Skip to main content
SpringerLink
Log in

An Optimal Choice of Characteristic Polynomial Roots for Pole Placement Control Design

  • TOPICAL ISSUE
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The problem of finding the arrangement of closed-loop control system poles that minimizes an objective function is considered. The system optimality criterion is the value of the H∞ norm of the frequency transfer function relative to the disturbance with constraints imposed on the system pole placement and the values of the H∞ norm of the sensitivity function and the transfer function from measurement noise to control. An optimization problem is formulated as follows: the vector of variables consists of the characteristic polynomial roots of the closed loop system with the admissible values restricted to a given pole placement region; in addition to the optimality criterion, the objective function includes penalty elements for other constraints. It is proposed to use a logarithmic scale for the moduli of the characteristic polynomial roots as elements of the vector of variables. The multi-extremality problem of the objective function is solved using the multiple start procedure. A coordinate descent modification with a pair of coordinates varied simultaneously is used for search.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

REFERENCES

  1. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmu-shcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems Subject to Exogenous Disturbances: Linear Matrix Inequalities Technique), Moscow: LENAND, 2014.

    Google Scholar 

  2. Besekerskii, V.A. and Popov, E.P., Teoriya sistem avtomaticheskogo regulirovaniya (The Theory of Automatic Regulation Systems), Moscow: Nauka, 1966.

    Google Scholar 

  3. Aleksandrov, A.G., Stability Margins of the Systems of Optimal and Modal Control, Autom. Remote Control, 2007, vol. 68, no. 8, pp. 1296–1308.

    Article  MathSciNet  Google Scholar 

  4. Chestnov, V.N., Design of Controllers of Multidimensional Systems with a Given Radius of Stability Margins Based on the H ∞-Optimization Procedure, Autom. Remote Control, 1999, vol. 60, no. 7, pp. 986–993.

    MathSciNet  Google Scholar 

  5. Astrom, K.J. and Murray, R.M., Feedback Systems: an Introduction for Scientists and Engineers, New Jersey: Princeton University Press, 2008.

    Google Scholar 

  6. Alexandrov, V.A., Chestnov, V.N., and Shatov, D.V., Stability Margins for Minimum-Phase SISO Plants: A Case Study, Proc. Eur. Control Conf., 2020, St. Petersburg, pp. 2068–2073.

  7. Gahinet, P. and Apkarian, P., A Linear Matrix Inequality Approach to H ∞-Control, Int. J. Robust Nonlin. Control, 1994, vol. 4, no. 4, pp. 421–448.

    Article  MathSciNet  Google Scholar 

  8. Apkarian, P. and Noll, D., Nonsmooth H ∞ Synthesis, IEEE Trans. Autom. Control, 2006, vol. 51, no. 1, pp. 71–86.

    Article  MathSciNet  Google Scholar 

  9. Polyak, B.T. and Khlebnikov, M.V., Static Controller Synthesis for Peak-to-Peak Gain Minimization As an Optimization Problem, Autom. Remote Control, 2021, vol. 82, no. 9, pp. 1530–1553.

    Article  MathSciNet  Google Scholar 

  10. Polyak, B.T. and Khlebnikov, M.V., New Criteria for Tuning PID Controllers, Autom. Remote Control, 2022, vol. 83, no. 11, pp. 1724–1741.

    Article  MathSciNet  Google Scholar 

  11. Shatov, D.V., Synthesis of Parameters of Proportionally-Integral and Proportionally-Integral-Differential Controllers for Stationary Linear Objects with Nonzero Initial Conditions, J. Comput. Syst. Sci. Int., 2023, vol. 62, pp. 17–26.

    Article  MathSciNet  Google Scholar 

  12. Alexandrov, V.A., Pole Placement Optimization for SISO Control System, Autom. Remote Control, 2021, vol. 82, no. 6, pp. 1013-1029.

    Article  MathSciNet  Google Scholar 

  13. Ugray, Z., Lasdon, L., Plummer, J., Glover, F., Kelly, J., and Marti, R., Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization, INFORMS J. Computing, 2007, vol. 19, no. 3, pp. 328–340.

    Article  MathSciNet  Google Scholar 

  14. Astrom, K.J. and Wittenmark, B., Computer-Controlled Systems. Theory and Design, Upper Saddle River, NJ: Prentice Hall, 1984.

    Google Scholar 

  15. Gaiduk, A.R., Teoriya i metody analiticheskogo sinteza sistem avtomaticheskogo upravleniya (polinomial’nyi podkhod) (Theory and Methods for the Analytic Synthesis of Automatic Control Systems (A Polynomial Approach)), Moscow: Fizmatlit, 2012.

  16. Alexandrov, V., Shatov, D., Abramenkov, A., and Abdulov, A., Position Control of Maneuverable Underwater Vehicle Based on Model Identification, Proc. of the 5th International Conference on Control Systems, Mathematical Modeling, Automation, and Energy Efficiency (SUMMA), 2023, Lipetsk, pp. 76–81.

  17. Wie, B. and Bernstein, D.S., A Benchmark Problem for Robust Control Design, Proc. Amer. Control Conf., San Diego, CA, 1990, pp. 961–962.

  18. Apkarian, P., Gahinet, P., and Buhr, C., Multi-Model, Multi-Objective Tuning of Fixed-Structure Controllers, Proc. Eur. Control Conf., 2014, Strasbourg, pp. 856–861.

Download references

Funding

The research presented in Sections 2 and 3 was supported by the Russian Science Foundation (project no. 23-29-00588, https://rscf.ru/en/project/23-29-00588/).

Author information

Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia

    V. A. Alexandrov

Authors
  1. V. A. Alexandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Alexandrov.

Additional information

This paper was recommended for publication by P.S. Shcherbakov, a member of the Editorial Board

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alexandrov, V.A. An Optimal Choice of Characteristic Polynomial Roots for Pole Placement Control Design. Autom Remote Control 85, 411–421 (2024). https://doi.org/10.1134/S0005117924050023

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117924050023

Keywords:

Search

Navigation