Stability (Jan) Flashcards
Why is stability so important aka what can’t an unstable system be designed for?
a specific transient response or steady state response
What does a response consist of?
Forced response (steady state) Natural Response
Define stability
A linear time-invariant system
where the natural response approaches 0
as time approached infinity
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
Define instability
A linear time-invariant system
where the natural response grows without bound (approached infinity)
as time approached infinity
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
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
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)
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
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
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
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
What does a row of zeros mean?
poles that are located on the imaginary axis hence contains an undamped oscillation in the system response
