Stability Analysis

Poles and Exponentials

• -ve pole => -ve exponential -> stable
  • roots in the negative half plane
  • output decays or approaches a final value
• neutral pole => 0 exponentials -> neutral/marginally stable
  • pure imaginary roots
  • output is neither magnified nor decayed
  • system output becomes unbounded if an input with the same frequency as these roots is fed into the system
• +ve pole => +ve exponentials -> unstable
  • roots in the right half plane
  • output explodes

Poles vs. Zeros

• Poles => affect system stability
• Zeros => affect system dynamics

Stability Criteria

• Why? Solving higher order transfer functions is hard

Necessary Condition

• What is this? It is a must-to-have if the system is stable. However, it is not sufficient to confirm that the system is stable)

Coefficients are non-zero and have the same sign

We need more reliable criteria.

Routh-Hurwitz Stability Criteria

These are both necessary and sufficient conditions

How to make the array?

• Put the coefficients in a zig zag started from the top left with the highest order coefficient (then go down, then top right and repeat)
• Find the determinant of a matrix that has the first column elements and the second column elements, multiply this by negative one, then divide by the first element in the second row (then use a new matrix that has the same first column elements with the next column elements to find the next term)

Case 1

• No zero terms in first column
• Zero sign changes

Case 2

• Single zero in the row -> replace with $0<ϵ⋘1$
• Zero sign changes

Case 3

• More than one zero in the row; this means that the poles are symmetric about the origin; we need to find out whether they are symmetric about the y-axis (unstable) or x-axis (marginally stable)
• Use the previous row to find get the polynomial (order indicates root pairs; order must be even, because they must provide root pairs)
• Use the auxiliary polynomial to find the symmetric roots

Case 4

• Same as case 3 but there are more than one pair of imaginary repeated roots; this moves the system from being marginally stable to being unstable