| Optimal disturbance rejection for the nonlinear delay systems is of important meaning in theory and practice. Actual systems are nonlinear inherently. The analysis and synthesis for nonlinear systems is not only hot in the academic field, but also necessary in the industry, aerospace and ocean engineering. The optimal control law for general nonlinear system with quadratic index is given in terms of the solution to the Hamilton-Jocabi-Bellman (HJB) equation or nonlinear Two Point Boundary Value (TPBV) problem. But for general nonlinear systems with the exception of the simplest cases, the analytical solution does not exist. This has inspired researchers to present some approaches to obtain an approximate solution to the HJB equation or the nonlinear TPBV problem as well as obtaining an approximate optimal control for the nonlinear systems. External persistent disturbances affect the performance of actural systems seriously, and the disturbance rejection has a wide application area. The optimal disturbance rejection for nonlinear systems has drawn serious attention. In actual systems, time delay exists widely and inevitably. The time delay causes unsynchronized application of the control forces, and in general the unsynchronization not only degrades the system performance, but also causes instability of the system. However, only limited studies are devoted to the area of optimal disturbance rejection for discrete-time nonlinear time-delay systems.Offshore platforms are typical time-delay systems with persistent external disturbances. Offshore platforms, especially the oil and gas production platforms play a more and more important role in the economy. Offshore platforms are generally used on-site, inevitably undergoing the environmental loads such as wind, waves and earthquake etc., which will induce continuous vibration of offshore platforms. The excessive vibration, on one hand, causes fatigue damage and decreases the reliability...
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