11 Differential equations as feedback systems

The differential equation

$$a_n{y}^{(n)}(t)+\mathrm{\dots }+a_{0}y=b_m{u}^{(m)}+b_{m-1}{u}^{(m-1)}(t)+\mathrm{\dots }+b_{0}u(t)$$

with constant coefficients can be realised as a feedback system involving taps, amplifiers, summing junction, integrators and differentiations. Proof. When $a_n\ne 0$, we integrate $n$ times and divide by $a_n$ to get

$$y=-\frac{a_{n-1}}{a_n}\int y-\mathrm{\dots }-\frac{a_0}{a_n}{\int }^{(n)}y+\frac{b_m}{a_n}{\int }^{(n)}{u}^{(m)}+\mathrm{\dots }+\frac{b_0}{a_n}{\int }^{(n)}u.$$

Then it is straightforward to realize the system.