On a Tracking Problem for Linear Systems and Its Computational Algorithm

Abstract

This paper gives the way to determine the optimal control on a tracking problem for plants with known weighting functions. In the tracking problem, the input function of time u(t) (t₀≤t≤t_f), where t₀, t_f are the starting-up time and the final time of control respectively, is so chosen that the output of the plant should exactly ‘track’ the specified time function over the time interval [t₁, t_f], where t₁ is a specified time in (t₀, t_f), and at the same time the total input energy is kept minimum. The equation expressing the constraint is transformed into a Volterra's integral equation of the second kind which facilitates the solution of the problem. Then the problem is formulated and solved in the framework of the function space approach.
Further, considering the fact that it is not easy to get optimal inputs in an analytic form in general, a computational algotithm which gives inputs sufficiently near to the optimal ones is presented. The proposed method gives approximate optimal inputs by matrix manipulations which are suited for digital computers. Several computed results are presented to illustrate the validity of the method.