Time domain response specifications

1. Time domain response specifications • Defined based on unit step response • Defined for closed-loop system

2. Prototype 2nd order system:

3. Settling time:

4. Stability • BIBO-stable: Def: A system is BIBO-stable if any bounded input produces bounded output. Otherwise it’s not BIBO-stable.

5. 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. A system is marginally stable if for all initial conditions, all variables in the system remain finite, but for some initial conditions, some variable does not converge to 0 as t→∞. A system is unstable if there are initial conditions that can cause some variables in the system to diverge to infinity. A.S., M.S. and unstable are mutually exclusive.

6. Asymptotically Stable

7. Asymptotically Stable vs BIBO-stable Thm: If a system is A.S., then it is BIBO-stable If a system is not BIBO-stable, then it cannot be A.S., it has to be either M.S. or unstable. But BIBO-stable does not guarantee A.S. in general. If there is no pole/zero cancellation, then BIBO-stable  Asymp Stable

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

9. 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

10. 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)

11. Routh Table

12. 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

13. Example: ←has roots:3,2,-1

14. (1*3-2*5)/1=-7 (1*10-2*0)/1=10 (-7*5-1*10)/-7

16. Remember this

17. Remember this

18. e.g.

19. 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 to replace 0-row 4) continue as usual

20. Example ←whole row=0

27. Replace by e

29. 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

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

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