Experimental verification of nonminimum phase endpoint feedback of coordinate measuring machines using asymmetric order doubling

A new pole placement method developed by modifying the symmetric root locus method of a class of linear quadratic regulators (LQR) is applied to the endpoint feedback of a coordinate measuring machine, and its performance and robustness properties are examined. Similar to the symmetric root locus, the new method places the same number of additional poles and zeros as that of the original plant, and the root locus is drawn for the order-doubled system to obtain closed loop poles. Unlike the symmetric root locus method, however, the new method allows us to place poles and zeros at asymmetric locations, i.e., not at the mirror image with respect to the imaginary axis. Despite the asymmetric root locus, 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 the left and right half planes. The design procedure to obtain a stable state feedback controller from the left half plane (LHP) portion of the asymmetric root locus has been developed. The endpoint feedback of coordinate measuring machines exhibit nonminimum phase behavior, having unstable zeros close to the imaginary axis. Although the symmetric root locus method incurs slow dynamics due to dominant poles drawn to the zeros near the imaginary axis, the proposed asymmetric method allows us to place them away from the imaginary axis. Significant improvements over the traditional LQR method are obtained in terms of speed of response and settling time while maintaining many of the robustness and performance features of LQR