Last week: analysis of pinion-rack w velocity feedback

Transcription

1 Last week: analysis of pinion-rack w velocity feedback Calculation of the steady state error Transfer function: V (s) V ref (s) = 0.362K s K Step input: V ref (s) = s Output: V (s) = s 0.362K s K Steady state: v( ) =lim s 0 sv (s) = 0.362K K 2 Steady state error: e( ) = v( ) = K. e( ) [for K =00] Lecture 6 Monday, Oct. 5

2 Today Analysis of steady-state errors Definition for step, ramp, parabola inputs Steady-state error in unity feedback systems System type and static error constants The role of integrators Steady-state error in the presence of disturbances Controller gain and step disturbance cancellation Wednesday & Friday: Root locus Lecture 6 Monday, Oct. 5

3 Inserting an integrator We ve changed our mind! Now the output is the rack position x(t). The feedback to the diff amp must also change to measure position Since v(t) = dx(t) dt x(t) = Z t 0 v(t 0 )dt 0 we don t need to re compute the plant TF. X meas Position meter, unit gain x(t) X(s) = s V (s), where V (s) is the output velocity of the previous system that we have already analyzed. differential amplifier plant integrator V ref (s) K s+2 V (s) s X(s) Lecture 6 Monday, Oct. 5

4 Inserting an integrator Calculation of the closed loop TF Plant TF: X(s) V s (s) = s(s +2) Closed loop TF: X(s) V ref (s) = 0.362K s 2 +2s K X meas Position meter, unit gain x(t) Calculation of the steady state error Step input: V ref (s) = s x( ) =lim s 0 s s 0.362K s 2 +2s K =. differential amplifier new plant e( ) = x( ) =0. V ref (s) E(s) error signal K V s (s) s+2 s X(s) Lecture 6 Monday, Oct. 5

5 Generalizing: different system inputs Images removed due to copyright restrictions. Please see: Table 7. and Fig. 7. in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, Lecture 6 Monday, Oct. 5

6 Generalizing: steady-state error for arbitrary input Images removed due to copyright restrictions. Please see Fig. 7.2 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, Unit step input: Steady-state error = unit step output as t Ramp input: Steady-state error = ramp output as t Generally, the steady state error is defined as i h i e( ) = lim t hr(t) c(t) =lim s 0 s R(s) C(s), where the last equality follows from the final value theorem. Lecture 6 Monday, Oct. 5

7 Generalizing: steady-state error for arbitrary system, unity feedback reference signal error signal plant & controller output signal From the definition of the steady state error, h i e( ) =lim s 0 s R(s) C(s) = lim s 0 se(s). From the block diagram we can also see that C(s) E(s) = G(s) E(s) = C(s) G(s). Lecture 6 Monday, Oct. 5

8 Generalizing: steady-state error for arbitrary system, unity feedback reference signal error signal plant & controller output signal Recall the closed loop TF of the unity feedback system C(s) R(s) = G(s) +G(s) C(s) = R(s)G(s) +G(s). Substituting into the two formulae from the previous page, E(s) = R(s) +G(s) e( ) =lim s 0 se(s) = lim s 0 sr(s) +G(s). Lecture 6 Monday, Oct. 5

9 Steady-state error and static error constants e( ) =lim s 0 sr(s) +G(s). Image removed due to copyright restrictions. Please see: Table 7. in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, e( ) = lim s 0 s where s +G(s) = lim s 0 +G(s) = +lim s 0 G(s) +K p K p =lim s 0 G(s). e( ) = lim s 0 s 2 s +G(s) where = lim s 0 s+sg(s) = lim s 0 sg(s) K v =lim s 0 sg(s). +K v e( ) = lim s 0 s 3 s +G(s) where =lim s 0 s 2 +s 2 G(s) = lim s 0 s 2 G(s) +K a K a =lim s 0 s 2 G(s). Note: the system must be stable (i.e., all poles on left-hand side or at the origin) for these calculations to apply Lecture 6 Monday, Oct. 5

10 System types and steady-state errors Table removed due to copyright restrictions. Please see: Table 7.2 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, L [u(t)] = s K p = lim s 0 G(s) L [tu(t)] = s 2 K v = lim s 0 sg(s) L 2 t2 u(t) = s 3 K a = lim s 0 s 2 G(s) n =0 n = n =2 Type0 Type Type2 Lecture 6 Monday, Oct. 5

11 Disturbances disturbance From the I O relationship of the plant, C(s) =E(s)G (s)g 2 (s)+d(s)g 2 (s). Equivalent block diagram with D(s) as input and E(s) asoutput. From the summation element, E(s) =R(s) C(s). Substituting C(s) and solving for E(s), E(s) =R(s) +G (s)g 2 (s) D(s) G 2 (s) +G (s)g 2 (s) Lecture 6 Monday, Oct. 5

12 Disturbances disturbance e( ) =lim s 0 se(s) =lim s 0 s R(s) +G (s)g 2 (s) D(s) G 2 (s) e R ( )+e D ( ), +G (s)g 2 (s) where sr(s) e R ( ) =lim s 0 +G (s)g 2 (s) sg 2 (s)d(s) e D ( ) = lim s 0 +G (s)g 2 (s). Lecture 6 Monday, Oct. 5

13 Unit step disturbance disturbance Special case: unit step disturbance D(s) = s. e D ( ) sg 2 (s) (/s) = lim s 0 +G (s)g 2 (s) lim s 0 G 2 (s) = +lim s 0 G (s)g 2 (s) =. lim s 0 G 2 (s) +lim s 0G (s) If K, K 2 are the gains of the controller and plant, respectively, then e( ) if K or K 2. Lecture 6 Monday, Oct. 5

14 Unit step disturbance: example G (s) G 2 (s) V ref (s) K s(s+2) X(s) DC motor with rack pinion load, position feedback, subject to unit step disturbance D(s) = s. e D ( ) = = lim s 0 G 2 (s) + lim s 0G (s) lim s 0 s(s +2) lim s 0K = K. Lecture 6 Monday, Oct. 5