# Why the bandwidth of the system with the minimum-order observer is higher than the full-order observer?

In Ogata's Modern Control Engineering (5th Ed., page 786) it says:

The bandwidth of the system with the minimum-order observer is higher than that of the system with the full-order observer, provided that the multiple observer poles are placed at the same place for both observers.

Can anyone explain me why this is so?

## 1 Answer

I think the key of that statement is the sentence

[...] provided that the multiple observer poles are placed at the same place for both observers.

Suppose that you have a Bode plot, with poles at $\omega_1,\omega_2,...,\omega_n$. If you leave those poles as they are (i.e. you don't change their locations) and you add more poles to the system, then the bandwidth will decrease.

Why? Well, suppose we add a single pole at $\omega_{n+1}$. In that case, the slope of the magnitude as a function of the frequency will decrease for all the frequencies such that $\omega>\omega_{n+1}$. If the magnitude was increasing, it will increase slower or will become constant (referring to the asymptotic Bode plot). If it was decreasing, it will decrease faster.

If you add even more poles and not just one, then, following the same reasoning, the bandwidth will decrease more abruptly.