Quadratic control problem with an energy constraint; approximate solutions

Abstract

The quadratic-error regulator problem with a control-energy constraint has been investigated from the viewpoint of having limited control energy available or as an indirect method of limiting the maximum control amplitude. The solution of this problem requires iteration on a parameter λ n + 1 with the required value of λ n + 1 being a function of the initial state. Thus the result is not as straightforward to apply as the well known Kalman result where no constraints are imposed; however, it is much simpler than the problem where control amplitude constraints are applied directly for which case one must iterate on an n dimensional vector. Also, for the systems studied it was found that λ n + 1 had a regular behavior as a function of initial state and that this behavior could be approximated by empirical equations. Therefore, an approximate control law could be computed for each initial state without iterating yielding real-time solutions. Three examples were considered in the numerical evaluation portion of the study. In addition to revealing the orderly behavior of the required value of λ n + 1 as a function of initial state, the numerical results also indicated that constraining control energy was quite effective in constraining maximum control amplitude.