What's the difference between non-minimum phase systems and minimum phase systems?

Question

I wonder if you can explain what's the difference between non-minimum phase systems and minimum phase systems? How can I recognize them in bode/time plots?

Is this a minimum phase system?

$$G(s) = \frac{\omega^2}{s^2 + 2\eta\omega s + \omega^2}$$

Answer 1

Cesareo's answer is incorrect because the inverse of your transfer function is not causal. The inverse of $G(s)$ has no poles but two zeros, hence it is not a causal system. Also, $\omega$ is not a zero of $G_(s)$ ($\omega^2$ just the gain), hence the sign of $\omega$ is relevant here.

$G(s)$ is not a minimum phase system and here is why:

A transfer function is minimum phase if it is stable and causal, and if the inverse is also stable and causal. This implies that the number of poles must be equal to the number of zeros, and all the poles and all the zeros must be within the unit circle for a discrete system (or lie on the left hand side of the s-plant for continuous transfer functions).

Since the inverse of your system is not causal, it is not a minimum phase system.

To get a better insight into the concept, consider two transfer functions $G_1(s)= \frac{s-1}{s+2}$ and $G_2(s)=\frac{s+1}{s+2}$. Both the transfer functions are causal and their inverses are also causal. Both the transfer functions are stable. Inverse of $G_1(s)$ is not stable and inverse of $G_2(s)$ is stable. Hence, $G_1(s)$ is not a minimum phase system whereas $G_2(s)$ is. Consider the Bode plots of $G_1(s)$ and $G_2(s)$. They have the same magnitude response but different phase responses.

Phase of $G_1(s)$ is $\phi_1 = -arctan(\omega)-arctan(\omega/2)$ and phase of $G_2(s)$ is $\phi_2 = arctan(\omega)-arctan(\omega/2)$. Because the first term in $\phi_2$ is positive, it always has lesser phase than that of $\phi_1$. It is also visible in their phase plots. Hence the name, minimum phase. .