Feedback Control in Distributed Parameter Gyroscopic Systems: A Solution of the Partial Eigenvalue A

Abstract This paper presents a novel solution to the partial eigenvalue assignment problem of an undamped gyroscopic distributed parameter system. The partial eigenvalue assignment problem is the problem of reassigning by feedback a few undesired eigenvalues of the open-loop operator pencil while leaving the remaining infinite number of eigenvalues unchanged. The distinctive practical features of our solution are (i) it requires the solution of only a small finite-dimensional linear algebraic system and knowledge of only a small finite number of eigenvalues and eigenvectors of the infinite-dimensional open-loop operator pencil, (ii) no spill-over occurs; that is, the remaining infinite number of eigenvalues and eigenvectors that are required to remain invariant will remain in their places and (iii) it is obtained completely in a distributed parameter setting and no discretisation to second-order system of differential equations is invoked so that vital inherent properties of the original system are fully preserved. Because of the above-mentioned practical features, the proposed solution is readily applicable to stabilise or to combat the effects of excessive vibrations in a large structure.

[1]  M. Keldysh,et al.  ON THE COMPLETENESS OF THE EIGENFUNCTIONS OF SOME CLASSES OF NON-SELFADJOINT LINEAR OPERATORS , 1971 .

[2]  Mark J. Balas,et al.  Trends in large space structure control theory: Fondest hopes, wildest dreams , 1982 .

[3]  Albrecht Böttcher,et al.  Lectures on Operator Theory and Its Applications , 1995 .

[4]  Biswa Nath Datta,et al.  PARTIAL EIGENSTRUCTURE ASSIGNMENT FOR THE QUADRATIC PENCIL , 2000 .

[5]  Amit Bhaya,et al.  On the design of large flexible space structures(LFSS) , 1985, 1985 24th IEEE Conference on Decision and Control.

[6]  Biswa Nath Datta,et al.  Spectrum Modification for Gyroscopic Systems , 2002 .

[7]  H. Langer,et al.  On some mathematical principles in the linear theory of damped oscilations of continua II , 1978 .

[8]  Suresh M. Joshi,et al.  Control of Large Flexible Space Structures , 1989 .

[9]  Biswa Nath Datta,et al.  Numerically robust pole assignment for second-order systems , 1996 .

[10]  P. Lancaster,et al.  Factorization of selfadjoint matrix polynomials with constant signature , 1982 .

[11]  Yitshak M. Ram,et al.  Pole assignment for the vibrating rod , 1998 .

[12]  Daniel J. Inman,et al.  Vibration: With Control, Measurement, and Stability , 1989 .

[13]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[14]  Peter Lancaster,et al.  Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum , 1991, Canadian Journal of Mathematics.

[15]  H. Langer,et al.  On some mathematical principles in the linear theory of damped oscillations of continua I , 1978 .

[16]  Biswa Nath Datta,et al.  Theory and computations of partial eigenvalue and eigenstructure assignment problems in matrix second-order and distributed-parameter systems , 2001 .

[17]  Peter Lancaster,et al.  Lambda-matrices and vibrating systems , 2002 .

[18]  Leiba Rodman.  An Introduction to Operator Polynomials , 1989 .

[19]  P. Lancaster,et al.  Strongly definitizable linear pencils in Hilbert space , 1993 .

[20]  B. Datta.  Numerical Linear Algebra and Applications , 1995 .

[21]  B. Datta,et al.  ORTHOGONALITY AND PARTIAL POLE ASSIGNMENT FOR THE SYMMETRIC DEFINITE QUADRATIC PENCIL , 1997 .

[22]  Sylvan Elhay,et al.  An Inverse Eigenvalue Problem for the Symmetric Tridiagonal Quadratic Pencil with Application to Damped Oscillatory Systems , 1996, SIAM J. Appl. Math..

[23]  Peter Lancaster,et al.  Damped vibrations of beams and related spectral problems , 1994 .

[24]  B.N. Datta,et al.  Multi-input partial eigenvalue assignment for the symmetric quadratic pencil , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[25]  Amit Bhaya,et al.  On the design of large flexible space structures (LFSS) , 1985 .

[26]  P. Lancaster,et al.  The theory of matrices : with applications , 1985 .