Stabilization of large-scale non-linear infinite delay systems

The problem of stabilizing a class of large-scale non-linear infinite delay systems is considered. It is shown that exponential stability can be achieved by a suitable choice of state feedback controls of large-scale systems with infinite delay, and that a Lyapunov functional can be constructed by using certain results from J. K. Hale's Theory of Functional Differential Equations (1977).     

Controllability of non-linear delay systems

In this paper we examine the approximate controllability in the space W² × L² of non-linear systems with delays in states and controls. If the uncontrolled system is asymptotically stable and if the linear part of the control system is controllable, then the non-linear delay system is approximately controllable. An upper bound on the magnitude of retardation for the system to be controllable is estimated. An example is given to demonstrate the practical verification of the new conditions     

Optimal controls of fractional impulsive partial neutral stochastic integro-differential systems with infinite delay in Hilbert spaces

In this paper, we establish the existence of optimal controls of systems governed by a class of fractional impulsive partial neutral stochastic integro-differential systems with infinite delay in a Hilbert space. The approaches used are fixed point theorem, stochastic analysis theory, and approximation technique combined with properties of the solution operator. Finally, an example is discussed to illustrate the efficiency of the results.     

Controllability of non-linear delay systems with an implicit derivative

Sufficient conditions are established for the controllability of non-linear delay systems with an implicit derivative. The results are obtained using the measure of the non-compactness of a set and the Darbo fixed-point theorem.     

Asymptotic behaviour for some large-scale systems with infinite delay

The aggregation-decomposition method is used to derive sufficient conditions for stability of large-scale systems with infinite delays under structural perturbations described by functional differential equations with finite lags appearing only in the interconnections. Electrical networks consisting of transmission lines interconnected with lumped linear and memoryless nonlinear elements can be represented by the dynamical systems considered in this paper.     

Exponential stability of switched stochastic delay systems with non-linear uncertainties

This article considers the robust exponential stability of uncertain switched stochastic systems with time-delay. Both almost sure (sample) stability and stability in mean square are investigated. Based on Lyapunov functional methods and linear matrix inequality techniques, new criteria for exponential robust stability of switched stochastic delay systems with non-linear uncertainties are derived in terms of linear matrix inequalities and average dwell-time conditions. Numerical examples are also given to illustrate the results.     

Approximate controllability of stochastic impulsive functional systems with infinite delay

This paper studies the approximate controllability of stochastic impulsive functional system with infinite delay in abstract space. By using the contraction mapping principle, some sufficient conditions are given with no compactness condition imposed on the semigroup generated by the linear part of the system. Then with the help of the Nussbaum fixed point theorem, the restriction of the combination of system parameters is dropped. Finally, an example is shown to illustrate our results.     

The stability of linear partial differential systems with infinite delay

This paper studies the stability of linear partial differential systems with infinite delay. Some simple criteria for stability are proved via the methods of Fourier transformation and inequality analysis.     

Euclidean controllability of linear delay systems with limited controls

For linear delay systems with limited controls the notion of a proper control system is introduced. If the uncontrolled systemdot{x}(t)= Ax(t)+ Bx(t-h)is uniformly asymptotically stable and the control equationdot{x}(t)=Ax(t)+Bx(t-)+Cu(t)is proper, then the control system is Euclidean null-controllable. We note that controllability is equivalent to the system being proper for delay systems with unlimited power.     

Adaptive non-linear compensation control based on neural networks for non-linear systems with time delay

This article investigates a new adaptive non-linear compensation controller for a class of time-delay non-linear systems with partly known dynamics. First, a non-linear neural-network(NN)-based identification model that includes a prior knowledge about the plant dynamics is discussed by using the approximation capabilities of NNs. Then, the adaptive non-linear compensation controller is developed to produce the desired tracking performance. The proposed controller based on the NN can reduce the effect of modelling uncertainties and provide the time-delay compensation, while stability of the closed-loop system is guaranteed. The effectiveness of the proposed scheme is demonstrated through the application to the control of a continuous stirred tank reactor.     

Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary

In this work, we study the null-controllability properties of linear parabolic systems with constant coefficients in the case where several controls are acting on different distributed subdomains and/or on the boundary. We prove a Kalman rank condition in the one-dimensional case. In the case where only distributed controls are considered, we also establish related results such as a Carleman estimate.     

一类非线性切换时延系统的鲁棒稳定性

本文通过利用平均驻留时间方法,研究一类具有不确定性非线性切换时延系统的指数稳定性问题。给出非切换系统的候选李雅普诺夫函数的衰减估计分析,然后以线性矩阵不等式的形式给出使系统保持指数稳定及鲁棒指数稳定的充分条件,同时也给出了系统状态指数衰减的具体的估计形式。     

On the equivalent bilinearization of non-linear controlled delay systems

Taylor polynomials are used to obtain an equivalent bilinear system for a non-linear analytic delay equation with control and the optimal control of the resulting bilinear system is considered.     

Approximate optimal controls for a class of Volterra systems with nonlinear delay

Necessary and sufficient conditions of approximate optimal controls for integral linear in state Volterra systems with nonlinear delays are derived. They are used for studying approximate optimal controls obtained by the control parameterization method as an important application.     

Initial condition issues on iterative learning control for non-linear systems with time delay

Most of the available results on iterative learning control address trajectory tracking problem for systems without time delay. The role of the initial function in tracking performance of iterative learning control for systems with time delay is not yet fully understood. In this paper, asymptotic properties of a conventional learning algorithm are examined for a class of non-linear systems with time delay in the presence of initial function errors. It is shown that a non-zero initial function deviation can cause a lasting tracking error on the entire operation. Impulsive action is one method to eliminate such lasting tracking error but it is not a practical approach. As an alternative, an initial rectifying action is introduced in the learning algorithm. The initial rectifying action is finite and used over a specified interval. It is shown to be effective in the improvement of tracking performance, in particular robustness and uniform convergence. The results are further extended to systems with multiple time delays. An example is given and computer simulations are presented to demonstrate the performance of the proposed approach.     

Robust sliding mode control of non-linear systems with delay: a design via polytopic formulation

This paper outlines the possibility to control non-linear uncertain systems with single, possibly time-varying state-delay. These systems are submitted to additive perturbations (possibly non-vanishing). The control is of sliding mode type and the sliding surface is designed to maximize the calculable set of delays which do not destabilize the relay–delay system. The way to deal with the non-linear parts of the model relies on a polytopic formulation, i.e. a decomposition into a non-linear combination of linear models, whose weighting coefficients satisfy a convex property. The conditions for the existence of the sliding regime are studied by using both a Lyapunov–Krasovskii functional and a Lyapunov–Razumikhin function. The optimization procedure for the choice of the sliding surface is expressed in terms of LMIs. An example illustrates the proposed method.     

Synthesis of controllers for continuous-time delay systems withsaturating controls via LMIs

The stabilization of linear continuous-time systems with time delay in the state and subject to saturating controls is addressed. Sufficient conditions obtained via a linear matrix inequality (LMI) formulation are stated to guarantee both the local stabilization and the satisfaction of some performance requirements. The method of synthesis consists in determining simultaneously a state feedback control law and an associated domain of safe admissible states for which the stability of the closed-loop system is guaranteed when control saturations effectively occur     

Optimal controls and their discontinuities in quadratic problems of delay systems

We construct and study quadratic optimal controls in delay systems of neutral type. In these systems we allow functional lags in the derivative, the state, the control, and in the cost index. We point out that, typically, optimal controls of time lag systems are discontinuous: discontinuities might occur as a consequence of delays in the control action and of the neutrality of the system. We exhibit computed examples of jumps in optimal controls whose origins are in each of the two types. Our techniques are based on the method of steps which we generalize to fit systems with functional lags. The work includes an analysis ofR^{n-}controllability in time lag systems.     

Numerical approach to the non-linear diofantic equations with applications to the controllability of infinite dimensional dynamical systems

The article is devoted to the analysis of the approximate controllability of a given type of second order infinite dimensional system, defined by the hyperbolic type partial differential equation with Dirichlet-like type zero boundary conditions defined in the n-dimensional rectangular prism. Following this aim, spectral theory for linear unbounded operators is involved. New numerical algorithms for solving some kind of non-linear diofantic equations, corresponding with the spectral properties of the unbounded operators, are presented and proved. An algorithm is optimized for eliminating the symmetrical solutions, not interesting in the scope of verification of the controllability. Next the limit which shows that the direct application of the analytical methods of verifying the controllability in the numerical approach is impossible is proved. Finally so-called partial approximate controllability is defined and the numerical algorithm for its verification is presented. Finally, proven theorems are applied to one particular infinite dimensional dynamical system.