Exponential Stability Analysis Research Articles

A novel result on stability analysis for uncertain neutral stochastic time-varying delay systems

This paper is concerned with the mean-square exponential stability analysis for uncertain neutral linear stochastic time-varying delay systems. By Lyapunov–Krasovskii theory and linear matrix inequality method, under the generalized Finsler lemma (GFL) framework, delay-dependent mean-square exponential stability criteria are established without involving model transformation, cross-terms bounding technique or additional free-weighting matrix. Moreover, GFL is also employed to obtain stability criteria for a class of uncertain linear stochastic neutral systems with different discrete and neutral delays. Numerical examples are presented to verify that the proposed approach is both less conservative and less computationally complex than the existing results.

Global Exponential Stability Analysis of Discrete‐Time Genetic Regulatory Networks with Time Delays

AbstractThis paper focuses on the study of global robust exponential stability of discrete‐time genetic regulatory networks (GRNs) with time‐invariant/time‐varying delays and parameter uncertainties. Many existing results on this problem are based on the linear matrix inequality (LMI) approach, which needs to verify whether there exists a feasible solution of a set of LMIs. Along with the increase in the number of genes, dimensions of LMIs will increase accordingly, which will lead to a large amount of calculation. Based on M‐matrix theory, sufficient conditions ensuring the global robust exponential stability of a class of discrete‐time GRNs with time‐invariant/time‐varying delays and parameter uncertainties are presented. These given conditions are to check whether a constructed constant matrix is a nonsingular M‐matrix. Simulation results of several examples are given to demonstrate the validity of the proposed method.

Robustness analysis of global exponential stability of non‐linear systems with time delays and neutral terms

The global stability of non-linear dynamical systems has been investigated extensively in recent decades. It is well known that time delay and neutral term could derail the stability of non-linear systems. This study presents new results on the robustness of the global exponential stability of non-linear systems with respect to time delay and neutral term. Given globally exponentially stable non-linear systems, the problems to be addressed herein are how much time delay and neutral term contraction coefficient are allowed so that the non-linear systems can remain to be globally exponentially stable, in the presence of time delay and neutral term. Upper bounds of allowable time delay and neutral term contraction coefficient will be derived for non-linear systems to sustain their global exponential stability. A numerical example is provided to illustrate the results.

Exponential stability analysis of linear systems with multiple successive delay components

A general class of linear systems with multiple successive delay components is considered in this article. The delays are assumed to vary in intervals, and delay-dependent exponential stability conditions are derived in terms of linear matrix inequalities. To reduce conservativeness, a new Lyapunov–Krasovskii functional is designed to contain more complete state information, so that a derivation procedure with time-varying delays treated as uncertain parameters can be adopted. Usage of slack variables and inequalities are refrained as much as possible when bounds on the Lyapunov derivative are sought. The stability criteria are tested by two popular numerical examples, with less conservative results obtained in all the checked cases. Besides, a practical application of the derived conditions is illustrated.

Exponential stability of discrete-time recurrent neural networks with time-varying delays in the leakage terms and linear fractional uncertainties

A class of discrete-time recurrent neural networks model is presented with the introduction of time-varying delays in the leakage terms and linear fractional uncertainties in parameters. The exponential stability analysis problem is studied, for the first time, for such kind of neural networks. In terms of linear matrix inequality approach, some delay-dependent stability criteria are established for the considered neural networks via a Lyapunov–Karasovskii functional. Three numerical examples are given to show the effectiveness of the theoretical results.

Exponential Stability Analysis for Discrete-Time Singular Systems with Randomly Occurring Delay

This paper investigates the problem of exponential stability analysis for discrete-time singular systems with time-varying delay. The delay enters into the singular systems in a random way, and such randomly occurring delay (ROD) obeys a certain Bernoulli distributed white noise sequence. By employing the linear matrix inequality (LMI) approach, a sufficient condition is established to ensure the considered systems exponentially mean-square stable. Two numerical examples are given to show the usefulness of the obtained result.

Exponential stability analysis for discrete-time impulsive delay neural networks with and without uncertainty

The purpose of this paper is to investigate the exponential stability for discrete-time impulsive delay neural networks and the robust exponential stability for discrete-time impulsive delay neural networks with uncertainty. By using Lyapunov functionals, first some new results on exponential stability for neural networks without uncertainty are presented, and then some results on robust exponential stability for neural networks with uncertainty are provided. Both the stability results that impulses act as perturbations and the stability results that impulses act as stabilizer are given. The obtained results have the virtue that they can deal with neural networks with any fixed time delay. Moreover, the impulsive interval is larger than 2 or the time delay is not needed in the main results. Some examples together with their simulations are also presented to show the effectiveness and the advantage of the obtained results.

Exponential stability for positive systems with bounded time-varying delays and static output feedback stabilization

This paper investigates the problem of exponential stability analysis and static output feedback stabilization for discrete-time and continuous-time positive systems with bounded time-varying delays. Based on the relationship between the solution to the system with time-varying delay and that to the corresponding system with constant delay under specific conditions, the equivalence between the α-exponential stability of such two types of systems is established. Then some necessary conditions and sufficient conditions are provided for α-exponential stability of positive systems with bounded time-varying delays. It is shown that, for such systems, the exponential stability with given decay rate is closely related to the bound of the delay. Then by using the singular value decomposition approach, sufficient conditions for the existence of static output feedback controllers are established in terms of linear programming (LP) problems. Some illustrative examples are given to show the correctness of the obtained theoretical results.

Robust Control of Collaborative Manipulators - Flexible Object System

Abstract In many manufacturing and automobile industries, flexible components need to be positioned with the help of coordinated operations of manipulators. This paper deals with the robust design of a control system for two planar rigid manipulators moving a flexible object in the prescribed trajectory while suppressing the vibration of the flexible object. Dynamic equations of the flexible object are derived using the Hamiltonian principle, which is expressed as a partial differential equation (PDE) with appropriate boundary conditions. Then, a combined dynamics is formulated by combining the manipulators and object dynamics without any approximation. The resulting dynamics are thus described by the PDEs, having rigid as well as flexible parameters coupled together. This paper attempts to develop a robust control scheme without approximating the PDE in order to avoid measurements of flexible coordinates and their time derivatives. For this purpose, the two subsystems, namely slow and fast subsystems, are identified by using the singular perturbation technique. Specific robust controllers for both the subsystems are developed. In general, usage of the singular perturbation technique necessitates exponential stability of both subsystems, which is evaluated by satisfying Tikhnov's theorem. Hence, the exponential stability analysis is performed for both subsystems. Focusing on two three-link manipulators holding a flexible beam, simulations are performed and simulation results demonstrate the versatility of the proposed robust composite control scheme.

Approximation of stochastic equilibria for dynamical systems with parametrical noise

A problem of the approximation for stochastic equilibria for the system with parametrical noise is considered. Our approach is based on first approximation stochastic systems technique. For these systems, we use a spectral theory of positive operators for the analysis of exponential mean square stability. Using this method we approximate a dispersion of random states in stochastic equilibrium of nonlinear dynamical system with parametrical noise.

Sampled-Data Control of Spacecraft Rendezvous with Discontinuous Lyapunov Approach

This paper investigates the sampled-data stabilization problem of spacecraft relative positional holding with improved Lyapunov function approach. The classical Clohessy-Wiltshire equation is adopted to describe the relative dynamic model. The relative position holding problem is converted into an output tracking control problem using sampling signals. A time-dependent discontinuous Lyapunov functionals approach is developed, which will lead to essentially less conservative results for the stability analysis and controller design of the corresponding closed-loop system. Sufficient conditions for the exponential stability analysis and the existence of the proposed controller are provided, respectively. Finally, a simulation result is established to illustrate the effectiveness of the proposed control scheme.

New Exponential Stability Conditions of Switched BAM Neural Networks with Delays

The exponential stability problem is considered in this paper for discrete-time switched BAM neural networks with time delay. The average dwell time method is introduced to deal with the exponential stability analysis of the systems for the first time. By constructing a new switching-dependent Lyapunov-Krasovskii functional, some new delay-dependent criteria are developed, which guarantee the exponential stability. A numerical example is provided to demonstrate the potential and effectiveness of the proposed algorithms.

On the stability issues of switched singular time‐delay systems with slow switching based on average dwell‐time

SUMMARYThe issue of exponential stability analysis of continuous‐time switched singular systems consisting of a family of stable and unstable subsystems with time‐varying delay is investigated in this paper. It is very difficult to analyze the stability of such systems because of the existence of time‐delay and unstable subsystems. In this regard, on the basis of the free‐weighting matrix approach, by constructing the new Lyapunov‐like Krasovskii functional, and using the average dwell‐time approach, delay‐dependent sufficient conditions are derived and formulated in terms of LMIs to check the exponential stability of such systems. This paper also highlights the relationship between the average dwell‐time of the switched singular time‐delay system, its stability, exponential convergence rate of differential states, and algebraic states. Finally, a numerical example is given to confirm the analytical results and illustrate the effectiveness of the proposed strategy. Copyright © 2012 John Wiley & Sons, Ltd.

Robustness analysis of global exponential stability of neural networks with Markovian switching in the presence of time-varying delays or noises

In this paper, we analyze the robustness of global exponential stability of neural networks with Markovian switching (NNwMS) subject to random disturbances or time-varying delays. Given a globally exponentially stable neural network with Markovian switching, the problems to be addressed herein are how much noises or time delays that the neural networks can remain to be globally exponentially stable. We characterize the upper bounds of the time delays or noise intensity for the NNwMS to sustain global exponential stability. Two numerical examples are provided to illustrate the theoretical results.

Exponential stability analysis of memristor-based recurrent neural networks with time-varying delays

This paper investigates the exponential stability problem about the memristor-based recurrent neural networks. Having more rich dynamic behaviors, neural networks based on the memristor will play a key role in the optimistic computation and associative memory, therefore, stability analysis of memristor-based neural networks are quite important. Based on the knowledge of memristor and recurrent neural network, the model of the memristor-based recurrent neural networks is established; and the stability of memristor-based neural networks with time-varying delays is studied. Several sufficient conditions for the global exponential stability of these neural networks are presented. These results ensure global exponential stability of memristor-based neural networks in the sense of Filippov solutions. In addition to providing criteria for memristor-based neural networks with time-varying delays, these stability conditions can also be used for memristor-based neural networks with constant time delays or without time delays. Furthermore, it is convenient to estimate the exponential convergence rates of this neural network by using the results. An illustrative example is given to show the effectiveness of the obtained results.

An improved exponential stability criterion for discrete-time switched non-linear systems with time-varying delays

This paper is concerned with the exponential stability analysis for a class of discrete-time switched non-linear systems with time-varying delays. By constructing an appropriate Lyapunov–Krasovskii function, a new criterion for checking the exponential stability of the addressed switched systems is established and then formulated in terms of linear matrix inequalities. It is shown that this new criterion can provide less conservative results than some existing ones. Two numerical examples are given to illustrate the effectiveness of the proposed results.

Exponential stability analysis for positive systems with delays

This study investigates the exponential stability analysis problem for continuous-time and discrete-time positive linear systems with delays. For given decay rate, necessary conditions for exponential stability of such systems are presented based on the asymptotical stability results of positive linear time-delay systems, and sufficient conditions are provided by using Lyapunov–Krasovskii functional method. It is shown that, the exponential stability with given decay rate of positive linear time-delay systems has to do with the delays. Some illustrative examples are given to show the correctness of the obtained theoretical results.

Asymptotic and exponential stability of uncertain system with interval delay

This paper considers the problem of asymptotic and exponential stability analysis of the continuous system with interval time-varying delay and uncertainties. By constructing a new Lyapunov–Krasovskii functional, several delay-range-dependent conditions are derived in terms of the linear matrix inequalities (LMIs). Two novel integral equalities are employed to overcome the disadvantages of the methods in existing references. The merits of the proposed criteria lie in the exponential stability being a free value and the less conservativeness. Finally, some numerical examples are included to illustrate the effectiveness and the improvement of the proposed method.

Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks

In this paper, we examine the stability of a linearized inverted pendulum system with two delayed position feedbacks. The semigroup approach is adopted in investigation for the well-posedness of the closed loop system. We prove that the spectrum of the system is located in the left complex half-plane and its real part tends to???? when the feedback gains satisfy some additional conditions. The asymptotic eigenvalues of the system is presented. By estimating the norm of the Riesz spectrum projection of the system operator that does not have the uniformly upper bound, we show that the eigenfunctions of the system do not form a basis in the state Hilbert space. Furthermore, the spectrum determined growth condition of the system is concluded and the exponential stability of the system is then established. Finally, numerical simulation is presented by applying the MATLAB software.

Exponential stability analysis of planar piecewise-linear systems: An integral function approach

This paper studies the exponential stability of continuous-time planar piecewise-linear systems (PPLS). By introducing a novel conception of integral function of PPLS and showing its properties, a sufficient and necessary condition for the exponential stability is derived. Furthermore, the exponential growth rate of system trajectories can be obtained accurately by computing the convergence radius of integral function. The algorithm for computing the integral function is developed and two examples are given to demonstrate the proposed approach.