Optimal Monotonicity-Preserving Perturbations of a Given Runge–Kutta Method

Higueras, Inmaculada; Ketcheson, David I.; Kocsis, Tihamér A.

doi:10.1007/s10915-018-0664-3

Cited by 10 publications

(9 citation statements)

References 27 publications

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“…For verifying the effectiveness of the aforementioned controller, a numerical simulation experiment was achieved in MATLAB/Simulink. The real-time state value of the manipulator system is solved by ode4 (Runge-Kutta) (Higueras et al, 2018). Through the feedback of the current state value, the current control torque of the system is obtained.…”

Section: Methodsmentioning

confidence: 99%

Self-Tuning Control of Manipulator Positioning Based on Fuzzy PID and PSO Algorithm

Liu

Yun

et al. 2022

Front. Bioeng. Biotechnol.

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With the manipulator performs fixed-point tasks, it becomes adversely affected by external disturbances, parameter variations, and random noise. Therefore, it is essential to improve the robust and accuracy of the controller. In this article, a self-tuning particle swarm optimization (PSO) fuzzy PID positioning controller is designed based on fuzzy PID control. The quantization and scaling factors in the fuzzy PID algorithm are optimized by PSO in order to achieve high robustness and high accuracy of the manipulator. First of all, a mathematical model of the manipulator is developed, and the manipulator positioning controller is designed. A PD control strategy with compensation for gravity is used for the positioning control system. Then, the PID controller parameters dynamically are minute-tuned by the fuzzy controller 1. Through a closed-loop control loop to adjust the magnitude of the quantization factors–proportionality factors online. Correction values are outputted by the modified fuzzy controller 2. A quantization factor–proportion factor online self-tuning strategy is achieved to find the optimal parameters for the controller. Finally, the control performance of the improved controller is verified by the simulation environment. The results show that the transient response speed, tracking accuracy, and follower characteristics of the system are significantly improved.

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Section: Methodsmentioning

confidence: 99%

Self-Tuning Control of Manipulator Positioning Based on Fuzzy PID and PSO Algorithm

Liu

Yun

et al. 2022

Front. Bioeng. Biotechnol.

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show abstract

“…which we then evolve in time using an explicit Runge-Kutta method of the form (6). This approach is known as a Lawson-type method [11].…”

Section: Ssp Theory For Explicit Integrating Factor Runge-kutta Methodsmentioning

confidence: 99%

“…Thus, if α i,j ≥ 0, all the intermediate stages u (i) in (6) are convex combinations of backward in time Euler and forward Euler operators, with ∆t replaced by |β i,j | α i,j ∆t. Following the same reasoning as above, any strong stability bound satisfied by the backward in time and forward in time Euler methods will then be preserved by the Runge-Kutta method (6) where F is replaced byF whenever the corresponding β is negative.…”

Section: Introductionmentioning

confidence: 99%

Downwinding for preserving strong stability in explicit integrating factor Runge–Kutta methods

Isherwood

Grant

2018

Pure and Applied Mathematics Quarterly

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Strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case for L 2 linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor Runge-Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge-Kutta methods where the transformed problem is evolved with an explicit SSP Runge-Kutta method with non-decreasing abscissas was recently established. However, these methods typically have smaller SSP coefficients (and therefore a smaller allowable time-step) than the optimal SSP Runge-Kutta methods, which often have some decreasing abscissas. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge-Kutta methods where the Runge-Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some cases the integrating factor approach with explicit SSP Runge-Kutta methods with non-decreasing abscissas performs nearly as well, if not better, than with explicit SSP Runge-Kutta methods with downwinding. In conclusion, while the downwinding approach can be rigorously shown to guarantee the SSP property for a larger time-step, in practice using the integrating factor approach by including downwinding as needed with optimal explicit SSP Runge-Kutta methods does not necessarily provide significant benefit over using explicit SSP Runge-Kutta methods with non-decreasing abscissas.

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“…• Strong stability preserving (SSP) Runge-Kutta methods (Hadjimichael et al, 2013;David I. Ketcheson et al, 2009;David I. Ketcheson, 2008) • SSP general linear methods (Bresten et al, 2017; • Low-storage Runge-Kutta methods (David I. Ketcheson, 2010) • Additive and downwind SSP Runge-Kutta methods (Higueras et al, 2018;David I. Ketcheson, 2011) • High-order parallel extrapolation and deferred correction methods (David I. Ketcheson & bin Waheed, 2014) • SSP linear multistep methods (Hadjimichael et al, 2016;Hadjimichael & Ketcheson, 2018) • Dense output formulas for Runge-Kutta methods (David I. Ketcheson et al, 2017) • Internal stability theory for Runge-Kutta methods (David I.…”

Section: Related Research and Softwarementioning

confidence: 99%