Value-Set-Based Approach to Robust Stability Analysis for Ellipsoidal Families of Fractional-Order Polynomials with Complicated Uncertainty Structure
Source abstract
Abstract
This paper presents the application of a value-set-based graphical approach to robust stability analysis for the ellipsoidal families of fractional-order polynomials with a complex structure of parametric uncertainty. More specifically, the article focuses on the families of fractional-order linear time-invariant polynomials with affine linear, multilinear, polynomic, and general uncertainty structure, combined with the uncertainty bounding set in the shape of an ellipsoid. The robust stability of these families is investigated using the zero exclusion condition, supported by the numerical computation and visualization of the value sets. Four illustrative examples are elaborated, including the comparison with the families of fractional-order polynomials having the standard box-shaped uncertainty bounding set, in order to demonstrate the applicability of this method.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Advanced Control Systems Design · Fractional Differential Equations Solutions · Probabilistic and Robust Engineering Design
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Centre for Security, Information and Advanced Technologies (CEBIA-Tech), Faculty of Applied Informatics, Tomas Bata University in Zlín, nám. T. G. Masaryka 5555, 76001 Zlín, Czech Republic
Department of Computer Engineering, Faculty of Engineering, Inonu University, 44280 Malatya, Turkey
Department of Automation and Control Engineering, Faculty of Applied Informatics, Tomas Bata University in Zlín, nám. T. G. Masaryka 5555, 76001 Zlín, Czech Republic
Funding
European Regional Development Fund · Ministry of Education, Youth and Sports of the Czech Republic
References (55)
Show source-provided references
1. Simani, S., and Castaldi, P. (2018). Robust Control Examples Applied to a Wind Turbine Simulated Model. Appl. Sci., 8.10.5772/intechopen.71526 ↗
2. Sato, T., Hayashi, I., Horibe, Y., Vilanova, R., and Konishi, Y. (2019). Optimal Robust PID Control for First- and Second-Order Plus Dead-Time Processes. Appl. Sci., 9.10.3390/app9091934 ↗
3. Kurokawa, R., Sato, T., Vilanova, R., and Konishi, Y. (2019). Discrete-Time First-Order Plus Dead-Time Model-Reference Trade-off PID Control Design. Appl. Sci., 9.10.3390/app9163220 ↗
4. Robust Controller Design for a Heat Exchanger Using H2, H∞, H2/H∞, and μ-Synthesis Approaches
5. Barmish, B.R. (1994). New Tools for Robustness of Linear Systems, Macmillan.
6. Robust control under parametric uncertainty: An overview and recent results10.1016/j.arcontrol.2017.05.001 ↗
7. Graphical analysis of robust stability for systems with parametric uncertainty: An overview10.1177/0142331210361409 ↗
8. Robust Stability Analysis for Systems with Real Parametric Uncertainty: Implementation of Graphical Tests in Matlab
9. Matušů, R., Şenol, B., and Pekař, L. (2017). Robust stability of fractional order polynomials with complicated uncertainty structure. PLoS ONE, 12.10.1371/journal.pone.0180274 ↗
10. Matušů, R. (2019). Value Sets of Ellipsoidal Polynomial Families with Affine Linear Uncertainty Structure. Cybernetics and Automation Control Theory Methods in Intelligent Algorithms: Proceedings of the 8th Computer Science On-Line Conference 2019, Volume 3—Advances in Intelligent Systems and Computing (Volume 986), Springer Nature Switzerland AG.10.1007/978-3-030-19813-8_26 ↗
11. Matušů, R., and Şenol, B. (2019). Application of Value Set Concept to Ellipsoidal Polynomial Families with Multilinear Uncertainty Structure. Computational Statistics and Mathematical Modeling Methods in Intelligent Systems: Proceedings of 3rd Computational Methods in Systems and Software 2019, Volume 2—Advances in Intelligent Systems and Computing (Volume 1047), Springer Nature Switzerland AG.10.1007/978-3-030-31362-3_9 ↗
12. Hurák, Z., and Šebek, M. (, 2000). New Tools for Spherical Uncertain Systems in Polynomial Toolbox for Matlab. Proceedings of the Technical Computing Prague 2000, Prague, Czech Republic. Available online: https://www.humusoft.cz/archive/events/tcp2000/.
13. Tesi, A., Vicino, A., and Villoresi, F. (1995, January 21–23). Robust Stability of Spherical Plants with Unstructured Uncertainty. Proceedings of the American Control Conference, Seattle, Washington, WA, USA.
14. Polyak, B.T., and Shcherbakov, P.S. (2000, January 12–15). Random Spherical Uncertainty in Estimation and Robustness. Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia.
15. Spherical Families of Polynomials: A Graphical Approach to Robust Stability Analysis
16. Matušů, R. (2014, January 17–21). Families of spherical polynomials: Description and robust stability analysis. Proceedings of the 18th International Conference on Systems, Santorini, Greece.
17. Matušů, R., and Prokop, R. (2015). Robust Stability Analysis for Families of Spherical Polynomials. Intelligent Systems in Cybernetics and Automation Theory: Proceedings of the 4th Computer Science On-line Conference 2015 (CSOC2015), Volume 2—Advances in Intelligent Systems and Computing (Volume 348), Springer International Publishing.
18. Quality-by-design by skewed spherical structured singular value10.1049/iet-cta.2014.1235 ↗
19. Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties10.1007/s11081-014-9255-9 ↗
20. Robust pole-placement technique for plants with ellipsoidally uncertain parameters10.1016/0009-2509(95)00398-3 ↗
21. Robust identification and control based on ellipsoidal parametric uncertainty descriptions10.1016/S0947-3580(00)71132-0 ↗
22. Identification and control: Joint input design and H∞ state feedback with ellipsoidal parametric uncertainty via LMIs10.1016/j.automatica.2007.06.025 ↗
23. Henrion, D., Šebek, M., and Kučera, V. (2001, January 11–14). LMIs for robust stabilization of systems with ellipsoidal uncertainty. Proceedings of the 13th International Conference on Process Control, Štrbské Pleso, Slovakia.
24. Fixed-order robust H∞ control and control-oriented uncertainty set shaping for systems with ellipsoidal parametric uncertainty10.1002/rnc.1614 ↗
25. Fixed-order H∞ controller design for systems with ellipsoidal parametric uncertainty10.1080/00207179.2010.540040 ↗
26. Identification and robust control for systems with ellipsoidal parametric uncertainty by convex optimization10.1002/asjc.437 ↗
27. Robust output feedback control for discrete-time systems with ellipsoidal uncertainty10.1093/imamci/dnv014 ↗
28. On the stability properties of polynomials with perturbed coefficients10.1109/TAC.1985.1103807 ↗
29. Robust stability with structured real parameter perturbations10.1109/TAC.1987.1104648 ↗
30. On the spectral set for a family of polynomials10.1109/9.62276 ↗
31. Frequency domain criteria for lp-robust stability of continuous linear systems10.1109/9.106161 ↗
32. Robust Stability of Thermal Control Systems with Uncertain Parameters: The Graphical Analysis Examples10.1016/j.applthermaleng.2017.07.089 ↗
33. Robust stability analysis of fractional order interval polynomials10.1016/j.isatra.2009.01.002 ↗
34. Şenol, B., and Yeroğlu, C. (2012, January 3–5). Computation of the Value Set of Fractional Order Uncertain Polynomials: A 2q Convex Parpolygonal Approach. Proceedings of the 2012 IEEE International Conference on Control Applications, Dubrovnik, Croatia.10.1109/CCA.2012.6402466 ↗
35. Şenol, B., and Yeroğlu, C. (2012, January 14–17). Robust Stability Analysis of Fractional Order Uncertain Polynomials. Proceedings of the 5th IFAC Workshop on Fractional Differentiation and its Applications, Nanjing, China.
36. Investigation of robust stability of fractional order multilinear affine systems: 2q-convex parpolygon approach10.1016/j.sysconle.2013.06.005 ↗
37. Oldham, K.B., and Spanier, J. (1974). Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press.
38. Podlubný, I. (1999). Fractional Differential Equations, Academic Press.
39. Recent history of fractional calculus10.1016/j.cnsns.2010.05.027 ↗
40. Gutiérrez, R.E., Rosário, J.M., and Machado, J.A.T. (2010). Fractional Order Calculus: Basic Concepts and Engineering Applications. Math. Probl. Eng., 2010.10.1155/2010/375858 ↗
41. Axtell, M., and Bise, M.E. (1990, January 21–25). Fractional calculus applications in control systems. Proceedings of the 1990 National Aerospace and Electronics Conference, Dayton, OH, USA.
42. Fractional-Order Systems and PIλDμ-Controllers10.1109/9.739144 ↗
43. Chen, Y., Petráš, I., and Xue, D. (2009, January 10–12). Fractional Order Control—A Tutorial. Proceedings of the 2009 American Control Conference, St. Louis, MO, USA.10.1109/ACC.2009.5160719 ↗
44. Stability of fractional-order systems with rational orders: A survey
45. Xue, D. (2017). Fractional-Order Control Systems: Fundamentals and Numerical Implementations, De Gruyter.10.1515/9783110497977 ↗
46. Modeling and Control of Quantum Systems: An Introduction10.1109/TAC.2012.2195830 ↗
47. Quantum control theory and applications: A survey10.1049/iet-cta.2009.0508 ↗
48. Chen, C., Wang, L.-C., and Wang, Y. (2013). Closed-Loop and Robust Control of Quantum Systems. Sci. World J., 2013.10.1155/2013/869285 ↗
49. Robust stability of uncertain linear quantum systems10.1098/rsta.2011.0527 ↗
50. Matušů, R., Şenol, B., and Vašek, V. (2019, January 23–26). Robust Stability of Fractional-Order LTI Systems with Independent Structure of Ellipsoidal Parametric Uncertainty. Proceedings of the 30th DAAAM International Symposium, Zadar, Croatia.10.2507/30th.daaam.proceedings.021 ↗
51. Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties10.2478/s13540-013-0010-2 ↗
52. Šebek, M. (2002). Robustní řízení (Robust Control), Czech Technical University in Prague. PDF slides for the course on Robust Control.
53. Matušů, R., and Prokop, R. (2015). Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach. Math. Probl. Eng., 2015.10.1155/2015/847210 ↗
54. Tempo, R., Calafiore, G., and Dabbene, F. (2013). Randomized Algorithms for Analysis and Control of Uncertain Systems: With Applications, Springer.10.1007/978-1-4471-4610-0 ↗
55. The uniform distribution: A rigorous justification for its use in robustness analysis10.1007/BF01211503 ↗