Use of negative weights in linear quadratic regulator synthesis

Abstract

An inverse problem of optimal linear quadratic regulators (LQR) is examined for single-input systems, and the selection of the weighting matrices that achieve a specified pole location is also discussed. In particular, the Kessler polynomial is used as a desirable pole location, and the weighting matrices are derived in an analytical form. Although this pole specification results in the use of some negative diagonal weights in the performance index, the existence and uniqueness of the Riccati solution are guaranteed by Molinari's theorem. Through the sacrifice of the circle condition, it is shown that some of the deficiencies of the LQR controllers are avoided, and that several characteristics (which classical controllers provide, but which modern methods cannot) are retained. An application to roll autopilot systems for missiles is given to illustrate and substantiate the proposed method, as well as to compare it with the conventional LQR.