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Synthesis of Invariant Nonlinear Single-Channel Sigmoid Feedback Tracking Systems Ensuring Given Tracking Accuracy

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Abstract

We consider the tracking problem under exogenous and parametric disturbances for nonlinear single-channel plants with mathematical model representable in a triangular input–output form. Within the framework of the block approach, we develop a decomposition procedure for nonlinear feedback synthesis that provides tracking of the target signal by the output variable with given accuracy in given time. We formalize a new type of sigmoid local feedbacks in the class of smooth everywhere bounded S-shaped functions that ensure invariance with respect to uncontrolled bounded disturbances not belonging to the control space without any assumptions about their smoothness. The results of numerical simulation of our algorithms for an inverted pendulum control system are presented.

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REFERENCES

  1. Wonham, W.M., Linear Multivariable Control: a Geometric Approach, New York: Springer-Verlag, 1979.

    Book  Google Scholar 

  2. Nikiforov, V.O., Adaptivnoe i robastnoe upravlenie s kompensatsiei vozmushchenii (Adaptive and Robust Control with Disturbance Compensation), St. Petersburg: Nauka, 2003.

    Google Scholar 

  3. Krasnova, S.A. and Utkin, A.V., Sigma function in observer design for states and perturbations, Autom. Remote Control, 2016, vol. 77, no. 9, pp. 1676–1688.

    Article  MathSciNet  Google Scholar 

  4. Malikov, A.I., Synthesis of state unknown input observers for nonlinear Lipschitz systems with uncertain disturbances, Autom. Remote Control, 2018, vol. 79, no. 3, pp. 406–424.

    Article  MathSciNet  Google Scholar 

  5. Krasnova, S.A., Estimating the derivatives of external perturbations based on virtual dynamic models, Autom. Remote Control, 2020, vol. 81, no. 5, pp. 897–910.

    Article  MathSciNet  Google Scholar 

  6. Andrievsky, B.R. and Furtat, I.B., Disturbance observers: methods and applications. I. Methods, Autom. Remote Control, 2020, vol. 81, no. 9, pp. 1563–1610.

    Article  MathSciNet  Google Scholar 

  7. Emel’yanov, S.V. and Korovin, S.K., Novye tipy obratnoi svyazi (Novel Feedback Types), Moscow: Nauka. Fizmatlit, 1997.

    Google Scholar 

  8. Edwards, S. and Spurgeon, S., Sliding Mode Control: Theory and Applications, Taylor & Francis, 1998.

  9. Utkin, V.I., Guldner, J., and Shi, J., Sliding Mode Control in Electromechanical Systems, New York: CRC Press, 2009.

    Google Scholar 

  10. Utkin, V.A. and Utkin, A.V., Problem of tracking in linear systems with parametric uncertainties under unstable zero dynamics, Autom. Remote Control, 2014, vol. 75, no. 9, pp. 1577–1592.

    Article  MathSciNet  Google Scholar 

  11. Krstic, M., Kanellakopoulos, I., and Kokotovic, P., Nonlinear and Adaptive Control Design, New York: Wiley, 1995.

    MATH  Google Scholar 

  12. Ebrahim, A. and Murphy, G.V., Adaptive backstepping controller design of an inverted pendulum, Proc. Thirty-Seventh Southeast. Symp. Syst. Theory (2005), pp. 172–174.

  13. Feng, H., Qiao, W., Yin, C., Yu, H., and Cao, D., Identification and compensation of non-linear friction for a electro-hydraulic system, Mech. Mach. Theory, 2019, vol. 141, pp. 1–13.

    Article  Google Scholar 

  14. Hidalgo, M. and Garcia, C., Friction compensation in control valves: nonlinear control and usual approaches, Control Eng. Pract., 2018, vol. 58, pp. 42–53.

    Article  Google Scholar 

  15. Antipov, A.S. and Krasnova, S.A., Block-based synthesis of a tracking system for a twin-rotor electromechanical system with constraints on state variables, Mech. Solids, 2021, vol. 56, no. 7, pp. 43–56.

    Article  Google Scholar 

  16. Kochetkov, S.A. and Utkin, V.A., Providing the invariance property on the basis on oscillation modes, Dokl. Math., 2013, vol. 88, no. 2, pp. 618–623.

    Article  MathSciNet  Google Scholar 

  17. Krasnova, S.A. and Mysik, N.S., Cascade synthesis of a state observer with nonlinear correction influences, Autom. Remote Control, 2014, vol. 75, no. 2, pp. 263–280.

    Article  MathSciNet  Google Scholar 

  18. Krasnova, S.A., Utkin, V.A., and Utkin, A.V., Block approach to analysis and design of the invariant nonlinear tracking systems, Autom. Remote Control, 2017, vol. 78, no. 12, pp. 2120–2140.

    Article  MathSciNet  Google Scholar 

  19. Antipov, A.S., Krasnov, D.V., and Utkin, A.V., Decomposition synthesis of the control system of electromechanical objects in conditions of incomplete information, Mech. Solids, 2019, vol. 54, no. 5, pp. 47–60.

    Article  Google Scholar 

  20. Luk’yanov, A.G., A block method of synthesis of nonlinear systems at sliding modes, Autom. Remote Control, 1998, vol. 59, no. 7. Part 1, pp. 916–933.

    MATH  Google Scholar 

  21. Krasnova, S.A., Sirotina, T.G., and Utkin, V.A., A structural approach to robust control, Autom. Remote Control, 2011, vol. 72, no. 8, pp. 1639–1666.

    Article  MathSciNet  Google Scholar 

  22. Utkin, V.A., Invariance and independence in systems with separable motion, Autom. Remote Control, 2001, vol. 62, no. 11, pp. 1825–1843.

    Article  MathSciNet  Google Scholar 

  23. Tsypkin, Y. and Polyak, B., High-gain robust control, Eur. J. Control, 1999, vol. 5, pp. 3–9.

    Article  Google Scholar 

  24. Polyak, B.T., Tremba, A.A., Khlebnikov, M.V., Shcherbakov, P.S., and Smirnov, G.V., Large deviations in linear control systems with nonzero initial conditions, Autom. Remote Control, 2015, vol. 76, no. 6, pp. 957–976.

    Article  MathSciNet  Google Scholar 

  25. Slotine, J.E. and Li, W., Applied Nonlinear Control, Englewood Cliffs, N.J.: Prentice-Hall, 1991.

    MATH  Google Scholar 

  26. Pesterev, A.V., Attraction domain estimate for single-input affine systems with constrained control, Autom. Remote Control, 2017, vol. 78, no. 4, pp. 581–594.

    Article  MathSciNet  Google Scholar 

  27. Goodfellow, I., Bengio, Y., and Courville, A., Deep Learning, MIT Press, 2016.

  28. Shumikhin, A.G. and Boyarshinova, A.S., Identification of a complex control object with frequency characteristics obtained with its dynamic neural network model, Autom. Remote Control, 2015, vol. 76, no. 4, pp. 650–657.

    Article  MathSciNet  Google Scholar 

  29. Krasnova, S.A. and Antipov, A.S., Hierarchical design of sigmoidal generalized moments of manipulator under uncertainty, Autom. Remote Control, 2018, vol. 79, no. 3, pp. 554–570.

    Article  MathSciNet  Google Scholar 

  30. Kochetkov, S.A., Krasnova, S.A., and Antipov, A.S., Cascade synthesis of electromechanical tracking systems with respect to restrictions on state variables, IFAC-PapersOnLine, 2017, vol. 50, no. 1, pp. 1042–1047.

    Google Scholar 

  31. Antipov, A.S. and Krasnova, S.A., Stabilization system of convey-crane position via sigmoid function, Mekhatron. Avtom. Upr., 2019, vol. 20, no. 10, pp. 41–54.

    Google Scholar 

  32. Kokunko, Yu. and Krasnova, S., Synthesis of a tracking system with restrictions on UAV state variables, Math. Eng. Sci. Aerospace (MESA), 2019, vol. 10, no. 4, pp. 695–705.

    Google Scholar 

  33. Fomichev, V.V., Kraev, A.V., and Rogovskii, A.I., On the zero dynamics equations of some nonlinear systems affine in control, Differ. Equations, 2018, vol. 54, no. 12, pp. 1654–1668.

    Article  MathSciNet  Google Scholar 

  34. Živanović, M.M. and Lazarević, M.P., Using the decomposition principle to stabilize the nominal motion of a mechanical system with given accuracy, Autom. Remote Control, 2012, vol. 73, no. 12, pp. 2001–2020.

  35. Yurkevich, V.D., Multi-channel control system design for a robot manipulator based on the time-scale method, Optoelectron. Instrum. Data Process., 2016, vol. 52, no. 2, pp. 196–202.

    Article  Google Scholar 

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Funding

This work was financially supported in part by the Russian Foundation for Basic Research, project no. 20-01-00363A.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997, Russia

    A. S. Antipov, S. A. Krasnova & V. A. Utkin

Authors
  1. A. S. Antipov
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  2. S. A. Krasnova
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  3. V. A. Utkin
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Correspondence to A. S. Antipov, S. A. Krasnova or V. A. Utkin.

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Translated by V. Potapchouck

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Antipov, A.S., Krasnova, S.A. & Utkin, V.A. Synthesis of Invariant Nonlinear Single-Channel Sigmoid Feedback Tracking Systems Ensuring Given Tracking Accuracy. Autom Remote Control 83, 32–53 (2022). https://doi.org/10.1134/S0005117922010039

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  • DOI: https://doi.org/10.1134/S0005117922010039

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