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.     

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.     

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 general stability criterion for time-varying systems using amodified detectability condition

Uniform asymptotic stability for general nonlinear time-varying systems is investigated from the output-to-state viewpoint. A unified criterion is proposed using a modified detectability condition and certain integral inequalities. The criterion extends the Krasovskii-LaSalle theorem to nonperiodic time-varying systems. An example taken from the tracking control of mobile robots is given to illustrate that, much as the invariance principle of LaSalle is used in studying the stability of time-invariant systems, our results can be used to study the stability of nonperiodic time-varying systems by checking the modified detectability condition     

Parametric state feedback design of linear distributed-parameter systems

The parametric approach for the design of state feedback controllers has been formulated so far only for linear lumped-parameter systems. It yields an explicit parametric expression for the state feedback gain given the closed-loop eigenvalues and the set of corresponding parameter vectors. This contribution presents a parameterisation of state feedback controllers for linear distributed-parameter systems with scalar state and distributed control. By introducing the closed-loop eigenvalues and the parameter vectors as design parameters, an explicit expression for the state feedback is obtained. In contrast to the pure eigenvalue assignment, the parameterisation allows the assignment not only of the closed-loop eigenvalues but also of the closed-loop eigenfunctions. The usefulness of the proposed parametric approach is demonstrated by decoupling the transfer behaviour of a MIMO diffusion system with respect to its dominant modes.     

An improved frequency time domain stability criterion for autonomous continuous systems

A sufficient condtion given for the asymptotic stability of a system having a single monotonic nonlinearity with slope confined to[0, k_{2}]and a transfer functionG(jomega), isRe(1 + X(jomega) + Y(jomega) + alphajomega)(G(jomega) + 1/k_{2}) geq 0wherealpha>0 , x(t)leq 0fort leq 0andx(t)=0fort>0 , y(t)leq0fort>0andy(t) = 0fort < 0, andintmin{-infty}max{infty}(| x(t)| + | y(t) | )dt < 1. The improvement consists of the addition of theX(jomega)term which corresponds to a nonzero time function fort<0, resulting inZ(jomega)multipliers whose phase angle is capable of varying from +90° to -90° any desired number of times. As is shown by examples, the new criterion gives better results than existing criteria. Also developed is an improved criterion for an odd monotonic nonlinearity.     

A Nyquist stability criterion for distributed parameter systems

A Nyquist graphical stability criterion is developed for distributed parameter, possible unstable, single-loop systems. Practical conditions are presented for existence, uniqueness, causality, and asymptotic or exponential stability of the closed-loop impulse response. The hypotheses are given for the transfer function only and do not require any knowledge of its time-domain impulse response     

A stability criterion for asynchronous multirate linear systems

A stability criterion is developed for linear, asynchronous multirate sampled-data systems. Such systems typically include a linear, time variant, continuous-time plant and several unsynchronized digital controllers. The analysis method uses a time-domain approach based on the closed-loop state transition matrix. The result is sufficient stability criterion which gives an objective measure of long-term and short-term stability. A simple example illustrates the method     

The Popov criterion for multiple-loop feedback systems

Although analytic versions of the Popov criterion are available for multiple-loop feedback systems, it is difficult to get a graphical representation which is as useful as the single-loop one. This paper presents a graphical representation which provides a sufficient condition for the Popov criterion to hold in the multiple-loop case. Two versions are given: the first is suited to design methods based on diagonal dominance and the second is suited to those based on characteristic loci. When the linear part of the system is normal, the second version is necessary as well as sufficient for the criterion to hold.     

A relaxed stability criterion for T-S fuzzy discrete systems

Stability conditions for Tanaka-Sugeno (T-S) fuzzy discrete systems based on a single quadratic Lyapunov function or a weighting dependent Lyapunov function have been mentioned in lots of literature. The existence of a common matrix P is required in the former, and r (rules' number) positive matrices satisfying r2 Lyapunov inequalities are necessary in the latter. In this paper, the weighting dependent Lyapunov function is used again. Moreover according to the idea of the firing rule group and the distance estimation between two successive states of the system, the relaxed stability criterion of T-S fuzzy discrete system is proposed.     

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 criterion for jump resonance of nonlinear control systems

A new criterion for jump resonance of nonlinear control systems is presented. The jump resonance phenomenon is classified into various types according to the form of the frequency response curve, some of which are new and not found in the previous publications. A simple graphical method to predict each type of jump resonance is studied. Experimental results by analog simulation are shown.     

A general stability criterion for congestion control with diverse communication delays

A unified duality model is proposed for describing the current Internet congestion control algorithms. Based on this model, the problem of the local asymptotic stability of the congestion control with heterogeneous propagation delays is formulated and solved. A general stability criterion is proved by using the stability theory for quasi-polynomials.     

Observer theory for distributed-parameter systems

This paper examines the problem of the approximate reconstruction of the unknown state variables in distributed-parameter systems. New results on the observer theory for important classes of linear and non-linear operator, partial differential, and partial differential-integral equations in describing distributed-parameter systems are presented. The specific developments employ the recent results on Lyapunov stability theory, along with the theory of linear and non-linear semigroup operators, and their infinitesimal generators. The questions of observability, stability of the state reconstruction error dynamics associated with the proposed observer structure are discussed. The theoretical results are illustrated with some applications to problems of the kinetic lumping of complex distributed-parameter chemical reaction systems, as well as the observer design for linear and non-linear distributed-parameter diffusion systems.     

中立型时滞系统的新的绝对稳定性的法则

研究了具有扇形有界非线性的中立型时滞系统的绝对稳定性. 利用新的Lyapunov-Krasovskii泛函, 基于线性矩阵不等式得到了保守性低的稳定性结果. 从而给出了该系统的新的绝对稳定的法则. 最后用数值算例来演示所得到的结论的可行性与较低的保守性. 文中也考虑了不确定性中立型时滞系统, 他们的不确定性必须是范数有界的, 但可以是时变的.     

A new robust stability measure for state feedback systems

This brief paper proposes the use of sensitivities of each individual system pole to measure robust stability of state feedback systems. Compared with the existing measures of this kind (M₁ and M₂), this measure is much easier to work with than M₁, while at least as accurate, and is definitely more accurate than M₂. The upper and lower bounds of this measure are also derived and proved to be the same as those of M₁. This measure can be maximized numerically using existing explicit and systematic procedures. Some analytical guidelines which relate explicitly and directly the state feedback system's individual pole selection, performance, and decoupling to this new measure are also available and presented.     

Elliptic ball stability criterion for discrete-time systems

The elliptic ball criterion is proved for the discrete-time non-linear feedback control equation p(6) x+ BM(t)q(B)x = 0, in which θxlpar;t) = x( t + 1). It is a geometrical condition for stability and instability which reduces to the well-known circle criterion in. the special case when the matrix M(t) is a scalar. For a certain class of equations the elliptic ball criterion is shown to be both necessary and sufficient for the existence of a special kind of quadratic Lyapunov function. The analogy with differential equations is found to fail in one respect