A unified framework for hybrid control: model and optimal control theory

• Published in: IEEE transactions on automatic control Vol. 43; no. 1; pp. 31 - 45
• Main Authors: Branicky, M.S., Borkar, V.S., Mitter, S.K.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.1998; Institute of Electrical and Electronics Engineers
• Subjects: Aerospace control; Applied sciences; Automata; Automata. Logic controller; Automatic control; Computer science; control theory; systems; Control systems; Control theory. Systems; Decision making; Differential equations; Electric variables control; Exact sciences and technology; Intelligent control; Optimal control; State-space methods; Value function; Automaton; Differential equation; Optimal control; Control system; State space method; Dynamic programming; Hierarchical system; Hysteresis; Hybrid system
• ISSN: 0018-9286
• DOI: 10.1109/9.654885

We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and near-optimal (precise) controls and derive "generalized quasi-variational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.