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Structural Methods in the Study of Complex Systems pp 57–90Cite as

Stability and the Kleinian View of Geometry

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 482))

Abstract

Youla parametrization of stabilizing controllers is a fundamental result of control theory: starting from a special, double coprime, factorization of the plant provides a formula for the stabilizing controllers as a function of the elements of the set of stable systems. In this case the set of parameters is universal, i.e., does not depend on the plant but only the dimension of the signal spaces. Based on the geometric techniques introduced in our previous work this paper provides an alternative, geometry based parametrization. In contrast to the Youla case, this parametrization is coordinate free: it is based only on the knowledge of the plant and a single stabilizing controller. While the parameter set itself is not universal, its elements can be generated by a universal algorithm. Moreover, it is shown that on the parameters of the strongly stabilizing controllers a simple group structure can be defined. Besides its theoretical and educative value the presentation also provides a possible tool for the algorithmic development.

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Acknowledgements

The authors would like to thank Prof. Tibor Vámos for the inspirative and influential discussions that led us to start the research on the relation between geometry and control.

Author information

Authors and Affiliations

  1. Institute for Computer Science and Control, Hungarian Academy of Sciences, Kende u. 13-17, Budapest, Hungary

    Zoltán Szabó & József Bokor

Authors
  1. Zoltán Szabó
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  2. József Bokor
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Corresponding author

Correspondence to Zoltán Szabó .

Editor information

Editors and Affiliations

  1. Department of Electrical, Electronic and Information Engineering “G. Marconi”, Alma Mater Studiorum Università di Bologna, Bologna, Bologna, Italy

    Elena Zattoni

  2. Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, ANCONA, Ancona, Italy

    Anna Maria Perdon

  3. Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, ANCONA, Ancona, Italy

    Giuseppe Conte

Appendix

Appendix

A state space realization for the sum of systems is given by

$$\begin{aligned} \left[ \begin{array}{c|c}A_1 &{} B_1 \\ \hline C_1 &{} D_1 \end{array}\right] + \left[ \begin{array}{c|c}A_2 &{} B_2 \\ \hline C_2 &{} D_2 \end{array}\right] = \left[ \begin{array}{cc|c}A_1 &{} 0 &{} B_1 \\ 0 &{} A_2 &{} B_2 \\ \hline C_1 &{} C_2 &{} D_1+D_2 \end{array}\right] , \end{aligned}$$
(2.41)

while the product of the systems can be expressed as:

$$\begin{aligned} \left[ \begin{array}{c|c}A_1 &{} B_1 \\ \hline C_1 &{} D_1 \end{array}\right] \left[ \begin{array}{c|c}A_2 &{} B_2 \\ \hline C_2 &{} D_2 \end{array}\right] = \left[ \begin{array}{cc|c}A_1 &{} B_1C_2 &{} B_1D_2 \\ 0 &{} A_2 &{} B_2 \\ \hline C_1 &{} D_1C_2 &{} D_1D_2 \end{array}\right] . \end{aligned}$$
(2.42)

Note that these realizations are not necessarily minimal. If D is invertible then a realization of the inverse system is

$$\begin{aligned} \left[ \begin{array}{c|c}A &{} B \\ \hline C &{} D \end{array}\right] ^{-1}= \left[ \begin{array}{c|c}A-BD^{-1}C &{} -BD^{-1} \\ \hline D^{-1}C &{} D^{-1} \end{array}\right] . \end{aligned}$$
(2.43)

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Szabó, Z., Bokor, J. (2020). Stability and the Kleinian View of Geometry. In: Zattoni, E., Perdon, A., Conte, G. (eds) Structural Methods in the Study of Complex Systems. Lecture Notes in Control and Information Sciences, vol 482. Springer, Cham. https://doi.org/10.1007/978-3-030-18572-5_2

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