Deterministic and stochastic time delay systems
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. Different modifications of the system (1) are considered. For each of them corresponding results are obtained. The second part of the monograph is devoted to similar problems for stochastic systems. Markov processes are considered. Accordingly, definitions of stochastic stability and stochastic stabilizability are introduced. Each section has numerous examples that illustrate the results of the obtained theorems. The monograph can be useful for solving concrete problems of stabilization and control theory. Examples of such problems are given at the beginning of the book.
scientific article; zbMATH DE number 1862448