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