Simone Bertoni’s Post

Classic Runge-Kutta method implemented in C. Let's see how to use it to run a simulation. Classic Runge-Kutta or RK4 is the most known method of the Runge-Kutta family. We can use it to simulate the evolution of a dynamic system, as an alternative to more basic methods like Euler. Let's take the simple model of a car described in https://buff.ly/3fsMaGL and refreshed in the picture below. We have shown that its motion is described by: dv/dt = (F - b * v^2) / m The RK4 provides an approach to compute y(t_n+1) starting from the knowledge of y_n and f(t, y) = dv(t)/dt. Here's the method: y(t_n+1) = y_n + T*(k1 + 2*k2 + 2*k3 + k4)/6, where t_n+1 = t_n + T Also: k1 = f(t_n, y_n) k2 = f(t_n + T/2, y_n + T*k1/2) k3 = f(t_n + T/2, y_n + T*k2/2) k4 = f(t_n + T, y_n + T*k3) In the picture below you can find: - A recap of the method - The model of the car above implemented in C with forward Euler and with the classic Runge-Kutta method - The result of a simulation that compares the continuous model implemented in Xcos (max integration time 0.01s) and the two implementations above (using T = 5 seconds) The parameters used are: m = 1000 kg b = 1 N*s^2/m^2 I limited the integration time in Xcos to 0.01s so that the result is very accurate and I chose T = 5s to push the two iterative methods to be in trouble, to see which one performs better. In this case, we can observe from the simulation result that RK4 is a lot more accurate (closer to Xcos) than Euler. A clarification on the implementation of the functions in C. The transient nature of the system is obtained by running the functions N times to generate N values of v, starting from N values of F. For example: v[i] = Car_FE(F[i], 1000, 1, 5); where i = 0 to N-1. If you enjoyed this follow me for more tips on control and embedded software engineering. Hit the 🔔 on my profile to get a notification for all my new posts. #controlsystems #embeddedsystems #softwareengineering #embeddedsoftware #controltheory #coding