1 / 38

Stability

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

dara-salinas
Download Presentation

Stability

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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. All variables include y & its derivatives all state variables but y=Cx+Du=Cx if x→0 then y→0 only need to check x→0 0

  3. Criterion for A.S.

  4. If there is no pole/zero cancellation, BIBO-stable A.S. If system is C.C. & C.O. no pole/zero cancellation BIBO-stable 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

  5. Thm: If a system is A.S. then it is BIBO-stable But BIBO-stable A.S. (mathematically)

  6. 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 all eigenvalues<0

  7. 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)

  8. Routh Table

  9. 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

  10. Example: ←has roots:3,2,-1

  11. (1x3-2x5)/1=-7 (1x10-2x0)/1=10 (-7x5-1x10)/-7

  12. Remember this

  13. Remember this

  14. e.g.

  15. 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

  16. Example ←whole row=0

  17. replace

  18. 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

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

  20. =6

  21. k>0.5

  22. + K(s+z) s+p 1 s(s2+2s+8) -

More Related