Optimal control of hybrid systems : theory and algorithms

Many complex systems are hybrid in the sense that: (i) the state set possesses continuous and discrete components, and (ii) system evolution may occur in both continuous and discrete time. One important class of hybrid systems is that characterized by a feedback configuration of a set of continuous controlled low level systems and a high level discrete controller; such systems appear frequently in engineering and are particularly evident when a system is required to operate in a number of distinct modes. Other classes of hybrid systems are found in such diverse areas as (i) air traffic management systems, (ii) chemical process control, (iii) automotive engine-transmission systems, and (iv) intelligent vehicle-highway systems.In this thesis we first formulate a class of hybrid optimal control problems (HOCPs) for systems with controlled and autonomous location transitions and then present necessary conditions for hybrid system trajectory optimality. These necessary conditions constitute generalizations of the standard Minimum Principle (MP) and are presented for the cases of open bounded control value sets and compact control value sets. These conditions give information about the behaviour of the Hamiltonian and the adjoint process at both autonomous and controlled switching times.Such proofs of the necessary conditions for hybrid systems optimality which can be found in the literature are sufficiently complex that they are difficult to verify and use; in contrast, the formulation of the HOCP given in Chapter 2 of this thesis, together with the use of (i) classical variational methods and more recent needle variation techniques, and (ii) a local controllability condition, called the small time tubular fountain (STTF) condition, make the proofs in that chapter comparatively accessible. We note that the STTF condition is used to establish the adjoint and Hamiltonian jump conditions in the autonomous switchings case.A hybrid Dynamic Programming Principle (HDPP) generalizing the standard dynamic programming principle to hybrid systems is also derived and this leads to hybrid Hamilton-Jacobi-Bellman (HJB) equation which is then used to establish a verification theorem within this framework. (Abstract shortened by UMI.