A Review of Some Subtleties of Practical Relevance

This paper reviews some subtleties in time-delay systems of neutral type that are believed to be of particular relevance in practice. Both traditional formulation and the coupled differential-difference equation formulation are used. The discontinuity of the spectrum as a function of delays is discussed. Conditions to guarantee stability under small parameter variations are given. A number of subjects that have been discussed in the literature, often using different methods, are reviewed to illustrate some fundamental concepts. These include systems with small delays, the sensitivity of Smith predictor to small delay mismatch, and the discrete implementation of distributed-delay feedback control. The framework prsented in this paper makes it possible to provide simpler formulation and strengthen, generalize, or provide alternative interpretation of the existing results

External Direct Sum Invariant Subspace and Decomposition of Coupled Differential-Difference Equations

This article discusses the invariant subspaces that are restricted to be external direct sums. Some existence conditions are presented that facilitate finding such invariant subspaces. This problem is related to the decomposition of coupled differential-difference equations, leading to the possibility of lowering the dimensions of coupled differential-difference equations. As has been well documented, lowering the dimension of coupled differential-difference equations can drastically reduce the computational time needed in stability analysis when a complete quadratic Lyapunov-Krasovskii functional is used. Most known ad hoc methods of reducing the order are special cases of this formulation

Strong stability of a class of difference equations of continuous time and structured singular value problem

This article studies the strong stability of scalar difference equations of continuous time in which the delays are sums of a number of independent parameters tau_i, i = 1, 2, . . . ,K. The characteristic quasipolynomial of such an equation is a multilinear function of exp(-tau_i s). It is known that the characteristic quasipolynomial of any difference equation set in the form of one-delayper- scalar-channel (ODPSC) model is also in such a multilinear form. However, it is shown in this article that some multilinear forms of quasipolynomials are not characteristic quasipolynomials of any ODPSC difference equation set. The equivalence between local strong stability, the exponential stability of a fixed set of rationally independent delays, and the stability for all positive delays is shown, and relations with the structured singular value problem are presented. A procedure to determine strong stability in the special case of up to three independent delay parameters in finite steps is developed. This procedure means that the structured singular value problem in the case of up to three scalar complex uncertain blocks can be solved in finite steps

Strong stability of a class of difference equations of continuous time and structured singular value problem

This article studies the strong stability of scalar difference equations of continuous time in which the delays are sums of a number of independent parameters τi, i = 1, 2, . . . , K. The characteristic quasipolynomial of such an equation is a multilinear function of e−τis. It is known that the characteristic quasipolynomial of any difference equation set in the form of one-delay-per-scalar-channel (ODPSC) model is also in such a multilinear form. However, it is shown in this article that some multilinear forms of quasipolynomials are not characteristic quasipolynomials of any ODPSC difference equation set. The equivalence between local strong stability, the exponential stability of a fixed set of rationally independent delays, and the stability for all positive delays is shown, and relations with the structured singular value problem are presented. A procedure to determine strong stability in the special case of up to three independent delay parameters in finite steps is developed. This procedure means that the structured singular value problem in the case of up to three scalar complex uncertain blocks can be solved in finite steps

Delay-Independent Stability Analysis of Linear Time-Delay Systems Based on Frequency

This paper studies strong delay-independent stability of linear time-invariant systems. It is known that delay-independent stability of time-delay systems is equivalent to some frequency-dependent linear matrix inequalities. To reduce or eliminate conservatism of stability criteria, the frequency domain is discretized into several sub-intervals, and piecewise constant Lyapunov matrices are employed to analyze the frequency-dependent stability condition. Applying the generalized Kalman–Yakubovich–Popov lemma, new necessary and sufficient criteria are then obtained for strong delay-independent stability of systems with a single delay. The effectiveness of the proposed method is illustrated by a numerical example

Transforming time-varying multivariable systems into block companion canonical forms

AbstractThe problem of transforming a class of linear time-varying continuous time systems into controllable and observable block companion canonical forms is considered. In terms of system block controllability (observability) matrix, this paper generalizes the results of Shieh et al. [3] and provides systematic and straightforward algorithms for obtaining block companion canonical forms. An example is provided to illustrate this transformation technique

Some remarks on Smith predictors A geometric point of view

International audienceIn this paper we develop a method to obtain the stability crossing curves of a Smith Predictor control scheme. More explicitly, we compute the crossing set, which consists of all frequencies corresponding to all points on the stability crossing curve, and we give their complete classification. Furthermore, the directions in which the zeros cross the imaginary axis are explicitly expressed

Some remarks on the delay stabilizing effect in SISO systems

This note addresses the stabilization problem of a class of SISO systems with a time delay in the input, and explore the stabilizing effect of time delay. More precisely, for a fixed feedback gain such that the closed loop system is unstable when the delay is set to zero, we present necessary and sufficient conditions for the delays such that the stability in closed-loop is achieved, and provide an explicit construction of the controllers. Next, we analyze conditions for preserving the closed-loop stability if parametric or time-varying delay uncertainties are present in the control law. Illustrative examples are also proposed

A geometric description of the set of stabilizing PID controllers

This article developed a new method to described the set of stabilizing PID control. The method is based on D-parameterization with natural description of the set. It was found that the stability crossing surface is a ruled surface that is completely determined by a curve known as discriminant. The discriminant is divided into sectors at the cusps. Corresponding to the sectors, the stability crossing surface is divided into positive and negative patches. A systematic study is conducted to identify the regions with a fixed number of right half-plane characteristic roots. The crossing directions of characteristic roots for positive patches and negative patches are also studied. As a result, a systematic method is developed to identify the regions of PID parameter such that the system is stabilized