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Quasidifferentiability and Uniform Observability of Linear Time-Varying Singularly Perturbed Systems

  • CONTROL THEORY
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Abstract

For linear time-varying singularly perturbed systems (LTVSPS) with quasidifferentiable coefficients and a small parameter multiplying some derivatives, the problem of uniform observability is considered. Necessary and sufficient conditions for the quasidifferentiability of the set of output functions independent of the small parameter are proved, observability matrices independent of the small parameter for the slow subsystem and the family of fast subsystems associated with the LTVSPS are constructed, and a connection is established between them and the observability matrix of the original system. On the basis of the complete decomposition of the original LTVSPS with respect to the action of the group of linear nonsingular transformations, we prove rank sufficient conditions for the uniform observability of the LTVSPS that are independent of the small parameter and valid for all sufficiently small values of it. The conditions are expressed in terms of the observability matrices of the slow subsystem and the family of fast subsystems of smaller dimensions than the original LTVSPS.

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REFERENCES

  1. Kalman, R., On the general theory of control systems, in Tr. I kongressa IFAK. T. 2 (Proc. I IFAC Congr. Vol. 2), Moscow: Akad. Nauk SSSR, 1961, pp. 521–547.

    Google Scholar 

  2. Kalman, R.E., Falb, P.L., and Arbib, M.A., Topics in Mathematical System Theory, New York–San Francisco–St. Louis–Toronto–London–Sydney: McGraw-Hill, 1969. Translated under the title: Ocherki po matematicheskoi teorii sistem, Moscow: Mir, 1971.

    MATH  Google Scholar 

  3. Krasovskii, N.H., Teoriya upravleniya dvizheniem (Motion Control Theory), Moscow: Nauka, 1968.

    Google Scholar 

  4. Gaishun, I.V., Vvedenie v teoriyu lineinykh nestatsionarnykh sistem (Introduction to the Theory of Linear Time-Varying Systems), Minsk: Inst. Mat. NAN Belarusi, 1999.

    Google Scholar 

  5. Astrovskii, A.I., Nablyudaemost’ lineinykh nestatsionarnykh sistem (Observability of Linear Time-Varying systems), Minsk: Izd. MIU, 2007.

    Google Scholar 

  6. Astrovskii, A.I. and Gaishun, I.V., State estimation for linear time-varying observation systems, Differ. Equations, 2019, vol. 55, no. 3, pp. 363–373.

    Article  MathSciNet  MATH  Google Scholar 

  7. Astrovskii, A.I. and Gaishun, I.V., Observability of linear time-varying systems with quasiderivative coefficients, SIAM J. Control Optim., 2019, vol. 57, no. 3, pp. 1710–1729.

    Article  MathSciNet  MATH  Google Scholar 

  8. Astrovskii, A.I. and Gaishun, I.V., Lineinye sistemy s kvazidifferentsiruemymi koeffitsientami: upravlyaemost’ i nablyudaemost’ dvizhenii (Linear Systems with Quasidifferentiable Coefficients: Controllability and Observability of Motions), Minsk: Belarus. Navuka, 2013.

    Google Scholar 

  9. Astrovskii, A.I. and Gaishun, I.V., Quasidifferentiability and observability of linear nonstationary systems, Differ. Equations, 2009, vol. 45, no. 11, pp. 1602–1611.

    Article  MathSciNet  MATH  Google Scholar 

  10. Astrovskii, A.I. and Gaishun, I.V., Uniform and approximate observability of linear time-varying systems, Autom. Remote Control, 1998, vol. 59, no. 7, pp. 907–915.

    MATH  Google Scholar 

  11. Derr, V.Ya., Nonoscillation of solutions of a linear quasidifferential equation, Izv. Inst. Mat. Inf. Udmurtsk. Gos. Univ., 1999, no. 1 (16), pp. 3–105.

  12. Vasil’eva, A.B. and Dmitriev, M.G., Singular perturbations in optimal control problems, J. Sov. Math., 1986, vol. 34, no. 3, pp. 1579–1629.

    Article  MATH  Google Scholar 

  13. Kalinin, A.I., Asimptoticheskie metody optimizatsii vozmushchennykh dinamicheskikh sistem (Asymptotic Methods for Optimizing Perturbed Dynamical Systems), Minsk: BGU, 2000.

    Google Scholar 

  14. Kokotović, P.V., Khalil, H.K., and O’Reilly, J., Singular Perturbations Methods in Control: Analysis and Design, New York–London: Academic Press, 1999.

  15. O’Reilly, J., Full-order observers for a class of singularly perturbed linear time-varying systems, Int. J. Control, 1979, vol. 30, no. 5, pp. 745–756.

    Article  MathSciNet  MATH  Google Scholar 

  16. Kopeikina, T.B., Observability of linear time-invariant singularly perturbed systems in the state space, Prikl. Mat. Mekh., 1993, vol. 57, no. 6, pp. 22–32.

    MathSciNet  Google Scholar 

  17. Kopeikina, T.B., Relative observability of linear time-dependent singularly perturbed systems with delay, Differ. Equations, 1998, vol. 34, no. 1, pp. 22–28.

    MathSciNet  MATH  Google Scholar 

  18. Kopeikina, T.B. and Tsekhan, O.B., On the theory of observability of linear singularly perturbed systems, Vestsi NAN Belarusi. Ser. Fiz.-Mat. Navuk, 1999, no. 3, pp. 22–27.

  19. Glizer, V.Y., Observability of singularly perturbed linear time-dependent differential systems with small delay, J. Dyn. Control Syst., 2004, no. 10, pp. 329–363.

  20. Tsekhan, O.B., Conditions for complete observability of linear time-invariant singularly perturbed second-order systems with delay, Vesn. Grodnensk. Dzyarzh. Univ. im. Ya. Kupaly. Ser. 2. Mat. Fiz. Inf. Vylich. Tekh. Kiravanne, 2014, no. 1 (170), pp. 53–64.

  21. Tsekhan, O.B., Conditions for pointwise controllability and pointwise observability of linear time-invariant singularly perturbed systems with delay, Tr. Inst. Mat. NAN Belarusi, 2021, vol. 29, no. 1–2, pp. 138–148.

  22. Tsekhan, O. and Pawluszewicz, E., Observability of singularly perturbed linear time-varying systems on time scales, 26th Int. Conf. Methods Models Autom. Robot. (MMAR) (2022), pp. 116–121.

  23. Dmitriev, M.G. and Kurina, G.A., Singular perturbations in control problems, Autom. Remote Control, 2006, vol. 67, no. 1, pp. 1–43.

    Article  MathSciNet  MATH  Google Scholar 

  24. Gaishun, I.V. and Goryachkin, V.V., Robust and interval observability of two-parameter discrete systems with interval coefficients, Vestsi NAN Belarusi. Ser. Fiz.-Mat. Navuk, 2016, no. 2, pp. 6–9.

  25. Kopeikina, T.B., Some approaches to the controllability investigation of singularly perturbed dynamic systems, Syst. Sci., 1995, vol. 21, no. 1, pp. 17–36.

    MathSciNet  MATH  Google Scholar 

  26. Horn, R. and Johnson, C., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1985. Translated under the title: Matrichnyi analiz, Moscow: Mir, 1989.

    Book  MATH  Google Scholar 

  27. Tikhonov, A.N., Systems of differential equations containing small parameters multiplying derivatives, Mat. Sb., 1952, vol. 31 (73), no. 3, pp. 575–586.

    Google Scholar 

  28. Chang, K., Singular perturbations of a general boundary value problem, SIAM J. Math. Anal., 1972, vol. 3, no. 3, pp. 520–526.

    Article  MathSciNet  MATH  Google Scholar 

  29. Zuber, I.E., Synthesis of an exponentially stable observer for linear time-varying systems with one output, Autom. Remote Control, 1995, vol. 56, no. 5, pp. 644–650.

    MATH  Google Scholar 

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ACKNOWLEDGMENTS

The author is grateful to Prof. A.I. Astrovskii for valuable advice and comments made during the preparation of this work.

Funding

The work was supported by the Ministry of Education of the Republic of Belarus, State Research Program “Convergence-2025,” project no. 1.2.04.

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Authors and Affiliations

  1. Yanka Kupala State University of Grodno, Grodno, 230023, Belarus

    O. B. Tsekhan

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  1. O. B. Tsekhan
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Correspondence to O. B. Tsekhan.

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Translated by V. Potapchouck

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Tsekhan, O.B. Quasidifferentiability and Uniform Observability of Linear Time-Varying Singularly Perturbed Systems. Diff Equat 59, 1130–1146 (2023). https://doi.org/10.1134/S0012266123080116

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  • DOI: https://doi.org/10.1134/S0012266123080116

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