Control de sistemas con incertidumbres paramétricas empleando el análisis de intervalos

Pedro Teppa Garran; Carlos  Acosta Farkass

doi:10.62876/tekhn.v28i1.7032

Authors

Pedro Teppa Garran Universidad Metropolitana, Venezuela https://orcid.org/0000-0001-6384-3185
Carlos Acosta Farkass Universidad Metropolitana, Venezuela https://orcid.org/0009-0004-6906-6339

DOI:

https://doi.org/10.62876/tekhn.v28i1.7032

Keywords:

Robust control, Parametric uncertainties, Interval analysis, Tracking, Disturbance rejection, United Nations SDG 9

Abstract

A method that combines the algebraic pole-assignment approach, the internal model principle, and the mathematical theory of interval analysis is proposed to design a fixed-order robust controller that guarantees tracking of a reference input with zero steady-state error, and the asymptotic rejection of perturbation signals for plants with parametric uncertainty. The model selected for the plant is an interval transfer function where the extremes of each interval are assumed to be known. The design is formulated through a system of interval linear equations whose solution is carried out using the INTLAB program, obtaining an interval controller. The method is applied in three cases of practical interest, such as a heat flow process, a coupled tank system, and an independently excited direct current motor. The applications were selected to consider different sources of uncertainty and the performance achieved in each case shows the effectiveness of the proposed method.

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References

Shumafov, M.M., (2019). Stabilization of Linear Control Systems and Pole Assignment Problem: A Survey. Vestnik St.Petersb. Univ.Math.52, pp. 349–367. DOI: https://doi.org/10.1134/S1063454119040095

Chu, E.K., (2001). Optimization and pole assignment in control system design. International Journal of Applied Mathematics and Computer Science, 11(5), pp. 1035-1053

Ackermann, J. E. (2009). Pole placement control. Control System, Robotics and Automation, 8(2011), 74-101.

Chen, C.T., (1987). Introduction to the linear algebraic method for control system design. IEEE Control Systems Magazine, 7(5), pp. 36-42. DOI: https://doi.org/10.1109/MCS.1987.1105378

Chen, C., (1999). Linear System: Theory and Design. Oxford University Press.

Åström, K. J., Wittenmark, B. (2013). Computer-controlled systems: theory and design. Courier Corporation.

Tu, Y. W., Ho, M. T. (2011). Robust second-order controller synthesis for model matching of interval plants and its application to servo motor control. IEEE Transactions on Control Systems Technology, 20(2), 530-537. DOI: https://doi.org/10.1109/TCST.2011.2118758

Francis, B. A. (Ed.). (1987). A course in H∞ control theory. Berlin, Heidelberg: Springer Berlin Heidelberg. DOI: https://doi.org/10.1007/BFb0007371

Chilali, M., & Gahinet, P. (1996). ℋ∞design with pole placement constraints: an lmi approach. IEEE Transactions on automatic control, 41(3), 358-367. DOI: https://doi.org/10.1109/9.486637

Badran, S. M., & Emam, A. S. (2012, May). H-infinity and Mixed H2/H-infinity with Pole-Placement Design via ILMI Method for Semi-active Suspension System. In 2012 Sixth Asia Modelling Symposium(pp. 150-155). IEEE. DOI: https://doi.org/10.1109/AMS.2012.23

Benyamina, M., Bouhamida, M., Denai, M., & Taleb, R. (2018). Modeling and Control of a UPFC System Using Pole-Placement and Hinf Robust Control Techniques. International Journal of Engineering and Technology (IJET). DOI: https://doi.org/10.21817/ijet/2018/v10i2/181002066

Wajdi, S., Anis, S., & Garcia,G. (2015). Robust sliding mode control approach for systems affected by unmatched uncertainties using H∞ with pole clustering constraints. Optimal Control Applications and Methods, 36(6), 919-935 DOI: https://doi.org/10.1002/oca.2147

Barmish, B. R., & Jury, E. I. (1994). New tools for robustness of linear systems. IEEE Transactions on Automatic Control, 39(12), 2525-2525.

Soylemez, M. T., & Munro, N. (1997). Robust pole assignment in uncertain systems. IEE Proceedings-Control Theory and Applications, 144(3), 217-224. DOI: https://doi.org/10.1049/ip-cta:19970892

Le, X., & Wang, J. (2013).Robust pole assignment for synthesizing feedback control systems using recurrent neural networks. IEEE transactions on neural networks and learning systems, 25(2), 383-393. DOI: https://doi.org/10.1109/TNNLS.2013.2275732

Soliman, H. M., & El Metwally, K. A. (2017). Robust pole placement for power systems using two‐dimensional membership fuzzy constrained controllers. IET Generation, Transmission & Distribution, 11(16), 3966-3973. DOI: https://doi.org/10.1049/iet-gtd.2016.2064

Chang, W. J., Lin, Y. H., Du, J., & Chang, C. M. (2019). Fuzzy control with pole assignment and variance constraints for continuous-time perturbed Takagi-Sugeno fuzzy models: Application to ship steering systems. International Journal of Control, Automation and Systems, 17(10), 2677-2692. DOI: https://doi.org/10.1007/s12555-018-0917-9

Dincel, E., & Söylemez,M. T. (2022). Robust PID controller design via dominant pole assignment for systems with parametric uncertainties. Asian Journal of Control, 24(2), 834-844. DOI: https://doi.org/10.1002/asjc.2484

Kazemi, M. H., & Tarighi, R. (2024). PID-based attitude control of quadrotor using robust pole assignment and LPV modeling. International Journal of Dynamics and Control, 1-13. DOI: https://doi.org/10.1007/s40435-023-01372-6

Francis, B. A., Wonham, W. M. (1976). The internal model principle of control theory. Automatica, 12(5), 457-465. DOI: https://doi.org/10.1016/0005-1098(76)90006-6

Hedberg, E., Löfberg, J., Helmersson, A. (2020). A pedagogical path from the internal model principle to Youla-Kučera parametrization. IFAC-PapersOnLine, 53(2), 17374-17379. DOI: https://doi.org/10.1016/j.ifacol.2020.12.2090

Persson, P., Åström, K. (1992) Dominant pole design a unified view of PID controller tuning, IFAC Proceedings Volumes, 25(14), 377-382. DOI: https://doi.org/10.1016/S1474-6670(17)50763-6

Moore, R. E., Kearfott, R. B., & Cloud, M. J. (2009). Introduction to interval analysis. Society for Industrial and Applied Mathematics. DOI: https://doi.org/10.1137/1.9780898717716

Alefeld, G., & Herzberger, J. (2012). Introduction to interval computation. Academic press.

Jaulin, L., Kieffer, M., Didrit, O., Walter, E., Jaulin, L., Kieffer, M., & Walter, É. (2001). Interval analysis. Springer London. DOI: https://doi.org/10.1007/978-1-4471-0249-6

Dorf, R., Bishop, R., (2017). Modern control systems. Pearson Prentice Hall.

Oettli, W., & Prager, W. (1964). Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numerische Mathematik, 6, 405-409. DOI: https://doi.org/10.1007/BF01386090

Rohn, J. (1989). Systems of linear interval equations. Linear algebra and its applications, 126, 39-78. DOI: https://doi.org/10.1016/0024-3795(89)90004-9

Rohn, J. (1996). Enclosing solutions of overdetermined systems of linear interval equations. Reliab. Comput., 2(2), 167-171. DOI: https://doi.org/10.1007/BF02425920

Nirmala, T., Datta, D., Kushwaha, H. S., & Ganesan, K. (2011). Inverse interval matrix: A new approach. Applied Mathematical Sciences, 5(13), 607-624.

Nirmala, T., Datta, D., Kushwaha, H. S., & Ganesan, K. (2013). The determinant of an interval matrix using Gaussian elimination method. International Journal of Pure and Applied Mathematics, 88(1), 15-34. DOI: https://doi.org/10.12732/ijpam.v88i1.2

Surya, S. H., Nirmala, T., & Ganesan, K. (2023). Jordan canonical form of interval matrices and applications. Aust. J. Math. Anal. Appl.20(2), 17pp

Nirmala, T., & Ganesan, K. (2019, June). Solution of interval linear system of equations-an iterative approach. In The11th National Conference on Mathematical Techniques and Applications(Vol. 2112, No. 1, p. 020105). DOI: https://doi.org/10.1063/1.5112290

Nirmala, T., & Ganesan, K. (2021, April). Solving system of interval linear equations by Gauss Jordon method using generalized interval arithmetic. In IOP Conference Series: Materials Science and Engineering(Vol. 1130, No. 1, p. 012052). IOP Publishing. DOI: https://doi.org/10.1088/1757-899X/1130/1/012052

Rump, S. M. (1999). INTLAB—interval laboratory. In Developments in reliable computing. Dordrecht: Springer Netherlands. DOI: https://doi.org/10.1007/978-94-017-1247-7_7

Hargreaves, G. I. (2002). Interval analysis in MATLAB. Numerical Algorithms, (2009.1).

Al-Saggaf, U., Mehedi, I., Bettayeb, M., and Mansouri, R. (2016). Fractional-order controller design for a heat flow process. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 230(7), 680-691. DOI: https://doi.org/10.1177/0959651816649917

Teppa-Garran, P. y El Gharib, G. (2024). Computer-assisted optimal tuning of PI controllers for nonlinear systems with amplitude constraints on the actuator. Ciencia e Ingeniería. 45(1), 1-10.

Peralta, A. M., Bautista, J. D. M., de Souza, R. U., & Arboleda, T. J. O. (2020). Modelación de Motor/generador de corriente continua conexión independiente con MATLAB/SIMULINK. Dominio de las Ciencias, 6(5), 361-377.