Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

Asymptotical synchronization for chaotic Lur’e systems using sampled-data control

This paper presents a new method for the asymptotical synchronization of two identical chaotic Lur’e systems using sampling control. The method is based on a new Lyapunov–Krasovskii functional (LKF) in the framework of an input delay method. Compared with existing works, the new LKF makes full use of the information on the nonlinear part of the system and introduces a novel term, which waives the common positive requirement of each LKF term to guarantee the positive of the whole LKF. A typical Chua’s circuit is given to verify the effectiveness of the proposed method.     

A new method for quantized sampled-data synchronization of delayed chaotic Lur’e systems

We design a quantized sampled-data controller for synchronization of delayed chaotic Lur’e systems. A new approach, extended Wirtinger-inequality-based Lyapunov–Krasovskii functional, is firstly proposed. This approach grasps more sampling information by introducing more free matrices in comparison with some existing methods. Using the system information at the dynamic partitioning point, a zero equality is formulated to fully utilize the inner sampling information. Based on the new approach and zero equality, some novel synchronization criteria are established. In the meantime, the desired quantized sampled-data control gain is obtained with larger sampling period than those in the existing works. Finally, two numerical examples illustrate the merits of the method.     

Delay-range-dependent synchronization criterion for Lur’e systems with delay feedback control

In this paper, a master–slave synchronization scheme is investigated by using feedback control mechanism with time-varying delay. The time-delay is assumed to be a time-varying continuous function belonging to a given range. By constructing a novel Lyapunov–Krasovskii functional, which includes the information of the range, new delay-range-dependent synchronization criterion is established in term of LMI. It is shown that the new criterion improve some of the previous results in the earlier references. Simulation example is given to show the effectiveness and less conservatism of the proposed criterion.     

Type-2 T–S fuzzy impulsive control of nonlinear systems

A novel impulsive control approach based on interval Type-2 T–S fuzzy model has been presented for nonlinear systems in this paper. This approach makes up for the drawback of Type-1 fuzzy impulsive control, which cannot fully handle the uncertainties in describing the complex nonlinear systems by Type-1 fuzzy membership functions and cannot give rigorous fuzzy rules. Further more, this approach uses the “broad band” effect of the Type-2 membership functions to solve the noise of training data and exterior disturbance of the Type-1 fuzzy impulsive control. By using Lyapunov theory and Lipschitz condition, which is combined with integrated approaches such as comparison methods and linear matrix inequalities, the Type-2 fuzzy impulsive controller is designed and the general asymptotical stability analysis of the systems is given. Finally, the simulation of the inverted pendulum model demonstrates the validity and superiority of the proposed method by easily determining the membership functions and choosing minimum number of fuzzy rules and the method can handle random disturbance and data uncertainties very well.     

Impulsive robust fault-tolerant feedback control for chaotic Lur’e systems

This paper considers the problem of impulsive robust fault-tolerant feedback control for chaotic Lur’e systems. The sufficient condition of the uncertain Lur’e systems possessing integrity against actuator failures is given by using linear matrix inequalities (LMIs), and at the same time, the closed-loop system with impulsive effects is globally asymptotically stable, and it has robustness of parameter uncertainties. An example is given to illustrate obtained result.     

Dupire It^{o}’s formula for the exponential synchronization of stochastic semi-Markov jump systems with mixed delay under impulsive control

This paper emphasizes the exponential synchronization for a class of stochastic semi-Markov jump systems with mixed delay via stochastic hybrid impulsive control. The impulsive sequence includes synchronous and asynchronous impulses with the impulsive gains being a sequence of stochastic variables. Inspired by the idea of average, a concept of ``average stochastic impulsive gain\" is used to qualify the impulse intensity. Our approach expands Dupire functional It^{o}$\"s formula to the stochastic semi-Markov jump systems with mixed delay for the first time. Moreover, in view of the established Lyapunov functional, graph theory, and stochastic analysis theory, some exponential synchronization criteria for the systems are derived. The theoretical results are applied to a class of Chua\"s circuit systems with semi-Markov jump and mixed delay. Some synchronization criteria for the circuit systems are provided. The simulation results verify the effectiveness of the theoretical results.     

Synchronization of chaotic Lur’e systems with quantized sampled-data controller

In this paper, we consider the master–slave synchronization problem of chaotic Lur’e systems. It is assumed that only quantized sampled measurements are available for the controller. By modeling the synchronization error system as an input-delay system and constructing a new Lyapunov functional, a new sufficient condition and feedback controller design method for global exponential asymptotical synchronization of master and slave system are obtained. The proposed approach has taken the feature of sample-induced delay into consideration and simulation results show the less conservativeness.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

Design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities

In this paper, design PD controller for master–slave synchronization of chaotic Lur’e systems with sector and slope restricted nonlinearities is presented. A new synchronization criterion is proposed based on Lyapunov functions with quadratic form of states and nonlinear functions of the systems. Sector and slope bounds are employed to the Lyanunov–Krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained. The criteria is given in terms of linear matrix inequalities (LMIs) by using Finsler’s lemma. A numerical example is provided to illustrate the effectiveness of the method.     

Projective synchronization of chaotic fractional-order energy resources demand–supply systems via linear control

A fractional-order energy resources demand–supply system is proposed. A projective synchronization scheme is proposed as an extension on the synchronization scheme of Odibat et al. (2010). The scheme is applied to achieve projective synchronization of the chaotic fractional-order energy resource demand–supply systems. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

Global analysis of an impulsive delayed Lotka–Volterra competition system

In this paper, a retarded impulsive n-species Lotka–Volterra competition system with feedback controls is studied. Some sufficient conditions are obtained to guarantee the global exponential stability and global asymptotic stability of a unique equilibrium for such a high-dimensional biological system. The problem considered in this paper is in many aspects more general and incorporates as special cases various problems which have been extensively studied in the literature. Moreover, applying the obtained results to some special cases, I derive some new criteria which generalize and greatly improve some well known results. A method is proposed to investigate biological systems subjected to the effect of both impulses and delays. The method is based on Banach fixed point theory and matrix’s spectral theory as well as Lyapunov function. Moreover, some novel analytic techniques are employed to study GAS and GES. It is believed that the method can be extended to other high-dimensional biological systems and complex neural networks. Finally, two examples show the feasibility of the results.     

Robust L ∞ reliable control for impulsive switched nonlinear systems with state delay

This paper investigates the problem of robust L _∞ reliable control for a class of uncertain impulsive switched nonlinear systems with time-delay in the presence of actuator failure. Based on the dwell time approach, we firstly obtain a sufficient condition of exponential stability for the impulsive switched nonlinear system with time-delay, and L _∞ performance for the considered system is also analyzed. Then, based on above results, a state feedback controller, which guarantees the exponential stability with L _∞ performance of the corresponding closed-loop system, is constructed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Robust H∞ nonlinear modeling and control via uncertain fuzzy systems

In theory, an Algebraic Riccati Equation (ARE) scheme applicable to robust H^∞ quadratic stabilization problems of a class of uncertain fuzzy systems representing a nonlinear control system is investigated. It is proved that existence of a set of solvable AREs suffices to guarantee the quadratic stabilization of an uncertain fuzzy system while satisfying H^∞-norm bound constraint. It is also shown that a stabilizing control law is reminiscent of an optimal control law found in linear quadratic regulator, and a linear control law can be immediately discerned from the stabilizing one. In practice, the minimal solution to a set of parameter dependent AREs is somewhat stringent and, instead, a linear matrix inequalities formulation is suggested to search for a feasible solution to the associated AREs. The proposed method is compared with the existing fuzzy literature from various aspects.     

Global exponential stability of impulsive Cohen–Grossberg neural network with time-varying delays

In this paper, the impulsive Cohen–Grossberg neural network model with time-varying delays is considered. Applying the idea of vector Lyapunov function, M-matrix theory and inequality technique, several new sufficient conditions are obtained to ensure global exponential stability of equilibrium point for impulsive Cohen–Grossberg neural network with time-varying delays. These results generalize a few previous known results and remove some restrictions on the neural network. An example is given to show the effectiveness of the obtained results. It is believed that these results are significant and useful for the design and applications of the Cohen–Grossberg neural network.     

Resilient dissipative dynamic output feedback control for uncertain Markov jump Lur’e systems with time-varying delays

The resilient dissipative dynamic output feedback control problem for a class of uncertain Markov jump Lur’e systems with piecewise homogeneous transition probabilities and time-varying delays in the discrete-time domain are examined in this study. The designed controller can tolerate additive uncertainties in the controller gain matrix, which result from controller implementations. The time-varying delays are also supposed to be mode-dependent with lower and upper bounds known a priori. By constructing a Lyapunov–Krasovskii functional candidate, the sufficient conditions regarding the existence of desired resilient dissipative controllers are obtained in terms of linear matrix inequalities, thereby ensuring that the resulting closed-loop system is stochastically stable and strictly dissipative. Two numerical examples were established to illustrate the effectiveness of the proposed theoretical results.     

Finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay via periodically intermittent control

In this paper, the issue of finite-time lag synchronization of coupled reaction–diffusion systems with time-varying delay (CRDSTD) is considered. A periodically intermittent controller is designed such that drive system and corresponding response system can achieve finite-time lag synchronization. By using graph theory and Lyapunov method, two sufficient criteria are presented to guarantee the finite-time lag synchronization of CRDSTD. Moreover, the time of achieving lag synchronization of CRDSTD is estimated. Finally, a numerical example is given to show the effectiveness of the proposed results.     

A novel feedforward–feedback suboptimal control of linear time-delay systems

This paper presents a new approach for solving the optimal control problem of linear time-delay systems with a quadratic cost functional. In this approach, a method of successive substitution is employed to convert the original time-delay optimal control problem into a sequence of linear time-invariant ordinary differential equations (ODEs) without delay and advance terms. The obtained optimal control consists of a linear state feedback term and a forward term. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving recursively the above-mentioned sequence of linear non-delay ODEs. A fast-converging iterative algorithm for this purpose is presented which provides a promising possible reduction of computational efforts. Numerical examples demonstrating the efficiency, simplicity and high accuracy of the suggested technique have been included. Simulation results reveal that just a few iterations of the proposed algorithm are required to find an accurate enough feedforward–feedback suboptimal control.     

Synchronization Criterion for Lur’e Systems via Delayed PD Controller

In this paper, the effects of a time varying delay on a chaotic drive-response synchronization are considered. Using a delayed feedback proportional-derivative (PD) controller scheme, a delay-dependent synchronization criterion is derived for chaotic systems represented by the Lur’e system with sector and slope restricted nonlinearities. The derived criterion is a sufficient condition for the absolute stability of the error dynamics between the drive and the response systems. By the use of a convex representation of the nonlinearity and the discretized Lyapunov-Krasovskii functional, stability condition is obtained via the LMI formulation. The condition represented in the terms of linear matrix inequalities (LMIs) can be solved by the application of convex optimization algorithms. The effectiveness of the work is verified through numerical examples.