Mixed control of hidden Markov jump systems

Abstract

Summary In this work, we study the mixed H2false/H∞ control for Markov jump linear systems with hidden Markov parameters. The hidden Markov process is denoted by false(normalθfalse(kfalse),trueθ^false(kfalse)false), where the nonobservable component θ(k) represents the mode of operation of the system, whereas trueθ^false(kfalse) represents the observable component provided by a detector. The goal is to obtain design techniques for mixed H2false/H∞ control problems, with the controllers depending only on the estimate trueθ^false(kfalse), for problems formulated in 3 different forms: (i) minimizing an upper bound on the H2 norm subject to a given restriction on the H∞ norm; (ii) minimizing an upper bound on the H∞ norm, while limiting the H2 norm; and (iii) minimizing a weighted combination of upper bounds of both the H2 and H∞ norms. We propose also new conditions for synthesizing robust controllers under parametric uncertainty in the detector probabilities and in the transition probabilities. The so‐called cluster case for the mixed H2false/H∞ control problem is also analyzed under the detector approach. The results are illustrated by means of 2 numerical examples.

Mixed H2/H∞ Fault Detection Filter for Markovian Jump Linear Systems

“…Along the lines of the present paper, a possible next step would be to consider that the Markov chain mode is not accessible, as considered in [22], which is an assumption that brings new challenges to the fault detection problem.…”

An iterative approach for the discrete‐time dynamic control of Markov jump linear systems with partial information

“…For the control ( f ( ς , υ )= υ in , without imposing the restriction on the norm), we obtain K through Lemma 2 of Oliveira et al bearing in mind the aforementioned cases ρ =0.5 and ρ =1.0. We use the following notation: υ (0) is the cost obtained in the first step of Algorithm 1, υ ∗ is the cost after applying Algorithm 1 for the chosen values ϵ and t max , and is the square of the norm of the closed‐loop system.…”

“…For the control, we calculate K through Lemma 1 of Oliveira et al for ρ =0.5 and ρ =1.0 and use it as an input to Algorithm 1. The notation is as follows.…”

“…We also consider that the states are subject to noise by setting , for all . We set We assume that the detector follows the conditional probability matrix: Solving the pure and controls by using Lemmas 1 and 2, respectively, in the work of Oliveira et al and Algorithm 1 with ϵ =2.5 e −4 and k max =2000, we get ς (0) =112.98, ς ∗ =112.60, , and for the …”

“…Now, for the mixed control, we set f ( ς , υ )= ς in and solve the conditions of Theorem 6 in the work of Oliveira et al and use the calculated K as an input to Algorithm 1 for . The result is shown in Figure .…”