Randomized algorithms for robust stability and guaranteed cost control of stochastic jump parameter systems with uncertain switching policies
type: Item
instance of: scholarly article
title: Randomized algorithms for robust stability and guaranteed cost control of stochastic jump parameter systems with uncertain switching policies
zbMATH Open document ID: 1078.93053
published in: Journal of Optimization Theory and Applications
publication date: 2005-04-29T00:00:00Z
review text: The author considers a linear jump control system described by the following differential equation: \[ \begin{aligned} \dot{x}(t) & = A( n(t)) x(t) + B (\eta(t)) u(t, \eta (t)) ,\\ x(0) & = x_0, \qquad \eta (0) = \eta_0 \end{aligned}\tag{1} \] where \(x(t) \in \mathbb R^n\) denotes the state of the controlled plant, \(u(t,i) \in \mathbb R^m\) denotes the control input relevant to the \(i\)th form of the plant, \(\{ \eta (t), t\geq 0\}\) is a continuous-time Markov process defined on the finite state space \(N = \{1, 2, \ldots, N\}.\) The transition probability matrix of this process \(P(t)\) is assumed to satisfy the Kolmogorov-Chapman equation \[ (d/ dt) P(t) = \Theta P(t) \tag{2} \] where the transition probability rate matrix \(\Theta\) satisfies the following conditions: \(q_{ij} \geq 0\), \(q_{ii} \leq 0.\) Let \(x(t)\) be a trajectory of the system (1) corresponding to an initial condition \(x(0) = x_0\), \(\eta (0) =i.\) The author considers state-feedback control policies \[ u(t) = K(t, x(t), \eta (t)), \quad K\: [0, \infty) \times \mathbb R^n \times N \to \mathbb R^m \tag{3} \] and presents the definition the robustly stabilizing control system of the form (1) with the control law (3). The problem is to find a controller that attains the minimum of the cost functional \(\inf_n J (u, x_0, i)\) where \[ J(u, x_0, i) = {\mathcal E}^{x_0, i} \biggl\{ \int_0^\infty [ x(t)' R(\eta(t)) x(t) + u(t)' G(\eta(t)) u(t)]\biggr\}\,dt. \] The author presents a decision of this problem in particular cases. An example is presented that illustrates the application of the results to design of stabilizing controllers for unstable power systems.
Mathematics Subject Classification ID: 93D09
Mathematics Subject Classification ID: 93D21
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 2162408
zbMATH Keywords: Markovian jump parameter systems
zbMATH Keywords: randomized algorithms
zbMATH Keywords: guaranteed cost control
zbMATH Keywords: robust control
zbMATH Keywords: matrix inequalities
author: Valery Ugrinovskii
reviewed by: Yu. V. Kozachenko
MaRDI profile type: Publication
full work available at URL: s10957-004-6474-9
OpenAlex ID: W2334982753
Wikidata QID: Q115146369
cites work: Q4222667
cites work: Q5690203
cites work: On stabilization of uncertain linear systems with jump parameters
cites work: On avoiding vertexization of robustness problems: the approximate feasibility concept
cites work: Probabilistic robust design with linear quadratic regulators
cites work: Stochastic algorithms for exact and approximate feasibility of robust LMIs
cites work: An ellipsoid algorithm for probabilistic robust controller design
cites work: Common Lyapunov Functions and Gradient Algorithms
cites work: Stochastic stability properties of jump linear systems
cites work: Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control
cites work: Randomized algorithms for probabilistic robustness with real and complex structured uncertainty.
DOI: 10.1007/S10957-004-6474-9
description: scientific article; zbMATH DE number 2162408