Analysis of Stability &

Transcription

1 INC 34 Feedback Control Sytem Analyi of Stability & Steady-State Error S Wonga arawan.won@kmutt.ac.th Summary from previou cla Firt-order & econd order ytem repone τ ωn ζω ω n n ζ ζ. ζ.5 ζ ζ.5 ct Timeec 8 T r.8 ω n % OS e ζπ / ζ, ζ < T 4 ζω n T p π ω d Block diagram Cacade Form Parallel Form Feedback Form

2 Today goal Stability analyi - Definition of tability - Stability condition - Routh-Hurwitz criterion Steady- tate error analyi Part I : Stability analyi

3 Definition of tability xt LTI ytem yt y natural t y forced t Initial condition Example: unit tep repone of a firt-order ytem ut yt force repone: /5ut natural repone:3/5exp-5t Y 5 / 5 3/ 5 5 y t 5 3 e 5 5t A pole of the input function forced repone A pole of the tranfer function natural repone Timeec Definition of tability xt LTI ytem yt y natural t y forced t Initial condition A ytem i tableif -the natural repone approache zero a t -for every bounded input the output i alo bounded a t A ytem i untableif -the natural repone approache infinity a t -for every bounded input the output i unbounded a t A ytem i marginally tableif - the natural repone remain contant or ocillate -there i at leat one bounded input for which the output ocillate

4 Stability condition A LTI ytem i aid to be tableif all the pole are in the LHP. A LTI ytem i aid to be untableif the ytem ha any pole in the RHP. A LTI ytem i aid to be marginally tableif the ytem ha nonrepeated jωaxipole. Source: R.C. Dorf & R.H. Bihop, Modern Control Sytem, 9 th Ed.. I thi cloed-loop ytem table?

5 Routh-Hurwitz criterion Routh Table Routh-Hurwitz criterion Routh table can tell how many ytem pole are in the LHP, in the RHP, and on the jω axi. The number of root of the polynomial that are in the RHP i equal to the number of ign change in the firt column. Any row of the Routh table can be multiplied by a poitive contant without changing the value of the row below.

6 Example pole are at 3.446,.768 ± j Try Skill-aement Exercie 6. Zero only in the firt column Replace the zero with an epilon, ε, then analye the ign change by taking ε - /. T

7 Zero only in the firt column Replace the zero with an epilon, ε, then analye the ign change by taking ε - /. T pole are at.349 ± j ± j.7 Entire row i zero An entire row of zero tell u that there i an even polynomial a a factor of the original polynomial. Even polynomial only have root that are ymmetrical about the origin.

8 Example T A factor of T 4 P 6 8 dp 3 4 d Example T jω, LHP Ued to conider the remaining root Ued to conider only the root of the even polynomial 4

9 Example T Example T Try Skill-aement Exercie 6.

10 Review Quetion. Where do ytem pole have to be to enure that a ytem i not untable?. What doe the Routh-Hurwitz criterion tell u? 3. What caue an entire row of zero to how up in the Routh table? 4. If a Routh table ha two ign change above the even polynomial and five ign change below the even polynomial, how many right-half-plane pole doe the ytem have? 5. If a even-order ytem ha a row of zero at the 3 row and two ign change below the 4 row, how many jωpole doe the ytem have? Part II : Steady-State Error Analyi

11 Definition of Steady-State Error Steady-tate error i the difference between the input and the output for a precribed tet input a t. Steady-tate error for unity feedback ytem Applying the final value theorem give E R C R e E R

12 Steady-tate error v. input type R e p e v e a e 3 Sytem type Sytem type i the value of nin the denominator, i.e. the number of pure integration in the forward path. n Type n Type n Type and o on

13 Steady-tate error v. ytem type where p v a i know a the poition contant. i know a the velocity contant. i know a the acceleration contant. Try Skill-aement Exercie 7. Example Find the value of o that there i % error in the teady tate. Since the ytem i of Type, the finite error i poible only for a ramp input. e v. Thu, 67. v Try Skill-aement Exercie 7.3

14 What if the feedback gain i not unity? Well, make it unity then. H H / See Example 7.8 & 7.9 and Try Skill-aement Exercie 7.5 Review Quetion. Name of the tet input ued to evaluate teady-tate error.. Define ytem type. 3. How many integration in the forward path are required in order for there to be zero teady-tate error for each of the tet input lited in Q.? 4. Increaing ytem gain ha what effect upon the teady-tate error?

15 Summary Stability analyi Routh Table Steady-tate error analyi