Robust MIMO LQR Control with Integral Action for Differential Drive Robots: A Lyapunov-Cost Function Approach
Received: 18 April 2025 | Revised: 27 May 2025 | Accepted: 1 June 2025 | Online: 7 June 2025
Corresponding author: Anh-Minh Duc Tran
Abstract
This study introduces an LQR-enhanced MIMO state feedback system with integral action to ensure precise trajectory tracking by eliminating persistent offsets. Unlike standard LQR methods, the proposed approach integrates integral action without requiring separate disturbance observers, reducing computational complexity for embedded systems. A unique Lyapunov-based analysis, leveraging the LQR cost function, confirms stability with uniform ultimate boundedness despite external perturbations. MATLAB simulations reveal RRMSE values under 10% for longitudinal speed and yaw rate, even when subjected to step and stochastic disturbances. This method outperforms conventional PID and LQR-feedforward techniques, offering a lightweight, efficient solution for DDWMR applications in industrial logistics, field exploration, and real-time robotic systems.
Keywords:
MIMO control, differential drive wheeled mobile robot, LQR, integral action, Lyapunov stabilityDownloads
References
R. Siegwart, I. R. Nourbakhsh, and D. Scaramuzza, Introduction to Autonomous Mobile Robots, 2nd ed. MIT Press, 2011.
G. Campion, G. Bastin, and B. Dandrea-Novel, "Structural properties and classification of kinematic and dynamic models of wheeled mobile robots," IEEE Transactions on Robotics and Automation, vol. 12, no. 1, pp. 47–62, Oct. 1996. DOI: https://doi.org/10.1109/70.481750
W. He, Y. Chen, and Z. Yin, "Adaptive Neural Network Control of an Uncertain Robot With Full-State Constraints," IEEE Transactions on Cybernetics, vol. 46, no. 3, pp. 620–629, Mar. 2016. DOI: https://doi.org/10.1109/TCYB.2015.2411285
X. Gao, R. Gao, P. Liang, Q. Zhang, R. Deng, and W. Zhu, "A Hybrid Tracking Control Strategy for Nonholonomic Wheeled Mobile Robot Incorporating Deep Reinforcement Learning Approach," IEEE Access, vol. 9, pp. 15592–15602, 2021. DOI: https://doi.org/10.1109/ACCESS.2021.3053396
Z. Ge et al., "Robust adaptive sliding mode control for path tracking of unmanned agricultural vehicles," Computers and Electrical Engineering, vol. 108, May 2023, Art. no. 108693. DOI: https://doi.org/10.1016/j.compeleceng.2023.108693
T. Yuan and R. Zhao, "LQR-MPC-Based Trajectory-Tracking Controller of Autonomous Vehicle Subject to Coupling Effects and Driving State Uncertainties," Sensors, vol. 22, no. 15, Jan. 2022, Art. no. 5556. DOI: https://doi.org/10.3390/s22155556
K. J. Astrom and T. Hagglund, PID Controllers: Theory, Design and Tuning. ISA, 1995.
B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. Dover Publications, 2007.
H. Ye and S. Wang, "Trajectory Tracking Control for Nonholonomic Wheeled Mobile Robots with External Disturbances and Parameter Uncertainties," International Journal of Control, Automation and Systems, vol. 18, no. 12, pp. 3015–3022, Dec. 2020. DOI: https://doi.org/10.1007/s12555-019-0643-y
H. Huang, J. Zhou, Q. Di, J. Zhou, and J. Li, "Robust neural network–based tracking control and stabilization of a wheeled mobile robot with input saturation," International Journal of Robust and Nonlinear Control, vol. 29, no. 2, pp. 375–392, 2019. DOI: https://doi.org/10.1002/rnc.4396
Y. Cao, K. Ni, T. Kawaguchi, and S. Hashimoto, "Path Following for Autonomous Mobile Robots with Deep Reinforcement Learning," Sensors, vol. 24, no. 2, Jan. 2024, Art. no. 561. DOI: https://doi.org/10.3390/s24020561
E. F. Camacho and C. B. Alba, Model Predictive Control. Springer, 2013.
Z. Sun, R. Wang, X. Meng, Y. Yang, Z. Wei, and Q. Ye, "A novel path tracking system for autonomous vehicle based on model predictive control," Journal of Mechanical Science and Technology, vol. 38, no. 1, pp. 365–378, Jan. 2024. DOI: https://doi.org/10.1007/s12206-023-1230-y
J. Peng, H. Xiao, and G. Lai, "Nonlinear Disturbance Observer Incorporated Model Predictive Strategy for Wheeled Mobile Robot’s Trajectory Tracking Control," International Journal of Control, Automation and Systems, vol. 22, no. 7, pp. 2251–2262, Jul. 2024. DOI: https://doi.org/10.1007/s12555-023-0207-z
V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electro-Mechanical Systems, 2nd ed. CRC Press, 2017. DOI: https://doi.org/10.1201/9781420065619
X. Ji, X. Wei, A. Wang, B. Cui, and Q. Song, "A novel composite adaptive terminal sliding mode controller for farm vehicles lateral path tracking control," Nonlinear Dynamics, vol. 110, no. 3, pp. 2415–2428, Nov. 2022. DOI: https://doi.org/10.1007/s11071-022-07730-x
S. L. Dai, S. He, X. Chen, and X. Jin, "Adaptive Leader–Follower Formation Control of Nonholonomic Mobile Robots With Prescribed Transient and Steady-State Performance," IEEE Transactions on Industrial Informatics, vol. 16, no. 6, pp. 3662–3671, Jun. 2020. DOI: https://doi.org/10.1109/TII.2019.2939263
H. Medjoubi, A. Yassine, and H. Abdelouahab, "Design and Study of an Adaptive Fuzzy Logic-Based Controller for Wheeled Mobile Robots Implemented in the Leader-Follower Formation Approach," Engineering, Technology & Applied Science Research, vol. 11, no. 2, pp. 6935–6942, Apr. 2021. DOI: https://doi.org/10.48084/etasr.3950
J. Doyle, "Robust and optimal control," in Proceedings of 35th IEEE Conference on Decision and Control, Kobe, Japan, 1996, vol. 2, pp. 1595–1598. DOI: https://doi.org/10.1109/CDC.1996.572756
C. C. Tsui, "High-performance state feedback, robust, and output feedback stabilizing control-a systematic design algorithm," IEEE Transactions on Automatic Control, vol. 44, no. 3, pp. 560–563, Mar. 1999. DOI: https://doi.org/10.1109/9.751350
M. Saeedi, J. Zarei, M. Saif, and A. Montazeri, "Event-Based Robust Control Techniques for Wheel-Based Robots Under Cyber-Attack and Dynamic Quantizer," in Mobile Robot: Motion Control and Path Planning, A. T. Azar, I. Kasim Ibraheem, and A. Jaleel Humaidi, Eds. Springer International Publishing, 2023, pp. 163–196. DOI: https://doi.org/10.1007/978-3-031-26564-8_6
Y. Wang, Z. Miao, H. Zhong, and Q. Pan, "Simultaneous Stabilization and Tracking of Nonholonomic Mobile Robots: A Lyapunov-Based Approach," IEEE Transactions on Control Systems Technology, vol. 23, no. 4, pp. 1440–1450, Jul. 2015. DOI: https://doi.org/10.1109/TCST.2014.2375812
C. Liu, J. Gao, and D. Xu, "Lyapunov-based model predictive control for tracking of nonholonomic mobile robots under input constraints," International Journal of Control, Automation and Systems, vol. 15, no. 5, pp. 2313–2319, Oct. 2017. DOI: https://doi.org/10.1007/s12555-016-0350-x
M. Chen, "Robust tracking control for self-balancing mobile robots using disturbance observer," IEEE/CAA Journal of Automatica Sinica, vol. 4, no. 3, pp. 458–465, 2017. DOI: https://doi.org/10.1109/JAS.2017.7510544
M. Fouzia, N. Khenfer, and N. E. Boukezzoula, "Robust Adaptive Tracking Control of Manipulator Arms with Fuzzy Neural Networks," Engineering, Technology & Applied Science Research, vol. 10, no. 4, pp. 6131–6141, Aug. 2020. DOI: https://doi.org/10.48084/etasr.3648
N. T. T. Vu, N. P. Tran, and N. H. Nguyen, "Recurrent Neural Network-based Path Planning for an Excavator Arm under Varying Environment," Engineering, Technology & Applied Science Research, vol. 11, no. 3, pp. 7088–7093, Jun. 2021. DOI: https://doi.org/10.48084/etasr.4125
A. M. D. Tran, T. V. Vu, and Q. D. Nguyen, "A Study on General State Model of Differential Drive Wheeled Mobile Robots," Journal of Advanced Engineering and Computation, vol. 7, no. 3, Sep. 2023, Art. no. 174. DOI: https://doi.org/10.55579/jaec.202373.417
H. Khalil, Nonlinear Systems. Upper Saddle River, NJ, USA: Pearson, 2002.
R. Fareh, M. R. Saad, M. Saad, A. Brahmi, and M. Bettayeb, "Trajectory Tracking and Stability Analysis for Mobile Manipulators Based on Decentralized Control," Robotica, vol. 37, no. 10, pp. 1732–1749, Oct. 2019. DOI: https://doi.org/10.1017/S0263574719000225
A. M. D. Tran and T. V. Vu, "A Dynamic Controller Architecture for Wheeled Mobile Robot Trajectory Tracking Utilizing Feedback Linearization and State Feedback," Journal of Advanced Engineering and Computation, vol. 8, no. 2, pp. 119–129, Jun. 2024. DOI: https://doi.org/10.55579/jaec.202482.456
Downloads
How to Cite
License
Copyright (c) 2025 Anh-Minh Duc Tran, Tri Vien Vu
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
