Search

This article discusses invariant subspaces of a matrix with a given partition structure. The existence of a nontrivial structured invariant subspace is equivalent to the possibility of decomposing the associated system with multiple feedback blocks such that the feedback operators are subject to a given constraint. The formulation is especially useful in the stability analysis of time‐delay systems using the Lyapunov–Krasovskii functional approach where computational efficiency is essential in order to achieve accuracy for large scale systems. The set of all structured invariant subspaces are obtained (thus all possible decompositions are obtained as a result) for the coupled differential‐difference equations (DDE) associated with the DDE of retarded and neutral types, as well as systems with a time‐varying delay. It was shown that the known ad hoc methods of reducing the dimensions of delay channels can be considered as special cases of decomposition where one subsystem has trivial dynamics. The reduction of computational cost is demonstrated by a numerical example. For the general case, a recursive procedure is developed to obtain the set of all structured invariant subspaces. Based on this procedure, a method is presented to obtain a nontrivial structured invariant subspace that considers computational efficiency and increased possibility of terminating in a finite number of steps. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This note is concerned with the stability problem of linear delay-differential systems of neutral type. A discretized Lyapunov functional approach is developed. The resulting stability criterion is formulated in the form of a linear matrix inequality. For nominal systems, the analytical results can be approached with fine discretization. Numerical examples show significant improvement over approaches in the literature. Index Terms--Linear matrix inequality (LMI), Lyapunov functional, stability, time-delay, uncertainty.

The robust stability problem of linear time-delay systems with norm-bounded uncertainty is further investigated by using a refined discretized Lyapunov functional approach. A new stability criterion is derived. The computational requirement is reduced for the same discretization mesh. Examples show that the results obtained by this new criterion significantly improve the estimate of the stability limit over some existing results in the literature. Index Terms--Linear matrix inequality (LMI), Lyapunov functional, stability, time-delay, uncertainty.

This paper gives a broad overview of the stability and control of time-delay systems. Emphasis is on the more recent progress and engineering applications. Examples of practical problems, mathematical descriptions, stability and performance analysis, and feedback control are discussed. Keywords: Time Delay, Stability, Feedback Control Lyapunov-Krasovskii Functional Linear Matrix Inequality

In a recent paper, it was proved that the frequency response arc and the inner frequency response set associated with D-stable polynomial is convex. This note points out that clockwise property and positive curvature is sufficient for convexity based on a result in classical differential geometry and provides an alternative proof of the convexity based on curvature. It is felt that this line of proof may be more easily generalized to study the rational function cases.

The design of stabilizing linear output feedback control of uncertain systems is a computationally demanding optimization problem as local minima may exist which are distinct from the global minimum. However, in some special cases, quasiconvexity can be proven through a simple redefinition of variables and/or a reformulation of the problem in the dual form. Such special cases include state feedback control for both discrete and continuous time systems.

The stability of linear systems subject to possibly fast time-varying uncertainties is analyzed. A method is developed to compute the upper bound of uncertainty size allowed for retaining quadratic stability. The algorithm consists of a two-level optimization. The inner level of the algorithm consists of choosing an extremum among a finite number of values. It is proved that although the outer level of the algorithm is a nonconvex optimization problem, no local minimum distinct from the global minimum can exist. An illustrative example is presented.

This paper concerns [H.sub.[infinity]] control of systems under norm bounded uncertainties in all the system matrices. This is an extension of the work by Xie et al. where only the A matrix is allowed to be uncertain. It is found that the problem shares the same formulation with the [H.sub.[infinity]] control problem for systems without uncertainties. It can also be viewed as reducing the problem of dealing with systems with both structured uncertainties and unstructured uncertainties to one with unstructured uncertainties only.

This note considers the robust stability problem for a class of time-delay systems with norm-bounded, and possibly time-varying, uncertainty. Based on the discretized Lyapunov functional approach, a stability criterion is derived. The time-delay is assumed constant and known. Numerical examples show that the results obtained by this new criterion significantly improve the estimate of the stability limit over some existing results in the literature. Index Terms--Linear matrix inequality (LMI), Lyapunov functional, stability, time-delay, uncertainty.

This note focuses on the analysis of conservatism introduced due to some model transformations used in the control literature. This work is an extension of a previous paper by the ,same authors on additional dynamics. The additional dynamics of the second order and parameterized model transformations are examined. The neutral model transformation introduces additional constraints for stability, which can be analyzed in a very similar manner to additional dynamics. Index Terms--Delay, model transformation, stability.

In studying the stability of time delay systems, many published results use a transformation to transform a system with single time delay to a system with distributed delay. In this article, the inherent limitations of such approaches are studied. Specifically, it is shown that such a transformation incurs additional dynamics that can be characterized by appropriate additional eigenvalues. The critical delay values when such additional eigenvalues cross the imaginary axis can be explicitly calculated. If the smallest of such delays is less than the stability delay limit of the original system, then any stability criteria obtained using such transformation will be conservative. Some examples are also included. Index Terms--Razumikhin theorem, stability, time-delay systems.

This paper studies the migration of double imaginary roots of the systems’ characteristic equation when two parameters are subjected to small deviations. The proposed approach covers a wide range of models. Under the least degeneracy assumptions, we found that the local stability crossing curve has a cusp at the point that corresponds to the double root, and it divides the neighborhood of this point into an S-sector and a G-sector. When the parameters move into the G-sector, one of the roots moves to the right half-plane, and the other moves to the left half-plane. When the parameters move into the S-sector, both roots move either to the left half-plane or the right half-plane depending on the sign of a quantity that depends on the characteristic function and its derivatives up to the third order. [ABSTRACT FROM AUTHOR]

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 τ i , i = 1 , 2 , … , K . The characteristic quasipolynomial of such an equation is a multilinear function of e − τ i s . 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. [ABSTRACT FROM AUTHOR]

Abstract: In general, a system with time delay may have multiple stable delay intervals. Especially, a stable delay interval does not always contain zero. Asymptotically accurate stability conditions such as discretized Lyapunov–Krasovskii functional (DLF) method and sum-of-square (SOS) method are especially effective for such systems. In this article, a DLF-based method is proposed to estimate the maximal stable delay interval accurately without using bisection when one point in this interval is given. The method is formulated as a generalized eigenvalue problem (GEVP) of linear matrix inequalities (LMIs), and an accurate estimate may be reached by iteration either in a finite number of steps or asymptotically. The coupled differential–difference equation formulation is used to illustrate the method. However, the idea can be easily adapted to the traditional differential–difference equation setting. [Copyright &y& Elsevier]

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 differentialdifference 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. [ABSTRACT FROM AUTHOR]

Abstract: Many practical systems have a large number of state variables, but only a few components involve time delays. These components are often scalar or low dimensional, and typically only one delay is present in each such component. A special form of coupled differential–difference equations with one delay per channel proposed recently is well suited to formulate such systems. This article extends the discretized Lyapunov–Krasovskii functional method to this class of systems. In addition to generality, this formulation also drastically reduces the computational cost for a typical system, and therefore is appropriate to use even for time-delay systems of retarded type. The discretized formulation is also simpler than the previous formulation for systems with multiple delays. [Copyright &y& Elsevier]

Abstract: A new form of coupled differential–difference equations with one delay in each channel is proposed to model systems with multiple delays. This formulation has significant advantage over the traditional differential–difference equations or the prevailing form of coupled differential–difference equations with multiple delays. Fundamental solutions, general solutions, and the construction of the Lyapunov–Krasovskii functional are discussed. For systems with a large number of state variables with multiple low-dimensional delay elements, this formulation allows a drastic reduction of computational cost as compared to the traditional differential–difference equation formulation when the discretized Lyapunov–Krasovskii functional method is used. [Copyright &y& Elsevier]

Abstract: This article discusses the Lyapunov–Krasovskii functional approach for the stability problem of coupled differential-functional equations. Such systems include, as special cases, many types of time-delay systems, including the lossless propagation model, some neutral time-delay systems and singular time-delay systems. After the general stability theory, the special case of coupled differential-difference equations is discussed, and the necessity for the existence of quadratic Lyapunov–Krasovskii functional is established. Discretization is used to render the stability criterion to an LMI form. [Copyright &y& Elsevier]

In this paper, we present a geometric method for describing the effects of the ‘delay-induced uncertainty’ on the stability of a standard Smith predictor control scheme. The method consists of deriving the ‘stability crossing curves’ in the parameter space defined by the ‘nominal delay’ and ‘delay uncertainty’, respectively. More precisely, we start by computing the ‘crossing set’, which consists of all frequencies corresponding to all points on the stability crossing curve, and next we give their ‘complete classification’, including also the explicit characterization of the ‘directions’ in which the zeros cross the imaginary axis. This approach complements existing algebraic stability tests, and it allows some new insights in the stability analysis of such control schemes. Several illustrative examples are also included. [ABSTRACT FROM PUBLISHER]

The discretized Lyapunov functional method for the stability problem of time-delay systems is further refined using a combination of integral inequality and variable elimination technique. As a result, the computational requirement is further reduced for the same discretization mesh. For systems without uncertainty, the convergence to the analytical solution is greatly accelerated. For uncertain systems, the new method is much less conservative. Numerical examples are presented to illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]