I'm solving a problem related with Laplace Transformation and and Transfer Function for a Linear systems Class, but I found a definition that is not clear to me. This is the full problem:
Consider a 2 input-2 output system denoted by:
$D_{11}(p)y_{1}(t)+D_{12}(p)y_{2}(t)=N_{11}(p)u_{1}(t)+N_{12}(p)u_{2}(t)$
$D_{21}(p)y_{1}(t)+D_{22}(p)y_{2}(t)=N_{21}(p)u_{1}(t)+N_{22}(p)u_{2}(t)$
where $N_{ij}$ and $D_{ij}$ are polynomials of $p=d/dt$. Whats the Transfer Matrix of the system?
So, my question is What does the polynomial $p=d/dt$ mean?
My approach for solving this is multiply $y(s)=D^{-1}(s)*N(s)*u(s)=G(s)u(s)$, but still no idea about the sense of $p$.
Presumably what is meant is polynomial functions of $\dfrac d{dt}.$
For example, suppose $f(x)=x^3-5x^2 + 2x +9.$
Then $f\left( \dfrac d {dt} \right) = \dfrac {d^3} {dt^3} - 5 \dfrac{d^2}{dt^2} + 2 \dfrac d {dt} + 9,$ so that, for example, $$ f\left( \frac d{dt} \right) u(t) = u'''(t) - 5u''(t) + 2u'(t) + 9u(t). $$