Generalized synchronization of discrete systems

Generalized synchronization of two discrete systems was discussed. By constructing appropriately nonlinear coupling terms, some sufficient conditions for determining the generalized synchronization between the drive and response systems were derived. In a positive invariant and bounded set, many chaotic maps satisfy the sufficient conditions. The effectiveness of the sufficient conditions is illustrated by three examples.

Distance-Stability of Nonlinear Discrete Systems

In this paper,the distance-sability of nonlinear discrete system is investigated by means of the Gauss-Seidel iteration method.Some algebric criteria of the distance-stability are ob-tained.Construction of Lyapunov function is avoided.

LINEAR ACTIVE STRUCTURES AND MODES (Ⅱ) —DISCRETE SYSTEMS AND BEAMS

The basic concepts about the active structures and some attributes of the modes were presented in paper “Liner Active Structures and Modes]( I) ". The characteristics of the active discrete systems and active beams were discussed, especially, the stability of the active structures and the orthogonality of the eigenvectors. The notes about modes were portrayed by a model of a seven-storeyed building with sensors and actuators. The concept of the adjoint active structure was extended from the discrete systems to the beams that were the representations of the continuous structures. Two types of beams with different placements of the measuring and actuating systems were discussed in detail. One is the beam with the discrete sensors and actuators, and the other is the beam with distributed sensor and actuator function. The orthogonality conditions were derived with the modal shapes of the active beam and its adjoint active beam. An example shows that the variation of eigenvalues with feedback amplitude for the homo-configuration and non-homo-configuration active structures.

A Modified Stability Analysis of Two-Dimensional Linear Time Invariant Discrete Systems within the Unity-Shifted Unit Circle

This paper proposes a method to ascertain the stability of two dimensional linear time invariant discrete system within the shifted unit circle which is represented by the form of characteristic equation. Further an equivalent single dimensional characteristic equation is formed from the two dimensional characteristic equation then the stability formulation in the left half of Z-plane, where the roots of characteristic equation f(Z) = 0 should lie within the shifted unit circle. The coefficient of the unit shifted characteristic equation is suitably arranged in the form of matrix and the inner determinants are evaluated using proposed Jury’s concept. The proposed stability technique is simple and direct. It reduces the computational cost. An illustrative example shows the applicability of the proposed scheme.

Certain Algebraic Test for Analyzing Aperiodic Stability of Two-Dimensional Linear Discrete Systems

This paper addresses the new algebraic test to check the aperiodic stability of two dimensional linear time invariant discrete systems. Initially, the two dimensional characteristics equations are converted into equivalent one-dimensional equation. Further Fuller’s idea is applied on the equivalent one-dimensional characteristics equation. Then using the co-efficient of the characteristics equation, the routh table is formed to ascertain the aperiodic stability of the given two-dimensional linear discrete system. The illustrations were presented to show the applicability of the proposed technique.

Noise-Tolerant ZNN-Based Data-Driven Iterative Learning Control for Discrete Nonaffine Nonlinear MIMO Repetitive Systems

Aiming at the tracking problem of a class of discrete nonaffine nonlinear multi-input multi-output(MIMO) repetitive systems subjected to separable and nonseparable disturbances, a novel data-driven iterative learning control(ILC) scheme based on the zeroing neural networks(ZNNs) is proposed. First, the equivalent dynamic linearization data model is obtained by means of dynamic linearization technology, which exists theoretically in the iteration domain. Then, the iterative extended state observer(IESO) is developed to estimate the disturbance and the coupling between systems, and the decoupled dynamic linearization model is obtained for the purpose of controller synthesis. To solve the zero-seeking tracking problem with inherent tolerance of noise,an ILC based on noise-tolerant modified ZNN is proposed. The strict assumptions imposed on the initialization conditions of each iteration in the existing ILC methods can be absolutely removed with our method. In addition, theoretical analysis indicates that the modified ZNN can converge to the exact solution of the zero-seeking tracking problem. Finally, a generalized example and an application-oriented example are presented to verify the effectiveness and superiority of the proposed process.

Model Order Reduction Methods for Discrete Systems via Discrete Pulse Orthogonal Functions

This paper explores model order reduction(MOR)methods for discrete linear and discrete bilinear systems via discrete pulse orthogonal functions(DPOFs).Firstly,the discrete linear systems and the discrete bilinear systems are expanded in the space spanned by DPOFs,and two recurrence formulas for the expansion coefficients of the system’s state variables are obtained.Then,a modified Arnoldi process is applied to both recurrence formulas to construct the orthogonal projection matrices,by which the reduced-order systems are obtained.Theoretical analysis shows that the output variables of the reducedorder systems can match a certain number of the expansion coefficients of the original system’s output variables.Finally,two numerical examples illustrate the feasibility and effectiveness of the proposed methods.

Controllability of Nonlinear Discrete Systems with Degeneracy

This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems,in which linear parts might admit certain degeneracy.By introducing Fredholm operators and coincidence degree theory,sufficient conditions for nonlinear discrete systems to be controllable are presented.In addition,applications are given to illustrate main results.

Robust Control for Interval Type-2 T-S Fuzzy Discrete Systems with Input Delays and Cyber Attacks

This paper focuses on the robust control issue for interval type-2 Takagi-Sugeno(IT2 T-S)fuzzy discrete systems with input delays and cyber attacks.The lower and upper membership functions are first utilized to IT2 fuzzy discrete systems to capture parameter uncertainties.By considering the influences of input delays and stochastic cyber attacks,a newly fuzzy robust controller is established.Afterward,the asymptotic stability sufficient conditions in form of LMIs for the IT2 closed-loop systems are given via establishing a Lyapunov-Krasovskii functional.Afterward,a solving algorithm for obtaining the controller gains is given.Finally,the effectiveness of the developed IT2 fuzzy method is verified by a numerical example.

Optimal Guaranteed Cost Control via Static Output Feedback for Uncertain Discrete Time Systems

Aim To study the optimal guaranteed cost control problem via static output feedback for uncertain linear discrete time systems with norm bounded parameter uncertainty in both the state and the control input matrices of the state space model. Methods\ An upper bound on a quadratic cost index was found for all admissible parameter uncertainties and minimized by using Lagrange multiplier approach. Results and Conclusion\ Sufficient conditions are given for the existence of a controller guaranteeing the closed loop system quadratic stability and providing an optimized bound. A numerical algorithm for solving the output feedback gain is also presented.

Stabilization of Discrete-time 2-D T-S Fuzzy Systems Based on New Relaxed Conditions

This paper is concerned with the problem of stabilization of the Roesser type discrete-time nonlinear 2-D system that plays an important role in many practical applications. First, a discrete-time 2-D T-S fuzzy model is proposed to represent the underlying nonlinear 2-D system. Second, new quadratic stabilization conditions are proposed by applying relaxed quadratic stabilization technique for 2-D case. Third, for sake of further reducing conservatism, new non-quadratic stabilization conditions are also proposed by applying a new parameter-dependent Lyapunov function, matrix transformation technique, and relaxed technique for the underlying discrete-time 2-D T-S fuzzy system. Finally, a numerical example is provided to illustrate the effectiveness of the proposed results.

Stabilization of discrete nonlinear systems based on control Lyapunov functions

The stabilization of discrete nonlinear systems is studied. Based on control Lyapunov functions, a sufficient and necessary condition for a quadratic function to be a control Lyapunov function is given. From this condition, a continuous state feedback law is constructed explicitly. It can globally asymptotically stabilize the equilibrium of the closed-loop system. A simulation example shows the effectiveness of the proposed method.

BOUNDEDNESS IN TERMS OF TWO MEASURES FOR IMPULSIVE DISCRETE SYSTEMS

This paper concerns a class of impulsive discrete systems and offers sufficient conditions of boundedness in terms of two measures for such systems, by employing vector Lyapunov functions and a new comparison principle. The result obtained generalizes those in known literatures.

NECESSARY AND SUFFICIENT CONDITIONS FOR QUADRATIC STABILITY OF UNCERTAIN DISCRETE TIME SYSTEMS

The robust stability analysis of discrete time systems with fast time varying uncertainties is considered in this paper. The necessary and sufficient conditions for quadratic stability are presented. Moreover, the stability robustness index is introduced as the measurement of the stability robustness. For the systems with given uncertain parameter bounds, checking the necessary and sufficient conditions and calculating the stability robust index are converted to solving minimax problems. It is shown that the maximization can be reduced to comparisons between the functional values of the corners when the parameter region is bounded by hyperpolydredon, and any local minimum value in the minimization is exactly the global minimum.

The Stability of Nonlinear Time-Delay Discrete Systems

In this paper, the stability of nonlinear time-delay discrete systems is investigated bymeans of the Gauss-Seidel iteration method. Some algebraic criteria of the stability are obtained. Theconstruction of Lyapunov function is avoided.

A simple approach for stability margin of discrete systems

In this paper, a technique is presented to determine the stability margin of the discrete systems using recursive algorithm for power of companion matrix and Gerschgorin Theorem and hence sufficient condition of stability is obtained. The method is illustrated with an example and it is compared with other methods proposed in the literature. The results have applications in the filter design.

Design of fuzzy sliding mode controller for SISO discrete-time systems

According to a class of nonlinear SISO discrete systems, the fiizzy sliding mode control problem is considered. Based on Takagi-Sugeno fuzzy model method, a fuzzy model is designed to describe the local dynamic performance of the given nonlinear systems. By using the sliding mode control approach, the global controller is constructed by integrating all the local state controllers and the global supervisory sliding mode controller. The tracking problem can be easily dealt with by taking advantage of the combined controller,and the robustness performance is improved finally. A simulation example is given to show the effectiveness and feasibility of the method proposed.

Stability Analysis of Uncertain Discrete Time-Delay Control Systems

Based on Lyapunov stability theory, a less conservative sufficient condilions for the stabih＇lies of uncertain discrete delayindependent and delay-dependent control systems are obtained by using the linear matrix inequality （LMI） approach. Judgement of the stability of time-delay systems is transformed to judgement of the feasible solution of an LMI, and hence is solved by use of MATLAB. Numerical simulations verify the validity of the proposed method.

POSITIVE PERIODIC SOLUTIONS OF THE FIRST-ORDER SINGULAR DISCRETE SYSTEMS

In this paper, we study two kinds of first-order singular discrete systems. By the fixed point index theory, we investigate the existence and multiplicity of positive periodic solutions of the systems.

Robust Optimal H∞ Control for Uncertain 2-D Discrete State-Delayed Systems Described by the General Model

This paper investigates the problem of robust optimal H∞ control for uncertain two-dimensional (2-D) discrete state-delayed systems described by the general model (GM) with norm-bounded uncertainties. A sufficient condition for the existence of g-suboptimal robust H∞ state feedback controllers is established, based on linear matrix inequality (LMI) approach. Moreover, a convex optimization problem is developed to design a robust optimal state feedback controller which minimizes the H∞ noise attenuation level of the resulting closed-loop system. Finally, two illustrative examples are given to demonstrate the effectiveness of the proposed method.