There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Recent years have witnessed a resurgence of research interests in analysing the stability of time-delay systems. Many results have been reported using a variety of approaches and techniques. However, much of the focus has been laid on the use of the Lyapunov-Krasovskii theory to derive sufficient stability conditions in the form of linear matrix inequalities. The purpose of this article is to survey the recent results developed to analyse the asymptotic stability of time-delay systems. Both delay-independent and delay-dependent results are reported in the article. Special emphases are given to the issues of conservatism of the results and computational complexity. Connections of certain delay-dependent stability results are also discussed.

A survey of linear matrix inequality techniques in stability analysis of delay systems

This paper focuses on the hybrid effects of parameter uncertainty, stochastic perturbation, and impulses on global stability of delayed neural networks By using the Ito formula, Lyapunov function, and Halanay inequality, we established several mean-square stability criteria from which we can estimate the feasible bounds of impulses, provided that parameter uncertainty and stochastic perturbations are well-constrained Moreover, the present method can also be applied to general differential systems with stochastic perturbation and impulses

Robust Exponential Stability of Uncertain Delayed Neural Networks With Stochastic Perturbation and Impulse Effects

This note addresses the stability problem of continuous-time positive systems with time-varying delays. It is shown that such a system is asymptotically stable for any continuous and bounded delay if and only if the sum of all the system matrices is a Hurwitz matrix. The result is a time-varying version of the widely-known asymptotic stability criterion for constant-delay positive systems. A numerical example illustrates the correctness of our result.

Stability Analysis for Continuous-Time Positive Systems With Time-Varying Delays

In this paper, we investigate the problem of robust static output feedback (SOF) control for networked control systems (NCSs) subject to network-induced delays and missing data. The uncertain system matrices are assumed to lie in a convex polytope. The network-induced delays are time varying but within a given interval. The random data missing is characterized by the Bernoulli random binary distribution. Delay-dependent conditions for the exponential mean-square stability are first established in terms of matrix inequalities. Then, for the robust stabilization problem, the design of an SOF controller is presented by solving bilinear matrix inequalities (BMIs). In order to efficiently solve a nonconvex BMI, we propose an approach based on the linear matrix inequality technique. Furthermore, the developed approach is employed to design the remote proportional-integral-derivative (PID) controller for NCSs. The design of a digital PID controller is formulated as a synthesis problem of the SOF control via an augmentation method. Simulation examples illustrate the effectiveness of the proposed methods.

Robust Static Output Feedback Control and Remote PID Design for Networked Motor Systems

This paper deals with sufficient conditions of asymptotic stability and stabilization for nonlinear discrete-time systems represented by a Takagi–Sugeno-type fuzzy model whose state variables take only nonnegative values at all times t for any nonnegative initial state. This class of systems is called positive systems. The conditions of stabilizability are obtained with state feedback control. This work is based on multiple Lyapunov functions. The results are presented in linear matrix inequalities form. A real plant is studied to illustrate this technique. Copyright © 2010 John Wiley & Sons, Ltd.

Stabilization of controlled positive discrete‐time T‐S fuzzy systems by state feedback control

This article deals with sufficient conditions of asymptotic stability for discrete two-dimensional (2D) time switching systems represented by a model of Roesser type with state feedback control. This class of systems can correspond to 2D state space or 2D time space switching systems. This work is based on common and multiple Lyapunov functions. The results are presented in LMI form. Two examples are given to illustrate the results.

Stabilisation of discrete 2D time switching systems by state feedback control

This paper deals with the problem of delay-dependent robust \(H_{\infty }\) filtering for uncertain two-dimensional (2-D) continuous systems described by Roesser state space model with time-varying delays, with the uncertain parameters assumed to be of polytopic type. A sufficient condition for \(H_{\infty }\) noise attenuation is derived in terms of linear matrix inequalities, so a robust \(H_{\infty }\) filter can be obtained by solving a convex optimization problem. Finally, some examples are provided to illustrate the effectiveness of the proposed methodology.

Robust H_{\infty } filtering for uncertain two-dimensional continuous systems with time-varying delays

The problem of stabilizability of the 2D continuous-time saturated systems under state-feedback control is solved in this paper. Two cases are considered: the first one, the control may saturate and limits may be attained. The second one, the control does not saturate and limits are avoided. Sufficient conditions of asymptotic stability are presented. The synthesis of the required controllers is given under LMIs form. Illustrative examples are treated.

Stabilization of 2D saturated systems by state feedback control

The stabilization problem for continuous-time fractional linear systems with the additional condition of non negativity of the states is solved. The obtained results are based on a direct Lyapunov function. In particular, the synthesis of state-feedback controllers is obtained by giving conditions in terms of Linear Programs. The solution is also extended to stabilization by asymmetric bounded control. Illustrative examples are provided to show the usefulness of the results.

Abdelaziz Hmamed

Papers

Stabilization of controlled positive discrete‐time T‐S fuzzy systems by state feedback control

Stabilisation of discrete 2D time switching systems by state feedback control

Robust H_{\infty } filtering for uncertain two-dimensional continuous systems with time-varying delays

Stabilization of 2D saturated systems by state feedback control

Stabilization of Continuous-Time Fractional Positive Systems by Using a Lyapunov Function