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. in general If system is C. C. & C. O. , then no pole/zero cancellation In this case, BIBO stable Asymp Stable
Routh Table
Routh 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 2 nd order: stable iff all coeff same sign 3 rd order: stable iff a) all coeff same sign b) b*c>a*d
Routh Criteria Regular case: 1 st col all non zero (1) A. S. 1 st col. all same sign (2) #sign changes in 1 st col. = #unstable-roots Special case 1: one whole row=0 1) get aux. poly. A(s), 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: 1 st col =0 but whole row≠ 0 1) replace 0 by a small e >0 2) continue as usual
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
Q: find region of stability in K-a plane.
- Slides: 13