Stability and stabilization of nonlinear discrete-time stochastic systems
Xiushan Jiang
School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Search for more papers by this authorCorresponding Author
Senping Tian
School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Senping Tian, School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China.
Email: [email protected]
Search for more papers by this authorTianliang Zhang
School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Search for more papers by this authorWeihai Zhang
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, China
Search for more papers by this authorXiushan Jiang
School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Search for more papers by this authorCorresponding Author
Senping Tian
School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Senping Tian, School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China.
Email: [email protected]
Search for more papers by this authorTianliang Zhang
School of Automation Science and Engineering, South China University of Technology, Guangzhou, China
Search for more papers by this authorWeihai Zhang
College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, China
Search for more papers by this authorSummary
This paper mainly studies the locally/globally asymptotic stability and stabilization in probability for nonlinear discrete-time stochastic systems. Firstly, for more general stochastic difference systems, two very useful results on locally and globally asymptotic stability in probability are obtained, which can be viewed as the discrete versions of continuous-time Itô systems. Then, for a class of quasi-linear discrete-time stochastic control systems, both state- and output-feedback asymptotic stabilization are studied, for which, sufficient conditions are presented in terms of linear matrix inequalities. Two simulation examples are given to illustrate the effectiveness of our main results.
REFERENCES
- 1Mao X. Stochastic Differential Equations and Their Applications. 2nd ed. Cambridge, UK: Horwood; 2007.
- 2Kushner HJ. Stochastic stability. In: Stability of Stochastic Dynamical Systems. Berlin, Germany: Springer; 1972: 97-124. Lecture Notes in Mathematics; vol. 294.
10.1007/BFb0064937 Google Scholar
- 3Has'minskii RZ. Stochastic Stability of Differential Equations. Alphen aan den Rijn, The Netherlands: Sijthoff and Noordhoff; 1980.
10.1007/978-94-009-9121-7 Google Scholar
- 4Elaydi S. An Introduction to Difference Equation. New York, NY: Springer; 2005.
- 5Zhang W, Xie L, Chen BS. Stochastic H2/H∞ Control: A Nash Game Approach. Boca Raton, FL: CRC Press; 2017.
10.1201/9781315117706 Google Scholar
- 6Mao X. Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions. Appl Math Comput. 2011; 217(12): 5512-5524.
- 7Zhang W, Zheng WX, Chen B-S. Detectability, observability and Lyapunov-type theorems of linear discrete time-varying stochastic systems with multiplicative noise. Int J Control. 2017; 90(11): 2490-2507.
- 8El Bouhtouri A, Hinrichsen D, Pritchard AJ. H∞-type control for discrete-time stochastic systems. Int J Robust Nonlinear Control. 1999; 13: 923-948.
10.1002/(SICI)1099-1239(199911)9:13<923::AID-RNC444>3.0.CO;2-2 Google Scholar
- 9Berman N, Shaked U. H∞ control for discrete-time nonlinear stochastic systems. IEEE Trans Autom Control. 2006; 51(6): 1041-1046.
- 10Xiao N, Xie L, Qiu L. Feedback stabilization of discrete-time networked systems over fading channels. IEEE Trans Autom Control. 2012; 57(9): 2176-2189.
- 11Lin X, Zhang W. A maximum principle for optimal control of discrete-time stochastic systems with multiplicative noise. IEEE Trans Autom Control. 2015; 60(4): 1121-1126.
- 12Peng S. General stochastic maximum principle for optimal control problems. SIAM J Control Optim. 1990; 28(4): 966-979.
- 13Zhang W, Lin X, Chen B-S. LaSalle-type theorem and its applications to infinite horizon optimal control of discrete-time nonlinear stochastic systems. IEEE Trans Autom Control. 2016; 62(1): 250-261.
- 14LaSalle JP. Stability theory of ordinary differential equations. J Differ Equ. 1968; 4: 57-65.
- 15Mao X. Stochastic versions of the LaSalle theorem. J Differ Equ. 1999; 153(1): 175-195.
- 16Elvira-Ceja S, Sanchez EN. Inverse optimal control for discrete-time stochastic nonlinear systems stabilization. Paper presented at: 2013 American Control Conference; 2013; Washington, DC.
- 17Kek SL, Teo KL, Ismail AAM. An integrated optimal control algorithm for discrete-time nonlinear stochastic system. Int J Control. 2010; 83(12): 2536-2545.
- 18Hernandez-Gonzalez M, Basin MV. Discrete-time optimal control for stochastic nonlinear polynomial systems. Int J Gen Syst. 2014; 43: 359-371.
- 19Kubrusly CS, Costa OLV. Mean square stability conditions for discrete stochastic bilinear systems. IEEE Trans Autom Control. 2003; 30(11): 1082-1087.
- 20Li Y, Zhang W, Liu X. Stability of nonlinear stochastic discrete-time systems. J Appl Math. 2013: 1-8.
- 21Saif M, Liu B, Fan H. Stabilisation and control of a class of discrete-time nonlinear stochastic output-dependent system with random missing measurements. Int J Control. 2017; 90(8): 1678-1687.
- 22Liu D, Wang L, Pan Y, Ma H. Mean square exponential stability for discrete-time stochastic fuzzy neural networks with mixed time-varying delay. Neurocomputing. 2016; 171: 1622-1628.
- 23Zhao P, Zhao Y, Guo R. Input-to-state stability for discrete-time stochastic nonlinear systems. Paper presented at: 2015 34th Chinese Control Conference (CCC); 2015; Hangzhou, China.
- 24Yin J. Asymptotic stability in probability and stabilization for a class of discrete-time stochastic systems. Int J Robust Nonlinear Control. 2015; 25(15): 2803-2815.
- 25Gao Z, Ahmed NU. Feedback stabilizability of non-linear stochastic systems with state-dependent noise. Int J Control. 1987; 45(2): 729-737.
- 26Zhang T, Deng F, Zhang W. Study on stability in probability of general discrete-time stochastic systems. Sci China Inf Sci. 2019. https://doi.rg/10.1007/s11432-018-9570-8
- 27Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM; 1994.
10.1137/1.9781611970777 Google Scholar
- 28Kallenberg O. Foundations of Modern Probability. New York, NY: Springer-Verlag; 2002.
10.1007/978-1-4757-4015-8 Google Scholar
- 29Revus D, Yor M. Continuous Martingales and Brownian Motion. New York, NY: Springer-Verlag; 1999.
10.1007/978-3-662-06400-9 Google Scholar
