Analysis of optimal control systems with time-varying elements via series expansion

Abstract

In this dissertation, a method of analyzing optimal control systems with time-varying elements via series expansion has been proposed. Up to the present, many attempts has been made to obtain an analytic form of optimal control input for linear time-varying systems via series expansion approach. However, there remain still two significant problems. The first is an requirement of the inverse of a matrix with large dimension such as nmxnm, where n is the dimension of system and m is the number of basis functions employed. The second is the abscene of an explicit or closed form, which is desirable form in calculation, for unknown solution vector to be found. Therefore, as the first problem, this thesis presents a method of finding explicit form formulas, which do not include the inverse of a matrix with large dimension, for unknown solution vector of several classes of time-varying dynamic systems such as linear time-varying systems, bilinear time-varying systems, time-varying scaled systems. Then, as a result of finding an analytic form for state transition matrix via series expansion approach, we find an analytic form of optimal control input time-varying system, which includes the inverse of a matrix each of whose entries is a linear combination of basis functions employed. here it is obvious that the inverse of such a matrix should be available desirably in an analytic form in order to use the feedback gains. It is pointed out that as the dimension n of the system and/or the number m of basis functions employed increases, hand calculation of the inverse of such a matrix is very tedious, and becomes almost impossible for the case when n and m are large. However, no indication was given concerning how to find the inverse of such a matrix. As the second problem, this thesis shows an effective method of obtaining an analytic form for the inverse of a matrix whose elements are linear combinations of basis functions adopted. By using the special property of each bas...