Time Response. First Order Systems. Time Constant, T c We call 1/a the time constant of the response. Chapter 4 Time Response

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1 Time Repoe Chapter 4 Time Repoe Itroductio The output repoe of a ytem i the um of two repoe: the forced repoe ad the atural repoe. Although may techique, uch a olvig a differetial equatio or takig the ivere Laplace traform, eable u to evaluate thi output repoe, thee techique are laboriou ad timecoumig. Productivity i aided by aalyi ad deig techique that yield reult i a miimum of time. If the techique i o rapid that we feel we derive the deired reult by ipectio, we ometime ue the attribute qualitative to decribe the method. The ue of pole ad zero ad their relatiohip to the time repoe of a ytem i uch a techique. Learig thi relatiohip give u a qualitative hadle o problem. The cocept of pole ad zero, fudametal to the aalyi ad deig of cotrol ytem, implifie the evaluatio of a ytem repoe. Firt Order Sytem A firt-order ytem without zero ca be decribed by the trafer fuctio how i Figure 4.4(a). If the iput i a uit tep, where R() = /, the Laplace traform of the tep repoe i C(), where a C( ) G( ) R( ) (4.5) a Takig the ivere traform, the tep repoe i give by at ct c f ( c ( e (4.6) where the iput pole at the origi geerated the forced repoe c f (=, ad the ytem pole at -a, a how i Figure 4.4(b), geerated the atural repoe c (=e -at. Equatio (4.6) i plotted i Figure 4.5. Let u examie the igificace of parameter a, the oly parameter eeded to decribe the traiet repoe. Whe t= /a, at e e 0.37 (4.7) t / a Or c t (4.8) t / a We ow ue Eq. (4.6), (4.7), ad (4.8) to defie three traiet repoe performace pecificatio. Time Cotat, T c We call /a the time cotat of the repoe.

2 T c a From Eq. (4.7), the time cotat ca be decribed a the time for e -at to decay to 37% of it iitial value. Alterately, from Eq. (4.8) the time cotat i the time it take for the tep repoe it fial value (ee Figure 4.5). FIGURE 4.5 Firt-order ytem repoe to a uit tep. The reciprocal of the time cotat ha the uit (/ecod), or frequecy Thu, we ca call the parameter a the expoetial frequecy. Sice the derivative of e -at i -a whe t=0, a i the iitial rate of chage of the expoetial at t=0. Thu the time cotat ca be coidered a traiet repoe pecificatio for a firt order ytem, ice it i related to the peed at which the ytem repod to a tep iput. The time cotat ca alo be evaluated from the pole plot (ee Figure 4.4(b)). Sice the pole of the trafer fuctio i at -a, we ca ay the pole i located at the reciprocal of the time cotat, ad the farther the pole from the imagiary axi, the fater the traiet repoe. Let u look at other traiet repoe pecificatio, uch a rie time, T r, ad ettlig time, T, a how i Figure 4.5. Rie Time, T r Rie time i defied a the time for the waveform to go from 0. to 0.9 of it fial value. Rie time i foud by olvig Eq. (4.6) for the differece i time at c(= 0.9 ad c(=0.. Hece, 0.9a.3 c( e t 0.9 a 0.a 0. c( e t 0. a T r (4.9) a a a

3 Settlig Time, T Settlig time i defied a the time for the repoe to reach, ad tay withi, % of it fial value. Lettig c(=0:98 i Eq. (4.6) ad olvig for time, t, we fid the ettlig time to be at e T (4.0) a Firt-Order Trafer Fuctio via Tetig Ofte it i ot poible or practical to obtai a ytem trafer fuctio aalytically. Perhap the ytem i cloed, ad the compoet part are ot eaily idetifiable. Sice the trafer fuctio i a repreetatio of the ytem from iput to output, the ytem tep repoe ca lead to a repreetatio eve though the ier cotructio i ot kow. With a tep iput, we ca meaure the time cotat ad the teady-tate value, from which the trafer fuctio ca be calculated. Coider a imple firt-order ytem, G()=K/(+a), whoe tep repoe i K K / a K / a C( ) G( ) R( ) (4.a) a a K at c( e (4.b) a FIGURE 4.6 Laboratory reult of a ytem tep repoe tet. If we ca idetify K ad a from laboratory tetig, we ca obtai the trafer fuctio of the ytem. For example, aume the uit tep repoe give i Figure 4.6. We determie that it ha the firtorder characteritic we have ee thu far, uch a o overhoot ad ozero iitial lope. From the repoe, we meaure the time cotat, that i, the time for the amplitude to reach 63% of it fial value.

4 Sice the fial value i about 0.7, the time cotat i evaluated where the curve reache =0.45, or about 0.3 ecod. Hece, a=/0.3=7.7. To fid K, we realize from Eq. (4.) that the forced repoe reache a teady-tate value of K/a=0.7. Subtitutig the value of a, we fid K=5.54. Thu, the trafer fuctio for the ytem i G()=5.54/(+7.7). It i iteretig to ote that the repoe of Figure 4.6 wa geerated uig the trafer fuctio G()=5/(+7). Secod Order Sytem To become familiar with the wide rage of repoe before formalizig our dicuio i the ext ectio, we take a look at umerical example of the ecod-order ytem repoe how i Figure 4.7. All example are derived from Figure 4.7(a), the geeral cae, which ha two fiite pole ad o zero. The term i the umerator i imply a cale or iput multiplyig factor that ca take o ay value without affectig the form of the derived reult. By aigig appropriate value to parameter a ad b, we ca how all poible ecod-order traiet repoe. The uit tep repoe the ca be foud uig C()=R()G(), where R()=/, followed by a partial-fractio expaio ad the ivere Laplace traform. Overdamped Repoe, Figure 4.7(b) For thi repoe, 9 9 C ( ) 9 9 ( 7.854)(.46 (4.) ) Thi fuctio ha a pole at the origi that come from the uit tep iput ad two real pole that come from the ytem. The iput pole at the origi geerate the cotat forced repoe; each of the two ytem pole o the real axi geerate a expoetial atural repoe whoe expoetial frequecy i equal to the pole locatio. Hece, the output iitially could have bee writte a 7.854t.46t c( K Ke K3e Thi repoe, how i Figure 4.7(b), i called overdamped (So amed becaue overdamped refer to a large amout of eergy aborptio i the ytem, which ihibit the traiet repoe from overhootig ad ocillatig about the teady-tate value for a tep iput. A the eergy aborptio i reduced, a overdamped ytem will become uderdamped ad exhibit overhoot.). We ee that the pole tell u the form of the repoe without the tediou calculatio of the ivere Laplace traform. Uderdamped Repoe, Figure 4.7 (c) For thi repoe, 9 9 C( ) 9 ( j 8)( j 8 (4.3) Where, c( K K K ) e K t K t 4e co co K4 K K3 8t K3 i 8t 8t K3 ; ta K

5 FIGURE 4.7 Secod-order ytem, pole plot, ad tep repoe. Thi fuctio ha a pole at the origi that come from the uit tep iput ad two complex pole that come from the ytem. We ow compare the repoe of the ecod-order ytem to the pole that geerated it. Firt we will compare the pole locatio to the time fuctio, ad the we will compare the pole locatio to the plot. From Figure 4.7(c), the pole that geerate the atural repoe are at j 8. Comparig thee value to c( i the ame figure, we ee that the real part of the pole matche the expoetial decay frequecy of the iuoid amplitude, while the imagiary part of the pole matche the frequecy of the iuoidal ocillatio.

6 Let u ow compare the pole locatio to the plot. Figure 4.8 how a geeral, damped iuoidal repoe for a ecod-order ytem. The traiet repoe coit of a expoetially decayig amplitude geerated by the real part of the ytem pole time a iuoidal waveform geerated by the imagiary part of the ytem pole. The time cotat of the expoetial decay i equal to the reciprocal of the real part of the ytem pole. The value of the imagiary part i the actual frequecy of the iuoid, a depicted i Figure 4.8. Thi iuoidal frequecy i give the ame damped frequecy of ocillatio, d. Fially, the teady-tate repoe (uit tep) wa geerated by the iput pole located at the origi. We call the type of repoe how i Figure 4.8 a uderdamped repoe, oe which approache a teadytate value via a traiet repoe that i a damped ocillatio. Example 4.: By ipectio, write the form of the tep repoe of the ytem i Figure 4.9. Solutio: Firt we determie that the form of the forced repoe i a tep. Next we fid the form of the atural repoe. Factorig the deomiator of the trafer fuctio i Figure 4.9, we fid the pole to be =-5j3.3. The real part, -5, i the expoetial frequecy for the dampig. It i alo the reciprocal of the time cotat of the decay of the ocillatio. The imagiary part, 3.3, i the radia frequecy for the iuoidal ocillatio. Uig our previou dicuio ad Figure 4.7(c) a a guide, we obtai 5t t c( K e K co3.3t K3 i3.3t K K4e co3.3t K3 Where, K K4 K K3 ; ta, ad c( i a cotat plu a expoetially K damped iuoid.

7 Udamped Repoe, Figure 4.7(d) 9 ( j3)( j3) Chapter 4 Time Repoe For thi repoe, 9 9 C( ) (4.4) Thi fuctio ha a pole at the origi that come from the uit tep iput ad two imagiary pole that come from the ytem. The iput pole at the origi geerate the cotat forced repoe, ad the two ytem pole o the imagiary axi at j3 geerate a iuoidal atural repoe whoe frequecy i equal to the locatio of the imagiary pole. Hece, the output ca be etimated a c( K K4 co3. 3t hi type of repoe, how i Figure 4.7(d), i called udamped. Note that the abece of a real part i the pole pair correpod to a expoetial that doe ot decay. Mathematically, the expoetial i e -0t =. Critically Damped Repoe, Figure 4.7 (e) For thi repoe, 9 9 C ( ) (4.5) 6 9 ( 3) Thi fuctio ha a pole at the origi that come from the uit tep iput ad two multiple real pole that come from the ytem. The iput pole at the origi geerate the cotat forced repoe, ad the two pole o the real axi at -3 geerate a atural repoe coitig of a expoetial ad a expoetial multiplied by time, where the expoetial frequecy i equal to the locatio of the real pole. Hece, the output ca be etimated a 3t 3t c( K Ke K3te Thi type of repoe, how i Figure 4.7(e), i called critically damped. Critically damped repoe are the fatet poible without the overhoot that i characteritic of the uderdamped repoe. We ow ummarize our obervatio. I thi ectio we defied the followig atural repoe ad foud their characteritic:. Overdamped repoe Pole: Two real at -, - Natural repoe: Two expoetial with time cotat equal to the reciprocal of the pole locatio, or t t c t K e ( ) K e. Uderdamped repoe Pole: Two complex at - d j d Natural repoe: Damped iuoid with a expoetial evelope whoe time cotat i equal to the reciprocal of the pole real part. The radia frequecy of the iuoid, the damped frequecy of ocillatio, i equal to the imagiary part of the pole, or t c( Ae d co dt

8 3. Udamped repoe Pole: Two imagiary at j Natural repoe: Udamped iuoid with radia frequecy equal to the imagiary part of the pole, or c( Aco t 4. Critically damped repoe Pole: Two real at - Natural repoe: Oe term i a expoetial whoe time cotat i equal to the reciprocal of the pole locatio. Aother term i the product of time, t, ad a expoetial with time cotat equal to the reciprocal of the pole locatio, or t t c t K e ( ) K te The tep repoe for the four cae of dampig dicued i thi ectio are uperimpoed i Figure 4.0. Notice that the critically damped cae i the diviio betwee the overdamped cae ad the uderdamped cae ad i the fatet repoe without overhoot. Exercie 4.3: For each of the followig trafer fuctio, write, by ipectio, the geeral form of the tep repoe: Trafer Fuctio Awer 400 6t G ( ) c( A Be co9.08t t.46t G ( ) c( A Be Ce t 5t G ( ) c( A Be Cte c( A B co5t G ( ) 65

9 4.5 The Geeral Secod-Order Sytem Now that we have become familiar with ecod-order ytem ad their repoe, we geeralize the dicuio ad etablih quatitative pecificatio defied i uch a way that the repoe of a ecod-order ytem ca be decribed to a deiger without the eed for ketchig the repoe. I thi ectio, we defie two phyically meaigful pecificatio for ecod-order ytem. Thee quatitie ca be ued to decribe the characteritic of the ecod-order traiet repoe jut a time cotat decribe the firt-order ytem repoe. The two quatitie are called atural frequecy ad dampig ratio. Let u formally defie them. Natural Frequecy, The atural frequecy of a ecod-order ytem i the frequecy of ocillatio of the ytem without dampig. For example, the frequecy of ocillatio of a erie RLC circuit with the reitace horted would be the atural frequecy. Dampig Ratio, Available defiitio for thi quatity i oe that compare the expoetial decay frequecy () of the evelope to the atural frequecy ( ). Thi ratio i cotat regardle of the time cale of the repoe. Alo, the reciprocal, which i proportioal to the ratio of the atural period to the expoetial time cotat, remai the ame regardle of the time bae. We defie the dampig ratio,, to be Expoetial decay frequecy, ( σd ) Natural period ( ) ξ Natural frequecy ( rad/), ( ω ) π Expoetial Time co tat The geeral ecod-order ytem ca be traformed to how the quatitie ad. Coider the geeral ytem b G( ) (4.6) a b Without dampig, the pole would be o the j-axi, ad the repoe would be a udamped iuoid. For the pole to be purely imagiary, a=0. Hece, b G( ) (4.7) b By defiitio, the atural frequecy,, i the frequecy of ocillatio of thi ytem. Sice the pole of thi ytem are o the j-axi at jb, b (4.8) Hece b (4.9) Now what i the term a i Eq. (4.6)? Aumig a uderdamped ytem, the complex pole have a real part,, equal to a/. The magitude of thi value i the the expoetial decay frequecy. Hece, Expoetial decay frequecy, ( σ) a / a ξ (4.0) Natural frequecy( rad/), ( ω ) from which a ξ (4.) Our geeral ecod-order trafer fuctio fially look like thi:

10 G( ) (4.) Now that we have defied ad v, let u relate thee quatitie to the pole locatio. Solvig for the pole of the trafer fuctio i Eq. (4.) yield, (4.4) FromEq. (4.4) we ee that the variou cae of ecod-order repoe are a fuctio of, they are ummarized i Figure 4.. Example 4.3: Give the trafer fuctio of Eq. (4.3), fid ad. 36 G ( ) Solutio: Here, a=4., b=36; b 36 6 ; a 4. a ξ i.e. ξ Comparig Eq. (4.3) to (4.), 36, from which =6. Alo, ξ 4.. Subtitutig the value of, ξ

11 Example 4.4: For each of the ytem how i Figure 4., fid the value of ad report the kid of repoe expected. Solutio: Firt match the form of thee ytem to the form how i Eq. (4.6) ad (4.). Sice a= ad =b, a ξ (4.5) b Uig the value of a ad b from each of the ytem of Figure 4., we fid =.55 for ytem (a), which i thu overdamped, ice > ; = for ytem (b), which i thu critically damped; ad = for ytem (c), which i thu uderdamped, ice <. Exercie 4.4: For each of the followig trafer fuctio, do the followig: () Fid the value of ad, () characterize the ature of the repoe. Awer Trafer Fuctio () () Nature of Repoe 400 G ( ) Uderdamped 900 G ( ) Overdamped 5 G ( ) Critically damped 65 G ( ) Udamped

12 Step Repoe of a Uderdamped Secod-Order Sytem Let u begi by fidig the tep repoe for the geeral ecod-order ytem of Eq. (4.). The traform of the repoe, C(), i the traform of the iput time the trafer fuctio, or ) ( K K K C (4.6) where it i aumed that < (the uderdamped cae). Expadig by partial fractio, uig the method decribed i Sectio., Cae 3, Eq. (4.6) yield ) ( C Takig the ivere Laplace traform, which i left a a exercie for the tudet, produce t t e t c t i co ) ( (4.8a) t e t c t co ) ( (4.8b) Where, ta A plot of thi repoe appear i Figure 4.3 for variou value of, plotted alog a time axi ormalized to the atural frequecy. We ow ee the relatiohip betwee the value of ad the type of repoe obtaied: The lower the value of, the more ocillatory the repoe. The atural frequecy i a time-axi cale factor ad doe ot affect the ature of the repoe other tha to cale it i time.

13 We have defied two parameter aociated with ecod-order ytem, ad. Other parameter aociated with the uderdamped repoe are rie time (T r ), peak time (T p ), percet overhoot (%OS), ad ettlig time (T ). Thee pecificatio are defied a follow (ee alo Figure 4.4):. Rie time, T r : The time required for the waveform to go from 0. of the fial value to 0.9 of the fial value.. Peak time, T P : The time required to reach the firt, or maximum, peak. 3. Percet overhoot, %OS: The amout that the waveform overhoot the teady-tate, or fial, value at the peak time, expreed a a percetage of the teady-tate value. 4. Settlig time, T : The time required for the traiet damped ocillatio to reach ad tay withi % of the teady-tate value. Evaluatio of Peak Time, T P T P i foud by differetiatig c( i Eq. (4.8) ad fidig the firt zero croig after t=0. Thi tak i implified by differetiatig i the frequecy domai by uig Item 7 of Table.. Aumig zero iitial coditio ad uig Eq. (4.6), we get L c ( C( ) (4.9) Completig quare i the deomiator, we have L c ( (4.30) Therefore, c t e t ( ) i t (4.3) Settig the derivative equal to zero yield

14 or t (4.3) t (4.33) Each value of yield the time for local maxima or miima. Lettig =0 yield t=0, the firt poit o the curve i Figure 4.4 that ha zero lope. The firt peak, which occur at the peak time, Tp, i foud by lettig = i Eq. (4.33): T P (4.34) Evaluatio of %OS From Figure 4.4 the percet overhoot, %OS, i give by cmax cfial % OS 00 (4.35) c fial The term c max i foud by evaluatig c( at the peak time, c(t P ).Uig Eq. (4.34) for T P ad ubtitutig ito Eq. (4.8) yield cmax c( TP ) e / co i / c max c( TP ) e (4.36) For the uit tep ued for Eq. (4.8), c fial (4.37) Subtitutig Eq. (4.36) ad (4.37) ito Eq. (4.35), we fially obtai / % OS e 00 (4.38) Notice that the percet overhoot i a fuctio oly of the dampig ratio,. Wherea Eq. (4.38) allow oe to fid %OS give, the ivere of the equatio allow oe to olve for give %OS. The ivere i give by l% OS /00 (4.38) l % OS /00 l % OS /00 l % OS /00 l % OS /00 l% OS /00 (4.39) l % OS /00

15 The derivatio of Eq. (4.39) i left a a exercie for the tudet. Equatio (4.38) (or, equivaletly, (4.39)) i plotted i Figure 4.5. Evaluatio of T I order to fid the ettlig time, we mut fid the time for which c( i Eq. (4.8) reache ad tay withi % of the teady-tate value, c fial. Uig our defiitio, the ettlig time i the time it take for the amplitude of the decayig iuoid i Eq. (4.8) to reach 0.0, or t e 0.0 (4.40) Thi equatio i a coervative etimate, ice we are aumig that co t at the ettlig time. Solvig Eq. (4.40) for t, the ettlig time i l 0.0 T (4.4) You ca verify that the umerator of Eq. (4.4) varie from 3.9 to 4.74 a varie from 0 to 0.9. Let u agree o a approximatio for the ettlig time that will be ued for all value of ; let it be 4 T (4.4) Evaluatio of T r A precie aalytical relatiohip betwee rie time ad dampig ratio,, caot be foud. However, uig a computer ad Eq. (4.8), the rie time ca be foud. We firt deigate t a the

16 ormalized time variable ad elect a value for. Uig the computer, we olve for the value of t that yield c(=0.9 ad c(=0.. Subtractig the two value of t yield the ormalized rie time, T r, for that value of. Cotiuig i like fahio with other value of, we obtai the reult plotted i Figure 4.6. Figure 4.6 ca be approximated by the followig polyomial: 3 T r (maximum error le tha /% for 0<<0:9), ad =0.5( T r ) ( T r ) + :504( T r )-.738 (maximum error le tha 5% for 0.<<0.9). The polyomial were obtaied uig MATLAB polyfit fuctio. FIGURE 4.6 Normalized rie time veru dampig ratio for a ecod-order uderdamped repoe. Example 4.5: Give the trafer fuctio 400 G ( ) 400 fid T p, %OS, T, ad T r. Solutio: ad are calculated a 0 ad 0.75, repectively. Now ubtitute ad ito Eq. (4.34), (4.38), ad (4.4) ad fid, repectively, that T P = ecod, %OS =.838, ad T =0.533 ecod. Uig the table i Figure 4.6, the ormalized rie time i approximately.3 ecod. Dividig by yield T r =0.3 ecod.

17 Evolutio of Peak Time, Percet Over-hoot, ad Settlig Time Relate to the Locatio of the Pole The pole plot for a geeral, uderdamped ecod-order ytem, i how i Figure 4.7. We ee from the Pythagorea theorem that the radial ditace from the origi to the pole i the atural frequecy,, ad the co=. Figure 4.7 Pole plot for a uderdamped ecod-order ytem. Now, comparig Eq. (4.34) ad (4.4) with the pole locatio, we evaluate peak time ad ettlig time i term of the pole locatio. Thu, T P (4.44) d 4 4 T (4.45) d where d i the imagiary part of the pole ad i called the damped frequecy of ocillatio,ad d i the magitude of the real part of the pole ad i the expoetial dampig frequecy. Equatio (4.44) how that T P i iverely proportioal to the imagiary part of the pole. Sice horizotal lie o the -plae are lie of cotat imagiary value, they are alo lie of cotat peak time. Similarly, Eq. (4.45) tell u that ettlig time i iverely proportioal to the real part of the pole. Sice vertical lie o the -plae are lie of cotat real value, they are alo lie of cotat ettlig time. Fially, ice =co, radial lie are lie of cotat. Sice percet overhoot i oly a fuctio of, radial lie are thu lie of cotat percet overhoot, %OS. Thee cocept are depicted i Figure 4.8, where lie of cotat T P, T, ad %OS are labeled o the -plae. Depicted i Figure 4.9(a) are the tep repoe a the pole are moved i a vertical directio, keepig the real part the ame. A the pole move i a vertical directio, the frequecy icreae, but the evelope remai the ame ice the real part of the pole i ot chagig. The figure how a cotat expoetial evelope, eve though the iuoidal repoe i chagig frequecy. Sice all curve fit uder the ame expoetial decay curve, the ettlig time i virtually the ame for all waveform. A overhoot icreae, the rie time decreae. Let u move the pole to the right or left. Sice the imagiary part i ow cotat, movemet of the pole yield the repoe of Figure 4.9(b). Here the frequecy i cotat over the rage of variatio of the real part. A the pole move to the left, the repoe damp out more rapidly, while the frequecy remai the ame. Notice that the peak time i the ame for all waveform becaue the imagiary part remai the ame.

18 FIGURE 4.8 Lie of cotat peak time, T P, ettlig time, T, ad percet overhoot, %OS. Here: T <T ; T P <T P ; %OS < %OS. Figure 4.9 Step repoe of ecod-order uderdamped ytem a pole move: (a) with cotat real part; (b) with cotat imagiary part; (c) with cotat dampig ratio.

19 Movig the pole alog a cotat radial lie yield the repoe how i Figure 4.9(c). Here the percet overhoot remai the ame. Notice alo that the repoe look exactly alike, except for their peed. The farther the pole are from the origi, the more rapid the repoe. Example 4.6: Give the pole plot how i Figure 4.0, fid ; ; T P ; %OS, ad T. Solutio: The dampig ratio i give by =co=co[arcta(7/3)]= The atural frequecy,, i the radial ditace from the origi to the pole, or =(7 +3 )=7.66. The peak time i T P ec (4.44) d The percet overhoot i / % OS e 00 6% (4.38) The approximate ettlig time i 4 4 T.333 ec 3 (4.45) Example 4.7: Give the ytem how i Figure 4., fid J ad D to yield 0% overhoot ad a ettlig time of ecod for a tep iput of torque T(. Solutio: Firt, the trafer fuctio for the ytem i / J G( ) (4.49) D K J J From the trafer fuctio, K J (4.50) Ad D J (4.5) But, from the problem tatemet, 4 T (4.5) Or hece

20 D 4 (4.53) J Alo, from Eq. (4.50) ad (4.5), 4 J (4.54) K From Eq. (4.39), a 0% overhoot implie = Therefore, from Eq. (4.54), J (4.55) K Hece, J K 0.05 (4.56) From the problem tatemet, K = 5 N-m/rad. Combiig thi value with Eq. (4.53) ad (4.56), D=.04 N-m-/rad, ad J = 0.6 kg-m. Exercie 4.5: Fid ; ; T ; T P ; T r, ad %OS for a ytem whoe trafer fuctio i 36 G ( ) 6 36 Solutio: Here, a = 6, b=36; b 36 9 ; a 6 a ξ i.e. ξ T 0.5 ec 0.49 (4.45) T P 0.8 ec (4.44) 9 (0.4) 3 Tr / ec / 0.4 / 0.4 % OS e 00 e % (4.38) Dampig frequecy: d 9 (0.4) Expoetial dampig frequecy: d rad

21 Sytem Repoe with Additioal Pole Let u ow look at the coditio that would have to exit i order to approximate the behavior of a three-pole ytem a that of a two-pole ytem. Coider a three-pole ytem with complex pole ad a third pole o the real axi. Aumig that the complex pole are at j ad the real pole i at - r, the tep repoe of the ytem ca be determied from a partial-fractio expaio. Thu, the output traform i A B( ) Cd D C( ) ( ) d r (4.57) Or, i the time domai, t t C( Au( e B co r dt C idt De (4.58) The compoet part of c( are how i Figure 4.3 for three cae of r. For Cae I, r = a r ad i ot much larger tha ; for Cae II, r = a r ad i much larger tha ; ad for Cae III, r =. Figure 4.3 Compoet repoe of a three-pole ytem: (a) pole plot; (b) compoet repoe: Nodomiat pole i ear domiat ecod-order pair (Cae I), far from the pair (Cae II), ad at ifiity (Cae III)

22 Let u direct our attetio to Eq. (4.58) ad Figure 4.3. If r >> (Cae II), the pure expoetial will die out much more rapidly tha the ecod-order uderdamped tep repoe. If the pure expoetial term decay to a iigificat value at the time of the firt overhoot, uch parameter a percet overhoot, ettlig time, ad peak time will be geerated by the ecod-order uderdamped tep repoe compoet. Thu, the total repoe will approach that of a pure ecod-order ytem (Cae III). If r i ot much greater tha (Cae I), the real pole traiet repoe will ot decay to iigificace at the peak time or ettlig time geerated by the ecod-order pair. I thi cae, the expoetial decay i igificat, ad the ytem caot be repreeted a a ecod-order ytem. The ext quetio i, How much farther from the domiat pole doe the third pole have to be for it effect o the ecod-order repoe to be egligible? The awer of coure deped o the accuracy for which you are lookig. However, thi book aume that the expoetial decay i egligible after five time cotat. Thu, if the real pole i five time farther to the left tha the domiat pole, we aume that the ytem i repreeted by it domiat ecod-order pair of pole. What about the magitude of the expoetial decay? Ca it be o large that it cotributio at the peak time i ot egligible? We ca how, through a partial-fractio expaio, that the reidue of the third pole, i a three-pole ytem with domiat ecod-order pole ad o zero, will actually decreae i magitude a the third pole i moved farther ito the left half-plae. Aume a tep repoe, C(), of a three-pole ytem: bc A B C D C( ) (4.59) a b c a b c where we aume that the odomiat pole i located at-c o the real axi ad that the teady-tate repoe approache uity. Evaluatig the cotat i the umerator of each term, ca c A ; B (4.60a) c b ca ca c a bc b C ; D (4.60b) c b ca c b ca A the odomiat pole approache ; or c, A ; B ; C a; D 0 (4.6) Thu, for thi example, D, the reidue of the odomiat pole ad it repoe, become zero a the odomiat pole approache ifiity. I thi cae, the cotrol ytem egieer ca ue the five time rule of thumb a a eceary but ot ufficiet coditio to icreae the cofidece i the ecod-order approximatio durig deig, but the imulate the completed deig. Example 4.8: Fid the tep repoe of each of the trafer fuctio how i Eq. (4.6) through (4.64) ad compare them T ( ) (4.6) 4.54 T ( ) (4.63) T 3( ) (4.64)

23 Solutio: The tep repoe, C i (), for the trafer fuctio, T i (), ca be foud by multiplyig the trafer fuctio by /, a tep iput, ad uig partial-fractio expaio followed by the ivere Laplace traform to fid the repoe, c i (. With the detail left a a exercie for the tudet, the reult are t C(.09e co 4.53t 3.8 (4.65) C( 0.9e 0t 3t.89e t t co 4.53t C(.4e 0.707e co 4.53t The three repoe are plotted i Figure 4.4. (4.66) (4.67) Figure 4.4 Step repoe of ytem T (), ytem T (), ad ytem T 3 (). Notice that c (, with it third pole at -0 ad farthet from the domiat pole, i the better approximatio of c (, the pure ecod-order ytem repoe; c 3 (, with a third pole cloe to the domiat pole, yield the mot error.

24 Sytem Repoe With Zero Now that we have ee the effect of a additioal pole, let u add a zero to the ecod-order ytem. We aw that the zero of a repoe affect the reidue, or amplitude, of a repoe compoet but do ot affect the ature of the repoe expoetial, damped iuoid, ad o o. Startig with a two-pole ytem with pole at (- j:88), we coecutively add zero at -3, -5, ad -0. The reult, ormalized to the teady-tate value, are plotted i Figure 4.5. We ca ee that the cloer the zero i to the domiat pole, the greater it effect o the traiet repoe. A the zero move away from the domiat pole, the repoe approache that of the two-pole ytem. Thi aalyi ca be reaoed via the partial-fractio expaio. If we aume a group of pole ad a zero far from the pole, the reidue of each pole will be affected the ame by the zero. Hece, the relative amplitude remai appreciably the ame. Figure 4.5 Effect of addig a zero to a two-pole ytem. A iteretig pheomeo occur if zero (a) i egative, placig the zero i the right half-plae. From Eq. (4.70) we ee that the derivative term, which i typically poitive iitially, will be of oppoite ig from the caled repoe term. Thu, if the derivative term, C(), i larger tha the caled repoe, ac(), the repoe will iitially follow the derivative i the oppoite directio from the caled repoe. The reult for a ecod-order ytem i how i Figure 4.6, where the ig of the iput wa revered to yield a poitive teady-tate value. Notice that the repoe begi to tur toward the egative directio eve though the fial value i poitive. A ytem that exhibit thi pheomeo i kow a a omiimum-phae ytem. Figure 4.6 Step repoe of a omiimum-phae ytem.

25 Effect of Noliearitie Upo Time Repoe Effect of Saturatio (a) (b) Figure 4.9 (a) Simulik block diagram, (b) Effect of amplifier aturatio o load agular velocity repoe.

26 Effect of Deadzoe (a) (b) Figure 4.30 (a) Simulik block diagram, (b) Effect of deadzoe o load agular diplacemet repoe; (figure cotiue).

27 Effect of backlah (a) (b) Figure 4.3 (a) Simulik block diagram, (b) Effect of backlah o load agular diplacemet repoe. Referece [] Norma S. Nie, Cotrol Sytem Egieerig, Sixth Editio, Joh Wiley ad So, Ic, 0.