CHAPTER HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG. 7.1 SECOND-ORDER SYSTEM Transfer Function

Transcription

1 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 7. SECOND-ORDER SYSTEM Tranfer Funcion Thi ecion inroduce a baic yem called a econd-order yem or a quadraic lag. Second-order yem are decribed by a econd-order differenial equaion ha relae he oupu variable y o he inpu variable x (he forcing funcion) wih ime a he independen variable. A d y B dy Cy x() (7.) d d A econd-order yem can arie from wo fir-order yem in erie, a we aw in Chap. 6. Some yem are inherenly econd-order, and hey do no reul from a erie combinaion of wo fir-order yem. Inherenly econd-order yem are no exremely common in chemical engineering applicaion. Mo econd-order yem ha we encouner will reul from he addiion of a conroller o a fir-order proce. Le examine an inherenly econd-order yem and develop ome erminology ha will be ueful in our analyi of he conrol of chemical procee. Conider a imple manomeer a hown in Fig. 7. The preure on boh leg of he manomeer i iniially he ame. The lengh of he fluid column in he manomeer i L. A ime, a preure difference i impoed acro he leg of he manomeer. Auming he reuling flow in he manomeer o be laminar and he eady-ae fricion law for drag force in laminar flow o apply a each inan, we will deermine he ranfer funcion beween he applied preure difference P and he manomeer reading h. If we perform a momenum balance on he fluid in he manomeer, we arrive a he following erm: ( Sum of force cauing fluid omove) ( Raeof change of momenum of fluid) (7.) 37

2 38 PART LINEAR OPEN-LOOP SYSTEMS P = P = P P D Reference level = h h/ h/ L Before (Iniial) Afer (Final) FIGURE 7 Manomeer. where Sum of force Un cauing fluid o move balanced preure force cauing moion Fricional force oppoing moion Unbalanced preure force p D p cauing moion ( P P ) gh D r 4 4 Fricional force Skin oppoing moion fricion Shear re a wall a wall Area in conac wih wall Fricional force Wal oppoing moion l p DL 8mV 8m dh p DL p DL D D d ( ) ( ) ( ) The erm for he kin fricion a he wall i obained from he Hagen-Poieuille relaionhip for laminar flow (McCabe, Smih and Harrio, 4). Noe ha V i he average velociy of he fluid in he ube, which i alo he velociy of he inerface, which i equal o dh/ d (ee Fig. 7 ).

3 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 39 P P V V h h/ h/ Reference level = FIGURE 7 Average velociy of he fluid in he manomeer. The rae of change of momenum of he fluid [he righ ide of Eq. (7.)] may be expreed a d ( Rae of change of momenum) ( ma velociy momenum correcion facor) d r p D 4 L ( b ) dv d ( ) r p D d h L b 4 d The momenum correcion facor b accoun for he fac ha he fluid ha a parabolic velociy profile in he ube, and he momenum mu be expreed a b m V for laminar flow (ee McCabe, Smih and Harrio, 4). The value of b for laminar flow i 4/3. Subiuing he appropriae erm ino Eq. (7.) produce he deired force balance equaion for he manomeer. r pd 4 4 d h L P P p 3 D p d ( ) gh D 8m dh r p DL 4 4 D d ( (7.3) Rearranging Eq. (7.3), we obain r pd 4 4 d h 8 L m 3 d + D dh p ( DL) gh D pd p r ( P P) d 4 4 and finally, dividing boh ide by r g ( p D /4), we arrive a he andard form for a econd-order yem.

4 4 PART LINEAR OPEN-LOOP SYSTEMS L d h 6mL dh P P P h 3g d rdgd rg rg (7.4) (A more deailed verion of he analyi of he manomeer can be found in Bird e al., 96). Noe ha a wih fir-order yem, andard form ha a coefficien of on he dependen variable erm, h in hi cae. Second-order yem are decribed by a econdorder differenial equaion. We may rewrie hi Eq. (7.4) in general erm a where dy dy z Y X() (7.5) d d L (7.6) 3g z 6mL rdg (7.7) X () P rg Solving for and z from Eq. (7.6) and (7.7) give and Y h (7.8) L 3g (7.9) z m 8 3 L rd g dimenionle (7.) By definiion, boh and z mu be poiive. The reaon for inroducing and z in he paricular form hown in Eq. (7.5) will become clear when we dicu he oluion of Eq. (7.5) for paricular forcing funcion X ( ). Equaion (7.5) i wrien in a andard form ha i widely ued in conrol heory. If he fluid column i moionle ( dy / d ) and locaed a i re poiion ( Y ) before he forcing funcion i applied, he Laplace ranform of Eq. (7.4) become From hi, he ranfer funcion follow: Y () zy () Y () X () (7.) Y () X () z (7.) The ranfer funcion given by Eq. (7.) i wrien in andard form, and we will how laer ha oher phyical yem can be repreened by a ranfer funcion having he

5 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 4 denominaor of z. All uch yem are defined a econd-order. Noe ha i require wo parameer, and z, o characerize he dynamic of a econd-order yem in conra o only one parameer for a fir-order yem. We now dicu he repone of a econd-order yem o ome of he common forcing funcion, namely, ep, impule, and inuoidal. Sep Repone If he forcing funcion i a uni-ep funcion, we have X () (7.3) In erm of he manomeer hown in Fig. 7, hi i equivalen o uddenly applying a preure difference [uch ha X ( ) P / r g ] acro he leg of he manomeer a ime. Superpoiion will enable u o deermine eaily he repone o a ep funcion of any oher magniude. Combining Eq. (7.3) wih he ranfer funcion of Eq. (7.) give Y () z (7.4) The quadraic erm in hi equaion may be facored ino wo linear erm ha conain he roo z a z b z z (7.5) (7.6) Equaion (7.4) can now be wrien / Y () ( a)( b) (7.7) The repone of he yem Y ( ) can be found by invering Eq. (7.7). The roo a and b will be real or complex depending on value of he parameer z. The naure of he roo will, in urn, affec he form of Y ( ). The problem may be divided ino he hree cae hown in Table 7.. Each cae will now be dicued. TABLE 7 Sep repone of a econd-order yem Cae y Naure of roo Decripion of repone I < Complex Underdamped or ocillaory II Real and equal Criically damped III > Real Overdamped or nonocillaory

6 4 PART LINEAR OPEN-LOOP SYSTEMS CASE I STEP RESPONSE FOR y <. For hi cae, he inverion of Eq. (7.7) yield he reul z / z Y () e in z an z z (7.8) To derive Eq. (7.8), ue i made of he echnique of Chap. 3. Since z <, Eq. (7.5) o (7.7) indicae a pair of complex conjugae roo in he lef half-plane and a roo a he origin. In erm of he ymbol of Fig. 3, he complex roo correpond o and * and he roo a he origin o 6. The reader hould realize ha in Eq. (7.8), he argumen of he ine funcion i in radian, a i he value of he invere angen erm. By referring o Table 3., we ee ha Y ( ) ha he form z / Y () C e C co z C3 in z (7.9) The conan C, C, and C 3 are found by parial fracion. The reuling equaion i hen pu in he form of Eq. (7.8) by applying he rigonomeric ideniy ued in Chap. 4, Eq. (4.6). The deail are lef a an exercie for he reader. I i eviden from Eq. (7.8) ha Y ( ) a. The naure of he repone can be underood mo clearly by ploing he oluion o Eq. (7.7) a hown in Fig. 7 3, where Y ( ) i ploed again he dimenionle variable / for everal value of z, including hoe above uniy, which will be conidered in he nex ecion. Noe ha for z < all he repone curve are ocillaory in naure and become le ocillaory a z i increaed. The lope a he origin in Fig. 7 3 i zero for all value of z. The repone of a econd-order yem for z < i aid o be underdamped. Wha i he phyical ignificance of an underdamped repone? Uing he manomeer a an example, if we ep-change he preure difference acro an underdamped manomeer, he liquid level in he wo leg will ocillae before abilizing. The ocillaion are characeriic of an underdamped repone. CASE II STEP RESPONSE FOR y. For hi cae, he repone i given by he expreion Y () e / (7.) Thi i derived a follow: Equaion (7.5) and (7.6) how ha he roo and are real and equal. By referring o Fig. 3 and Table 3., i i een ha Eq. (7.) i he correc form. The conan are obained, a uual, by parial fracion. The repone, which i ploed in Fig. 7 3, i nonocillaory. Thi condiion, z, i called criical damping and allow he mo rapid approach of he repone o Y wihou ocillaion.

7 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG Y()..8.6 ζ = ζ = FIGURE 7 3 Repone of a econd-order yem o a uni-ep forcing funcion. / CASE III STEP RESPONSE FOR z >. For hi cae, he inverion of Eq. (7.7) give he reul Y () z / z e coh z inh z z (7.) where he hyperbolic funcion are defined a e inh a e e e coh a The procedure for obaining Eq. (7.) i parallel o ha ued in he previou cae. The repone ha been ploed in Fig. 7 3 for everal value of z. Noice ha he repone i nonocillaory and become more luggih a z increae. Thi i known a an overdamped repone. A in previou cae, all curve evenually approach he line Y. a a a a

8 44 PART LINEAR OPEN-LOOP SYSTEMS Acually, he repone for z > i no new. We aw i previouly in he dicuion of he ep repone of a yem conaining wo fir-order yem in erie, for which he ranfer funcion i Y () X () ( )( ) (7.) Thi i rue for z > becaue he roo and are real, and he denominaor of Eq. (7.) may be facored ino wo real linear facor. Therefore, Eq. (7.) i equivalen o Eq. (7.) in hi cae. By comparing he linear facor of he denominaor of Eq. (7.) wih hoe of Eq. (7.), i follow ha z z ( ) (7.3) z z ( ) (7.4) Noe ha if, hen and z. The reader hould verify hee reul. Uing MATLAB/Simulink o Deermine he Sep Repone of he Manomeer Conider a manomeer a illuraed in Fig. 7. The manomeer i being ued o deermine he preure difference beween wo inrumen ap on an air line. The working fluid in he manomeer i waer. Deermine he repone of he manomeer o a ep change in preure acro he leg of he manomeer. Daa L cm g 98 cm/ m cp. g/ ( cm ) 3 r. g/cm P rg for cm for D. cm,. cm,.3 cm (Three Cae) for h eworking fluid, waer Soluion. From Eq. (7.4), we have he governing differenial equaion for he manomeer: L 3g d h 6mL dh P P P h d rdgd rg rg

9 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 45 In erm of ranformed deviaion variable, hi become Y () zy () Y () X () P P where Y h h and X rg rg 3L 8m and z g rd 3L g Le calculae he ime conan for he manomeer. L ( cm) g 3( 98 cm /) and he damping coefficien for he hree differen ube diameer z 8m 3L 8 [. g/ ( cm )] r 3 D g (. g/cm )( D ) 3( cm) ( 98 cm/ ). 443 D Diameer (cm) Clearly we have one underdamped yem ( z < ), one criically damped yem ( z ), and one overdamped yem ( z > ). One mehod of obaining he repone i o ubiue he value of and z ino Eq. (7.8), (7.), and (7.) and plo he reuling equaion, realizing ha he forcing funcion i ime a uni ep. Anoher way o obain he repone i o ue MATLAB and Simulink o obain he repone of he ranfer funcion Y/X o he forcing funcion inpu X. Y () X () z and X The hree neceary ranfer funcion are a follow: Diameer (cm) z z Tranfer funcion (coninued)

10 46 PART LINEAR OPEN-LOOP SYSTEMS The Simulink model for imulaing he ranfer funcion i hown in Fig. 7 4, and he repone i hown in Fig Overdamped manomeer Preure forcing funcion / Criically damped manomeer Scope Underdamped manomeer FIGURE 7 4 Simulink diagram for manomeer imulaion. Underdamped 8 Criically damped Overdamped Heigh Time FIGURE 7 5 Manomeer repone o ep inpu.

11 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 47 Subiuing he value for and z ino Eq. (7.8), (7.), and (7.), we ge Y () z z / e in z z an z 5..3e in 4. an.93.9 rad Y () e / Y () z e / coh z underdamped manomeer 369 e /. criical. 369 ly damped manomeer z inh z 99. { [ ]} e coh(9.54 ).4 inh(9.54 ) z overdamped manomeer Ploing hee repone give he ame reul a he Simulink model. On a pracical noe, noice ha increae wih he oal lengh of he fluid column and ha z increae wih he vicoiy of he fluid. If he damping coefficien z i mall (< <.), he repone of he manomeer o a change in preure can be very ocillaory, and i become difficul o obain accurae reading of he preure. To dampen he ocillaion, i i common pracice o place a rericion on he bend of he ube. Thi increae he drag force of he fluid and i equivalen o increaing m in he equaion for z. Such a rericion (a parially open valve) i called a nubber. Term Ued o Decribe an Underdamped Syem Of hee hree cae, he underdamped repone occur mo frequenly in conrol yem. Hence a number of erm are ued o decribe he underdamped repone quaniaively. Equaion for ome of hee erm are lied below for fuure reference. In general, he erm depend on z and/or. All hee equaion can be derived from he ime repone a given by Eq. (7.8); however, he mahemaical derivaion are lef o he reader a exercie.. Overhoo. Overhoo i a meaure of how much he repone exceed he ulimae value following a ep change and i expreed a he raio A/B in Fig The overhoo for a uni ep i relaed o z by he expreion Overhoo exp pz z (7.5) Thi relaion i ploed in Fig The overhoo increae for decreaing z.

12 48 PART LINEAR OPEN-LOOP SYSTEMS Period T. A C Repone ime limi Y() B T r Rie ime Repone ime FIGURE 7 6 Term ued o decribe an underdamped econd-order repone...8 f n f.6.4 Overhoo. Decay raio ζ FIGURE 7 7 Characeriic of a ep repone of an underdamped econd-order yem. Why are we concerned abou overhoo? Perhap he emperaure in our chemical reacor canno be allowed o exceed a pecified emperaure o proec he caaly from deacivaion, or if i a level conrol yem, we don wan he ank o overflow. If we know hee phyical limiaion, we can deermine allowable value of z and chooe our conrol yem parameer o be ure o ay wihin hoe limi.. Decay raio. The decay raio i defined a he raio of he ize of ucceive peak and i given by C/A in Fig The decay raio i relaed o z by he expreion pz Decay raio exp ( overhoo) z (7.6) which i ploed in Fig Noice ha larger z mean greaer damping, hence greaer decay.

13 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG Rie ime. Thi i he ime required for he repone o fir reach i ulimae value and i labeled r in Fig The reader can verify from Fig. 7 3 ha r increae wih increaing z. 4. Repone ime. Thi i he ime required for he repone o come wihin 5 percen of i ulimae value and remain here. The repone ime i indicaed in Fig The limi 5 percen are arbirary, and oher limi can be ued for defining a repone ime. 5. Period of ocillaion. From Eq. (7.8), he radian frequency (radian/ime) i he coefficien of in he ine erm; hu, radian frequency w z (7.7) Since he radian frequency w i relaed o he cyclical frequency f by w p f, i follow ha f T p z (7.8) where T i he period of ocillaion (ime/cycle). In erm of Fig. 7 6, T i he ime elaped beween peak. I i alo he ime elaped beween alernae croing of he line Y. 6. Naural period of ocillaion. If he damping i eliminaed [ B in Eq. (7.), or z ], he yem ocillae coninuouly wihou aenuaion in ampliude. Under hee naural or undamped condiion, he radian frequency i /, a hown by Eq. (7.7) when z. Thi frequency i referred o a he naural frequency w n : wn (7.9) The correponding naural cyclical frequency f n and period T n are relaed by he expreion fn T p (7.3) n Thu, ha he ignificance of he undamped period. From Eq. (7.8) and (7.3), he naural frequency i relaed o he acual frequency by he expreion f z f n which i ploed in Fig Noice ha for z <.5 he naural frequency i nearly he ame a he acual frequency. In ummary, i i eviden ha z i a meaure of he degree of damping, or he ocillaory characer, and i a meaure of he period, or peed, of he repone of a econd-order yem.

14 5 PART LINEAR OPEN-LOOP SYSTEMS Impule Repone If a uni impule d ( ) i applied o he econd-order yem, hen from Eq. (7.) and (3A.) he ranform of he repone i Y () z (7.3) A in he cae of he ep inpu, he naure of he repone o a uni impule will depend on wheher he roo of he denominaor of Eq. (7.3) are real or complex. The problem i again divided ino he hree cae hown in Table 7., and each i dicued below. CASE I IMPULSE RESPONSE FOR y <. The inverion of Eq. (7.3) for z < yield he reul Y () z z / e in z (7.3) which i ploed in Fig The lope a he origin in Fig. 7 8 i. for all value of z. A imple way o obain Eq. (7.3) from he ep repone of Eq. (7.8) i o ake he derivaive of Eq. (7.8) wih repec o (remember from App. 3A ha he derivaive of he uni-ep funcion i he impule funcion). Comparion of Eq. (7.4) and (7.3) how ha Y () Y () (7.33) impule ep ζ = τy(). = / FIGURE 7 8 Repone of a econd-order yem o a uni-impule forcing funcion.

15 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 5 The preence of on he righ ide of Eq. (7.33) implie differeniaion wih repec o in he ime repone. In oher word, he invere ranform of Eq. (7.3) i d Y () impule Y () d ( ) (7.34) Applicaion of Eq. (7.34) o Eq. (7.8) yield Eq. (7.3). Thi principle alo yield he reul for he nex wo cae. CASE II IMPULSE RESPONSE FOR y. For he criically damped cae, he repone i given by ep which i ploed in Fig Y () e / (7.35) CASE III IMPULSE RESPONSE FOR y >. For he overdamped cae, he repone i given by Y () z e z / inh z (7.36) which i alo ploed in Fig To ummarize, he impule repone curve of Fig. 7 8 how he ame general behavior a he ep repone curve of Fig However, he impule repone alway reurn o zero. Term uch a decay raio, period of ocillaion, ec., may alo be ued o decribe he impule repone. Many conrol yem exhibi ranien repone uch a hoe of Fig Sinuoidal Repone If he forcing funcion applied o he econd-order yem i inuoidal X ()Ainw hen i follow from Eq. (7.) and (4.3) ha Y () Aw ( w )( z ) (7.37) The inverion of Eq. (7.37) may be accomplihed by fir facoring he wo quadraic erm o give Aw / Y () ( jw )( jw)( a)( b) (7.38)

16 5 PART LINEAR OPEN-LOOP SYSTEMS ( Here a and b are he roo of he denominaor of he ranfer funcion and are given by Eq. (7.5) and (7.6). For he cae of an underdamped yem ( z < ), he roo of he denominaor of Eq. (7.38) are a pair of pure imaginary roo ( j w, j w ) conribued by he forcing funcion and a pair of complex roo z / j z /, z / j z / We may wrie he form of he repone Y ( ) by referring o Fig. 3 and Table 3.; hu z / Y () C C e cow inw C3 co z C4 in z (7.39) The conan are evaluaed by parial fracion. Noice in Eq. (7.39) ha a, only he fir wo erm do no become zero. Thee remaining erm are he ulimae periodic oluion; hu Y () Ccow Cinw (7.4) The reader hould verify ha Eq. (7.4) i alo rue for z. From hi lile effor, we ee already ha he repone of he econd-order yem o a inuoidal driving funcion i ulimaely inuoidal and ha he ame frequency a he driving funcion. If he conan C and C are evaluaed, we ge from Eq. (4.6) and (7.4) Y () A ( w) zw ( ) in ( w f) (7.4) where f zw an ( w ) By comparing Eq. (7.4) wih he forcing funcion i i een ha X () Ainw. The raio of he oupu ampliude o he inpu ampliude i oupu ampliude Ampliude raio inpu ampliude ( w ) zw ( ) I will be hown in Chap. 5 ha hi may be greaer or le han, depending upon he value of z and w. Thi i in direc conra o he inuoidal repone of he fir-order yem, where he raio of he oupu ampliude o he inpu ampliude i alway le han.. The oupu lag he inpu by phae angle f. f phae angle an zw ( w )

17 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 53 I can be een from Eq. (7.4), and will be hown in Chap. 5, ha f approache 8 aympoically a w increae. The phae lag of he fir-order yem, on he oher hand, can never exceed 9. Dicuion of oher characeriic of he inuoidal repone will be deferred unil Chap. 5. We now have a our dipoal coniderable informaion abou he dynamic behavior of he econd-order yem. I happen ha many conrol yem ha are no ruly econd-order exhibi ep repone very imilar o hoe of Fig Such yem are ofen characerized by econd-order equaion for approximae mahemaical analyi. Hence, he econd-order yem i quie imporan in conrol heory, and frequen ue will be made of he maerial in hi chaper. 7. TRANSPORTATION LAG A phenomenon ha i ofen preen in flow yem i he ranporaion lag. Synonym for hi erm are dead ime and diance velociy lag. A an example, conider he yem hown in Fig. 7 9, in which a liquid flow hrough an inulaed ube of uniform cro-ecional area A and lengh L a a x() q Cro-ecional area = A L FIGURE 7 9 Syem wih ranporaion lag. y() q conan volumeric flow rae q. The deniy r and he hea capaciy C are conan. The ube wall ha negligible hea capaciy, and he velociy profile i fla (plug flow). The emperaure x of he enering fluid varie wih ime, and i i deired o find he repone of he oule emperaure y ( ) in erm of a ranfer funcion. A uual, i i aumed ha he yem i iniially a eady ae; for hi yem, i i obviou ha he inle emperaure equal he oule emperaure; i.e., x y (7.4) If a ep change were made in x ( ) a, he change would no be deeced a he end of he ube unil laer, where i he ime required for he enering fluid o pa hrough he ube. Thi imple ep repone i hown in Fig. 7. If he variaion in x ( ) were ome arbirary funcion, a hown in Fig. 7, he repone y ( ) a he end of he pipe would be idenical wih x ( ) bu again delayed by x() x() y() τ y() τ τ (a) (b) FIGURE 7 Repone of ranporaion lag o variou inpu.

18 54 PART LINEAR OPEN-LOOP SYSTEMS uni of ime. The ranporaion lag parameer i imply he ime needed for a paricle of fluid o flow from he enrance of he ube o he exi, and i can be calculaed from he expreion volume of ube volumeric flow rae or AL q I can be een from Fig. 7 ha he relaionhip beween y ( ) and x ( ) i (7.43) y () x( ) (7.44) Subracing Eq. (7.4) from Eq. (7.44) and inroducing he deviaion variable X x x and Y y y give Y () X ( ) (7.45) If he Laplace ranform of X ( ) i X ( ), hen he Laplace ranform of X ( ) i e X ( ). Thi reul follow from he heorem on ranlaion of a funcion, which wa dicued in App. 3A. Equaion (7.45) become Y () e X () or Y () e X () (7.46) Therefore, he ranfer funcion of a ranporaion lag i e. The ranporaion lag i quie common in he chemical proce indurie where a fluid i ranpored hrough a pipe. We hall ee in a laer chaper ha he preence of a ranporaion lag in a conrol yem can make i much more difficul o conrol. In general, uch lag hould be avoided if poible by placing equipmen cloe ogeher. They can eldom be enirely eliminaed. APPROXIMATION OF TRANSPORT LAG. The ranpor lag i quie differen from he oher ranfer funcion (fir-order, econd-order, ec.) ha we have dicued in ha i i no a raional funcion (i.e., a raio of polynomial.) A hown in Chap. 3, a yem conaining a ranpor lag canno be analyzed for abiliy by he Rouh e. The ranpor lag can alo be difficul o imulae by compuer. For hee reaon, everal approximaion of ranpor lag ha are ueful in conrol calculaion are preened here. One approach o approximaing he ranpor lag i o wrie e a / e and o expre he denominaor a a Taylor erie; he reul i e e 3 3 / / 3!

19 CHAPTER 7 HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG 55 Keeping only he fir wo erm in he denominaor give e (7.47) Thi approximaion, which i imply a fir-order lag, i a crude approximaion of a ranpor lag. An improvemen can be made by expreing he ranpor lag a / e e / e Expanding numeraor and denominaor in a Taylor erie and keeping only erm of fir-order give e / / fir-order Padé (7.48) Thi expreion i alo known a a fir-order Padé approximaion. Anoher well-known approximaion for a ranpor lag i he econd-order Padé approximaion: e / / / / econd-order Padé (7.49) Equaion (7.48) i no merely he raio of wo Taylor erie; i ha been opimized o give a beer approximaion. Y () () (3) FIGURE 7 Sep repone o approximaion of he ranpor lag e : () ; () fir-order Padé; (3) econd-order Padé ; (4) e. (4) /τ The ep repone of he hree approximaion of ranpor lag preened here are hown in Fig. 7. The ep repone of e i alo hown for comparion. Noice ha he repone for he fir-order Padé approximaion drop o before riing exponenially oward. The repone for he econdorder Padé approximaion jump o and hen decend o below before reurning gradually back o. Alhough none of he approximaion for e i very accurae, he approximaion for e i more ueful when i i muliplied by everal fir-order or econd-order ranfer funcion. In hi cae, he oher ranfer funcion filer ou he high-frequency conen of he ignal paing hrough he ranpor lag, wih he reul ha he ranpor lag approximaion, when combined wih oher ranfer funcion, provide a aifacory reul in many cae. The accuracy of a ranpor lag can be evaluaed mo clearly in erm of frequency repone, a opic covered laer in hi book.