I am reading the book "Autotamic Control Systems" by Farid Golnaraghi and Benjamin C.Kuo, Tenth Edition.
In the book:
Figure 1
Consider the nonunity-feedback system above, where r(t) is the input (but not the reference signal), u(t) is the actuating signal, b(t) is the feedback signal, and y(t) is the output. In this case, input and output may not have same units or dimensions. In order to establish a proper formula for the error, we must first obtain a clear picture of what the reference signal is. For a stable system, the steady-state output will be tracking the reference signal. The error of the system at all times is the difference between the reference signal and the output. In order to establish the reference signal, let us modify the system above by first factoring out the feedback gain H(s):
Figure 2
Error in the system is the difference between the output and the desired value of the output—or the reference signal. In this case, the reference signal in Laplace domain is R(s)G1(s), as shown in figure 2. The value of G1(s) is obviously is related to 1/H(s), and it may be obtained based on the time response characteristics of the original system in Fig 1. Obviously, the reference signal reflects the desired value of the system output based on a given input, and it cannot contain additional transient behavior due to H(s).
There are two possible scenarios based on the value of H(s). Case 1: $$G_1(s) = \frac{1}{\lim_{s\to 0}H(s)} = constant $$ which means that H(s) cannot have poles at s = 0. Hence, the reference signal becomes: Reference signal = $R(s)[1/H(0)] = R(s)G_1(s)$. Case 2: H(s) has Nth-order zero at s = 0 $$G_1(s) = (\frac{1}{s^N})\frac{1}{\lim_{s\to 0}H(s)} = \frac{1}{s^NH(0)}$$ Reference signal = $$\frac{R(s)}{s^NH(0)} = R(s)G_1(s)$$
Firstly, I cannot understand the bold texts. Secondly, why are there two scenarios of H(s)? What if H(s) has poles at s = 0? Why in case 1:$$G_1(s) = \frac{1}{\lim_{s\to 0}H(s)}?$$ Is it final-value theorem? I understand the theorem but I can't relate this. Similary, in case 2, why: $$G_1(s) = (\frac{1}{s^N})\frac{1}{\lim_{s\to 0}H(s)} = \frac{1}{s^NH(0)} ?$$