A survey of some recent results in the stability and instability analyses of time-varying systems

After formulating the problem of stability and instability of linear and non-linear time varying feedback systems, we give a concise survey of a few known stability results for non-linear time invariant and time varying systems. This serves as motivation for the establishment of general frequency domain stability and instability criteria for time varying linear and non-linear feedback systems. All the details are omitted in an attempt to give an overall view. Finally some comments are made on the stability analysis of systems governed by partial differential equations.     

New power law inequalities for fractional derivative and stability analysis of fractional order systems

Three new power law inequalities for fractional derivative are proposed in this paper. We generalize the original useful power law inequality, which plays an important role in the stability analysis of pseudo state of fractional order systems. Moreover, three stability theorems of fractional order systems are given in this paper. The stability problem of fractional order linear systems can be converted into the stability problem of the corresponding integer order systems. For the fractional order nonlinear systems, a sufficient condition is obtained to guarantee the stability of the true state. The stability relation between pseudo state and true state is given in the last theorem by the final value theorem of Laplace transform. Finally, two examples and numerical simulations are presented to demonstrate the validity and feasibility of the proposed theorems.     

Robust stability for a type of uncertain time-delay systems

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Mean stability of stochastic difference systems

In this paper the mean stability of linear and non-linear stochastic difference systems is considered. For linear systems the relationship between mean stability and other stability definitions is explored. For the non-linear system explicit criteria for mean stability are derived when the non-linear term satisfies a certain realistic condition.     

LAGRANGE'S THEOREM FOR A CLASS OF NONHOLONOMICSYSTEMS AND ITS APPLICATION

I.IntroductionSinceE.T.Whittaker.proposedfoestabilit}'problellll'lofnonholononlicsystemsin1904forthefirsttime,thescholarsathomeandabroad11a\'emadealotofresearchesontheequilibriunlstabilityoflinearand11olllinearnonllolollolnicsystems,andhaveobtainedaseriesofimportantresultslZ--7].Hobbled'er,theexpositionandapplicationrelatedtoLagrange'stheorenlinthestabilityanalysisfornonholonomicsystemsisseldonlseenuptonow.Althoughitwasmentionedinreference[3].aspecialdiscussionhasnotbeencarriedoutyet.Asafam…     

Stability of Singularly Perturbed Systems with Structural Perturbations. Some Applications

Singularly perturbed systems with structural perturbations are analyzed for stability on the basis of matrix-valued Lyapunov functions. Sufficient conditions of stability and uniform asymptotic stability for automatic control and stabilization systems of an orbital observatory are established     

Stability of Singularly Perturbed Systems with Structural Perturbations. General Theory

Singularly perturbed systems with structural perturbations are analyzed for stability. New sufficient conditions of asymptotic stability and uniform asymptotic stability are established. The systems in question are widely used in control theory, engineering, etc     

非完整系统稳定性的若干进展

介绍非完整系统稳定性理论的某些近代进展,包括非完整系统平衡位置关于全部变量的稳定性,平衡状态流形的稳定性,平衡位置关于部分变量稳定性及其与关于全部变量的稳定性的关系,平稳运动的稳定性,以及非完整控制系统的镇定．同时,讨论非完整系统稳定性的几个主要应用,并给出几个未来研究方向的建议      

Stability and stabilization of the family of pseudolinear differential systems

New methods for the analysis of the robust stability of equilibrium states are developed for some classes of nonlinear differential systems. We formulate sufficient conditions for the stability of the trivial solutions of the families of pseudolinear controlled systems with undetermined coefficient matrices and feedback from the measured output. A method is developed for the analysis of stability according to the first approximation to the family of nonlinear systems. The application of the obtained results is reduced to the solution of systems of linear differential matrix inequalities. We present an example of a system of stabilization of a double inverted pendulum.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.     

Stability Analysis of Continuous Systems with Structural Perturbations

The paper is a survey of the development of the direct Lyapunov method based on matrix-valued functions for systems with structural perturbations. Sufficient conditions for stability in the large, uniform asymptotic stability in the large, exponential stability in the large, and instability of nonlinear continuous systems under structural perturbations are presented.     

Technical stability of nonlinear time-varying systems with small parameters

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On the stability of multiple parameter time-varying dynamic systems

A technique is presented for determaining the stability of lumped-pararaeter, time-varying, dynamic systems with aperiodic coefficients. An “energy like” function is used to develop stability conditions which are direct in terms of the coefficient matrices. The significance of what is presented here is twofold. First, it gives stability conditions applicable to systems which are not necessarily periodic. Second, is allows for a systematic categorization of the effects of the parameter changes on system response and stability, in order to provide a better understanding of the behavior of this class of dynamic systems as they arise in various areas of engineering.     

Lyapunov stability for continuous-time multidimensional nonlinear systems

This paper deals with the stability of continuous-time multidimensional nonlinear systems in the Roesser form. The concepts from 1D Lyapunov stability theory are first extended to 2D nonlinear systems and then to general continuous-time multidimensional nonlinear systems. To check the stability, a direct Lyapunov method is developed. While the direct Lyapunov method has been recently proposed for discrete-time 2D nonlinear systems, to the best of our knowledge what is proposed in this paper are the first results of this kind on stability of continuous-time multidimensional nonlinear systems. Analogous to 1D systems, a sufficient condition for the stability is the existence of a certain type of the Lyapunov function. A new technique for constructing Lyapunov functions for 2D nonlinear systems and general multidimensional systems is proposed. The proposed method is based on the sum of squares (SOS) decomposition, therefore, it formulates the Lyapunov function search algorithmically. In this way, polynomial nonlinearities can be handled exactly and a large class of other nonlinearities can be treated introducing some auxiliary variables and constrains.     

STABILITY FOR THE EQUILIBRIUM STATE OF CHAPLYGIN’S SYSTEMS

The stability for the equitibrium states of Chaplygin’s systems is considered. Theequations of motion of Chaplygin’s systems and the existence conditions of their equilibrium states are given. Some criteria of stability for the equilibrium.states of Chaplygin’s systems are obtained. Two examples are finally given.     

Technical Stability Conditions for Systems of Variable Structure with Derivatives of the Control Function

We derive sufficient conditions for the technical stability of forced states of automatic control systems with a variable structure depending on the derivatives of the control function. For arbitrary admissible initial perturbations from a measurable set of initial states of a control system with filtering, the technical stability conditions do not depend on the sliding conditions in the given domain of the system parameters. We find how the properties of the eigenvalues of the quadratic forms of the Lyapunov functions are related to the technical stability conditions for automatic control systems of variable structure with filtering. The technical stability conditions are analyzed for automatic control systems with third-order filtering where the forming signal creates eight structures     

Stability of nonlinear comparison equations for discrete large-scale systems

On the stability analysis of large-scale systems by Lyapunov functions, it is necessary to determine the stability of vector comparison equations. For discrete systems, only the stability of linear autonomous comparison equations was studied in the past. In this paper, various criteria of stability for discrete nonlinear autonomous comparison equations are completely established. Among them, a criterion for asymptotic stability is not only sufficient, but also necessary, from which a criterion on the function class C ₁ is derived. Both of them can be used to determine the unexponential stability, even in the large, for discrete nonlinear (autonomous or nonautonomous) systems. All the criteria are of simple algebraic forms and can be readily used.Projects Supported by the National Natural Science Foundation of China, 1880359.     

A method of stability analysis of nonlinear large-scale systems

Results of stability analysis of some classes of perturbed equations of motion are presented. Sufficient conditions for the asymptotic stability in the large and instability of the equilibrium state of large-scale systems are established using a quadratic Lyapunov function. Some results are illustrated by examples of fourth-order systems     

On stability boundary of linear multi-parameter Hamiltonian systems

In this paper an approximate equation is derived to describe smooth parts of the stability boundary for linear Hamiltonian systems, depending on arbitrary number of parameters. With this equation, we can obtain parameters corresponding to the stability boundary, as well as to the stability and instability domains, provided that one point on the stability boundary is known. Then differential equations describing the evolution of eigenvalues and eigenvectors along a curve on the stability boundary surface are derived. These equations also allow us to obtain curves belonging to the stability boundary. Applications to linear gyroscopic systems are considered and studied with examples. The project supported by the National Science Foundations of Russia and China (10072012)     

多自由度非线性随机系统的响应与稳定性

响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.