Skip to main content
SpringerLink
Log in

Static Feedback Design in Linear Discrete-Time Control Systems Based on Training Examples

  • LINEAR SYSTEMS
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

The problem of static feedback design in linear discrete time-invariant control systems is considered. The desired behavior of the system is defined by a set of its output variation laws (training examples) and by a requirement to the degree of its stability. Controller’s structural constraints are taken into account. Explicit relations are obtained and an iterative method based on these relations is proposed to find a good initial approximation of the desired gain matrix and to refine it sequentially. In the general case, simple-structure gain matrices are found: in such matrices, only those components are nonzero that are necessary and sufficient to give the system the desired properties. Some examples are provided to illustrate the method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. Polyak, B.T., Khlebnikov, M.V., and Rapoport, L.B., Matematicheskaya teoriya avtomaticheskogo upravleniya (Mathematical Theory of Automatic Control), Moscow: Lenand, 2019.

  2. Sadabadi, M.S. and Peaucelle, D., From Static Output Feedback to Structured Robust Static Output Feedback: A Survey, Ann. Rev. Control, 2016, vol. 42, pp. 11–26.

    Article  Google Scholar 

  3. Syrmos, V.L., Abdallah, C.T., Dorato, P., and Grigoriadis, K., Static Output Feedback—a Survey, Automatica, 1997, vol. 33, no. 2, pp. 125–137.

    Article  MathSciNet  Google Scholar 

  4. Toker, O. and Ozbay, H., On the Np-Hardness of Solving Bilinear Matrix Inequalities and Simultaneous Stabilization with Static Output Feedback, IEEE American Control Conference, Seattle, 1995, pp. 2525–2526.

  5. Toscano, R., Structured Controllers for Uncertain Systems: A Stochastic Optimization Approach, New York: Springer-Verlag, 2013.

    Book  Google Scholar 

  6. Rosinova, D., Vesely, V., and Kucera, V., A Necessary and Sufficient Condition for Static Output Feedback Stabilizability of Linear Discrete-Time Systems, Kybernetika, 2003, vol. 39, pp. 447–459.

    MathSciNet  Google Scholar 

  7. Cao, Y.Y., Lam, J., and Sun, Y.X., Static Output Stabilization: An ILMI Approach, Automatica, 1998, vol. 34, no. 12, pp. 1641–1645.

    Article  Google Scholar 

  8. Wang, X., Pole Placement by Static Output Feedback, J. Math. Syst. Estim. Control, 1992, vol. 2, no. 2, pp. 205–218.

    MathSciNet  Google Scholar 

  9. Pakshin, P.V. and Ryabov, A.V., A Static Output Feedback Control for Linear Systems, Autom. Remote Control, 2004, vol. 65, no. 4, pp. 559–566.

    Article  MathSciNet  Google Scholar 

  10. Agulhari, C.M., Oliveira, R.C., and Peres, P.L., LMI Relaxations for Reduced-Order Robust H∞-control of Continuous-Time Uncertain Linear Systems, IEEE Trans. Autom. Control, 2012, vol. 57, no. 6, pp. 1532–1537.

    Article  MathSciNet  Google Scholar 

  11. Ebihara, Y., Tokuyama, K., and Hagiwara, T., Structured Controller Synthesis Using LMI and Alternating Projection Method, Int. J. Control, 2004, vol. 77, no. 12, pp. 1137–1147.

    Article  MathSciNet  Google Scholar 

  12. Grigoriadis, K.M. and Beran, E.B., Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints, in Advances in Linear Matrix Inequality Methods in Control, Philadelphia: SIAM, 2000, pp. 251–267.

    Google Scholar 

  13. Leibfritz, F., An LMI-Based Algorithm for Designing Suboptimal Static H2 /H∞ Output Feedback Controllers, SIAM J. Control Optim., 2001, vol. 39, no. 6, pp. 1711–1735.

    Article  MathSciNet  Google Scholar 

  14. Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Sparse Feedback in Linear Control Systems, Autom. Remote Control, 2014, vol. 75, no. 12, pp. 2099–2111.

    Article  MathSciNet  Google Scholar 

  15. Bykov, A.V. and Shcherbakov, P.S., Sparse Feedback Design in Discrete-Time Linear Systems, Autom. Remote Control, 2018, vol. 79, no. 7, pp. 1175–1190.

    Article  MathSciNet  Google Scholar 

  16. Lin, F., Fardad, M., and Jovanoviґc, M.R., Design of Optimal Sparse Feedback Gains via the Alternating Direction Method, IEEE Trans. Autom. Control, 2013, vol. 58, no. 9, pp. 2426–2431.

    Article  MathSciNet  Google Scholar 

  17. Belozyorov, V.Y., New Solution Method of Linear Static Output Feedback Design Problem for Linear Control Systems, Linear Algebra Appl., 2016, vol. 504, pp. 204–227.

    Article  MathSciNet  Google Scholar 

  18. Blumthaler, I. and Oberst, U., Design, Parametrization, and Pole Placement of Stabilizing Output Feedback Compensators via Injective Cogenerator Quotient Signal Modules, Linear Algebra Appl., 2012, vol. 436, pp. 963–1000.

    Article  MathSciNet  Google Scholar 

  19. Johnson, T. and Athans, M., On the Design of Optimal Constrained Dynamic Compensators for Linear Constant Systems, IEEE Trans. Autom. Control, 1970, vol. 15, pp. 658–660.

    Article  Google Scholar 

  20. Moerder, D. and Calise, A., Convergence of a Numerical Algorithm for Calculating Optimal Output Feedback Gains, IEEE Trans. Autom. Control, 1985, vol. 30, pp. 900–903.

    Article  Google Scholar 

  21. Choi, S. and Sirisena, H., Computation of Optimal Output Feedback Gains for Linear Multivariable Systems, IEEE Trans. Autom. Control, 1974, vol. 19, pp. 254–258.

    Article  Google Scholar 

  22. Kreisselmeier, G., Stabilization of Linear Systems by Constant Output Feedback Using the Riccati Equation, IEEE Trans. Autom. Control, 1975, vol. 20, pp. 556–557.

    Article  MathSciNet  Google Scholar 

  23. Toivonen, H.T., A Globally Convergent Algorithm for the Optimal Constant Output Feedback Problem, Int. J. Control, 1985, vol. 41, no. 6, pp. 1589–1599.

    Article  Google Scholar 

  24. Geromel, J., Peres, P., and Souza, S., Convex Analysis of Output Feedback Structural Constraints, Proc. IEEE Conf. on Decision and Control, San Antonio, 1993, pp. 1363–1364.

  25. Iwasaki, T. and Skelton, R., All Controllers for the General H∞ Control Problem: LMI Existence Conditions and State Space Formulas, Automatica, 1994, vol. 30, pp. 1307–1317.

    Article  MathSciNet  Google Scholar 

  26. Paraev, Yu.I. and Smagina, V.I., Problems of Simplifying the Structure of Optimal Controllers, Avtomat. i Telemekh., 1975, no. 6, pp. 180–183.

  27. Mozzhechkov, V.A., Prostye struktury v teorii upravleniya (Simple Structures in Control Theory), Tula: Tula State University, 2000.

  28. Mozzhechkov, V.A., Design of Simple-Structure Linear Controllers, Autom. Remote Control, 2003, vol. 64, no. 1, pp. 23–36.

    Article  MathSciNet  Google Scholar 

  29. Mozzhechkov, V.A., Design of Simple Robust Controllers for Time-Invariant Dynamic Systems, J. Comput. Syst. Sci. Int., 2021, vol. 60, pp. 353–363.

    Article  MathSciNet  Google Scholar 

  30. Mozzhechkov, V.A., Synthesis of Simple Relay Controllers in Self-oscillating Control Systems, Autom. Remote Control, 2022, vol. 83, no. 9, pp. 1393–1403.

    Article  MathSciNet  Google Scholar 

  31. Mozzhechkov, V.A., Simple Structures in Problems of Control Theory: Formalization and Synthesis, J. Comput. Syst. Sci. Int., 2022, vol. 61, pp. 295–312.

    Article  MathSciNet  Google Scholar 

  32. Vapnik, V.N., An Overview of Statistical Learning Theory, Transactions on Neural Networks, 1999, vol. 10, no. 5, pp. 988–999.

    Article  Google Scholar 

  33. Vorontsov, K.V., Combinatorial Bounds for the Quality of Learning by Precedents, Dokl. Akad. Nauk, 2004, vol. 394, no. 2, pp. 175–178.

    MathSciNet  Google Scholar 

  34. Mohri, M., Rostamizadeh, A., and Talwalkar, A., Foundations of Machine Learning, Massachusetts: MIT Press, 2012.

    Google Scholar 

  35. Schmidhuber, J., Deep Learning in Neural Networks, Neural Networks, 2015, vol. 61, pp. 85–117.

    Article  Google Scholar 

  36. Ikramov, Kh.D., Chislennoe reshenie matrichnykh uravnenii (Numerical Solution of Matrix Equations), Moscow: Nauka, 1984.

  37. Gill, Ph.E., Murray, W., Wright, M.H., Practical Optimization, London: Academic Press, 1981.

    Google Scholar 

  38. Bertsekas, D.P., Convex Optimization Algorithms, Belmont: Athena Scientific, 2015.

    Google Scholar 

  39. Polyak, B., Introduction to Optimization, Optimization Software, 1987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tula State University, Tula, Russia

    V. A. Mozzhechkov

Authors
  1. V. A. Mozzhechkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Mozzhechkov.

Additional information

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

Publisher’s Note.

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

APPENDIX

APPENDIX

Proof of Statement 1. Let the matrix K be the solution of system (16). Equations (16) and (6) are equivalent if all the desired trajectories Yγ, γ ∈ {\(\overline {1,\,\,q} \)}, belong to the set of solutions of system (1)–(4); see the considerations above. Hence, under all other hypotheses of the proposition, choosing the matrix K based on equalities (16) ensures the requirements (6). This proves the sufficiency part of Proposition 1. If the matrix K is not the solution of system (16), violating equations (16) will also violate conditions (6). If some of the desired trajectories Yγ, γ ∈ {\(\overline {1,\,\,q} \)}, do not belong to the set of solutions of system (1)–(4), the equality yk = \(y_{k}^{\gamma }\) will not hold for them at each time instant k ∈ {\(\overline {1,\,\,N} \)}. Therefore, conditions (6) will fail as well. This proves the necessity part of Proposition 1.

Proof of Statement 2. This result follows from the equivalence of conditions (1)–(4) and (7)–(9) (on the one hand) and conditions (4), (8), (17), and (18) (on the other hand).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mozzhechkov, V.A. Static Feedback Design in Linear Discrete-Time Control Systems Based on Training Examples. Autom Remote Control 84, 947–955 (2023). https://doi.org/10.1134/S0005117923090047

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117923090047

Keywords:

Search

Navigation