Model Reduction for Dynamic Systems with Time Delays: A Linear Matrix Inequality Approach

This dissertation, "Model Reduction for Dynamic Systems With Time Delays: a Linear Matrix Inequality Approach" by Qing, Wang, 王卿, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled Model Reduction for Dynamic Systems with Time Delays: A Linear Matrix Inequality Approach submitted by Qing Wang for the degree of Doctor of Philosophy at the University of Hong Kong April 2007 Studies of dynamic systems with time delays have received considerable research atten- tions in recent years because of their wide applicability in modeling a variety of physical systems. Thisstudyfocusesonderivingloworderdynamicsystemsfromhigherorderones accordingtocertainapproximationcriteria, withaparticularemphasisonthosewithtime delays. The proposed reduced order models guarantee the same structural characteristics astheoriginalones. Theapproximationcriterion is chosento bethe energy-to-peak gain, which is a new performance index for dynamic systems with time delays, or H norm of theerrorsystemsbetweentheoriginalsystemsandthereducedones. Themodelreduction problems are then formulated as sequential linear matrix inequality (LMI) minimization problems. Four di(R)erent kinds of dynamic systems are considered in this research: (a) retarded systems with time-varying delay; (b) neutral systems with time-varying delays; (c) polytopic systems with time-varying delays; and (d) discrete-time Markovian jump systems with mode-dependent time delays. Systematic studies are carried out to o(R)er the general solution to these problems and they are outlined as follows: (1) Performancecriteria. Delay-dependentperformanceconditionsrelatedtoenergy- to-peak gain or H norm of dynamic systems with time delays are given in terms of LMIs, which are crucial to the development of the proposed model reduction problems. The energy-to-peak gain performance of above systems is considered for the rst time and is also used as an approximation criterion for the proposed model reduction problems. (2) Characterization of reduced order models. The reduced order models with undetermined system matrices must assure the same structure as those of the orig- inal systems. The state variables of the error systems compose the states of the original and reduced order systems, and the output variables of the error systems are denoted by the di(R)erence between the outputs of the original systems and those of the reduced ones. With the aid of the projection lemma, delay-dependent su- cient conditions are derived for the characterization of the reduced order models in terms of LMIs with inverse constraints. They ensure that the reduced order models are stable and the energy-to-peak gain or H norm of the error systems is less than some given value. (3) Model reduction algorithms. Ecient algorithms designed based on the cone complementarity linearization (CCL) technique are applied to solve LMI problems under inverse constraints, which are further transformed into LMI minimization problems. Furthermore, the proposed algorithms are successfully implemented on the platform of Robust Control Toolbox of MATLAB. Detailed algorithms to con- struct the reduced order models are provided. (4) Numericalexamples. Todemonstratethee(R)ectivenessoftheCCLalgorithmand the validity of the theoretical results, numerical examples are provided to generate reduced order models for di(R)erent kinds of dynamic systems with time delays. DOI: 10.5353/th_b3864543 Subjects: Matrix inequalities Time delay systems System theory - Mathematical models