Control from computer science

This paper presents some of the principles underlying verification and controller synthesis techniques for discrete dynamical systems developed within Computer Science along with some ideas to extend them to continuous and hybrid systems. Hopefully, this will provide control theorists and engineers with an additional perspective of their discipline as seen by a sympathetic outsider, uncommitted to the customs and traditions of the domain. Inter-cultural experience can be frustrating but sometimes fun. 1. WHAT AM I DOING HERE? Being one of those who have chosen to study computer science partly due to an inability to understand differential equations, I feel a bit uncomfortable to publish a paper in a journal whose pages are probably full of occurrences of that terrifying ∫ symbol. The scientific reason for my presence here is perhaps being one of those few computer scientists interested in the so-called hybrid systems research which was supposed to bring together the Computer Science and Control communities. So let me first speak about what I understand. 2. WHAT IS VERIFICATION? Verification 2 like Control is concerned with a model-based design of systems. That is, we want to build something (a “controller”) that makes some part of the real world (the “environment” 1 This research was supported in part by the European Community projects 26270 VHS (Verification of Hybrid Systems) and IST-2001-33520 CC (Control and Computation). 2 The term “verification” is used as a short approximation for the disciplines and communities interested in “modeling, design and analysis of reactive systems” or “formal methods in system design”. This is not “the” mainstream in Computer Science — I am not sure there is a mainstream in such a diverse domain. or “plant”) behave in a certain desired way. Instead of using trial-and-error methods we build a mathematical model which describes the combined dynamics of the environment and the controller. On this model we can make “gedanken experiments”, e.g. manipulation of formulae or numerical simulations, to convince ourselves that the controller indeed makes the environment behave as required. If the model is a good approximation of the real world, there is a chance that a controller validated on the model will work properly when implemented. 3 The description just given does not specify the type of dynamical models considered. In classical control these are models of continuous dynamical systems in either continuous or discrete time, and since examples of such systems appear in every decent control textbook, I will move directly to discrete systems of the type treated by the verification community and illustrate them via an example. 2.1 The Coffee Machine Suppose we want to build a machine that distributes various hot drinks to customers who pay 3 I mention this trivial fact because mathematicians, discrete and continuous alike, who spend much of their time in the abstract world, sometimes tend to forget it.