Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control

Klaus Röbenack; Daniel Gerbet

doi:10.52846/stccj.2022.2.1.32

Authors

Klaus Röbenack TU Dresden https://orcid.org/0000-0002-3347-0864
Daniel Gerbet TU Dresden https://orcid.org/0000-0002-8527-8727

DOI:

https://doi.org/10.52846/stccj.2022.2.1.32

Keywords:

Linear time-invariant, eigenvalue placement, static output feedback, polynomial ideals, Gröbner bases, quantifier elimination, norm minimization

Abstract

The controller design for linear time-invariant state space systems seems to be straightforward and well established. This is not true for static output feedback control, which is still a challenging task. This paper deals with controller design based on eigenvalue assignment. We consider the placement of distinct as well as multiple real eigenvalues or complex conjugate pairs. The desired eigenvalue configurations are characterised in terms of algebraic divisibility of the characteristic polynomial of the closed-loop system. We also consider the problem of partial eigenvalue placement, where not all eigenvalues are fixed by feedback. Degrees of freedom in the controller design are used for the minimization of various matrix norms of the feedback gain matrix.

References

H. Kimura, “On pole assignment by output feedback,” International Journal of Control, vol. 28, no. 1, pp. 11–22, 1978.

X. Wang, “Pole placement by static output feedback,” Journal of Math. Systems, Estimation, and Control, vol. 2, pp. 205–218, 1992.

J. Rosenthal and J. Willems, “Open problems in the area of pole placement,” in Open Problems in Mathematical Systems and Control Theory, ser. Communication and Control Engineering Series, V. D. Blondel, E. D. Sontag, M. Vidyasagar, and J. C. Willems, Eds. London: Springer–Verlag, 1999, pp. 181–191.

M. Franke, “Eigenvalue assignment by static output feedback – on a new solvability condition and the computation of low gain feedback matrices,” International Journal of Control, vol. 87, no. 1, pp. 64–75, 2014.

A. G. Yannakoudakis, “The static output feedback from the invariant point of view,” IMA Journal of Mathematical Control and Information, vol. 33, no. 3, pp. 639–668, 2016.

U. Konigorski, “Parametric eigenvalue assignment by constant output feedback–a cascaded approach,” Automatisierungstechnik, vol. 68, no. 10, pp. 817–825, 2020.

K. R¨obenack and D. Gerbet, “Minimum norm partial eigenvalue placement for static output feedback control,” in International Conference on System Theory, Control and Computing (ICSTCC 2021), Ias¸i, Romania, Oct. 2021, pp. 212–219.

Y. Saad, “Projection and deflation method for partial pole assignment in linear state feedback,” IEEE Transactions on Automatic Control, vol. 33, no. 3, pp. 290–297, 1988.

B. N. Datta and D. R. Sarkissian, “Partial eigenvalue assignment in linear systems: existence, uniqueness and numerical solution,” in Proceedings of the Mathematical Theory of Networks and Systems (MTNS), Notre Dame, 2002.

W. M. Wonham, Linear Multivariable Control - A Geometric Approach, ser. Lecture Notes in Economics ans Mathematical Science. Springer- Verlag, 1974, vol. 101.

V. Mehrmann and H. Xu, “An analysis of the pole placement problem. II. The multi-input case,” Electronic Transactions on Numerical Analysis, vol. 5, pp. 77–97, Sept. 1997, http://eudml.org/doc/119610.

R. Brockett and C. Byrnes, “Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint,” IEEE Transactions on Automatic Control, vol. 26, no. 1, pp. 271–284, 1981.

V. A. Zaitsev, “Modal control of a linear differential equation with incomplete feedback,” Differential Equations, vol. 39, no. 1, pp. 145–148, 2003.

Z. Kukelova, P. Krsek, V. Smutny, and T. Pajdla, “Gr¨obner basis solutions to satellite trajectory control by pole placement,” in Proceedings of the Advanced Maui Optical and Space Surveillance Technologies Conference, 2013, pp. 748–757.

D. A. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, 4th ed. Switzerland: Springer International Publishing, 2015.

T. Becker and V. Weispfenning, Gr¨obner Bases, 2nd ed. New York: Springer-Verlag, 1998.

D. Plaumann, Einf¨uhrung in die Algebraische Geometrie. Springer Spektrum, 2020.

L. H. Keel, J. A. Fleming, and S. P. Bhattacharyya, “Minimum norm pole assignment via Sylvester’s equation,” Contemp. Math, vol. 47, pp. 265–272, 1985.

A. Varga, “Robust and minimum norm pole assignment with periodic state feedback,” IEEE Transactions on Automatic Control, vol. 45, no. 5, pp. 1017–1022, 2000.

——, “Robust pole assignment via Sylvester equation based state feedback parametrization,” in CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design (Cat. No. 00TH8537), 2000, pp. 13–18.

B. N. Datta, W.-W. Lin, and J.-N. Wang, “Robust and minimum norm partial pole assignment in vibrating structures with aerodynamics effects,” in 42nd IEEE International Conference on Decision and Control (CDC), vol. 3, 2003, pp. 2358–2363.

B. F. Caviness and J. R. Johnson, Eds., Quantifier Elimination and Cylindical Algebraic Decomposition. Wien: Springer, 1998.

A. Tarski, “A decision method for elementary algebra and geometry,” in Quantifier elimination and cylindrical algebraic decomposition, B. F. Caviness and J. R. Johnson, Eds. Wien: Springer, 1998, pp. 24–84.

G. Roppenecker, “Complete modal synthesis of constant output feedback controllers (in german),” Automatisierungstechnik, vol. 37, no. 1-12, pp. 304–311, 1989.

J. Deutscher and P. Hippe, “A frequency domain approach to complete modal synthesis,” IFAC Proceedings Volumes, vol. 33, no. 13, pp. 55–60, 2000.

L. Gr¨oll, Methodik zur Integration von Vorwissen in die Modellbildung, ser. Schriftenreihe des Instituts f¨ur Angewandte Informatik / Automatisierungstechnik am Karlsruher Institut f¨ur Technologie. KIT Scientific Publishing, 2015, vol. 52.

K. R¨obenack and R. Voßwinkel, “Solution of control engineering problems by means of quantifier elimination (in german),” Automatisierungstechnik, vol. 67, no. 9, pp. 714–726, 2019.

——, “Eigenvalue placement by quantifier elimination — the static output feedback problem,” Acta Cybernetica, vol. 24, no. 3, pp. 409–427, 2020. [Online].

H. Anai and S. Hara, “A robust control system design by a special quantifier elimination methods using a Sturm-Habicht sequence,” Proc. IMACS-ACA’99, 1999.

J. B. Lasserre, “Polynomials with all zeros real and in a prescribed interval,” Journal of Algebraic Combinatorics, vol. 16, no. 3, pp. 231–237, 2002.

K. R¨obenack and R. Voßwinkel, “On real stable pole placement for structured systems using Sturm and Sturm-Habicht sequences,” IFACPapersOnLine, vol. 53, no. 2, pp. 4546–4552, 2020.

R. A. Freeman and P. V. Kokotovi´c, “Design of ’softer’ robust nonlinear control laws,” Automatica, vol. 29, pp. 1425–1437, 1993.

G. H. Golub and C. F. van Loan, Matrix Computations, 3rd ed. Baltimore: John Hopkins University Press, 1996.

K. Reinschke, Lineare Regelungs- und Steuerungstheorie, 2nd ed. Berlin, Heidelberg: Springer-Verlag, 2014.

R. Voßwinkel, L. Pyta, F. Schr¨odel, ˙I. Mutlu, D. Mihailescu-Stoica, and N. Bajcinca, “Performance boundary mapping for continuous and discrete time linear systems,” Automatica, vol. 107, pp. 272–280, 2019.

K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, New Jersey: Prentice Hall, 1998.

D. Liberzon, Calculus of variations and optimal control theory: A Concise Introduction. Princeton, New Jersey: Princeton Univerity Press, 2021.

“Maxima, a Computer Algebra System,” http://maxima.sourceforge.net.

The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.0), 2020, https://www.sagemath.org.

O. L. Mangasarian and S. Fromivitz, “The fritz john necessary optimality conditions in the presence of equality and inequality constraints,” Journal of Mathematical Analysis and Applications, vol. 17, pp. 37–47, Jan. 1967.

R. Henrion, “On constraint qualifications,” Journal of Optimization Theory and Applications, vol. 72, pp. 187–197, Jan. 1992.

G. Giorgi, “A note on the guignard constraint qualification and the guignard regularity condition in vector optimization,” Applied Mathematics, vol. 4, no. 4, pp. 734–740, 2013.

B. Anderson, N. Bose, and E. Jury, “Output feedback stabilization and related problems-solution via decision methods,” IEEE Trans. on Automatic Control, vol. 20, no. 1, pp. 53–66, Feb. 1975.

J. H. Davenport and J. Heintz, “Real quantifier elimination is doubly exponential,” Journal of Symbolic Computation, vol. 5, no. 1, pp. 29–35, 1988.

A. Seidenberg, “A new decision method for elementary algebra,” Annals of Mathematics, vol. 60, no. 2, pp. 365–374, 1954.

G. E. Collins, “Quantifier elimination for real closed fields by cylindrical algebraic decompostion,” in Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, May 20–23, 1975. Springer, 1975, pp. 134–183.

V. Weispfenning, “The complexity of linear problems in fields,” Journal of Symbolic Computation, vol. 5, no. 1-2, pp. 3–27, 1988.

T. Sturm, “Thirty years of virtual substitution: Foundations, techniques, applications,” in Proc. of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation. ACM, 2018, pp. 11–16.

L. Gonz´alez-Vega, T. Recio, H. Lombardi, and M.-F. Roy, “Sturm-Habicht sequences, determinants and real roots of univariate polynomials,” in Quantifier Elimination and Cylindrical Algebraic Decomposition, B. F. Caviness and J. R. Johnson, Eds. Vienna: Springer, 1998, pp. 300–316.

L. Yang, X. Hou, and Z. Zeng, “Complete discrimination system for polynomials,” Science in China Series E Technological Sciences, vol. 39, no. 6, p. 628–646, 1996.

G. E. Collins and H. Hong, “Partial cylindrical algebraic decomposition for quantifier elimination,” Journal of Symbolic Computation, vol. 12, no. 3, pp. 299–328, 1991.

A. Dolzmann and T. Sturm, “Redlog: Computer algebra meets computer logic,” ACM SIGSAM Bulletin, vol. 31, no. 2, pp. 2–9, 1997.

M. Koˇsta, “New concepts for real quantifier elimination by virtual substitution,” Dissertation, Universit¨at des Saarlandes, Fakult¨at f¨ur Mathematik und Informatik, Saarbr¨ucken, Germany, 2016, https://publikationen.sulb.uni-saarland.de/handle/20.500.11880/26735.

H. Yanami and H. Anai, “Development of SyNRAC,” in Computational Science – ICCS 2006, V. N. Alexandrov, G. D. van Albada, P. M. A. Sloot, and J. Dongarra, Eds. Berlin, Heidelberg: Springer, 2006, pp. 462–469.

C. Chen and M. M. Maza, “Quantifier elimination by cylindrical algebraic decomposition based on regular chains,” Journal of Symbolic Computation, vol. 75, pp. 74–93, 2016.

C. W. Brown and C. Gross, “Efficient preprocessing methods for quantifier elimination,” in CASC, ser. Lecture Notes in Computer Science, vol. 4194. Springer, 2006, pp. 89–100.

C. W. Brown, “QEPCAD – Quantifier Elimination by Partial Cylindrical Algebraic Decomposition.” [Online]. Available: https://www.usna.edu/CS/qepcadweb/B/QEPCAD.html

“Minimum Norm Static Output Feedback.” [Online]. Available: https://github.com/TUD-RST/minimum-norm-output-feedback

S. Guney and A. Atasoy, “An approach to pole placement method with output feedback,” in UKACC Control Conference, 2011. [Online]. Available: https://www.researchgate.net/publication/267424630_An_Approach_to_Pole_Placement_Method_with_Output_Feedback

T. H. Lee, Q. G. Wang, and E. K. Koh, “An iterative algorithm for pole placement by output feedback,” IEEE Trans. on Automatic Control, vol. 39, no. 3, pp. 565–568, Mar. 1994.

K. R¨obenack, R. Voßwinkel, and M. Franke, “On the eigenvalue placement by static output feedback via quantifier elimination,” in Mediterranean Conference on Control and Automation (MED’18), Zadar, Croatia, June 2018, pp. 133–138.