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Piecewise Linear Control Systems

Johansson, Mikael LU (1999) In PhD Theses TFRT-1052.
Abstract
This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much... (More)
This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much more powerful than the commonly used quadratic Lyapunov functions. It is shown how piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. The computations are based on a compact parameterization of continuous piecewise quadratic functions and conditional analysis using the S-procedure. A unifying framework for computation of a variety of Lyapunov functions via convex optimization is established based on this parameterization. Systems with attractive sliding modes and systems with bounded regions of attraction are also treated. Dissipativity analysis and optimal control problems with piecewise quadratic cost functions are solved via convex optimization. The basic results are extended to fuzzy systems, hybrid systems and smooth nonlinear systems. It is shown how Lyapunov functions with a discontinuous dependence on the discrete state can be computed via convex optimization. An automated procedure for increasing the flexibility of the Lyapunov function candidate is suggested based on linear programming duality. A Matlab toolbox that implements several of the results derived in the thesis is presented. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Professor Boyd, Stephen, Stanford University
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Piecewise quadratic functions, Linear matrix inequalities, Convex optimization, Optimal control, Performance analysis, Piecewise linear Lyapunov functions, Piecewise quadratic Lyapunov functions, Quadratic stability, Lyapunov stability, Piecewise linear systems, Nonlinear systems, Fuzzy systems, Hybrid systems., Automation, robotics, control engineering, Automatiska system, robotteknik, reglerteknik
in
PhD Theses
volume
TFRT-1052
pages
199 pages
publisher
Department of Automatic Control, Lund Institute of Technology (LTH)
defense location
Room E:1406, E-building, Lund Institute of Technology
defense date
1999-03-26 10:15
ISSN
0280-5316
language
English
LU publication?
yes
id
87eefd7f-6d0f-4dbf-8546-949a2d8fc0f7 (old id 19355)
date added to LUP
2007-05-24 14:39:26
date last changed
2016-09-19 08:44:56
@phdthesis{87eefd7f-6d0f-4dbf-8546-949a2d8fc0f7,
  abstract     = {This thesis treats analysis and design of piecewise linear control systems. Piecewise linear systems capture many of the most common nonlinearities in engineering systems, and they can also be used for approximation of other nonlinear systems. Several aspects of linear systems with quadratic constraints are generalized to piecewise linear systems with piecewise quadratic constraints. It is shown how uncertainty models for linear systems can be extended to piecewise linear systems, and how these extensions give insight into the classical trade-offs between fidelity and complexity of a model. Stability of piecewise linear systems is investigated using piecewise quadratic Lyapunov functions. Piecewise quadratic Lyapunov functions are much more powerful than the commonly used quadratic Lyapunov functions. It is shown how piecewise quadratic Lyapunov functions can be computed via convex optimization in terms of linear matrix inequalities. The computations are based on a compact parameterization of continuous piecewise quadratic functions and conditional analysis using the S-procedure. A unifying framework for computation of a variety of Lyapunov functions via convex optimization is established based on this parameterization. Systems with attractive sliding modes and systems with bounded regions of attraction are also treated. Dissipativity analysis and optimal control problems with piecewise quadratic cost functions are solved via convex optimization. The basic results are extended to fuzzy systems, hybrid systems and smooth nonlinear systems. It is shown how Lyapunov functions with a discontinuous dependence on the discrete state can be computed via convex optimization. An automated procedure for increasing the flexibility of the Lyapunov function candidate is suggested based on linear programming duality. A Matlab toolbox that implements several of the results derived in the thesis is presented.},
  author       = {Johansson, Mikael},
  issn         = {0280-5316},
  keyword      = {Piecewise quadratic functions,Linear matrix inequalities,Convex optimization,Optimal control,Performance analysis,Piecewise linear Lyapunov functions,Piecewise quadratic Lyapunov functions,Quadratic stability,Lyapunov stability,Piecewise linear systems,Nonlinear systems,Fuzzy systems,Hybrid systems.,Automation,robotics,control engineering,Automatiska system,robotteknik,reglerteknik},
  language     = {eng},
  pages        = {199},
  publisher    = {Department of Automatic Control, Lund Institute of Technology (LTH)},
  school       = {Lund University},
  series       = {PhD Theses},
  title        = {Piecewise Linear Control Systems},
  volume       = {TFRT-1052},
  year         = {1999},
}