This article explores and compares the use of the linear quadratic regulator (LQR) theory to control a single-degree-of-freedom flexible link robot (FLR) using an integer-order model and a fractional-order model. The fractional-order model is obtained from the integer-order one by introducing new “fictitious” fractional states. The LQR technique determines feedback gains to optimally control a system by considering its dynamics and cost function. Two LQR schemes are designed here: one for the integer-order model using a linear matrix inequality reformulation and the other for the fractional-order model involving analytical tuning and numerical optimization. A genetic algorithm is used to optimize the fractional-order LQR by minimizing the same cost function as the integer-order model. The resulting control system minimizes disturbances, ensuring accurate tracking, stability, and robustness in flexible robotic systems. The complete scheme integrates active disturbance rejection control (ADRC) and Luenberger observers to estimate internal states. Simulations and experiments demonstrate that the fractional-order LQR outperforms the integer-order version in terms of energy efficiency, evaluated using total variation (TV) and the integral absolute control signal (IACS). Then we have shown that feeding back fictitious fractional states besides the standard integer-order ones reduces the LQR cost index and improves the robot performance.
AlandoliEALeeTS (2020) A critical review of control techniques for flexible and rigid link manipulators. Robotica38(12): 2239–2265.
2.
AlandoliEARashidMZASulaimanM (2017) A comparison of pid and lqr controllers for position tracking and vibration suppression of flexible link manipulator. Journal of Theoretical & Applied Information Technology95(13): 2949–2955.
3.
AlandoliEALeeTSLinYJ, et al. (2021) Dynamic model and intelligent optimal controller of flexible link manipulator system with payload uncertainty. Arabian Journal for Science and Engineering46: 7423–7433.
4.
BenftimaSBatlleVFBenattiaS, et al. (2022) A fast online estimator of the main vibration mode of mechanisms from a biased slightly damped signal. In: IECON– 48th annual conference of the IEEE industrial electronics society, Brussels, Belgium, 17–20 October 2022. IEEE.
5.
BenftimaSGharabSFeliu BatlleV (2023) Fractional modeling and control of lightweight 1 dof flexible robots robust to sensor disturbances and payload changes. Fractal and Fractional7(7): 504.
6.
BilgicHHSenMAYapiciA, et al. (2021) Meta-heuristic tuning of the lqr weighting matrices using various objective functions on an experimental flexible arm under the effects of disturbance. Arabian Journal for Science and Engineering46(8): 7323–7336.
7.
BoscariolPGasparettoAZanottoV (2010) Model predictive control of a flexible links mechanism. Journal of Intelligent and Robotic Systems58: 125–147.
8.
ChenLYanYMuC, et al. (2016) Characteristic model-based discrete-time sliding mode control for spacecraft with variable tilt of flexible structures. IEEE/CAA Journal of Automatica Sinica3(1): 42–50.
9.
De LucaABookWJ (2016) Robots with Flexible Elements. Berlin, Heidelberg: Springer Handbook of Robotics, 243–282.
10.
FeliuVRattanKSBrownHB (1990) Adaptive control of a single-link flexible manipulator. IEEE Control Systems Magazine10(2): 29–33.
11.
FeliuVRattanKSBrownHBJr (1992) Modeling and control of single-link flexible arms with lumped masses. ASME Journal of Dynamic Systems, Measurement, and Control114: 59–69.
12.
FeliuVCastilloFJRamosF, et al. (2012) Robust tip trajectory tracking of a very lightweight single-link flexible arm in presence of large payload changes. Mechatronics22(5): 594–613.
13.
Feliu-TalegonDFeliu-BatlleV (2024) Vibration control of very large and lightweight structures mounted on a four-legged platform. Journal of Vibration and Control30: 104–115.
14.
Feliu-TalegonDFeliu-BatlleVCastillo-BerrioC (2016) Motion control of a sensing antenna with a nonlinear input shaping technique. Revista Iberoamericana de Automática e Informática industrial13(2): 162–173.
15.
GaoHYuZHuJ (2023) A survey on modeling and control methods for flexible systems. In: 2023 6th international symposium on autonomous systems (ISAS), Nanjing, China, 23–25 June 2023. IEEE, 1–6.
16.
GeMChiuM-SWangQ-G (2002) Robust pid controller design via lmi approach. Journal of Process Control12(1): 3–13.
17.
GharabSBenftimaSBatlleVF (2021) Fractional control of a lightweight single link flexible robot robust to strain gauge sensor disturbances and payload changes. Actuators10(12): 317.
18.
LiYChenYQ (2008) Fractional order linear quadratic regulator. In: 2008 IEEE/ASME international conference on mechatronic and embedded systems and applications, Beijing, China, 12–15 October 2008. IEEE, 363–368.
19.
LochanKSuklabaidyaSRoyBK (2015) Sliding mode and adaptive sliding mode control approaches of two link flexible manipulator. In: Proceedings of the 2015 conference on advances in robotics, Istanbul, Turkey, 27–31 July 2015, 1–6.
20.
MamaniGBecedasJFeliuV (2012) Sliding mode tracking control of a very lightweight single-link flexible robot robust to payload changes and motor friction. Journal of Vibration and Control18(8): 1141–1155.
MohammadNNAzmanAAMarzakiMH, et al. (2018) Comparison of the performance and energy consumption between pid control and self-tuning fuzzypid in compact hydro distillation process. In: 2018 IEEE international conference on automatic control and intelligent systems (I2CACIS), Shah Alam, Malaysia, 20 October 2018. IEEE, pp. 7–12.
23.
MonjeCAChenYQVinagreBM, et al. (2010) Fractional-order Systems and Controls: Fundamentals and Applications. Springer Science & Business Media.
24.
NassefAMAbdelkareemMAMaghrabieHM, et al. (2023) Metaheuristic-based algorithms for optimizing fractional-order controllers—a recent, systematic, and comprehensive review. Fractal and Fractional7(7): 553.
25.
NguyenVBMorrisAS (2007) Genetic algorithm tuned fuzzy logic controller for a robot arm with two-link flexibility and two-joint elasticity. Journal of Intelligent and Robotic Systems49: 3–18.
26.
OmisoreOMHanSXiongJ, et al. (2020) A review on flexible robotic systems for minimally invasive surgery. IEEE Transactions on Systems, Man, and Cybernetics: Systems52(1): 631–644.
27.
PetrášI (2011) Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation. Berlin: Springer Science + Business Media.
28.
PodlubnyI (1998) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press.
29.
ReisJCPSá da CostaJ (2012) Motion planning and actuator specialization in the control of active-flexible link robots. Journal of Sound and Vibration331(14): 3255–3270.
30.
SayahkarajyMMohamedZMohd FaudziAA (2016) Review of modelling and control of flexible-link manipulators. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering230(8): 861–873.
31.
SierociukDVinagreBM (2010) Infinite horizon state-feedback lqr controller for fractional systems. In: 49th IEEE conference on decision and control (CDC), Atlanta, Georgia, 15–17 December 2010. IEEE, pp. 6674–6679.
32.
SkogestadS (2003) Simple analytic rules for model reduction and pid controller tuning. Journal of Process Control13(4): 291–309.
33.
SunCHeWHongJ (2016) Neural network control of a flexible robotic manipulator using the lumped spring-mass model. IEEE Transactions on Systems, Man, and Cybernetics: Systems47(8): 1863–1874.
34.
SunCGaoHHeW, et al. (2018) Fuzzy neural network control of a flexible robotic manipulator using assumed mode method. IEEE Transactions on Neural Networks and Learning Systems29(11): 5214–5227.
35.
TahirNMBatureAABatureUI, et al. (2016) Vibration and tracking control of a single link flexible manipulator using lqr and command shaping. Journal of Multidisciplinary Engineering Science and Technology3: 4252–4258.
36.
TakeshitaAYamashitaTKawaguchiN, et al. (2021) Fractional-order lqr and state observer for a fractional-order vibratory system. Applied Sciences11(7): 3252.
37.
VogelJHertkornKMenonRU, et al. (2016) Flexible, semi-autonomous grasping for assistive robotics. In: 2016 IEEE international conference on robotics and automation (ICRA), Stockholm, Sweden, 16–21 May 2016. IEEE, pp. 4872–4879.
38.
WaiR-JLeeM-C (2004) Intelligent optimal control of single-link flexible robot arm. IEEE Transactions on Industrial Electronics51(1): 201–220.
39.
WangF-YGaoY (2003) Advanced Studies of Flexible Robotic Manipulators: Modeling, Design, Control and Applications. Singapore: World Scientific, Vol. 4.
40.
WeiYChenYChengS, et al. (2017) A note on short memory principle of fractional calculus. Fractional Calculus and Applied Analysis20(6): 1382–1404.
