The dominant approach for solving ODE in control systems books is ode45 since the majority of these books use Matlab. I'm not acquainted with how the ode45 works but lately I started reading about Euler's method in this book Numerical Methods for Engineers. If the step size is very small, then the results are satisfactory. For simulation, one can actually set the step size to be very small value. I've used ode45 in herehere for regulation and tracking problems. I faced some difficulties for using ode45 for tracking problem since the step size is not fixed. Now for the same experiment, I've used the Euler's method with step size 0.001 sec. The results are amazing and so friendly in comparison with ode45. This is a snapshot from the result
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Euler’s Method Or ode45 for solving ODE for control systems
The dominant approach for solving ODE in control systems books is ode45 since the majority of these books use Matlab. I'm not acquainted with how the ode45 works but lately I stared reading about Euler's method in this book Numerical Methods for Engineers. If the step size is very small, then the results are satisfactory. For simulation, one can actually set the step size to be very small value. I've used ode45 in here for regulation and tracking problems. I faced some difficulties for using ode45 for tracking problem since the step size is not fixed. Now for the same experiment, I've used the Euler's method with step size 0.001 sec. The results are amazing and so friendly in comparison with ode45. This is a snapshot from the result
And this is the code
clear all; clc; dt = 0.001; t = 0; % initial values of the system a = 0; % angular displacement da = 0; % angular velocity % PID tuning Kp = 50; Kd = 18.0; Ki = 0.08; error = 0; %System Parameters: m = 0.5; % mass (Kg) d = 0.0023e-6; % viscous friction coefficient L = 1; % arm length (m) I = 1/3*m*L^2; % inertia seen at the rotation axis. (Kg.m^2) g = 9.81; % acceleration due to gravity m/s^2 % Generate Desired Trajectory y = 0:dt:(3*pi)/2; AngDes = y; % Ang: angle , Des: desired AngDesPrev = 0; for i = 1:numel(y) % get the first derviative of the desired angle using Euler method. dAngDes = ( AngDes(i) - AngDesPrev )/ dt; AngDesPrev = AngDes(i); % torque input u = Kp*( AngDes(i) - a ) + Kd*( dAngDes - da ) + Ki*error; % accumulated error error = error + ( AngDes(i) - a ); %store the erro E(i) = ( a - AngDes(i) ); T(i) = t; dda = 1/I*(u - d*da - m*g*L*sin(a)); % get the function and its first dervative da = da + dda*dt; a = a + da*dt; %store data for furhter investigation A(i) = a; dA(i) = da; t = t + dt; end plot(T, AngDes, 'b', T, A, 'g', 'LineWidth', 1.0) h = legend('$\theta_{d}(t)$', '$\theta(t)$'); set(h, 'Interpreter','LaTex') My question is why ode45 is preferred in many control books assuming the step size is very small.