I have a dynamic system governed by the following equation:

$\displaystyle K(s)+\frac{1}{\mu_g}\int_0^s\frac{1}{2\pi}C_{L_\alpha}(s-s_1)K(s_1)ds_1=\int_0^s\frac{1}{2\pi}C_{L_g}(s-s_1)\frac{d\left(\frac{u(s_1)}{U}\right)}{ds_1}ds_1$ ,

where $u(s)/U$ is:

$$ \displaystyle \frac{u(s)}{U}=\left\{ \begin{array}{l} \frac{1}{2}\left(1-\cos\left(\frac{\pi s}{H}\right)\right)\quad&\left(0\leq s\leq 2H\right)\\ 0\quad&\left(0>s>2H\right)\ \end{array} \right. , $$

$\mu_g$, $H$ are known parameters and both $C_{L_\alpha}(s)/2\pi$ and $C_{L_g}(s)/2\pi$ are known exponential functions.

I would like to solve this dynamic system in Simulink, where $K(s)$ is the unknown output and $s$ is my time variable. However the two integrals of the system equation are convolution integrals, and I am not sure how to simulate those within my Simulink model.

Below is my attempt. I generate u_U, C_L_alpha and C_L_g in my workspace by means of the timeseries Matlab function. For the convolution integrals I use the convolution block of Simulink library. However the results I obtain are wrong (I have a reference to compare to).

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How can I correctly represent the convolution integrals of my system in my Simulink model?


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