My system is very dynamic and du/dt would cause errors.
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Two remarks here:
1. If you draw the Bode plot of the transfer function of the derivator, it looks like a line that ramps up with 20 dB/decade. In other words, for high frequencies it amplifies the signal (most likely it is a noise in this region). So, what you want is to avoid this phenomena, however keeping the derivator behavior for lower frequencies. In order to achieve this, replace your derivative block with a transfer function block:
where you have to choosetau sufficiently low. Notice that if tau = 0, then you end up with the derivator H(s) = s.
2. You say something about using integrators. This is possible if you solve your equation the other way around, namely you know x and want to calculate y. The dynamics of y is described by:
dy/dt = (1/L) (2y-x)
For this, you need to know the initial condition y(0) of your system, together with x(t). You code this equation in matlab or Simulink using an integrator block.
It must be due to the limitation of calculating du/dt. If you are using all the default setting, you'll notice some warning message in the Command Window regarding step size.
There is no perfect way of calculating the theoretical du/dt in numerical simulation. Doing the du/dt first and then the integration is the problem. If you move the two integrators ahead of the du/dt block (right after the negative feedback summation block), you'll find that the results match perfectly.
You would pass a function handle to trainDynamics to the ode solver, e.g. ode45, along with a time span and initial value vector. The xdot = xdot' line simply turns xdot into a column vector for output. See the example in the doc for solving 2nd order ode's:
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