Applications of Mathematics, Vol. 62, No. 1, pp. 37-47, 2017


Exact controllability of linear dynamical systems: A geometrical approach

María Isabel García-Planas

Received December 3, 2015.  

Abstract:  In recent years there has been growing interest in the descriptive analysis of complex systems, permeating many aspects of daily life, obtaining considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Concretely, for complex systems it is of interest to study the exact controllability; this measure is defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state. In this paper, we focus the study on the obtention of the set of all $B$ making the system $(A,B)$ exact controllable.
Keywords:  controllability; exact controllability; eigenvalue; eigenvector; linear system
Classification MSC:  93B05, 93B27, 93B60


References:
[1] J. Assan, J. F. Lafay, A. M. Perdon: Computation of maximal pre-controllability submodules over a Noetherian ring. Syst. Control Lett. 37 (1999), 153-161. DOI 10.1016/S0167-6911(99)00015-8 | MR 1751260 | Zbl 0917.93024
[2] F. Cardetti, M. Gordina: A note on local controllability on Lie groups. Syst. Control Lett. 57 (2008), 978-979. DOI 10.1016/j.sysconle.2008.06.001 | MR 2465147 | Zbl 1148.93005
[3] C. Chen: Introduction to Linear System Theory. Holt, Rinehart and Winston Inc., New York (1970).
[4] M. I. García-Planas, J. L. Domínguez-García: Alternative tests for functional and pointwise output-controllability of linear time-invariant systems. Syst. Control Lett. 62 (2013), 382-387. DOI 10.1016/j.sysconle.2013.02.003 | MR 3038418 | Zbl 1276.93016
[5] A. Heniche, I. Kamwa: Using measures of controllability and observability for input and output selection. IEEE International Conference on Control Applications 2 (2002), 1248-1251. DOI 10.1109/CCA.2002.1038784
[6] P. Kundur: Power System Stability and Control. McGraw-Hill, New York (1994).
[7] C.-T. Lin: Structural controllability. IEEE Trans. Autom. Control 19 (1974), 201-208. DOI 10.1109/TAC.1974.1100557 | MR 0452870 | Zbl 0282.93011
[8] Y. Liu, J. Slotine, A. Barabási: Controllability of complex networks. Nature 473 (2011), 167-173. DOI 10.1038/nature10011
[9] R. W. Shields, J. B. Pearson: Structural controllability of multiinput linear systems. IEEE Trans. Autom. Control 21 (1976), 203-212. DOI 10.1109/TAC.1976.1101198 | MR 0462690 | Zbl 0324.93007
[10] Z. Yuan, C. Zhao, Z. R. Di, W. X. Wang, Y. C. Lai: Exact controllability of complex networks. Nature Communications 4 (2013), 1-12. DOI 10.1038/ncomms3447
[11] Z. Yuan, C. Zhao, W. X. Wang, Z. R. Di, Y. C. Lai: Exact controllability of multiplex networks. New J. Phys. 16 (2014), 103036, 24 pages. DOI 10.1088/1367-2630/16/10/103036 | MR 3275862

Affiliations:   María Isabel García-Planas, Matemàtica Aplicada I, (MA1), Universitat Politècnica de Catalunya UPC, Av. Diagonal, 647, Pl. 2, 08028 Barcelona, Spain, e-mail: maria.isabel.garcia@upc.edu


 
PDF available at: