Exponential Stability Of Linear Systems Research Articles

Exponential stability of linear systems under a class of Desch–Schappacher perturbations

In this paper, we investigate the uniform exponential stability of the system where the unbounded operator is the infinitesimal generator of a linear semigroup of contractions in a Hilbert space and is a Desch–Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to prove the uniform exponential stabilization of some bilinear partial differential equations.

On the Exponential Stability of Neutral Linear Systems with Variable Delays

We investigate the exponential stability of a linear system of neutral-type with variable time lags. By using the Newton–Leibniz formula and a Lyapunov–Krasovskii functional, we prove two results on the exponential stability of solutions. The stability criteria are stated in the form of linear matrix inequalities. By using the MATLAB-Simulink software, we present two numerical examples illustrating the applicability of our assumptions. The obtained results extend and generalize the already known results available from the related literature.

Solutions of systems with the Caputo–Fabrizio fractional delta derivative on time scales

Caputo–Fabrizio fractional delta derivatives on an arbitrary time scale are presented. When the time scale is chosen to be the set of real numbers, then the Caputo–Fabrizio fractional derivative is recovered. For isolated or partly continuous and partly discrete, i.e., hybrid time scales, one gets new fractional operators. We concentrate on the behavior of solutions to initial value problems with the Caputo–Fabrizio fractional delta derivative on an arbitrary time scale. In particular, the exponential stability of linear systems is studied. A necessary and sufficient condition for the exponential stability of linear systems with the Caputo–Fabrizio fractional delta derivative on time scales is presented. By considering a suitable fractional dynamic equation and the Laplace transform on time scales, we also propose a proper definition of Caputo–Fabrizio fractional integral on time scales. Finally, by using the Banach fixed point theorem, we prove existence and uniqueness of solution to a nonlinear initial value problem with the Caputo–Fabrizio fractional delta derivative on time scales.

Novel condition for exponential stability of linear system with mixed non-differential time-varying delay with feedback control

The problem of exponential stability for linear system with mixed time-varying delay is investigated. The stability of linear system are proposed with feedback control. Based on the constructing of improved Lyapunov-Krasovskii functionals combined with Leibniz-Newton’s formula and by utilizing the integral inequality, new sufficient condition for the stabilization of the systems is first established in terms of LMIs. A numerical example is given to illustrate that the effectiveness of the proposed feedback control.

A Note on the Exponential Stability of Linear Systems with Variable Retardations

A Note on the Exponential Stability of Linear Systems with Variable Retardations

Stability and Control Design for Time-Varying Systems with Time-Varying Delays using a Trajectory-Based Approach

A recent work by Mazenc and Malisoff provides a trajectory-based approach for proving stability of time-varying systems with time-varying delays. Here, we provide several significant applications of their approach. In two results, we use a Lyapunov function for a corresponding un-delayed system to provide a new method for proving stability of linear continuous-time time-varying systems with bounded time-varying delays. We allow uncertainties in the coefficient matrices of the systems. Our main results use upper bounds on an integral average involving the delay. The results establish input-to-state stability with respect to disturbances. We also provide a novel reduction model approach that ensures global exponential stabilization of linear systems with a time-varying pointwise delay in the input, which allows the delay to be discontinuous and uncertain. Finally, we provide an alternative to the reduction model method, based on a different dynamic extension. Our examples demonstrate the usefulness of our findings in several settings.

Exponential Stability of Linear Systems with Multiple Time Delays

Exponential Stability of Linear Systems with Multiple Time Delays

Exponential Stability of Linear Systems with Multiple Time Delays

Exponential Stability of Linear Systems with Multiple Time Delays

Simple uniform exponential stability conditions for a system of linear delay differential equations

Uniform exponential stability of linear systems with time varying coefficientsẋi(t)=-∑j=1m∑k=1rijaijk(t)xj(hijk(t)),i=1,…,mis studied, where t⩾0,m and rij,i,j=1,…,m are natural numbers, aijk:[0,∞)→R and hijk:[0,∞)→R are measurable functions. New explicit result is derived with the proof based on Bohl–Perron theorem. The resulting criterion has advantages over some previous ones in that, e.g., it involves no M-matrix to establish stability. Several useful and easily verifiable corollaries are deduced and examples are provided to demonstrate the advantage of the stability result over known results.

Synthesis of Razumikhin and Lyapunov–Krasovskii approaches to stability analysis of time-delay systems

In this paper, a necessary and sufficient condition for the exponential stability of linear systems with several time-delays is presented. Such a condition is based on the construction of quadratic lower bounds for the Lyapunov–Krasovskii functionals on the special Razumikhin-type set of functions. The result reveals a constructive procedure for the stability analysis whose application is illustrated with examples.

Necessary stability conditions for linear delay systems

We present necessary conditions for the exponential stability of linear systems with multiple delays. They are expressed in terms of the delay Lyapunov matrix of the Lyapunov–Krasovskii functionals of complete type approach. New properties of independent interest, that establish connections of the system fundamental matrix with its Lyapunov matrix, are crucial elements of our proof. We illustrate our work with a number of examples.

Exponential Stability for Linear Systems with Interval TimeVarying Delays

Exponential Stability for Linear Systems with Interval TimeVarying Delays

Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment

We consider the problem of controlling a linear system when the state is available with a known time-varying delay (delayed-state feedback control) or the actuator is affected by a delay. The solution proposed in this paper consists in partially assigning the spectrum of the closed-loop system to guarantee the exponential zero-state stability with a prescribed decay rate by means of a finite-dimensional control law. A non conservative bound on the maximum allowed delay for the prescribed decay rate is presented, which holds for both cases of constant and time-varying delays. An advantage over recent and similar approaches is that differentiability or continuity of the delay function is not required. We compare the performance of our approach, in terms of delay bound and input signal, with another recent approach.

Exponential Stability Criteria for Nonautonomous Difference Systems

The aim of this paper is to characterize the exponential stability of linear systems of difference equations with slowly varying coefficients. Our approach is based on the generalization of the freezing method for difference equations combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entries of a matrix-valued function, instead of bounds on its eigenvalues. By this method, new explicit stability criteria for linear nonautonomous systems are derived.

New stability conditions for systems with distributed delays

In the present paper, sufficient conditions for the exponential stability of linear systems with infinite distributed delays are presented. Such systems arise in population dynamics, in traffic flow models, in networked control systems, in PID controller design and in other engineering problems. In the early Lyapunov-based analysis of systems with distributed delays (Kolmanovskii & Myshkis, 1999), the delayed terms were treated as perturbations, where it was assumed that the system without the delayed term is asymptotically stable. Later, for the case of constant kernels and finite delays, less conservative conditions were derived under the assumption that the corresponding system with the zero-delay is stable (Chen & Zheng, 2007). We will generalize these results to the infinite delay case by extending the corresponding Jensen’s integral inequalities and Lyapunov–Krasovskii constructions. Our main challenge is the stability conditions for systems with gamma-distributed delays, where the delay is stabilizing, i.e. the corresponding system with the zero-delay as well as the system without the delayed term are not asymptotically stable. Here the results are derived by using augmented Lyapunov functionals. Polytopic uncertainties in the system matrices can be easily included in the analysis. Numerical examples illustrate the efficiency of the method. Thus, for the traffic flow model on the ring, where the delay is stabilizing, the resulting stability region is close to the theoretical one found in Michiels, Morarescu, and Niculescu (2009) via the frequency domain analysis.

LMI approach to exponential stability of linear systems with interval time-varying delays

This paper addresses exponential stability problem for a class of linear systems with time-varying delay. The time delay is assumed to be a continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a set of augmented Lyapunov–Krasovskii functional combined with the Newton–Leibniz formula technique, new delay-dependent sufficient conditions for the exponential stability of the systems are first established in terms of linear matrix inequalities (LMIs). An application to exponential stability of uncertain linear systems with interval time-varying delay is given. Numerical examples are given to show the effectiveness of the obtained results.

A Delay-dependent Condition for the Exponential Stability of Switched Linear Systems with Time-varying Delay

We use a Lyapunov-Krasovskii functional approach to establish the exponential stability of linear systems with timevaryingdelay. Our delay-dependent condition allows to compute simultaneously the two bounds that characterize theexponential stability rate of the solution. A simple procedure for constructing switching rule is also presented.

Exponential stability and stabilization of sampled-data systems with time-varying period

This article proposes a novel approach to assess the exponential stability of linear systems with sampled-data inputs. The paper considers both uncertainties in the model parameters and in the sampling period. Inspired by the input-delay approach and the stability of impulsive systems, the proposed method provides easy tractable stability conditions. Sufficient stability and stabilization conditions are provided to deal with both cases of constant and time-varying sampling periods. The period-dependent conditions are expressed using computable linear matrix inequalities. Several examples show the efficiency and the limitation of such stability criteria.

Discontinuous Lyapunov Functional for Linear Systems with Sawtooth Delays*

Abstract Exponential stability of linear systems with time-varying piecewise-continuous delays is studied. It is assumed that the delay function has a form of a sawtooth with a constant delay derivative ≠ 0. In the recent paper (Fridman, 2009) piecewise-continuous (in time) Lyapunov-Krasovskii Functionals (LKFs) have been suggested for the stability analysis of sampled-data systems (with ≠ = 1) in the framework of input delay approach. Differently from the existing time-independent LKFs for systems with time-varying delays, the discontinuous ones can guarantee the stability under the sampling which may be greater than the analytical upper bound on the constant delay that preserves the stability. The objective of the present paper is to extend the piecewise-continuous LKF method to systems with a general sawtooth delay. The discontinuous terms of LKFs improve the results for all values of, though the most essential improvement corresponds to = 1.

An improved criterion for exponential stability of linear systems with multiple time delays

Exponential stability of linear systems with multiple time delays is studied in this paper. By using an improved Lyapunov functional, we have obtained an improved criterion, which is strictly less conservative than the very recent criterion in Ren and Cao [Fengli Ren, Jinde Cao, Novel α -stability of linear systems with multiple time delays, Appl. Math. Comput. 181 (2006), 282–290], and an estimation of Lyapunov factor. A numerical example is also given to shown the superiority of our result to those in the literature.