State Estimation and Tracking Control for Hybrid Systems by Gluing the Domains

Abstract

In this dissertation, we propose a new observer and tracking controller design approach for a class of hybrid dynamical systems with state jumps. The hybrid dynamical system exhibits characteristics typical of both continuous-time dynamical system and discrete-time dynamical system. Therefore, it can be modeled as differential equation of the continuous-time dynamics, difference equation of the discrete-time dynamics, the interaction between them. Since the interaction of continuous-time and discrete-time dynamics in a hybrid system leads to rich dynamical behavior and unfamiliar phenomena, several challenges are encountered when we deal with this system.

The observer design considered in this dissertation is to construct a dynamical system called an observer that estimates the state of a given hybrid dynamical system (without any input), from an output of the given system. In addition, the tracking controller design is to construct a dynamical system called a tracking controller that makes an input for a given hybrid dynamical system (with an input) such that the state of the given system tracks a given reference. There many results of the observer and tracking controller designs for the continuous-time and discrete-time dynamical systems, but the results for the hybrid dynamical systems are insufficient. Moreover, the results are applied to some classes of hybrid systems (switched systems, hormone systems, powertrain systems, and so on) rather than general hybrid dynamical systems.

The proposed idea dealing with the hybrid dynamical system is to "glue" the jump set (a part of the domain where the jumps take place) onto its image. Then, on the "glued" domain, the hybrid dynamical system becomes a continuous-time dynamical system without any jumps. Especially, for some class of the system, the continuous-time dynamical system has a smooth vector field via some notion, "smoothing". Furthermore, we specify this concept of gluing as a map and investigate the essential conditions of the map.

By this map, we obtain the "glued" hybrid dynamical system (which is a continuous-time dynamical system) and it may be possible to construct an observer and/or a tracking controller through conventional methods for continuous-time dynamical systems. From these constructions, we obtain the observer and tracking controller for the hybrid system. Especially, the proposed observer does not require any detection of the state jumps while many previous results does. Furthermore, the proposed tracking controller does not need to make the state jump whenever the jumps of the reference happen. Simulation results for examples including mechanical system with impacts and ripple generator in AC/DC converter illustrate the effectiveness of the proposed approach.