The ‘unreachable poles’ defect in LQR theory: analysis and remedy

Abstract

In virtually every application of optimum linear-quadratic regulator (LQR) theory there exists a hidden region of ‘unreachable poles’ (in the left half-plane) which cannot be realized as optimum closed-loop poles. These regions of unreachable closed-loop poles are not visible using the solution procedures ordinarily employed in LQR applications and their lurking presence has (apparently) been overlooked by many professors, textbook writers and industrial users of LQR control theory for the past 25 years. The existence of these regions of unreachable poles represents a serious defect in the LQR method because those regions may (and often do!) contain closed-loop pole patterns which are considered highly desirable by classical control engineering standards, i.e. by ITAE and other classical standards of ‘ideal’ transient response. We first show how one can identify the regions of unreachable poles in an LQR problem. Then, it is shown how one can modify conventional LQR theory to overcome this defect and make all unreachable poles (in the left half-plane) become reachable. By this means, an explicit formula is derived for the LQR state-weighting matrix Q which will automatically produce ITAE or any other arbitrarily prescribed closed-loop pole patterns in the left half-plane.