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On the minimum number of general or dedicated controllers required for system controllability

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  • Portal, Alberto
  • Zufiria, Pedro J.

Abstract

The paper determines the minimum number of general or dedicated controllers which are required to guarantee the controllability of a class of dynamical systems. For this purpose, we begin by reformulating Kalman controllability condition for linear systems (A, B) in the context of the similarity equivalence class associated with A, and then we provide new alternative characterizations of such controllability condition based on the cases when matrix A is in Jordan or Frobenius form, so that theoretical aspects and computational requirements are comparatively commented. In the second part of the paper, given a system state matrix A, we solve the optimal design problem of determining the minimum number of controllers (columns of B) required for making (A, B) controllable, resorting also to the Jordan and Frobenius canonical forms in the similarity equivalence class associated with A. These canonical forms are proven to be also fundamental to provide bounds on the minimum number of dedicated controllers (corresponding to columns of B with a single non-zero element) required to make (A, B) controllable. Some examples serve to illustrate the fundamental results.

Suggested Citation

  • Portal, Alberto & Zufiria, Pedro J., 2019. "On the minimum number of general or dedicated controllers required for system controllability," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 417-427.
    • Handle: RePEc:eee:apmaco:v:355:y:2019:i:c:p:417-427
    DOI: 10.1016/j.amc.2019.03.015
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    References listed on IDEAS

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    1. Yang-Yu Liu & Jean-Jacques Slotine & Albert-László Barabási, 2011. "Controllability of complex networks," Nature, Nature, vol. 473(7346), pages 167-173, May.
    2. J. S. Devitt & R. A. Mollin, 1986. "The rational canonical form of a matrix," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 9, pages 1-8, January.
    3. Alberto Portal & Pedro J. Zufiria, 2015. "Controlling Systems with a Single Actuator," Springer Books, in: Zhenji Zhang & Zuojun Max Shen & Juliang Zhang & Runtong Zhang (ed.), Liss 2014, edition 127, pages 1197-1202, Springer.
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