A stability criterion for nonlinear feedback systems

A frequency domain criterion is formulated to prove global asymptotic stability of feedback control system with more than one nonlinear element. The input-output characteristics of the nonlinearities are confined to sectors of the input-output planes. Sufficient conditions for global asymptotic stability are derived in terms of a number of inequalities equal to the number of nonlinear elements.     

A stability criterion for discrete nonlinear systems with timedelayed feedback

A new criterion on the non-Lyapunov method of stability analysis is developed for a large class of discrete time nonlinear systems. The amenable class is broad and includes systems mainly encountered in practice, for example, all plants described by functions of Lipschitz class, linear, bilinear, and polynomial functions, etc. Unlike other publications on the stability of discrete nonlinear systems, the authors consider the output feedback with time delays, and the control law from a broader class of functions. The derived sufficient criterion is simple and easily applicable     

A stability inequality for nonlinear feedback systems with slope-restricted nonlinearity

Using Lyapunov's direct method, a stability inequality in frequency domain is derived for nonlinear feedback systems with slope-restricted nonlinearity. In the present approach, a transformed system that involves the slope of the nonlinearity is considered, thus leading to a stability inequality that incorporates the slope information (but not the sector information) of the nonlinearity. The approach is essentially an extension of a recent idea of the author.     

A stability criterion of nonlinear control systems

In this correspondence an alternative form of the frequency domain stability criterion of nonlinear control systems is shown. A graphical method of performing this criterion using the gain-phase plane is also discussed.     

A general stability criterion for non-linear time-varying feedback systems†

In this paper a stability theorem is proved for systems with a non-linear time-varying feedback element. It involves a trade-off between restrictions on the non-linear characteristic, on its rate of variation, and on the transfer function of the forward path element. It clarifies the relationship between many existing frequency domain stability criteria, such as the Popov criterion, the Nyquist criterion, and the circle criteria.     

Delay-range-dependent stability criterion for interval time-delay systems with nonlinear perturbations

In this paper, we consider the problem of robust stability for a class of linear systems with interval time-varying delay under nonlinear perturbations using Lyapunov-Krasovskii (LK) functional approach. By partitioning the delay-interval into two segments of equal length, and evaluating the time-derivative of a candidate LK functional in each segment of the delay-interval, a less conservative delay-dependent stability criterion is developed to compute the maximum allowable bound for the delay-range within which the system under consideration remains asymptotically stable. In addition to the delay-bi-segmentation analysis procedure, the reduction in conservatism of the proposed delay-dependent stability criterion over recently reported results is also attributed to the fact that the time-derivative of the LK functional is bounded tightly using a newly proposed bounding condition without neglecting any useful terms in the delay-dependent stability analysis. The analysis, subsequently, yields a stable condition in convex linear matrix inequality (LMI) framework that can be solved non-conservatively at boundary conditions using standard numerical packages. Furthermore, as the number of decision variables involved in the proposed stability criterion is less, the criterion is computationally more effective. The effectiveness of the proposed stability criterion is validated through some standard numerical examples.     

A stability criterion for a class of discrete nonlinear systems with random parameters

In this paper, discrete nonlinear models with random parameters are studied. A new frame to classify and analyze discrete stochastic nonlinear systems has been developed from deterministic nonlinear systems to stochastic nonlinear systems. This frame is broad and includes a large class of stochastic nonlinear systems. A stability criterion developed for this frame is a non-Lyapunov method of stability analysis and is easily applied. In addition, this derived sufficient condition of stability is obtained without the assumption of stationarity for the random noise as frequently assumed in the literature.     

具有非线性扰动的变时滞中立型系统鲁棒稳定性新判据

研究一类具有非线性扰动的时变时滞中立型系统鲁棒稳定性问题。基于直接Lyapunov Krasovskii泛函并结合自由权矩阵方法的分析方法,建立了线性矩阵不等式(LMI)形式的离散时滞和中立时滞均相关稳定性判据。与以往方法不同,在处理泛函导数时,该方法不包含任何模型变换和涉及交叉项的处理,只是通过引入相关项自由权矩阵,充分考虑各项之间的相互关系,降低了结论的保守性。最后,利用Matlab的LMI工具箱进行了的数值仿真, 算例仿真表明所提出的判据的有效性。     

A general frequency stability criterion for multi-input-output,lumped and distributed-parameter feedback systems

The original Nyquist criterion is based on the comparison of the encirclement of the frequency plot of the return ratio function with the number of poles and the number of zeros of the same function to determine the closed-loop stability of a feedback system. The extensions of the return ratio idea to the stability study of multi-variable feedback systems have used the same terminology and followed a similar course. For the multi-input—output case, the use of the Nyquist criterion or its extension is by no means a simple matter. This paper establishes a new frequency stability criterion which converts the Nyquist criterion from a return ratio oriented approach to a return difference oriented one. Instead of examining the encirclement of the return ratio function to a critical point, we examine the phase change of the positive frequency of the return difference function, and the number of zeros of the positive frequency of the return difference function. This result simplifies the stability study of multi-input—output lumped systems tremendously, and covers multi-input-output distributed-parameters systems naturally. For illustration, several typical examples—single-input-output feedback systems with minimum phase or non-minimum phase open-loop transfer functions, multi-input-output feedback systems with stable or unstable open-loop transfer matrices, multi-input-output feedback systems with irrational or transcendental type distributed-parameter open-loop transfer matrices—are included.     

A stability criterion for general systems

The direct method of Liapunov characterizes stability properties of sets in dynamical systems in terms of the existence of corresponding real-valued Liapunov functions. The traditional limitation of Liapunov functions to real values has blocked the extension of this approach to more general systems. In this paper, a stability concept analogous to classical uniform stability is defined as a relationship between a quasiorder and a uniformity on the same set, and it is shown that stability in this sense occurs if and only if there exists a Liapunov function taking values in a certain partially ordered uniform space associated with the given uniformity and called its retracted scale. A few general properties of scales and retracted scales are discussed, and the continuity of the Liapunov functions is briefly considered.     

Application of Yakubovich criterion for stability of nonlinear stochastic systems

Frequency domain criteria for stability of a class of multiloop nonlinear stochastic systems are presented in this paper. The Yakubovich criterion is used in the investigation of asymptotic stability with probability one. The obtained results are illustrated by the example.     

A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems

This paper deals with absolute stability for a class of nonlinear neutral systems using a discretized Lyapunov functional approach. A delay-dependent absolute stability criterion is obtained and formulated in the form of linear matrix inequalities (LMIs). The criterion is valid not only for systems with small delay, but also for systems with non-small delay. Neither model transformation nor bounding technique for cross terms, nor free weighting matrix method is involved through derivation of the stability criterion. Numerical examples show that for small delay case, the results obtained in this paper significantly improve the estimate of the stability limit over some existing result in the literature; for non-small delay case, the ideal results can also be achieved.     

On one stability criterion for linear systems of differential equations with periodic coefficients

To check stability in linear systems of differential equations with periodic coefficients, we use a weighted Lozinski logarithmic norm. As the weights, we propose to take positive components of the Perron eigenvector for a constant off-diagonally nonnegative matrix that results by averaging the original system with diagonal elements replaced by their real parts and all off-diagonal coefficients replaced by their absolute values.     

On vanishing stability regions in nonlinear systems with high-gain feedback

When local stabilization of nonlinear systems is achieved by linear feedback, the resulting stability region may vanish as the feedback gains increase. This is demonstrated by examples in which neglected nonlinearities create an unstable limit cycle around an asymptotically stable equilibrium. As the feedback gains tend to infinity the unstable limit cycle shrinks to the equilibrium. If a feedback linearization design is applied, the same instability mechanism may occur when the nonlinearities are not precisely known.     

Input-output stability of time-varying nonlinear multiloop feedback systems

Sufficient conditions for the input-output stability (BIBO stability) of time-varying nonlinear multiloop feedback systems are established. The present objective is to analyze large-scale systems in terms of their lower order subsystems (subloops) and in terms of their interconnecting structure. Both time-domain and frequency-domain results are presented. In order to demonstrate the usefulness of the present approach, two specific examples are considered.     

Absolute stability of an interval family of nonlinear dynamic systems with nonlinear feedback

Based on the direct Lyapunov method, the sufficient conditions of the absolute stability of an interval family of Lurie nonlinear dynamic systems are obtained. Checking of these conditions requires small computational costs.     

A new asymptotic stability criterion for nonlinear time-variantdifferential equations

A new sufficient condition for asymptotic stability of ordinary differential equations is proposed. Unlike classical Lyapunov theory, the time derivative along solutions of the Lyapunov function may have positive and negative values. The classical Lyapunov approach may be regarded as an infinitesimal version of the present theorem. Verification in practical problems is harder than in the classical case; an example is included in order to indicate how the present theorem may be applied