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Output‐based H 2 optimal controllers for a class of discrete‐time stochastic linear systems with periodic coefficients
Author(s) -
Dragan Vasile,
Morozan Toader,
Stoica AdrianMihail
Publication year - 2014
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.3173
Subject(s) - mathematics , markov chain , multiplicative noise , multiplicative function , riccati equation , linear system , norm (philosophy) , discrete time and continuous time , type (biology) , control theory (sociology) , state space , white noise , stochastic control , optimal control , mathematical optimization , computer science , differential equation , control (management) , mathematical analysis , transmission (telecommunications) , ecology , statistics , signal transfer function , artificial intelligence , political science , analog signal , law , biology , telecommunications
Summary The aim of the paper is to present a design procedure of the optimal controller minimizing the H 2 ‐type norm of discrete‐time stochastic linear systems with periodic coefficients simultaneously affected by a nonhomogeneous but periodic Markov chain and state and control multiplicative white noise perturbations. Firstly, two H 2 ‐type norms for the linear stochastic systems under consideration were introduced. These H 2 ‐type norms may be viewed as measures of the effect of the additive white noise perturbations on the regulated output of the considered system. Before deriving of the state space representation of the optimal controller, some useful formulae of the two H 2 ‐type norms were obtained. These formulae are expressed in terms of periodic solutions of some suitable linear equations and are derived in the absence of some additional assumptions regarding the Markov chain other than the periodicity of the sequence of the transition probability matrices. Further, it is shown that the optimal H 2 controller depends on the stabilizing solutions of some specific systems of coupled Riccati equations, which generalize the well‐known control and filtering equations from linear time invariant case. For the readers convenience, the paper presents iterative numerical algorithms for the computations of the stabilizing solutions of these Riccati type systems. The theoretical developments are illustrated by numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.