Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie 4. ontext an iretion In Leon we performe a material balane on a mixing tank an erie a firt-orer ytem moel. We ue that moel to preit the open-loop proe behaior an it loe-loop behaior, uner feebak ontrol. In thi leon, we ompliate the proe, an fin that ome aitional analyi tool will be ueful. DYNAMI SYSTEM BEHAVIOR 4. math moel of ontinuou blening tank We onier two tank in erie with ingle inlet an outlet tream.,, A olume V, A olume V Our omponent A ma balane i written oer eah tank. V V A A A A A (4.-) A in Leon, we hae reognize that eah tank operate in oerflow: the olume i ontant, o that hange in the inlet flow are quikly upliate in the outlet flow. Hene all tream are written in term of a ingle olumetri flow. Again, we will regar the flow a ontant in time. Alo, eah tank i well mixe. Putting (4.-) into tanar form A A A A A (4.-) reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie we ientify two firt-orer ynami ytem ouple through the ompoition of the intermeiate tream, A. If we iew the tank a eparate ytem, we ee that A i the repone ariable of the firt tank an the input to the eon. If intea we iew the pair of tank a a ingle ytem, A beome an intermeiate ariable. The pee of repone epen on two time ontant, whih (a before) are equal to the ratio of olume for eah tank an the ommon olumetri flow. We write (4.-) at a teay referene onition to fin A,r,r A,r A,r (4.-) We ubtrat the referene onition from (4.-) an thu expre the ariable in eiation form. A A A A A (4.-4) 4. oling the ouple equation - a eon-orer ytem A uual, we will take the initial onition to be zero (repone ariable at their referene onition). We may ole (4.-4) in two way: Beaue the firt equation ontain only A, we may integrate it iretly to fin A a a funtion of the input. Thi olution beome the foring funtion in the eon equation, whih may be integrate iretly to fin A. That i A t t t e e (4.-) t t t t t e e e t A e (4.-) On efining a peifi iturbane we an integrate (4.-) to a olution. Alternatiely, we may eliminate the intermeiate ariable A between the equation (4.-4) an obtain a eon-orer equation for A a a funtion of. The tep are () ifferentiate the eon equation () ole the firt equation for the eriatie of A reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie The reult i () ole the original eon equation for A (4) ubtitute in the equation of the firt tep. A A ( ) A (4.-) Two ma torage element le to two firt-orer equation, whih hae ombine to proue a ingle eon-orer equation. A homogeneou olution to (4.-) an be foun iretly, but the partiular olution epen on the nature of the iturbane: A t t e A e A A,part ( ) (4.-4) where the ontant A an A are foun by inoking initial onition after the partiular olution i etermine. 4. repone of ytem to tep iturbane Suppoe a tep hange Δ our in the inlet onentration at time t. Either (4.-) or (4.-4) yiel (t t ) (t t ) U t t e e A ( ) Δ Eah tank ontribute a firt-orer repone bae on it own time ontant. Howeer, thee repone are weighte by fator that epen on both time ontant. The reult in igure 4.- look omewhat ifferent from the firt-orer repone we hae een. We hae plotte the tep repone of a eonorer ytem with an.5 in arbitrary unit. At uffiiently long time, the initial onition ha no influene an the outlet onentration will beome equal to the new inlet onentration; in thi repet it look like the firt-orer ytem repone. Howeer, the initial behaior iffer: the outlet onentration rie graually intea of abruptly. Thi S-hape ure, often alle igmoi, i a feature of ytem of orer greater than one. Phyially, we an unertan thi by realizing that the hange in inlet onentration mut prea through two tank, an it reahe the eon tank only after being ilute in the firt. (4.-) reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie * /.5 4 5 6.8 * A/.6.4. 4 5 t igure 4.-: Repone to tep hange in inlet ompoition 6 4.4 introuing the Laplae tranform We bother with the Laplae tranform for two reaon: after the initial learning pain, it atually make the math eaier, o we will ue it in eriation ome of the terminology in linear ytem an proe ontrol i bae on formulating the equation with Laplae tranform. Definition: the Laplae tranform turn a funtion of time y(t) into a funtion of the omplex ariable. Variable ha imenion of reiproal time. All the information ontaine in the time-omain funtion i preere in the Laplae omain. y () L{ y(t) } y(t)e t (4.4-) (In thee note, we ue the notation y() merely to iniate that y(t) ha been tranforme; we o not mean that y() ha the ame funtional epenene on that it oe on t.) untional tranform: textbook (for example, Marlin, Se. 4.) uually inlue table of tranform pair, o thee eriation from efinition (4.4-) are primarily to emontrate how the table ame to be. reie 6 Mar 6 4
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie L { U(t t )} at { } L e e e a a U(t t t e t e t e at e t t t ( a)t )e t (4.4-) (4.4-) Operational tranform: thi allow u to tranform entire equation, not jut partiular funtion. f (t) L L f (t) e f (t)e f (t)e t t t { f (t)} f () t ( fe ) f (t)e t (4.4-4) L t f ( ξ) ξ f ( ξ) t f L L t () t e f ( ξ) { f () t } f ( ξ) { f () t } ξ e t t t ξ e t ξ e t (4.4-5) reie 6 Mar 6 5
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie Inerting tranform: ue the table to inert imple Laplae-omain funtion to their time-omain equialent. To implify the polynomial funtion often foun in ontrol engineering we may ue partial fration expanion. The ompliate ratio in (4.4-6) an be inerte if i expane into a erie of impler fration. N() N()...... D() n m n n m ( ) ( )... ( ) ( ) ( ) (4.4-6) In (4.4-6), an are repeate root of the enominator. The inere tranform of eah term will inole an exponential funtion of the root i. f () ( ) n n t t e f (t) (n )! (4.4-7) Variety in the alue of the oeffiient i ome from the numerator funtion N(). how to write the expanion Arrange the enominator o that the oeffiient of eah i. If there are no repeate root, eah root appear in one term. N() ( )( )( ) ( ) ( ) ( ) (4.4-8) If a root i repeate, it require a term for eah repetition. N() ( )( ) ( ) ( ) ( ) (4.4-9) Some root may appear a omplex onjugate pair, o that, for example a jb a jb (4.4-) where j i the quare root of -. how to ole for the oeffiient - it only algebra ) or eah of the real, itint root, multiply the expanion by eah RH enominator an ubtitute the alue of the root for to iolate the oeffiient. Thi alo work for the highet power of a repeate root. ) With ome oeffiient etermine, it may be eaiet to ubtitute arbitrary alue for to get equation in the unknown oeffiient. reie 6 Mar 6 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie ) or repeate root, either (a) multiply the expanion by an take the limit a. Howeer, thi will not iolate oeffiient aoiate with repeate omplex root. (b) multiply the expanion by the RH enominator of highet power. Differentiate thi equation with repet to, an ubtitute the alue of the root for. ontinue ifferentiating in thi manner to iolate ueie oeffiient. 4) or omplex root, oling for one oeffiient i enough. The other oeffiient will be the omplex onjugate. 4.5 oling linear ODE with Laplae tranform We return to (4.-4), the two equation that eribe onentration in the tank. A A A A A (4.-4) We perform the Laplae tranform on the entire firt equation. It itribute aro aition, an ontant may be fatore out. L L A A A { } { } { A L } L L We next perform an operational tranform on the eriatie. Beaue the funtional form of the ariable A an are not yet known, we imply iniate a ariable in the Laplae omain. { () ()} A A () A A () A () () () (4.5-) (4.5-) We an eaily ole (4.5-) for A. If we imilarly treat the eon equation in (4.-4), we arrie at the equialent formulation in the Laplae omain. reie 6 Mar 6 7
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie A A () () A () () Equation (4.5-) are not olution - we hae not ole anything! We merely hae a new formulation of problem (4.-4), a formulation that i more abtrat (what on earth i Laplae omain?) an yet impler, by irtue of being algebrai. It i important to remember that all the information hel in the ifferential equation (4.-4) i preere in the Laplae omain formulation (4.5-). We proee towar olution by eliminating the intermeiate ariable in (4.5-). We fin A (4.5-) () () (4.5-4) With (4.5-4) we hae gone a far a we an without knowing more about the iturbane. That i, we annot inert the right-han ie of (4.5-4) until we an atually ubtitute a funtional tranform for the ariable (). In thi ene, (4.5-4) reemble (4.-) an (4.-4): a olution neeing more peifiation. A in Setion 4., uppoe a tep hange Δ our in the inlet onentration at time t. ( t t ) Δ (t) U (4.5-5) We mut take the Laplae tranform, () Δ e t (4.5-6) whih we may ubtitute into (4.5-4). A () Δ t e (4.5-7) Thi IS the olution, the tep repone of the two tank in erie. Of oure, it really mut be inerte to the time omain. We treat the polynomial enominator from either the table or partial fration expanion: reie 6 Mar 6 8
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie L e t e t (4.5-8) Next we apply the time elay (t t ) (t t ) t L e U( t t ) e e (4.5-9) Remembering the ontant fator, we omplete the inere tranform of (4.5-7). (t t ) (t t ) A U(t t ) Δ e e (4.5-) whih, of oure, i iential to (4.-), erie in the time omain. 4.6 eribing ytem with tranfer funtion In Setion 4. we erie a ytem moel to eribe tranient behaior in a tank-in-erie proe. Then in Setion 4.5 we ue Laplae tranform to ole it. Let u now o the ame proeure in the abtrat. Begin with the firt-orer lag, written in eiation ariable: y y x (t) y () (4.6-) After taking Laplae tranform, we relate input an output by an algebrai equation: y () x () (4.6-) The ratio in (4.6-) multiplie x () (the tranform of iturbane x (t)) an in the proe onert that ignal into y () (the tranform of the repone y (t)). We all thi ratio the tranfer funtion G(). G () y () (4.6-) x () G() ontain all the information about the ODE (4.6-). We houl from now reognize it, when we ee it, a a firt-orer lag. Shoul we want to know how the firt-orer lag behae in repone to ome iturbane, we tranform the iturbane, multiply it by the firt-orer lag tranfer funtion, an then take the inere tranform of the reult. Let u generalize (4.5-4), whih eribe two firt-orer lag in erie: reie 6 Mar 6 9
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie y () x () (4.6-4) The tranfer funtion for thi eon-orer ytem i a prout of two firtorer lag y () G ()G () G() x () (4.6-5) We hall onier a tranfer funtion to be a ompletely atifatory eription of a ynami ytem. We hall learn to notie it gain (longterm teay-tate relationhip between y an x ), time ontant, an pole (root of the enominator). Table 4.6-: harateriti of ytem we hae tuie type equation tranfer funtion t orer lag y y (t) x (t) n orer y y oerampe* ( ) y x (t) ( )( ) *why oerampe? There are other n orer form to be enountere. pole teay tate gain - - - - - - 4.7 eribing ytem with blok iagram The blok iagram i a graphial iplay of the ytem in the Laplae omain. x () G() y () It omprie blok an arrow, an thu reemble many other type of flow iagram. In our ue with ontrol ytem, howeer, the arrow repreent ignal, ariable that hange in time, whih are not neearily atual flow tream. The blok ontain the tranfer funtion, whih may be a imple a a unit onerion between x an y, or a eription of more ompliate ynami behaior. Remember that the tranfer funtion inorporate all the ynami information in the ytem equation. Thi iagram implie the Laplae omain relationhip y () G()x () (4.7-) The real alue of blok iagram i to repreent the flow of ignal among multiple blok. The Blok Diagram Rule (ee Marlin, Se.4.4): reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie exluion: only one input an output to a blok. input ytem output umming: two ignal may be umme at an expliit umming juntion. The algebrai ign i iniate at the juntion (if omitte, i preume to be poitie). x () x () G () G () y () - y () x () G () y () multiple aignment: a ingle ignal may fee it alue to multiple blok. Thi oe NOT iniate that the ignal i iie up among the blok. x () G () y () G () y () y () G () y () Blok iagram may be turne into equation by imple algebra. It i uually mot onenient to tart with an output an work bakwar by ubtitution. In the umming iagram y () G ()x G G () () () ( y() y() ) ( G ()x () ()x () G ()G ()x () G ()G ()x () G ) (4.7-) In the multiple aignment iagram y () G ()y () G y () G ()y () G ()G ()G ()x ()x () () (4.7-) reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie Similarly, equation may be turne into blok iagram. Sytem (4.7-) ha two input an thu require at leat blok. x () G () G () x () G () G () - y () Sytem (4.7-) ha two output for one input. Input x * i not plit it full alue i ent to eah of two blok. x () G () G () y () G () G () y () Thi pair of blok iagram i equialent to the pair from whih they were erie. A a further illutration, we apply the blok iagram rule to the two-tank ytem in (4.5-): () A() A() () A() OR 4.8 frequeny repone from the tranfer funtion In Setion.5 we erie the ytem repone to a ine input by integrating the ifferential equation. We learne that the frequeny repone - that i, the long-term oillation - oul be haraterize by it amplitue ratio an phae angle; thee quantitie were expree on a Boe plot. Alternatiely, we may erie the frequeny repone iretly from the tranfer funtion by ubtituting jω for, where j i the quare root of - an ω i the raian frequeny of the ine input. or the eon-orer tranfer funtion (4.6-5), reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie G(jω) ( ωj)( ωj) ( ωj)( ωj) ( ω )( ω ) ω ( ) ( ω )( ω ) ω The funtion G(jω), although perhap aunting to behol, i imply a omplex number. A uh, it ha real an imaginary part, an a magnitue an a phae angle. It turn out that the magnitue of G(jω) i the amplitue ratio of the frequeny repone. j (4.8-) G(jω) ( ω ) ω( ) ( ω )( ω ) 4 ω ( ) ω ( ω )( ω ) ω ω ( ) (4.8-) urthermore, the phae angle of G(jω) i the frequeny repone phae angle. G(jω) tan tan ( ω) ) ( G(jω) ) ( ) Im G(j Re ω ω (4.8-) In igure 4.8-, the Boe plot abia ha been normalize by the quare root of the prout of the time ontant. We ee that the amplitue ratio of a eon-orer ytem eline more wiftly than that of a firt-orer ytem: the lope of the high-frequeny aymptote i -. Unbalaning the time ontant further ereae the amplitue ratio. The phae angle an reah -8º. It i ymmetri about -9º. reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie. amplitue ratio/gain. phae angle (eg).... - 5-6 -9 - -5-8.. ω( ).5 (raian) igure 4.8-: Boe plot for oerampe eon orer ytem 4.9 tability of the two-tank ytem, an linear ytem in general The tep repone (4.5-) how no term that grow with time, o long a the time ontant are poitie. urthermore, the amplitue ratio in igure 4.8- i boune. Thu, a eon-orer oerampe ytem appear to be table to boune input. In pratial term, a onentration iturbane at the inlet houl not prooke a runaway repone at the outlet. We hae een firt- an eon-orer linear ytem. Let u generalize to arbitrary orer: y y y a n n n a n n a n y x (4.9-) The funtion x repreent all manner of boune iturbane, expree in eiation form. The ytem propertie, howeer, reie on the left-han ie of (4.9-), an it tability behaior houl be inepenent of the partiular nature of the iturbane x. Hene, we may examine the homogeneou equation reie 6 Mar 6 4
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie a n y n n y y a n a y (4.9-) n n The olution to (4.9-) i the um of n term, eah ontaining a fator e i t, where i i a real root, or the real part of a omplex root, of the harateriti equation. n n a n r a n r ar (4.9-) Hene if all the i are negatie, the olution annot grow with time an will thu be table. If we take the Laplae tranform of (4.9-), y () x () G() n a n a n n a (4.9-4) we ee that the enominator of the tranfer funtion i iential to the harateriti equation. Hene, the tability of the ytem i etermine by the pole of the tranfer funtion: pole with zero or poitie real part iniate a ytem untable to boune iturbane. Table 4.6- how that firt-orer lag an oerampe eon-orer ytem are table if their time ontant are poitie. ONTROL SHEME 4. tep - peify a ontrol objetie for the proe Our ontrol objetie i to maintain the outlet ompoition A at a ontant alue. 4. tep - aign ariable in the ynami ytem The ontrolle ariable i learly A. The inlet ompoition i a iturbane ariable. The ompoition A i an intermeiate ariable. We hae no aniate manipulate ariable; hene we eie to a a onentrate make-up tream a we i with the ingle tank in Leon. We oul a the tream to the firt or the eon tank - we eek aie: One aior ay, A it to the firt tank, where the iturbane enter. That way, the manipulation an interat thoroughly with the iturbane; it a matter of ue through ooperation. A eon aior ay, A it to the eon tank. That way the manipulate ariable an affet the ontrolle ariable more iretly, through one tank intea of two. Thi aie bring to min the ifferene we hae een between firt- an eon-orer repone. Yet the firt aior i better-ree, frienlier, an ha a omforting manner - we eie to a the make-up tream to the firt tank. reie 6 Mar 6 5
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie Material balane on the olute gie V V A A A A A A (4.-) We expet to ue a relatiely mall make-up flow of onentration A ; hene we make the approximation that ~. Hene V V A A A A A A (4.-) A i our utom, we write (4.-) at a teay referene onition an ubtrat thi referene to leae the ariable in eiation form. After taking Laplae tranform, we eliminate intermeiate ariable A() to fin A () A () () (4.-) ( )( ) ( )( ) The time ontant are the uual olume-to-flow ratio. The pole of the two tranfer funtion are negatie, o aing a make-up tream ha not mae the ytem untable. ompare (4.-) to (4.5-4), whih eribe the two tank without makeup flow. igure 4.-, the blok iagram of (4.-), emphaize that ontrolle ariable A i influene by both iturbane an manipulate ariable. () () ( )( ) ( )( ) A () A igure 4.-: Blok iagram of two-tank mixing proe 4. tep - proportional ontrol We will ue proportional ontrol. Although we reognize the iaantage of offet, a emontrate in Leon, we feel onfient in reie 6 Mar 6 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie our ability to manage it at an aeptable leel by inreaing the ontroller gain. The proportional algorithm i bia gain ( ) (4.-) A,etpt A Shoul the outlet ompoition fall below the et point, the error will beome poitie. A poitie ontroller gain gain will inreae make-up flow aboe the bia alue, whih will at to inreae the outlet ompoition. 4. tep 4 - hooe et point an limit A in Setion.6, we ientify any appliable afety limit, hooe the eire operating point, peify limit of tolerable ariation about it, an upply make-up flow in uffiient quantity to ounterat the antiipate iturbane. EQUIPMENT 4.4 omponent of the feebak ontrol loop We houl onier the equipment more realitially. igure 4.4- i a blok iagram howing the four omponent foun in mot proe ontrol appliation: proe, enor, ontroller, an final ontrol element. Eah blok ontain a tranfer funtion that relate output to input. The ignal between blok are Laplae tranform of eiation ariable. We reognize that the proe will typially omprie multiple blok for iturbane an manipulate ariable input. An example i the mixing tank proe that hown in igure 4.-. proe x () x m () G () G m () y () final ontrol element G () y o () ontroller ε () G () G () - enor y () y p,e () G p () y p () igure 4.4-: General blok iagram of feebak ontrol loop reie 6 Mar 6 7
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie The output ignal that we hae eignate a the ontrolle ariable y i meaure by a enor. In our mixing tank example, we nee ome meaurement that reliably iniate ompoition; epening on the nature of the olution, it might be a hromatograph, onutiity meter, petrometer, enitometer, et. The ontroller ubtrat the enor ignal y from the et point y p an exeute the ontroller algorithm G () on the error. The ubtration i ometime ai to take plae in the omparator. The ontroller output y o rie a final ontrol element to proue manipulate ariable x m. In hemial proe inutrie thi i mot often a ale in a pipe arrying ome flui tream, but it oul alo be a motor, a heater ontrol, et. 4.5 tranfer funtion for loop omponent Although we are not yet reay to are harware, we an improe our eription of equipment behaior by poing tranfer funtion for eah of the loe loop omponent iniate in igure 4.4-. Notie that a tranfer funtion ha two main part: teay an ynami. The teay part i the gain; gain iniate the magnitue of the effet of the tranfer funtion on the input ignal, an it perform the unit onerion between input an output. In (4.-), the tranfer funtion that onert make-up flow into outlet onentration A ha a gain of A /. The gain epen on thee two proe parameter, an the unit are hoen to be onitent with thoe of the input an output ignal. The other tranfer funtion in (4.-) ha a imenionle unity gain, inepenent of proe parameter. The ynami part i eerything ele - all the ytem time ontant an the funtion of the Laplae ariable. The ynami part haraterize the way that an input ignal i proee in time. In (4.-), both tranfer funtion feature eon-orer ynami. enor Let u preume that the enor i fat - really fat - o that negligible time elape between a hange in the ontrolle ariable y an it meaurement y. Then the tranfer funtion i y y (4.5-) Being really fat mean that the tranfer funtion ha NO ynami part. Suh a tranfer funtion iniate a pure gain proe, one in whih hange in the input are intantaneouly een in the output. The imenion of the gain are reie 6 Mar 6 8
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie ontroller _ in ( ) (4.5-) ontrolle ar iable where the enor i preume to elier the meaurement to the ontroller in uitable ontroller_in unit. ontroller We ee that a proportional ontroller i alo a pure gain proe between error ignal an ontroller repone. y o ε (4.5-) where the error i onentionally efine with the et point a poitie: ε yp y (4.5-4) or now we will leae the ontroller ignal imenion unpeifie. Howeer, we an be ure that the gain ha imenion of ontroller _ out ( ) (4.5-5) ontroller _ in et point The error ignal ha imenion uitable for the ontroller, whih implie that y p an y hae the ame unit. Howeer, the operator might prefer to hae the et point expree in the unit of the ontrolle ariable (oalle engineering unit), whih implie that y p,e an y hae the ame unit. The et point tranfer funtion perform thi unit onerion; it i a pure gain proe with gain iential to that of the enor. That i G p (4.5-6) final ontrol element The ale i a mehanial eie that take ome time to moe. We might imagine that a ale an hange more quikly than a large hemial proe eel, an thu that for many ontrol appliation the ale ynami an be neglete. In our ae, howeer, we will aume that the ale operate with firt-orer ynami, uh that x m yo (4.5-7) The ale gain ha imenion of reie 6 Mar 6 9
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie flow ( ) (4.5-8) ontroller _ out LOSED LOOP BEHAVIOR 4.6 aembling the omponent into a loe loop The loe loop blok iagram in igure 4.4- how how the loop omponent are arrange. The tranfer funtion for the ariou omponent are efine in Setion 4.5. We will now ue the blok iagram rule to erie the equation for the loe loop, in three tep: in general, goo for any appliation of igure 4.4- applying the hoie we mae in Setion 4.5 aapting the general nomenlature of Setion 4.5 to the two-tank mixing proe We begin with the ontrolle ariable, whih i the output of the loe loop ytem, an work bakwar through the iagram until all path are trae an the input appear. y () G x () G x G x () G G x () G G m m m G G ε () G m () ( G y () G y ()) p p,e (4.6-) At thi point, we ollet the ontrolle ariable on the left-han ie. y () y () ( G G G G ) m G G x () G x () G G G ( G G G G ) ( G G G G ) m G m m G G G m p p y p,e y () p,e () (4.6-) Equation (4.6-) how how a ontrolle ariable repon to a iturbane an et point input. It i erie from igure 4.4- an applie to any ytem that an be repreente by the figure. We now peialize (4.6-) with tranfer funtion we efine in Setion 4.5. Thee ue the general nomenlature of igure 4.4-, but epen on aumption we mae about fat enor an firt-orer ale. G m G y() x () yp,e() (4.6-) G m G m reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie Equation (4.6-) begin to how how the general tranfer funtion beome peifi funtion of the Laplae ariable. Now we further peialize (4.6-) to the two-tank problem by ubtituting the proe tranfer funtion an peifi nomenlature from (4.-) or, equialently, igure 4.-. ( )( ) ( )( ) ( )( ) ( )( ) () () () A,p A A A A (4.6-4) Beneath it apparent omplexity, (4.6-4) imply tell how the outlet onentration reat to iturbane an to a et point input. Thi loeloop repone i ompare in igure 4.6- to the open-loop repone of the proe. ( )( ) ( )( ) () () () A A ( )( ) ( )( ) ( )( ) ( )( ) () () () A,p A A A A tranfer funtion for iturbane other input ( )( ) ( )( ) () () () A A ( )( ) ( )( ) ( )( ) ( )( ) () () () A,p A A A A tranfer funtion for iturbane other input igure 4.6-: omparing open- an loe-loop repone It i lear that the iturbane repone ha a ifferent harater, beaue the tranfer funtion ha hange. I we mae a goo eiion on ontrol algorithm, an I we tune the ontroller properly, the loe-loop repone houl be better. 4.7 ome perpetie on how we erie the loe-loop repone Remember what we i: we propoe a blok iagram of feebak ontrol an erie the aoiate tranfer funtion between input an output. Then we ubtitute the omponent tranfer funtion appropriate to our partiular problem. Intea of the blok iagram algebra, we oul hae ombine the Laplae omain equation of Setion 4.5 iretly, eliminating intermeiate ariable until we arrie at (4.6-4). reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie urthermore, we oul hae proeee entirely in the time omain, a we i in Leon. That i, the eon-orer proe ODE oul hae been ombine with the firt-orer ale ODE an algebrai equation for enor an ontroller to arrie at an ODE for the ontrolle ariable with iturbane an et point foring funtion. We hae ue new tool - the Laplae tranform an the blok iagram - but the unerlying objetie, an the relationhip between input an output, were the ame a working in the time omain. Thi i not myteriou. 4.8 alulating loe-loop repone But how oe the loe-loop perform? We approah thi by implifying (4.6-4). A () A A A () A,p () (4.8-) learly we i not uee at that... A with (4.5-4) an (4.-), we woul like to ubtitute a partiular iturbane for () in (4.8-) an inert the reult to obtain the time repone of A. Here we enounter an obtale: our tranform table o not feature anything a ompliate a (4.8-). urthermore, to ue partial fration expanion we mut fin the root of the ubi equation; howeer, we are unlikely to fin an analytial expreion for thee. That i unfortunate, beaue it woul be helpful to know how the tranfer funtion pole epen on the ontroller gain. We reort to numerial metho. Here i our plan: o a partial-fration expanion of eah tranfer funtion in (4.8-) in term of the pole multiply eah term in the expanion by the iturbane of interet an inert to fin the repone for a partiular alue of ontroller gain, fin the pole numerially repeat for ifferent alue of to map out the behaior Thi expeient of uing numerial alulation oe not how u the funtional epenene of the repone on the parameter, but it oe get the job one. reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie We will begin with the iturbane tranfer funtion in (4.8-). The polynomial ratio part of the tranfer funtion i written a the um of fration. ( )( )( ) A (4.8-) The pole of the tranfer funtion are i, an the oeffiient i epen on thee pole, a well a the numerator. We will keep in min that both i, an i epen on the ytem time ontant an gain, a well a the ontroller gain. Soling for the oeffiient i an algebra problem. The reult are ( )( ) ( )( ) ( )( ) (4.8-) Thu numerial alue of oeffiient i an be ompute for eah et of pole i. Now we ue expanion (4.8-) to rewrite (4.8-) for a iturbane repone. With no hange in et point, A,p () i ientially zero. () () () A (4.8-4) Now, a an example, we poe a tep iturbane in the inlet ompoition. reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie (t) ΔU(t t Δ () e t ) (4.8-5) We ubtitute the tep iturbane (4.8-5) into the ytem moel (4.8-4) to obtain A () Δ t e (4.8-6) Eah term inert to a tep repone. Remembering to apply the time elay, we fin the reult ΔU(t t ) A(t) ( e ) ( e ) ( e (t t ) (t t ) (t t ) Examining (4.8-7) we learn that the exponential term ontribute aoring to the magnitue of the pole i : mall pole (larger time ontant) aue the term to perit. We ee that there will be offet, beaue A oe not go to zero at long time. The amount of offet will epen on the magnitue of the oeffiient i ; our experiene in Leon woul ugget that thee will beome maller a the ontroller gain inreae. 4.9 alulating the repone for a partiular example We begin with imilar parameter alue to our example in Leon :. m min -,r 6 - m min - V 6 m (thu 5 min) V 4 m (thu. min),r 8 kg m - Ao,r kg m - A 4 kg m -. min. m min - ontroller_out -.5 ontroller_in m kg - We may alulate root in (4.8-) with alulator, preaheet, or omputer oe. or example, uing matlab we obtain root of polynomial 4 by >> root ([ 4]) ) (4.8-7) reie 6 Mar 6 4
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie an -.5.9i -.5 -.9i Table 4.9-: pole of loe loop iturbane tranfer funtion (out in - ) (min - ) (min - ) (min - ) - -. -.. -. -.4 -.856.4 -. -.47 -.56 Notie that zero ontroller gain lea to pole equal to the negatie inere of the three ytem time ontant. Thu our loe-loop tranfer funtion reue to eribe the behaior of the proe alone, uner open-loop onition. After uing the pole in Table 4.9- to ompute the olution (4.8-7) we obtain a plot of the repone behaior. Inee ontroller gain an be inreae to reue the effet of the input iturbane. reie 6 Mar 6 5
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie inlet omp eiation (kg m - ) make-up eiation ( -6 m - ).5.5 -.5. -.5 -. -.5 -. -.5..4 4 4. outlet ompoition eiation (kg m - )..8.6.4.. time (min) 4 igure 4.9-: Step repone of two tank uner proportional ontrol 4. urprie - inreaing gain introue oillation! We hae been ery tentatie with the gain etting, o we at more aggreiely to uppre offet. The pole beome omplex! reie 6 Mar 6 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie Table 4.-: omplex pole of tranfer funtion (out in - ) (min - ) (min - ) (min - ). -. -.49 j.88 -.49 - j.88 -. -.98 j. -.98 - j. 5 -.48 -. j. -. - j. We mut moify (4.8-7) to aommoate omplex number. The root an are omplex onjugate, an (realling our iuion of partial fration expanion) o are oeffiient an. We efine four new real quantitie to replae the omplex one: a jb A jb a jb A jb (4.-) We ubtitute thee efinition into (4.8-7), realling Euler relation, to obtain ( a ) e jb t at e ( o bt jin bt ) (4.-) A ΔU(t t ) (t) (t t ) a(t t ) ( e ) A e ( A o b(t t ) Bin b(t t )) (4.-) Parameter A, B, a, an b are not ariable with time, in the ene of A, but they o epen on the alue of the ontroller gain. Parameter a an b are foun (ia the root-fining proeure) in Table 4.-. A an B ome ia omplex algebra from (4.8-). b A b B ( a b ) b ab [ ] ( a b )( a) b ( a b )( a) a( a) b( a b )( a) b [ ] b (4.-4) Taking ata from Table 4.- an uing (4.-4), we an plot repone (4.-). reie 6 Mar 6 7
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie inlet omp eiation (kg m - ) make-up flow eiation ( -5 m - ).5.5 -.5 - -4-6 -8 - - -4 4 outlet ompoition eiation (kg m - ).9.8.7.6.5.4.... 5 4 time (min) igure 4.-: Oillatory tep repone. or the ingle tank of Leon, the loe loop behaior wa qualitatiely the ame a that of the proe itelf. Here, howeer, loing the loop ha introue behaior we woul NOT ee in the proe alone: the repone ariable oillate in repone to a teay input. The key i the thir-orer harateriti equation, whih an amit omplex root. The tranfer funtion i thir orer beaue the eon-orer proe wa plae in a feebak loop with a firt-orer ale. If the ytem mathemati proie reie 6 Mar 6 8
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie a true repreentation of the proe equipment, we will ee oillation in operation. 4. loe-loop tability by the Boe riterion igure 4.- how u that, epite the onet of oillation, inreaing gain uppree the offet. Howeer, we mut till be minful of the poibility of intability. In igure 4.- we plot the pole on a omplex plane. We ee that inreaing gain bring the loe loop to the point of oillation an then inreae the imaginary (oillatory) omponent. We obere alo that the real omponent approahe zero. Realling that tability epen on the real part of the pole being negatie, we ee that the loe loop will beome untable at a ontroller gain between 5 an 6. 4.. imaginary (min - )... -. arrow how inreaing from to -. -. -4. - - -8-6 -4 - real (min - ) igure 4.-: Root lou plot Thi root lou plot proie a map of the tability limit; thi i partiularly helpful, beaue we were unable to erie a ingle expreion that howe the effet of ontroller gain on the real part of the pole. We might imagine that fining the pole will only beome more iffiult a we onier more ompliate proee an ontroller. Hene we introue an alternatie mean of preiting tability: the Boe riterion. We eelop the riterion intuitiely; realling igure 4.4-, we begin by realizing that intability in a preiouly table ytem happen beaue of feebak in a loe loop. Suppoe that the output ignal y ontain ome flutuating omponent at a partiular frequeny ω. Thi omponent i inerte in the omparator by being ubtrate from the et point, an i reie 6 Mar 6 9
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie fe to the ontroller. If the loop (proe, ontroller, enor, ale, et.) ontribute a phae elay of -8 at frequeny ω, the inerte ignal return to the output in phae to reinfore the flutuation. If in aition the amplitue ratio at ω i greater than one, the flutuation will grow. We are not ontriing a irumtane here. In any realiti proe there will be mall iturbane, flutuation oer a wie omain of frequeny. The loop proee thee flutuation aoring to it frequeny repone harateriti. Depening on the amplitue ratio, ignal at ome frequenie may be amplifie. Aurely, ignal at high frequenie will be elaye by at leat 8. If thee onition oerlap, there will be an oillating ignal that will grow. We nee not upply it; the ytem will elet it from the petrum of bakgroun noie. Thi intuitie eelopment i eribe in more etail by Marlin (Se..6). To turn our intuition into a metho, we return to the general eription of the loe loop tranfer funtion in (4.6-). Reall that the pole are the root of the harateriti equation, whih i the enominator of the tranfer funtion. Thi harateriti equation i alway of the form plu the prout of the tranfer funtion aroun the loop. or oneniene, we will all thi prout the loop tranfer funtion G L. harateriti equation G G G G L G m (4.-) It i the amplitue ratio an phae angle, that i, the frequeny repone, of G L that etermine whether ignal will grow in the loop. irt we fin ω, the frequeny at whih the phae elay i -8. At thi rooer frequeny, we inpet the amplitue ratio; if it i le than one, the ytem will attenuate reinfore iturbane, an thu be table. Thu the Boe riterion ealuate the tability of ( G L ) - from the frequeny repone of G L. or our proe, (4.6-4) gie G L () A ( )( ) (4.-) The phae angle of G L i the um of the phae angle of the ariou element in (4.-). reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie G L (jω) A tan ( ω ) tan ( ω ) tan ( ω ) Equation (4.-) may be ole for the rooer frequeny ω ; that i, the frequeny at whih the loop elay the ignal by -8º. (4.-) ( ω ) tan ( ω ) tan ( ω ) 8 tan (4.-4) The amplitue ratio of G L i the prout of the amplitue ratio of the ariou element in (4.-). G L (jω) A A ω ω ω (4.-5) We are partiularly interete in the amplitue ratio at the rooer frequeny, R A. R A A (4.-6) ω ω ω Uing the ata in Setion 4.9, we fin the rooer frequeny from (4.-4) to be.5 raian minute -. We notie that phae lag in the loop epen only on the tank an ale; the proportional ontroller, being a pure gain ytem, ontribute no lag to the ynami repone of the loop. Hene the rooer frequeny oe not ary with the ontroller gain etting. Uing the rooer frequeny an further ata from Setion 4.9, we fin from (4.-6) that the rooer amplitue ratio will be when the ontroller gain i 5.55. The effet of ontroller gain i to amplify the ignal in the loop. Aroun 5.55, therefore, the ytem output will oillate unabate at frequeny ω. At higher gain etting, the amplitue of the oillation will grow in time. (The frequeny of thee oillation epen on the pole of the tranfer funtion.) igure 4.- i a Boe plot for the loop tranfer funtion G L, howing gain below, at, an aboe the intability threhol. The tability threhol (amplitue ratio, phae angle -8 ) i hown by a ingle point at the rooer frequeny reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie amplitue ratio. 5 5.55... phae angle (eg) -5 - -5 - -5 rooer point -.. ω (raian min - ) igure 4.-: Boe plot for loop tranfer funtion igure 4.- how untable tep repone at gain of 6 an. The latter repone quikly get out of han. Notie how the make-up flow arie in repone to the inreaing error in the outlet ompoition. reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie inlet omp eiation (kg m - ) make-up flow eiation ( -5 m - ).5.5 -.5 5 4 - - - -4-5 5 5 5. outlet ompoition eiation (kg m - )... -. -. 6 -. 5 5 5 time (min) igure 4.-: Step repone for two-tank proe at high gain 4. tuning bae on tability limit - gain an phae margin We tune a ontroller eeking goo performane, omewhere between the extreme of no ontrol an intability. One metho of tuning i imply to maintain a reaonable itane from the intability limit an preume that the reult i an improement oer haing no ontrol. Thu, reie 6 Mar 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie we fin the intability threhol an tune the ontroller to leae margin between thee onition an normal operation. The margin are iniate in igure 4.-, whih how a Boe plot for the loop tranfer funtion G L at ome arbitrary ontroller etting. R A () GM > R A () φ φ -8 PM > () (4) ω ω igure 4.-: Illutration of gain margin an phae margin at a ingle ontroller etting The gain margin an phae margin are the itane hown on the orinate. Their efinition are ω GM R A PM 8 φ ( > for tability) ( > for tability) (4.-) The ontroller etting etermine the amplitue ratio an phae angle ure. rom thoe ure we then alulate the margin to ee if they are atifatory: () ue a phae angle of -8 to fin the rooer frequeny ω () ue an amplitue ratio of to fin the frequeny ω () ue ω to fin the amplitue ratio R A, an thu GM (4) ue ω to fin the phae angle φ, an thu PM In igure 4.-, the ytem i table. Howeer, a the ontroller gain i inreae, the Boe plot will hift o that ω an ω approah eah other. At the intability threhol, ω equal ω, the gain margin i, an phae margin i zero. reie 6 Mar 6 4
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie A proeure for tuning proportional ontroller by tability margin i: () ue a phae angle of -8 to fin the rooer frequeny ω () at ω, fin the gain that make R A (tability limit) () at ω, reue the gain to make R A /GM (gain margin) (4) ue a phae angle of PM -8 to fin the frequeny ω (5) at ω, fin the gain that make R A (phae margin) Marlin (Se..8) reommen tuning to maintain GM ~ an PM ~. Typially one or the other will be limiting. igure 4.- how the reult of alulation for our tank example. Earlier, we foun the rooer frequeny from (4.-4). Then (4.-6) wa ole for the ontroller gain that gae R A.5 (thu GM ). The reult wa 6.. Then (4.-) wa ole for the frequeny ω to gie a phae angle of -5 (thu PM ). Then (4.-5) wa ole for the ontroller gain that gae an amplitue ratio of at frequeny ω. The reult wa a muh lower gain of 7. Therefore for our ytem, PM i limiting, an the lower gain woul be hoen. or referene, igure 4.- alo how the tability limit etermine earlier. The gain an phae margin hae gien u a tuning riterion for eleting a ontroller gain. Uing the hoen gain, we an now preit the performane in repone to iturbane an et point hange. The alulation woul be imilar to thoe illutrate in Setion 4.9 an 4.: a partial fration expanion leaing to an expreion for the repone, with parameter alue bae on numerial root-fining. reie 6 Mar 6 5
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie amplitue ratio phae angle (eg).... -5 - -5-5.55 6. 7-5 -.. ω (raian min - ) igure 4.-: Boe plot illutrating GM an PM limit on gain 4. onluion We hae omplete ynami analyi an ontrol of a more ompliate proe than in Leon. In oing o we hae introue new tool for analyi - the Laplae tranform an blok iagram - an eelope tability an tuning riteria. Wa it a goo iea to liten to the appealing aior an put the make-up flow into the firt tank? A goo way to examine the quetion woul be to repeat the full analyi for the other ae. Een without oing that, howeer, we might reflet how remoing one lag from the ytem might affet the Boe tability riterion for the loe loop 4.4 referene Marlin, Thoma E. Proe ontrol. n e. Boton, MA: MGraw-Hill,. ISBN: 796. 4.5 nomenlature a ontant reie 6 Mar 6 6
Spring 6 Proe Dynami, Operation, an ontrol.45 Leon 4: Two Tank in Serie A ontant A,A ontant of integration b ontant B ontant ontant ontant in partial fration expanion A intermeiate tream onentration of olute A A exit tream onentration of olute A A make-up tream onentration of olute A inlet tream onentration of olute A A referene onentration of olute A at teay tate Δ hange in olute onentration olumetri flowrate olumetri flowrate of make-up tream f funtion G tranfer funtion Im operator that take imaginary part of omplex number j quare root of - gain (time-inepenent part of tranfer funtion) L Laplae operator N() polynomial in r ummy ariable in polynomial harateriti equation Re operator that take real part of omplex number R A the amplitue ratio of the loop tranfer funtion R A the amplitue ratio of the loop tranfer funtion at the rooer frequeny omplex Laplae omain ariable t time t time at whih iturbane our U unit tep funtion V olume of tank V olume of tank x(t) input ignal to ytem y(t) output ignal from ytem root of polynomial in ε error; et point minu ontrolle ariable time ontant of tank time ontant of tank time ontant of ale ξ ummy ariable of integration ω raian frequeny (ha imenion of raian time - ) ω rooer frequeny, at whih loop tranfer funtion lag i -8 reie 6 Mar 6 7