Deterministic and stochastic time delay systems
type: Item
instance of: scholarly article
title: Deterministic and stochastic time delay systems
zbMATH Open document ID: 1056.93001
published in: Control Engineering
publication date: 2003-02-02T00:00:00Z
review text: The monograph is devoted to problems of stability and stabilization of systems with deviating arguments. It consists of two parts. The first part considers deterministic systems of the form \[ \begin{aligned} \dot x(t)&= A(t) x(t)+ \sum^m_{j=1} A_{dj}(t) x(t-\tau)+ B_1(t) \overline\omega(t) + B(t) u(t),\\ z(t)&= C_1(t) x(t)+ D_{11}(t)+ D_{12}(t) u(t),\\ y(t)&= C_2(t) x(t)+ D_{21}(t)\overline\omega(t),\\ x(t)&= \phi(t),\;t\in [-\tau,0],\end{aligned}\tag{1} \] where \(x(t)\in \mathbb{R}^n\) is the state vector, \(u(t)\in \mathbb{R}^m\) is the control input vector, \(\overline\omega(t)\in\mathbb{R}^l\) is the square-integrable disturbance input vector, \(z(t)\in\mathbb{R}^p\) is the controlled output vector, \(y(t)\in \mathbb{R}^q\) is the measured output vector, \(\tau\) is the time delay, \(\phi(t)\) is the initial function, and \(A(t)\), \(A_d(t)\), \(B_1(t)\), \(B(t)\), \(C_1(t)\), \(C_2(t)\), \(D_{11}(t)\), \(D_{12}(t)\) and \(D_{21}(t)\) have the form of sums of constant matrices and ``disturbing terms''. As an instrument of investigation, the second Lyapunov method with Lyapunov-Krasovskij functionals \[ V(x_t)= x^T(t) Px(t)+ \int^t_{t-\tau} x^T(s) Qx(s)\,ds, \] is used. Conditions of stability, stabilizability, \(H_\infty\) control, robust stabilizability, and robust \(H_\infty\) control for systems in this form are obtained. One should note that the concepts of robust stabilizability and robust \(H_\infty\) control appeared in the last decade and are concerned with uncertainty of mathematical models. The obtained results are based on standard second Lyapunov methods and are concerned with condition of positive definiteness of the functional and negative definiteness of its full derivative. These conditions are formulated in the form of existence of positive definite matrices included in the Lyapunov-Krasovskij functional and of negative definiteness of an extended matrix of quadratic form, obtained by combining current phase coordinates and prehistory. Differe…
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 93C23
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 93B36
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 93D21
Mathematics Subject Classification ID: 34K50
zbMATH DE Number: 1862448
zbMATH Keywords: stability
zbMATH Keywords: stabilization
zbMATH Keywords: disturbance
zbMATH Keywords: time delay
zbMATH Keywords: second Lyapunov method
zbMATH Keywords: Lyapunov-Krasovskij functionals
zbMATH Keywords: robust stabilizability
zbMATH Keywords: robust \(H_\infty\) control
zbMATH Keywords: Markov processes
zbMATH Keywords: stochastic stability
author: E. K. Boukas
author: Z. K. Liu
reviewed by: Denis Khusainov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1862448