Differential Time-delay Systems Research Articles

Stability Analysis of Some Types of Singularly Perturbed Time-Delay Differential Systems: Symmetric Matrix Riccati Equation Approach

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented.

An Integro-differential-delay Operator Approach to Transformations of Linear Differential Time-delay Systems

In this paper, we further develop the study of rings of integro-differential-delay operators considered as noncommutative polynomial algebras satisfying standard calculus identities. Within the algebraic analysis approach, we show that transformations and reductions of linear differential time-delay systems can be interpreted as homomorphisms and isomorphisms of finitely presented left modules over an algebra of integro-differential-delay operators. In particular, we show how Fiagbedzi-Pearson's transformation can be found again and generalized. This transformation maps the solutions of a first-order differential linear system with state and input delays to the solutions of a purely state-space linear system. Fiagbedzi-Pearson's transformation reduces to the well-known Artstein's reduction when the system has no state delay and yields an isomorphism of the solution spaces.

Stability analysis of time-delay differential systems with impulsive effect suffered by logic choice

In the field of artificial intelligence, nonlinear systems with logic function are everywhere. This paper investigates a time-delay differential system with impulsive effects suffered by logic choice. By the semi-tensor product theory of matrices and constructing Lyapunov function, we analyze the stability of the new system and obtain some stability criteria. The numerical simulations illustrate the validity of these criteria.

Symbolic Computation for Integro-Differential-Time-Delay Operators with Matrix Coefficients

The purpose of this paper is to algebraize and automatize computations with linear differential time-delay systems and their solutions. To this end, we explain an algebraic construction of the ring of integro-differential operators with linear substitutions having (noncommutative) matrix coefficients, which contains the ring of integro-differential-time-delay operators. Based on a reduction system for this ring, we show how such operators can be uniquely expanded into irreducible terms.Symbolic computations with these operators and their normal forms are implemented in a Mathematica package. This even allows for computations with systems having generic size and/or undetermined matrix coefficients. We illustrate how, by elementary computations with operators in this framework, results like the method of steps can be found and proven in an automated way. Normal form computations with our package can be used to partly automatize solving operator equations. As an example, we recover a generalization of Artstein’s reduction, which solves an equivalence problem of a class of differential time-delay control systems.

Model Reduction of Neutral Linear and Nonlinear Time-Invariant Time-Delay Systems With Discrete and Distributed Delays

The problem of model reduction by moment matching for linear and nonlinear differential time-delay systems is studied. The class of models considered includes neutral differential time-delay systems with discrete-delays and distributed-delays. The description of moment is revisited by means of a Sylvester-like equation for linear time-delay systems and by means of the center manifold theory for nonlinear time-delay systems. In addition the moments at infinity are characterized for both linear and nonlinear time-delay systems. Parameterized families of models achieving moment matching are given. The parameters can be exploited to derive delay-free reduced order models or time-delay reduced order models with additional properties, e.g. , interpolation at an arbitrary large number of points. Finally, the problem of obtaining a reduced order model of an unstable system is discussed and solved.

The Modeling and Control of a Singular Biological Economic System with Time Delay in a Polluted Environment

This paper brings up the idea of a biological economic system with time delay in a polluted environment. Firstly, by proper linear transformation and parametric method, the singular time-delay systems are transformed to differential time-delay systems. Then, using center manifold theory and Poincare normal form method, the direction of Hopf bifurcation and the stability and period of its periodic orbits are analysed. At last, we have performed numerical simulation to support the analytical results.

Robust stability of positive linear time delay systems under time-varying perturbations

Abstract By a novel approach, we get explicit robust stability bounds for positive linear time-invariant time delay differential systems subject to time-varying structured perturbations or non-linear time-varying perturbations. Some examples are given to illustrate the obtained results. To the best of our knowledge, the results of this paper are new.

Special issue on symbolic methods in multidimensional systems theory

In the seventies, the study of transfer matrices of linear time-invariant systems led to the development of the polynomial approach (Kailath, Rosenbrock, et al.). In this approach, univariate polynomial matrices play a central role (e.g., Hermite and Smith normal forms, Bezout equations etc.). When generalizing linear systems given by ordinary differential/difference equations to differential time-delay systems, systems with parameters, 2-D or 3-D filters and circuits, one had to face the case of systems described by means of matrices with entries in multivariate polynomial rings. These new systems were called 2-D or 3-D and, more generally, multidimensional systems (Bose, Lin, Pommaret, Oberst, Youla, Wood, et al.). For such systems, no normal forms such as Hermite and Smith forms exist. In order to handle these problems, the concept of Grobner bases (developed by Buchberger) was introduced in multidimensional systems theory. In many ways, the computation of these bases can be seen as an extension of both Gaussian elimination and the Euclidean algorithm. Grobner bases have been implemented in nearly all modern computer algebra systems. Another difficulty with multidimensional systems is the lack of a canonical first order representation. Several models (Roesser, Fornasini-Marchesini, reduced Spencer form etc.) have been studied extensively. The geometric approach to systems and control (Basile,Marro, Wonham, et al.) has recently been extended to these classes of multidimensional systems yielding promising results. The reduction to first order representations (realization problem) is of great importance to many applications of multidimensional systems theory such as uncertainty modeling for robust control. Special computer algebra packages have been developed for the effective manipulation of linear fractional transformations. In his pioneering work, Kalman developed a module-theoretic approach to the realization problem. Independently, the Japanese school of Sato, Kashiwara, and Kawai proposed an

Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.

RobustH∞Control for Linear Stochastic Partial Differential Systems with Time Delay

This paper investigates the problems of robust stochastic mean square exponential stabilization and robustH∞for stochastic partial differential time delay systems. Sufficient conditions for the existence of state feedback controllers are proposed, which ensure mean square exponential stability of the resulting closed-loop system and reduce the effect of the disturbance input on the controlled output to a prescribed level ofH∞performance. A linear matrix inequality approach is employed to design the desired state feedback controllers. An illustrative example is provided to show the usefulness of the proposed technique.

Further Results on the Decomposition and Serre's Reduction of Linear Functional Systems

Given a linear functional system (e.g., ordinary/partial differential system, differential time-delay system, difference system), the decomposition problem aims at studying when it can be decomposed as a direct sum of subsystems. This problem was constructively studied in [4] and the corresponding algorithms were implemented in the OreMorphisms package [5]. Using the OreMorphisms package, many classical linear differential time-delay systems were proved to be decomposable, which highly simplifies the study of their structural properties. Serre's reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. It was constructively studied in [1, 6] and successfully applied to different classical examples of differential time-delay systems. Serre's reduction can be seen as a particular case of the decomposition problem. The goal of the present paper is to explicitly provide the links between these two problems. We illustrate the different results with an explicit example of a differential time-delay system.

Stability radius of linear parameter-varying systems and applications

In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459–472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278–2295]. Finally, we give two examples to illustrate the obtained results.

A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices

In this work, we give an extension of the classical Perron–Frobenius theorem to positive quasi-polynomial matrices. Then the result obtained is applied to derive necessary and sufficient conditions for the exponential stability of positive linear time-delay differential systems.

Effective algorithms for parametrizing linear control systems over Ore algebras

In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying systems of ordinary differential equations (ODEs), differential time-delay systems, underdetermined systems of partial differential equations (PDEs), multidimensional discrete systems, multidimensional convolutional codes, etc. We give effective algorithms which check whether or not a linear control system over some Ore algebra is controllable, parametrizable, flat or π-free.

Linear control systems over ore algebras: Effective algorithms for the computation of parametrizations

In this paper, we study linear control systems over Ore algebras. Within this mathematical framework, we can simultaneously deal with different classes of linear control systems such as time-varying ordinary differential systems, differential time-delay systems, partial differential equations, multidimensional discrete systems, etc. We give effective algorithms which check whether or not a linear system over some Ore algebra is controllable, parametrizable or flat.