On the Stability of Equilibrium of a Distributed Parameter Feedback Control System

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.

Aizerman's method for investigating asymptotic stability in the large, which was extended by Kodama to encompass nonlinear sampled-data systems, is further extended in this paper. The concept is developed with specific reference to pulse-width-modulated feedback systems, but the method is applicable to a more general class of nonlinear discrete-time systems. The significant difference between the class of systems treated here and the class treated by Kodama is that in the latter class the nonlinear factor is a scalar, whereas in the former, it is a vector. This difference is not trivial; the presence of the vector nonlinearity requires a substantial change in the procedure for determining the range of a parameter over which the system is asymptotically stable in the large. By application of the method presented in this paper, one can obtain a sufficient condition for asymptotic stability in the large of pulse-width-modulated feedback systems. The method is not limited to cases in which the poles of the transfer function of the plant are real and/or simple and the result obtained, at least in those cases investigated to date, is not highly conservative.

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

On the stability of certain systems under resonance conditions

Theory of nonlinear control

THERE IS AN important class of feedback control systems known as sampled-data systems in which the data at one or more points consist of trains of pulses or sequences of numbers. Such systems may be analyzed by the use of classical difference equation techniques or by employing the Laplace and Fourier transform methods. It is the purpose of this article to unify and extend the latter methods of analysis of sampled-data systems, and to formulate the input-output relationships for such systems in the frequency domain.

The paper addresses bifurcations of limit cycles for a class of feedback control systems depending on parameters. A set of simple approximate analytical conditions characterizing all the generic limit cycle bifurcations is determined via a first-order harmonic balance analysis in a suitable frequency band. Based on the results of this analysis, an approach to limit cycle bifurcation control is proposed. In particular, an example concerning a biological delay model is developed, where a flip bifurcation control is designed via a modified Pyragas technique.

THERE is an important class of feedback control systems known as sampled-data systems or sampling servomechanisms in which the data at one or more points consist of trains of pulses or sequences of numbers. Such systems may have a variety of forms, a common example of which is shown in Figure 1. In the case illustrated, the sampling is performed on the control error by a so-called sampler which is indicated as a mechanical switch which closes momentarily every T seconds. The data at the output of such a switch consist of a train of equally spaced pulses of short duration whose envelope is the control error function. In some practical systems, the separation between successive pulses is controlled by some characteristic of the input signal and consequently is not constant. Such systems will not be considered in this paper.

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.

The modeling and control of a horizontally slewing inextensible Timoshenko-beam including the correct centrifugal stiffening effect and a tip payload are considered. Partial differential equations of motion and orthogonality conditions for the constrained modes are derived. A finite dimensional dynamic model simplified by using the orthogonality conditions is obtained. This nonlinear dynamic model places no restrictions on slenderness ratio, slewing angle, and rotational speed provided that the flexure deflections remain small. To achieve a joint angle trajectory tracking with rapid suppression of elastic vibrations, a nonlinear controller is designed using input-output linearization and elastic mode stabilization. The input-output linearization is used to generate a desired joint angle trajectory command. The elastic mode stabilization is designed based on the linear time-invariant part of the partially linearized system. Stability of the closed-loop system is investigated, and a sufficient condition for asymptotic stability is established. This stability condition justifies the existence of joint angle trajectory commands for accurate joint angle trajectory following and simultaneous vibration suppression.

A resolvent operator of a linear parabolic differential equation with infinite delay is used for investigation of asymptotic behaviour of a nonlinear equation. A method based on a variation of parameters formula yields sufficient conditions for asymptotic stability. A semigroup approach allows to examine a bifurcation of the Hopf type. As an example the Volterra population model with diffusion is treated.

SUMMARYA system is considered underactuated if the number of the actuator inputs is less than the number of degrees of freedom for the system. Sliding mode control for underactuated systems has been shown to be an effective way to achieve system stabilization. It involves exponentially stable sliding surfaces so that when the closed‐loop system trajectory reaches the surface, it moves along the surface while converging to the origin. In this paper, a general framework that provides sufficient conditions for asymptotic stabilization of underactuated nonlinear systems using sliding mode control in the presence of system uncertainties is presented. Specifically, it is shown that the closed‐loop system trajectories reach the sliding surface in finite time, and a constructive methodology to determine exponential stability of the closed‐loop system on the sliding surface is developed, which ensures asymptotic stability of the overall closed‐loop system. Furthermore, the aforementioned framework provides the basis to determine an estimate of the domain of attraction for the closed‐loop system with uncertainties. Finally, the results developed in the paper are experimentally validated using a linear inverted pendulum testbed to show a good match between the actual domain of attraction of the upward equilibrium state of the pendulum and its analytical estimate.Copyright © 2012 John Wiley & Sons, Ltd.

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.

This paper proposes a general method for the synthesis of non-linear output feedback controllers for single-input singleoutput quasi-linear parabolic partial differential difference equation (PDDE) systems, for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Initially, a non-linear model reduction scheme which is based on combination of Galerkin's method with the concept of approximate inertial manifold is employed for the derivation of differential difference equation (DDE) systems that describe the dominant dynamics of the PDDE system. Then, these DDE systems are used as the basis for the explicit construction of non-linear output feedback controllers through combination of geometric and Lyapunov techniques. The controllers guarantee stability and enforce output tracking in the closed-loop parabolic PDDE system independently of the size of the state delay, provided that the separation of the slow and fast eigenvalues of the spatial differential operator is sufficiently large and an appropriate matrix is positive definite. The methodology is successfully employed to stabilize the temperature profile of a tubular reactor with recycle at a spatially non-uniform unstable steadystate.

Criteria of asymptotic stability of differential and difference inclusions encountered in control theory