(1) i. In this paper it is assumed that the process dynamics can be described with reasonable accuracy by a second-order-plusdead-time

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1 IEEE-TTTC Int Conf on Automaton, Qualty and Testng, Robotcs AQTR, -7, (Eds Mclea and Stoan, ISBN X, Cluj-Naoca, Romana, 006 Develoment and Evaluaton of a PID Auto-Tunng Controller Ioan Naşcu, Robn De Keyser, Slvu Folea, Tudor Buzdugan Techncal Unversty of Cluj Naoca, Deartment of Automaton, Ioan.Nascu@aut.utcluj.ro, Tudor.Buzdugan@aut.utcluj.ro, Slvu.Folea@aut.utcluj.ro Ghent Unversty, EeSA-deartment of Electrcal Energy, Systems & Automaton, rdk@autoctrl.ugent.be Abstract-Ths aer resents comaratve studes regardng a recently develoed extenson of the wdely used relay-feedback PID auto-tuner. The roosed method conssts of two stes: rocess dentfcaton and controller desgn. Frst, a non-teratve rocedure s suggested for dentfcaton of two onts on the rocess Nyqust curve. A second-order-lus-dead-tme model s obtaned and then used for PID controller desgn based on the nternal model rncle (IMC. For the dentfcaton of the two onts on the Nyqust curve a ure relay n the feedback loo (as used n standard auto-tunng and a relay whch oerates on the ntegral of the error are used. The aer comares the erformance of autotunng methods wth exermental studes on a commercal auto-tunng PID controller - a Shmaden PID autotuner. Keywords: auto-tunng, PID controller, relay exerment, Nyqust lot, commercal PID controllers. I. INTRODUCTION At the begnnng of the new mllennum the PID controller contnues to be a key comonent of ndustral control. Ther oularty s justfed by the followng advantages: they have a smle structure, ther rncle s well understood by nstrumentaton engneers and ther control caabltes have roven to be adequate for most control loos. Moreover, due to rocess uncertantes, a more sohstcated control scheme s not necessarly more effcent n real-lfe alcatons than a well-tuned PID controller. However, t s common that PID controllers are oorly tuned n ractce because the choce of controller arameters requres rofessonal secalzed knowledge by the user. To smlfy ths task and to reduce the tunng-tme (rocess startng u erod, PID controllers can ncororate auto-tunng caabltes. The auto-tuners are equed wth a mechansm caable of comutng the correct arameters automatcally when the regulator s connected to the feld [Aström, et al., 984; Aström, et al., 99; Aström, et al., 995; Leva, et al., 00]. Auto-tunng s a very desrable feature and almost every ndustral PID controller rovdes t nowadays. These features rovde easy-to-use controller tunng and have roven to be well acceted among rocess engneers. For the automatc tunng of the PID controllers, several methods have been roosed. Some of these methods are based on dentfcaton of one ont of the rocess frequency resonse, whle the others are based on the knowledge of some characterstc arameters of the oen-loo rocess ste resonse. The dentfcaton of a ont of the rocess frequency resonse can be erformed ether usng a roortonal regulator, whch brngs the closed-loo system to the stablty boundary, or by a relay forcng the rocess outut to oscllate. Åström and Hägglund [Åström, et al., 984] reort an mortant and nterestng aroach. Ther method s based on the Zegler-Nchols frequency doman desgn formula. A relay connected n a feedback loo wth the rocess s used n order to determne the crtcal ont. Ths contrbuton brefly descrbes the develoment of two auto-tunng methods based on the dentfcaton of two onts on the rocess Nyqust curve [Nascu and De Keyser, 003]. The erformance of both autotunng methods are comared wth those of Åström-Hägglund method and wth exermental studes on a commercal auto-tunng PID controller - a Shmaden PID auto-tuner. II. THE EXPERIMENTAL SET-UP A standard closed loo system wth sngle nut sngle outut, as shown n fgure, s consdered. To erform relay feedback exerments, the rocess s frst brought to steady-state condtons n manual control or wth any relmnary tuned PID controller. The rocess H s assumed to be lnear, stable and roer. The PID controller has a non-nteractng structure cascaded wth a frst order flter: H c (s Kc( + + std ( st T s + In ths aer t s assumed that the rocess dynamcs can be descrbed wth reasonable accuracy by a second-order-lusdead-tme (SOPDT model as: f /06/$ IEEE

2 H θ s k e (s, θ 0 and τ, ζ>0 ( τ s + ζτs + In order to tune the PID controller we wll aroxmate e -θs by (-θs and usng the well-known IMC-PID desgn method we then obtan: ζτ K c k (Tc + θ (3 τ Tc T ζτ, Td, T f ζτ T + θ thus resultng n the closed-loo transfer functon: H s (s Hc(s θ e H0(s + H (s H c (s ( T c s + The arameter T c s the closed-loo tme constant and s a desgn arameter whch can be selected by the user n order to tune the controller aggresveness. The arameters of the rocess transfer functon ( are assumed to be unknown and - n order to dentfy them - onts of the frequency resonse have to be estmated usng two relay feedback exerments: the frst exerment uses a relay wth hysteress or an ntegratng relay ( a relay wthout hysteress whch has as nut the ntegral of the error and the second exerment uses a ure relay (wthout hysteress. After erformng these two exerments the values of the rocess frequency resonse at two dfferent frequences ω and ω are obtaned: H (j a + jb and H (jω a + jb c (4 ω (5 The rocedure how to erform these exerments and how to obtan the values for (a,b ;, s further exlaned n secton III. III. THE TUNING METHOD Note from ( that H (jω can be wrtten as follows: wth H k H (jω, τ ω + ζτ jω (jω H (jω H (jω, (6 H (jω ( θ jω If we defne - for a certan ont on the Nyqust lot - the magntude, the hase angle and the real and magnary comonents of H (jω, (jω and (jω to be M, α, H H a, b, resectvely M, α, a,b and M, α, a, b, then: M a + b M M ( a + b ( a + b (7 α α + α b tg( α tg( α tg( α tg( α α a + tg( α tg( α b b a a a b ab b b aa + bb + a a (8 a b ab aa + bb b a a b (9 aa + bb a b ab From (9 - and afterwards (7 - t can be wrtten: a b ab ( a + b a + aa bb + ( a + b ( a + b ( a + b ( a + b a a ( aa + bb ( aa + bb thus leadng to: and usng (9 agan also to: a a + bb a ( a + b a M a b ab b M Thus for the two onts obtaned from the relay feedback exerments we have: aa + bb a M (0a a b a b b M a b a a + b b a M a b b M (0b The arameters a and b (, can be obtaned usng the two relay feedback exerments. In order to dentfy a ont on the rocess Nyqust curve, a relay connected n a feedback loo wth the rocess s used, forcng the rocess outut to oscllate. Therefore assume that d s the relay amltude and ε s the relay hysteress wdth. For the gven values d and ε we wll obtan oscllaton wth amltude h and erod T n the rocess outut. In the frst exerment there are two ossbltes: usng a relay wth hysteress, a ont n the thrd quadrant on the

3 rocess Nyqust curve, P h (a,jb, s dentfed [Aström, et al., 995]: π ω, T a π h 4d ε, (a πε b, M a + b 4d or, usng an ntegratng relay (ε 0 the ont gven by the ntersecton of the rocess Nyqust curve and the negatve magnary axs, P (a,jb, s dentfed: π πh ω M b (b, a 0, b, T 4d In the second exerment, usng a ure relay (ε 0, the ont gven by the ntersecton of the rocess Nyqust curve and the negatve real axs, P (a,jb, s dentfed: π πh ω M a (, a, b 0, T 4d The real and magnary comonents of H (jω and H (jω can be wrtten as: ( ω k ( τω ( τω + ( ζτω ( ω ( τω + ( ζτω ( ( ( τω M k a Re H (j k ( ζτ ω b Im H (j k ζτ ωm a Re H (j ω b Im H (j ω θω (3 Insertng n (0 the relatons (3 for a, b, a, b ;, gves 4 equatons wth 4 unknown arameters: k, θ, τ and ζτ: ( τ ω M a b θω (4.a k ζτωm b + a θω (4.b a k ( τ ω M b θω (4.c k b ζτωm + a θω (4.d Usng n the frst exerment a relay wth hysteress and n the second exerment a ure relay (the HR case then the frst dentfed ont on the rocess Nyqust curve s n the thrd quadrant and the second one s the ntersecton wth the negatve real axs (thus b 0. From (4.b and (4.d results: From (4.c and (4.a results: k ab θ ω ( M a a (5.a ( ω ω M M k (5.b ω M ( a bθω aω M Now, (4.a and (4.b gve: τ ω k b a + (5.c ( θω ω M k ( b + aθω ωm ζτ (5.d These values can then be used n (3 to calculate the PIDarameters. Usng n the frst exerment an ntegratng relay and n the second exerment a ure relay (the IR case then the frst dentfed ont on the rocess Nyqust curve s the ntersecton wth the negatve magnary axs (thus a 0 and the second one s the ntersecton wth the negatve real axs (thus b 0. In ths case we obtan the followng arameters whch can then be used n (3 to tune the PID: a θ ω b k b τ ω k ζτ ω b ( ω ω ab ω ( aθω + b k a + b ω (6

A relay wthout hysteress can be used only f the rocess Nyqust curve crosses the negatve real axs.

4 To erform relay feedback exerments, the rocess s frst brought to steady-state condtons n manual control or wth any relmnary tuned PI controller. For rocesses of frst or second order wthout tme delay, the value of θ n the model ( wll be zero. Strctly alyng theory, these rocesses cannot be forced to oscllate by a ure relay. A relay wthout hysteress can be used only f the rocess Nyqust curve crosses the negatve real axs. In fact, consderng that n any dgtal controller mlementaton the samlng rocess tself ntroduces a hase lag and that n real stuatons the rocess outut s fltered, t can be assumed that all the rocesses n ractcal cases wll oscllate when a relay controller s connected. Unfortunately, n absence of hysteress, the relay exerment gves small values for the amltude of oscllaton, dffcult to measure n the resence of measurement nose. Thus, n case of small amltude oscllatons, n the ure relay exerment a and θ are fxed to zero, k s comuted from steady-state condtons and τ and ζτ are comuted usng measurements only from the frst exerment (a, b, ω. If there are no others restrctons, the amltude of the relay wll be set to 0.u 0 and the hysteress to 0.0y 0. Better results can be obtaned usng IR method. IV. EXPERIMENTS The aer comares the three auto-tunng methods based on takng the rocess to ts stablty lmt wth the auto-tunng method mlemented n the numercal controller SR73A. A MATLAB test-bed for smulatng the auto-tunng algorthms and for studyng the behavor of the SR73A controller wth smulated rocesses has been mlemented. The erformances of the methods have been comared on three tyes of rocesses and the advantages of usng one method or another for each stuaton has been hghlghted. Sluggsh rocesses have been selected because ths numercal controller has a mnmum samlng tme of second. For testng the erformances of the auto-tunng mechansm mlemented n the SR73A controller a test-bed has been set u, and conssts of the PC runnng the erformance evaluaton mechansm, an RS3-RS485 converter for udatng the arameters of the controller from the PC and a NI data acquston board for rovdng the ndustral controller wth smulated lant analog sgnals and readng ts commands, as resented n Fgure. For usng the Natonal Instruments PC-00 data acquston board wth MATLAB, a seres of MEX-fles have been develoed n C. MEX stands for MATLAB Executable. MEX-fles are dynamcally lnked subroutnes roduced from C or Fortran source code that, when comled, can be run from wthn MATLAB n the same way as MATLAB M-fles or bult-n functons. The external nterface functons rovde functonalty to transfer data between MEX-fles and MATLAB, and the ablty to call MATLAB functons from C or Fortran code. The source code for a MEX-fle conssts of two dstnct arts: A comutatonal routne that contans the code for erformng the comutatons that you want mlemented n the MEX-fle. Comutatons can be numercal comutatons as well as nuttng and oututtng data. A gateway routne that nterfaces the comutatonal routne wth MATLAB by the entry ont mexfuncton and ts arameters rhs, nrhs, lhs, nlhs, where rhs s an array of rght-hand nut arguments, nrhs s the number of rght-hand nut arguments, lhs s an array of left-hand outut arguments, and nlhs s the number of left-hand outut arguments. The gateway calls the comutatonal routne as a subroutne. Fgure. The setu of the test-bed and the grahcal nterface of the alcaton testng the SR73A controller. The lant s smulated n MATLAB but t s controlled by the ndustral controller through the NI PC-00 DAQ. At each samlng tme the data acquston board measures the duty factor of the PWM control sgnal ssued by the controller and va the MEX routnes s fed nto the model. The outut of the smulated lant s evaluated and sent back to the controller as an analog sgnal. The roosed auto-tunng algorthms have been tested by smulated examles. Three dfferent models of the rocess transfer functon has been consdered:

5 H P (s (7 (40s + (0s + H P (s (8 3 (0s + (0s + H P 3(s (9 (40s + (40s + Setont changes have been analyzed. For smulatons, the amltude of the setont s. When usng the ndustral controller, the outut of the lant has been normalzed and lotted. The closed loo ste resonses usng IR (curve and HR (curve methods, Zegler-Nchols method (curve 3 and SR73 (curve 4 dgtal controller are resented. Oen loo rocess ste resonse (curve 5 has been also resented. The desgn arameter T c can be secfed by the user or can be chosen automatcally as a ercent of ζτ or ζτ+θ. Here T c s fxed at (ζτ/ for the uer lots and resectvely (ζτ/5 for lower lots. In fgure 3 a second order rocess gven by (7 s consdered. It can be observed that the IR ( and HR ( autotunng methods gve smlar results and better erformance than the classcal Zegler-Nchols (fast resonse but bg overshoot auto-tunng method or SR73 (4 dgtal controller (slow resonse and overshoot. The change of the user-defned arameter T c n IR and HR methods s nfluencng the closedloo behavor (uer lots slower resonse lower lots faster resonse. The oen loo behavor (5 s sgnfcantly mroved for the IR ( and HR ( auto-tunng methods by reducng the settlng tme. The dfference between the classcal Zegler-Nchols (3 and the roosed IR ( and HR ( methods s vsble. However, by choosng the rght value for the desgn-arameter T c n IR and HR methods t s ossble to obtan smlar erformance as the ZN auto-tuner (that has no desgn arameter. Fgure 4. Performance comarson between: oen loo (5, IR method (, HR method (, Zegler-Nchols method (3 and SR73 (4 dgtal controller. Ste resonse, rocess gven by (8, T c fxed at (ζτ/ for the uer lots and (ζτ/5 for lower lots. Fgure 3. Performance comarson between: oen loo (5, IR method (, HR method (, Zegler-Nchols method (3 and SR73 (4 dgtal controller. Ste resonse, rocess gven by (7, T c fxed at (ζτ/ for the uer lots and (ζτ/5 for lower lots. Smlar results (fgure 4 are obtaned consderng a thrd order rocess gven by (8. Consderng the rocess gven by (9 the results (fgure 5 are dfferent for IR method ( and HR method ( (better results for IR. Also the results wth SR73 (4 dgtal controller are better than for revous cases.

6 dentfyng two onts on the rocess frequency resonse and usng the IMC-PID desgn method. The methods are smle and as other auto-tunng technques, does not requre an advanced knowledge of control engneerng theory. These auto-tunng methods yeld good PID arameters for a secfc class of rocesses (stable rocesses wth selfregulaton; t s not a general methodology for arbtrary rocess models. The controller develoed wth the IR and HR auto-tunng method gves good erformance for ste resonse (fast dynamcs, no overshoot, n most of the cases the results beng better than n the case of Zegler-Nchols method or SR73 dgtal controller. The change of the user-defned arameter T c s nfluencng the closed-loo behavour. REFERENCES Fgure 5. Performance comarson between: oen loo (5, IR method (, HR method (, Zegler-Nchols method (3 and SR73 (4 dgtal controller. Ste resonse, rocess gven by (9, T c fxed at (ζτ/ for the uer lots and (ζτ/5 for lower lots. V. CONCLUSION Some relay-based algorthms for auto-tunng of PID controllers have been resented, assumng a smle rocess model structure and achevng the controller tunng by [] Åström, K.J. and T. Hägglund. Automatc tunng of smle regulators wth secfcatons on hase and amltude margns, Automatca, 0, 984,.645. [] Åström, K.J., C.C. Hang, P. Persson and W.K. Ho. Towards ntellgent PID control. Automatca, 8, 99,. [3] Åström, K.J. and T. Hägglund. PID Controllers: Theory, Desgn and Tunng. Instrument Socety of Amerca, Research Trangle Park, NC, USA, 995 [4] Leva, A., C. Cox and A. Ruano. Hands-on Pd autotunng: a gude to better utlsaton. IFAC Professonal Bref, 00 [5] Nascu I., De Keyser R., A Novel Alcaton of Relay-feedback for PID Auto-tunng, IFAC CSD 03 Conference on Control Systems Desgn, 003, Bratslava, Slovak Reublc, CDaer d08767, 6 [6] Ogata, K. Modern Control Engneerng. Prentce-Hall, 990 [7] Ho, W.K., C.C. Hang and L.S. Cao. Tunng of PID controllers based on gan and hase margn secfcatons, Automatca, 3, No.3, 995,.497. [8] Pouln, E., A. Pomerleau, A. Desbens and D. Hodoun. Develoment and evaluaton of an auto-tunng and adatve PID controller, Automatca, 3, 996,.7. [9] Sche,T.S.. Automatc tunng of PID controllers based on transfer functon estmaton. Automatca, 30, 994,.983 [0] SR73A and SR74A Seres Dgtal Controler Instructon Manual [] Lab-PC-00 / AI User Manual [] Lab-PC-00 / AI Regster-Level Programmer Manual [3] NI-DAQ User Manual for PC Comatbles [4] NI-DAQ Functon Reference Hel (february 00 edton [5] MATLAB - The Language of Techncal Comutng [6] MATLAB - Creatng Grahcal User Interfaces

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