Linear Time-invariant Delay Systems Research Articles

A complete characterization of minima of the spectral abscissa and rightmost roots of second-order systems with input delay

Abstract The numerical minimization of the spectral abscissa function of linear time-invariant time-delay systems, an established approach to compute stabilizing controllers with a fixed structure, often gives rise to minima characterized by active characteristic roots with multiplicity higher than one. At the same time, recent theoretical results reveal situations where the so-called multiplicity induced dominancy property holds, i.e., a sufficiently high multiplicity implies that the root is dominant. Using an integrative approach, combining analytical characterizations, computation of characteristic roots and numerical optimization, a complete characterization of the stabilizability of second-order systems with input delays is provided, for both state feedback and delayed output feedback. The level sets of the minimal achievable spectral abscissa are also characterized. These results shed light on the complex relations between (configurations involving) multiple roots, the property of being dominant roots and the property of corresponding to (local/global) minimizers of the spectral abscissa function.

On controllers performance for a class of time-delay systems: Maximum decay rate

This paper states some results aiming at finding the parameters of a controller that provides the maximum decay rate of the closed-loop response for a linear time-invariant time-delay system. These results help to understand the behavior of dominant roots and the spectral abscissa under parametric variations, including the delay. The controller parameters or points of best performance are obtained for compact subsets of the system’s stability region. In addition, it is shown that these points are not always related to obtaining a multiple dominant root or the existence of a root of maximal multiplicity. To illustrate the effectiveness and diversity of application of the proposed results, four key examples from the literature were chosen and the points of best performance were found.

Linear Time-Invariant Discrete Delay Systems in Laguerre Domain

This article provides the formulas connecting the Laguerre spectrum of the output signal of a linear discrete-time time-invariant delay system to the Laguerre spectrum of its input signal. The Laguerre-domain system representation is meaningful when the input signal is square summable, and the system is stable. A certain type of polynomials arising in the evaluation of the output spectrum due to the presence of time delay is defined, and key properties of these are investigated. The polynomials are characterized by a three-term recurrence relation that also facilitates their numerically reliable calculation.

Discretization of the Functionals With Prescribed Derivative

There are a number of recent results aimed at developing finite necessary and sufficient stability tests for linear time-invariant time delay systems within the framework of Lyapunov-Krasovskii functionals with prescribed derivative approach. These tests are based on the concept of Lyapunov matrix. They usually require verification of positive definiteness of a certain block matrix whose blocks correspond to the Lyapunov matrix evaluated at several discretization points. In this work, we present a new test of this kind which employs a combination of the discretized Lyapunov functionals method of K. Gu and the functionals with prescribed derivative approach. Unlike existing combinations of those techniques, we avoid discretization of the functional's derivative and use the methodology of the necessary and sufficient stability tests development instead. We show a connection between our test and some of the existing results.

Observer Design in LTI Time-Delay Systems using Partial Pole Placement with Applications

In order to estimate the state of a dynamical system, we address the problem of designing an observer for linear time-invariant (LTI) dynamical systems including a time-delay. Sufficient conditions for the existence of the proposed observer are given using partial placement of the poles for the error. Namely, we exploit the multiplicity-induced-dominance property of the characteristic function corresponding to the system's error. The effectiveness of the proposed observer design is shown in both state lag and input lag respectively through the problems of Mach number control in a wind tunnel and stabilization of the inverted pendulum on a cart by using the delay.

Robust Controller Design for Descriptor-type Time-delay Systems

Linear time-invariant descriptor-type time-delay systems are considered. A robust stabilizing controller design approach for such systems is introduced. Uncertainties both in the time-delays and in other system parameters are considered. A frequency-dependent scalar bound on such uncertainties is first derived. Once this bound is found, the controller design is completely based on the nominal model. However, satisfying a scalar frequency-dependent condition, which uses the derived bound, guarantees robust stability. An example is also presented to illustrate the proposed approach

Delay independence of modes, controllability, observability, and decentralised fixed modes

The most general form of linear time-invariant time-delay systems described by delay-differential-algebraic equations is considered. Necessary and sufficient conditions for a mode of such a system to be delay-independent are derived. Then, such a system under centralised control is considered, and some necessary and sufficient conditions for a mode of such a system to be spectrally uncontrollable and/or spectrally unobservable independent of any time-delays are derived. Finally, such a system under a decentralised control structure is considered, and some necessary and sufficient conditions for a decentralised fixed mode to be independent of any time-delays are derived. An example to illustrate the use of the results is also presented.

Necessary stability conditions for linear systems with incommensurate delays

The Lyapunov matrix is a key element of the construction of Lyapunov–Krasovskii functionals with prescribed derivative for linear time-invariant time delay systems. Moreover, stability criteria which are based exclusively on the Lyapunov matrix have been developed recently. A weak point of the approach is a lack of effective techniques to compute the Lyapunov matrix when the delays are incommensurate, i.e. at least two of them are rationally independent. To overcome this difficulty, we present in this paper necessary stability conditions for systems with incommensurate delays based on the Lyapunov matrix of a “close” auxiliary system with commensurate delays, which can be computed by known techniques. An explicit condition for the choice of delays of the auxiliary system is provided. Illustrative examples are given.

Complete type functionals for homogeneous time delay systems

For linear time-invariant time delay systems, the so-called Lyapunov–Krasovskii functionals of complete type (Kharitonov and Zhabko, 2003) are known to be effective in the stability analysis and a number of applications. More precisely, there exist the necessary and sufficient asymptotic stability and instability conditions expressed in terms of these functionals. The case excluded from consideration in the theory (since the functionals either do not exist or are not uniquely defined) is violation of the Lyapunov condition, i.e. the case of systems with the eigenvalues placed symmetrically with respect to the origin of the complex plane. In this paper, an analogue of this theory for a class of nonlinear time delay systems with homogeneous right-hand sides of degree greater than one and a constant delay is developed. An explicit expression for the Lyapunov–Krasovskii functionals as well as necessary and sufficient conditions for the asymptotic stability and instability of the trivial solution based on these functionals are given. An important assumption, which constitutes an analogue of the Lyapunov condition for linear systems, is existence of the Lyapunov function for a delay free system, obtained from the original one setting the delay equal to zero. Furthermore, this Lyapunov function is a key element in the construction of the functional. The functionals are applied to estimating the region of attraction.

Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a property usually referred to as multiplicity-induced-dominancy (MID). In this paper, we investigate the MID property for one of the simplest cases of systems with two delays, a scalar delay-differential equation of first order with two delayed terms of order zero. We discuss the standard approach based on the argument principle for establishing the MID property for single-delay systems and some of its limitations in the case of our simple system with two delays, before proposing a technique based on crossing imaginary roots that allows to conclude that the MID property holds in our setting.

Model Reference Adaptive Controller for LTI Systems with Time-variant Delay

In this paper, a new Direct Model Reference Adaptive Control Procedure (DMRAC) for Linear Time-Invariant (LTI) delay systems is presented with the use of the concept of the command generator tracker which expands the class of processes that can now be controlled with zero output error. The stability of the error between the system and the model is guaranteed by the Lyapunov theory. The new algorithm is applied to control a perturbed delay system. Matlab simulation examples are given to demonstrate the usefulness of the algorithm.

Design of [formula omitted] stable fixed-order decentralised controllers in a network of sampled-data systems with time-delays

A methodology is proposed for the design of sampled-data fixed-order decentralised controllers for Multiple Input Multiple Output (MIMO) Linear Time-Invariant (LTI) time-delay systems. Imperfections in the communication links between continuous-time plants and controllers arising due to transmission time-delays, aperiodic sampling, and asynchronous sensors and actuators are considered. We model the errors induced due to the control imperfections using an operator approach leading to a simple L2 stability criterion. A frequency domain-based direct optimisation approach towards controller design is proposed in this paper. This approach relies on the minimisation of cost functions, for stability and robustness against control imperfections, as a function of the controller or design parameters. First, the proposed method towards controller design is applied to generic MIMO LTI systems with time-delays. Second, when the delay system to be controlled has the structure of a network of coupled quasi-identical subsystems, we use a scalable algorithm to design identical decentralised controllers through network structure exploitation. Quasi-identical subsystems are identical subsystems that have non-identical uncertainties or control imperfections. By exploiting the structure, we improve the computational efficiency and scalability with the number of subsystems. The methodology has been implemented in a publicly available software, which supports system models in terms of delay differential algebraic equations. Finally, the effectiveness of the methodology is illustrated using a numerical example.

Robust stabilisation of linear time‐invariant time‐delay systems via first order and super‐twisting sliding mode controllers

This study presents a novel scheme for the synthesis of first-order and super-twisting sliding mode controllers for the robust stabilisation of a class of linear time-invariant time-delay systems subject to matched disturbances. Starting from a stability analysis of the system to guarantee that the resulting sliding mode dynamics is asymptotically stable, linear matrix inequality conditions of reduced conservatism are derived by using the Lyapunov-Krasovskii approach. Based on the stability analysis, the sliding mode controllers are synthesised to force the evolution of the closed-loop system trajectories to converge onto a prescribed sliding surface and to ensure that they remain there for all subsequent time. Unlike existing results, the implementation of the proposed approach does not involve strong requirements on the system structure. A numerical and a practical example along with a comparative analysis prove the effectiveness of the proposal and highlight its benefits.

A Novel Method to Compute the Structured Distance to Instability for Combined Uncertainties on Delays and System Matrices

This paper presents a novel method to compute the distance to instability for linear time-invariant delay systems with multiple delays, subjected to combined uncertainties on the system matrices and the delay parameters. The method is able to fully exploit structure and possible interdependencies between the uncertainties affecting the coefficient matrices, the property that realistic perturbations are real valued, and the structure of the delay equation. The resulting algorithm relies on a characterization in terms of the pseudospectral abscissa and on techniques from optimization over manifolds of matrices.

Imaginary characteristic roots of neutral systems with commensurate delays

This paper concentrates on stability analysis of neutral linear time-invariant (LTI) delay systems with multiple commensurate time-delays. A new numerical procedure is offered for the determination of purely imaginary characteristic roots of neutral delay systems, which plays a crucial role in assessing the system stability. Based on simple linear algebra it is shown that the imaginary characteristic roots of such a system can be found by calculating the generalized eigenvalues of an associated matrix pair. If the system is of retarded type we just need to determine the eigenvalues of a single matrix. The results extend previously known work on neutral delay systems subject to a single delay for neutral systems subject to multiple commensurate delays. In the light of the main result of this paper we present a new method to determine the largest first time-delay interval for which a neutral delay system preserves its stability. The paper is closed by showing numerical examples that illustrate the applicability and effectiveness of the proposed method.

Computing the robust H-infinity norm of time-delay LTI systems with real-valued and structured uncertainties

In this paper a new method to calculate the robust (worst-case) H∞-norm of a linear time-invariant time-delay system with an arbitrary number of delays with uncertainties both in the system matrices and the delay parameters is presented. The proposed approach fully exploits the real-valued and structured nature of the uncertainties. As in real-life applications, uncertainties can influence more than one system matrix. The algorithm utilizes the relation between the robust H∞-norm and the pseudo-spectrum of an associated non-linear (delay) eigenvalue problem with both real and complex perturbations. More precisely it uses the Newton-bisection method to find the robust complex distance to instability of this eigenvalue problem which will be shown to be equal to the reciprocal of the robust H∞-norm.

Parameter-Space Stability Analysis of LTI Time-Delay Systems With Parametric Uncertainties

A novel approach to stability analysis of linear time-invariant (LTI) time-delay systems of retarded type with incommensurate time delays and polynomial dependence on parametric uncertainties is presented. Using a branch and bound algorithm, which relies on Taylor models and polynomials in Bernstein form, we first determine the stability-crossing set in the delay/parameter space and then evaluate stability for each disjoint region. The novel approach can be used to either nonconservatively check stability of time-delay systems with interval parameters or to map stable regions to a low-dimensional parameter space while taking additional interval parameters into account. The algorithm is applied to several examples with parametric uncertainties and/or incommensurate time delays.

A Suboptimal Shifting Based Zero-pole Placement Method for Systems with Delays

An appropriate setting of eventual controller parameters for a derived controller structure represents an integral part of many control design approaches for dynamical systems. This contribution is aimed at a practically applicable and uncomplicated controller tuning method for linear time-invariant time delay systems (LTI-TDSs). It is based on placing the dominant poles as well as zeros of the given infinite-dimensional feedback control system by matching them with the desired ones given by known dynamical properties of a simple fixed finite-dimensional model. The desired placing is done successively by applying the Quasi-Continuous Shifting Algorithm (QCSA) first such that poles and zeros are forced to be as close as possible to those of the model. Concurrently, rests of both system spectra are shifted to the left as far as possible to minimize the spectral abscissa. The obtained results are then enhanced by a non-convex optimization technique applied to a selected objective function reflecting the distance of desired model roots from the eventual system ones and the spectral abscissae. Retarded LTI-TDS are primarily considered; however, systems with neutral delays are touched as well. The efficiency of the proposed method is proved via numerical examples in Matlab/Simulink environment. Some drawbacks and possible improvements or extensions of the algorithm for the future research are also concisely suggested to the reader.

A comparative study of two tuning rules for delayed compensation controllers

The focus of this contribution is on the use of two controller tuning techniques for delayed controllers designed by an algebraic approach for linear time-invariant time delay systems. The well-known Chien-Hrones-Reswick (CHR) method and the Equalization Method (EM) are used. The tuning procedure is applied to compensation-type controllers that include internal delays, and hence there can be found a link to the habitual Smith predictor structure. This study considers two typical representatives of controlled plants with both input-output and internal delays; namely, the first- and second-order (of derivatives) systems are taken. Numerical comparative experiments are presented and discussed.

Stabilization of Lossless Propagation Time-Delay Systems * *This work is supported by the Scientific and Technical Research Council of Turkey (TÜBİTAK) under grant number 115E379.

Abstract Problem of constructing finite-dimensional stabilizing controllers for a class of linear time-invariant neutral time-delay systems, namely lossless propagation time-delay systems, is considered while the effect of small time-delay perturbations on stability is taken into account. For this purpose, a dynamic output feedback controller design approach which provides more flexible and automated design of the controller than previously proposed design approaches based on the continuous pole placement algorithm is proposed. A design example is also presented, to demonstrate the proposed approach.