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Hybrid Systems - A Control Engineering Perspective


M. Morari

Piecewise Smooth Dynamical Systems: Analysis, Numerics and Applications, University of Bristol, Bristol, UK

Hybrid systems - loosely defined as systems comprised of continuous and discrete/switched components - are prevalent in all domains of engineering. Over the last few years this system class has attracted much attention and various tools have emerged for studying and affecting its behavior. In this presentation I will describe a recently developed approach for modeling, analysis and controller synthesis that is built on mixed integer mathematical programming. I will illustrate the merits of the technique on a wide range of examples from the automotive, the electrical power and the biomedical domains. I will start by describing a new framework for modeling, analyzing and controlling systems whose behavior is governed by interdependent physical laws, logic rules, and operating constraints, denoted as Mixed Logical Dynamical (MLD) systems. They are described by linear dynamic equations subject to linear inequalities involving real and integer variables. MLD models are equivalent to various other system descriptions like Piece-Wise Affine (PWA) systems and Linear Complementarity (LC) systems. They have the advantage, however, that all problems of system analysis (like controllability, observability, stability and verification) and all problems of synthesis (like controller design and filter design) can be readily expressed as mixed integer linear or quadratic programs, for which many commercial software packages exist. In the second part of the talk I recall some concepts of mathematical programming and show their connections with optimal control. In particular, I point out that finite time optimal control problems with constraints can be expressed as mathematical programs that depend on the initial state as a parameter, so called multi-parametric programs. "Solving" a multi-parametric program is synonymous with finding the solution of the mathematical program as an explicit function of the parameter. In the control context, solving the multi-parametric program is synonymous with finding the optimal state feedback controller. I will review the various algorithms that have emerged for the solution of multi-parametric (mixed integer) linear and quadratic programs and describe the broad range of controller synthesis problems that can be addressed with these new tools. In the final part of the presentation I will discuss in detail some practical applications that have been tackled with these new tools: Traction control for automobiles (Ford), optimal control of co-generation power plants, the control of voltage collapse in power grids, Direct Torque Control of electrical machines (all with ABB), electronic throttle control (with Ford and U. of Zagreb), and Driver Assistance Systems (with Daimler-Chrysler).


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