Development of modeling and control analysis of jumpless hybrid systems

本研究的目的是针对一类混杂系统，即无跳跃现象的分段仿射系统，发展一系列从建模到控制的系统控制理论。研究结果如下：首先，将分段仿射系统的离散状态看作0-1开关，提出了离散转移规则的新概念--基于开关的转移规则。该规则对应于具有时延的切换现象的理想化。其次，我们给出了解的定义和这类系统的适定性，然后给出了适定系统的一个刻画。这种方法甚至为更一般的混杂系统的适定性分析提供了基础。第三，提出了反馈适定性的概念，给出了双峰系统通过状态反馈达到适定性的充要条件，这是系统可镇定的必要条件。此外，基于这种反馈适定性分析，我们得到了所有反馈适定双峰系统的典范形式，得到了系统可镇定的一个充分条件。最后给出了双模系统和四模系统的镇定方法，证明了通过一定的反馈变换，可以将一类分段仿射系统的控制问题归结为一般线性系统的控制问题。

The purpose of this research project is to develop a series of system control theory from modeling to control for a class of hybrid systems, i.e., piecewise affine systems with no jump phenomena. The result is as follows : first, we have proposed a new concept of discrete transition rule, named switch-based transition rule by regarding the discrete state of the piecewise affine system as 0-1 switches. This rule corresponds to an idealization of switching phenomena with time-delays. Next, we have presented a definition of solutions and well-posedness of such systems, and then a characterization of well-posed systems. This approach gives a basis of the well-posedness analysis even for more general class of hybrid systems. Third, the concept of feedback well-posedness has been proposed and a necessary and sufficient condition for the bimodal system to be rendered well-posed via state feedback, which is a necessary condition for stabilizability. Furthermore, based on this feedback well-posedness analysis, we have derived a canonical form of all feedback well-posed bimodal systems to obtain a sufficient condition for the system to be stabilizable. Finally we have proposed a stabilization method of bimodal systems and 4-modal systems, which shows that the control problem of some class of piecewise affine systems can be reduced to that of the usual linear systems by some feedback transformation.