有限频域内Markov跳变系统的控制问题研究

As a kind of stochastic hybrid systems, Markov jump systems have been applied to many fields, such as electric systems, aerospace system, and economy systems. To satisfy the disturbance attenuation performance for engineering systems in some given frequency interval, this project is concerned with the controller design problem for Markov jump systems for disturbances in finite frequency interval. Moreover, transition probabilities are assumed to to known, uncertain with known bounds and completely unknown. There are two main reasons which lead the considered problem is more complex. One is that the conditions for the system performance in finite frequency are non-convex; the other is coming from the unknown transition probabilities. To overcome the difficulties induced by the above reasons and solve the controller design problem, combining the new finite frequency technique with the robust control method, we will pay our main attention to the non-convex coupling mechanism and convert the considered problem to be solvable and optimizable. Firstly, based on the assumption that the system state is measurable, considering the disturbance with frequency constraint, the state feedback control problem of Markov jump systems with partly unknown transition probabilities is considered. To solve the considered problem by the convex method, a transformation is employed. Based on the obtained results, we further investigate the dynamic output control of the considered system. Lastly, some extensions will be carried out. Namely, the above results are extended to deal with the cases that system with uncertainties and input saturation respectively. This research is significant to promote the Markov jump theory and its application in engineering.

Markov跳变系统是一类复杂的混杂系统，在电力、航空航天诸领域具有广泛应用。为满足实际控制系统对某个频率区间的干扰抑制要求，本项目针对转移概率部分未知的Markov跳变系统研究其在有限频域内的控制器设计问题。由于该系统的有限频性能刻画条件非凸，并与未知转移概率非线性耦合，使得该问题不易求解。为克服非线性给控制器设计及系统性能优化带来的困难，将深入研究它们的耦合机制，结合有限频分析新技术及鲁棒凸优化方法，将其转化为可解的优化问题。首先，基于系统状态可测，考虑外部干扰为有限频率信号，研究转移概率部分未知Markov跳变系统有限频状态控制器设计问题，给出解决该问题的凸优化方法。在此基础上，研究其有限频动态输出反馈控制器设计问题。进一步，将上述结果推广至系统带有参数不确定性以及输入饱和的情形。开展该研究对完善Markov跳变系统理论及其在工程中的应用具有重要意义。

Markov跳变系统是一类复杂的混杂系统，在电力、航空航天诸领域具有广泛应用。为满足实际控制系统对某个频率区间的干扰抑制要求，本项目针对转移概率部分未知的 Markov 跳变系统，考虑外部扰动为有限频信号，在频域内设计控制器保证闭环系统的稳定性，并考虑优化闭环系统的干扰抑制性能。针对转移速率已知、不确定但边界已知和完全未知部分未知，充分利用转移速率性质，分别从确定性和随机的角度给出了未知转移速率估计新策略，解除了未知转移速率和Lyapunov变量之间的耦合非线性；提出新的不等式放缩措施，给出处理不确定转移速率产生的非线性的线性化方法；针对系统状态不可测，从输出控制的结构特征出发，提出新的放缩分离策略来解除控制器增益与Lyapunov矩阵之间的耦合非线性，给出可以优化闭环系统性能的输出反馈控制方案。针对放缩技术带来的保守性，从信号不变性原理出发，依据控制器结构，采用系统增广技术，引入新的自由变量，消除系统矩阵、控制增益与Lyapunov矩阵之间的耦合关系，给出静态输出反馈控制器设计的无损的分离解耦方案。针对有限频性能结构的非线性挑战，基于Parseval定理结合构造分离策略，给出有限频域内控制器设计凸优化条件；针对传感器非线性问题，将其转化为带有扇区界的非线性结构，然后结合S过程和Finsler引理，引入新的矩阵参数，实现设计参数与辅助变量之间的分离，进而给出非特殊结构的滤波器设计的统一框架；最后，将上述研究过程中所获得的方法用于网络控制系统中，给出事件触发下的Markov跳变系统控制器和触发机制共同优化设计方法。开展该研究对完善Markov跳变系统理论及其在工程中的应用具有重要意义。