AUTO-TUNING OF PID CONTROLLERS BASED ON THE RELAY FEEDBACK TEST FOR MULTI-MODE LOW DAMPING SYSTEMS

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1 AUTO-TUNING OF PID CONTROLLERS BASED ON THE RELAY FEEDBACK TEST FOR MULTI-MODE LOW DAMPING SYSTEMS Patrck Magalhães Cardoso Federal Unversty o Uberlânda FEMEC School o Mechancal Engneerng Av. João Naves de Ávla, 6 Bloco M Sala, CEP Uberlânda MG patrck@mecanca.uu.br José Francsco Rbero jrbero@mecanca.uu.br Abstract: The strong presence o PID (Proportonal-Integral-Dervatve) controllers n the ndustry and the dcultes o tunng these controllers have motvated the development o several auto-tunng technques. The relay eedback methodology s the most mportant o these technques. In the present lterature, although contemplatng a great number o systems, t does not contemplate the auto-tunng o PIDs or oscllatory systems wth low dampng and several vbraton modes. Ths consttutes the man motvaton o ths research work. Ths work presents a new methodology o automatc tunng o PIDs based on the relay eedback test: the complete method. The methodology adds to the relay test a compensator and an automatc and varable reerence sgnal. The requency response uncton o the system s estmated wth condence at low requences (DC) and close to the rst natural requency o the system. The PID gans are obtaned by ttng the requency response o the dynamc system (mechancal system + controller) to a reerence response. The method s evaluated numercally or varous dynamcal systems and expermentally on a cantlever beam wth pezoelectrcs actuators. The work concludes that the proposed methodologes are ecent or a wde spectrum o systems, especally or oscllatory systems wth low dampng and several vbraton modes n the band o nterest. Keywords: PID controller, relay eedback, auto-tunng.. Introducton Nowadays a technologcally modern socety s not bult wthout the presence o automatc control systems. In the control systems stand out those whose actons control (system nputs) depends o the responses (outputs) o the system to be controlled. Such controllers are denomnated eedback controllers. The eedback controllers more spread n the technologcal socety are the Proportonal-Integral-Dervatve type (PID). The smple archtecture o the PID controllers make o these a good tool n the systems control, beng ts responsble nowadays or more than 9% o all the closed-loop controllers n the ndustry (Huang, ). Such use and versatlty has wakened great attenton o researchers n the last years (Preace, ). Essentally, PID controller's project requres the choce o three parameters: the gans proportonal, ntegral and dervatve. It s denomnated tunng o a PID the task or methodology used to nd such parameters. The tunng can be made on-lne or o-lne, n way automatc or manual. In the ndustry, n general, the tunng o PID s accomplshed ndvdually or each process, mprovng the perormance and the robustness o the group o processes. When the gans o PID are ound n a manual way, the tunng task becomes monotonous and slow, and the controller's perormance depends on experence and on knowledge that the operators have about the process. It s recognzed that n practce many ndustral controllers are tuned poorly. For ths reason, the researchers and control engneers have destned attenton more and more to the technques o automatc tunng o controllers. The auto-tunng allows ndng the gans o the controller n an automatc way, startng rom a requrement put by an operator, (Hagglund and Aström, 988, Aström et al, 993 and Cardoso, ). Industral experences ndcate that the auto-tunng controllers are more advantageous n the perormance and n the tme nvolved n the search o the controller's parameters (Hang et al, ). Aström and Hagglund () dscusses the PID controllers' use n the current tmes, and concludes or the longevty o the same ones, observng that such controllers wll be part o the ndustral parks or many years, even wth the appearance o new control technques. The method more known o PID controllers' tunng was proposed by Zegler and Nchols n 94 (Zegler and Nchols, 94). They proposed two tunng alternatves: one or systems that present growng responses when subjects to step nput and other or systems that present unstable behavor or hgh gan n untary eedback loop. Ths methodology, although o smple concepton, t s revealed, n practce, mprecse or many systems (Huang, ). Astrom and Hagglund, recognzng the lmtatons o the Zegler and Nchols method, proposed the use o a relay n the eedback loop o the system to be tuned, creatng the "relay method or tunng o PID" (Astrom and Hagglund, 984; Hagglund and Aström, 988; Huang, ; Hang et al, and Preace, ). In the relay test appear oscllatons n the output o the system n a very close requency to the crtcal requency, (requency where the phase o the output s delayed o -8 degrees o the nput), and once known ths oscllaton pont (gan and perod crtcs) s derved expressons or the PID'S gans. Ths methodology was one o the rst ones to be marketed and your success s due to your smplcty and robustness (Hagglund and Aström, 988; Hang and Sn, 99

2 and Hagglund and Aström, 99). Several researchers developed methodologes that mody the relay eedback test n some way, makng possble your use n a larger range o processes. In Cardoso () two new methodologes o automatc tunng o the PID controllers, based n the relay eedback test, are presented: the complete method and the smpled method. In the methods proposed by Cardoso () stand out the hgh automaton level o these methodologes and the expressve generalty, beng applcable to several system types, besdes to oscllatory systems wth low dampng and several natural requences, what cannot be ound n most o the methods presents n the current lterature. The smpled method s detaled n Cardoso (), and the complete method s presented n ths work. Ths work s organzed lke ths: rstly a short ntroducton s accomplshed about the relay eedback test and latter the complete method s detaled. Soon aterwards the proposed method s appled numercally to several systems and expermentally to a contnuous system consttuted by a cantlever beam wth pezoelectrcs actuators. Fnally, are presented the conclusons.. The Relay Feedback Test The automatc tunng o the PID controllers remounts to the year o 94, when Zegler and Nchols proposed a methodology capable to nd PID controller's gan startng rom dented characterstcs o the system to be controlled. Bascally they are three the methodologes proposed by Zegler and Nchols: the Step Response Method, the Frequency Response Method and the Moded Method o Zegler-Nchols. Larger detals o the methodologes proposed by Zegler and Nchols can be ound n Cardoso (), Zegler and Nchols (94), Hagglund and Aström (988) and Wttenmark and Aström (989). The development o the relay eedback test proposed by Astrom and Hagglund (984) let, ntally, o the Zegler- Nchols Frequency Response Method lmtatons on dentyng the crtcal pont (crtcal gan and crtcal perod where the phase o the system s -8 o ). Although the experment proposed by Zegler and Nchols n the Frequency Response Method s smple n the characterzaton o the system and tunng o PID, t s o dcult automaton, once the oscllaton ampltude must be mantaned under control, snce the operaton o systems close to the unstable area s dangerous. Besdes ths lmtaton, the accurate determnaton o the crtcal gan s an arduous work n practcal condtons. Aström and co-workers, recognzng the lmtatons o the Zegler-Nchols Frequency Response Method, moded the procedure o determnaton o the crtcal pont's parameters, addng an element o the type relay n the eedback loop o the system (Aström and Hagglund, 984 and Hagglund and Aström, 988). Ths procedure promotes statonary oscllatons very close to the crtcal requency or a great range o processes (Huang, ). Relay Relé re(t) e(t) u(t) y(t) G(s) PID Fgure. Scheme o the relay eedback test. System Processo Relay Output System Output Tme Fgure. System output sgnal and output o relay eedback. The Fgure shows the relay eedback test dagram. The swtch can alternate among the relay test - when the PID tunng s accomplshed - and the control o the system wth PID prevously tuned n. The relay eedback test, Fg., produces n stead state a square wave type n the crtcal requency o the system, or most o the processes, and the system responses n a snusodal way, as show the Fg.. These sgnals, showed n Fg., are n phase opposton and the oscllaton ampltude o the system s proportonal to the ampltude o the relay. It s desred to obtan the value o the crtcal gan (Ku) and o the crtcal perod (Tu), lke n the Zegler-Nchols Frequency Response Method. Expandng the statonary relay output sgnal, that s a square wave wth requency w u and ampltude h, n Fourer seres and, assumng that the system attenuates the superor order harmonc eects, and stll admttng that the oscllaton o the system output, y(t), have ampltude a and that ths sgnal s n phase opposton wth the nput sgnal, u(t), t can be determned the crtcal gan by the equaton (Wttenmark and Aström, 989): 4h Ku = = G( j w ) πa u ()

3 Determned the values o the crtcal gan and o the crtcal perod, t can be used the rules o the Zegler-Nchols Frequency Response Method (Astrom and Hagglund, 984; Hagglund and Aström, 988; Cardoso, ) to nd PID controller's gans. Wth the tuned controller the swtch o the scheme o the Fg. can be commuted, puttng the system on the PID acton. Ths method, also called o deal relay method, produces satsactory results or a wde spectrum o processes, however t presents two mportant lmtatons the representaton o the relay output sgnal (that s a square wave) or the rst term o the Fourer seres t s an approach that, or many processes (systems o hgh order, wth sgncant delays, etc.), t compromse the tuned controller's perormance (Wang et al, 997). the method makes possble the dentcaton o just a pont o the system requency response uncton, what can be nsucent to descrbe the process satsactorly, resultng n errors o the controller's tunng. Several methodologes o auto-tunng o PID exst derved o the relay eedback test proposed by Astrom and Hagglund (984). However, such methodologes are not presented n ths work. The man methodologes presents n the current lterature are presented n Cardoso (). 3. The Complete Method The methodology proposed here possesses a derent conguraton or the relay eedback test (Fg. 3). In ths test stand out the smple relay, the compensator, Q(s), the reerence sgnal, Re(t) and the system to be controlled, G p (s). Re(t) e(t) u(t) y(t) Gp(s) Relay Relé Processo System yq(t) Q(s) Compensador Compensator Fgure 3. Scheme o the proposed dentcaton test. The compensator Q(s) nserted n the eedback loop o the system wll must delay the nput sgnal o the relay, delayng the output sgnal o system o -9 o. Besdes ths uncton the compensator stll has the property o rejectng hgh requences, protectng the test aganst nosy ntererences. To be reached these speccatons proposes that the compensator s a pure ntegrator or a rst order low-pass lter wth cut requency very neror to the rst natural requency o the system. The reerence sgnal, Re(t), must be derent rom zero to promote an asymmetry n the relay output so that the output sgnal has a level DC. Ths reerence sgnal s varable wth the tme and t depends, n the magntude and n the requency, o the output sgnal o the compensator. It proposes to calculate the reerence sgnal n the ollowng way: Re () t = nre [ pc( t ) vl( t )] + vl( t ) () The values pc(t) and vl(t) are the maxmum and mnmum o the output sgnal o the compensator, yq(t), dented n the prevous perod and nre (reerence level) assumes values between.6 and.9.when nre s.5 the value o the reerence sgnal s same that the medum value o the oscllaton sgnal o the compensator, wth the relay oscllatng n a symmetrcal way, not nsertng DC sgnal on the system. When nre s. the nput sgnal o the relay have lttle oscllatons or, s saturated n the pc(t) value at the statonary oscllatons, not permttng the relay oscllatons and nsertng just DC sgnal on the system. Thus, choosng nre value between.6 and.9 be assure asymmetrc relay oscllatons, nsertng on the system a DC sgnal + square wave n the rst natural requency. Wth the conguraton proposed above or the relay test, t s assured that the oscllaton requency s very close o the rst natural requency o the system and, due to the ntroducton o the reerence sgnal derent rom zero, t s had two very dened areas n the requency response uncton o the system: the regon o the rst natural requency and the regon o statc level (low requences). 3.. Identcaton o the system's FRF Assumng that the system s ntally n steady state, the test begns and are acqured, or a known nterval o tme, the sgnals o the relay output (u(t)) and o the system output (y(t)), (see Fg. 3). The sgnals y(t) and u(t) are multpled

4 by a decreasng exponental term, accordng to Eq. 3. At the end o the test, the values o y( % t ) and u( t ) zero. At the nal tme, the system must be under statonary oscllaton. % are closed to y(t) % = y(t)e u( % t ) = u( t )e where α >. αt αt (3) As the sgnals modulated by the exponental tend to zero at the end o the test, can be appled the Fast Fourer Transorm (FFT) as shown to ollow: N % jwkt % % k= k= N % jwkt % % k= k= jwkt jwkt ( α ) Y( jw ) T y(kt )e T y(kt )e = Y jw + ( α ) U( jw ) T u( kt )e T u( kt )e = U jw + (4) where: w π NT = ; =,,...,m; N m = ; N s the number o the samples o each sgnal and T s the sample tme. The moded Frequency Response Functon (FRF) o the system s: Y( jw + α ) Y( % jw ) G( jw + α ) = = = G% jw U( jw + α ) U( % jw ) where: =,,...,m ( ) (5) The moded requency response o the system, Eq. (5), s enough or PID controller's project, not needng larger computatonal eorts or the estmate the requency response G(jw ). However, necessary, s possble to nd t (Wang et al, 997 e 999, e Hang et al, ). Although a good approach or the requency response o the system s obtaned wth a sngle test, to mprove the perormance and to reduce the random errors provoked by external dsturbances, the procedure s accomplshed Na G% jw, n the requency doman s gven by the average o all values tmes, and the moded nal transer uncton, ( ) o the system's transer uncton, ( ) G jw %, n each requency: p Na G% jw = G% jw ( ) ( ) p β Na β = where: β ndcates the test, Na s the number o accomplshed tests and =,,...,m. (6) Once several tests are accomplshed or the determnaton o the requency response uncton o the system, s possble to nd the value o the coherence uncton n each requency o nterest, by applyng the expresson (Bendat and Persol, 986; Cardoso, ): γ uy Na * Uβ ( jw + α) Yβ ( jw + α) β = + = Na Na * * Uβ ( jw + α) Uβ ( jw + α) Yβ ( jw + α) Yβ ( jw + α) β= β= ( jw α) where: β ndcates the sample; Na the total number o accomplshed samples; U(jw +α) and Y(jw +α) are the FFTs o the sgnals u(t) and y(t) modulated by the exponental declne (α); =,,...,m beng m hal o the ponts number o each sample and, the symbol * denotes the complex conjugated. (7)

5 The Equaton (7) allows determnng n whch requences the moded system FRF, calculated by the Eq. (6), s relable, n other words, n whch requences the dentcaton was well succeeded. Ths normaton s used n the choce o the areas o PID controller's tunng. 3.. The Tunng o the PID Controller Wth the requency response uncton o the system, or the moded FRF, the next step s to accomplsh the controller's tunng. Bascally ths tunng method conssts o adjustng the gans (Kp, K, Kd) o PID controller, mnmzng the error among FRF o the system+controller n open-loop and desred FRF n open-loop, that reaches the requrements put by the operator. Lke ths, ntally s speced a desred requency response o the system+controller n closed-loop that, or nterpretaton easness and knowledge, can be the classc second order system: w H n r () s = (8) s + ξ wns+ wn where: w n s the desred natural requency and ξ s the desred dampng actor and, must be speced by the operator. O the Equaton (8) s derved the open-loop moded requency response uncton, desred or the controlled system: r ( ) r( α ) H ( jw + α ) r ( α ) r wn = + = = H jw + n G% jw G jw where: declne. (jw + α ) + ξ w(jw + α ) w denotes the requences where the controller wll be tuned and α s the coecent o the exponental Consderng PID controller gven by the ollowng structure: (9) K Gc () s = Kp + + Kd s s or: ( ) ( jw + α ) c ( α ) K G% jw = Kp+ + Kd jw + () () The gans (Kp, K, Kd) o PID controller are ound by ttng the open-loop FRF o the system+controller, G% p( jw ) G% c( jw ), to the desred open-loop requency response uncton, ( G% ) r jw. The proceedng s accomplshed by mnmzng the mean square error dened n the ollowng equaton: N mnϑ = G % p( jw ) G % c( jw ) G % r( jw ) () Kp,K,Kd = where: N s the number o requences o ttng. The Equaton () provdes a system wth N equatons, where each equaton corresponds at the one requency n that the controller wll be tunng. Ths system o N equatons and only three varables (the PID gans) can be solved by the least square method. The requences w are ound takng the requences n that the coherence uncton s greater than.95, or be where the dentcaton o the system was accomplshed wth success. These areas are, n general, the statc area (low requences) and the area close to the rst natural requency.

6 4. Numercal Evaluaton The proposed methodology was numercally evaluated, beng used as reerences other works o PID tunng ound n the lterature (see Cardoso,). An oscllatory system wth low dampng and several natural requences was also evaluated. In all numercal tests were consdered: () nose n the nput and output o the system o the order o % o the ampltude o the sgnals; () quantzaton errors n the nput and output, lookng or to smulate the sgnals acquston; () untary ampltudes or the eedback relay; (v) desred response speced wth a dampng actor ξ=.77 (that assure a phase margn greater than 6 o ) and a natural requency (w n ) equal hal the requency o relay oscllaton, or hal the rst natural requency. Were dented wth success the DC gan o the system and the pont o smaller requency where the phase o the system s -9 o, n all tests. To ollow some numercal smulatons are presented. Low order non-oscllatory system:.57 G( s ) = e ( 8.6s + ) Man characterstcs: monotonc non-oscllatory long dead tme low order 8.7s PID controller ound: Kp =.585 K =.7 Kd = 5.8 Fgure 4. Low order non-oscllatory system. - Hang et al () Ampltude Unt Step Response desred desejado obtaned obtdo Tme [s] Hgh order non-oscllatory system:.8 G( s ) = e 3 ( s+ ) ( s+ ) s - Wang et al ().8 Unt Step Response Man characterstcs: monotonc non-oscllatory long dead tme hgh order PID controller ound: Kp =.88 K =.85 Kd =.85 Fgure 5. Hgh order non-oscllatory system. Ampltude Tme [s] desred desejado obtaned obtdo

7 Hgh order oscllatory system: G( s ) = e ( s + s+ 3)( s+ 3).3s - Hang et al ().8 Unt Step Response Man characterstcs: monotonc oscllatory hgh dampng hgh order PID controller ound: Kp =.8 K = 3.3 Kd =.73 Fgure 6. Hgh order oscllatory system. Ampltude.6.4. desejado obtdo 5 5 Tme [s] desred obtaned Non-mnmal phase system: ( 5s) G( s ) = e ( 5s + )( 4s + s + ) Man characterstcs: non-mnmal phase oscllatory hgh dampng PID controller ound: Kp =.796 K =.38 Kd =.893 Fgure 7. Non-mnmal phase system. 5s - Wang et al (999) Ampltude Unt Step Response desred desejado obtaned obtdo Tme [s] Oscllatory system wth low dampng and several natural requences: G() s = e s +.87s s +.99s s +.35s Natural requences and ts dampng: = 4.5 Hz - ξ =.5 = 4.63 Hz - ξ =.5 3 = 8.5 Hz - ξ =..s Unt Step Response Man characterstcs: monotonc oscllatory low dampng hgh order several natural requences very close two natural requences Ampltude desred obtaned PID controller ound: Kp =.73 K =.67 Kd =.69 Fgure 8. Oscllatory system wth low dampng and several natural requences Tme [s]

8 5. Expermental Evaluaton The proposed methodology was appled to a contnuous system, wth nntes degrees o reedom. Ths system s consttuted o a cantlever beam wth pezoelectrc actuators ncorporated (Fg. 9). An electromagnetc proxmty sensor s used to detect the beam traverse dsplacements n a certan pont. The sgnals o ths sensor go by a sgnal condtoner and are sent to the acquston system. Wth the sgnal o the proxmty sensor, the computer accomplshes the necessary calculatons and, through the acquston system, sends a voltage sgnal or the power ampler o the pezoelectrc actuator (PZT), sendng the condtoned sgnal to PZT (one n each ace o the beam) that s deormed. Once PZT s xed to beam, t orces the beam provokng ts movement. Data o the cantlever beam wth pezoelectrc actuators ncorporated: Dmensons: Length x Wdth x Thckness o the beam: 4 x 34.5 x. mm Poston o the PZT rom base: 5 mm Poston o the sensor rom base: 4 mm Pezoelectrc actuator: Actuator ACX QPN PZT Sensor Beam Pezoelectrc Actuator PZT Dsabled Sensor Ampler Condtoner Fgure 9. Cantlever beam system wth pezoelectrc actuators ncorporated. In the Fgure s showed the requency response uncton (FRF) o the beam system obtaned wth a sgnal analyzer (n red/contnuous) and the dented FRF wth the proposed test (n blue/dashed). In ths gure, the nverted trangle ndcates the requency o relay oscllaton, whch s very close to the rst natural requency o the system. For the proposed method, the coherence uncton norms a good condence o estmaton at low requences and at the area close to the rst natural requency o the system. In the dentcaton test be used a samplng tme o 3 ms and, was accomplshed tests wth seconds o tme each. The relay ampltude was. V, the reerence level (nre) equal to.9 and the compensator o the ntegrator type. Coherence Phase [ o ] Magntude [db] relay test analyzer Frequency [Hz] Fgure. Frequency response uncton o the cantlever beam system: reerence FRF and dented FRF. Once the two areas o nterest (low requences DC gan and area close to the rst natural requency o the system) were dented by proposed methodology, the desred response was speced wth a dampng actor (ξ ) o.77 and a natural requency (w n ) o. Hz. Fttng the open-loop response o the system+controller to the desred response n open-loop, was obtaned the ollowng PID controller's gans: Kp = -.383, K = e Kd =.76.

9 In the Fgure the response o the beam system s showed when s speced a square type reerence sgnal wth ampltude o. V and requency o. Hz. Ampltude [V] [V] Reerence Snal de sgnal reerenca and output e sada o do the sstema controlled controlado system reerênca reerence controlado controlled Tempo [s] Fgure. Controlled beam system when s speced a square reerence sgnal. 6. Conclusons The technques o auto-tunng o PIDs controllers have showed more and more attractve, once the tuned controllers present a good perormance wth a smple and ast project. Several works exst regardng the auto-tunng o PID controllers. However, the present methods n the current lterature are just appled to an or other system type and, t are not applcable to oscllatory systems wth low dampng and several natural requences. The complete method proposed can be appled wth success to a large range o processes, even to oscllatory systems wth low dampng and several natural requences. Although demands certan computatonal eort, manly n the FRFs' calculaton, the complete method can be mplemented n dgtal sgnal processors (DSPs), as or example the TMS3C47, or stll n mcroprocessors o low processng capacty, snce the acqured data are stored and processed ater the dentcaton test. The method hghlghts by the hgh automaton level, once ew user nterventons are make on the tunng process o PID controller. Although the objectve o ths work was to evaluate n whch systems the proposed method could be appled, the tuned PIDs controllers presented good perormance. Specyng other values or the desred dampng actor (ξ) and or the desred natural requency (w n ), the controllers' perormance can be mproved. In the non-mnmal phase system, although the PID controller's structure s not capable to elmnate the nonmnmal phase characterstcs, t kept the response o the controlled system very close to the desred response. The proposed method showed very attractve, wth good perormance and, ast and smple project or the tested systems, even to oscllatory systems wth low dampng and several natural requences. The proposed method can be moded to project a PI controller or a PD controller, as also s possble to specy several types o desred responses. 7. Acknowledgement The authors are grateul to CNPq (Conselho Naconal de Desenvolvmento Centíco e Tecnológco) or nancal support provded to ths work. 8. Reerences Tme [s] Aström, K. J. and Hagglund, T.,, The Future o PID Control, Control Engneerng Practce, Vol. 9, pp Aström, K. J., Hagglund, T., Hang, C. C. and Ho, W. K., 993, "Automatc Tunng and Adaptaton or PID Controllers A Survey", Control Engneerng Practce, Vol., pp Aström, K. J. and Hagglund, T., 984, Automatc Tunng o Smple Regulators wth Speccatons on Phase and Ampltude Margns, Automatca, Vol., pp Bendat, J. S. and Persol, A. G., 986, "Random Data: Analyss and measurements Procedures", nd edton, Wley NY. Cardoso, P. M.,, "Estudo, Proposta e Avalação de Novas Metodologas de Sntona Automátca de Controladores PID Baseadas no Ensao do Relé Realmentado", Dssertação de Mestrado, Unversdade Federal de Uberlânda, Uberlânda-MG, Brasl. Hagglund, T. and Aström, K. J., 99, "Industral Adaptve Controllers Based on Frequency Response Technques", Automatca, Vol. 7, pp Hagglund, T. and Aström, K. J., 988, "Automatc Tunng o PID Controllers", Instrument Socety o Amerca.

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