102 Proof of realization

Let the transfer function be $T(s)=p(s)/q(s)$ where $p(s)={\sum }_{j=0}^n{a}_j{s}^j$ and $q(s)={\sum }_{k=0}^m{b}_k{s}^k$ are polynomials with $b_m=1$. Consider the differential equation

$$y=\sum _{j=0}^n{a}_j\frac{d^{j}u}{d{t}^j}$$

which has Laplace transform

$$Y(s)=p(s)U(s).$$

Also, we can realise the proper rational function $1/q(s)$ as the transfer function of a SISO system, as in Proposition 57. By combining these in series, we realise a system with transfer function $T(s)=p(s)/q(s).$