Analysis of Feedback Control Systems
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1 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Analyi of Feedbak Control Sytem ntrodution to Feedbak Control Sytem 1 Cloed oo Reone 3 Breaking Aart the Problem to Calulate the Overall Tranfer Funtion 5 Shortut for Calulating Overall Tranfer Funtion 5 nner Feedbak oo Examle 6 Feedforward Examle 7 nternal Feedforward Examle 8 Develoing Blok Diagram from Proe Equation 9 Tyial ontroller trategie 11 Effet of Controller Strategie on Firt Order Proe 13 Effet of Proortional Control 14 Effet of P Control 16 Effet of PD Control 19 ntrodution to Feedbak Control Sytem Blok diagram of generalized roe & orreonding feedbak ontrol loo Final Control Element (Atuator) Controller Proe y C M a y y m m Meauring Element The roe ha 2 inut, the diturbane (alo known a the load or the roe load) and a meaurable variable M, and one outut y (the ontrolled variable) The diturbane hange unreditably Our goal i to adjut the meaurable variable M o that we kee the outut variable y a teady a oible Feedbak ontrol take the following te: John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
2 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Meaure the value of the outut, y m Comare y m to the et oint, y Determine the deviation y y m Deviation roeed by ontroller to give an outut ignal C to the final ontrol element The final ontrol element make hange to the meaurable ontrol variable M The roe itelf i referred to a oen loo a ooed to when the ontrol i turned on when it i referred to a loed loo Tye of feedbak ontrol ytem: FC flow ontrol PC reure ontrol C liquid-level ontrol TC temerature ontrol CC omoition ontrol Tyial meauring devie: Temerature: thermooule Preure: reure tranduer Flow: orifie late, venturi tube, turbine flow meter, hot-wire anemometer iquid-level: float-atuated devie Comoition: hromatograh, R analyzer, UV analyzer, H meter Final ontrol element are tyially valve of ome ort Deending uon ituation, eified a fail oen or fail loe John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
3 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Cloed oo Reone Final Control Element (Atuator) Controller Proe y C M a y y m m Meauring Element Above i a blok diagram for a generalized loed-loo ytem We have equation for: Proe: y M Meauring Element: y m m y Comarator: y y Controller Outut: C Final Control Element: M C a m We would like a et of tranfer funtion that relate the outut y to the two inut y & (whih i the overall box around the roe & feedbak ontrol loo) The tranfer funtion hould have the form: y y SetPoint oad We have individual tranfer funtion that will make u thee overall tranfer funtion We jut have to ombine them uing tandard rule of algebra Starting with the equation for the final ontrol element: M C a M nert equation for ontroller outut a John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
4 Colorado Shool of Mine CHEN403 Feedbak Control Sytem M y y nert equation for omarator a m M y y nert equation for meauring devie a m and then inert thi into the roe equation: y y y a m Algebraially olving for y : y y y a m a 1 m a a y y a y y 1 1 m a m a a o: SetPoint 1 and oad 1 m a Uually look at two tye of roblem: m a Servo roblem No diturbane & ontroller at to kee the outut near the et oint: y y SetPoint Regulator roblem Set oint remain the ame & ontroller at to mooth out diturbane: y oad John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
5 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Breaking Aart the Problem to Calulate the Overall Tranfer Funtion y + - a y + + m Z Thi i a lot of math We an get the ame thing by tarting with a roblem where there are THREE inut and everything feed in a forward diretion Conider the blok flow diagram above The relationhi between the three inut i: y a y ma Z However, note that thi blok diagram i imly the firt one we looked at with Z y So we an make thi ubtitution & do a bit of algebra to get: y a y ma y 1 ma y a y a y y 1 1 m a m a whih i what we determined before Shortut for Calulating Overall Tranfer Funtion Evaluating the overall tranfer funtion between an inut & outut an get quite omliated, eeially if there are everal load and loo For a ytem with a ingle feedbak loo, the tranfer funtion between an inut Y and an outut Y i: in out John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
6 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Y Y out in n f n f where f i the rodut of the tranfer funtion between Y in and Y out, i the rodut of all tranfer funtion within the loo, and n f and n are then number of negative ign within the forward ath & the loo, reetively For a imle feedbak ontrol loo whih only ha a negative ign in the omarator the loo law i: Y Y out in f 1 f there are multile loo, then the ituation get more omliated f the loo are all embedded and do not ro boundarie then thi loo formula an be alied equentially nner Feedbak oo Examle inner R C m2 m1 Thi blok diagram i an embedded, multi-loo examle (ie, aade ontrol) To get the tranfer funtion between R and C we mut firt relae the inner loo with an overall tranfer funtion & then an take are of the outer loo The inner loo tranfer funtion will be: Y Y out John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017 f 22 inner 1 1 in 2 2 m2 Now, the overall tranfer funtion will be:
7 Colorado Shool of Mine CHEN403 Feedbak Control Sytem 22 C 1 R 1 1 1inner 1 m f 1 inner m2 C R m m1 1 1 m1 2 2 m2 Feedforward Examle f R v C m Thi blok diagram i for a ituation where the load information i ombined with the outut information to give a ombined feedforward-feedbak ontrol To get the tranfer funtion between and C we mut onider both forward ath The outut C for the two earate ath involving will give: f v C 1 1 m v m v Now, the overall tranfer funtion will be: C f v 1 m v John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
8 Colorado Shool of Mine CHEN403 Feedbak Control Sytem nternal Feedforward Examle R M v + C - + m Thi blok diagram i for a ituation where the information for the maniulated variable goe through an internal model (See Chater 12) Now there are two feedbak loo We an lit off one with the following blok diagram We ve added a new inut (well, kind of, ine we really know that Z C ) but we only have one feedbak loo R M v C Z m The relationhi of the outut ( C ) to eah of the inut will be: (Note that v m v C R Z 1 1 v m v m i not art of the feedbak loo!) 1 1 vm C v m v R mv Z Now we take into aount that Z C : John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
9 Colorado Shool of Mine CHEN403 Feedbak Control Sytem 1 1 vm C v m v R mv C 1 1 mv v m C vm v R 1 1 mv C v m v R 1 v m v C R 1 mv 1 mv So, the overall tranfer funtion are: load C 1 v m 1 m v C v R 1 m v Develoing Blok Diagram from Proe Equation et' draw a blok diagram for level ontrol on a ingle tank A the maniulated variable we an ue either the effluent flow rate, F 1, or in the inlet flow rate, F 0 When F 0 i the maniulated (ie, ontrol) variable then let ue F1 C1h 1 The overall material balane beome: 1 dh C 1 A F 0 C1h h F0 F0 dt A 1 1 C The roe itelf look like the following 1 F 0 1 h f we meaure the liquid level & ontrol it value with the inlet flowrate then roe look like the following Note that there i a maniulated variable but no load: John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
10 Colorado Shool of Mine CHEN403 Feedbak Control Sytem h + - a F 0 1 h m et ontrol the liquid level by maniulating the outlet flow (uh a with a um) in uh a way a to make the outlet flow indeendent of the liquid level So now F 1 i the ontrol variable and F 0 will be the diturbane variable The overall material balane beome: dh 1 1 A F 0 F1 h F0 F1 F0 F1 dt A A and the blok diagram i: F 0 h F a h m or it an alo be drawn a: F 0 h F a h m John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
11 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Before we go any further, notie the ign hange at the omarator Normally we define h R h m, but here we have hanged the ign! Thi i beaue uing F 1 to ontrol the liquid level give what an be thought of a an invere ontrol tye Normally, if the meaured variable i too mall, then the maniulated variable mut be inreaed (eg, if the temerature in a tank i too low, then the heat to the tank i inreaed) For the 1 t ae, ontrol uing F 0, if the level i too high, then the flow in mut be dereaed; if the level i too low, then flow in mut be inreaed However, here we mut go in the ooite diretion f the level i too high, then the flow out mut be inreaed; if the level i too low, then flow out mut be dereaed n the blok diagram, thi logi an be aommodated either by making the ontrol tranfer funtion negative or by hanging the ign at the omarator For the 1 t ae, uing the inlet flow a the maniulated variable, the overall tranfer funtion between the et oint and the liquid level will be: v h v 1 v h vm 1 R v m v m For the 2 nd ae, uing the outlet flow a the maniulated variable, the overall tranfer funtion will be: h v v v h 1 R v m vm 1 vm Notie that the oition of the negative ign have hanged The other major differene i the form of the roe tranfer funtion, Tyial ontroller trategie Tyial ontroller trategie and arameter value (SEM g 197): Proortional (P) ontrol Controller outut will be: P t E t P P t E t John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
12 Colorado Shool of Mine CHEN403 Feedbak Control Sytem where i the ontroller gain and P i the ontroller bia The gain i ometime referred to a roortional band PB where PB 100/ and tyially ket in range 1 PB 1000 Thi ontroller tranfer funtion i: Proortional-integral (P) ontrol Controller outut will be: t t P t E t E d P P t E t E d 0 0 where i the integral time ontant or reet time Thi i tyially et within the range min Thi ontroller tranfer funtion i: 1 1 Proortional-integral-derivative (PD) ontrol Controller outut will be: t de P t E t E d D P dt 0 where D i the derivative time ontant The derivative ortion of the ontrol antiiate what the error will be in the immediate future ometime referred to a antiiatory ontrol Thi i tyially et within the range 01 D 10 min Thi ontroller tranfer funtion i: 1 1 D Derivative ontrol an give a udden kik when te hange are introdued To get around thi, indutrial ontroller will atually imlement derivative ontrol in an aroximate manner: 1 D 1 D 1 where i a ontant between 005 and 02, mot tyially 01 Another way to eliminate the derivative kik i to aly the derivative ation to the meaured value of the outut, not the error n thi ae the ignal out of the ontroller will be: 1 D C 1 ym D 1 John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
13 Colorado Shool of Mine CHEN403 Feedbak Control Sytem The loed loo blok diagram for thi tye of ontroller an be exreed like that in the following diagram y 1 C M a + y y m D 1 D m Sometime, eeially with neumati tranmiion line, there may be a time delay due to ignal tranmiion Thi will normally be ignored However, if the time delay i large enough, then the time delay tranfer funtion will be: P0 d e P 1 i Effet of Controller Strategie on Firt Order Proe The ontroller trategie will have different harateriti effet on a roe A firt order roe with one maniulated variable and one load will be ued to how thee effet Both tranfer funtion will ue the ame time ontant,, but different roe gain The underlying ODE and reulting tranfer funtion will be: dy y M y M dt 1 1 M Another imlifiation ued here will be to neglet areiable dynmi from the meauring devie & the final ontrol elelment, ie, 1 m a John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
14 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Effet of Proortional Control For roortional (P) ontrol: And the overall tranfer funtion will be: where: 1 1 y y y What are the imliation of thi? y y y 1 1 The reone of the ytem remain 1 t order The time ontant ha been dereaed ( ) meaning that the reone of the ytem i fater The roe gain have dereaed There will be an offet at the new ultimate value of the reone John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
15 Colorado Shool of Mine CHEN403 Feedbak Control Sytem The lat item i not immediately obviou from the reone exreion Firt, let u define the offet a the differene between a teady tate reone, y, and the orreonding et oint: * * Offet y y y y y y y y where the exreion an be ut in term of deviation variable if the initial teady tate i at the initial et oint For a hange in the et oint and/or the load we an determine the new teady tate value by alying the Final Value Thereom For a te hange in the et oint of y y then the et oint dynami funtion will be: y y, the dynami reone of the outut will be: y y y 1 1 the ultimate value will be: y y lim y lim y lim y t and the offet will be: Offet y y 1 y, 1 1 y y 1 1, We would like the offet to be zero, but thi i not oible unle For a te hange in the load without a hange in et oint then y y 0 and The load dynami funtion will be:, John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
16 Colorado Shool of Mine CHEN403 Feedbak Control Sytem the dynami reone of the outut will be: y, 1 1 the ultimate value will be: y lim y lim y lim, t and the offet will be: Offet 1 Again, we would like the offet to be zero, but again thi i not oible unle The offet i harateriti of P ontrol The only time when there will be no offet i when the roe tranfer funtion ha an integrating fator (ie, a 1/ fator) For examle, if the firt order roe i atually a ure integrator, then /, the tranfer funtion between the et oint & the outut will be: y y y y, 1 1 and the ultimate value of the reone to a te hange in the et oint will be: y y lim y lim y y 0 0 whih lead to a zero offet Effet of P Control For roortional-integral (P) ontrol: John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
17 Colorado Shool of Mine CHEN403 Feedbak Control Sytem 1 1 then: y y y y y Note the tranfer funtion have inreaed by an order of 1 (from 1 t order to 2 nd order) The arameter for the 2 nd order ytem are: o the tranfer funtion ould alo be exreed a: John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, y y Both of the tranfer funtion have an term in the numerator o they are more omliated than what we have been dealing with u to now But thee term lead to the
18 Colorado Shool of Mine CHEN403 Feedbak Control Sytem roerty that P ontrol ha zero offet for hange in both the et oint and the load For examle, for a te hange in the et oint of y y then the dynami reone of the outut will be: y the ultimate value will be: y, 1 y y lim y lim y lim y t and the offet will be: Offet y y 0 The an term in the numerator will hange the exeted form of the reone urve from tandard 2 nd order ytem reone For the load, the reone will be the derivative of the tandard 2 nd order reone to the driving funtion For a ram hange to the load, the reone will look like the tandard reone to a te-hange driving funtion For a te hange in the load, the reone will look like the tandard reone to an imule driving funtion: y For the et oint, the reone have two art: the tandard reone with a gain of one & the derivative of the tandard 2 nd order reone to the driving funtion The derivative art will die out after a hort eriod of time leaving the tandard reone a the long-time olution For a te hange in the et oint thi will look like a te-hange reone lu an imule reone: 1 y 1 y y y , John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
19 Colorado Shool of Mine CHEN403 Feedbak Control Sytem Deending uon the ombination of and / the ytem will be overdamed, underdamed, or ritially damed The following figure how the relationhi of the daming fator to thee arameter Note that for a given value there i a minimum for an adjutment of / = ' / = 25 / = 10 / = 05 / = 025 / = Effet of PD Control For roortional-integral-derivative (PD) ontrol: 1 1 D then: John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
20 Colorado Shool of Mine CHEN403 Feedbak Control Sytem 1 1 D 1 1 y y D 1 1 D D y D 1 1 D 2 D 1 y D D 2 D 1 y D 2 1 D Note that the integral ation ha inreaed the order of the tranfer funtion by 1 (from 1 t order to 2 nd order); the derivative ation doe not affet thi The arameter for the 2 nd order ytem are: D / / D D o the tranfer funtion ould alo be exreed a: John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, D 1 y y The derivative ation will inreae the harateriti time but dereae the daming fator The firt ation will low down the reone but the eond will eed it u Both affet mut be ombined to determine the overall affet Both of the tranfer funtion have term in the numerator that lead to zero offet (rimarily from the integral ation) The form of the reone urve to a load hange will be idential to that for P ontrol The reone for a et oint hange, however, ha an
21 Colorado Shool of Mine CHEN403 Feedbak Control Sytem additional hort-time ontribution that look like the 2 nd derivative of the foring funiton reone For examle, for a te hange in the et oint of y y the dynami reone of the outut an be determined from: y 2 D 1 y y y D y John Jehura (jjehura@mineedu) Coyright 2017 Aril 23, 2017
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