Development of modeling and control analysis of jumpless hybrid systems

本课题的研究目的是针对一类混杂系统，即，无跳跃现象的分段仿射系统。结果如下：首先，将分段仿射系统的离散状态看作0-1开关，提出了一种新的离散转移规则概念，称为基于开关的转移规则。这条规则对应于一个理想化的切换现象的时间延迟。接下来，我们给出了这类系统的解和适定性的定义，并给出了适定性的刻画。该方法为更一般的混杂系统的适定性分析奠定了基础。第三，提出了反馈适定性的概念，并给出了双模系统通过状态反馈而成为适定性的充要条件，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，我们推导了所有反馈适定性双模系统的标准形式，以获得系统可镇定的充分条件。最后，我们提出了一种双模态系统和四模态系统的镇定方法，表明通过某种反馈变换，可以将一类分段仿射系统的控制问题归结为通常的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.