Quiz on Stability Analysis

1. What is the common name used specifically for the roots of the denominator in a Laplace transfer function?

A. Zero
Incorrect. A zero is a root of the numerator
B. Discontinuity
Incorrect. The poles are the roots of the denominator
C. Pole
Correct. A pole is a root of the denominator. It gives information about stability and oscillations.
D. Locus root
Incorrect. A root locus plot displays the roots of the closed loop system with changing controller gain.

2. The Final Value Theorem (FVT) suggests the steady state value approaches 2. The roots of the denominator are -2, -3, and 1 +/- 3i. What conclusion can be made about the system?

A. FVT proves that the system will converge to a value of 2
Incorrect. The FVT cannot be applied because the system is not stable.
B. FVT does not apply in this case, so no conclusion can be drawn
Incorrect. From the positive real roots, the system is unstable
C. The positive real roots show that the system is unstable. The FVT is not applicable
Correct. The FVT cannot be applied for unstable systems
D. The system will diverge from an initial value of 2
Incorrect. The FVT cannot be applied for unstable systems. The Initial Value Theorem (IVT) doesn't answer the question.

Information for Questions 3 and 4: A closed transfer function is

$$G(s) = \frac{3K_c}{2s^3 + 2s^2 + 3s + 5 + 3K_c}$$

The coefficients of the denominator determine the stability range for the controller gain `K_c` with a Routh array with `n=3`.

$$a_n s^n + a_{n-1} s^{n-1} + a_{n-2} s + a_{n-3}$$

Routh Array

`a_n=2``a_{n-2}=3`
`a_{n-1}=2``a_{n-3}=5 + 3 K_c`
`b_{1}={a_{n-1}a_{n-2}-a_n a_{n-3}}/{a_{n-1}``b_{2}=0`
`c_{1}=5+3 K_c`0

3. Find the value of `b_1` from the Routh array.

A. `2 + 3K_c`
Incorrect. The terms of the numerator are used in the wrong order
B. `4 + 6K_c`
Incorrect. Divide by `a_{n-1}`
C. `-2 - 3K_c`
Correct. `b_{1}={a_{n-1}a_{n-2}-a_n a_{n-3}}/{a_{n-1}}={6-2(5+3 K_c)}/{2} = -2-3 K_c`
D. `5 + 3K_c`
Incorrect. See the equation for `b_1`

4. What do the values of `b_1` and `c_1` reveal about the range of stable controller gains for the system?

A. `0 < K_c < 2`
Incorrect. The limits are negative.
B. `-{2}/{3} < K_c < {2}/{3}`
Incorrect. Flip from > to < when dividing by a negative
C. `0 < K_c < {2}/{3}`
Incorrect. The sign is incorrect
D. `-{5}/{3} < K_c < - {2}/{3}`
Correct. The leading edge must not change sign. Because the two values above are positive, these must also be positive with `-2-3 K_c>0` and `5+3 K_c>0`.
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