Time-varying Lyapunov Matrix Research Articles

Stability and stabilization analysis of periodic switching stochastic systems based on Lyapunov function with continuous time-varying matrix polynomials

In this paper, a Lyapunov function with continuous time-varying Lyapunov matrix polynomials is constructed to ensure the exponential stability in mean square of a periodic switched linear stochastic system, where the Lyapunov matrix of this Lyapunov function is no longer a constant matrix, but a continuous time-varying matrix polynomial. Then, based on this stability study, we design an aperiodic intermittent control for unstable periodic switching stochastic systems to solve the influence of unstable subsystems and switching rules on stability, and the corresponding control gains are continuous time-varying matrix polynomial. Finally, an example is demonstrated to illustrate the effectiveness of the proposed control strategy.

Finite-time zeroing neural networks with novel activation function and variable parameter for solving time-varying Lyapunov tensor equation

Time-varying Lyapunov tensor equation (TV-LTE) is an extension of time-varying Lyapunov matrix equation (TV-LME), which represents more dimensions of data. In order to solve the TV-LTE more effectively, this paper proposes two improved zeroing neural network (ZNN) models based on a novel activation function and variable parameter, which have shorter convergence time and computation time. The novel activation function is composed of an exponential function and a sign-bi-power (SBP) function, which is mentioned as the exponential SBP (ESBP) function. Then, based on the ESBP activation function and the standard ZNN design method, an ESBP zeroing neural network (ES-ZNN) model is first provided. In addition, considering the relationship among the error matrix, design parameter and computational efficiency, this paper further designs an exponential parameter that varies dynamically with time and the error matrix. Replacing the fixed parameter with the proposed exponential variable parameter, an exponentially variable parameter ES-ZNN (EVPES-ZNN) model is provided to enhance the computational efficiency and convergence performance of the ES-ZNN model. Furthermore, the upper bounds on convergence time of such two ZNN models are theoretically calculated. Simulation experiments demonstrate the theoretical conclusion that the ES-ZNN and EVPES-ZNN models are able to solve the TV-LTE in finite-time.

Observer-based tracking control design for periodic piecewise time-varying systems

In this study, the problem of robust tracking problem for continuous-time periodic piecewise time-varying delay systems with disturbances and uncertainties has been examined by using observer-based H∞ tracking control protocol. Precisely, the state space representation of the system is formulated to reconstruct the unmeasurable states via the available informations of input/output dynamics. Based on the suitable Lyapunov-Krasovskii functional with periodic piecewise time-varying Lyapunov matrix, a set of sufficient conditions in the form of linear matrix inequalities is derived which affirms the robust asymptotic tracking performance and satisfactory disturbance attenuation for the addressed model. Particularly, the matrix polynomial approach is utilized which makes the analysis to be more flexible and amenable to convex optimization tool. Accordingly, the period time-varying controller and observer gains are computed through the feasible solution of derived matrix inequalities. Conclusively, the simulation results are presented to confirm the efficiency and applicability of the theoretical outcomes.

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.

Non-fragile H ∞ control of periodic piecewise time-varying systems based on matrix polynomial approach

This paper investigates the problem of non-fragile control for periodic piecewise time-varying systems. Based on a Lyapunov function with continuous time-varying Lyapunov matrix polynomial, and combining with the positiveness and negativeness properties of matrix polynomials, the performance analysis is first accomplished. Then consider two types of controller gain perturbations that are formulated by time-varying matrix parameters and norm-bounded uncertainties. The additive and multiplicative non-fragile controllers to guarantee the performance of the system are formed, of which the controller gain could be solved with linear matrix inequalities directly. The designed non-fragile controller is desirable in applications. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

Finite-Time Non-Fragile Extended Dissipative Control of Periodic Piecewise Time-Varying Systems

This paper studies the problem of finite-time non-fragile extended dissipative control for the periodic piecewise time-varying systems. First, based on the time-varying Lyapunov matrix and the matrix polynomial condition, a sufficient condition of finite-time boundedness for periodic piecewise time-varying subsystem is derived, and the finite-time extended dissipative performance is then analyzed. Furthermore, considering two types of time-varying controller perturbations, the non-fragile controllers are developed, which can guarantee the finite-time boundedness of periodic piecewise time-varying systems and satisfy the extended dissipative performance at the same time. Finally, numerical examples demonstrate the effectiveness of the proposed methods.

New Conditions of Analysis and Synthesis for Periodic Piecewise Linear Systems With Matrix Polynomial Approach

In this paper, new conditions of the stability, stabilization and L 2 -gain performance of periodic piecewise systems are proposed. Both the continuous and discontinuous Lyapunov functions with dwell-time related time-varying Lyapunov matrix polynomial are adopted, and methods guaranteeing the positive and negative definiteness of a matrix polynomial are introduced. Exponential stability conditions are derived based on continuous and discontinuous Lyapunov functions, respectively. A stabilizing controller with time-varying controller gain is designed with continuous Lyapunov function and the weighted L 2 -gain performance based on the discontinuous Lyapunov matrix polynomial is studied as well. Numerical examples are used to verify the effectiveness of the proposed methods.

Stability analysis problems of periodic piecewise polynomial systems

This paper studies the stability analysis problems of periodic piecewise systems, in which subsystems are given in the time-varying polynomial forms. A Lyapunov function with continuous time-varying Lyapunov matrix is adopted, which relaxes constraints on the variation of Lyapunov function in each subsystem. Using the time interval information, the exponential stability and stabilizing controller synthesis are studied. The results provide a possible alternative method to study the general periodic time-varying systems, which may further support the analysis and synthesis of general periodic systems. The effectiveness of the proposed method is validated and illustrated via numerical examples.

Stability and $L_2$ Synthesis of a Class of Periodic Piecewise Time-Varying Systems

In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function

In this paper, the H∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method.

Event-Triggered Controller Design With Varying Gains for T-S Fuzzy Systems.

This paper investigates the problem of designing event-triggered (ET) fuzzy controllers with varying gains for T-S fuzzy systems. New types of fuzzy controllers named ET fuzzy controllers with varying gains are proposed, in which the gains are automatically updated according to an online iteration algorithm at different triggering instants. By designing discontinuous Lyapunov function with time-varying Lyapunov matrix, sufficient conditions in terms of linear matrix inequalities (LMIs) are obtained to ensure the stability of the closed-loop system. It is shown that the proposed method achieves better system performances than the existing design methods with a quadratic Lyapunov function or ET fuzzy controllers with fixed gains. An example is provided to illustrate the effectiveness of the proposed method.

Time-varying H∞ controller design for periodic piecewise vibration systems

In this article, a kind of time-varying H∞ controller, which could be easily obtained by solving linear matrix inequalities, is proposed to attenuate the vibration of periodic piecewise vibration systems. The Lyapunov function with continuous time-varying Lyapunov matrix is adopted to develop a general condition of H∞ performance index for periodic piecewise vibration systems. An H∞ controller with continuous time-varying controller gain is designed to attenuate the system vibration. A numerical example is used to verify the merit of the proposed controller.

H∞ control of periodic piecewise vibration systems with actuator saturation

In this paper, an H∞ controller with actuator saturation consideration is proposed to attenuate the vibration of periodic piecewise vibration systems. Based on a continuous Lyapunov function with a time-varying Lyapunov matrix, the H∞ performance index of periodic piecewise vibration systems is studied first. On the basis of the obtained H∞ criterion, the conditions of designing a state-feedback active vibration controller are proposed in matrix inequality form with actuator saturation taken into account. Because of the nonconvexity of the conditions, a corresponding algorithm to compute the controller gain is developed as well. A representative numerical example is used to verify the effectiveness of the proposed method.

Stability, stabilization and [formula omitted]-gain analysis of periodic piecewise linear systems

In this paper, the stability, stabilization and L2-gain problems are investigated for periodic piecewise linear systems, in which not all subsystems are Hurwitz. First, some sufficient and necessary conditions for the exponential stability are established. By employing a discontinuous Lyapunov function with time-varying Lyapunov matrix, stabilization and L2-gain conditions of periodic piecewise linear systems are proposed by allowing the corresponding Lyapunov function to be possibly non-monotonically decreasing over a period. A state-feedback periodic piecewise controller is developed to stabilize the system, and the corresponding algorithm is proposed to compute the controller gain. The L2-gain criteria with continuous time-varying Lyapunov matrix and piecewise constant Lyapunov matrices are studied as well. Numerical examples are given to show the validity of the proposed techniques.

Online Solution of Time-Varying Lyapunov Matrix Equation by Zhang Neural Networks

By constructing a matrix-valued unbounded error-function, this paper develops and exploits a new type of recurrent neural networks, named as Zhang neural networks, for the time-varying Lyapunov matrix equation with accuracy and effectiveness. In general, a scalar-valued norm-based energy function is defined for the design and development of the conventional gradient-based neural networks, which could only solve the time-invariant matrix equation exactly. Comparison with some recent patents on the neural networks designed originally for the time-invariant problems solving, the patents relevant to Zhang neural networks is designed for the solution of time-varying problems based on the matrix/ vector-valued error function. An illustrative example substantiates that the presented Zhang neural networks can effectively solve such matrix equation with time-varying coefficients, while the conventional gradient-based neural networks could only approximately approach to the theoretical solution.