Transfer function of system is $H(s)=\frac{(s-p')}{s(s-p)(s^2+2\lambda \omega_0 s+\omega_o^2)}$
In lecture notes the state space equivalent is
$$x_1'=x_2$$ $$x_2'=-\omega_0^2 x_1 - 2 \lambda \omega_0 x_2 + x_3 + x_4 + u$$ $$x_3'=px_3 + x_4 +u$$ $$x_4'=-p'u$$
Can someone explain how this works? All exaples I've come across so far show $x_n' = x_{n+1}'$ apart from largest derivative of x. How can $x_2'$ reference $x_3$ and $x_4$ ?
This is called Realizations and you can find information here:
https://en.wikipedia.org/wiki/Realization_(systems)
Or an reference more formally:
Linear Systems Theory and Design
Chi-Tsong Cheng
Third Edition
Chapter 4: State-Space Solutions and Realizations.