Last week: analysis of pinionrack w velocity feedback


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1 Last week: analysis of pinionrack 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 steadystate errors Definition for step, ramp, parabola inputs Steadystate error in unity feedback systems System type and static error constants The role of integrators Steadystate 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: steadystate 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: Steadystate error = unit step output as t Ramp input: Steadystate 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: steadystate 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: steadystate 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 Steadystate 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 lefthand side or at the origin) for these calculations to apply Lecture 6 Monday, Oct. 5
10 System types and steadystate 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
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