Analysis and synthesis of singular systems with time-delays
After a detailed description of predecessor results on singular nonlinear systems and especially on Markov jump systems (MJSs) and singular MJSs (SMJSs), and a short presentation of the obtained results in the chapters, the authors turn to part I Singular systems (SSs) with time delays and part II SMJSs. The basic content of this monograph are recent advances in analysis and synthesis of singular time-delayed systems with or without Markov jumping parameters. In Ch. 2 using the method of linear matrix inequalities (LMI) and a finite sum inequality the delay-dependent and independent robust control problems for discrete time SSs with time delay and norm bounded parametric uncertainties are studied with the aim to obtain a state feedback controller for the system to be regular causal and stable for all uncertainties. The obtained results are illustrated by two numerical examples. In Ch. 3 the continuous time SSs with time delay of the following form are considered \[ E \dot{x}(t) = Ax(t)+A_{d}x(t-d)+B_{w}w(t), \quad x(t)=\varphi (t), t\in[-d,0] \] where \(z(t) = Cx(t)+Du(t)\) is the controlled output, \(u(t)\) is the control input, and \(\text{rank}\,E = r \leq n\), \(A, A_{d}, B, B_{w}, C, D\) are constant matrices. In using the LMI approach and an augmented Lyapunov functional the problem of delay-dependent state feedback \(H_{\infty}\) control is solved. In Ch. 4 similar systems but with time-varying delays \(d=d(t)\) and actuator failures are studied with the aim to establish the time-dependent condition for the system would be regular, impulse free, exponentially stable and passive. Numerical examples are given. As in Ch. 3 an explicit expression for the desired controller is obtained. Ch. 5 is devoted to dissipativity analysis for SSs with time-varying delays of the previous form and also for discrete time SSs with discrete and distributed time-varying delays. The definitions of dissipativity property are introduced using the notion of energy supply function. In studying the delay partitioning technique several delay dependent dissipativity conditions are derived. In Ch. 6 the SS with time-delays is considered \[ E \dot{x}(t) = A\,x(t)+A_{d}x(t-d)+B_{w}w(t), \quad y(t)=C\,x(t) + C_{d}(x-d) + D_{w}w(t), t\in[-d,0] \] \[ Z(t) = Lx(t), x(t)=\varphi(t), t \in [-d, 0] \] where \(x(t) \in \mathbb R^n\) is the state, \(y(t)\in\mathbb R^s\) is the measurement, \(z(t) \in\mathbb R^q\) is the signal to be estimated, \(w(t) \in \mathbb R^p\) is the disturbance input belonging to \(L_2[0, \infty)\), \(\text{rank}\, E = r \leq n\) with constant matrices of the relevant dimensions. The following full-order filter is introduced \[ E_{f}\dot{\hat{x}}(t)=A_{f}\hat{x}(t) + B_{f} y(t),\quad \hat{z}(t) = C_{f} \hat{x}(t), \] where \(\hat{x}\in\mathbb R^n\), \(\hat{z}(t) \in\mathbb R^q\), with constant filter matrices of appropriate dimensions, which are to be designed. The problem is stated to design a full-order filter such that the relevant to the filter filtering error system would be exponentially admissible with a prescribed \(l_{2}\)-\(l_{\infty}\) performance index. A delay dependent sufficient condition for the solvability of the stated problem is obtained in terms of matrix inequalities. In Ch. 7 the \(H_{\infty}\) filtering problem for SSs with time-varying delays is solved at the usage of LMI approach. A delay-dependent condition is proposed for the existence of \(H_{\infty}\) filter. Ch. 8 solves the problem of \(H_{\infty}\) filter design for discrete time networked SSs with both multiple stochastic time-varying communication delays and probabilistic missing measurements. Part II of the book is devoted to singular Markov jump systems (SMJSs) with time delays and contains the following chapters: Ch. 9. Stability Analysis for discrete-time SMJSs with time-varying delays; Ch. 10 \(H_{\infty}\) Control for SMJSs with time delays; Ch.11. Passivity analysis for SMJSs with time-varying delays; Ch. 12. \(L_{2} - L_{\infty}\) Filtering for discrete-time SMJSs with time-varying delays; Ch. 13. Mode-Independent \(H_{\infty}\) filter design for SMJSs with constant delays; Ch. 14. Mode-Dependent \(H_{\infty}\) filtering for SMJSs with time-varying delays.
scientific article; zbMATH DE number 6167988