A stability criterion for pulse-width-modulated feedback control systems

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

On the stability of certain systems under resonance conditions

In this paper the stability problem for n general class of non-linear feedback control systems governed by parabolic partial differential equations is considered. Sufficient conditions for asymptotic stability are derived by the method of comparison functions. Two classes of feedback control systems are considered. The basic tool in the analysis is the Westphal-Priodi theorem and an extension originally suggested by Szarski. The main results are applied to an illustrative example.

The goal of this contribution is presenting a Theorem which states the asymptotic stability of a feedback controlled system with a Linear Generalized Model Predictive Control (LGMPC). Concerning the asymptotic stability, a sufficient and constructive condition on the weight matrices of the cost function used in the optimization problem in LGMPC for one step prediction horizon is demonstrated. The condition consists of a lower bound for one of these matrices. The obtained condition is explained and discussed by means of some physical considerations. The second part of this contribution is devoted to the saturation case and proves a sufficient condition for obtaining asymptotic stability and saturation avoidance.

Robust analysis and control for a class of uncertain nonlinear discrete-time systems

A secondary controller which uses a number of Annular Momentum Control Devices (AMCD's) for enhancement of structural damping in large flexible space structures is investigated. The closed-loop secondary controller is shown to be equivalent to a certain closed-loop optimal linear regulator, and the necessary and sufficient conditions for asymptotic stability are obtained. Sufficient conditions for asymptotic stability are also obtained which are much less restrictive than those obtained in an earlier paper. Design of a robust stable primary controller is also discussed.

In this article, using the method of stochastic Lyapunov functions (SLF), we obtain new algebraic criteria and sufficient conditions for the asymptotic stability with probability 1 for the solutions of a system of linear stationary difference equations with random (of the type of a vector "white" sequence of random variables) coefficients, which are discrete analogs of the conditions established earlier by several authors for stochastic Ito equations with continuous time. We assume that in the absence of parametric random perturbations, an unperturbed determined system of difference equations is asymptotically Lyapunov stable (the matrix A of the system is convergent). The conditions are formulated either in terms of the negative definiteness of certain matrix relations, involving the matrix A and the matrices of the parametric perturbations Bi, i = i, 2 .... , r (sufficient conditions, which in many cases are nearly necessary), or in terms of the existence of a positive-definite solution H of the Sylvester matrix algebraic equation e

This paper deals with sufficient conditions of asymptotic stability and stabilization for nonlinear discrete‐time systems represented by a Takagi–Sugeno‐type fuzzy model whose state variables take only nonnegative values at all timestfor any nonnegative initial state. This class of systems is called positive systems. The conditions of stabilizability are obtained with state feedback control. This work is based on multiple Lyapunov functions. The results are presented in linear matrix inequalities form. A real plant is studied to illustrate this technique. Copyright © 2010 John Wiley & Sons, Ltd.

This paper modifies the prototype model-free adaptive control (PMFAC) approach in which system output fluctuation and overshoot (SOFO) is easy to occur. Targeting a class of nonlinear discrete-time single-input single-output systems, historical input/output sensor data are utilized to online construct equivalent explicit data model for control design, making the controller model-free. Then, a novel compensation input term including a compensation gain and a signum function parameterized by incremental output is proposed to soften system inputs, which can in turn restrain the SOFO. Numerical simulations and a speed control experiment for a direct current motor verify that the modified approach can restrain the SOFO successfully and reduce response time compared with the PMFAC.

A sufficient condition for asymptotic stability in the large is proposed for nonlinear systems. It is applicable if the system in question can be decomposed into subsystems, if appropriate Lyapunov functions are obtained for the subsystems, and if the connections between subsystems have bounded dc gains. It is generally less restrictive than the condition previously presented by Bailey for similar systems. An estimate of transient behavior, together with the stability condition, is also given.

We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function.  &nbsp We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets.

Mathematical models of dynamic processes described by systems of differential-difference equations of delay type with a linearly increasing aftereffect are considered. Such a class of systems has been investigated significantly worse than the class of systems with limited aftereffect. However, in recent times many new applications have appeared in the controlled dynamic processes described by such systems. This paper is devoted to the study of the asymptotic stability of the zero solution of homogeneous differential-difference systems with several concentrated linearly increasing delays. The theoretical basis of the study is the approach of B.S. Razumikhin, which made it possible to obtain coefficient sufficient conditions for asymptotic stability. Further analysis of the asymptotic stability of nonlinear systems with unlimitedly increasing delay can be based on an adaptation of the Lyapunov-Krasovsky approach. As an application, we consider a dynamic model of the span of a large family of UAVs over a limited tunnel, which is described by a system of differential-difference equations with concentrated constant and linearly increasing delays. The approach used in the work can be applied to the analysis of stability, including systems with distributed delay.

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by dX(t)=A(ξ(t))X(t)dt+H(ξ(t))X(t)dw(t), where ξ(t) is a finite‐valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a sufficient asymptotic stability condition.

This paper addresses the problems of the robust stability and stabilization for fractional order systems based on uncertain Takagi–Sugeno fuzzy model. A sufficient condition of asymptotical stability for fractional order uncertain T–S fuzzy model is given, and a parallel distributed compensating fuzzy controller is designed to asymptotically stabilize the model. The sufficient conditions are formulated in the format of linear matrix inequalities. The fractional order T–S fuzzy model of a chaotic system, which has complex nonlinearity, is developed as a test bed. The effectiveness of the approach is tested on fractional order Rossler system and fractional order uncertain Lorenz system.

In this work, the asymptotic stability bounds are identified for a class of linear quasi-periodic dynamical systems with stochastic parametric excitations and nonlinear perturbations. The application of a Lyapunov–Perron (L-P) transformation converts the linear part of such systems to a linear time-invariant form. In the past, using the Infante’s approach for linear time-invariant systems, stability theorem and corollary were derived and demonstrated for time periodic systems with variation in stochastic parameters. In this study, the same approach is extended toward linear quasi-periodic with stochastic parameter variations. Furthermore, the Lyapunov’s direct approach is employed to formulate the stability conditions a for quasi-periodic system with nonlinear perturbations. If the nonlinearities satisfy a bounding condition, sufficient conditions for asymptotic stability can be derived for such systems. The applications of stability theorems are demonstrated with practical examples of commutative and noncommutative quasi-periodic systems.