# zbMATH — the first resource for mathematics

Nonregular decoupling with stability of two-output systems. (English) Zbl 1265.93203
Summary: In this paper we present a solution to the decoupling problem with stability of linear multivariable systems with 2 outputs, using nonregular static state feedback. The problem is tackled using an algebraic-polynomial approach, and the main idea is to test the conditions for a decoupling compensator with stability to be feedback realizable. It is shown that the problem has a solution if and only if Morse’s list $$I_{2}$$ is greater than or equal to the infinite and unstable structure of the proper and stable part of the stable interactor of the system. A constructive procedure to find state feedback, which achieves decoupling with stability, is also presented.
##### MSC:
 93D15 Stabilization of systems by feedback 93C35 Multivariable systems, multidimensional control systems
##### Keywords:
linear multivariable system; decoupling; stability
Full Text:
##### References:
 [1] Descusse J., Dion J. M.: On the structure at infinity of linear square decoupled systems. IEEE Trans. Automat. Control AC-27 (1982), 971-974 · Zbl 0485.93042 · doi:10.1109/TAC.1982.1103041 [2] Descusse J., Lafay J. F., Malabre M.: Solution of the static-state feedback decoupling problem for linear systems with two outputs. IEEE Trans. Automat. Control AC-30 (1985), 914-918 · Zbl 0566.93010 · doi:10.1109/TAC.1985.1104089 [3] Descusse J., Lafay J. F., Malabre M.: Solution to Morgan’s problem. IEEE Trans. Automat. Control 33 (1988), 732-739 · Zbl 0656.93018 · doi:10.1109/9.1289 [4] Dion J. M., Commault C.: The minimal delay decoupling problem: Feedback implementation with stability. SIAM J. Control Optim. 26 (1988), 66-82 · Zbl 0646.93049 · doi:10.1137/0326005 [5] Falb P. L., Wolovich W. A.: Decoupling in the design and synthesis of multivariable control systems. IEEE Trans. Automat. Control AC-12 (1967), 651-659 · doi:10.1109/TAC.1967.1098737 [6] Herrera A.: Sur le decouplage des systemes lineaires par des lois statiques non regulieres. PhD Thesis, Université de Nantes, Ecole Centrale Nantes 1991 [7] Herrera A., Torres J. A., Ruiz-León J.: The nonregular Morgan’s problem: A polynomial solution for the case of two outputs. Proc. European Control Conference (ECC’93), Groningen 1993, pp. 2275-2278 [8] G. J. C. Martínez, Malabre M.: The row by row decoupling problem with stability: A structural approach. IEEE Trans. Automat. Control 39 (1994), 2457-2460 · Zbl 0825.93252 · doi:10.1109/9.362849 [9] Morse A. S.: Structural invariants of linear multivariable systems. SIAM J. Control 11 (1973), 446-465 · Zbl 0259.93011 · doi:10.1137/0311037 [10] Ruiz-León J., Zagalak, P., Eldem V.: On the Morgan problem with stability. Kybernetika 32 (1996), 425-441 · Zbl 0885.34055 · www.kybernetika.cz · eudml:27972 [11] Vidyasagar M.: Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, MA 1985 · Zbl 0655.93001 [12] Wolovich W. A., Falb P. L.: Invariants and canonical forms under dynamic compensation. SIAM J. Control Optim. 14 (1976), 996-1008 · Zbl 0344.93019 · doi:10.1137/0314063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.