Perturbation analysis of internal balancing for lightly damped mechanical systems with gyroscopic and circulatory forces

Abstract

Approximate expressions are developed for internally balanced singular values corresponding to the modes of mechanical systems with gryoscopic forces, light damping, and small circulatory forces. The singular values involve input and output coupling, modal frequency, and modal damping, and they serve as a guide for model reduction by modal truncation. The derivation of these singular values is based on perturbation analysis, and the satisfaction of a frequency separation condition is required to insure their validity. Nomenclature A= system matrix B-input matrix C = output matrix A 0 = system matrix for conservative forces A l = system matrix for nonconservative forces T = transformation matrix W c = system controllability grammian WQ = system observability grammian W c^ = controllability grammian for two-mode subsystem € = small parameter associated with nonconservative forces a, = ith balanced singular value d t = ith approximate balanced singular value a>j = ith system frequency C/.. = ith system damping ratio Q = diagonal matrix of system frequencies

Robust linear quadratic Gaussian control for flexible structures

“…The authors of Ref. 4 derive the important result that if certain similarity and minimum-phase conditions are satisfied, the closed-loop poles can be made asymptotically insensitive to Presented as Paper 86-2051 at the AIAA Guidance, Navigation, and Control Conference, Williamsburg, VA, Aug. [18][19][20]1986; received March 29, 1988; revision received Sept. 16,1988 parameter uncertainties by allowing particular weighting matrices in the linear quadratic regulator (LQR) problem and covariance matrices in the Kalman-Bucy filter (KBF) problem to approach infinity. Because of the way these matrices are chosen, the regulator and filter gains will approach infinity as well.…”