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We investigate feedback control of linear quantum systems subject to feedback-loop time delays. In particular, we examine the relation between the potentially achievable control performance and the time delays, and provide theoretical guidelines for the future experimental setup in two physical systems, which are typical in this research field. The evaluation criterion for the analysis is given by the optimal control performance formula, the derivation of which is from the classical control theoretic results about the input-output delay systems.

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The stability of feedback systems consisting of linear time-delay plants and PID controllers has been investigated for many years by means of several methods, of which the Nyquist criterion, a generalization of the Hermite-Biehler Theorem, and the root location method are well known. The main purpose of these researches is to determine the range of controller parameters that allow stability. Explicit and complete expressions of the boundaries of these regions and computation procedures with a finite number of steps are now available only for first-order plants, provided with one time delay. In this note, the same results, based on Pontryagin's studies, are presented for arbitrary-order plants.

This paper deals with linear time-varying, delay systems. Extensions of the concept of differential flatness \cite{Fliess_95} to this context have been first proposed in \cite{Mounier_95,Fliess_96} (see also \cite{Rudolph_03,Chyzak_05}), by the introduction of $\pi$-flat output. Roughly speaking, it means that every system variable may be expressed as a function of a particular output $y$, a finite number of its time derivatives, time delays, and predictions, the latter resulting from the prediction operator $\pi^{-1}$. We propose a simple and constructive algorithm for the computation of $\pi$-flat outputs based on concepts of polynomial algebra, in particular Smith-Jacobson decomposition of polynomial matrices. Some examples are provided to illustrate the proposed methodology.

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parametrization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.

In this paper, we report the identification of global and partial phase synchronizations in linear arrays of unidirectionally coupled piecewise linear time-delay systems with two different coupling configurations. In particular, in a linear array with open end boundary conditions, global phase synchronization (GPS) is achieved by a sequential synchronization as a function of the coupling strength (a second order transition). Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. On the other hand, in a linear array with closed end boundary conditions (ring topology), partial phase synchronization (PPS) is achieved by forming different groups of phase synchronized clusters above some threshold value of the coupling strength (a first order transition) where they continue to be in a stable PPS state. We confirm the occurrence of both global and partial phase synchronizations in two different piecewise linear time-delay systems using various numerical methods.

In this paper, we report the identification of global and partial phase synchronizations in linear arrays of unidirectionally coupled piecewise linear time-delay systems with two different coupling configurations. In particular, in a linear array with open end boundary conditions, global phase synchronization (GPS) is achieved by a sequential synchronization as a function of the coupling strength (a second order transition). Further, the asynchronous ones in the array with respect to the main sequentially synchronized cluster organize themselves to form clusters before they achieve synchronization with the main cluster. On the other hand, in a linear array with closed end boundary conditions (ring topology), partial phase synchronization (PPS) is achieved by forming different groups of phase synchronized clusters above some threshold value of the coupling strength (a first order transition) where they continue to be in a stable PPS state. We confirm the occurrence of both global and partial phase synchronizations in two different piecewise linear time-delay systems using various numerical methods.

In this paper, we study the cooperative output regulation problem for the discrete linear time-delay multi-agent systems by distributed observer approach. In contrast with the same problem for continuous-time linear time-delay multi-agent systems, the problem has two new features. First, in the presence of time-delay, the regulator equations for discrete-time linear systems are different from those for continuous-time linear systems. Second, under the standard assumption on the connectivity of the communication graph, a distributed observer for any continuous-time linear leader system always exists. However, this is not the case for discrete-time systems, and the behavior of a distributed observer is much more complicated. Thus, we will first study the existence of the discrete distributed observer, and then present the solvability of the problem by distributed dynamic output feedback control law.

In this paper, we study the cooperative output regulation problem for the discrete linear time-delay multi-agent systems by distributed observer approach. In contrast with the same problem for continuous-time linear time-delay multi-agent systems, the problem has two new features. First, in the presence of time-delay, the regulator equations for discrete-time linear systems are different from those for continuous-time linear systems. Second, under the standard assumption on the connectivity of the communication graph, a distributed observer for any continuous-time linear leader system always exists. However, this is not the case for discrete-time systems, and the behavior of a distributed observer is much more complicated. Thus, we will first study the existence of the discrete distributed observer, and then present the solvability of the problem by distributed dynamic output feedback control law.

In this paper, we study the cooperative robust output regulation problem for linear uncertain multi-agent systems with both communication delay and input delay by the distributed internal model approach. The problem includes the leader-following consensus problem of linear multi-agent systems with time-delay as a special case. We first generalize the internal model design method to systems with both communication delay and input delay. Then, under a set of standard assumptions, we have obtained the solution of the problem via both the state feedback control and the output feedback control. In contrast with the existing results, our results apply to general uncertain linear multi-agent systems, accommodate a large class of leader signals, and achieve the asymptotic tracking and disturbance rejection at the same time.

After a brief introduction on the parameter plane methods, the behavior of simple systems with time delay has been initially studied by the root-loci method and then the state of the art of parameter plane methods with time delay has been presented. A variation of the solution in the state of the art presentation has been derived and an appropriate digital computer program has been constructed. As an extension of studies in the parameter plane, a three variable parameter system has been approached by using basic descriptive geometry properties and simple illustrative examples have been worked out in the parameter space. Finally, a real life system with time delay has been analysed and redesigned by parameter plane methods.

For linear stochastic systems with time delay, the optimal control is derived that minimizes the ensemble average of a quadratic (in states and control) performance measure. The optimal control obtained is functionally dependent upon the expected values of the state variables conditioned on the measurements. It is shown that the optimal control and estimation can be performed independently; i.e., the separation theorem holds for the class of problems considered. The optimal control is linearly dependent upon the best estimates which minimize the expected value of the estimation error squared.