I am trying to implement a worked example from the paper "AN ALGEBRAIC FRAMEWORK FOR LINEAR IDENTIFICATION" by Michel Fliess. The objective is to estimate the value of unknown parameter "a" from known y(t) and u(t). The signals and the constant are related by the equation
$$ s^2 Y(s) = a s Y(s) + s U(s) +s y(0) + \gamma $$
This equation is differentiated twice and divided by $s^2$ to remove the differentiations with respect to time when we go for inverse laplace transform.
$$ \left[ s^{-1} \frac{d^2y}{ds^2} + 2s^{-2}\frac{dy}{ds}\right]a = \frac{d^2y}{ds^2} + 4s^{-2} \frac{dy}{ds} + 2s^{-2} y - \left( s^{-1} \frac{d^2u}{ds^2} + 2s^{-2} \frac{du}{ds}\right) $$
I have simulated the system in simulink and got y(t) and u(t) signals correctly(same as the plots given in the paper).
$$ a = \frac{t^2y(t) - \int_0^t \left(4\sigma y(\sigma) + \sigma^2u(\sigma)\right)d\sigma +2 \int_0^t\int_0^\sigma \left( \lambda u(\lambda) + y(\lambda)\right) d\lambda d\sigma}{\int_0^t \sigma^2 y(\sigma) d\sigma -2\int_0^t\int_0^\sigma \lambda y(\lambda) d\lambda d\sigma} $$
When the inverse laplace transform is taken, the numerator is a vector (due to the first term $t^2y(t)$). For this term, I am doing element wise multiplication of the time vector and y vector in Matlab and I get a vector in return. The denominator is a constant, as both are closed integrations. In the publication, both the numerator and denominator are given as vectors. My question is how do I get the denominator as a vector as in the publication when it is clearly a constant coming from a closed integration.
Signals generated by my Simulink model and the image from the paper are pasted below for quick reference.
Signals, Numerator and Denominator from reference paper.
Input and output signals from my simulink model, matching the paper figure
