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Complete parametric approach for eigenstructure assignment in a class of second-order linear systems. (English) Zbl 1009.93036
This paper considers eigenstructure assignment via proportional-plus-derivative feedback control in a class of second-order linear systems in the form of $$\ddot q- A\dot q- Cq= Bu$$. Under the controllability condition of the matrix pair $$[A,B]$$, simple complete parametric expressions for both the closed-loop eigenvector matrices and the feedback gain matrices are established in terms of the closed-loop eigenvalues and a group of parameter vectors. Both the closed-loop eigenvalues and the group of parameters can be properly chosen to produce a closed-loop system with some additional desired specifications. The main computations involved are the Smith form reductions of two polynomial matrices, or two sets of singular value decompositions when the closed-loop eigenvalues are known a priori. A third-order illustrative example is presented.

##### MSC:
 93B55 Pole and zero placement problems 93B60 Eigenvalue problems
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##### References:
 [1] Balas, M.J., Trends in large space structure control theory: fondest hopes, wildest dreams, IEEE transactions on automatic control, 27, 522-535, (1982) · Zbl 0496.93007 [2] Bhaya, A.; Desoer, C., On the design of large flexible space structures (LFSS), IEEE transactions on automatic control, 30, 1118-1120, (1985) · Zbl 0574.93044 [3] Chu, E.K.; Datta, B.N., Numerically robust pole assignment for second-order systems, International journal control, 64, 4, 1113-1127, (1996) · Zbl 0850.93318 [4] Datta, B.N.; Rincon, F., Feedback stabilisation of a second-order system: A nonmodal approach, Linear algebra applications, 188-189, 135-161, (1993) · Zbl 0778.65047 [5] Diwekar, A.M.; Yedavalli, R.K., Stability of matrix second-order systems: new conditions and perspectives, IEEE transactions on automatic control, 44, 9, 1773-1776, (1999) · Zbl 0958.93081 [6] Duan, G.R., Solutions to matrix equation AV+BW=VF and their application to eigenstructure assignment in linear systems, IEEE transactions on automatic control, AC-38, 2, 276-280, (1993) · Zbl 0775.93098 [7] Duan, G. R., Thompson, S., & Liu, G. P. (1999). On solution to the matrix equation AV+EVJ=BW+G. Proceedings of 1999 IEEE Conference on Decision and Control pp. 2742-2743. Crowne Plaza Hotel, 100N. 1st Street, Phoenix, Arizona, USA, December 7-10, 1999. [8] Juang, J.N.; Lim, K.B.; Junkins, J.L., Robust eigensystem assignment for flexible structures, Journal of guidance, control and dynamics, 12, 3, 381-387, (1989) · Zbl 0682.93023 [9] Kim, Y.; Kim, H.S.; Junkins, J.L., Eigenstructure assignment algorithm for mechanical second-order systems, Journal of guidance, control and dynamics, 22, 5, 729-731, (1999) [10] Meirovitch, L.; Baruh, H.J.; Oz, H., A comparison of control techniques for large flexible systems, Journal of guidance, control and dynamics, 6, 302-310, (1983) · Zbl 0512.93012
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