Small gain problem in coupled differential‐difference equations,time‐varying delays,and direct Lyapunov method

This article presents a Lyapunov–Krasovskii formulation of scaled small gain problem for systems described by coupled differential‐difference equations. This problem includes H_∞ problem with block‐diagonal uncertainty as a special case. A discretization may be applied to reduce the conditions into linear matrix inequalities. As an application, the stability problem of systems with time‐varying delays is transformed into the scaled small gain problem through a process of either one‐term approximation or two‐term approximation. The cases of time‐varying delays with and without derivative upper‐bound are compared. Finally, it is shown that similar conditions can also be obtained by a direct Lyapunov–Krasovskii functional method for coupled differential‐functional equations. Numerical examples are presented to illustrate the effectiveness of the method in tackling systems with time‐varying delays. Copyright © 2010 John Wiley & Sons, Ltd.     

Delay‐range‐dependent control of nonlinear time‐delay systems under input saturation

This paper describes a delay‐range‐dependent local state feedback controller synthesis approach providing estimation of the region of stability for nonlinear time‐delay systems under input saturation. By employing a Lyapunov–Krasovskii functional, properties of nonlinear functions, local sector condition and Jensen's inequality, a sufficient condition is derived for stabilization of nonlinear systems with interval delays varying within a range. Novel solutions to the delay‐range‐dependent and delay‐dependent stabilization problems for linear and nonlinear time‐delay systems, respectively, subject to input saturation are derived as specific scenarios of the proposed control strategy. Also, a delay‐rate‐independent condition for control of nonlinear systems in the presence of input saturation with unknown delay‐derivative bound information is established. And further, a robust state feedback controller synthesis scheme ensuring L₂ gain reduction from disturbance to output is devised to address the problem of the stabilization of input‐constrained nonlinear time‐delay systems with varying interval lags. The proposed design conditions can be solved using linear matrix inequality tools in connection with conventional cone complementary linearization algorithms. Simulation results for an unstable nonlinear time‐delay network and a large‐scale chemical reactor under input saturation and varying interval time‐delays are analyzed to demonstrate the effectiveness of the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and L∞‐gain analysis for a class of nonlinear positive systems with mixed delays

This paper addresses the asymptotic stability and L_∞‐gain analysis problem for a class of nonlinear positive systems with both unbounded discrete delays and distributed delays. With the assumption that the nonlinear function is strictly increasing, we first give a characterization on the positivity of the nonlinear system. Then, with some mild assumptions on the delays, a necessary and sufficient condition to ensure the asymptotic stability is presented. Moreover, an explicit expression of the L_∞‐gain of such nonlinear positive systems is given in terms of the system matrices. Finally, a numerical example is given to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

L2 − L∞ Consensus Control for High‐Order Multi‐Agent Systems with Nonuniform Time‐Varying Delays

This paper considers consensus problem for high‐order multi‐agent systems with dynamically changing topologies and nonuniform time‐varying delays. By means of the tree‐type transformation approach, the model transformation is conducted and the consensus problem is converted into an L₂ ? L_∞ control problem of equivalent reduced‐order systems. Furthermore, a Lyapunov‐Krasovkii function is constructed for stability analysis and sufficient conditions in terms of linear matrix inequalities are derived to ensure the consensus with the prescribed L₂ ? L_∞ performance. A numerical simulation is provided to verify the correctness of the theoretical results.     

Robust H∞ Filtering For Uncertain Stochastic Time‐Delay Systems

This paper deals with the problem of robust H_∞ filtering for uncertain stochastic systems. The system under consideration is subject to time‐varying norm‐bounded parameter uncertainties and unknown time delays in both the state and measurement equations. The problem we address is the design of a stable filter that ensures the robust stochastic stability and a prescribed H_∞ performance level for the filtering error system irrespective of all admissible uncertainties and time delays. A suffient condition for the solvability of this problem is proposed and a linear matrix inequality approach is developed for the design of the robust H_∞ filters. An illustrative example is provided to demonstrate the effctiveness of the proposed approach.     

Delay‐range‐dependent H∞ control for Markovian jump systems with mode‐dependent time delays

This paper considers mean‐square exponential stability and H_∞ control problems for Markovian jump systems (MJSs) with time delays which are time‐varying in an interval and depend on system mode. By exploiting a novel Lyapunov‐Krasovskii functional which takes into account the range of delay, and by making use of some techniques, new delay‐range‐dependent stability result and bounded real lemma for MJSs are obtained, where the introduction of the lower bound of delay is shown to be advantageous for reducing conservatism. Moreover, a sufficient condition for the solvability of the H_∞ control problem is derived in terms of linear matrix inequalities. Finally, illustrative examples are presented to show the advantage and effectiveness of the proposed approaches. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Delay‐dependent stability criterion and H∞ analysis for Markovian jump systems with time‐varying delays

This paper deals with the problems of stochastic stability and H_∞ analysis for Markovian jump linear systems with time‐varying delays. In terms of linear matrix inequalities, a less conservative delay‐dependent stability criterion for Markovian jump systems is proposed by constructing a different Lyapunov‐Krasovskii functional and introducing improved integral‐equalities approach, and a sufficient condition is derived from the H_∞ performance. Numerical examples are provided to demonstrate the efficiency and reduced conservatism of the results in this paper. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

H∞ Non‐Fragile Synchronous Guaranteed Control of Uncertainty Complex Dynamic Network with Time‐Varying Delay

A kind of H_∞ non‐fragile synchronization guaranteed control method is put forward for a class of uncertain time‐varying delay complex network systems with disturbance input. The network under consideration includes unknown but bounded nonlinear coupling functions f(x) and the coupling term and node system with time‐varying delays. The nonlinear vector function f(x) need not be differentiable but should satisfy the norm bound. A non‐fragile state feedback controller of the gain with sufficiently large regulation margin is designed. It is ensured that the parameters of the controller could still be effective under small perturbation. The sufficient conditions for the existence of H_∞ synchronous non‐fragile guaranteed control of this system have been obtained by constructing a suitable Lyapunov‐Krasovskii functional, adopting matrix analysis, using the theorem of Schur complement and linear matrix inequalities (LMI). These conditions can guarantee robust asymptotic stability for each node of network with disturbance as well as achieve a prescribed robust H_∞ performance level. Finally, the feasibility of the designed method is verified by a numerical example.     

H∞ Control of Linear Discrete‐Time Systems With Norm‐Bounded Nonlinear Uncertainties

In this paper, robust H_∞ control of a class of discrete‐time uncertain systems in state‐space form with linear nominal parts and norm‐bounded nonlinear uncertainties in both state and output equations is discussed. Such systems have a unique characterisic; that is, the two norm‐bounded nonlinear uncertainties have the equivalent representation by means of time‐varying and norm‐bounded linear uncertainties. To overcome the conservativenss of [5], the two nonlinear uncertainty sets are considered to be different. Then, by converting such systems into related discrete‐time linear systems with time‐varying and norm‐bounded linear uncertainties, we obtain that a sufficient condition for robust H_∞ control of such systems is equivalent to the solvability of the same problem of the related linear uncertain systems, which is solvable by means of a linear algebraic Riccati inequality.     

L2‐Gain Analysis for Synchronization of Stochastic Complex Network with Switched Topology and Time‐Varying Delay

We investigate the exponential stability and L₂‐gain analysis for the synchronization of stochastic complex networks under average dwell time switched topology with consideration of external disturbance, internal noise and fast time‐varying delay in the synchronized process. Based on the proposed stochastic network, a new L₂‐gain synchronization is proposed to solve the mean‐square exponential stable under switched topology with an H_∞ performance from the extrinsic disturbances to the synchronization error. The obtained results are applicable for the fast time‐varying case with larger‐than‐1 delay derivative. Finally, numerical simulations are performed to demonstrate the effectiveness of our strategies.     

Finite‐time boundedness of switched systems with time‐varying delays via sampled‐data control

In the framework of sampled‐data control, finite‐time boundedness (FTB) of switched systems with time‐varying delays is investigated. Sufficient conditions for FTB of switched systems with time‐varying delays via sampled‐data control are proposed. Moreover, considering the relationship between the sampling period and the mode‐dependent average dwell time, switching signals are designed. In addition, finite‐time weighted L₂‐gain (FTW‐L₂‐gain) of switched systems with time‐varying delays is proposed to measure their disturbance tolerance capacity within a finite‐time interval. Multiple Lyapunov‐Krasovskii functionals are applied to complete subsequent proofs in detail. Simulation results are exemplified to verify the proposed method.     

LMI APPROACH TO ROBUST FILTERING FOR DISCRETE TIME‐DELAY SYSTEMS WITH NONLINEAR DISTURBANCES

This paper investigates the problem of robust filtering for a class of uncertain nonlinear discrete‐time systems with multiple state delays. It is assumed that the parameter uncertainties appearing in all the system matrices reside in a polytope, and that the nonlinearities entering into both the state and measurement equations satisfy global Lipschitz conditions. Attention is focused on the design of robust full‐order and reduced‐order filters guaranteeing a prescribed noise attenuation level in an H∞ or l₂‐l∞ sense with respect to all energy‐bounded noise disturbances for all admissible uncertainties and time delays. Both delay‐dependent and independent approaches are developed by using linear matrix inequality (LMI) techniques, which are applicable to systems either with or without a priori information on the size of delays.     

LMI OPTIMIZATION APPROACH FOR DELAY‐DEPENDENT H∞ CONTROL OF TIME‐VARYING DELAY SYSTEMS

In this paper, the robust delay‐dependent H_∞ control for a class of uncertain systems with time‐varying delay is considered. An improved state feedback H_∞ control is proposed to minimize the H_∞‐norm bound via the LMI optimization approach. Based on the proposed result, delay‐dependent criteria are obtained without using the model transformation technique or bounded inequalities on cross product terms. The linear matrix inequality (LMI) optimization approach is used to design the robust H_∞ state feedback control. Some numerical examples are given to illustrate the effectiveness of the approach.     

Networked‐based stabilization via discontinuous Lyapunov functionals

This paper presents a new stability and L₂‐gain analysis of linear Networked Control Systems (NCS). The new method is inspired by discontinuous Lyapunov functions that were introduced by Naghshtabrizitextitet al. (Syst. Control Lett. 2008; 57 :378–385; Proceedings 26th American Control Conference, New York, U.S.A., July 2007) in the framework of impulsive system representation. Most of the existing works on the stability of NCS (in the framework of time delay approach) are reduced to some Lyapunov‐based analysis of systems with uncertain and bounded time‐varying delays. This analysis via time‐independent Lyapunov functionals does not take advantage of the sawtooth evolution of the delays induced by sample‐and‐hold. The latter drawback was removed by Fridman (Automatica 2010; 46 :421–427), where time‐dependent Lyapunov functionals for sampled‐data systems were introduced. This led to essentially less conservative results. The objective of the present paper is to extend the time‐dependent Lyapunov functional approach to NCS, where variable sampling intervals, data packet dropouts, and variable network‐induced delays are taken into account. The Lyapunov functionals in this paper depend on time and on the upper bound of the network‐induced delay, and these functionals do not grow along the input update times. The new analysis is applied to the state‐feedback and to a novel network‐based static output‐feedback H_∞ control problems. Numerical examples show that the novel discontinuous terms in Lyapunov functionals essentially improve the results. Copyright © 2011 John Wiley & Sons, Ltd.     

Delay‐range‐dependent robust H∞ filtering for uncertain systems with interval time‐varying delays

This paper is concerned with the problem of delay‐range‐dependent robust H_∞ filtering for systems with time‐varying delays in a range. The aim of this problem is to design a filter such that, for all admissible uncertainties, the filtering error system is robustly asymptotically stable with a prescribed H_∞ level. The desired filter can be constructed by solving a set of linear matrix inequalities (LMIs). An illustrative numerical example is provided to demonstrate the effectiveness of the proposed method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Nonlinear H∞ observer design for one‐sided Lipschitz discrete‐time singular systems with time‐varying delay

This paper investigates the H_∞ observer design problem for a class of nonlinear discrete‐time singular systems with time‐varying delays and disturbance inputs. The nonlinear systems can be rectangular and the nonlinearities satisfy the one‐sided Lipschitz condition and quadratically inner‐bounded condition, which are more general than the traditional Lipschitz condition. By appropriately dealing with these two conditions and applying several important inequalities, a linear matrix inequality–based approach for the nonlinear observer design is proposed. The resulting nonlinear H_∞ observer guarantees asymptotic stability of the estimation error dynamics with a prescribed performance γ. The synthesis condition of H_∞ observer design for nonlinear discrete‐time singular systems without time delays is also presented. The design is first addressed for one‐sided Lipschitz discrete‐time singular systems. Finally, two numerical examples are given to show the effectiveness of the present approach.     

H∞ Control Synthesis for Lurie Networked Control Systems with Multiple Delays Based on the Non‐Uniform Characteristic

This paper investigates the H _∞ control synthesis problem for Lurie networked control systems (NCSs) with multiple time‐varying delays. With the consideration of both network‐induced delays and data packet dropout, Lurie NCSs discussed in this work can effectively be transformed into Lurie control systems with multiple time‐varying delays. In addition, with the non‐uniform distribution characteristics of network delays, based on the delay probability distribution, the stable controller design, and H _∞ synthesis, approaches are derived in the form of linear matrix inequalities (LMIs). Finally, a set of numerical examples are studied, and the results demonstrate the applicability and effectiveness of the suggested approaches.     

Delay‐dependent adaptive reliable H∞ control of linear time‐varying delay systems

This paper considers the problem of delay‐dependent adaptive reliable H_∞ controller design against actuator faults for linear time‐varying delay systems. Based on the online estimation of eventual faults, the parameters of adaptive reliable H_∞ controller are updating automatically to compensate the fault effects on the system. A new delay‐dependent reliable H_∞ controller is established using a linear matrix inequality technique and an adaptive method, which guarantees the stability and adaptive H_∞ performance of closed‐loop systems in normal and faulty cases. A numerical example and its simulation results illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.