1. Stability BIBO stability:
Def: A system is BIBO-stable if any bounded input produces bounded output.
otherwise it’s not BIBO-stable.

2. Asymptotically Stable
A system is asymptotically stable if for any arbitrary initial conditions, all variables in the system converge to 0 as t→∞ when input=0.
Thm: If a system is A.S.
then it is BIBO-stable
But BIBO-stable does not guarantee A.S.in general
If there is no pole/zero cancellation,
BIBO stable  Asymp Stable

3. Characteristic polynomials
Three types of models:
Assume no p/z cancellation
System characteristic polynomial is:

4. Routh Table

5. Routh Stability criterion:
d(s) is A.S. iff 1st col have same sign
the # of sign changes in 1st col
= # of roots in right half plane
Necessary for AS: All coeff same sign
2nd order: stable iff all coeff same sign
3rd order: stable iff a) all coeff same sign
b) b*c>a*d

6. Routh Criteria Regular case: 1st col all non zero
(1) A.S.: 1st col. all same sign
(2) #sign changes in 1st col. = #roots in RHP
Special case 1: one whole row=0
1) get aux. poly. A(s) from row above,
2) use coeff of A’(s) to replace 0-row
3) continue, and roots of A(s) are roots of d(s)
Special case 2: 1st col =0 but whole row≠0
1) replace 0 by a small e >0
2) continue as usual

7. Example
←whole row=0

14. Routh criteria Useful case: parameter in d(s)
How to use: 1) form table as usual
2) set 1st col. >0
3) solve for parameter range for A.S.
2’) set one in 1st col=0
3’) solve for parameter that leads to M.S. or leads to sustained oscillation

15. Example: Proportional control+
s+3
s(s+2)(s+1) KP

16. =6

18. PD (proportional +derivative) control
s+3
s(s+2)(s+1)
KP+KDs

19. KP KD -3

23. Q: find region of stability in K-a plane.