一般时滞系统最优控制的逐步优化方法(英文)

Stage-by-Stage optimization approach to optimal control for general time-delay systems

上世纪60年代以来,借助于动态规划,无时滞系统的最优控制已经得到了很好的解决,然而,经典的动态规划没有像无时滞系统中逐步导出控制器一样,成功地应用于时滞系统。在本文中,针对一般时滞系统的最优控制问题,通过引入一对偶的倒向随机系统并应用内积理论,将提出一个逐步优化算法。借助于该方法,我们可以逐步对线性二次型(LQ)进行完全平方,进而得到控制器的解析解。有趣的是,时滞系统的控制器设计等价于对偶的倒向随机时滞系统不同信号的估计器设计。所提出的一般方法是行之有效的,因为借助于此方法,我们可以将时滞系统的最优控制从确定整个控制序列旋即变为逐一确定序列的元。可以相信所给出的逐步优化算法也可以用于解决其他相关的复杂控制问题,如控制问题解的充要条件等。

It is well known that the optimal control of delay-free systems has been well solved with dynamic programming since 60s of the last century, however the classical dynamic programming has not yet been successfully applied to time delay systems so as to derive the controller step by step as for delay-free systems. In this paper, we proposed a stage-by-stage optimization approach to the optimal control problem for general delayed systems by introducing an associated dual backward stochastic system and applying inner-product theory. The proposed approach allows the square of linear quadratic （LQ） form step by step for the general time delay systems to be completed and then the analytical solu- tion to the controller is presented. It is interesting to show that the controller design for time delay systems is equivalent to the estimator design of different signals from dual backward stochastic time delay systems. The proposed general approach is powerful as it allows us to reduce the optimal control for time delay systems from one of determining an entire control sequence at once to one of determining the elements of the sequences singly and recursively. It is hoped that the proposed stage-by-stage approach can be applied to solve other related complex control problems such as the necessary and sufficient solution to H∞ control.