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
Sys characteristic polynomial is:

4. A polynomial
is said to be Hurwitz
or stable
if all of its roots are in O.L.H.P
A system is stable if its char. polynomial is Hurwitz
A nxn matrix is called Hurwitz or stable
if its char. poly det(sI-A) is Hurwitz, or
if all eigenvalues have real parts<0

5. Routh-Hurwitz Method
From now on, when we say stability we mean A.S. / M.S. or unstable.
We assume no pole/zero cancellation,
A.S. BIBO stable
M.S./unstable not BIBO stable
Since stability is determined by denominator, so just work with d(s)

6. Routh Table

7. Repeat the process until s0 row
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
Note: if highest coeff in d(s) is 1,
A.S. 1st col >0
If all roots of d(s) are <0, d(s) is Hurwitz

8. Example:
←has roots:3,2,-1

9. (1x3-2x5)/1=-7
(1x10-2x0)/1=10
(-7x5-1x10)/-7

11. Remember this

12. Remember this

13. e.g.

14. Routh Criteria Regular case: (1) A.S. 1st col. all same sign
(2)#sign changes in 1st col.
=#roots with Re(.)>0
Special case 1: one whole row=0
Solution: 1) use prev. row to form aux. eq. A(s)=0
2) get
3) use coeff of in 0-row
4) continue

15. Example
←whole row=0

22. Replace by e

24. 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

25. Example +
s+3
s(s+2)(s+1) Kp

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