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Mixed Robustness: Analysis of Systems with Uncertain Deterministic and Random Parameters by the Example of Linear Systems

  • LINEAR SYSTEMS
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Abstract

The robustness of linear systems with constant coefficients is considered. There are methods and tools for analyzing the stability of systems with random or deterministic uncertainties. At the same time, there are no approaches to the analysis of systems containing both types of parametric uncertainty. The article classifies the types of robustness and introduces a new type—“mixed parametric robustness”—which includes several variations. The proposed statements of mixed robustness problems can be viewed as intermediate versions between the classical deterministic and probabilistic approaches to robustness. Several cases are listed in which the problems are easy to solve. In the general case, stability tests based on the scenario approach can be applied to robust systems; however, these tests can be computationally costly. A simple graph-analytical approach based on robust \(D \)-decomposition (robust \(D \)-partition) is proposed to calculate the desired stability probability. This method is suitable for the case of a small number of random parameters. The final stability probability estimate is calculated in a deterministic way and can be found with arbitrary precision. Approximate methods for solving the above problems are described. Examples and a generalization of mixed robustness to other types of systems are given.

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Notes

  1. In general, the proposed problem statements are suitable for any parameter-dependent systems; see the final section 6 for details. One can also consider nonparametric uncertainty, but the formal description of the corresponding problems is difficult. The generalization of mixed robustness to other system properties or performance criteria besides stability is trivial.

  2. The English term robustness (robust) is interpreted even more diversely. In Russian, the term robustness in relation to control systems was systematically used in the monograph [1].

  3. Here “disturbance” pertains to changes in system parameters or other characteristics rather than external influences. Historically, robustness is a system’s ability to maintain a given property under varying characteristics of external signals rather than to resist exogenous disturbances or suppress external noise with given specific characteristics. The border between the two cases is blurred, and the term “robustness” has no well-established scope.

  4. Strictly speaking, the distribution is given by the triple \( (\Delta ,\mathcal {F},\mathcal {P})\) with the elementary event set \(\Delta \), the sigma algebra of its subsets \(\mathcal {F}\), and the probability measure \(\mathcal {P}:\mathcal {F}\rightarrow [0, 1] \), but for brevity, redundant notation is omitted. In what follows, the measure is denoted by symbol \(\mu \). The traditional Lebesgue measure is used, and the distribution notation is associated with the set \(\Delta \). The uncertainty sets considered are assumed to be measurable. One can consider both absolutely continuous distributions given by the distribution density function and discrete distributions, or their mixture. In the context of robust analysis, from the probabilistic point of view, their support and the (cumulative) distribution function are important.

  5. Up to sets of zero probability. Depending on the problem, this difference can be negligible or, on the opposite, significant.

  6. The value of the performance indicator in the statement is replaced by the stability criterion for brevity. The mentioned chapter contains many other ideas of robustness analysis, for example, the study of degradation of the performance criterion (with random parameters), etc.; see also a similar approach to “scaled” robustness in [13].

  7. Extensions to other types of systems are presented in Sec. 6.

  8. This method is especially suitable for analyzing the robustness of any property of the system determined by the location of the roots of the characteristic polynomial.

  9. The first and last intervals can be of infinite length, for example, \((-\infty ,b_1]\). In addition, these intervals can be open or half-open, for example, \([a_j,b_j] \) for some \(j \) and \((a_j,b_j) \) for other \(j \), etc. The difference between these cases has measure zero and does not affect the calculation of the probability for an absolutely continuous distribution \( \delta \). However, these subtleties must be taken into account in the case of a discrete or mixed distribution.

REFERENCES

  1. Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.

    Google Scholar 

  2. Discussion on robustness problems in control systems, Autom. Remote Control, 1992, vol. 53, no. 1, pp. 134–142.

  3. Weinmann, A., Uncertain Models and Robust Control, Vienna: Springer, 1991.

    Book  Google Scholar 

  4. Barmish, B.R., New Tools for Robustness of Linear Systems, New York: Macmillan Coll Div., 1993.

    MATH  Google Scholar 

  5. Bhattacharyya, S.P., Chapellat, H., and Keel, L.H., Robust Control: The Parametric Approach, New Jersey: Prentice Hall, 1995.

    MATH  Google Scholar 

  6. Polyak, B.T. and Tsypkin, Ya.Z., Robust stability of linear systems, Dokl. Math., 1991, vol. 36, no. 2, pp. 111–113.

    MathSciNet  MATH  Google Scholar 

  7. Ershov, A.A., Stable methods of estimating parameters (survey), Autom. Remote Control, 1979, vol. 39, no. 8, pp. 1152–1181.

    MATH  Google Scholar 

  8. Braverman, M.E. and Rozonoer, L.I., Robustness of linear dynamic systems. I, Autom. Remote Control, 1991, vol. 52, no. 11, pp. 1493–1498.

    Google Scholar 

  9. Kharitonov, V.L., On asymptotic stability of an equilibrium of a family of systems of linear differential equations, Differ. Uravn., 1978, vol. 14, no. 11, pp. 2086–2088.

    MathSciNet  Google Scholar 

  10. Nemirovskii, A., Several NP-hard problems arising in robust stability analysis, Math. Control Signals Syst., 1993, vol. 6, no. 2, pp. 99–105.

    Article  MathSciNet  Google Scholar 

  11. Poljak, S. and Rohn, J., Checking robust nonsingularity is NP-hard, Math. Control Signals Syst., 1993, vol. 6, no. 1, pp. 1–9.

    Article  MathSciNet  Google Scholar 

  12. Tempo, R., Calafiore, G., and Dabbene, F., Randomized Algorithms for Analysis and Control of Uncertain Systems: with Applications, London: Springer, 2013.

    Book  Google Scholar 

  13. Garatti, S. and Campi, M.C., Modulating robustness in control design: principles and algorithms, IEEE Control Syst. Mag., 2013, vol. 33, no. 2, pp. 36–51.

    Article  MathSciNet  Google Scholar 

  14. Tempo, R., Bai, E.W., and Dabbene, F., Probabilistic robustness analysis: explicit bounds for the minimum number of samples, Proc.35th IEEE Conf. Dec. Control, 1996, vol. 3, pp. 3424–3428.

    Article  Google Scholar 

  15. Barmish, B.R. and Lagoa, C.M., The uniform distribution: a rigorous justification for its use in robustness analysis, Math. Control Signals Syst., 1997, vol. 10, no. 3, pp. 203–222.

    Article  MathSciNet  Google Scholar 

  16. Bai, E.W., Tempo, R., and Fu, M., Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis, Proc. 1997 Am. Control Conf., 1997, vol. 1, pp. 861–865.

    Google Scholar 

  17. Calafiore, G.C. and Campi, M.C., The Scenario Approach to Robust Control Design, IEEE Trans. Autom. Control, 2006, vol. 51, no. 5, pp. 742–753.

    Article  MathSciNet  Google Scholar 

  18. Gryazina, E. and Polyak, B., Random sampling: billiard walk algorithm, Eur. J. Oper. Res., 2014, vol. 238, no. 2, pp. 497–504.

    Article  MathSciNet  Google Scholar 

  19. Fan, M.K.H., Tits, A.L., and Doyle, J.C., Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Trans. Autom. Control, 1991, vol. 36, no. 1, pp. 25–38.

    Article  MathSciNet  Google Scholar 

  20. Jayasuriya, S., Frequency domain design for robust performance under parametric, unstructured, or mixed uncertainties, J. Dyn. Syst. Meas. Control, 1993, vol. 115, no. 2B, pp. 439–451.

    Article  MathSciNet  Google Scholar 

  21. Reinelt W., Ljung L., Robust control of identified models with mixed parametric and non-parametric uncertainties, Proc. 2001 Eur. Control Conf., 2001, pp. 3564–3569.

  22. Vitus, M.P., Zhou, Z., and Tomlin, C.J., Stochastic control with uncertain parameters via chance constrained control, IEEE Trans. Autom. Control, 2016, vol. 61, no. 10, pp. 2892–2905.

    Article  MathSciNet  Google Scholar 

  23. da Silva de Aguiar, R.S., Apkarian, P., and Noll, D., Structured Robust Control against Mixed Uncertainty, IEEE Trans. Control Syst. Tech., 2018, vol. 26, no. 5, pp. 1771–1781.

    Article  Google Scholar 

  24. Chapellat, H., Dahleh, M.A., and Bhattacharyya, S.P., Robust stability under structured and unstructured perturbations, IEEE Trans. Autom. Control, 1990, vol. 35, no. 10, pp. 1100–1108.

    Article  MathSciNet  Google Scholar 

  25. Fujisaki, Y., Oishi, Y., and Tempo, R., Mixed deterministic/randomized methods for fixed order controller design, IEEE Trans. Autom. Control, 2008, vol. 53, no. 9, pp. 2033–2047.

    Article  MathSciNet  Google Scholar 

  26. Wu, D., Gao, W., Song, C., and Tangaramvong, S., Probabilistic interval stability assessment for structures with mixed uncertainty, Struct. Safety, 2016, vol. 58, pp. 105–118.

    Article  Google Scholar 

  27. Keel, L.H. and Bhattacharyya, S.P., Robust, fragile, or optimal?, IEEE Trans. Autom. Control, 1997, vol. 42, no. 8, pp. 1098–1105.

    Article  MathSciNet  Google Scholar 

  28. Jury, E.I., Robustness of discrete systems, Autom. Remote Control, 1990, vol. 51, no. 5, pp. 571–592.

    MathSciNet  MATH  Google Scholar 

  29. Petrov, N.P. and Polyak, B.T., Robust D-partition, Autom. Remote Control, 1991, vol. 52, no. 11, pp. 1513–1523.

    MathSciNet  MATH  Google Scholar 

  30. Tremba, A.A., Robust D-decomposition under \(l_p \)-bounded parametric uncertainties, Autom. Remote Control, 2006, vol. 67, no. 12, pp. 1878–1892.

    Article  MathSciNet  Google Scholar 

  31. Siljak, D.D., A robust control design in the parameter space, Proc. 1992 IEEE Int. Symp. Circuits Syst., 1992, vol. 6, pp. 2723–2727.

    Article  Google Scholar 

  32. Hwang, C., Hwang, L., and Hwang, J., Robust D-partition, J. Chin. Inst. Eng., 2010, vol. 33, no. 6, pp. 811–821.

    Article  Google Scholar 

  33. Mihailescu-Stoica, D., Schrödel, F., and Adamy, J., All stabilizing PID controllers for interval systems and systems with affine parametric uncertainties, 11th Asian Control Conf. (ASCC) (2017), pp. 576–581.

  34. Matsuda, T. and Mori, T., Stability feeler: a tool for parametric robust stability analysis and its applications, IET Control Theory Appl., 2009, vol. 3, no. 12, pp. 1625–1633.

    Article  MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

The author is keenly grateful to B.T. Polyak for his support in the preparation of the article and also thanks the anonymous referee, whose comments have led to a significant improvement of the article.

Funding

This work was supported in part by the Russian Science Foundation, project no. 16-11-10015.

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

    A. A. Tremba

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Translated by V. Potapchouck

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Tremba, A.A. Mixed Robustness: Analysis of Systems with Uncertain Deterministic and Random Parameters by the Example of Linear Systems. Autom Remote Control 82, 410–432 (2021). https://doi.org/10.1134/S0005117921030036

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