Bounds for the solution of the Lyapunov matrix equation — A unified approach

In recent years, several bounds have been reported for the solution of the continuous and the discrete Lyapunov equations. Using the unified Lyapunov equation, we give in this paper bounds for the solution of this equation. In the limiting cases, the bounds reduce to existing bounds for both the continuous and discrete Lyapunov equations.     

Delay-dependent stabilisation of systems with time-delayed state and control: application to a quadruple-tank process

This work presents a study on the stabilisation of linear systems with delays in the control and states, motivated by a benchmark problem from the literature (a quadruple-tank system). A methodology for the stabilisation of this kind of systems is presented, expressing the conditions as LMIs. Using several examples from the literature, it is shown that the proposed stabilisation theorem is less conservative than previous results. Finally, the technique is applied to the benchmark problem, showing how it is possible to derive efficient controllers for realistic problems, using the proposed technique.     

Stabilization of 2D saturated systems by state feedback control

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.     

Analysis and Design of $$H_{\infty }$$ Controllers for 2D Singular Systems with Delays

The \(H_{\infty }\) control design problem is solved for the class of 2D discrete singular systems with delays. More precisely, the problem addressed is the design of state-feedback controllers such that the acceptability, internal stability and causality of the resulting closed-loop system are guaranteed, while a prescribed \(H_\infty \) performance level is simultaneously fulfilled. By establishing a novel version of the bounded real lemma, a linear matrix inequality condition is derived for the existence of these \(H_\infty \) controllers. They can then be designed by solving an iterative algorithm based on LMI optimizations. An illustrative example shows the applicability of the algorithm proposed.     

Stabilisation of discrete 2D time switching 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.     

Comments on “Vector norms as Lyapunov functions for linearsystems” [with reply]

The purpose of this note is to give a counterexample to a possible implication of the main result of the above paper by Kiendl-Adamy-Stelzner (ibid., vol.37, no.6, p.839-42 (1992)), to modify this result, to restate its theorem accordingly, and to supply a proof for it. In reply, Kiendl et al. point out that they did not formulate the condition as necessary for a vector norm being a Lyapunov function. Due to Hmamed's remarks, they have reviewed their statements for imprecise formulations     

Componentwise stability of linear systems: a non-symmetrical case

In this paper, we present an extension of the concept of componentwise asymptotic (exponential) stability of linear systems in the non-symmetrical case. The main motivation for these results is the need for a more detailed evaluation of the dynamical behaviour of linear systems in electrical engineering and biology. Necessary and sufficient conditions for componentwise asymptotic (exponential) stability in the non-symmetrical case are given. The symmetrical case is obtained as a particular case. © 1997 by John Wiley & Sons, Ltd.     

Delay-dependent stability and stabilisation of continuous 2D delayed systems with saturating control

This paper deals with the stabilisation problem of continuous two-dimensional (2D) delayed systems, in the presence of saturations on the control signals. For this, a new delay decomposition approach is proposed to deal with the stability and stabilisation issues. The idea is that the range of variation of each delay is divided into segments, and a specific Lyapunov– Krasovskii functional is used that contains different weight matrices in each segment. Then, based on this approach, new delay-dependent stability and stabilisation criteria for continuous 2D delayed systems are derived. These criteria are less conservative and include some existing results as special cases. Some numerical examples are provided to show that a significant improvement is achieved using the proposed approach.