New summation inequalities and their applications to discrete-time delay systems

This paper provides new summation inequalities in both single and double forms to be used in stability analysis of discrete-time systems with time-varying delays. The potential capability of the newly derived inequalities is demonstrated by establishing less conservative stability conditions for a class of linear discrete-time systems with an interval time-varying delay in the framework of linear matrix inequalities. The effectiveness and least conservativeness of the derived stability conditions are shown by academic and practical examples.Comment: 15 pages, 01 figur

An Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems

A general framework is presented for analyzing the stability and performance of nonlinear and linear parameter varying (LPV) time delayed systems. First, the input/output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints (IQCs). A constant delay is a linear, time-invariant system and this leads to a simple, intuitive interpretation for these frequency domain constraints. This simple interpretation is used to derive new IQCs for both constant and varying delays. Second, the performance of nonlinear and LPV delayed systems is bounded using dissipation inequalities that incorporate IQCs. This step makes use of recent results that show, under mild technical conditions, that an IQC has an equivalent representation as a finite-horizon time-domain constraint. Numerical examples are provided to demonstrate the effectiveness of the method for both class of systems

Time-delay systems : stability, sliding mode control and state estimation

University of Technology, Sydney. Faculty of Engineering and Information Technology.Time delays and external disturbances are unavoidable in many practical control systems such as robotic manipulators, aircraft, manufacturing and process control systems and it is often a source of instability or oscillation. This thesis is concerned with the stability, sliding mode control and state estimation problems of time-delay systems. Throughout the thesis, the Lyapunov-Krasovskii (L-K) method, in conjunction with the Linear Matrix Inequality (LMI) techniques is mainly used for analysis and design. Firstly, a brief survey on recent developments of the L-K method for stability analysis, discrete-time sliding mode control design and linear functional observer design of time-delay systems, is presented. Then, the problem of exponential stability is addressed for a class of linear discrete-time systems with interval time-varying delay. Some improved delay-dependent stability conditions of linear discrete-time systems with interval time-varying delay are derived in terms of linear matrix inequalities. Secondly, the problem of reachable set bounding, essential information for the control design, is tackled for linear systems with time-varying delay and bounded disturbances. Indeed, minimisation of the reachable set bound can generally result in a controller with a larger gain to achieve better performance for the uncertain dynamical system under control. Based on the L-K method, combined with the delay decomposition approach, sufficient conditions for the existence of ellipsoid-based bounds of reachable sets of a class of linear systems with interval time-varying delay and bounded disturbances, are derived in terms of matrix inequalities. To obtain a smaller bound, a new idea is proposed to minimise the projection distances of the ellipsoids on axes, with respect to various convergence rates, instead of minimising its radius with a single exponential rate. Therefore, the smallest possible bound can be obtained from the intersection of these ellipsoids. This study also addresses the problem of robust sliding mode control for a class of linear discrete-time systems with time-varying delay and unmatched external disturbances. By using the L-K method, in combination with the delay decomposition technique and the reciprocally convex approach, new LMI-based conditions for the existence of a stable sliding surface are derived. These conditions can deal with the effects of time-varying delay and unmatched external disturbances while guaranteeing that all the state trajectories of the reduced-order system are exponentially convergent to a ball with a minimised radius. Robust discrete-time quasi-sliding mode control scheme is then proposed to drive the state trajectories of the closed-loop system towards the prescribed sliding surface in a finite time and maintain it there after subsequent time. Finally, the state estimation problem is studied for the challenging case when both the system’s output and input are subject to time delays. By using the information of the multiple delayed output and delayed input, a new minimal order observer is first proposed to estimate a linear state functional of the system. The existence conditions for such an observer are given to guarantee that the estimated state converges exponentially within an Є-bound of the original state. Based on the L-K method, sufficient conditions for Є-convergence of the observer error, are derived in terms of matrix inequalities. Design algorithms are introduced to illustrate the merit of the proposed approach. From theoretical as well as practical perspectives, the obtained results in this thesis are beneficial to a broad range of applications in robotic manipulators, airport navigation, manufacturing, process control and in networked systems

New H∞ control design for polytopic systems with mixed time-varying delays in state and input

This paper concerns with the problem of state-feedback H∞ control design for a class of linear systems with polytopic uncertainties and mixed time-varying delays in state and input. Our approach can be described as follows. We first construct a state-feedback controller based on the idea of parameter-dependent controller design. By constructing a new parameter-dependent Lyapunov-Krasovskii functional (LKF), we then derive new delay-dependent conditions in terms of linear matrix inequalities ensuring the exponential stability of the corresponding closed-loop system with a H∞ disturbance attenuation level. The effectiveness and applicability of the obtained results are demonstrated by practical examples

Robust H

The robust filtering problem for a class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay is investigated. The parameter uncertainties are assumed to be time varying norm bounded in both the state and measurement equations. By using the Lyapunov stability theory and some new relaxed techniques, sufficient conditions are proposed to guarantee the robustly stochastic stability with a prescribed H∞ performance level of the filtering error system for all admissible uncertainties, sensor nonlinearities, and time-varying delays. These conditions are dependent on the lower and upper bounds of the time-varying delays and are obtained in terms of a linear matrix inequality (LMI). Finally, two simulation examples are provided to illustrate the effectiveness of the proposed methods

Dynamical analysis of particular class of time-delay control systems

U disertaciji su razmatrani problemi dinamike analize posebnih klasa sistema sa istim vremenskim kašnjenjem. Prošireni su osnovni rezultati na polju ljapunovske stabilnosti linearnih, vremenski diskretnih sistema sa istim vremenskim kašnjenjem. Data je Ljapunov–Krasovski metoda za vremenski diskretne sisteme sa istim vremenskim kašnjenjem. Prezentovani su potrebni i dovoljni uslovi asimptotske stabilnosti, zavisne od isto vremenskog kašnjenja, linearnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem. Dati su dovoljni uslovi asimptotske stabilnosti, nezavisne od isto vremenskog kašnjenja, klase linearnih, perturbovanih sistema sa višestrukim vremenskim kašnjenjem. Prezentovani su dovoljni uslovi D–stabilnosti klase linearnih, vremenski diskretnih sistema sa istim vremenskim kašnjenjem. Dati su dovoljni uslovi eksponencijalne stabilnosti vremenski diskretnih sistema sa istim vremenskim kašnjenjem i perturbacijama. Prezentovani su potrebni i dovoljni uslovi kvadratne stabilnosti linearnih, vremenski diskretnih sistema sa istim vremenskim kašnjenjem u stanju i neodreenostima. Potrebni i dovoljni uslovi asimptotske stabilnosti, zavisni od isto vremenskog kašnjenja, velikih, linearnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem, su dati. Prouena je stabilnost velikih, intervalnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem. Izvedeni su novi dovoljni kriterijumi, zavisni i nezavisni od isto vremenskog kašnjenja, stabilnosti na konanom vremenskom intervalu i atraktivne praktine stabilnosti linearnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem, kao i odgovarajui rezultati koji se tiu problema praktine nestabilnosti. Istražen je problema stabilnosti na konanom vremenskom intervalu za klasu linearnih, vremenski diskretnih sistema sa vremenski promenljivim kašnjenjem. Numeriki primeri su dati da demonstriraju primenu prezentovanih metoda.control systems are considered. Some of the basic results in the area of Lyapunov stability of linear, discrete time–delay systems are extended. A Lyapunov–Krasovskii method for discrete time–delay systems is gived. Necessary and sufficient conditions for delay–dependent asymptotic stability of linear, continuous and discrete time–delay systems is offered. Sufficient conditions, independent of delay, for asymptotic stability of a particular class of linear perturbed time–delay systems with multiple delays are gived. New sufficient conditions for the D–stability of a particular class of linear, discrete time–delay systems are established. Sufficient conditions for the exponential stability of discrete time–delay systems with perturbations are gived. Necessary and sufficient conditions for quadratic stability of uncertain linear discrete systems with state delay are presented. New necessary and sufficient conditions for delay–dependent asymptotic stability of a particular class of large–scale, linear, continuous and discrete time–delay systems are established. The stability of continuous and discrete large–scale time–delay interval systems are considered. A new sufficient delay–dependant and delay–independent criteria for the finite time stability and attractive practical stability of linear continuous and discrete time–delay systems has been derived, as well as corresponding results concerning instability problems. Finite–time stability problem has been investigated for a class of linear discrete time–varying delay systems. Numerical examples are given to demonstrate the application of the proposed methods

Stability of Discrete-Time Systems with Time-Varying Delay: Delay Decomposition Approach

This article deals with the problem of obtaining delay-dependent stability conditions for a class of discrete-time systems with interval time-varying delay. Using the decomposition the delay interval into two unequal subintervals by tuning parameter α, a new interval delay-dependent Lyapunov-Krasovskii functional is constructed to derive novel delay-dependent stability conditions which are expressed in terms of linear matrix inequalities. This leads to reduction of conservatism in terms of the upper bounds of the maximum time-delay. The numerical examples show that the obtained result is less conservative than some existing ones in the literature