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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 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]

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]

The stability of uncertain systems with multiple time-delay is studied using a quadratic Lyapunov functional. By choosing piecewise linear parameters for the kernel of a quadratic expression, the guaranteed stability condition is formulated in the form of linear matrix inequality. The extent of conservatism depends on the grid size. However, numerical results show that even a very coarse discretization can produce quite satisfactory results. The possibility of non-uniform grid size and more general Lyapunov functional is of interest even for a single delay system. Numerical examples are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]