Stability BIBO stability Def A system is BIBOstable
Stability • BIBO stability: Def: A system is BIBO-stable if any bounded input produces bounded output. otherwise it’s not BIBO-stable.
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.
If there is no pole/zero cancellation, BIBO-stable A. S. If system is C. C. & C. O. BIBO-stable no pole/zero cancellation A. S. Exact pole/zero cancellation only happens mathematically, not in real systems. From now on, assume no p/z cancellation BIBO stable A. S. all char. val<0 all eigenvalues<0 all poles<0
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(s. I-A) is Hurwitz all Re(eigenvalues)<0
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)
Routh Table
Repeat the process until s 0 row Stability criterion: 1) d(s) is A. S. iff 1 st col have same sign 2) the # of sign changes in 1 st col = # of roots in right half plane Note: if highest coeff in d(s) is 1, A. S. 1 st col >0 If all roots of d(s) are <0, d(s) is Hurwitz
Example: ←has roots: 3, 2, -1
(1 x 3 -2 x 5)/1=-7 (1 x 10 -2 x 0)/1=10 (-7 x 5 -1 x 10)/-7
Remember this
Remember this
e. g.
Routh Criteria Regular case: (1) A. S. 1 st col. all same sign (2)#sign changes in 1 st 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
Example ←whole row=0
replace
Useful case: parameter in d(s) How to use: 1) form table as usual 2) set 1 st col. >0 3) solve for parameter range for A. S. 2’) set one in 1 st col=0 3’) solve for parameter that leads to M. S. or leads to sustained oscillation
Example + Kp s+3 s(s+2)(s+1)
=6
k>0. 5
- Slides: 29