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# Analysis and Control of Hybrid Systems

 Author(s):F. Borrelli, M. Morari Conference/Journal:International Conference on Motion and Vibration Control, Saitama, Japan Abstract: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 Tutorial we will describe a recently developed approach for the modeling, analysis and controller synthesis that is built on mixed integer mathematical programming. We will illustrate the merits of the technique on a wide range of examples including traction control (with Ford U.S.) and anesthesia control (with the University Hospital in Berne, Switzerland). We 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 PieceWise 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 Tutorial we recall some concepts of mathematical programming and show its connections with optimal control. In particular, we 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. We review the various algorithms which have emerged for the solution of multi-parametric (mixed integer) linear and quadratic programs and we describe the broad range of controller synthesis problems that can be addressed with these new tools. In the final part of the Tutorial we will discuss in detail some practical applications that have been tackled with these new tools. We will look at the traction control problem where the underlying hybrid model is piece-wise affine and various constraints must be obeyed. The synthesized controller is also piece-wise affine and can be implemented conveniently as a look-up table. The controller was tested successfully on a Ford Focus. The second example described in detail will be the control of pain relief (analgesia) during anesthesia. The modeling and controller design will be discussed and the test results obtained during operations on different patients will be shown. Finally, we will briefly review applications in other areas like optimal control of co-generation power plants and the control of voltage collapse in power grids (with ABB). Year:2002 Type of Publication: (10)Course Supervisor: No Files for download available. `% No recipe for automatically generating a BibTex entry for (10)Course` Permanent link

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