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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]

Abstract: This article studies the stability crossing set of linear time-invariant systems with two scalar-delay channels. This study is crucial to the complete stability analysis along the idea of D-subdivision method. The characteristic quasipolynomial of such systems contains an exponential term with the sum of two delays in its exponent (cross term). A complete parameterization and geometric characterization of the stability crossing set is conducted. It was found instrumental to relate it to an associated quasipolynomial without such a cross term through an elimination process. However, a spurious stability crossing set may arise in this process. A revised parameterization method is derived that automatically eliminates such a spurious stability crossing set. [Copyright &y& Elsevier]

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]