## 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`.