Parametric eigenvalue assignment by constant output feedback – a cascaded approach

Ulrich Konigorski

doi:10.1515/auto-2020-0050

Published by De Gruyter (O) September 22, 2020

Parametric eigenvalue assignment by constant output feedback – a cascaded approach

Parametrische Eigenwertvorgabe mittels konstanter Ausgangsrückführung – ein kaskadiertes Verfahren

Ulrich Konigorski
Prof. Dr.-Ing. Ulrich Konigorski is Head of the Control Systems and Mechatronics Lab at the Institute of Automatic Control and Mechatronics in the Department of Electrical Engineering and Information Technology at the Technical University of Darmstadt. His method-oriented main areas of research are the modeling and design of linear and non-linear MIMO-Systems while the application-oriented research focuses in particular on modeling, simulation and digital control of mechatronic systems as well as the analysis and design of chassis and vehicle dynamics control.

From the journal at - Automatisierungstechnik

https://doi.org/10.1515/auto-2020-0050

Showing a limited preview of this publication:

Abstract

In this paper a numerical efficient approach to the problem of eigenvalue assignment by constant output feedback is presented. It improves the well known Kimura’s condition by 2, i. e., it is shown that if m+p≥n−1 generically a solution to this design problem exists where n,m and p denote the dimensions of the system states, inputs and outputs, respectively. The algorithm is based on a cascaded control scheme with up to three design steps. The first two steps merely require standard methods from linear algebra while the last step only in case of m+p=n−1 demands for the numerical solution of a system of three polynomial equations each of order two. The design procedure explicitly embodies all degrees of freedom beyond eigenvalue assignment. Thus, they can be used to account for other design it is shown goals, e. g., to minimize the spectral condition number of the closed-loop system or a norm of the feedback gain as it is shown by numerical examples from literature.

Zusammenfassung

Der Beitrag beschreibt einen numerisch effizienten Zugang zum Polvorgabeentwurf mittels einer konstanten Ausgangsrückführung. Das vorgestellte Verfahren verbessert die bekannte Entwurfsbedingung von Kimura um 2, d. h. es wird gezeigt, dass generisch für m+p≥n−1 eine Lösung des Entwurfsproblems existiert, wobei n,m and p die jeweiligen Dimensionen des Zustandsraums sowie der Ein- und Ausgänge bezeichnen. Der Algorithmus basiert auf einer Kaskadenstruktur der Regelung mit bis zu drei Entwurfsschritten. Die beiden ersten Entwurfsschritte benötigen lediglich Standardverfahren der linearen Algebra und nur im Falle von m+p=n−1 erfordert der letzte Schritt die numerische Lösung eines Systems von drei Polynomgleichungen jeweils zweiter Ordnung. Über die Polvorgabe hinaus ermöglicht das Verfahren auch den expliziten Zugriff auf sämtliche weiteren verbleibenden Entwurfsfreiheitsgrade. Diese können somit zur Berücksichtigung weiterer Entwurfsziele verwendet werden. Wie anhand von numerischen Beispielen aus der Literatur gezeigt wird, lassen sich so beispielsweise die spektrale Kondition des geschlossenen Regelkreises oder die Reglernorm minimieren.

Keywords: output feedback; eigenvalue assignment; pole placement; minimal gain feedback; spectral condition number; robust pole placement

Schlagwörter: Ausgangsrückführung; Eigenwertvorgabe; Polvorgabe; minimale Reglernorm; spektrale Konditionszahl; robuste Polvorgabe

About the author

Ulrich Konigorski

Prof. Dr.-Ing. Ulrich Konigorski is Head of the Control Systems and Mechatronics Lab at the Institute of Automatic Control and Mechatronics in the Department of Electrical Engineering and Information Technology at the Technical University of Darmstadt. His method-oriented main areas of research are the modeling and design of linear and non-linear MIMO-Systems while the application-oriented research focuses in particular on modeling, simulation and digital control of mechatronic systems as well as the analysis and design of chassis and vehicle dynamics control.

Acknowledgment

The author appreciates the valuable hints and comments of the anonymous reviewers which helped to improve the reading of the paper.

References

1. B. C. Moore. On the flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment. IEEE Trans. Autom. Control, AC-21(5):689–692, 1976.10.1109/CDC.1975.270678Search in Google Scholar

2. W. M. Wonham. On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control, AC-12(6):660–665, 1967.10.1109/TAC.1967.1098739Search in Google Scholar

3. U. Konigorski. A new direct approach to the design of structurally constrained controllers. Preprints of the 10th IFAC World Congress, 8:293–297, 1987.10.1016/S1474-6670(17)55097-1Search in Google Scholar

4. T. H. Lee, Q. G. Wang and E. K. Koh. An iterative algorithm for pole placement by output feedback. IEEE Trans. Autom. Control, 39(3):565–568, 1994.10.23919/ACC.1993.4793494Search in Google Scholar

5. J. Leventides and N. Karcanias. Global asymptotic linearisation of the pole placement map: a closed-form solution for the constant output feedback problem. Automatica, 31(9):1303–1309, 1995.10.1016/0005-1098(95)00047-ZSearch in Google Scholar

6. X. A. Wang. Grassmannian, central projection, and output feedback pole assignment of linear systems. IEEE Trans. Autom. Control, AC-41(6):786–794, 1996.10.1109/9.506231Search in Google Scholar

7. G. Franzé and L. Carotenuto. A general formula for eigenvalue assignment by static output feedback with application to robust design. Systems & Control Letters, 49:175–190, 2003.10.1016/S0167-6911(02)00321-3Search in Google Scholar

8. K. Yang and R. Orsi. Static output feedback pole placement via a trust region approach. IEEE Trans. Autom. Control, 52(11):2146–2150, 2007.10.1109/TAC.2007.908339Search in Google Scholar

9. M. Franke. Eigenvalue assignment by static output feedback – on a new solvability condition and the computation of low gain feedback matrices. Int. J. Control, 87(1):64–75, 2014.10.1080/00207179.2013.822102Search in Google Scholar

10. J. Leventides, N. Karcanias, I. Meintanis and E. Milonidis. Reduced sensitivity solutions to global linearization of the pole assignment map. Proceedings of the 19th IFAC World Congress, pages 4727–4732, 2014.10.3182/20140824-6-ZA-1003.02389Search in Google Scholar

11. A. G. Yannakoudakis. The static output feedback from the invariant point of view. IMA Journal of Mathematical Control and Information, 33(13):639–668, 2016.10.1093/imamci/dnu057Search in Google Scholar

12. K. Röbenack, R. Voßwinkel and M. Franke. On the eigenvalue placement by static output feedback via quantifier elimination. 26th Mediterranean Conference on Control and Automation (MED), pages 133–138, 2018.10.1109/MED.2018.8442817Search in Google Scholar

13. X. Wang. Pole placement by static output feedback. Journal of Math. Systems, Estimation, and Contr., 2(2):205–218, 1992.Search in Google Scholar

14. M. Fu. Pole placement via static output feedback is np-hard. IEEE Trans. Autom. Control, AC-49(5):855–857, 2004.10.1109/TAC.2004.828311Search in Google Scholar

15. H. Kimura. Pole assignment by gain output feedback. IEEE Trans. Autom. Control, AC-20(8):509–516, 1975.10.1109/TAC.1975.1101028Search in Google Scholar

16. S. Srinathkumar. Eigenvalue/eigenvector assignment using output feedback. IEEE Trans. Autom. Control, AC-23(1):79–81, 1978.10.1109/TAC.1978.1101685Search in Google Scholar

17. L. R. Fletcher, J. Kautsky, G. K. G. Kolka and N. K. Nichols. Some necessary and sufficient conditions for eigenstructure assignment. Int. J. Control, 42(6):1457–1468, 1985.10.1080/00207178508933437Search in Google Scholar

18. B.-H. Kwon and M-J. Youn. Eigenvalue-generalized eigenvector assignment by output feedback. IEEE Trans. Autom. Control, AC-32(5):417–421, 1987.10.1109/TAC.1987.1104623Search in Google Scholar

19. G. Roppenecker and J. O’Reilly. Parametric output feedback controller design. Automatica, 25(2):259–265, 1989.10.1016/0005-1098(89)90079-4Search in Google Scholar

20. V. L. Syrmos and F. L. Lewis. Output feedback eigenstructure assignment using two sylvester equations. IEEE Trans. Autom. Control, 38(3):495–499, 1993.10.1109/9.210155Search in Google Scholar

21. V. L. Syrmos and F. L. Lewis. A bilinear formulation for the output feedback problem in linear systems. IEEE Trans. Autom. Control, 39(2):410–414, 1994.10.1109/9.272346Search in Google Scholar

22. A. T. Alexandridis and P. N. Paraskevopoulos. A new approach to eigenstructure assignment by output feedback. IEEE Trans. Autom. Control, 41(7):1046–1050, 1996.10.1109/9.508914Search in Google Scholar

23. L. R. Fletcher and J. F. Magni. Exact pole assignment by output feedback, Part 1. Int. J. Control, 45(6):1995–2007, 1987.10.1080/00207178708933862Search in Google Scholar

24. O. Bachelier, J. Bosche and D. Mehdi. On pole placement via eigenstructure assignment approach. IEEE Trans. Autom. Control, 51(9):1554–1558, 2006.10.1109/TAC.2006.880809Search in Google Scholar

25. U. Konigorski. Pole placement by parametric output feedback. Systems & Control Letters, 61:292–297, 2012.10.1016/j.sysconle.2011.11.015Search in Google Scholar

26. L. Kailath. Linear Systems. Prentice-Hall, New Jersey, 1980.Search in Google Scholar

27. W. Gröbner. Matrizenrechnung. B.I.-Wissenschaftsverlag, Bibliographisches Institut Mannheim/Wien/Zürich, 1966.Search in Google Scholar

28. A. J. Sommese and C. W. Wampler. Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, New Jersey, London, Singapore, 2005.10.1142/5763Search in Google Scholar

29. M. M. Fahmy and J. O’Reilly. Multistage parametric eigenstructure assignment by output-feedback control. Int. J. Control, 48(1):97–116, 1988.10.1080/00207178808906163Search in Google Scholar

30. R. Hermann and C. F. Martin. Applications of algebraic geometry to systems theory: Part I. IEEE Trans. Autom. Control, AC-22(1):19–25, 1977.10.1109/TAC.1977.1101395Search in Google Scholar

31. R. W. Brockett and C. I. Byrnes. Multivariable nyquist criteria, root loci, and pole placement: a geometric viewpoint. IEEE Trans. Autom. Control, AC-26(1):271–284, 1981.10.1109/TAC.1981.1102571Search in Google Scholar

32. John W. Eaton, David Bateman, Søren Hauberg and Rik Wehbring. GNU Octave version 5.2.0 manual: a high-level interactive language for numerical computations, 2020.Search in Google Scholar

33. O. Bachelier and D. Mehdi. Non-iterative pole placement technique: a step further. Journal of the Franklin Institute, 345:267–281, 2008.10.1016/j.jfranklin.2007.10.001Search in Google Scholar

34. J. Kautsky, N. K. Nichols and P. van Dooren. Robust pole assignment in linear state feedback. Int. J. Control, 41(5):1129–1155, 1985.10.1080/0020718508961188Search in Google Scholar

Received: 2020-04-07

Accepted: 2020-07-07

Published Online: 2020-09-22

Published in Print: 2020-10-25

Parametric eigenvalue assignment by constant output feedback – a cascaded approach

Abstract

Zusammenfassung

About the author

Acknowledgment

References

Journal and Issue

Articles in the same Issue