An almost optimal control design method for nonlinear time-delay systems

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. In this article, optimal control problems for a class of time-delay nonlinear systems subject to mixed control–state constraints are studied. The delays in the systems are on the state and/or on the control input. The theory of optimal control based on measure theory, functional analysis and linear programming (LP) is extended in order to optimise a definite objective function, and to design an appropriate optimal control for the nonlinear time-delay systems. Using an embedding process, the problem is first transferred to a new optimal measure problem and then this new problem is replaced with one in which minimising a linear form over a subset of linear equalities is required. It is shown that this minimisation is global and the embedding procedure allows one to develop a computational method to find a solution using a finite-dimensional LP techniques. Numerical examples are also presented to illustrate the procedure of the design method. 1. Introduction The dynamics of many control systems may be expressed by time-delay equations. The delay(s) may appear in the system state, control input and/or output (Fridman, Shaked, and Suplin 2005; Levaggi and Punta 2006). Delays in the control input arise in many chemical processes and radiation problems in physics. Delays may also appear because of physical properties of equipment used in the system, signal transmission or measurement of system variables. For example, actuators, sensors and field networks which are involved in feedback loops may exhibit delays. Time-delay systems are also used to model several different mechanisms in the dynamics of epidemics. and interactions across spatial distances or through complicated paths have been modelled by the introduction of time-delay systems (Busenberg and Cooke 1992; Boukas and Liu 2002). Whilst the delay equations were introduced in the eighteenth century …