Multivariable fuzzy PID controller using convex optimization and positive real lemma for nonlinear systems

Document Type : Original Manuscript

Authors

1 Department of Electrical engineering, University of Isfahan. Iran.

2 Department of Electrical Engineering, University of Isfahan, Iran.

Abstract

In this study, an algorithm is proposed for the design of a multivariable fuzzy PID controller using convex optimization  and linear matrix inequality (LMI) constraints. The considered fuzzy system is of the Takagi–Sugeno type and consists of  subsystems based on PID controllers. The parameters of these subsystems are optimized to achieve desirable  performance. The positive real lemma is employed to guarantee the stability of the overall fuzzy system. One of the most  significant applications of this approach lies in the control of industrial robots. Since robotic systems are typically  multivariable in nature, the utilization of advanced controllers such as multivariable fuzzy PID controllers can play a crucial role in enhancing their performance. In this work, the proposed control scheme will be implemented both on a simulation example and practical robot manipulator. 

Keywords

Main Subjects

References

[1] A. Agrawal, R. Verschueren, S. Diamond, S. P. Boyd, A rewriting system for convex optimization problems, Journal of Control and Decision, 5(1) (2018), 42-60. https://doi.org/10.1080/23307706.2017.1397554
[2] R. T. Bambang, Development of architectures for internet telerobotics systems, Journal of Bionic Engineering, 4(4) (2007), 291-297. https://doi.org/10.1016/S1672-6529(07)60043-2
[3] R. P. Borase, D. K. Maghade, S. Y. Sondkar, S. N. Pawar, A review of PID control, tuning methods and ap plications, International Journal of Dynamics and Control, 9(2) (2021), 818-827. https://doi.org/10.1007/ s40435-020-00665-4
[4] M. Boreale, L. Collodi, Linearization, model reduction and reachability in nonlinear ODEs, Formal Methods in System Design, 66 (2025), 419-454. https://doi.org/10.1007/s10703-025-00477-2
[5] S. Boyd, M. Hast, K. J. Åström, MIMO PID tuning via iterated LMI restriction, International Journal of Robust and Nonlinear Control, 26(8) (2016), 1718-1731. https://doi.org/10.1002/rnc.3376
[6] S. Boyd, L. Vandenberghe, Convex optimization, Cambridge University Press, Cambridge, UK, (2004), 142-144. https://doi.org/10.1017/CBO9780511804441
[7] B. Brogliato, R. Lozano, B. Maschke, O. Egeland, Dissipative systems analysis and control: Theory and applications, 3rd ed., Springer, Cham, (2020), 7-38. https://doi.org/10.1007/978-3-030-19420-8
[8] S. Fateh, M. M. Fateh, Adaptive fuzzy control of robot manipulators with asymptotic tracking performance, Journal of Control, Automation and Electrical Systems, 31 (2020), 52-61. https://doi.org/10.1007/s40313-019-00496-5
[9] D. Gontijo, J. M. Araújo, F. O. Souza, Robust multivariable PID controller design for MIMO second order systems: A linear matrix inequality approach, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 239(5) (2025), 870-880. https://doi.org/10.1177/09596518241300677
[10] M. C. Grant, S. P. Boyd, Graph implementations for nonsmooth convex programs, in Recent Advances in Learning and Control, Springer, (2008), 95-110. https://doi.org/10.1007/978-1-84800-155-8_7
[11] B. Kosko, Fuzzy systems as universal approximators, IEEE Transactions on Computers, 43(11) (1994), 1329-1333. https://doi.org/10.1109/12.324566
[12] Q. Li, X. Zhou, F. Lin, et al., Novel proportional-integral-derivative control framework on continuous-time positive systems using linear programming, Mathematics, 12(4) (2024), 617. https://doi.org/10.3390/math12040617
[13] T. M. Linhares, E. S. Tognetti, T. R. Calliero, Output feedback control with inexact premise variables of T-S fuzzy systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 55(3) (2025), 2380-2386. https: //doi.org/10.1109/TSMC.2024.3521369
[14] J. Liu, Intelligent control design and MATLAB simulation, Springer, Singapore, (2017), 60-65. https://doi.org/ 10.1007/978-981-10-5263-7
[15] J. Luo, J. Li, L. Xiao, et al., A novel proportional-integral-parameter zeroing neural network and its application to the quaternion-valued time-varying linear matrix inequality, IEEE Transactions on Emerging Topics in Computing, 13(4) (2025), 1565-1576. https://doi.org/10.1109/TETC.2025.3629357
[16] L. A. Mozelli, V. C. da S. Campos, An improved control law for TS fuzzy models: Less conservative LMI conditions by using membership functions derivative, Fuzzy Sets and Systems, 517 (2025), 109455. https://doi.org/10.1016/ j.fss.2025.109455
[17] Y. Quilles-Marinho, R. C. L. F. Oliveira, P. L. D. Peres, Output-feedback control design for Takagi-Sugeno fuzzy systems through Lyapunov functions depending polynomially on the states, Fuzzy Sets and Systems, 499 (2025), 109156. https://doi.org/10.1016/j.fss.2024.109156
[18] R. Sanchis, I. Peñarrocha, A PID tuning approach to find the optimal compromise among robustness, performance and control effort: Implementation in a free software tool, International Journal of Control, 97(1) (2024), 16-35. https://doi.org/10.1080/00207179.2021.1989491
[19] J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11(1-4) (1999), 625-653. https://doi.org/10.1080/10556789908805766
[20] Y. Su, C. Zheng, A new nonsingular integral terminal sliding mode control for robot manipulators, International Journal of Systems Science, 51(8) (2020), 1418-1428. https://doi.org/10.1080/00207721.2020.1764658
[21] H. H. Tang, N. S. Ahmad, Fuzzy logic approach for controlling uncertain and nonlinear systems: A comprehensive review of applications and advances, Systems Science and Control Engineering, 12(1) (2024), 2394429. https://doi.org/10.1080/21642583.2024.2394429
[22] V. Tinoco, M. F. Silva, F. N. Santos, et al., A review of advanced controller methodologies for robotic manipulators, International Journal of Dynamics and Control, 13 (2025), 36. https://doi.org/10.1007/s40435-024-01533-1
[23] R. H. Tütüncü, K. C. Toh, M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3, Mathematical Programming, 95 (2003), 189-217. https://doi.org/10.1007/s10107-002-0347-5
[24] H. F. Yamanaka, D. J. S. Oncoy, R. Cardim, et al., Robust control with feedback of positions for nonlinear mechanical systems represented by fuzzy T-S models, in Proceedings of the Brazilian Congress of Automation (CBA 2024), (2024). https://doi.org/10.20906/CBA2024/4413
[25] J. F. Zhang, J. W. Wang, H. K. Lam, H. X. Li, Boundary output tracking of nonlinear parabolic differential systems via fuzzy PID control, IEEE Transactions on Fuzzy Systems, 32(12) (2024), 6863-6877. https://doi.org/10.1109/ TFUZZ.2024.3432554
[26] Y. Zhu, R. Chen, S. Xiao, et al., Positive real lemmas for singular fractional order systems without equality constraints, Advances in Guidance, Navigation and Control, Lecture Notes in Electrical Engineering, 1349, Springer, Singapore, (2025), 474-482. https://doi.org/10.1007/978-981-96-2248-1_46
Iranian Journal of Fuzzy Systems
Volume 23, Issue 6
November and December 2026
Pages 189-207

Files

Share

How to cite

Statistics