In realizing optimum control using Pontryagin's maximum principle, it happens very often that the actual value of the performance index is not so good as expected, owing to the existence of control error. In this respect, the highest accuracy attainable is desired in designing control devices. On the contrary, the increase in weight or cost of a control system is usually undesirable. In these cases, it is necessary to know previously about the quantitative influence of the control function error on the value of performance index.
In order to demonstrate the practical way to estimate the necessitated accuracy, a quantitative method of evaluating the change of performance index due to the difference between the ideal optimum trajectory and the deviated ones is shown, adopting the maximization problem of the arrival distance of a moving vehicle with certain fuel-effectiveness characteristics as a typical example. The optimum solution is found to be composed of usual Bang-Bang control and singular solution control, the former being switched to the latter at an appropriate time. First the performance index values integrated along deviated trajectories parallel to the singular one are calculated. Then the same index values are examined changing the maximum value of the control function of the Bang-Bang control zone variously to give different deviated trajectories.
Consequently it is found for this example that the higher degree of accuracy is demanded in the singular control zone.
Also values of performance index for the same system using only the Bang-Bang control are discussed with the maximun value of the control function variously reduced.