Collaborative Research: Piecewise Linear Hybrid Systems

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This proposal deals with the design and analysis of efficient algorithms forthe class of hybrid systems called (discrete-time) piecewise linear (PL)systems. The area of hybrid systems concerns issues of modeling, computation andcontrol for systems which combine discrete and continuous components.The subclass of PL systems, which we investigate in this project,provides one systematic approachto discrete-time hybrid systems, naturally blending switching mechanisms withclassical linear components.PL systems are of interest as controllers as well as identification models,and they can be thought of as arbitrary interconnections of finite automataand linear systems. Tools from automata theory, logic and related areas of computerscience and finite mathematics are used in the study of PL systems, inconjunction with linear algebra techniques, all in the context of a ``PLalgebra'' formalism. In this project, we will study basic algorithmicand computational problems for these problems with applications tomodelling of biological systems.More concretely, basic questions for any class of systems are those of equivalence, and, inparticular, whether state spaces are equivalent under a change of variables.This project will address various algorithmic and computational challengesrelated to this state-space equivalence and isomorphism problem for PLsystems, as well as questions of learnability and representability for suchsystems. The project also suggests the use of PL models for differential geneexpression testing.