Almost Sure Stabilization for Feedback Controls of Regime-Switching Linear Systems With a Hidden Markov Chain
Almost Sure Stabilization for Feedback Controls of Regime-Switching Linear Systems With a Hidden Markov Chain
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DOI：
10.1109/tac.2009.2026938
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发表时间：
2009-08
影响因子：
6.8
通讯作者：
B. Bercu;F. Dufour;G. Yin
中科院分区：
文献类型：
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作者：
B. Bercu;F. Dufour;G. Yin
这项工作致力于涉及未知马尔可夫链的自适应控制系统的几乎确定的稳定性。控制系统显示由微分方程表示的连续动态和由隐马尔可夫链给出的离散事件。在之前对此类问题的研究中，使用了平均准则，它仅提供了某种期望意义上的系统行为。对系统行为的更仔细的审查必然需要考虑样本路径属性。与以前关于具有隐马尔可夫链的自适应控制系统的稳定性的研究不同，其中考虑了平均标准，这项工作侧重于基本过程的几乎确定的稳定性或样本路径稳定性。在简单条件下，结果表明，只要反馈控制的连续分量具有线性增长，所得过程就是规则的。此外，通过适当选择Lyapunov函数，表明自适应系统几乎肯定是稳定的。作为副产品，还确定受控过程是正循环的。
This work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown Markov chain. The control system displays continuous dynamics represented by differential equations and discrete events given by a hidden Markov chain. In the previous investigation on this class of problems, averaging criteria were used, which provides only the system behavior in some expectation sense. A closer scrutiny of the system behavior necessarily requires the consideration of sample path properties. Different from previous work on stabilization of adaptive controlled systems with a hidden Markov chain, where average criteria were considered, this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes. Under simple conditions, it is shown that as long as the feedback controls have linear growth in the continuous component, the resulting process is regular. Moreover, by appropriate choice of the Lyapunov functions, it is shown that the adaptive system is stabilizable almost surely. As a by-product, it is also established that the controlled process is positive recurrent.