Design of an LMI-based Linear Quadratic Pole Placement Controller for Quadcopters
DOI:
https://doi.org/10.63746/njtd.v22i4.3215Keywords:
Attitude Control, LMI, Optimization, Pole-placement, QuadcopterAbstract
Quadcopter systems have become part of the aviation industry with their applications seen in surveillance, search and rescue, photography and so on. Proportional-Integral-Derivative (PID) and Linear Quadratic Regulator (LQR) control techniques do not provide robust performance of quadcopters due to their extensive tuning requirements, and the non-linear and coupled nature of the system dynamics. This paper presents an approach which combines Linear Quadratic Regulator (LQR) and Pole-Placement (PP), coined Linear Quadratic Pole Placement (LQPP) to design the state feedback gain matrix for the attitude and position control of the quadcopter system. The pole placement region was selected as the intersection between a strip region, and a disk region within the stability region for the poles. The optimization problem was to minimize the cost function of the Algebraic Riccati equation, which penalizes the control effort, and the system states. The pole locations are then used as constraints and then cast as linear matrix inequalities (LMI). The mathematical model for the dynamics and kinematics of the quadcopter system is presented, and the LMI-based design concept for the LQPP is also formulated. The dominant poles of the quadcopter system are utilized for pole placement, while the remaining poles are used to minimize the cost function of the linear quadratic regulator. Simulation results in MATLAB/Simulink demonstrate that the optimized controller significantly reduces overshoot by 5.66% and 7.24% for the position and attitude responses respectively, and settling time by 9.05% and 7.93% for the position and attitude responses respectively compared to the unoptimized LQPP controller. The results demonstrate that the LQPP design markedly improves attitude and position control, offering a robust and efficient solution for quadcopter stabilization. This approach paves the way for deployment in precision-critical missions such as infrastructure inspection, search and rescue, and dynamic aerial mapping.
References
Ahmad, F., Kumar, P., Bhandari, A., & Patil, P. (2020) Simulation of the Quadcopter Dynamics with LQR based Control, Materials Today: Proceedings, 24, pp. 326–332. Available at: https://doi.org/10.1016/j.matpr.2020.04.282.
Ahmadi Dastgerdi, K., Asadi, D., & Nabavi Chashmi, S. Y. (2022) Fault tolerant control of a quadrotor based on incremental nonlinear dynamic inversion. International Journal of Aeronautics and Astronautics, 3(1), 28–47. https://doi.org/10.55212/ijaa.1033224
Amiruddin, B. P., Iskandar, E., Fatoni, A., & Santoso, A. (2021) Deep Learning based System Identification of Quadcopter Unmanned Aerial Vehicle.2020 3rd International Conference on Information and Communication Technology, Yogyakarta, Indonesia. 24th – 25th November, 2020. DOI: 10.1109/ICOIACT50329.2020.9332059
Andersson, O., Wzorek, M., & Doherty, P. (2017) Deep Learning Quadcopter Control via Risk-Aware Active Learning, Proceedings of the 31th Conference on Artificial Intelligence (AAAI 2017), pp. 3812–3818. Available at: http://aaai.org/ocs/index.php/AAAI/AAAI17/paper/view/14622/14104.
Aslam, M. S., & Bilal, H. (2024) Modeling a Takagi-Sugeno (T-S) fuzzy for unmanned aircraft vehicle using fuzzy controller. Ain Shams Engineering Journal, Vol. 15 (10) pg. 1-14. https://doi.org/10.1016/j.asej.2024.102984
Babawuro, A.Y., Tahir, N. M., Muhammed, M., & Sambo, A. U. (2020) Optimized state feedback control of quarter car active suspension system based on LMI algorithm’, Journal of Physics: Conference Series, 15(02). Available at: https://doi.org/10.1088/1742-6596/1502/1/012019.
Bose, S., Hote, Y. V., & Hanwate, S. (2019). Analysis of Initial Undershoot in Step Response of Type-1 Nonminimum Phase Systems. Proceedings of 2019 IEEE Region 10 Symposium, TENSYMP 2019, 28–32. https://doi.org/10.1109/TENSYMP46218.2019.8971176
Carrio, A., Sampedro, C., & Rodriguez-Ramos, A. (2017) A Review of Deep Learning Methods and Applications for Unmanned Aerial Vehicles, Journal of Sensors, 2017, pp. 1–13. Available at: https://doi.org/10.1155/2017/3296874.
Cedro, L., Wieczorkowski, K., & Szcze?niak, A. (2024) An Adaptive PID Control System for the Attitude and Altitude Control of a Quadcopter. Acta Mechanica et Automatica, 18(1), 29–39. https://doi.org/10.2478/ama-2024-0004
Datta, S. (2017) Feedback Controller Norm Optimization for Linear Time Invariant Descriptor Systems with Pole Region Constraint, IEEE Transactions on Automatic Control, 62(6), pp. 2794–2806.
Deng, X., Sun, X., Liu, R., & Wei, W. (2017) Optimal analysis of the weighted matrices in LQR based on the differential evolution algorithm, Proceedings of the 29th Chinese Control and Decision Conference, CCDC 2017, pp. 832–836. Available at: https://doi.org/10.1109/CCDC.2017.7978635.
De Oliveira, M. C., & Zheng, Y. (2022). ConvexParameterization of Stabilizing Controllers and its LMI-based Computation via Filtering. Proceedings of the IEEE Conference on Decision and Control, 5498–5504. https://doi.org/10.1109/CDC51059.2022.9993254
Duan, G. R., & Yu, H. H. (2013) LMIs in Control Systems: Analysis, Design and Applications. Taylor & Francis Group, CRC Press.
Fatkhullin, I., & Polyak, B. (2020) Optimizing static linear feedback: gradient method, SIAM Journal on Control and Optimization. Vol. 59 (5) pp. 1–21. 10.1137/20M1329858
Gonzalez-Villagomez, J., Rodriguez-Donate, C., Lopez-Ramirez, M., Mata-Chavez, R. I., & Palillero-Sandoval, O. (2023). Novel Iterative Feedback Tuning Method Based on Overshoot and Settling Time with Fuzzy Logic. Processes, 11(3), 694. https://doi.org/10.3390/pr11030694
Howard, D., & Merz, T. (2015) A platform for the direct hardware evolution of quadcopter controllers, IEEE International Conference on Intelligent Robots and Systems, 2015-Decem, pp. 4614–4619. Available at: https://doi.org/10.1109/IROS.2015.7354034.
Ka, C. (2002) SIMULTANEOUS LINEAR QUADRATIC POLE PLACEMENT (LQPP) CONTROL DESIGN, in 15th Triennial World Congress, Barcelona, Spain.
Karam, K., Mansour, A., Khaldi, M., Clement, B., & Ammad-Uddin, M. (2024). Quadcopters in Smart Agriculture?: Applications and Modelling. Applied Sciences, 14(19), p.g. 1–26.
Li, T. H. S., & Shieh, M. Y. (2000). Design of a GA-based fuzzy PID controller for non-minimum phase systems. Fuzzy Sets and Systems, 111(2), 183–197. https://doi.org/10.1016/S0165-0114(97)00404-1
Lin, C., Hu, B., Shao, C., Li, W., Li, C., & Xie, K. (2022). Delay-Dependent Optimal Load Frequency Control for Sampling Systems with Demand Response. IEEE Transactions on Power Systems, 37(6), 4310–4324. https://doi.org/10.1109/TPWRS.2022.3154429
Mardijah, D. P. (2022) Control Design of Quadcopter using Output Feedback Control Pole Placement. Proceeding of the International Conference on Computer Engineering, Network and Intelligent Multimedia, CENIM 2022. pp. 197-202. ISSN: 9781665476508. doi: 10.1109/CENIM56801.2022.10037547
Martins, L., Cardeira, C., & Oliveira, P. (2019) Linear quadratic regulator for trajectory tracking of a quadrotor, IFAC-PapersOnLine, 52(12), pp. 176–181. Available at: https://doi.org/10.1016/j.ifacol.2019.11.195.
Morbale, J., Vidyapeeth, B., Rohan, P., & Vidyapeeth, N. B. (2022) Quadcopter Drone with Face Recognition. August. https://www.researchgate.net/publication/362517060
Nguyen, N.P., & Hong, S.K. (2018) Sliding mode Thau observer for actuator fault diagnosis of quadcopter UAVs’, Applied Sciences (Switzerland), 8(10). Available at: https://doi.org/10.3390/app8101893.
Nguyen, N.P., & Hong, S.K. (2019) Fault-Tolerant control of quadcopter uavs using robust adaptive sliding mode approach, Energies, 12(1). Available at: https://doi.org/10.3390/en12010095.
Okyere, E., Bousbaine, A., Poyi, G. T., Joseph, A. K., & Andrade, J. M. (2019) LQR controller design for quad-rotor helicopters, in 9th International Conference on Power Electronics, Machines and Drives (PEMD 2018), pp. 4003–4007. Available at: https://doi.org/10.1049/joe.2018.8126.
Pounds, P., Mahony, R., & Corke, P. (2010) Modelling and control of a large quadrotor robot, Control Engineering Practice, 18(7), pp. 691–699. Available at: https://doi.org/10.1016/j.conengprac.2010.02.008.
Reizenstein, A. (2017) Position and Trajectory Control of a Quadcopter Using PID and LQ Controllers. [Unpublished] M.Sc Thesis, Division of Automatic Control, Department of Electrical Engineering, Linkoping University, Sweden
Rezende, J. C., Carvalho, L., Neto, J. R. L., Costa, M. V. S., Fortes, E. V., & Macedo, L. H. (2021). LQR design using LMIs and the robust D-stability criterion for low-frequency oscillation damping in power systems. 2021 14th IEEE International Conference on Industry Applications, INDUSCON 2021 - Proceedings, 188–195. https://doi.org/10.1109/INDUSCON51756.2021.9529623
Sahrir, N. H., & Mohd Basri, M. A. (2022). Modelling and Manual Tuning PID Control of Quadcopter. Lecture Notes in Electrical Engineering, 921 LNEE, 346–357. https://doi.org/10.1007/978-981-19-3923-5_30
Tengis, T., & Batmunkh, A. (2017) Quadcopter stabilization using state feedback controller by pole placement method, International Journal of Internet, Broadcasting and Communication, 9(1), pp. 1–8.
Thu, K.M., & Gavrilov, A.I. (2017) Designing and Modeling of Quadcopter Control System Using L1 Adaptive Control, Procedia Computer Science, 103(October 2016), pp. 528–535. Available at: https://doi.org/10.1016/j.procs.2017.01.046.
Wilt, E., & Sands, T. (2022). Microsatellite Uncertainty Control Using Deterministic Artificial Intelligence. Sensors, 22(22), 8723. https://doi.org/10.3390/s22228723
Xu, D., Whidborne, J.F., & Cooke, A. (2016) Fault tolerant control of a quadrotor using L1 adaptive control, International Journal of Intelligent Unmanned Systems, 4(1), pp. 43–66. Available at: https://doi.org/10.1108/IJIUS-08-2015-0011.
Yoon, J., & Doh, J. (2022). Optimal PID control for hovering stabilization of quadcopter using long short term memory. Advanced Engineering Informatics, Vol. 53(1) pg. 1-14. https://doi.org/10.1016/j.aei.2022.101679
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