Asymmetric order doubling: a pole placement method for nonminimum phase systems

A new pole placement method is developed by modifying the symmetric root locus method associated with a class of linear quadratic regulators (LQR). The method places the same number of additional poles and zeros as the original plant, and the root locus is drawn for the order-doubled system to obtain the closed loop poles. It allows us to place additional poles and zeros at asymmetric locations with respect to the imaginary axis. It is shown that a physically meaningful state feedback gain exists for an arbitrary root locus parameter as long as all the branches of the root locus are separated into left and right half planes. The design procedure to obtain a stable state feedback controller from the left half plane portion of the asymmetric root locus is developed. This method generalizes the LQR method and is, in particular, suitable for nonminimum phase systems when the symmetric choice of poles and zeros do not necessarily satisfy all the design criteria. The algorithm is implemented on a 3D-coordinate measuring machine (CMM) which exhibits a nonminimum phase behavior. A significant improvement over the traditional LQR method is obtained in terms of speed of response and settling time while maintaining many of the robustness and performance features of LQR