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Special Issue on Time-Delay Systems

Time-delay often appears in many control systems (such as aircraft, chemical or process control systems) either in the state, the control input, or the measurements. Unlike ordinary differential equations, delay systems are infinite dimensional in nature and time-delay is, in many cases, a source of instability. The stability issue and the performance of control systems with delay are, therefore, both of theoretical and practical importance. Delay equations were first considered in the literature in the XVIII century (e.g. the works of Bernoulli, Euler or Condorcet). A systematic study of such equations began in the 20’s of the XX century by V. Volterra, and in the end of 40’s of the same century by A. Myshkis and R. Bellman. Lyapunov’s second method for the stability of delay systems was developed at the end of 50’s by N. Krasovskii who introduced Lyapunov functionals and by B. Razumikhin who formulated the appropriate Lyapunov functions. Since 1950s, the subject of delay systems or functional differential equations has received a great deal of attention in Mathematics, Biology and Control Engineering. Over the past decade, much effort has been invested in the analysis and synthesis of uncertain systems with time-delay. Based on the Lyapunov theory of stability, various results have been obtained that provide, for example, finite-dimensional sufficient conditions for stability and stabilization. Departing from the classical linear finite-dimensional techniques which apply Smith predictor type designs, the new methods simultaneously allow for delays in the state equations and for uncertainties in both the system parameters and the time delays. During the early stages, delay-independent results were obtained which guarantee stability and prescribed performance levels of the resulting solutions. Recently, delay-dependent results have been derived that considerably reduce the overdesign entailed in the delay-independent solutions. In the present issue, new results are obtained for various control and identification problems for delay systems. These results are based either on Lyapunov methods or on frequency domain considerations. The nine papers included in this issue have been written by prominent specialists in the field of Systems and Control who have touched upon a variety of control and stability problems for linear and nonlinear systems with delay. Lyapunov-based stability methods for different classes of linear delay equations, either continuous or discrete time (difference equations), are presented in the first three papers: