SaM - ODE (Ordinary Differential Equation)

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ODE (Ordinary Differential Equation) : $e(t)+a_1{e}^{\prime }(t)+a_2{e}^{\prime \prime }(t)+\dots =b_{0}g(t)+b_1{g}^{\prime }(t)+\dots$ ⇒ Laplace Domain: \begin{align}&E(s) \kern2px + \\ & a_1sE(s) - a_1e(0^-) \kern2px + \\ & a_2s^2E(s) - a_2se(0^-)- a_2e'(0^-) = \\ &b_0G(s) \kern2px + \\ &b_1sG(s) - b_1g(0^-)\end{align} ⇒ Polynomials $A(s)$ (of $p$ order) and $B(s)$ (of $z$ order):

And then I define also a polynomial $I(s)$ which accounts for all the times related to initial condition:

Important:

Therefore, if I have no initial condition, I’ll have this response here:

You know very well that for real linear system, the pole of this response are all stable:
==The pole of the system, $A(s)$ has stable zeros. That means that the function $H(s)$, which is called transfer function we found has stable poles==. And we write again:
==So this eqaution has stable poles as well, and the poles are the same because we have the same polynomial at the denominator, $A(s)$. So the dynamics of the initial condition and of the transient part of the response to the input is the same==.