The effects of small parameter variations on the performance index of optimal control systems with initial and final target manifolds, free end time, and bang-bang control are analysed in this paper.A new approach to the sensitivity equation is presented. This approach takes into account the pulse-shaped variation produced by tho parameter change on the bangbang control. An expression, that relates the variations of the performance index, the trajectory, the final time, and the parameter, is derived. This expression extends to the class of optimal systems with bang-bang control, a result previously obtained by Courtin and Rootenberg (1971).
IntroductionPerformance index sensitivity, of optimal control systems to small parameter variations, received increasing attention in the teehnical literature; see, for example, Pagurek (I 965), Witsenhausen (1965) and Y oula and Dorato (196S).Kokotovic et al. (1969) analysed the performance index sensitivity of the optimal control problem with initial and final manifolds, fixed final time, and unconstrained control. Their result was extended by Courtin and Rootenberg (1971) to problems with free terminal time and with a terminal penalty that appears in the performance index. This paper deals with the sensitivity, to small parameter variations, of optimal systems where the control appears linearly in the plant dynamics and in the performance index, but with a bound on the control magnitude.For these systems, which occur frequently in practical applications (linear time-optimal systems for instance), the optimal control is bang-bang. Hence a small parameter variation produces finite control variations during infinitesimal intervals of time.In order to take into account this type of control variation, a new approach to the trajectory sensitivity equation of the system is used in this paper. In addition, thc problem of time-optimal system is considered.For thc general bang-bang problem, a relation between the first-order variation of the performance index and the errors on the boundaries is derived. This relation further extends the previously mentioned results to the case of optimal control systems with bang-bang control implementation.
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