Contributions to the Control of Hybrid and Switched Linear Systems

Abstract

Hybrid linear systems are a class of hybrid systems where the continuous time evolution is governed by a set of first order linear ordinary differential equations and the jump dynamics are described by a set of first order linear difference equations. Switched linear systems are a subclass of hybrid linear systems with a continuous evolution of the system states. Due to the large number of physical applications, control of hybrid and switched linear systems has received considerable attention over the past years. This dissertation provides novel contributions to the control of such systems and extends some existing results in this area. This work focuses on different problems regarding stability and stabilization of switched linear systems and optimal control of hybrid linear systems. The major part of this thesis concerns stability and stabilization of switched linear systems. The results are primarily presented in terms of conditions for stability of autonomous switched systems. Then, stabilization methods aim at designing a local state feedback for each mode of a controlled switched system to satisfy the stability criteria for the closed loop system modes. Parts of these results rely on the concept of common left eigenvectors and left eigenstructure assignment. To this end, several techniques for eigenstructure assignment in the context of linear systems are developed. Afterwards, these techniques are employed for characterizing exponential stability and for stabilization of a class of switched linear systems with state dependent switching and certain restrictions on the switching manifolds. In addition, they are used for quadratic stabilization of a class of controlled switched linear systems with arbitrary switching signals where the open loop constituent matrices share an invariant subspace to which a common quadratic Lyapunov function can be associated. Another stability approach makes use of the Kalman-Yakubovic-Popov lemma to demonstrate quadratic stability of a class of switched linear systems. This class is characterized by arbitrary switching between two modes, where the difference of the constitute matrices is of rank m ≥ 1, and to which a symmetric transfer function matrix can be associated. These results extend existing results in quadratic stability of rank-1 difference switched linear systems. The thesis also addresses problems in linear quadratic control of hybrid linear systems. In these problems, cost functions are quadratic, the final time and the number and sequence of switches are given. Constraints are specified by linear (in-)equalities in the state space. Switching between different dynamics may either occur at fixed or free time instances. A parameterization method with respect to initial and end points of each interval in a generalized time domain is employed. Numerical solutions for such problems are suggested.