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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 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]

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]

The stability problem for systems with distributed delay is considered using discretized Lyapunov functional. The coefficients associated with the distributed delay are assumed to be piecewise constant, and the discretization mesh may be non-uniform. The resulting stability criteria are written in the form of linear matrix inequality. Numerical examples are also provided to illustrate the effectiveness of the method. The basic idea can be extended to a more general setting with more involved formulation. [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]

The design problem of model-based adaptive control of robots with unknown parameters is considered. By introducing a nonholonomic transformation of the error dynamics, a general class of adaptive algorithms is obtained, which includes some previously derived results as special cases. The algorithms can be implemented through fast recursive schemes such as the Newton-Euler formulation without requiring excessive feedback gains. By setting certain terms, it is also possible to obtain an exponential convergence of the joint errors. [ABSTRACT FROM AUTHOR]