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

Approximate expressions are developed for internally balanced singular values corresponding to the modes of mechanical systems with gyroscopic forces, light damping, and small circulatory forces. A brief overview is first given of the balanced realization model reduction method, including a discussion of recent work. The models considered are defined, and a perturbation analysis is used to show that the modal representation becomes asymptotically balanced as damping reduces to zero. The approximate balanced singular values are calculated, and a simple example of a flexible, dual-spin spacecraft is given as an illustration of the results.

[1]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .

[2]  Thomas Kailath,et al.  On generalized balanced realizations , 1980 .

[3]  Robert E. Skelton,et al.  Comments on "Balanced gains and their significance for L 2 model reduction" , 1986 .

[4]  D. Bernstein,et al.  The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore , 1985 .

[5]  Robert E. Skelton,et al.  Another balanced controller reduction algorithm , 1984 .

[6]  L. Meirovitch.  Observer Modal Control of Dual-Spin Flexible Spacecraft , 1979 .

[7]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[8]  Leonard M. Silverman,et al.  Linear time-variable systems: Balancing and model reduction , 1983 .

[9]  E. Jonckheere.  Principal component analysis of flexible systems--Open-loop case , 1984 .

[10]  K. Fernando,et al.  Singular perturbational approximations for discrete-time balanced systems , 1983 .

[11]  R. Skelton,et al.  A note on balanced controller reduction , 1984 .

[12]  F. Janssens.  Solution of linear gyroscopic systems , 1982 .

[13]  B. Noble.  Applied Linear Algebra , 1969 .

[14]  Leonard Meirovitch,et al.  A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems , 1974 .

[15]  C. Gregory.  Reduction of large flexible spacecraft models using internal balancing theory , 1984 .

[16]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[17]  B. Anderson,et al.  Controller reduction via stable factorization and balancing , 1986 .

[18]  P. Kabamba.  Balanced gains and their significance for L^{2} model reduction , 1985 .

[19]  L. Meirovitch.  A Modal Analysis for the Response of Linear Gyroscopic Systems , 1975 .

[20]  L. Silverman,et al.  Singular value analysis of deformable systems , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[21]  Ph. Opdenacker,et al.  Singular Value Analysis, Balancing, and Model Reduction of Large Space Structures , 1984, 1984 American Control Conference.

[22]  Edmond A. Jonckheere,et al.  A new set of invariants for linear systems--Application to reduced order compensator design , 1983 .

[23]  B. Anderson,et al.  Model reduction for control system design , 1984 .