An Introduction to the Special Issue on Time Delay Systems: Modelling, Identification, Stability, Control and Applications

The special Issue “Time Delay Systems: Modelling, Identification, Stability, Control and Applications” intents to collect high-erudite papers aiming theoretical and/or practical matters dealing with time delay systems. In feedback control systems, delay as a generic part of many processes is the phenomenon which unambiguously deteriorates the quality of a control performance. Modern control theory has been dealing with this problem since its nascence – the well known Smith predictor has been known for longer than five decades. Systems with delays in technological and other processes are usually assumed to contain delay elements in input-output relations only, which results in shifted arguments on the right-hand side of differential equations. All the system dynamics is hence traditionally modelled by point accumulations in the form of a set of ordinary differential equations. However, this conception is a rather restrictive in effort to fit the real plant dynamics since inner feedbacks can be of timedistributed or delayed nature. Time delay (hereditary, anisochronic) models, contrariwise, offer a more universal dynamics description applying both derivatives and delay elements on the left-hand side of a differential equation, either in a lumped or distributed form. Modelling, identification, stability analysis, stabilization, control, etc. of time delay systems are challenging and fascinating tasks in modern systems and control theory as well as in academic and industrial applications. Many related problems are unsolved and many questions remain unanswered. The aim of this special issue is to highlight greatly significant recent developments on the topics of time delay systems, their estimation, modelling and identification, stability analysis, various (algebraic, adaptive and predictive) control strategies, relaybased autotuning and interesting academic and reallife applications. The papers included in the special issue are ordered from the most theoretical one to more directly applicable ones. The state estimation or filtering problem is of great importance in both theory and application, and in the last decades, this problem has gotten extensive concern and many solution schemes have been proposed and successfully put into action. Among them, Kalman filtering, which minimizes the variance of the estimation error, is the most famous one. Observer design for linear time delay systems is the matter of the first paper of Mohammad Ali Pakzad, entitled “Kalman Filter Design for Time Delay Systems”. An easy way to compute least square estimation error of an observer for time delay systems is derived, where the time delay terms exist in the state and output of the system. Based on the least square estimation error an optimization algorithm to compute a Kalman filter for time delay systems is proposed. By employing the finite characterization of a Lyapunov functional equation, the existence of sufficient conditions for obtaining the right solution and guaranteeing the proper convergence rate of the estimation error is evaluated. It is shown that this finite characterization can be calculated by means of a matrix exponential function. The desirable performance of the proposed observer is demonstrated through the simulation of several numerical examples. Fractional differential equations have gained considerable significance and most fractional systems may contain a delay term. The authors Mohammad Ali Pakzad and Sara Pakzad propose an exact method for the BIBO stability analysis of a large class of fractional order delay systems. In their paper entitled “Stability Map of Fractional Order Time-Delay Systems”, the stability robustness for linear time invariant fractional order systems with time delay against delay uncertainties is considered. The complexity arises due to the exponential type transcendental terms and fractional order in their characteristic equation. It is shown that this procedure numerically reveals all possible stability regions exclusively in the space of the delay. Using WSEAS TRANSACTIONS on SYSTEMS Libor Pekař, Filippo Neri