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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 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