Commensurate Delays Research Articles

Stability Analysis of Linear Time-Invariant Fractional Exponential Delay Systems

This brief presents a new approach for the stability analysis of linear fractional exponential delay systems with commensurate orders and multiple commensurate delays that enable us to decide on some cases that were previously open problems. In the proposed approach, first an auxiliary polynomial is generated by mapping the principal sheet of the Riemann surface and a pseudodelay transformation. Next, this auxiliary polynomial is employed in determining all possible purely imaginary characteristic roots for any positive time delay. Then, the concept of stability is expressed as a function of delay. Two illustrative examples are provided to demonstrate the effectiveness of the proposed method and to gain a better understanding of the problem, and the root-locus curve of these systems has been plotted as a function of time delay.

Robust non-fragile fractional order PID controller for linear time invariant fractional delay systems

A fractional order PID controller is designed to stabilize fractional delay systems with commensurate orders and multiple commensurate delays, where the time delays in the system may belong to several distinct intervals. Moreover, the controller parameters should belong to given intervals. In order to stabilize the system, the D-subdivision method is employed to choose the stabilizing set of the controller parameters from their available values. Furthermore, the nearest values of the obtained stabilizing set to their mean values are selected as the controller parameters so that a non-fragile controller is concluded. Two numerical examples evaluate the proposed control design method.

Observability of linear systems with commensurate delays and unknown inputs

This paper investigates the observability analysis for linear time systems whose outputs are affected by unknown inputs. Three different definitions of observability are proposed. By introducing the Smith form and comparing the invariant factors, a sufficient condition is deduced for each proposed observability definition. Three examples are given for the purpose of highlighting the effectiveness of the proposed approach.

Finite spectrum assignment problem for a delay type system

For a spectrally controlled linear time-invariant delay system with commensurable delays, we construct a static state feedback ensuring an arbitrary finite spectrum of the closedloop system. By an appropriate choice of the spectrum, one can make the closed-loop system asymptotically stable. The results are illustrated by examples.

Complete calming and stabilization of delay systems using spectral reduction

Spectrally controllable autonomous system with commensurable delays is reduced to a system with finite spectrum. For the spectrally reduced system, a state feedback is constructed such that it ensures complete calming of the original system and stability of the closed-loop system. The distributed delay in the closed-loop system can be only in one variable. The results are illustrated by an example.

Inversion of vector delay systems

We consider stationary linear vector systems with commensurable delays. We obtain sufficient conditions for the reducibility of such systems to canonical form with the extraction of null dynamics. A constructive algorithm for the reduction of a system to that form is presented. We suggest a method for estimating the unknown input for vector delay systems with given accuracy.

Stability analysis of time-delayed linear fractional-order systems

In this article, an exact and robust method for the determination of all the possible stability regions in the parametric space has been presented for the fractional order systems with time delay. Using the approach presented in this study, first, the transcendental terms have been eliminated from the characteristic equation and then, all the locations on the imaginary axis crossed by the roots and also their direction of crossing have been calculated. Finally, the concept of the stability as a function of the time delay has been described for a general class of linear fractional order systems with several commensurate delays. Prior to the conclusion, several examples have been given to validate the results of the proposed approach.

On Computing Puiseux Series for Multiple Imaginary Characteristic Roots of LTI Systems With Commensurate Delays

Obtaining the Puiseux series of multiple imaginary (characteristic) roots (MIRs) is a fundamental issue in the stability analysis of time-delay systems. However, to the best of the authors' knowledge, this issue has not been fully investigated up to date. This note focuses on the Puiseux series expansion of MIRs of linear time-invariant systems including commensurate delays. For an MIR of any multiplicity, we propose an algorithm for defining the structure of the Puiseux series, as well as the explicit computation of the corresponding coefficients. By using the proposed method, we can find all the Puiseux series corresponding to all the root loci.

Complete controllability and controllability for linear autonomous systems of neutral type

We solve the complete controllability problem for linear autonomous systems of neutral type with many commensurable delays in the case when a system solution is required to be an absolutely continuous functions. We propose a control method for the said objects if they do not have the complete controllability property.

Stability of linear time invariant fractional delay systems of retarded type in the space of delay parameters

This paper presents a new method for assessing the bounded input bounded output stability of a class of fractional delay systems with commensurate orders and multiple commensurate delays of retarded type. In the proposed method, first, by mapping the principal sheet of the Riemann surface and a pseudo-delay transformation, an auxiliary polynomial is generated. Then, this auxiliary polynomial is employed to find roots of the characteristic equation on the imaginary axis. The properties of the root path close to these roots are used to identify intervals of delay values, in which the system is stable. The obtained results are illustrated via some numerical examples.

Conversion of linear time-invariant time-delay feedback systems into delay-differential equations with commensurate delays

A new stability analysis method of time-delay systems (TDSs) called the monodromy operator approach has been studied under the assumption that a TDS is represented as a time-delay feedback system consisting of a finite-dimensional linear time-invariant (LTI) system and a pure delay. For applying this approach to TDSs described by delay-differential equations (DDEs), the problem of converting DDEs into representation as time-delay feedback systems has been studied. With regard to such a problem, it was shown that, under discontinuous initial functions, it is natural to define the solutions of DDEs in two different ways, and the above conversion problem was solved for each of these two definitions. More precisely, the solution of a DDE was represented as either the state of the finite-dimensional part of a time-delay feedback system or a part of the output of another time-delay feedback system, depending on which definition of the DDE solution one is talking about. Motivated by the importance in establishing a thorough relationship between time-delay feedback systems and DDEs, this paper discusses the opposite problem of converting time-delay feedback systems into representation as DDEs, including the discussions about the conversion of the initial conditions. We show that the state of (the finite-dimensional part of) a time-delay feedback system can be represented as the solution of a DDE in the sense of one of the two definitions, while its ‘essential’ output can be represented as that of another DDE in the sense of the other type of definition. Rigorously speaking, however, it is also shown that the latter representation is possible regardless of the initial conditions, while some initial condition could prevent the conversion into the former representation. This study hence establishes that the representation of TDSs as time-delay feedback systems possesses higher ability than that with DDEs, as description methods for LTI TDSs with commensurate delays.

YALTA: a Matlab toolbox for the H∞-stability analysis of classical and fractional systems with commensurate delays

In this paper we describe YALTA, a Matlab toolbox dedicated to the H∞-stability analysis of classical and fractional systems with commensurate delays given by their transfer function. Delay systems of both retarded and neutral type are considered. The asymptotic position of high modulus poles is given. For a fixed known delay, poles of small modulus of standard delay systems are approximated through a Padé-2 scheme. For a delay varying from zero to a prescribed positive value, stability windows as well as root loci are given. We deeply describe how we have circumvented the numerical issues of algorithms developed in Fioravanti et al. [2010a, 2012] as well as the limitations of this toolbox. Finally, several examples are given.

Critical simple imaginary characteristic roots and invariance properties for quasipolynomials: A missing link

This paper addresses some properties of simple characteristic roots of quasipolynomials including commensurate delays. Although such a problem seems easy and was largely treated in the literature, to the best of the authors' knowledge, the invariance and the ultimate stability properties have not been fully investigated. In this paper, we propose a new frequency-sweeping framework, which simultaneously considers the Taylor series of the critical imaginary roots as well as the Puiseux series of singular points of the frequency-sweeping curves. Through analyzing the algebraic properties of the corresponding frequency-sweeping curves, we are able to completely characterize the invariance property of the simple imaginary roots with respect to the corresponding critical delay values. As a consequence of the invariance property, the ultimate stability property can be easily derived. Finally, as a byproduct of the approach proposed in the paper, the complete stability for time-delay systems with only simple imaginary roots can be systematically derived. Some illustrative examples complete the presentation.

A bridge between Lyapunov-Krasovskii and spectral approaches for stability of difference equations

In this paper, we analyze the stability properties of a class of systems governed by linear difference equations. Lyapunov stability and asymptotic stability are analyzed using two different approaches, namely Lyapunov-Krasovskii techniques and spectral techniques. In the case of commensurable delays, we carry out an analytical form of the solution, allowing us to determine necessary and sufficient conditions for stability. A link with discrete-time systems is made, and some comments on time-varying delays are proposed.***Copyright©2013 IFAC

The asymptotic analysis of multiple imaginary characteristic roots for LTI delayed systems based on Puiseux–Newton diagram

The paper proposes a novel procedure for the asymptotic expansions of root loci around multiple imaginary roots of an exponential polynomial, which is necessary for the stability analysis of the LTI systems with commensurate delays. With the LTI delay systems given as exponential polynomials (also called quasi-polynomial), we seek to characterise the asymptotic behaviours of the characteristic roots of such systems in an algebraic way and determine whether the imaginary roots cross from one half-plane into another or only touch the imaginary axis. According to the Weierstrass preparation theorem, the quasi-polynomial equation is equivalent to an algebraic equation in the neighbourhood of a singular point. Furthermore, our result gives an explicit expression of the coefficients of the algebraic equation in infinite power series of delay parameter, and the determinations of such power series coefficients refer to the computation of residues of memorphic functions. Subsequently, the classic Puiseux–Newton diagram algorithm can be used to calculate the algebraic expansions of the reduced equation directly. Thus, the asymptotic behaviours of root loci around singular points of the quasi-polynomial equation are obtained. Some illustrative simulations are given to check the validity of the proposed method on asymptotic analysis with a powerful software.

When retarded nonlinear time-delay systems admit an input–output representation of neutral type

The paper shows that nonlinear retarded time-delay systems can admit an input–output representation of neutral type. This behaviour represents a strictly nonlinear phenomenon, for it cannot happen in the linear time-delay case where retarded systems always admit an input–output representation of retarded type. A necessary and sufficient condition for a nonlinear system to exhibit this behaviour is given, and a strategy for finding an input–output representation of retarded type is outlined. Some open problems that arise consequently are discussed as well. All the systems considered in this work are time-invariant and have commensurable delays.

Simultaneous stabilization: Construction of a universal stabilizer for linear plants with delay with the use of spectral reducibility

We consider the problem of the construction of a common stabilizing controller for a family of k linear stationary nth-order plants with commensurable delays in the controls and the state variables. Our approach to stabilization is based on the construction of a common two-loop stabilizer. The controller of the first (internal) loop solves the problem of reducing each plant of the family to a finite spectrum, and the controller of the second (external) contour solves the stabilization problem.

Spectral reducibility of delay differential controlled systems in the regular case

For pointwise controlled autonomous linear differential systems with commensurate delays, a dynamic controller is designed that reduces the original system to a difference-differential system with a finite spectrum. The results are illustrated by an example.

Complete controllability problem and its generalization for linear autonomous systems of neutral type

A solution to the complete controllability problem is given for linear autonomous systems of neutral type with multiple commensurable delays. A control method for the specified systems that do not possess the complete controllability property is proposed.

Complete damping of a linear autonomous differential-difference system by a controller of the same type

For a spectrally controllable linear autonomous system of retarded type with commensurable delays, we construct a dynamic state feedback ensuring a complete damping of the original system. The closed-loop system is of retarded type as well. The results are illustrated by an example.