Skip to main content
SpringerLink
Log in

Density Systems: Analysis and Control

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

Abstract

This paper considers a class of systems called density systems. For such systems, the derivative of a quadratic function depends on some function termed the density function. The latter function is used to define the properties of the space affecting the behavior of the systems under consideration. The role of density systems in control law design is shown. Control systems are constructed for plants with known and unknown parameters. The theoretical results are illustrated by numerical simulation.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.

REFERENCES

  1. Krasnosel’skii, M.A., Perov, A.I., Povolotskii, A.I., and Zabreiko, P.P., Plane Vector Fields, New York: Academic Press, 1966.

    Google Scholar 

  2. Zhukov, V.P., Necessary and Sufficient Conditions for Instability of Nonlinear Autonomous Dynamic Systems, Autom. Remote Control, 1990, vol. 51, no. 12, pp. 1652–1657.

    Google Scholar 

  3. Zhukov, V.P., On the Divergence Conditions for the Asymptotic Stability of Second-Order Nonlinear Dynamical Systems, Autom. Remote Control, 1999, vol. 60, no. 7, pp. 934–940.

    MathSciNet  Google Scholar 

  4. Rantzer, A., A Dual to Lyapunov’s Stability Theorem, Systems & Control Letters, 2001, vol. 42, pp. 161–168.

    Article  MathSciNet  Google Scholar 

  5. Furtat, I.B., Divergent Stability Conditions of Dynamic Systems, Autom. Remote Control, 2020, vol. 81, no. 2, pp. 247–257.

    Article  MathSciNet  Google Scholar 

  6. Furtat, I.B. and Gushchin, P.A., Stability Study and Control of Nonautonomous Dynamical Systems Based on Divergence Conditions, J. Franklin Institute, 2020, vol. 357, no. 18, pp. 13753–13765.

    Article  MathSciNet  Google Scholar 

  7. Furtat, I.B. and Gushchin, P.A., Stability/Instability Study and Control of Autonomous Dynamical Systems: Divergence Method, IEEE Access, 2021, no. 9, pp. 49088–49094.

  8. Furtat, I.B. and Gushchin, P.A., Divergence Method for Exponential Stability Study of Autonomous Dynamical Systems, IEEE Access, 2022, no. 10, pp. 49088–49094.

  9. Landau, L.D. and Lifshitz, E.M., Fluid Mechanics, 2nd ed., vol. 6 of Course of Theoretical Physics, Butterworth-Heinemann, 1987.

  10. Liberzon, D. and Trenn, S., The Bang-Bang Funnel Controller for Uncertain Nonlinear Systems with Arbitrary Relative Degree, IEEE Trans. Autom. Control, 2013, vol. 58, no. 12, pp. 3126–3141.

    Article  MathSciNet  Google Scholar 

  11. Bechlioulis, C. and Rovithakis, G., A Low-Complexity Global Approximation-Free Control Scheme with Prescribed Performance for Unknown Pure Feedback Systems, Automatica, 2014, vol. 50, no. 4, pp. 1217–1226.

    Article  MathSciNet  Google Scholar 

  12. Berger, T., Le, H., and Reis, T., Funnel Control for Nonlinear Systems with Known Strict Relative Degree, Automatica, 2018, vol. 87, pp. 345–357.

    Article  MathSciNet  Google Scholar 

  13. Furtat, I.B. and Gushchin, P.A., Control of Dynamical Plants with a Guarantee for the Controlled Signal to Stay in a Given Set, Autom. Remote Control, 2021, vol. 82, no. 4, pp. 654–669.

    Article  MathSciNet  Google Scholar 

  14. Furtat, I.B. and Gushchin, P.A., Nonlinear Feedback Control Providing Plant Output in Given Set, Int. J. Control, 2021. https://doi.org/10.1080/00207179.2020.1861336

  15. Filippov, A.F., Differential Equations with Discontinuous Righthand Sides, Dordrecht: Springer, 1988.

    Book  Google Scholar 

  16. Kolmogorov, A.N. and Fomin, S.V., Elements of the Theory of Functions and Functional Analysis, Dover, 1999.

    Google Scholar 

  17. LaSalle, J.P., Stability of Nonautonomous Systems, Nonlinear Analysis, Theory, Methods & Applications, 1976, vol. 1, no. 1, pp. 83–91.

    MathSciNet  Google Scholar 

  18. Demyanov, V.F. and Rubinov, A.M., Introduction to Constructive Nonsmooth Analysis, Frankfurt: Peter Lang Verlag, 1995.

  19. Polyakov, A., Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems, IEEE Trans. Autom. Control, 2012, vol. 57, no. 8, pp. 2106–2110.

    Article  MathSciNet  Google Scholar 

  20. Utkin, V.I., Sliding Modes and Their Applications in Variable Structure Systems, Moscow: Mir, 1978.

  21. Dolgopolik, M.V. and Fradkov, A.L., Nonsmooth and Discontinuous Speed-Gradient Algorithms, Non-linear Anal. Hybrid Syst., 2017, vol. 25, pp. 99–113.

    Article  MathSciNet  Google Scholar 

  22. Khalil, H.K., Nonlinear Systems, 3rd ed., Pearson, 2001.

    Google Scholar 

  23. Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nonlinear and Adaptive Control of Complex Systems, Dordrecht–Boston–London: Kluwer Academic Publishers, 1999.

    Google Scholar 

  24. Vasil’eva, A.B. and Butuzov, V.F., Asimptoticheskie razlozheniya reshenii singulyarno vozmushchennykh uravnenii (Asymptotic Decompositions of Solutions of Singularly Perturbed Equations), Moscow: Nauka, 1973.

  25. Arnold, V.I., Ordinary Differential Equations, Berlin–Heidelberg: Springer, 1992.

    Google Scholar 

  26. Carroll, S.M., Spacetime and Geometry: An Introduction to General Relativity, San Francisco: Addison Wesley, 2004.

    Google Scholar 

  27. Tee, K.P., Ge, S.S., and Tay, E.H., Barrier Lyapunov Functions for the Control of Output-Constrained Nonlinear Systems, Automatica, 2009, vol. 45, no. 4, pp. 918–927.

    Article  MathSciNet  Google Scholar 

  28. Azimi, V. and Hutchinson, S., Exponential Control Lyapunov-Barrier Function Using a Filtering-Based Concurrent Learning Adaptive Approach, IEEE Trans. on Automatic Control, 2022, vol. 67, no. 10, pp. 5376–5383.

    Article  MathSciNet  Google Scholar 

  29. Furtat, I.B., Gushchin, P.A., Nguyen, B.H., and Kolesnik, N.S., Adaptive Control with a Guarantee of a Given Performance, Upravl. Bol’sh. Sist., 2023, no. 102, pp. 44–57.

Download references

Funding

This work was performed in the Institute for Problems in Mechanical Engineering, the Russian Academy of Sciences, under the support of state order no. 121112500298-6 (The Unified State Information System for Recording Research, Development, Design, and Technological Work for Civilian Purposes).

Author information

Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

    I. B. Furtat

Authors
  1. I. B. Furtat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. B. Furtat.

Additional information

This paper was recommended for publication by A.M. Krasnosel’skii, 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

Furtat, I.B. Density Systems: Analysis and Control. Autom Remote Control 84, 1175–1190 (2023). https://doi.org/10.1134/S0005117923110024

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117923110024

Keywords:

Search

Navigation