On a differential geometric formulation of an optimal tracking problem

Concepts of differential geometry have been used in formulating an optimal tracking problem in state-space. For a given model of a linear time-invariant system and a reference trajectory satisfying appropriate dynamics, a class of system inputs can be selected such that the system trajectory asymptotically tracks the reference trajectory. The system response can be expressed as the sum of a transient response and a forced response in which the transient dynamics is governed by an arbitrary stable matrix and the forced dynamics is that of the reference trajectory. The system input is a function of the stable matrix which governs transient dynamics. These dynamical considerations are cast in the formalism of differential geometry and the following result is derived: the magnitude of the input at any instant is independent of the stable matrix for those stable matrices which satisfy a certain constraint. That is, for the same magnitude of the input, the system speed of response is variable via the choice of the stable matrix from a certain restricted class. This naturally leads to an optimal problem which is that of minimizing transient response effort subject to the above constraint. A class of solutions to the optimal problem is proposed.