Chapter 2 Two-Degree-of-Freedom PID Controllers Structures

Chapter 2 Two-Degree-of-Freedom PID Controllers Structures As n most of the exstng ndustral process control applcatons, the desred value of the controlled varable, or set-pont, normally remans constant regulatory control or dsturbance rejecton operaton) but needs to be changed servo-control or setpont trackng operaton) we are manly nterested n the two-degree-of-freedom 2DoF) mplementaton of the PID control algorthms. The extra parameter that the 2DoF control algorthm provdes s used to mprove ther servo-control behavor whle consderng the regulatory control performance and the closed-loop control system robustness [1 5]. Ths 2DoF feature can be ncorporated both nto a PI or a PID control algorthm. Although all the controllers wth a proportonal ntegral PI) control algorthm are mplemented n the same way, have the same transfer functon, ths s not the case wth commercal controllers wth proportonal ntegral dervatve PID) control algorthms. In fact, usually, the control algorthm mplementaton s manufacturer dependent and not all of ts varatons are avalable n the same controller. Even more, the controllers manufacturers use dfferent names for the same PID algorthm [6]. The dversty of the PID control algorthms s evdent n [7]. In addton, t would be the case that a tunng rule of nterest had been obtaned usng a control algorthm dfferent from the one mplemented n the controller to tune. In ths case, controller parameters converson s requred that wll also ndcate f the pursued equvalent controller exsts. On that bass, the most wdely used PID control algorthms are presented n ths chapter by also provdng converson formulae that allows to convert the parameters of one algorthm from those obtaned for another formulaton. As t wll be seen ths converson wll not always be possble, showng some formulatons are more general than others. Sprnger Internatonal Publshng Swtzerland 2016 V.M. Alfaro and R. Vlanova, Model-Reference Robust Tunng of PID Controllers, Advances n Industral Control, DOI 10.1007/978-3-319-28213-8_2 7

8 2 Two-Degree-of-Freedom PID Controllers Structures 2.1 Proportonal Integral Dervatve Control Algorthm Consder the general controller block dagram depcted n Fg. 2.1. The output or control effort of a proportonal P) ntegral I) and dervatve D) control algorthm s gven, n general, by Ut) = Acton {U P t) + U I t) + U D t) + U b }, 2.1) f 0 % Ut) 100 %, and 0 or 100 %, dependng on the controller acton f the controller output reaches one of ts lmts. In 2.1) U P s the proportonal term or proportonal control acton, gven by U P t) = K p Et) = K p [Rt) Y t)], 2.2) wth a proportonal gan K p ; U I s the ntegral term or ntegral control acton,gven by t t U I t) = K Eξ)dξ = K [Rξ) Y ξ)]dξ, 2.3) 0 wth an ntegral gan K ; and U D the dervatve term or dervatve control acton, gven by U D t) = K d det) dt 0 d[rt) Y t)] = K d, 2.4) dt wth a dervatve gan K d.the controller output bas U b s usually set to 50 %. In 2.1) 2.4) controller nputs Rt) and Y t), and output Ut) change n the range from 0 to 100 %. The controller Acton sgn, +1 Reverse) or 1 Drect), must be selected equal to the controlled process gan sgn to preserve the negatve feedback characterstc of the control loop. In the followng, we wll assume that the controller Acton has been selected correctly, that all the closed-loop control varables are wthn ther correspondng 0 100 % range, and that the control system s ntally at a steady-state stable operatng Fg. 2.1 Controller block dagram

2.1 Proportonal Integral Dervatve Control Algorthm 9 Fg. 2.2 Closed-loop control block dagram pont gven by {R o, Y o, U o }. Then, we only consder devaton varables {r, y, u} around ths operatng pont and then the controller output bas wll not be ncluded n followng controllers equatons. A lnear control system s based on a lnearzed process model descrpton that relates devaton varables from ts operatng pont values. On that bass, the lnearzed closed-loop control system for varable devatons rs), ys), us), and ds) s reduced as depcted n Fg. 2.2, where Ps) s the transfer functon of the controlled process model and C r s) and C y s) the controller aspects appled to the set-pont and the feedback sgnal, respectvely. The possble measurement nose ns) has been also ncluded. 2.2 Two-Degree-of-Freedom 2DoF) PID Control Algorthms The most wdely used proportonal ntegral dervatve or PID control algorthms are brefly descrbed below. Each formulaton s provded by a specfc notaton for ts parameters n order to dstngush the correspondng mplementatons when proceedng later on to provde the converson equatons from one algorthm to the other. 2DoF Standard PID The textbook 2DoF proportonal ntegral dervatve control algorthm s the Standard PID whose output s gven by the followng [8 10]: or ut) = K p { e p t) + 1 T t 0 } de d t) e ξ)dξ + T d, 2.5) dt { us) = K p e p s) + 1 T s e s) + T } ds αt d s + 1 e ds), 2.6)

10 2 Two-Degree-of-Freedom PID Controllers Structures Fg. 2.3 Two-degree-of-freedom Standard PID controller wth e p s) = βrs) y s), 2.7) e s) = rs) y s), 2.8) e d s) = γ rs) y s), 2.9) y s) = ys) + ns), 2.10) where K p s the controller gan, T the ntegral tme constant, T d the dervatve tme constant, β and γ the set-pont weghts, and α the dervatve flter constant. The 2DoF PID block dagram s depcted n Fg. 2.3. To avod an extreme nstantaneous change at the controller output sgnal when a set-pont step change occurs normally γ s set to zero [11, 12]. In ths case 2.6) reduces to { us) = K p βrs) y s) + 1 [ rs) y s) ] ) } Td s y s), 2.11) T s αt d s + 1 that wll be denoted as PID 2. In addton, n the followng t s assumed that the measurement nose s fltered, then y s) ys). The controller output 2.11) may be rearranged, for analyss purposes, as follows: us) = K p β + 1 ) rs) K p 1 + 1 T s T s + T ) ds ys), 2.12) αt d s + 1 where the C r s) and C y s) controller aspects read as C r s) = K p β + 1 ), 2.13) T s

2.2 Two-Degree-of-Freedom 2DoF) PID Control Algorthms 11 C y s) = K p 1 + 1 T s + T ) ds, 2.14) αt d s + 1 beng the controller parameters θ c = { K p, T, T d,α,β,γ = 0 }. Although the Standard form s the classcal mplementaton of the PID control algorthm, the followng forms are also found n the control lterature [10, 12, 13]. 2DoF Parallel PID The parallel or ndependent gans PID control algorthm s us) = β p K p + K ) rs) K p + K ) s s + K d s ys), 2.15) α p K d s + 1 where the C r s) and C y s) controller aspects read as C r s) = β p K p + K ), 2.16) s C y s) = K p + K ) s + K d s, 2.17) α p K d s + 1 wth parameters θ cp = { K p, K, K d,α p,β p,γ p = 0 }. K p s the proportonal gan, K the ntegral gan, and K d the dervatve gan. 2DoF Seres or Industral PID The 2DoF verson of the seres nteractng mplementaton of the PID algorthm s us) = K p β + 1 ) T s rs) K p 1 + 1 T s where the C r s) and C y s) controller aspects read as C r s) = K p C y s) = K p β + 1 T s 1 + 1 T s ) T d s + 1 ) α T d s + 1 ys), 2.18) ), 2.19) ) T d s + 1 α T d s + 1 ), 2.20) wth parameters θ c = { K p, T, T d,α,β,γ = 0 }. 2DoF Ideal PID wth Flter A commonly used PID mplementaton n Internal Model Control IMC)-based controller desgn s gven by the followng: us) = K p β + 1 ) T rs) K p 1 + 1 ) s T s + T d s 1 T f s + 1 ) ys), 2.21)

12 2 Two-Degree-of-Freedom PID Controllers Structures where the C r s) and C y s) controller aspects read as C r s) = K p C y s) = K p β + 1 T s 1 + 1 T s + T ), 2.22) ) d s 1 T f s + 1 ), 2.23) wth parameters θc = { K p, T, Td, T f,β,γ = 0 }. T f s the controller ntput flter tme constant. 2.3 PID Control Algorthms Converson Relatons As t can be observed from the presented PID forms, whereas the reference controller aspect takes the same form n all formulatons, t s the feedback part the one that prevents a drect translaton of the controller parameters from one formulaton to another. Ths s mportant because some of the exstng tunng rules have been conceved for a specfc PID formulaton. As an example, the dervatons of the celebrated SIMC tunng [14] are wth the Seres or Industral formulaton n mnd, whereas much of the other proposals are based on the Standard one. Due to the possblty that the control PID algorthm of the controller to tune be dfferent to the one consdered by the tunng rule to use t s necessary to have converson relatons to obtan equvalent parameters between two or more of them [15]. In what follows, we present converson formulae to get the controller parameters for one specfc PID formulaton startng from the parameters got for another dfferent one. Converson from a 2DoF Parallel PID to a Standard PID A PID 2 controller 2.11) equvalent to the 2DoF Parallel PID 2.15) s found usng the followng relatons: K p = K p, 2.24) T = K p, K 2.25) T d = K d, K p 2.26) α = α p K p, 2.27) β = β p, 2.28) γ = γ p = 0. 2.29) There s a drect relaton between the Standard and Parallel PID algorthms then ths last one wll not be further consdered.

2.3 PID Control Algorthms Converson Relatons 13 Converson from a 2DoF Seres PID to a Standard PID It s possble to obtan a Standard 2DoF PID controller 2.11) equvalent to the 2DoF Seres PID 2.18) usng the followng relatons: K p = F c K p, 2.30) T = F c T 2.31) T d = 1 α F c )T d, 2.32) F c α = F c α 1 α F c,α < 1 + T T d, 2.33) β = β, F c 2.34) γ = γ = 0, 2.35) F c = 1 + 1 α )T d. 2.36) T where F c 2.36) sthepid 2s to PID 2 converson factor. It takes nto account the dervatve flter constant α. The converson constrant n 2.33) usually holds then we may say that there s a Standard PID equvalent to a Seres one. Converson from a 2DoF Ideal PID wth Flter to a Standard PID A Standard 2DoF PID controller 2.11) equvalent to the Ideal PID wth flter 2.21), denoted by PID 2F, can be obtaned usng the followng relatons: K p = Fc K p, 2.37) T = Fc T, 2.38) T d = T d T Fc f, Td c T f, 2.39) Fc α = f Td c T, f 2.40) β = β, Fc 2.41) γ = γ = 0, 2.42) Fc = 1 T f, 2.43) T T f < T, for PI T f = 0. 2.44) where F c 2.44) sthepid 2F to PID 2 converson factor.

14 2 Two-Degree-of-Freedom PID Controllers Structures In ths case, an equvalent PID 2 controller cannot always be obtaned as shown n 2.39) and 2.44). Usng the converson factors presented above, exact equvalent feedback C y s)) and set-pont C r s)) controllers transfer functons for a PID 2 2.12) may be obtaned for 2DoF PID controllers gven by 2.15), 2.18), and 2.21). Exact equvalent controllers guarantee to obtan the same control system performance and robustness n case a 2DoF PID controller s replaced wth a PID controller wth a dfferent 2DoF algorthm. Converson from a 2DoF Standard PID to a Seres PID In the other drecton a 2DoF Seres PID controller equvalent to a 2DoF Standard one can be found usng the followng relatons: K p = F c K p, 2.45) T = F c T, 2.46) T d d, F c 2.47) α = αf c 1 + α, 2.48) β = β, F c 2.49) γ = γ = 0, 2.50) [ ] F c = 0.5 1 + αt d + 1 4 + 2α)T d + α2 Td 2. 2.51) T T Due to the constrant mposed by 2.51) there wll not always exst a Seres PID equvalent to a Standard PID. It wll only exst f ) 2 α 2 Td 4 + 2α) T Td T T 2 ) + 1 > 0. 2.52) If the PID 2 dervatve flter constant s taken as α = 0.1 there s a Seres equvalent PID controller only f T > 4.20 T d. Fgure 2.4 shows that ths constran ncreases as α ncreases. As can be seen n same fgure quadratc nequalty 2.52) can be approxmated by the followng straght lne for 0 α 1.0: T T d > 4.05 + 1.80 α. 2.53)

2.3 PID Control Algorthms Converson Relatons 15 Fg. 2.4 T /T d condton to obtan a Seres PID equvalent to a Standard PID 5.8 5.6 5.4 5.2 There s a Seres PID controller equvalent to a Standard PID. T /T d lmt 5 4.8 4.6 4.4 4.2 4 3.8 0 0.2 0.4 0.6 0.8 1 α Converson from a 2DoF Standard PID to an Ideal PID wth Flter The PID 2F s a more general control algorthm and, as ndcated above, not always an equvalent PID 2 controller may be obtaned from the PID 2F but t s always possble to obtan a PID 2F control algorthm equvalent to the PID 2 usng the followng relatons: K p = F cf K p, 2.54) T = F cf T, 2.55) ) 1 + α = T d, 2.56) F cf T d T f = αt d, 2.57) β = β, F cf 2.58) γ = 0, 2.59) F cf = 1 + αt d T. 2.60) where F cf 2.60) sthepid 2 to PID 2F converson factor. Consderng the above we may say that n the 2DoF PID controllers parametrc space θ c θ c θc. Controller parameters converson equatons show that the dervatve flter constant α, α p, α ) must be take nto account to obtan an equvalent controller wth a dfferent control algorthm from a gven one.

16 2 Two-Degree-of-Freedom PID Controllers Structures Fg. 2.5 2DoF PID controllers converson To summarze the above relatons a 2DoF PID controllers converson chart s shown n Fg. 2.5. The sold arrows ndcate drectons on where there are always equvalent controllers and the dashed arrows the drectons on where there are constrants to obtan equvalent controllers. As can be seen n ths chart the 2DoF Ideal PID wth flter s the most general proportonal ntegral dervatve control algorthm. 2.4 PID Controller wth Two Input Flters The dfferent sgnals that enter the PID controller are normally fltered n dfferent ways before they enter the controller. However, as ponted out n [16], a proper choce of these flters can mprove the performance of the feedback loop consderably. Therefore, t s mportant to keep these flters n mnd durng the desgn procedures. In order to nclude nto the controller desgn the measurement nose flter and also to have more freedom for the servo-control desgn, the control algorthm may be aggregated wth two nput flters as depcted n Fg. 2.6 [16, 17]. These flters should be consdered as an ntegral part of the desgn procedure. The control algorthm s of ndependent gans deal parallel PID mplementaton) whose output sgnal s gven by [8]: us) = K p [ r s) y s) ] + K s [ r s) y s) ] K d sy s), 2.61) where K p s the controller proportonal gan, K dervatve gan γ = 0). the ntegral gan, and K d the

2.4 PID Controller wth Two Input Flters 17 Fg. 2.6 Closed-loop control system of a controller wth two nput flters The set-pont r and feedback y sgnals are fltered before they enter the controller. Then r and y n 2.61) aregvenby r s) = F r s)rs), y s) = F y s) [ys) + ns)]. 2.62) Usng 2.62) nto2.61), t s obtaned that us) = K p + K ) F r s)rs) K p + K ) s s + K ds F y s) [ys) + ns)]. 2.63) In a compact form 2.63) s expressed as us) = C r s)f r s)rs) C y s)f y s) [ys) + ns)]. 2.64) The set-pont flter F r s) s selected strctly proper and gven by the transfer functon F r s) = σ T rs + 1 T r s + 1) 2, 2.65) where T r s ts tme constant and σ an adjustable parameter. Flter 2.65) avods to have a step change n the controller output when a set-pont step change s made. The feedback flter nose flter ) F y s) s selected of frst order for PI controllers, gven by 1 F y s) = D fy s) = 1 T f s + 1, 2.66) wth tme constant T f, and of second order for PID controllers, gven by F y s) = 1 D fy s) = 1 T f 2/2s2 + T f s + 1, 2.67) to provde hgh-frequency roll-off measurement nose attenuaton) wth ether controllers.

18 2 Two-Degree-of-Freedom PID Controllers Structures Input flters transfer functon gans are constraned to be equal, lm s 0 F r s) = lm s 0 F y s), to ensure that n steady state the controller ntegral acton operates on the error sgnal. Consderng F r and F y as part of the controller be desgned the selectable parameters of the set-pont controller are θ cr = { K p, K, T r,σ,γ = 0 }, and correspondng to the feedback controller θ cy = { } K p, K, K d, T f. Then, parameters of. the controller as a whole are θ c = θcr θcy = { K p, K, K d, T f, T r,σ,γ = 0 }. The set-pont and feedback sgnal flters combnaton wth the PID control algorthm s denoted as PID 2IF controller. For tunng rules comparson, n addton to the quanttatve performance and robustness ndces and the responses shapes, the process control-orented characterstcs of the PID 2IF controllers must brng to the front. Wth the PID 2IF controllers there s not any abrupt change at the controller output when a step change s made on the set-pont. To mmc ths characterstc wth a 2DoF PID controller ts proportonal set-pont weght β must be made zero. Wth ths, the second degree of freedom s lost and the servo-control response delayed. The other mportant characterstc of the PID 2IF controllers s ther frequency response roll-off. It s normal that n process control applcatons the feedback sgnal be corrupted wth hgh-frequency measurement nose. If ths nose s not properly fltered t wll generate hgh control sgnal varatons resultng n a deteroraton of the fnal control element. If a measurement nose flter s added to a Standard PID controller after ts tunng the flter dynamcs wll affect the control system robustness and performance. Then, t s essental that both these characterstcs be part of the controller desgn from the begnnng. Chapter Remarks The 2DoF) PID algorthm mplementatons are presented as well as the converson relatons between ther parameters. From the presented PID algorthms the Ideal PID wth flter s the more general one. The aggregaton of the deal PID control algorthm wth two nput flters allows to nclude two mportant ndustral features: hgh-frequency roll-off and lack of a control effort abrupt change on a step set-pont modfcaton. References 1. Arak, M.: On two-degree-of-freedom PID control system. Tech. rep., SICE research commttee on modelng and control desgn of real systems 1984) 2. Arak, M.: Two-degree-of-freedom control system I. Syst. Control 29, 649 656 1985) 3. Arak, M., Taguch, H.: Two-degree-of-freedom PID controllers. Int. J. Control Autom. Syst. 14), 401 411 2003) 4. Taguch, H., Arak, M.: Two-degree-of-freedom PID controllers ther functons and optmal tunng. In: IFAC Dgtal Control: Past, Present and Future of PID Control, Aprl 5 7. Terrassa, Span 2000)

References 19 5. Taguch, H., Arak, M.: Survey of researches on two-degree-of-freedom PID controllers. In: The 4th Asan Control Conference, Sept 25 27, Sngapore 2002) 6. Gerry, J.P.: A comparson of PID algorthms. Control Eng. 343), 102 105 1987) 7. O Dwyer, A.: Handbook of PI and PID Controller Tunng Rules, 3rd edn. Imperal College Press, London 2009) 8. Åström, K.J., Hägglund, T.: PID Controllers: Theory, Desgn and Tunng. Instrument Socety of Amerca, Research Trangle Park, NC 27709, USA 1995) 9. Seborg, D.E., Edgar, T.F., Mellchamp, D.A.: Process Dynamcs and Control, 2nd edn. Wley, Hoboken, NJ 07030, USA 2004) 10. Sung, S.W., Lee, I.B.: Process Identfcaton and PID Control. John Wley & Sons Asa) Pte Ltd, Sngapore 2009) 11. Tan, W., Yuan, Y., Nu, Y.: Tunng of PID controller for unstable process. In: IEEE Internatonal Conference on Control Applcatons. Kohala Cost-Island of Hawa, Hawa, USA, Aug 22 27 1999) 12. Vsol, A.: Practcal PID Control. Sprnger Verlag London Lmted 2006) 13. Johnson, M.A.: PID Control New Identfcaton and Desgn Methods, chap. PID Control Technology, pp. 1 46. Sprnger-Verlag London Ltd., U.K. 2005) 14. Skogestad, S.: Smple analytc rules for model reducton and PID controller tunng. J. Process Control 13, 291 309 2003) 15. Alfaro, V.M., Vlanova, R.: Converson formulae and performance capabltes of two-degreeof-freedom PID control algorthms. In: 17th IEEE Internatonal Conference on Emergng Technloges and Factory Automaton ETFA 2012) 2012). Sept 17 21, Kraków, Poland 16. Hägglund, T.: A unfed dscusson on sgnal flter n PID control. Control Eng. Pract. 21, 994 1006 2013) 17. Alfaro, V.M., Vlanova, R.: Performance and robustness consderatons for tunng of proportonal ntegral/proportonal ntegral dervatve controllers wth two nput flters. Ind. Eng. Chem. Res. 52, 18287 18302 2013)

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