I have a block diagram that looks like this:
Now, I am being asked about the following questions:
- Find the open-loop transfer function $L(s)$
- Find the closed-loop transfer function $G(s)$ from $U(s)$ to $Y(s)$
- Using Routh stability criterion to find the boundary of K using $T_1$ and $T_2$
- Draing the vector locus based on L(s)
- When K = 10, the vector locus of L(s) intersects with the real-axis at $(-0.2, 0)$. Using Nyquist stability criterion to find the stable boundary of K.
- When K increases, how do the stability, responsibility, and steady-state characteristics will change?
The following are the procedures that I took to solve the questions:
1. $$ L(s) = \dfrac{K}{s(T_1s + 1)(T_2s + 1)} $$
2. $$ G(s) = \dfrac{L(s)}{1 + L(s)} = \dfrac{K}{s(T_1s + 1)(T_2s + 1) + K} $$
3. $$ 1 + L(s) = T_1T_2s^3 + (T_1 + T_2)s^2 + s + K = 0 $$
| $s^3$ | $T_1T_2$ | 1 |
|---|---|---|
| $s^2$ | $T_1 + T_2$ | K |
| $s^1$ | $\dfrac{(T_1 + T_2) - KT_1T_2}{T1 + T2}$ | 0 |
| $s^0$ | K |
$$ 0 < K < \dfrac{T_1 + T_2}{T_1T_2} $$
4. $$ |L(j\omega)| = \dfrac{|K|}{|j\omega||1 + jT_1\omega||1 + jT_2\omega|} = \dfrac{K}{\omega \sqrt{1 + (T_1\omega)^2} \sqrt{1 + (T_2\omega}^2} $$
$$ \angle L(j\omega) = \angle K - \angle (j\omega) - \angle (1 + jT_1\omega) - \angle (1 + jT_2\omega) = -\dfrac{\pi}{2} - tan^{-1}(T_1\omega) - tan^{-1}(T_2\omega) $$
5. $$ L(j\omega) = \dfrac{-K(T_1 + T_2)\omega^2 - j(T_1T_2\omega^3 - \omega)}{\{(T_1 + T_2)\omega^2\}^2 + (T_1T_2\omega^3 - \omega)^2} $$
$$ T_1T_2\omega^3 - \omega = 0\\ \omega_0^2 = \dfrac{1}{T_1T_2} $$
$$ L(j\omega_0) = -\dfrac{KT_1T_2}{T_1 + T_2} $$
$$ 0 < K < \dfrac{T_1 + T_2}{T_1T_2} $$
- K responds to the error between the desired setpoint and the actual output. A high proportion gain K can make system responses quickly, but can introduce oscillation and instability. To sum up, with proper K, the steady-state error can be reduced and increase the responsibility of the system.
In question 4, I tried to draw the vector locus based on the open-loop transfer function. However, it includes variables that make it hard to draw the locus. Is there any other way to draw it easier? As far as I know, question 4 $L(s)$ can be written as question 5 $L(s)$, but this does not make things simpler.
Besides, I believe the boundary derived from Routh and Nyquist criterion should be consistent, and the results I got indeed are the same. However, I am not sure about the hint of point $(-0.2, 0)$.
As for question 6, is it fair enough to depict directly from the K itself?
Can anyone point out the detail that I missed? Thank you for your time and advice.