LQG balancing and reduced LQG compensation of symmetric passive systems

In this paper, a technique for balancing a system in a closed-loop fashion is developed. This technique is referred to as ‘LQG balancing’, for it assumes that the system to be balanced is closed up with a standard LQG feedback loop. This paper focuses on LQG balancing of symmetric passive systems; such systems are used to model large vibrating structures with collocated rate sensors and actuators. A balancing characterization of reciprocity, passivity and losslessness is provided. A new method for reducing both the plant and the LQG compensator is developed. It is shown, using a hyperstability argument, that the loop made up with the full plant and any reduced LQG compensator is stable and has acceptable feedback properties.

[1]  M. Kreĭn,et al.  ANALYTIC PROPERTIES OF SCHMIDT PAIRS FOR A HANKEL OPERATOR AND THE GENERALIZED SCHUR-TAKAGI PROBLEM , 1971 .

[2]  V. Belevitch,et al.  Classical network theory , 1968 .

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

[4]  Paul A. Fuhrmann,et al.  Mathematical Theory of Networks and Systems , 1984 .

[5]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[6]  B. Anderson.  A SYSTEM THEORY CRITERION FOR POSITIVE REAL MATRICES , 1967 .

[7]  J. Willems.  Dissipative dynamical systems Part II: Linear systems with quadratic supply rates , 1972 .

[8]  M. Balas,et al.  Feedback control of flexible systems , 1978 .

[9]  Vasile Mihai Popov,et al.  Hyperstability of Control Systems , 1973 .

[10]  Robert E. Skelton,et al.  Modal Truncation for Flexible Spacecraft , 1981 .

[11]  Louis Weinberg,et al.  Network Analysis and Synthesis , 1962 .

[12]  A. Laub,et al.  Feedback properties of multivariable systems: The role and use of the return difference matrix , 1981 .

[13]  P. Fuhrmann.  On symmetric rational transfer functions , 1983 .

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

[15]  Dean Karnopp.  Dynamic forces in multiport mechanics: Direct and Lagrangian formulations , 1982 .

[16]  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.

[17]  Huibert Kwakernaak,et al.  Linear Optimal Control Systems , 1972 .

[18]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .