一类离散时间无限状态马尔可夫跳跃系统H∞控制
H∞ Control for a Class of Discrete-Time Infinite State Markov Jump Systems
摘要: 研究了一类具有同时受乘性噪声和无限马尔可夫跳参数影响的离散时间随机系统的控制问题。首先，给出了一个关于黎卡提方程解的线性不等式，通过求解线性不等式，构造了一个控制器，其次，利用算子理论和随机分析等知识给出离散时间随机系统的无限时域的有界实引理，并且通过一个耦合的黎卡提方程，证明了线性不等式的解和有界实引理之间的等价性。最后关于随机系统的一个线性反馈控制方案以黎卡提方程稳定解的线性矩阵不等式形式被提出，保证了随机控制系统的内部均方稳定性。
Abstract: The control problem of a class of discrete-time stochastic systems affected by multiplicative noise and infinite Markov jump parameters is studied. Firstly, a linear inequality about the solution of Riccati equation is given, and a controller is constructed by solving the linear inequality. Secondly, the bounded real lemma in infinite time domain of discrete-time stochastic systems is given by using the knowledge of operator theory and stochastic analysis. Through a coupled Riccati equation, the equivalence between the solution of linear inequality and bounded real lemma is proved. Finally, a linear feedback control scheme for stochastic systems is proposed in the form of linear matrix inequality of the stable solution of Riccati equation, which ensures the internal mean square stability of stochastic control systems.
文章引用：何鑫, 严芳, 赵红霞, 贾亚琪, 张春梅. 一类离散时间无限状态马尔可夫跳跃系统H∞控制[J]. 动力系统与控制, 2023, 12(3): 139-148. https://doi.org/10.12677/DSC.2023.123015

参考文献

[1] Lam, J., Shu, Z., Xu, S.Y. and Boukas, E.-K. (2007) Robust H∞ Control of Descriptor Discrete-Time Markovian Jump Systems. International Journal of Control, 80, 374-385.
[Google Scholar] [CrossRef] 
[2] Kang, Y., Zhang, J.F. and Ge, S.S. (2008) Robust Output Feedback H∞ Control of Uncertain Markovian Jump Systems with Mode-Dependent Time-Delays. International Journal of Control, 81, 43-61.
[Google Scholar] [CrossRef] 
[3] Aberkane, S. and Dragan, V. (2012) H∞ Filtering of Periodic Markovian Jump Systems: Application to Filtering with Communication Constraints. Automatica, 48, 3151-3156.
[Google Scholar] [CrossRef] 
[4] Costa, O.L.V., Assumpção Filho, E.O., Boukas, E.K. and Marques, R.P. (1999) Constrained Quadratic Control of Markovian Jump Linear Systems. Automatica, 35, 617-626.
[Google Scholar] [CrossRef] 
[5] Boukas, E.K. (2010) H∞ Control of Discrete-Time Markov Jump Systems with Bounded Transition Probabilities. Optimal Control Applications & Methods, 30, 477-494.
[Google Scholar] [CrossRef] 
[6] 田恩刚, 岳东, 杨继全. 具有随机非线性和部分转移概率未知的马尔科夫系统的H∞控制[J]. 控制理论与应用, 2014, 31(3): 392-396.
[7] Wang, Z., Yang, F., Ho, D. and Liu, X. (2007) Robust Variance-Constrained H∞ Control for Stochastic Systems with Multiplicative Noises. Journal of Mathematical Analysis & Applications, 328, 487-502.
[Google Scholar] [CrossRef] 
[8] Kong, S.L. and Zhang, Z.S. (2012) Optimal Control of Stochastic System with Markovian Jumping and Multiplicative Noises. Acta Automatica Sinica, 38, 1113-1118.
[Google Scholar] [CrossRef] 
[9] Zhang, L. (2009) H∞ Estimation for Discrete-Time Piecewise Homogeneous Markov Jump Linear Systems. Automatica, 45, 2570-2576.
[Google Scholar] [CrossRef] 
[10] Graciani, C.C., Todorov, M.G. and Fragoso, M.D. (2021) Fast Switching Detector-Based H2 Control of Markov Jump Linear Systems with Multiplicative Noises. SIAM Journal on Control and Optimization, 59, 4243-4267.
[Google Scholar] [CrossRef] 
[11] Zhao, Y., Zhang, T. and Zhang, W. (2020) Asynchronous H∞ Control for Uncertain Singular Stochastic Markov Jump Systems with Multiplicative Noise Based on Hidden Markov Mode. Journal of the Franklin Institute, 357, 5226-5247.
[Google Scholar] [CrossRef] 
[12] Liu, Y. and Hou, T. (2018) Exponential Stability and Robust H∞ Control for Discrete-Time Time-Delay Infinite Markov Jump Systems. Discrete Dynamics in Nature and Society, 2018, Article ID: 3676083.
[13] Dragan, V., Morozan, T., Stoica, M., et al. (2010) Mathematical Methodsin Robust Control of Discrete-Time Linear Stochastic Systems. Springer, New York.
[Google Scholar] [CrossRef] 
[14] Hou, T., Jie, W. and Ma, H. (2016) Bounded Real Lemma for Discrete-Time Stochastic Systems with Infinite Markov Jumps. Proceedings of 2016 35th Chinese Control Conference (CCC), Chengdu, 27-29 July 2016, 1856-1861.
[Google Scholar] [CrossRef] 
[15] Morozan, T. and Dragan, V. (2018) Observability and Detectability of a Class of Discrete-Time Stochastic Linear Systems. IMA Journal of Mathematical Control & Information, 23, 371-394.
[Google Scholar] [CrossRef] 
[16] Todorov, M.G. and Fragoso, M.D. (2009) Output Feedback H∞ Control of Continuous-Time Infinite Markovian Jump Linear Systems via LMI Methods. Proceedings of 49th IEEE Conference on Decision & Control, Atlanta, 15-17 December 2010, 6505-6510.
[Google Scholar] [CrossRef] 
[17] Liu, Y., Hou, T. and Bai, X. (2017) Infinite Horizon H2/H∞ Optimal Control for Discrete-Time Infinite Markov Jump Systems with (x, u, v)-Dependent Noise. Proceedings of 2017 36th Chinese Control Conference (CCC), Dalian, 26-28 July 2017.
[18] Dragan, V. and Morozan, T. (2008) Discrete-Time Linear Equations Defined by Positive Operators on Ordered Hilbert Spaces. Revue Roumaine Des Mathematiques Pures Et Appliquees, 53, 131-166.

为你推荐