Initial Condition Sensitivity Functions and Their Applications

Abstract

The well-known method of obtaining sensitivity functions is usually restricted by the condition of continuity imposed on the functions of the coordinates in system equations. For the case of discontinuous functions, more elaborate procedures are required to give a good linear approximation. It is shown that sensitivity functions are discontinuous at the discontinuous points. Also, relations between elements of sensitivity functions at these points are shown to be linear for a given nominal solution of the system. The switching time of the desired Bang-Bang control can be estimated if variations in initial conditions are known. The changes in terminal states and cost function due to deviations in initial conditions can be determined, to first order, by the use of sensitivity functions. Bounds on the deviations in initial conditions can be found by a worst case approach so that the desired terminal conditions are satisfied within given tolerances. The fundamental importance of these techniques in a number of areas of application, for instance, guidance and control of aerospace vehicles, is well known.