Absolute Exponential Stability Research Articles

Absolute Exponential Stability Criteria for Some Classes of Nonlinear Time-Varying Systems with Delays and Sector Nonlinearities

Absolute Exponential Stability Criteria for Some Classes of Nonlinear Time-Varying Systems with Delays and Sector Nonlinearities

Absolute exponential stability of switching Lurie systems with time-varying delay via dwell time switching

This paper addresses the absolute exponential stability analysis for switching Lurie systems with time-varying delay. With the switching time-varying Lyapunov functions technique, sufficient conditions in terms of linear-matrix inequalities are presented to ensure the absolute exponential stability of switching Lurie systems with time-varying delay under mode-dependent dwell time switching signals. Furthermore, the conservatism of the proposed method for stability analysis is studied. Simulation examples illustrate the validity of the obtained results.

Exponential stability analysis for a class of switched nonlinear time-varying functional differential systems

This paper proposes a unified approach for studying global exponential stability of a general class of switched systems described by time-varying nonlinear functional differential equations. Some new delay-independent criteria of global exponential stability are established for this class of systems under arbitrary switching which satisfies some assumptions on the average dwell time. The obtained criteria are shown to cover and improve many previously known results, including, in particular, sufficient conditions for absolute exponential stability of switched time-delay systems with sector nonlinearities. Some simple examples are given to illustrate the proposed method.

Absolute Exponential Stability for Switching Positive Systems With Time-Delay Based on the Φ-Dependent ADT Strategy

The absolute exponential stability problem for the continuous- and discrete-time, respectively, switching positive nonlinear time-delay system is investigated under the nonlinearity of systems satisfying a certain sector condition in this paper. Two multiple co-positive Lyapunov-Krasovskii functionals (MCLKF) of such systems are constructed for the continuous- and discrete-time contexts respectively. By using the MCLKF and the Φ-dependent average dwell time switching strategy, more general stability conditions compatible with some existing works are established. Furthermore, the obtained results are generalized to other types of systems. A numerical example, finally, implies the validity and significance of the results proposed.

Absolute exponential L1-gain analysis and synthesis of switched nonlinear positive systems with time-varying delay

This paper investigates absolute exponential stability and stabilization with L1-gain performance of switched nonlinear positive systems with time-varying delay. Nonlinear functions considered in the paper are located in a sector field and the time-varying delay is unknown but bounded. The notion of absolute exponential L1 stability of switched nonlinear positive systems is first introduced. A nonlinear Lyapunov–Krasovskii functional is constructed for the underlying systems. By using the Lyapunov–Krasovskii functional, a sufficient condition for absolute exponential L1 stability of the systems is established in terms of linear programming. Then, absolute exponential L1 control synthesis of the systems is addressed. A feedback control law containing nonlinear functions and a state-feedback control law are designed, respectively. By comparing with existing results, it is shown that the present design is less conservative. Finally, an illustrative example is provided to illustrate the effectiveness of the results.

Absolute exponential stability of switched nonlinear time-delay systems

This paper investigates absolute exponential stability of switched nonlinear time-delay systems in both continuous-time and discrete-time contexts. The nonlinearities of the systems satisfy a certain sector condition. First, an improved Lyapunov–Krasovskii functional of switched nonlinear time-delay systems in the continuous-time form is constructed. By using the multiple Lyapunov–Krasovskii functionals and average dwell time switching approaches, two less conservative sufficient conditions for the absolute exponential stability of the systems with single/multiple delays are established, respectively. Then, the obtained results are extended to discrete-time systems. All proposed conditions are described via linear programming. By some comparisons, it is shown that the results in the paper are less conservative than those in the literature. Finally, two examples are given to show the effectiveness of the theoretical findings.

Absolutely Exponential Stability and Temperature Control for Gas Chromatograph System Under Dwell Time Switching Techniques.

This paper provides a design strategy for temperature control of the gas chromatograph. Usually gas chromatograph is modeled by a simple first order system with a time-delay, and a proportion integration (PI) controller is widely used to regulate the output of the gas chromatograph to the desired temperature. As the characteristics of the gas chromatograph varies at the different temperature range, the single-model based PI controller cannot work well when output temperature varies from one range to another. Moreover, the presence of various disturbance will further deteriorate the performance. In order to improve the accuracy of the temperature control, multiple models are used at the different temperature ranges. With a PI controller designed for each model accordingly, a delay-dependent switching control scheme using the dwell time technique is proposed to ensure the absolute exponential stability of the closed loop. Experiment results demonstrate the effectiveness of the proposed switching technique.

$p$th Moment absolute exponential stability of stochastic control system with Markovian switching

In this paper we discuss the $p$th moment absolute exponential stability of stochastic control system with Markovian switching. We first give a new concept of $p$th moment absolute exponential stability, then we establish some theorems under different hypotheses to guarantee the system $p$th moment absolutely exponentially stable. These sufficient conditions in our theorems are algebraic criteria in terms of matrix inequalities, and we introduce an $M$-method with MATLAB to compute them. Finally, some examples are given to illustrate our results.

Absolute stability of the Kirchhoff string with sector boundary control

This paper addresses the stability problem of the nonlinear Kirchhoff string with nonlinear boundary control. The nonlinear boundary control is the negative feedback of the transverse velocity of the string at one end, which satisfies a sector constraint. Employing the integral type multiplier method, we establish explicit absolute exponential stability of the Kirchhoff string system.

Absolute exponential stability and stabilization of switched nonlinear systems

This paper is concerned with the problems of absolute exponential stability and stabilization for a class of switched nonlinear systems whose system matrices are Metzler. Nonlinearity of the systems is constrained in a sector field, which is bounded by two odd symmetric piecewise linear functions. Multiple Lyapunov functions are introduced to deal with the stability of such nonlinear systems. Compared with some existing results obtained by the common Lyapunov function approach in the literature, the conservatism of our results is reduced. All present conditions can be solved by linear programming. Furthermore, the absolute exponential stabilization for the considered systems is designed by the state-feedback and average dwell time switching strategy. Two examples are also given to illustrate the validity of the theoretical findings.

Alternative Stability Conditions for Switched Discrete Time Linear Systems

Alternative necessary and sufficient conditions for absolute exponential stability are presented, for switched discrete–time linear systems. To reach these results, we exploit concepts from set theory and in specific the forward reachability set mappings. The developed theorem can be utilized to construct iterative numerical procedures for verifying stability. Two examples illustrate the effectiveness of the proposed methodology.

Group of Continuous Time Recurrent Neural Networks

This paper explain a different type process for absolute exponential stability (AEST) of a group of continuous time recurrent neural networks with locally Lipschitz continuous and monotone non decreasing activation function. The result extends and improves the existing the analysis of absolute stability (ABST) and absolute exponential stability (AEST). Keywords: Neural network; absolute exponential stability; global exponentially stability.

A Gel’fand-type spectral radius formula and stability of linear constrained switching systems

Using ergodic theory, in this paper we present a Gel’fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup S+ restricted to a subset that need not carry the algebraic structure of S+ This generalizes the Berger–Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to S.

Absolute exponential stability criteria for a class of nonlinear time-delay systems

This paper is concerned with the problem of absolute exponential stability analysis for a class of continuous-time systems with sector-bounded nonlinearities and interval time-varying delays. Absolute exponential stability criteria, which depend not only on the delay range but also on the decay rate, are derived by using a novel Lyapunov functional. These criteria are expressed in the form of linear matrix inequalities (LMIs) and they can easily be checked. By solving these LMIs, estimates of the exponential decay rate and decay coefficient are obtained. A numerical example is provided to demonstrate the effectiveness of the proposed method.

Absolute Exponential Stability of Recurrent Neural Networks With Generalized Activation Function

In this paper, the recurrent neural networks (RNNs) with a generalized activation function class is proposed. In this proposed model, every component of the neuron's activation function belongs to a convex hull which is bounded by two odd symmetric piecewise linear functions that are convex or concave over the real space. All of the convex hulls are composed of generalized activation function classes. The novel activation function class is not only with a more flexible and more specific description of the activation functions than other function classes but it also generalizes some traditional activation function classes. The absolute exponential stability (AEST) of the RNN with a generalized activation function class is studied through three steps. The first step is to demonstrate the global exponential stability (GES) of the equilibrium point of original RNN with a generalized activation function being equivalent to that of RNN under all vertex functions of convex hull. The second step transforms the RNN under every vertex activation function into neural networks under an array of saturated linear activation functions. Because the GES of the equilibrium point of three systems are equivalent, the next stability analysis focuses on the GES of the equilibrium point of RNN system under an array of saturated linear activation functions. The last step is to study both the existence of equilibrium point and the GES of the RNN under saturated linear activation functions using the theory of M-matrix. In the end, a two-neuron RNN with a generalized activation function is constructed to show the effectiveness of our results.

Comments on “Absolute exponential stability of a class of neural networks with unbounded delay”

Recently, Zhang, Suda and Iwasa [Zhang, Jiye, Suda, Yoshihiro, & Iwasa, Takashi (2004). Absolute exponential stability of a class of neural networks with unbounded delay, Neural Network, 17, 391-397] established the following: In this letter, we would like to point out that the necessary condition in above Result given by Zhang et al. (2004) is incorrect in general.

Absolute exponential stability analysis of delayed bi-directional associative memory neural networks☆

Abstract The problem of absolute exponential stability for delayed bi-directional associative memory neural networks with time delay is investigated via Lyapunov stability theory. A new sufficient condition ensuring existence and uniqueness of equilibrium and its absolute exponential stability is derived. An illustrative example is given to demonstrate the effectiveness of the obtained results.

Necessary and sufficient condition for the absolute exponential stability of a class of neural networks with finite delay

In this Letter, a necessary and sufficient condition is established for ascertaining the absolute exponential stability for a class of finite delayed neural networks with connection matrices A having nonnegative off-diagonal elements, delayed feedback matrix A τ having nonnegative elements and strictly increasing and Lipschitz activation functions.

Absolute exponential stability of a class of recurrent neural networks with multiple and variable delays

In this paper, we derive some new conditions for absolute exponential stability (AEST) of a class of recurrent neural networks with multiple and variable delays. By using the Holder's inequality and the Young's inequality to estimate the derivatives of the Lyapunov functionals, we are able to establish more general results than some existing ones. The first type of conditions established involves the convex combinations of column-sum and row-sum dominance of the neural network weight matrices, while the second type involves the p-norm of the weight matrices with p∈[1,+∞].

Absolute exponential stability analysis of delayed neural networks

In this Letter, we investigate the absolute exponential stability of a class of delayed neural networks. A new sufficient condition ensuring existence and uniqueness of equilibrium and its absolute exponential stability is derived. When the neural network model is simplified to one without delays, the present condition is reduced to the well-known additive diagonal stability condition of the interconnection weight matrix, which was previously established and proven to be general enough for ensuring stability of neural networks without delays in the literature. Thus, our condition generalizes the additive diagonal stability condition to the case of neural networks with delays.