Abstract

Based on the Roesser model with saturation nonlinearities (SNL) and time-varying delays (TVDs), this paper studies the global asymptotic stability (GAS) of two-dimensional (2-D) uncertain discrete systems (DSs). The underlying system involves norm-bounded parameter uncertainties. By utilizing the idea of Wirtinger-based inequality (WBI) with reciprocal convex inequality (RCI), a new criterion is derived to ensure the GAS of 2-D systems. Numerical examples demonstrate the advantages of the proposed method. With the MATLAB software and YALMIP 3.0, it is found that the obtained criterion provides less stringent results than an existing criterion.

Keyword

2-D discrete system, Reciprocal convex inequality, Saturation nonlinearity, Time-varying delay, Wirtinger-based.

Cite this article

Chaurasia D, Singh K, Kandanvli VK, Kar H

Refference

[1][1]Kaczorek T. Two-dimensional linear systems. In advances in control 1999 (pp. 283-4). Springer, London.

[2][2]Song J, Niu Y. Co-design of 2-D event generator and sliding mode controller for 2-D Roesser model via genetic algorithm. IEEE Transactions on Cybernetics. 2020; 51(9):4581-90.

[3][3]Fornasini E. A 2-D systems approach to river pollution modelling. Multidimensional Systems and Signal Processing. 1991; 2(3):233-65.

[4][4]Bors D, Walczak S. Application of 2D systems to investigation of a process of gas filtration. Multidimensional Systems and Signal Processing. 2012; 23(1):119-30.

[5][5]Butterweck HJ, Ritzerfeld JH, Werter MJ. Finite wordlength in digital filters: a review. 1988.

[6][6]Claasen TA, Mecklenbrauker W, Peek J. Effects of quantization and overflow in recursive digital filters. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1976; 24(6):517-29.

[7][7]Willson JAN. Some effects of quantization and adder overflow on the forced response of digital filters. Bell System Technical Journal. 1972; 51(4):863-87.

[8][8]Ahn CK. A new condition for the elimination of overflow oscillations in direct form digital filters. International Journal of Electronics. 2012; 99(11):1581-8.

[9][9]Chen SF. Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model. Applied Mathematics and Computation. 2010; 216(9):2613-22.

[10][10]Kar H. A new sufficient condition for the global asymptotic stability of 2-D state-space digital filters with saturation arithmetic. Signal Processing. 2008; 88(1):86-98.

[11][11]Kokil P. An improved criterion for the global asymptotic stability of 2-D discrete state-delayed systems with saturation nonlinearities. Circuits, Systems, and Signal Processing. 2017; 36(6):2209-22.

[12][12]Singh V. Improved LMI-based criterion for global asymptotic stability of 2-D state-space digital filters described by Roesser model using twoʼs complement arithmetic. Digital Signal Processing. 2012; 22(3):471-5.

[13][13]Ahn CK. Two-dimensional digital filters described by roesser model with interference attenuation. Digital Signal Processing. 2013; 23(4):1296-302.

[14][14]Lu X, Li H. Input-to-state stability theorem and strict Lyapunov functional constructions for time-delay systems on time scales. Journal of Control and Decision. 2021; 8(4):394-402.

[15][15]Chen SF, Fong IK. Delay-dependent robust H∞ filtering for uncertain 2-D state-delayed systems. Signal Processing. 2007; 87(11):2659-72.

[16][16]Malik SH, Tufail M, Rehan M, Rashid HU. Overflow oscillations‐free realization of discrete‐time 2D Roesser models under quantization and overflow constraints. Asian Journal of Control. 2022; 24(3):1416-25.

[17][17]Tadepalli SK, Kandanvli V, Kar H. A new delay-dependent stability criterion for uncertain 2-D discrete systems described by Roesser model under the influence of quantization/overflow nonlinearities. Circuits, Systems, and Signal Processing. 2015; 34(8):2537-59.

[18][18]Paszke W, Lam J, Gałkowski K, Xu S, Lin Z. Robust stability and stabilisation of 2D discrete state-delayed systems. Systems & Control Letters. 2004; 51(3-4):277-91.

[19][19]Feng ZY, Xu L, Wu M, He Y. Delay-dependent robust stability and stabilisation of uncertain two-dimensional discrete systems with time-varying delays. IET Control Theory & Applications. 2010; 4(10):1959-71.

[20][20]Huang S, Xiang Z. Delay-dependent stability for discrete 2D switched systems with state delays in the roesser model. Circuits, Systems, and Signal Processing. 2013; 32(6):2821-37.

[21][21]Paszke W, Lam J, Galkowski K, Xu S, Kummert A. Delay-dependent stability condition for uncertain linear 2-D state-delayed systems. In proceedings of the 45th IEEE conference on decision and control 2006 (pp. 2783-8). IEEE.

[22][22]Hua M, Zhang J, Chen J, Fei J. Delay decomposition approach to robust delay-dependent H∞ filtering of uncertain stochastic systems with time-varying delays. Transactions of the Institute of Measurement and Control. 2014; 36(8):1143-52.

[23][23]Park P, Ko JW, Jeong C. Reciprocally convex approach to stability of systems with time-varying delays. Automatica. 2011; 47(1):235-8.

[24][24]Jiang X, Han QL, Yu X. Stability criteria for linear discrete-time systems with interval-like time-varying delay. In proceedings of the 2005, American control conference, 2005. (pp. 2817-22). IEEE.

[25][25]Nam PT, Pathirana PN, Trinh H. Discrete Wirtinger-based inequality and its application. Journal of the Franklin Institute. 2015; 352(5):1893-905.

[26][26]Tadepalli SK, Kandanvli VK, Vishwakarma A. Criteria for stability of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities. Transactions of the Institute of Measurement and Control. 2018; 40(9):2868-80.

[27][27]He Y, Wang QG, Xie L, Lin C. Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control. 2007; 52(2):293-9.

[28][28]Badie K, Alfidi M, Chalh Z. Improved delay-dependent stability criteria for 2-D discrete state delayed systems. In international conference on intelligent systems and computer vision 2018 (pp. 1-6). IEEE.

[29][29]Roesser R. A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control. 1975; 20(1):1-10.

[30][30]Hinamoto T. 2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 1993; 40(2):102-10.

[31][31]Chen SF. Stability analysis for 2-D systems with interval time-varying delays and saturation nonlinearities. Signal Processing. 2010; 90(7):2265-75.

[32][32]Badie K, Alfidi M, Tadeo F, Chalh Z. Robust state feedback for uncertain 2-D discrete switched systems in the roesser model. Journal of Control and Decision. 2021; 8(3):331-42.

[33][33]Barrett P. Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In conference on the theory and application of cryptographic techniques 1986 (pp. 311-23). Springer, Berlin, Heidelberg.

[34][34]Akyildiz IF, Su W, Sankarasubramaniam Y, Cayirci E. Wireless sensor networks: a survey. Computer Networks. 2002; 38(4):393-422.

[35][35]Chen C, Shu M, Yang Y, Gao T, Bian L. Robust H∞ path tracking control of autonomous vehicles with delay and actuator saturation. Journal of Control and Decision. 2022; 9(1):45-57.

[36][36]Naghshtabrizi P, Hespanha JP, Teel AR. Stability of delay impulsive systems with application to networked control systems. Transactions of the Institute of Measurement and Control. 2010; 32(5):511-28.

[37][37]Boyd S, El GL, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. Society for Industrial and Applied Mathematics; 1994.

[38][38]Yang R, Zheng WX. Two-dimensional sliding mode control of discrete-time Fornasini–Marchesini systems. IEEE Transactions on Automatic Control. 2019; 64(9):3943-8.

[39][39]Li Z, Zhang T, Ma C, Li H, Li X. Robust passivity control for 2-D uncertain Markovian jump linear discrete-time systems. IEEE Access. 2017; 5:12176-84.

[40][40]Li F, Chen J, Zhang L, Wang Q, Jiang L. Improved delay-dependent stability analysis of fixed-point state-space digital filters with time-varying delay and generalized overflow arithmetic. IEEE Access. 2022; 10:9406-19.

[41][41]Malik SH, Tufail M, Rehan M, Ahmed S. State and output feedback local control schemes for nonlinear discrete-time 2-D roesser systems under saturation, quantization and slope restricted input. Applied Mathematics and Computation. 2022.

[42][42]Kanellakis A, Tawfik A. A new sufficient criterion for the stability of 2-D discrete systems. IEEE Access. 2021; 9:70392-5.

[43][43]Chesi G. Exact LMI conditions for stability and mathcal gain analysis of 2-D mixed continuous–discrete time systems via quadratically frequency-dependent Lyapunov functions. IEEE Transactions on Automatic Control. 2021; 67(3):1147-62.

[44][44]Chesi G. A novel LMI condition for stability of 2D mixed continuous-discrete-time systems via complex LFR and Lyapunov functions. In 12th Asian control conference (ASCC) 2019 (pp. 162-6). IEEE.

[45][45]Zhu Z, Lu JG, Zhang QH. LMI-based stability analysis of continuous-discrete fractional-order 2D roesser model. IEEE Transactions on Circuits and Systems II: Express Briefs. 2021; 69(6):2797-801.

[46][46]Bachelier O, Cluzeau T, Rigaud A, Silva AFJ, Yeganefar N. On exponential stability of a class of descriptor continuous linear 2D roesser models. International Journal of Control. 2022.

[47][47]Badie K, Alfidi M, Tadeo F, Chalh Z. Delay-dependent stability and H∞ performance of 2-D continuous systems with delays. Circuits, Systems, and Signal Processing. 2018; 37(12):5333-50.

[48][48]Wang M, Feng G, Qiu J. Finite-frequency fuzzy output feedback controller design for roesser-type two-dimensional nonlinear systems. IEEE Transactions on Fuzzy Systems. 2020; 29(4):861-73.

[49][49]Trinh H. Stability analysis and control of two-dimensional fuzzy systems with directional time-varying delays. IEEE Transactions on Fuzzy Systems. 2017; 26(3):1550-64.

[50][50]Chandra PV, Negi R. Based anti-windup controller for two-dimensional discrete delayed systems in presence of actuator saturation. IMA Journal of Mathematical Control and Information. 2018; 35(2):627-60.

[51][51]Qin W, Wang G, Li L, Shen M. Fault detection for 2-D continuous-discrete state-delayed systems in finite frequency domains. IEEE Access. 2020; 8:103141-8.

[52][52]Lofberg J. YALMIP: atoolbox for modeling and optimization in MATLAB. In international conference on robotics and automation 2004 (pp. 284-9). IEEE.

[53][53]Seuret A, Gouaisbaut F, Fridman E. Stability of discrete-time systems with time-varying delays via a novel summation inequality. IEEE Transactions on Automatic Control. 2015; 60(10):2740-5.

[54][54]Ahn CK. Elimination of overflow oscillations in 2-D digital filters described by roesser model with external interference. IEEE Transactions on Circuits and Systems II: Express Briefs. 2013; 60(6):361-5.