Exponential stability of linear distributed parameter systems with time-varying delays

Exponential stability analysis via the Lyapunov-Krasovskii method is extended to linear time-delay systems in a Hilbert space. The operator acting on the delayed state is supposed to be bounded. The system delay is admitted to be unknown and time-varying with an a priori given upper bound on the delay. Sufficient delay-dependent conditions for exponential stability are derived in the form of Linear Operator Inequalities (LOIs), where the decision variables are operators in the Hilbert space. Being applied to a heat equation and to a wave equation, these conditions are reduced to standard Linear Matrix Inequalities (LMIs). The proposed method is expected to provide effective tools for stability analysis and control synthesis of distributed parameter systems.     

Finite-time stability of linear time-varying systems with jumps

This paper deals with the finite-time stability problem for continuous-time linear time-varying systems with finite jumps. This class of systems arises in many practical applications and includes, as particular cases, impulsive systems and sampled-data control systems. The paper provides a necessary and sufficient condition for finite-time stability, requiring a test on the state transition matrix of the system under consideration, and a sufficient condition involving two coupled differential-difference linear matrix inequalities. The sufficient condition turns out to be more efficient from the computational point of view. Some examples illustrate the effectiveness of the proposed approach.     

The stability of linear time-varying systems

It is shown that for the system [xdot](t) = A(t)×(t) where the matrix A(t) is normal, the eigenvalues of A(t) characterize the stability of the system in a manner similar to when A (t) is a constant matrix. Also, for linear systems with time-varying feedback gains, simple, easily applicable analytic criteria are obtained for stability.     

Note on stability of linear systems with time-varying delay

This note considers the stability of linear systems with a time-varying delay. We are interested in a simple Lyapunov–Krasovskii functional (LKF) approach without delay decomposition. In this category, all recent tractable results had a fixed bound on the allowable maximum size of the delay for years. We propose a new simple LKF including the cross terms of variables and quadratic terms multiplied by a higher degree scalar function, and present a new result expressed in the form of LMIs. We show, by two well-known examples, that our result overcomes the previous allowable maximum size of delay and it is less conservative than the previous results having a relatively small upper bound in the derivative of time-delay.     

Input-output stability of sampled-data linear time-varying systems

In this paper we consider the input-output stability of feedback systems consisting of a continuous-time, linear, time-varying plant with a discrete time, linear time-varying controller. These results generalize some of the results of Chen and Francis (1991, 1995) where the plant is restricted to be time-invariant     

Covariance bounds for discrete-time linear systems with time-varying parameter uncertainty

This paper deals with the computation of upper bounds for the state covariance matrix of discrete-time linear systems subject to stochastic excitation and additive time-varying uncertainty in the system dynamic matrix. Such upper bounds are obtained as the stabilizing solutions of suitable H ∞ -type Riccati equations. A necessary and sufficient condition for the existence of such solutions is given in terms of the H ∞ -norm of a suitable transfer function. As for the computation of the optimal bound, it is demonstrated that the bounds are a convex function of a scalar parameter, so that efficient numerical schemes can be worked out.     

Robust exponential stability of time-varying linear systems

This paper investigates the robustness of time-varying linear systems under a large class of complex time-varying perturbations. Previous results⁸ which were restricted to bounded linear perturbations of output feedback type are generalized to unbounded and nonlinear perturbations of multi-output feedback type. We establish a lower bound for the stability radius of these systems and show how it may be possible to improve the bound using time-varying scalar transformations of the state, input and output variables. The results are applied to derive Gershgorin type stability criteria for time-varying linear systems.     

Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function

In this paper, the problem of exponential stability and stabilization for a class of uncertain linear time-varying systems is considered. The system matrix belongs to a polytope and the time-varying parameter as well as its time derivative are bounded. Based on a time-varying version of Lyapunov stability theorem, new sufficient conditions for the exponential stability and stabilization via parameter dependent state feedback controllers (i.e., a gain scheduling controllers) are given. Using parameter dependent Lyapunov function, the conditions are formulated in terms of two linear matrix inequalities without introducing extra useless decision variables and hence are simply verified. The results are illustrated by numerical examples.     

Small-gain stability conditions for linear systems with time-varying delays

In this paper we show that a variety of stability conditions, both existing and new, can be derived for linear systems subject to time-varying delays in a unified manner in the form of scaled small-gain conditions. From a robust control perspective, our development seeks to cast the stability problem as one of robust stability analysis, and the resulting stability conditions are also reminiscent of robust stability bounds typically found in robust control theory. The development is built on the well-known conventional robust stability analysis, requiring essentially no more than a straightforward application of the small gain theorem. The derived conditions have conceptual appeal, and they can be checked using standard robust control toolboxes.     

Robust stability for linear time-delay systems with linear parameter perturbations

A new robust state feedback controller design in the time domain is developed for parametrically perturbed linear systems with time delay. The properties of the matrix measure and comparison theorem are employed to investigate the robust stability conditions of systems with linear structured or unstructured perturbations. A method is proposed to design the robust state feedback controller to satisfy the robustness requirement. Examples are given to illustrate the proposed method.     

具概率分布变时滞随机系统鲁棒稳定性

本文研究了一类有非线性时变随机时滞的线性不确定系统的鲁棒稳定性．其中时变随机时滞表征为伯努利随机过程,具有已知的概率分布和变化范围．通过构造新泛函,建立了基于线性矩阵不等式的鲁棒均方指数稳定的充分条件,此条件易于用MATLABH2具箱来验证．本文所获得结果的主要特征是稳定性条件依赖时滞的概率分布和时滞导数的上界．同时也证明了允许时变随机时滞的时滞比之传统的确定性时滞有更大的变化范围,因此我们的条件比确定性时滞更为保守．算例表明了文中所提方法的有效性．     

Finite-time stability of linear time-varying singular systems with impulsive effects

In this paper, we introduce a new concept of finite-time stability of linear time-varying singular systems with impulses at fixed times, and present new results for the above-mentioned class of system in the form of sufficient conditions. A sufficient condition is given in terms of matrix inequalities, which gives the opportunity of fitting the finite-time control problem in the general framework via impulsive control. A numerical example is presented to illustrate the efficiency of the proposed result.     

New stability criteria of linear singular systems with time-varying delay

Most of the existing results on the stability problem of delayed singular systems only pertain to the case of constant delay. This is due to the fact that time-varying delay makes it hardly possible to explicitly express the fast variables. In this paper, aiming at dealing with the case of time-varying delay, we create a way to prove the stability by using a perturbation approach. Rather, we first get the decay rate for slow variables by using Lyapunov functional approach and, furthermore, guarantee that the fast variables fall into decay by characterising their effect on the derived decay rate. Also, we present a convexity technique in computing the constructed Lyapunov functional which contributes to the elimination of the possible conservatism caused by the varying rate of delay. Finally, we provide two numerical examples to demonstrate the effectiveness of the method.     

Reliable control for linear systems with time-varying delays and parameter uncertainties

In this paper, reliable control for linear systems with time-varying delays and parameter uncertainties is considered. By constructing newly augmented Lyapunov–Krasovskii functionals and utilizing some mathematical techniques such as Leibnitz's rule, Schur's complement, reciprocally convex combination, and so on, a reliable controller design method for linear systems with time-varying delays and parameter uncertainties will be suggested in Theorem 1. Based on the result of Theorem 1, a non-reliable stabilization criterion will be presented in Corollary 1. Theorem 1 and Corollary 1 are derived within the framework of linear matrix inequalities(LMIs) which can be easily solved by utilizing various optimization algorithms. Two numerical examples are included to show the effectiveness and necessity of the proposed results.     

On the stability of slowly time-varying linear systems

New conditions are given in both deterministic and stochastic settings for the stability of the system x=A(t)x when A(t) is slowly varying. Roughly speaking, the eigenvalues of A(t) are allowed to wander into the right half-plane as long as on average they are strictly in the left half-plane.This work was funded by the NSF under Grant ECS-8806063, and was completed while the author was with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, U.S.A.     

Sufficient conditions for stability of linear time-varying systems

In this paper we consider sufficient conditions for the exponential stability of linear time-varying systems of the form . Stability guaranteeing upper bounds for different measures of parameter variations are derived.     

On asymptotic stability of linear time-varying infinite-dimensional systems

It is shown that every asymptotically equistable linear time-varying infinite-dimensional discrete-time system x_k+1 = A_kx_k is uniformly asymptotically equistable, if A_k is a collectively compact sequence of bounded linear operators. Next, this result is used to prove that for a broad class of linear retarded functional differential equations, the notions of asymptotic equistability and uniform asymptotic equistability coincide.     

Robust stability of linear time-varying delta-operator formulateddiscrete-time systems

Some earlier results on robust stability of continuous-time and discrete-time control systems are extended to a class of linear time-varying delta-operator formulated discrete-time systems