Grve~ a mathematical model in the form of first order vector differential equations and specified initial values of state, the determination of the optimal control schedule of a system reduces to a particular problem of numerical analysis studied by several authors, for instance [I]. In practice, however, errors occur in the fitting of the mathematical model, and, more simply, in the measurement of the/n/tial state. In this paper only the effect of errors in the measurement of the initial state are considered.
The use of a control schedule calculated to be optimal for measured initial values of state for an inaccurately measured system leads, in general, to a loss of performance. If we have a sensitivity relation between this loss and some parameters describing the errors in measurement we are able to identify the values of these error parameters which give us an acceptable level of performance loss.
In this paper we take as error parameters the errors of measurement themselves, and derive the sensitivity of an optimal system to specified errors of measurement. This is not the only approach; in a second paper [2] we take as error parameters some sufficient statistics of the measurement errors and consider the sensitivity of the system to variations of the statistical description of the state.
In Section 2 we express the performance loss as a function of the initial value of state. We approximate this performance loss by a Taylor's expansion about the measured value of x. In Sections 3, 4 we give methods of determining the coefficients of this expansion. These we call the sensitivity coefficients. In Sections 5, 6 we discuss the properties of these coefficients and in Section 7 their computation. In Section 8 we give an example of the calculation of the sensitivity coefficients in a problem describing a missile and target on collision courses.
One computationally feasible method of representing the performance loss as a function of the measurement errors is to expand the performance loss as a truncated Taylor's series in the measurement errors. This will be valid as long as the measurement errors are small. It is shown in the next section that the first derivatives of the performance loss are zero, so it is necessary to calculate at least the second order partial derivatives to get a nontrivial approximation. A method of doing this is given in the following sections.