Numerical modelling of control time-delay system

Discrete-time control schemes are often used for realization of positional control strategies for continuous time control problems (Krasovskii and Subbotin, 1974; Osipov, 1971). However for general systems with delays (which are often called functional differential equations (FDE)) exact solutions in both continuous and discrete schemes can be found in exceptional cases. So, as rule, it is necessary to solve corresponding control problems using suitable numerical algorithms. In surveys (Cryer and Tavernini, 1972; Hall and Watt, 1976; Bellen, 1985) different numerical methods for FDE are discussed. In this paper positional implicit Runge-Kutta-like numerical methods of solving general FDE are presented. The elaborated methods are direct analogs of the corresponding numerical methods of ODE case. Also we present new theorem (which generalizes the results of (Kim and Pimenov, 1997; Kim and Pimenov, 1998)) on convergence order. The obtained results are based on distinguishing of finite dimensional and infinite dimensional components in the structure of FDE, on interpolation and extrapolation of the discrete prehistory of the model, on application of constructions of i-smooth analysis (Kim, 1996). The notion of an approximation order for an optimal strategy in the discrete scheme of realization is introduced. The conditions, which guarantee required approximation order of the optimal strategy, are presented. Copyright (C) 1998 IFAC.