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Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality

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Abstract

The research deals with the methodology intended to root robust quality indices in the interval control system, the parameters of which are affinely included in the coefficients of a characteristic polynomial. To determine the root quality indices we propose to depict on the root plane not all edges of the interval parametric polytope (as the edge theorem says), but its particular vertex-edge route. In order to define this route we need to know the angle sequence at which the edge branches depart from any integrated pole on the allocation area. It is revealed that the edge branches can integrate into the route both fully or partially due to intersection with other branches. The conditions which determine the intersection of one-face edge images have been proven. It is shown that the root quality indices can be determined by its ends or by any other internal point depending on a type of edge branch. The conditions which allow determining the edge branch type have been identified. On the basis of these studies we developed the algorithm intended to construct a boundary vertex-edge route on the polytope with the interval parameters of the system. As an illustration of how the algorithm can be implemented, we determined and introduced the root indices reflecting the robust quality of the system used to stabilize the position of an underwater charging station for autonomous unmanned vehicles.

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References

  1. S. P. Bhattacharyya. Robust control under parametric uncertainty: An overview and recent results. Annual Reviews in Control, vol.44, pp.45–77, 2017. DOI: https://doi.org/10.1016/j.arcontrol.2017.05.001.

    Article  Google Scholar 

  2. J. Ackermann. Robust Control: Systems with Uncertain Physical Parameters, London, UK: Springer-Verlag, pp. 57–76, 1993.

    Book  Google Scholar 

  3. D. Mihailescu-Stoica, F. Schrodel, R. Vobetawinkel, J. Adamy. On robustly stabilizing PID controllers for systems with a certain class of multilinear parameter dependency. In Proceedings of the 26th Mediterranean Conference on Control and Automation, IEEE, Zadar, Croatia, pp. 1–6, 2018. DOI: https://doi.org/10.1109/MED.2018.8442811.

    Google Scholar 

  4. Y. V. Hote. Necessary conditions for stability of Khari-tonov polynomials. IETE Technical Review, vol.28, no.5, pp. 445–448, 2011. DOI: https://doi.org/10.4103/0256-4602.85977.

    Article  Google Scholar 

  5. B. Y. Juang. Robustness of pole assignment of an interval polynomial using like λ-degree feedback gain based on the Kharitonov theorem. In Proceedings of SICE Annual Conference, IEEE, Taipei, China, pp. 3475–3484, 2010.

    Google Scholar 

  6. A. Karimi, A. Nicoletti, Y. M. Zhu. Robust H, controller design using frequency-domain data via convex optimization. International Journal of Robust and Nonlinear Control, vol.28, no. 12, pp.3766–3783, 2018. DOI: https://doi.org/10.1002/rnc.3594.

    Article  MathSciNet  Google Scholar 

  7. T. A. Bryntseva, A. L. Fradkov. Frequency-domain estimates of the sampling interval in multirate nonlinear systems by time-delay approach. International Journal of Control, pp. 1–8, 2018. DOI: https://doi.org/10.1080/00207179.2017.1423394.

    Google Scholar 

  8. Y. Hwang, Y. R. Ko, Y. Lee, T. H. Kim. Frequency-domain tuning of robust fixed-structure controllers via quantum-behaved particle swarm optimizer with cyclic neighborhood topology. International Journal of Control, Automation and Systems, vol. 16, no. 2, pp. 426–436, 2018. DOI: https://doi.org/10.1007/s12555-016-0766-3.

    Article  Google Scholar 

  9. B. Basu, A. Staino. Time-frequency control of linear time-varying systems using forward Riccati differential equation. In Proceedings of Indian Control Conference, IEEE, Kanpur, India, pp. 223–228, 2018. DOI: https://doi.org/10.1109/INDI-ANCC.2018.8307982.

    Google Scholar 

  10. J. Garcia-Tirado, H. Botero, F. Angulo. A new approach to state estimation for uncertain linear systems in a moving horizon estimation setting. International Journal of Automation and Computing, vol.13, no. 6, pp. 653–664, 2016. DOI: https://doi.org/10.1007/s11633-016-1015-1.

    Article  Google Scholar 

  11. A. Khalil, J. H. Wang, O. Mohamed. Robust stabilization of load frequency control system under networked environment. International Journal of Automation and Computing, vol.14, no. 1, pp.93–105, 2017. DOI: https://doi.org/10.1007/s1l633-016-1041-z.

    Article  Google Scholar 

  12. Z. Gao, L. R. Zhai, Y. D. Liu. Robust stabilizing regions of fractional-order PP controllers for fractional-order systems with time-delays. International Journal of Automation and Computing, vol. 14, no. 3, pp. 340–349, 2017. DOI: https://doi.org/10.1007/s1l633-015-0941-7.

    Article  Google Scholar 

  13. M. S. Sunila, V. Sankaranarayanan, K. Sundareswaran. Comparative analysis of optimized output regulation of a SISO nonlinear system using different sliding manifolds. International Journal of Automation and Computing, vol.14, no.4, pp.450–462, 2017. DOI: https://doi.org/10.1007/s1l633-017-1078-7.

    Article  Google Scholar 

  14. Y. Jiang, J. Y. Dai. An adaptive regulation problem and its application. International Journal of Automation and Computing, vol. 14, no. 2, pp. 221–228, 2017. DOI: https://doi.org/10.1007/s11633-015-0900-3.

    Article  Google Scholar 

  15. A. Zouhri, I. Boumhidi. Decentralized robust ft control of large scale systems with polytopic-type uncertainty. International Review on Automatic Control, vol.9, no.2, pp. 103–109, 2016. DOI: https://doi.org/10.15866/ireaco.v9i2.8728.

    Article  Google Scholar 

  16. M. Khadhraoui, M. Ezzine, H. Messaoud, M. Darouach. Full order ft filter design for delayed singular systems with unknown input and bounded disturbance: Time and frequency domain approaches. International Review on Automatic Control, vol.9, no. 1, pp.26–39, 2016. DOI: https://doi.org/10.15866/ireaco.v9il.7843.

    Article  Google Scholar 

  17. B. B. Alagoz, C. Yeroglu, B. Senol, A. Ates. Probabilistic robust stabilization of fractional order systems with interval uncertainty. ISA Transactions, vol.57, pp. 101–110, 2015. DOI: https://doi.org/10.1016/j.isatra.2015.01.003.

    Article  Google Scholar 

  18. H. S. Ahn, Y. Q. Chen. Necessary and sufficient stability condition of fractional-order interval linear systems. Auto-matica, vol.44, no. 11, pp.2985–2988, 2008. DOI: https://doi.org/10.1016/j.automatica.2008.07.003.

    MathSciNet  MATH  Google Scholar 

  19. B. Senol, A. Ates, B. B. Alagoz, C. Yeroglu. A numerical investigation for robust stability of fractional-order uncertain systems. ISA Transactions, vol.53, no.2, pp. 189–198, 2014. DOI: https://doi.org/10.1016/j.isatra.2013.09.004.

    Article  Google Scholar 

  20. J. G. Lu, Y. Q. Chen. Robust stability and stabilization of fractional-order interval systems with the fractional order a: The 0 < a < 1 case. IEEE Transactions on Automatic Control, vol.55, no. 1, pp. 152–158, 2010. DOI: https://doi.org/10.1109/TAC.2009.2033738.

    Article  MathSciNet  Google Scholar 

  21. I. N’Doye, M. Darouach, M. Zasadzinski, N. E. Radhy. Robust stabilization of uncertain descriptor fractional-order systems. Automatica, vol.49, no.6, pp. 1907–1913, 2013. DOI: https://doi.org/10.1016/j.automatica.2013.02.066.

    Article  MathSciNet  Google Scholar 

  22. P. M. Young, M. P. Newlin, J. C. Doyle. μ analysis with real parametric uncertainty. In Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, UK, vol.2, pp. 1251–1256, 1991. DOI: https://doi.org/10.1109/CDC.1991.261579.

    Google Scholar 

  23. S. Sumsurooah, M. Odavic, S. Bozhko. μ approach to robust stability domains in the space of parametric uncertainties for a power system with ideal CPL. IEEE Transactions on Power Electronics, vol.33, no. 1, pp.833–844, 2018. DOI: https://doi.org/10.1109/TPEL.2017.2668900.

    Article  Google Scholar 

  24. S. Sumsurooah, M. Odavic, S. Bozhko, D. Boroyevic. Toward robust stability of aircraft electrical power systems: Using a μ-based structural singular value to analyze and ensure network stability. IEEE Electrification Magazine, vol.5, no.4, pp.62–71, 2017. DOI: https://doi.org/10.1109/MELE.2017.2757383.

    Article  Google Scholar 

  25. K. Chaker, A. Moussaoui, B. Sbartai. ¼-synthesis control applied to counter the seismic load action on a building structure. International Review of Automatic Control, vol.10, no. 1, pp. 92–99, 2017. DOI: https://doi.org/10.15866/ireaco.vlOil.10617.

    Article  Google Scholar 

  26. B. K. Sahu, B. Subudhi, M. M. Gupta. Stability analysis of an underactuated autonomous underwater vehicle using extended-routh’s stability method. International Journal of Automation and Computing, vol.15, no. 3, pp. 299–309, 2018. DOI: https://doi.org/10.1007/s1l633-016-0992-4.

    Article  Google Scholar 

  27. X. Q. Zhang, X. Y. Li, J. Zhao. Stability analysis and anti-windup design of switched systems with actuator saturation. International Journal of Automation and Computing, vol.14, no. 5, pp. 615–625, 2017. DOI: https://doi.org/10.1007/s1l633-015-0920-z.

    Article  Google Scholar 

  28. Z. Liu, Y. Z. Wang. Regional stability of positive switched linear systems with multi-equilibrium points. International Journal of Automation and Computing, vol.14, no. 2, pp.213–220, 2017. DOI: https://doi.org/10.1007/s1l633-016-1003-5.

    Article  MathSciNet  Google Scholar 

  29. F. D. C. Da Silva, J. B. De Oliveira, A. D. De Araujo. Robust interval adaptive pole-placement controller based on variable structure systems theory. In Proceedings of the 25th International Conference on Systems Engineering, IEEE, Las Vegas, USA, pp. 45–54, 2017. DOI: https://doi.org/10.1109/IC-SEng.2017.73.

    Google Scholar 

  30. W. Wiboonjaroen, T. Sooknuan, M. Thumma. Robust pole placement by state-Pi feedback control for interval plants. In Proceedings of Computing Conference, IEEE, London, UK, pp. 1350–1356, 2017. DOI: https://doi.org/10.1109/SAI.2017.8252266.

    Google Scholar 

  31. L. H. Keel, S. P. Bhattacharyya. Robustness and fragility of high order controllers: A tutorial. In Proceedings of IEEE Conference on Control Applications, IEEE, Buenos Aires, Argentina, pp. 191–202, 2016. DOI: https://doi.org/10.1109/CCA.2016.7587837.

    Google Scholar 

  32. A. A. Nesenchuk. A method for synthesis of robust interval polynomials using the extended root locus. In Proceedings of American Control Conference, IEEE, Seattle, USA, pp. 1715–1720, 2017. DOI: https://doi.org/10.23919/ACC.2017.7963200.

    Google Scholar 

  33. Y. Chursin, D. Sonkin, M. Sukhodoev, R. Nurmuhametov, V. Pavlichev. Control system for an object with interval-given parameters: Quality analysis based on leading coefficients of characteristic polynomials. International Review of Automatic Control, vol.11, no. 4, pp.203–207, 2018. DOI: https://doi.org/10.15866/ireaco.vlli4.15727.

    Article  Google Scholar 

  34. B. Senol, C. Yeroglu. Robust stability analysis of fractional order uncertain polynomials. In Proceedings of the 5th IFAC Workshop on Fractional Differentiation and its Applications, Nanjing, China, pp. 1–6, 2012.

    Google Scholar 

  35. A. V. Egorov, C. Cuvas, S. Mondie. Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays. Automatica, vol.80, pp.218–224, 2017. DOI: https://doi.org/10.1016/j.automatica.2017.02.034.

    Article  MathSciNet  Google Scholar 

  36. S. A. Gayvoronskiy, T. Ezangina. The algorithm of analysis of root quality indices of high order interval systems. In Proceedings of the 27th Chinese Control and Decision Conference, IEEE, Qingdao, China, pp. 3048–3052, 2015. DOI: https://doi.org/10.1109/CCDC.2015.7162444.

    Google Scholar 

  37. O. S. Vadutov, S. A. Gayvoronskiy. Application of edge routing to the stability analysis of interval polynomials. Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, vol.6, pp.7–12, 2003.

    MathSciNet  Google Scholar 

  38. S. A. Gayvoronskiy, T. Ezangina, I. Khozhaev. The analysis of permissible quality indices of the system with affine uncertainty of characteristic polynomial coefficients. In Proceedings of International Automatic Control Conference, IEEE, Taichung, China, pp. 30–34, 2016. DOI: https://doi.org/10.1109/CACS.2016.7973879.

    Google Scholar 

  39. B. B. Alagoz. A note on robust stability analysis of fractional order interval systems by minimum argument vertex and edge polynomials. IEEE/CAA Journal of Automatica Sinica, vol.3, no.4, pp.411-121, 2016. DOI: https://doi.org/10.1109/JAS.2016.7510088.

  40. C. Othman, I. Ben Cheikh, D. Soudani. On the internal multi-model control of uncertain discrete-time systems. International Journal of Advanced Computer Science and Applications, vol.7, no.9, pp.88–98, 2016. DOI: https://doi.org/10.14569/IJACSA.2016.070912.

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Ministry of Education and Science of the Russian Federation (No. 2.3649. 2017/PCh).

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Authors and Affiliations

  1. School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Tomsk, 634050, Russia

    Sergey Gayvoronskiy, Tatiana Ezangina, Ivan Khozhaev & Viktor Kazmin

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  1. Sergey Gayvoronskiy
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Correspondence to Ivan Khozhaev.

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Recommended by Associate Editor James Whidborne

Sergey Gayvoronskiy received the Ph.D. degree in control systems engineering from the Tomsk Polytechnic University, Russia in 1990. He is presently an associated professor of the Division for Automation and Robotics at the School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Russia. He was repeatedly awarded by Ministry of Education and Science of Russian Federation, Russian Union of Young Scientists and Tomsk Polytechnic University for his educational and scientific achievements.

His research and teaching interests include analysis and synthesis of robust and adaptive control systems for control objects and processes with uncertain parameters.

Tatiana Ezangina received the Ph. D. degree in system analysis, control and data processing from Tomsk Polytechnic University, Russia in 2016. She is presently a junior researcher of Telecommunications, Electronics and Underwater Geology Laboratory, National Research Tomsk Polytechnic University, Russia. She was repeatedly awarded by the Government of Russian Federation, Ministry of Education and Science of Russian Federation, Tomsk Polytechnic University and other institutions for her scientific achievements.

Her research interests include robust and adaptive control system analysis and synthesis, tethered underwater vehicles development and software development.

Ivan Khozhaev received the B.Sc. and M.Sc. degrees (honors) in control systems engineering from the Tomsk Polytechnic University, Russia in 2014 and 2016, accordingly. He is presently a Ph.D. student at the Division for Automation and Robotics, the School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Russia.

His research interests include robust and adaptive control systems synthesis and analysis, unmanned underwater vehicles development and computational fluid dynamics.

Viktor Kazmin received the Ph.D. degree in control systems engineering from the Tomsk Polytechnic University, Russia in 1996. He is presently an associated professor of the Division for Automation and Robotics at the School of Computer Science and Robotics, National Research Tomsk Polytechnic University, Russia. He was repeatedly awarded by Tomsk Polytechnic University for his educational achievements.

His research and teaching interests include analysis and synthesis of feedback control systems for internal combustion engines, fundamentals of control theory and automated control.

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Gayvoronskiy, S., Ezangina, T., Khozhaev, I. et al. Determination of Vertices and Edges in a Parametric Polytope to Analyze Root Indices of Robust Control Quality. Int. J. Autom. Comput. 16, 828–837 (2019). https://doi.org/10.1007/s11633-019-1182-y

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