There are so many definitions of the minimum phase transfer function, and these are two of them.
- The transfer function of the system which has no zeros or poles at right half plane.
- The transfer function which has the minimum phase angle range among the systems which has the same magnitude characteristic.
And these two sentences are describing the same thing. So I want to prove that these statements are equivalent. How to prove it?
PS : Suppose that the system is continuous.
Answer
To your second definition it should be added that you only consider causal transfer functions, because it is not difficult to find a smaller phase lag with a non-causal system:
A minimum-phase system is a causal and stable system with a phase lag that is smaller than the phase lag of any other causal and stable system with the same magnitude response.
Note that a real-valued zero in the left half-plane contributes a phase change of $\pi/2$ to the total phase as we move along the frequency axis from $\omega=0$ to $\omega\rightarrow\infty$:
$$0\le\arg\{j\omega+a\}<\frac{\pi}{2},\qquad a>0,\quad\omega\in[0,\infty)\tag{1}$$
A complex conjugate pair of zeros contribute a phase change of $\pi$.
On the other hand, a real-valued zero in the right half-plane contributes a phase change of $-\pi/2$ as $\omega$ moves from zero to infinity:
$$-\frac{\pi}{2}<\arg\{a-j\omega\}\le 0,\qquad a>0,\quad\omega\in[0,\infty)\tag{2}$$
Note that I've chosen the sign of the term $s\pm a$ such that in both cases the phase is zero for $\omega=0$, which is necessary for a fair comparison between the two cases.
Consequently, exchanging a zero in the left half-plane for a zero in the right half-plane (without changing the magnitude of the frequency response) will always result in an additional phase lag of $\pi$ for $\omega\rightarrow\infty$.
As an example, consider two first-order transfer functions:
$$H_1(s)=\frac{s+2}{s+1}\quad\text{and}\quad H_2(s)=\frac{2-s}{s+1}\tag{3}$$
The signs of $H_1(s)$ and $H_2(s)$ were chosen such that their phases are both zero for $s=0$ (and not $\pm\pi$). The figure below shows the phase plots ($\arg\{H_1(j\omega)\}$ in blue, and $\arg\{H_2(j\omega)\}$ in green):
The pole contributes a phase change of $-\pi/2$ as $\omega$ moves from zero to infinity. The left half-plane zero of $H_1(s)$ contributes a phase change of $\pi/2$, resulting in a net phase change of zero, whereas the right half-plane zero of $H_2(s)$ contributes a phase change of $-\pi/2$, resulting in a total phase change of $-\pi$.
Another way to see the same thing is to note that any causal and stable transfer function can be written as the product of the minimum-phase transfer function with the same magnitude and a causal and stable allpass:
$$H(s)=H_m(s)H_a(s)\tag{4}$$
It can be shown that the phase of a causal and stable allpass is always non-positive for $\omega\in[0,\infty)$, and, consequently, the phase lag of the minimum-phase system is always less than or equal to the phase lag of any other causal and stable system with the same magnitude response.
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