various parts of the system [17, 20–22]. Such delays are another source of instability in discrete-time systems. The stability criteria for time delay systems are broadly classified into delay-independent and delay-dependent. In general, delay-dependent approach [15, 17, 18, 20, 22–37] leads to less conservative results as compared to delay-independent approach [16, 19, 20, 38]. The delay partitioning approach has been utilized in [30, 31] for the stability analysis of systems with interval-like time-varying delay. The stability analysis of discrete-time systems involving overflow nonlinearities, parameter uncertainties, and state delays is an important problem. Delay-independent stability criteria for a class of discrete-time state-delayed systems with saturation nonlinearities have been presented in [16, 19]. A delay-dependent approach for the stability analysis
References
[1]
T. A. C. M. Claasen, W. F. G. Mecklenbr？uker, and J. B. H. Peek, “Effects of quantization and overflow in recursive digital filters,” Institute of Electrical and Electronics Engineers. Transactions on Audio and Electroacoustics, vol. ASSP-24, no. 6, pp. 517–529, 1976.
[2]
H. J. Butterweck, J. H. F. Ritzerfeld, and M. J. Werter, “Finite wordlength in digital filters—a review,” EUT Report 88-E-205, Eindhoven University of Technology, Eindhoven, The Netherlands, 1988.
[3]
T. Ooba, “Stability of linear discrete dynamics employing state saturation arithmetic,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 48, no. 4, pp. 626–630, 2003.
[4]
P. M. Ebert, J. E. Mazo, and M. G. Taylor, “Overflow oscillations in digital filters,” Bell System Technical Journal, vol. 48, no. 9, pp. 2999–3020, 1969.
[5]
W. L. Mills, C. T. Mullis, and R. A. Roberts, “Digital filter realizations without overflow oscillations,” Institute of Electrical and Electronics Engineers. Transactions on Audio and Electroacoustics, vol. 26, no. 4, pp. 334–338, 1978.
[6]
I. W. Sandberg, “The zero-input response of digital filters using saturation arithmetic,” Institute of Electrical and Electronics Engineers. Transactions on Circuits and Systems, vol. 26, no. 11, pp. 911–915, 1979.
[7]
J. H. F. Ritzerfeld, “A Condition for the overflow stability of second-order digital filters that is satisfied by all scaled state-space structures using saturation,” IEEE Transactions on Circuits and Systems, vol. 36, no. 8, pp. 1049–1057, 1989.
[8]
T. Bose and M.-Q. Chen, “Overflow oscillations in state-space digital filters,” IEEE Transactions on Circuits and Systems, vol. 38, no. 7, pp. 807–810, 1991.
[9]
D. Liu and A. N. Michel, “Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters,” IEEE Transactions on Circuits and Systems I, vol. 39, no. 10, pp. 798–807, 1992.
[10]
H. Kar and V. Singh, “A new criterion for the overflow stability of second-order state-space digital filters using saturation arithmetic,” IEEE Transactions on Circuits and Systems I, vol. 45, no. 3, pp. 311–313, 1998.
[11]
H. Kar and V. Singh, “Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach,” IEEE Transactions on Circuits and Systems II, vol. 51, no. 1, pp. 40–42, 2004.
[12]
V. Singh, “Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic,” IEEE Transactions on Circuits and Systems, vol. 37, no. 6, pp. 814–818, 1990.
[13]
V. Singh, “Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic: an LMI approach,” Digital Signal Processing, vol. 16, no. 1, pp. 45–51, 2006.
[14]
H. Kar, “An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic,” Digital Signal Processing, vol. 17, no. 3, pp. 685–689, 2007.
[15]
S.-F. Chen, “Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1251–1257, 2009.
[16]
V. K. R. Kandanvli and H. Kar, “Robust stability of discrete-time state-delayed systems with saturation nonlinearities: linear matrix inequality approach,” Signal Processing, vol. 89, no. 2, pp. 161–173, 2009.
[17]
V. K. R. Kandanvli and H. Kar, “Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed systems using quantization/overflow nonlinearities,” International Journal of Robust and Nonlinear Control, vol. 21, no. 14, pp. 1611–1622, 2011.
[18]
V. K. R. Kandanvli and H. Kar, “Delay-dependent stability criterion for discrete-time uncertain state-delayed systems employing saturation nonlinearities,” Arabian Journal for Science and Engineering, vol. 38, no. 10, pp. 2911–2920, 2013.
[19]
X. Ji, T. Liu, Y. Sun, and H. Su, “Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities,” International Journal of Systems Science, vol. 42, no. 3, pp. 397–406, 2011.
[20]
M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, vol. 5 of Control Engineering, Marcel Dekker, New York, NY, USA, 2000.
[21]
S. Xu and J. Lam, “A survey of linear matrix inequality techniques in stability analysis of delay systems,” International Journal of Systems Science, vol. 39, no. 12, pp. 1095–1113, 2008.
[22]
S. Xu, J. Lam, and Y. Zou, “Improved conditions for delay-dependent robust stability and stabilization of uncertain discrete time-delay systems,” Asian Journal of Control, vol. 7, no. 3, pp. 344–348, 2005.
[23]
H. Huang and G. Feng, “Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay,” IET Control Theory & Applications, vol. 4, no. 10, pp. 2152–2159, 2010.
[24]
H. Shao and Q.-L. Han, “New stability criteria for linear discrete-time systems with interval-like time-varying delays,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 56, no. 3, pp. 619–625, 2011.
[25]
W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay,” IEE Proceedings Control Theory and Applications, vol. 150, no. 4, pp. 412–416, 2003.
[26]
Q.-L. Han, “Improved stability criteria and controller design for linear neutral systems,” Automatica, vol. 45, no. 8, pp. 1948–1952, 2009.
[27]
Y. Wu, Y. Wu, and Y. Chen, “Mean square exponential stability of uncertain stochastic neural networks with time-varying delay,” Neurocomputing, vol. 72, no. 10–12, pp. 2379–2384, 2009.
[28]
X. Jiang, Q.-L. Han, and X. Yu, “Stability criteria for linear discrete-time systems with interval-like time-varying delay,” in Proceedings of the American Control Conference (ACC '05), pp. 2817–2822, Portland, Ore, USA, June 2005.
[29]
J. Liu and J. Zhang, “Note on stability of discrete-time time-varying delay systems,” IET Control Theory & Applications, vol. 6, no. 2, pp. 335–339, 2012.
[30]
X. Meng, J. Lam, B. Du, and H. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems,” Automatica, vol. 46, no. 3, pp. 610–614, 2010.
[31]
P. Kokil, H. Kar, and V. K. R. Kandanvli, “Stability analysis of linear discrete-time systems with interval delay: a delay-partitioning approach,” ISRN Applied Mathematics, vol. 2011, Article ID 624127, 10 pages, 2011.
[32]
V. K. R. Kandanvli and H. Kar, “A delay-dependent approach to stability of uncertain discrete-time statedelayed systems with generalized over flow nonlinearities,” ISRN Computational Mathematics, vol. 2012, Article ID 171606, 8 pages, 2012.
[33]
Y. He, M. Wu, G.-P. Liu, and J.-H. She, “Output feedback stabilization for a discrete-time system with a time-varying delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 53, no. 10, pp. 2372–2377, 2008.
[34]
B. Zhang, S. Xu, and Y. Zou, “Improved stability criterion and its applications in delayed controller design for discrete-time systems,” Automatica, vol. 44, no. 11, pp. 2963–2967, 2008.
[35]
H. Gao and T. Chen, “New results on stability of discrete-time systems with time-varying state delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 52, no. 2, pp. 328–334, 2007.
[36]
H. Gao, J. Lam, C. Wang, and Y. Wang, “Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay,” IEE Proceedings Control Theory and Applications, vol. 151, no. 6, pp. 691–698, 2004.
[37]
O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “Improved delay-dependent stability criteria for discrete-time systems with time-varying delays,” Circuits, Systems, and Signal Processing, vol. 32, no. 4, pp. 1949–1962, 2013.
[38]
L. Bakule, J. Rodellar, and J. M. Rossell, “Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 51, no. 12, pp. 1943–1950, 2006.
[39]
P. Park, J. W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011.
[40]
L. Xie, M. Fu, and C. E. de Souza, “ -control and quadratic stabilization of systems with parameter uncertainty via output feedback,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 37, no. 8, pp. 1253–1256, 1992.
[41]
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994.
[42]
J. L？fberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in Proceedings of the IEEE International Symposium on Computer Aided Control System Design, pp. 284–289, Taipei, Taiwan, September 2004.
[43]
J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11/12, no. 1–4, pp. 625–653, 1999.
