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Flashcards in Stability (Jan) Deck (13):
1

Why is stability so important aka what can't an unstable system be designed for?

a specific transient response or steady state response

2

What does a response consist of?

Forced response (steady state)
Natural Response

3

Define stability

A linear time-invariant system
where the natural response approaches 0
as time approached infinity

4

Define Marginal stability

A linear time-invariant system
where the natural response neither decays nor grows but remains constant or oscillates
as time approached infinity

5

Define instability

A linear time-invariant system
where the natural response grows without bound (approached infinity)
as time approached infinity

6

What is a marginally stable system response

an undamped system
roots contained only imaginary parts and are of unit multiplicity
undamped as natural response is constant aka no energy loss

7

What does the Routh-Hurwitz Criterion allow us to do?

We can determine where the poles are on the complex s-plane and hence yield information about the stability of the system
- without the need to solve for closed loop system poles

8

How do you interpret the Routh-Hurwits table?

-look at first column
- sign change = presence of pole in RHP (unstable)
- no sign change = poles are in LHP or on the imaginary axis (stable/marginally stable)

9

What are the two special cases that can arise in the Routh-Hurwitz Table?

1) A zero will sometimes occur in the first column of a row
2) An entire row of zeros will sometimes occur

10

Why do they cause difficulty when finding stability?

A zero has no sign (therefore no sign change) and the RH table cannot be interpreted in the conventional manner

11

1) A zero will sometimes occur in the first column of a row
STEPS

1) replace zero with ϵ (epsilon) therefore allowing the rest of the table to be filled out
2) let ϵ be a small positive integer (e.g ϵ=1) and solve to fine sign of all elememts

12

2) Complete row of zeros
STEPS

1) create an auxiliary polynomial with the row above the row of zeros
- the polynomial is formed by skipping every second s power
2) differentiate auxiliary polynomial
3) sub into equation

13

What does a row of zeros mean?

poles that are located on the imaginary axis hence contains an undamped oscillation in the system response