Improved Robust Stability Criteria for Time‐Delay Lur'e System

This paper deals with the problem of the robustly absolute stability for neutral‐type Lur'e systems with mixed time‐varying delay. By combining the piecewise analysis theory with the reciprocally convex method and Wirtinger‐based inequality technology, some new delay‐dependent stability criteria are proposed via a modified Lyapunov‐Krasovskii functional (LKF) approach. The stability conditions can be solved by using standard linear matrix inequality (LMI) convex optimization solvers. The criteria are less conservative than some previous ones. Three numerical examples are presented to show the effectiveness of the proposed approach.     

Robust Absolute Stability Analysis of Multiple Time‐Delay Lur'e Systems With Parametric Uncertainties

The problem of robust absolute stability for time‐delay Lur'e systems with parametric uncertainties is investigated in this paper. The nonlinear part of the Lur'e system is assumed to be both time‐invariant and time‐varying. The structure of uncertainty is a general case that includes norm‐bounded uncertainty. Based on the Lyapunov–Krasovskii stability theory, some delay‐dependent sufficient conditions for the robust absolute stability of the Lur'e system will be derived and expressed in the form of linear matrix inequalities (LMIs). These conditions reduce the conservativeness in computing the upper bound of the maximum allowed delay in many cases. Numerical examples are given to show that the proposed stability criteria are less conservative than those reported in the established literatures.     

New delay-dependent absolute stability criteria for Lur'e systems with time-varying delay

In this article, the absolute stability problem is investigated for Lur'e systems with time-varying delay and sector-bounded nonlinearity. By employing the delay fractioning idea, the new augmented Lyapunov functional is first constructed. Then, by introducing some slack matrices and by reserving the useful term when estimating the upper bound of the derivative of Lyapunov functional, the new delay-dependent absolute stability criteria are derived in terms of linear matrix inequalities. Several numerical examples are presented to show the effectiveness and the less conservativeness of the proposed method.     

奇异Lur'e网络的自适应牵制聚类同步控制

本文分别研究了由奇异和非奇异Lur''e系统组成的具有多种耦合方式时滞复杂动态网络的聚类同步问题. 通过设计一类牵制反馈控制器, 仅控制当前聚类中与其它聚类有直接连接的Lur''e系统, 从而有效减少控制器个数同时降低控制成本. 考虑到网络具有多种耦合方式, 本文合理构造Lyapunov 泛函, 并有效利用扇形条件、非线性函数类概念、S过程以及Lyapunov 稳定性定理等方法, 分别给出奇异和非奇异Lur''e 动态网络实现聚类同步的判定条件. 此外, 为有效节省控制成本, 针对控制强度设计一类自适应更新定律, 从而得到网络实现聚类同步的最优控制强度. 最后, 为了验证所得聚类同步判定条件和控制器的有效性, 对一类典型Lur''e网络聚类同步问题进行了数值仿真.     

Absolute instability of Lur'e systems and its application to oscillation analysis of uncertain genetic networks

We derive instability criteria for Lur'e systems with sector‐bounded nonlinearities and uncertain external signals. First, we define absolute instability of an equilibrium and derive an absolute instability condition for a fixed equilibrium point in terms of a linear matrix inequality, which is analogous to the well‐known circle stability criterion. Then, the condition is extended to a parametric absolute instability condition, which is applicable to the instability test of a Lur'e system with an equilibrium point whose location is affected by uncertain nonlinearities and uncertain external signals. Finally, we show that the proposed analysis method is useful through the oscillation analysis of an uncertain genetic network model. Copyright © 2014 John Wiley & Sons, Ltd.     

Improved robust stability criteria for a class of Lur'e systems with interval time-varying delays and sector-bounded nonlinearity

This paper is concerned with the absolute and robust stability for a class of neutral-type Lur'e systems with an interval time-varying delay and sector-bounded nonlinearity. By discretising the delay interval into two segmentations with an unequal width, new delay-dependent sufficient conditions for the absolute and robust stability of neutral-type Lur'e systems are proposed in terms of linear matrix inequalities (LMIs) by employing a modified Lyapunov-Krasovskii functional (LKF). These conditions reduce the conservativeness in computing the maximum allowed delay bounds (MADBs) in many cases. Finally, several standard numerical examples are presented to show the effectiveness of the proposed approach.     

Robust Analysis of Discrete‐Time Lur'e Systems with Slope Restrictions Using Convex Optimization

This paper considers robust stability and robust performance analysis for discrete‐time linear systems subject to nonlinear uncertainty. The uncertainty set is described by memoryless, time‐invariant, sector bounded, and slope restricted nonlinearities. We first give an overview of the absolute stability criterion based on the Lur'e‐Postkinov Lyapunov function, along with a frequency domain condition. Subsequently, we derive sufficient conditions to compute the upper bounds of the worst case H₂ and worst case H∞ performance. For both robust stability testing and robust performance computation, we show that these sufficient conditions can be readily and efficiently determined by performing convex optimization over linear matrix inequalities.     

Robust absolute stability criteria for uncertain Lur'e systems of neutral type

This paper is concerned with robust absolute stability of uncertain Lur'e systems of neutral type. Some delay‐dependent stability criteria are obtained and formulated in the form of linear matrix inequalities. The criteria cover some existing results as their special cases. Neither model transformation nor bounding technique for cross terms is involved through derivation of the stability criteria. Numerical examples show the effectiveness of the criteria. Copyright © 2007 John Wiley & Sons, Ltd.     

Further results on delay‐dependent absolute and robust stability for time‐delay Lur'e system

In this paper, together with two improved Lyapunov–Krasovskii functionals, several novel sufficient conditions are derived to guarantee two kinds of time‐delay Lur'e system to be absolutely and robustly stable, in which the linear fractional uncertainties are involved, and both time‐delay and its variation rate can be fully utilized. Through using the combined convex technique, some previously ignored terms can be reconsidered when estimating the triple integral Lyapunov–Krasovskii functional terms’ derivative, and the criteria are presented in terms of LMIs, in which its solvability heavily depends on the information of addressed systems. Finally, the comparing results in the numerical examples demonstrate our idea can reduce the conservatism more efficiently than some present ones. Copyright © 2013 John Wiley & Sons, Ltd.     

Synchronization Criterion and Control Scheme for Lur'e Type Complex Dynamical Networks with Switching Topology and Coupling Time‐Varying Delay

In this paper, the synchronization problem is addressed in the context of Lur'e type complex switched network (CSN) with coupling time‐varying delay in which every node is a Lur'e system. Based on the Lyapunov–Krasovskii theory and linear matrix inequality (LMI) technique, a delay‐dependent synchronization criterion and a decentralized state feedback dynamic controller for synchronization of CSNs have been proposed. By choosing a common Lyapunov–Krasovskii functional and using the combined reciprocal convex technique, some previously ignored terms can be reconsidered and less conservative conditions can be obtained. In addition, by using an eigenvalue‐decoupling method and convex optimization theory, high‐dimension LMIs are decoupled into a set of low‐dimension ones and the computation complexity of the criterion can be significantly reduced. The effectiveness and applicability of the suggested control solution is verified and assessed through the analysis for two numerical examples.     

New absolute stability criteria for time‐delay Lur'e systems with sector‐bounded nonlinearity

This paper is concerned with the problem of absolute stability of time‐delay Lur'e systems with sector‐bounded nonlinearity. Several novel criteria are presented by using a Lur'e–Postnikov function. For a general Lur'e system with known time delay, the absolute stability of it is analyzed by solving a set of linear matrix inequalities (LMIs). The maximum upper bound of the allowable time delay for a general Lur'e system is derived by solving a convex optimization problem. The feasibility of the LMIs implies some frequency‐domain interpretations which are similar to the frequency‐domain inequalities in the circle criterion and the Popov criterion. Copyright © 2009 John Wiley & Sons, Ltd.     

On Designing Time‐delay Feedback Controller for Master‐Slave Synchronization of Lur'e Systems

This paper is concerned with the problem of designing a time‐delay output feedback controller for master‐slave synchronization of Lur'e systems. The time delay is divided into two intervals and different energy functions are defined in each interval, which together provide a new Lyapunov–Krasovskii functional and derive a new delay‐dependent synchronization criterion. A sufficient condition for the existence of such a feedback controller is given, and an explicit expression of such a controller is also achieved. These algorithms are formulated in terms of linear matrix inequalities that can be solved easily. Chua's circuit is used to illustrate the effectiveness of the design method.     

Novel observer design method for Lur'e differential inclusion systems

This paper proposes a new method to design observers for Lur'e differential inclusion systems. The feature of this method is that the designed observers do not contain any set-valued functions. single-input-single-output (SISO) systems are considered firstly, then the results are extended to multiple-input-multiple-output (MIMO) systems. For MIMO systems, the form of reduced-order observers is also presented. Simulations are given to show these observers work well.     

Master‐slave synchronization for delayed Lur'e systems using time‐delay feedback control

This paper deals with the problem of designing delayed feedback controllers of master‐slave synchronization for Lur'e systems with time‐variant delay (0≤τ₀≤τ(t)≤τ_m). Through partitioning the intervals [0, τ₀] and [τ₀, τ_m], respectively, and choosing two augmented Lyapunov‐Krasovskii functionals(LKFs), two delay‐dependent synchronization criteria are formulated in the form of linear matrix inequalities (LMIs), in which the conservatism can be greatly reduced by thinning the partitioning of delay intervals and employing convex combination. Thus the sufficient conditions can be derived for the existence of delayed feedback controllers, and the controller gain matrices can be achieved by solving the established LMIs. Finally, three numerical examples are given to illustrate the presented synchronization schemes. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Homogeneous control law design for output synchronization of networked Lur'e‐type systems with disturbances

In this paper, the homogeneous distributed control law is proposed such that the outputs of the networked Lur'e systems with disturbances are synchronized. A novel adaptive distributed state‐based disturbance compensator technique is utilized such that the disturbances can be offset asymptotically. The design of the distributed compensator is constructive in that some particular functions of the states of the Lur'e systems are included. The functions always exist for the considered Lur'e‐type system. Moreover, a theorem of networked Lur'e systems subject to disturbances is first presented to show the boundedness of the states of the closed‐loop system. Then, the synchronization is shown by the incremental passivity theory and graph theory. Finally, the effectiveness of the current method is illustrated by the networked Van der Pol oscillators with disturbances.     

Distributed impulsive synchronization of Lur'e dynamical networks via parameter variation methods

This paper is devoted to investigating the exponential synchronization of coupled Lur'e dynamical networks with multiple time‐varying delays and stochastic disturbance. The problem studied in this paper could be regarded as a kind of leader‐following synchronization issue. As the networks may suffer from certain impulsive disturbance, an effective distributed impulsive control protocol is proposed to synchronize the stochastic Lur'e dynamical networks. According to the comparison principle, the average impulsive interval, and the extended formula for the variation of parameters, sufficient conditions are derived for successful achievement of the network synchronization with consideration to different functions of impulsive effects. Furthermore, the exponential convergence rate is obtained based on the impulsive solution equation. In addition, finally, some numerical simulations are given to illustrate the validity of the control scheme and the theoretical analysis.     

Robust cooperative output regulation of heterogeneous Lur'e networks

In this paper, we study robust cooperative output regulation problems for a directed network of Lur'e systems that consist of a nominal linear dynamics with an unknown static nonlinearity around it through negative feedback. We assume that the linear part of each agent is identical, but the nonlinearities are allowed to be different for distinct agents. In this sense, the network is heterogeneous. As is common in the context of Lur'e systems, the unknown nonlinearities are assumed to be sector bounded within one given sector. The interconnection graph among these agents is assumed to contain a directed spanning tree. Similar to classical output regulation problems, there is a virtual exosystem generating a reference signal in which all the agents are required to track cooperatively. Our designed distributed dynamic state/output feedback protocol makes a copy of the reference signal at each agent asymptotically, and then the robust cooperative output regulation problem becomes a robust tracking problem that can be handled by each agent via local information. It turns out that our cooperative protocols are fully distributed. Sufficient conditions on the existence of output synchronization protocols are given along with some discussions on these conditions. Finally, two simulation examples illustrate our design. Copyright © 2016 John Wiley & Sons, Ltd.     

Observer design for the Lur'e differential inclusion system with Markovian jumping parameters

This article deals with the observer design for the Lur'e differential inclusion system with Markovian jumping parameters. The set-valued mapping in the system is assumed to be upper semi-continuous, closed, convex, bounded and monotone. The full-order and reduced-order observers are designed to make the corresponding error systems exponentially stable in the mean square, respectively. The criteria are given to guarantee the existence of both full-order and reduced-order observers. The rotor system is considered as an example to show the validity of the proposed method.