具有可列Markov跳变结构的随机乘积噪声系统的能观性、能检性及其在H2/H∞控制中的应用

Markov jump systems can be well used to capture the switching property of the state and parameters of physical plants caused by randomly abrupt phenomena. Therefore, the relevant theory and technique have long been hot topics in the field of control studies. While fruitful results have been contributed under the assumption that the underlying Markov chain takes values in a finite set, there is little progress concerning the countably infinite jump case in the available literature. Especially on the structural characteristics, the study of observability for countably infinite Markov jump systems remains untouched. Moreover, the existing notions of detectability are all introduced from the mathematical viewpoint due to the requirement of theoretical derivation, which can not reflect the dynamic behavior of real plants. In fact, the model of countably infinite jump systems is more general and its property is essentially different from that of the finite jump case. The first intention of this project is to properly define the concepts of exact observability and exact detectability for stochastic systems with countably jumping structures, according to the relationship between the measurement output and system state. Then, by use of the operator spectrum in an infinite dimensional Banach space, corresponding spectral criteria will be established for checking these two conditions, respectively. Further, we will address the stabilizing solution to a generalized algebraic Riccati equation under the condition of exact observability; When the prerequisite of exact detectability is fulfilled, we will prove that the considered system is stable if and only if a generalized Lyapunov equation admits a unique positive semidefinite solution. The two previous conclusions will be both applied in the design of infinite horizon H2/H∞ controller. Finally, the robust H2/H∞ control approach will be explored in the case that part information of the transition rate of Markov chain is unknown.

Markov跳变系统能够很好地描述由随机突发现象引起的物理系统状态和参数的跳变特性，其理论与技术一直是控制领域的研究热点。然而现有成果普遍是基于Markov链的状态空间是有限集的假设得到的，鲜见可列无穷集情形，尤其在结构特性方面，尚未涉及能观性的研究，已提出的能检性概念均是出于理论推导的需要从数学角度引进的，无法体现系统本身的动力学性能。事实上，可列跳变系统的模型更具一般性，其性质与有限跳变情形存在本质差异。本课题拟首先从量测输出和状态二者之间的联系角度定义这类系统的精确能观（检）性，借助无穷维Banach空间中的算子谱理论建立相应的谱判据。进而探讨精确能观假设下广义代数Riccati方程的镇定解的性质；在精确能检条件下，证明系统稳定等价于广义Lyapunov方程的半正定解存在且惟一，并将上述结论应用于无穷时域H2/H∞控制器的设计中。最后研究转移概率信息部分未知时H2/H∞控制器的设计。

Markov 跳变系统能够很好地描述由随机突发现象引起的物理系统状态和参数的跳变特性，其理论与技术一直是控制领域的研究热点。项目组首先研究了这类系统的稳定性、能检性和鲁棒控制。借助无穷维Banach空间的算子谱理论证明了系统指数均方稳定的谱判据，以及Lyapunov稳定性定理，即系统指数均方稳定等价于一族耦合广义Lyapunov方程正定解的存在性，若系统满足强能检，则上述条件降低为方程族半正定解的存在性，从而纠正了关于含可列Markov跳的随机乘积噪声系统的稳定性的一些错误知识，厘清了渐近均方稳定、随机稳定和指数均方稳定三者之间的关系。项目组还分别分析了具有可列Markov跳变结构的随机乘积噪声系统的有限/无穷时域鲁棒性能，证明了界实引理，采用半定规划法得到了混合H2/H∞反馈控制律，并通过数值例子说明了所提出方法的有效性。.通过引入关键分析工具——monodromy算子，项目组首次定义了含非齐次Markov链的随机周期时变Markov系统的谱并详细讨论了它的性质，进一步还通过谱的具体矩阵表示形式给出了系统能观性、能检性的判据。在应用部分，项目组讨论了周期系数Riccati方程反馈镇定解存在的充分条件，在系统满足能检的条件下，建立了系统的稳定性与广义Lyapunov方程的半正定解的存在惟一性之间的联系。.针对状态、控制和扰动均依赖噪声的带Markov跳的随机乘积噪声系统，项目组给出了无穷时域鲁棒H2/H∞控制器的设计方法，得到了反馈增益的形式，并提出了Riccati方程的倒向迭代算法。