Adaptive event‐triggered control for nonlinear discrete‐time systems

On the stability of certain systems under resonance conditions

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.

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

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.

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.

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.

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.

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.

This paper considers the stability and l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -gain analysis for positive two-dimensional (2D) linear systems with state delays described by the Roesser model based on Takagi-Sugeno(T-S) fuzzy logic rules. The co-positive type Lyapunov function method is applied to establish sufficient conditions of asymptotical stability for the addressed system. l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -gain index is used to analyze the noise attenuation performance of the positive 2D T-S fuzzy delayed systems. The obtained results are formulated in the form of linear matrix inequalities (LMIs). Finally, an illustrative example is given to verify the effectiveness of the proposed method.

Robust distributed synchronization of networked linear systems with intermittent information

MHE is an optimization based strategy for state estimation that explicitly allows for nonlinear models and inequality constraints. In this work we investigate strategies to guarantee the stability of moving horizon estimation (MHE). We begin our discussion by analyzing the stability of the abstract MHE problem. Sufficient conditions for asymptotic stability are established. Using forward dynamic programming to analyze MHE, we introduce the fundamental concept of arrival cost to bound the size of the estimation problem. The key result is if one can construct a global lower bound for the arrival cost, then asymptotic stability is guaranteed. In the second part of our discussion, we propose a strategy that circumvents the need for a global lower bound for the arrival cost. In particular, by including bounds to guarantee the existence of a decreasing nominal sequence, we circumvent the need for a global lower bound for the arrival cost. This result is significant, because, in general, we cannot calculate nor bound the arrival cost. Notable exceptions are linear systems, where the Kalman filter covariance generates a global lower bound for the arrival cost.

MHE is an optimization based strategy for state estimation that explicitly allows for nonlinear models and inequality constraints. In this work we investigate strategies to guarantee the stability of moving horizon estimation (MHE). We begin our discussion by analyzing the stability of the abstract MHE problem. Sufficient conditions for asymptotic stability are established. Using forward dynamic programming to analyze MHE, we introduce the fundamental concept of arrival cost to bound the size of the estimation problem. The key result is if one can construct a global lower bound for the arrival cost, then asymptotic stability is guaranteed. In the second part of our discussion, we propose a strategy that circumvents the need for a global lower bound for the arrival cost. In particular, by including bounds to guarantee the existence of a decreasing nominal sequence, we circumvent the need for a global lower bound for the arrival cost. This result is significant, because, in general, we cannot calculate nor bound the arrival cost. Notable exceptions are linear systems, where the Kalman filter covariance generates a global lower bound for the arrival cost.

Chapter 4 - On Unknown Input Observer Design for Linear Systems With Delays in States and Inputs

Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability

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.