Consider a dynamical system:

$$x_1 = h_a \star s(x_2-x_3)$$ $$x_2 = h_a \star s(n_1x_1 + p(t))$$ $$x_3 = h_b \star s(n_2x_1)$$

$\star$ is a convolution, and $n_1$,$n_2$ are constants
$h_a(t) = ate^{-at}$,
$h_b(t) = bte^{-bt},$
are impulse response functions,
$s(x)= \frac{1}{1+e^{-x}}$ is the sigmoid function.

Finally, let us consider $$x_2(t)-x_3(t)=y(t)$$, as the output of the system, and $$p(t)$$ is a function that serves as the system's input.

I wanted to derive the ODE describing the system with respect to the output variable $y$. I couldn't find a way to do so, since it seems impossible (at least to me) to express $y$ explicitly. I also tried using Laplace transform, but I stuck at some point, due to the fact that there are two different impulse response functions, which results in the following if the calculations are correct:

$$Y(s) = \frac{1}{(s+a)^2}L(s(n_1x_1 + p(t))) - \frac{1}{(s+b)^2}L(s(n_2x_1))$$ $$L^{-1}[(s+a)^2 (s+b)^2 Y(s)]=L^{-1}[(s+b)^2L(s(n_1x_1 + p(t)))]-L^{-1}[(s+a)^2L(s(n_2x_1))]$$

It seems that the input to the 4th order ODE would be in the form of convolutions.

Is there any way to represent such systems which have multiple feedback loops, in terms of a single block diagram with one input $p(t)$ and one output $y(t)$?


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