How to discretize a nonlinear system

Question

How can i discretize the following nonlinear system. Im using Matlab and Casadi for Model Predictive Control. The Constant C is betwenn 0 and 1.

dx/dt = C * x/(x^2 + 1)

Thank you for your time and Help.

Answer 1

Well if you want to use the file to discretize a differential equation, there are a lot of methods such as simple Euler, Runge-Kutta, and so on. Let me say how to use the Euler method. based on the differential definition:

dx/dt = (x(i+1) - x(i))/dt

here i is the discretization index and dt is sample time (typically 0.01). If I have to use your equation:

(x(i+1) - x(i))/dt = C*x(i)/(x(i)^2 + 1)

after simplification:

x(i+1) = x(i) + dt*(C*x(i)/(x(i)^2 + 1))

this is your discretized model. In matlab, just use the following code:

C = 0.5;
N = 100;
x(1) = 1;       % initial condition
dt = 0.01;

i = 1;
for i = 1:N
    x(i+1) = x(i) + dt*(C*x(i)/(x(i)^2 + 1));
end

Answer 2

If you are just looking to build it from blocks, something like this should work:

You basically need to invert the formula to:

x = int(C * x/(x^2 + 1))

Everything to the right of the equal sign then feeds the input to the integrator and the output of the integrator becomes x.