Globally Asymptotically Stable Nonlinear PID Controlwith a Generalized Saturation Function
NIU Guojun1, QU Cuicui2, PAN Bo3, FU Yili3
1. School of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China; 2. Hangzhou SIASUN Robot&Automation CO. LTD., Hangzhou 311200, China; 3. State Key Laboratory of iRobotics and System, Harbin Institute of Technology, Harbin 150001, China
Abstract:In order to solve the problem of the strict conditions of traditional saturation function, a saturation function is proposed and applied to the linear PD (proportional-differential) + nonlinear PI (proportional-integral) control law. The globally asymptotic stability conditions of nonlinear PID control law are derived using Lyapunov's stability theorem and LaSalle's invariance principle. In order to improve the accuracy of nonlinear PID control, the tuning of nonlinear PID control parameters is accomplished by the multi-objective genetic algorithm NSGA-II (non-dominated sorted genetic algorithm-II), taking both the time integral of the absolute value of position tracking error and the time integral of the absolute value of input torque error as the objective functions, regarding the globally asymptotic stability conditions and the rated driving torque of each motor as the constraint conditions. The saturation function with minimum time integral of position tracking error is selected, and then the robustness of the nonlinear PID control law with the saturation function to model uncertainty, input disturbance, and noise is studied. Compared with the traditional PID control law and the nonlinear PID control law with the traditional saturation function, the position tracking accuracy of the proposed method is improved by nearly two orders of magnitude and one order of magnitude, respectively. The proposed saturation function shows strong reaction near the equilibrium point, which makes the errors converge to the equilibrium point quickly. And it is helpful to improve the position tracking accuracy and the robustness of nonlinear PID control law.
[1] Kelly R. Global positioning of robot manipulators via PD control plus a class of nonlinear integral actions[J]. IEEE Transactions on Automatic Control, 1998, 43(7):934-938. [2] Salinas A, Moreno-Valenzuela J, Kelly R. A family of nonlinear PID-like regulators for a class of torque-driven robot manipulators equipped with torque-constrained actuators[J]. Advances in Mechanical Engineering, 2016, 8(2):1-14. [3] Yarza A, Santibanez V, Moreno-Valenzuela J. Global asymptotic stability of the classical PID controller by considering saturation effects in industrial robots[J]. International Journal of Advanced Robotic Systems, 2011, 8(4):34-42. [4] Sahib M A, Ahmed B S. A new multi-objective performance criterion used in PID tuning optimization algorithms[J]. Journal of Advanced Research, 2016, 7(1):125-134. [5] Omar M, Soliman M, Ghany A M A, et al. Optimal tuning of PID controllers for hydrothermal load frequency control using ant colony optimization[J]. International Journal on Electrical Engineering and Informatics, 2013, 5(3):348-360. [6] Can M S, Ozguven O F. PID tuning with neutrosophic similarity measure[J]. International Journal of Fuzzy Systems, 2017, 19(2):489-503. [7] Garrido J, Ruz M L, Morilla F, et al. Interactive tool for frequency domain tuning of PID controllers[J]. Processes, 2018, 6(10):197. [8] Ozyetkin M M, Onat C, Tan N. PID tuning method for integrating processes having time delay and inverse response[J]. IFAC-PapersOnLine, 2018, 51(4):274-279. [9] Kang M, Cheong J, Do H M, et al. A practical iterative PID tuning method for mechanical systems using parameter chart[J]. International Journal of Systems Science, 2017, 48(13):2887-2900. [10] Sun J, Zhou H, Ma X, et al. Study on PID tuning strategy based on dynamic stiffness for radial active magnetic bearing[J]. ISA Transactions, 2018, 80:458-474. [11] Khodja M A, Tadjine M, Boucherit M S, et al. Tuning PID attitude stabilization of a quadrotor using particle swarm optimization (experimental)[J]. International Journal for Simulation and Multidisciplinary Design Optimization, 2017, 8(A8):1-9. [12] Leva A, Papadopoulos A V, Seva S, et al. Explicit model-based real PID tuning for efficient load disturbance rejection[J]. Industrial & Engineering Chemistry Research, 2019, 58(51):1-40. [13] Yan W, Zhu Y. Identification-based PID tuning without external excitation[J]. International Journal of Adaptive Control and Signal Processing, 2018, 32(11):1529-1545. [14] Orrante-Sakanassi J, Santibánez V, Hernández-Guzmán V M. 六自由度运动模拟平台的分析及结构参数的优化[D].哈尔滨:哈尔滨工业大学,2014.Liu G J. Analysis and structural parameter optimization of 6-DOF motion simulation plaform[D]. Harebin:Harbin Institute of Technology, 2014. [16] Teel A R. Global stabilization and restricted tracking for multiple integrators with bounded controls[J]. Systems & Control Letters, 1992, 18(3):165-171. [17] Mendoza M, Zavalario A, Santibanez V, et al. Output-feedback proportional-integral-derivative-type control with simple tuning for the global regulation of robot manipulators with input constraints[J]. IET Control Theory and Applications, 2015, 9(14):2097-2106. [18] Zavala-Río A, Santibáñez V. Simple extensions of the PD-with-gravity-compensation control law for robot manipulators with bounded inputs[J]. IEEE Transactions on Control Systems Technology, 2006, 14(5):958-965. [19] Panda S. Multi-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting genetic algorithm-II[J]. International Journal of Electrical Power & Energy Systems, 2011, 33(7):1296-1308. [20] Ayala H V H, Dos Santos Coelho L. Tuning of PID controller based on a multiobjective genetic algorithm applied to a robo-tic manipulator[J]. Expert Systems with Applications, 2012, 39(10):8968-8974. [21] Taherkhorsandi M, Mahmoodabadi M J, Talebipour M, et al. Ship hull-propeller system optimization based on the multi-objective evolutionary algorithm[J]. Proceedings of the Institution of Mechanical Engineers, Part C:Journal of Mechanical Engineering Science, 2015, 231(1):175-192.