An estimation approach for autotuning of event-based PI control systems

Transcription

1 Acta de la XXXIX Jornada de Automática, Badajoz, 5-7 de Septiembre de 08 An etimation approach for autotuning of event-baed PI control ytem Joé Sánchez Moreno, María Guinaldo Loada, Sebatián Dormido Departamento de Informática y Automática, UNED, C/ Juan del Roal 6, 8040 Madrid {janchez, mguinaldo, dormido}@dia.uned.e Antonio Viioli Department of Mechanical and Indutrial Engineering, Univerity of Brecia, via Branze 38, 53 Brecia, Italy {Antonio.viioli@unib.it} Abtract An identification procedure deigned to be part of an autotuning method for event-baed proportionalintegral (PI) control ytem i propoed in thi contribution. The rationale of the identification method i baed on the information obtained from the limit cycle that the event baed ampler plu an adequate tuning of the PI controller can generate in the cloed loop. From the information of two limit cycle at different frequencie, the parameter of the common tranfer function ued for tuning of PI controller will be deduced. Simulation demontrate the effectivene of the method. Keyword: end-on-delta, limit cycle, event, identification, autotuning, PI controller. uch point correpond to the third and fifth harmonic of the output ytem). In the procedure decribed here, the PI controller work on-line during the identification a the proportional and integral part are taken into account to generate the limit cycle. Alo, the iue of providing accurate reult at low frequencie i worked out by adding an additional delay in order to reduce the frequency of the limit cycle. The paper i organied a follow. In Section the event-baed architecture i preented. The eventbaed identification procedure i decribed in Section 3. Section 4 explain how to improve the procedure. Finally, concluion are given in Section 5. Introduction Method for the identification of tranfer function parameter in event-baed PI control loop have been propoed in the lat year in everal publication. The firt invetigation wa decribed in [4]. In that work, the proce parameter are etimated conidering a limit cycle generated by a pre-tuned event-baed PI controller. Other two method are decribed in [5, 6]. In uch contribution, the rationale of the etimation method i baed on curving fitting and tate-pace approache. Contribution on pecific method for identification in an event-baed control loop have been recently reported in [0, ]; both method are baed on forcing a limit cycle. The identification approach decribed in thi paper i baed on [] but taking into account the full PI controller. In [], the integral part of the controller i deactivated during the identification and only the proportional part i ued to generate the limit cycle; it i alo neceary to add a bia to the ampler output to introduce aymmetry in the limit cycle to calculate the dc gain. However, one of the con decribed in [] i that in lower frequencie the identification of procee with integration can be not very accurate a ome of the critical point neceary to etimate the parameter are located in the firt and econd quadrant of the Nyquit plot (it i due to the fact that Fig.. Event-baed control architecture. Event-baed PI control architecture The control architecture conidered in thi contribution i hown in Figure. In thi event-baed control ytem, when the ampler detect an event, it end the information to the PI controller C(. Many logical condition have been propoed in the literature for the occurrence of an event. The one employed here i the Symmetric Send-On-Delta (SSOD) ampling [4]. It behaviour i decribed a * e ( t) ( i ) i ( i ) if e( t) ( i ) e*( t if e( t) [( i ), ( i ) ] e*( t if e( t) ( i ) e*( t ) i ) i () ) i With thi logical condition, the ampler receive a continuou ignal e(t) and generate a ampled ignal e (t) that i multiple of. The key of the relationhip between e(t) and e (t) i that it can be conidered a a 436

2 generalization of a relay with hyterei. Thi implie that it decribing function can be derived [7]. the parameter of the tranfer function ued for tuning a PI controller. The rationale of the procedure conit in forcing the ytem to illate at a frequency by the detuning of the PI controller C(. A aid before, the ytem will illate at a conequence of the interection of G( C( with the reciprocal of the SSOD ampler decribing function in the Nyquit map, that i, G( j ) (3) N ( ) Figure : Nyquit plot of N( ). The decribing function of the SSOD ampler i given by [9] N( ) A j A m A m m k k A () where A i the amplitude of a inuoidal input ignal, and m A. The portrait of N( ) i hown in Figure for A,. Each interection in Figure of the ytem G( C( with an arc of N( ) produce an illation (or limit cycle) of a different amplitude: Interection with the arc tarting in C produce illation with A,, interection with the arc tarting in C generate illation with A, 3, and o on. So, for example, the interection of a ytem G ( with the point C j in the Nyquit map repreent 4 4 the exitence of a limit cycle of amplitude A and frequency ; thi frequency atifie the expreion G( j ) C. 3 Identification procedure The identification method i baed on the table illation induced in the ytem G( thank to the exitence of the event-baed ampler. It mut be noticed that the current proce to identify mut interect the negative real axi (if not, it hould be added a certain delay). Once the ytem i in a table limit cycle, experimental meaurement derived from the illatory ignal are taken and ued to obtain A the condition for the exitence of limit cycle i given by (3), a convenient detuning of C( will produce an illatory behaviour of the ytem. So, if the Nyquit point where the ytem i illating at i meaured experimentally, that i, G( j ), it i feaible to derive the parameter of a given tranfer function model P ˆ( ). Thu, once the ytem i illating, the procedure for fitting a model i: (a) To meaure G( j ), (b) To get the experimental value of the proce at the illation frequency, that i, j ), by removing C( j ) from (3), (c) To obtain j ) and arg j ), (d) To equate the two value obtained in the previou tep to the magnitude and argument expreion of the tranfer function elected to fit, and (e) To olve the equation ytem and get the model parameter. Thee tep are now explained in a more detailed way. The olution adopted to get G( j ) during a tet i firt preented in [4] and i proved in []. A in a limit cycle, y(t) and u(t) are periodic and piecewie ignal, uing the Laplace tranform of both, it can be written / Y ( j y t e dt ) ( ) 0 G( j ),(4) U ( j / ) jt u( t) e dt 0 j t where y(t) and u(t) are meaured during a tet. It mut be noticed that (4) cannot be applied to determine the teady gain ( 0) becaue the illation produced by the SSOD block are ymmetric and the integration of the period will be zero. How the procedure i applied to get the teady gain will be 437

3 explained afterward, but we anticipate that the incluion of an additional delay will play a key role. A the PI control parameter and are known, it i poible to obtain the value of C j ), ( C( j ) K p (5) jti Uing (4) and (5), it i eay to obtain the experimental value of the proce at the illation frequency, G( j ) j ) (6) C( j ) The tranfer function model conidered in thi work to explain the procedure are: Model FOPTD: K L P ˆ ( e T (7) Model IFOPTD: K L P ˆ ( e T (8) Model SOPTD: K L P ˆ ( e ( T ) (9) and their argument and magnitude expreion are: Model FOPTD: K j ) (0) T arg j ) arctan( T ) L () Model IFOPTD: K j ) T () arg P ˆ( j ) L arctan( T ) (3) Model SOPTD: K j ) T (4) arg j ) arctan T (5) L To get K and T it i neceary to equate j ) with the magnitude of a tranfer function j ) and olve the ytem of equation. A there are two unknown, K and T, in the magnitude expreion, it i neceary to run two tet to get two experimental value, that i, j _ ) and j _ ). Notice that each tet will be run with a different et of control parameter to force the ytem to illate at different frequencie, that i, _ and _. It will be explained how to modify the PI parameter in the following paragraph depending on the proce and the model to identify. Once K and T are known, the delay L i obtained by equating arg j _ ) with the argument expreion of the elected tranfer function model to fit, that i, with (), (3) or (5). The following expreion are the reult of olving the equation for the three model. For the ake of implicity, repreent _ i, P i repreent i P ), and arg Pi correpond to ( _ i arg j ). _ i Model FOPTD: ω ω K P P (6) P ω P ω P ω P T (7) P P ω arg P arctan( Tω ) L (8) ω Model FOPTDI: 4 ω ω K P P ωω (9) P ω P ω 4 4 T P ω P ω (0) P ω P ω arg P arctan( Tω ) 0. 5 L ω () Model SOPTD: K ω ω P ω P ω P P () P P T (3) P ω P ω arg P arctan( Tω ) L (4) ω A aid before, it i neceary to run two tet to meaure G( j _ ) and G( j _ ). The firt tet i done jut by increaing the proportional gain K p until the ytem reache a limit cycle and illate at a frequency _. The econd tet i prepared by a 4 438

4 econd increae of K p to reach a new limit cycle at another frequency. However, the previouly defined procedure jut work when the current proce and the model to fit have the ame order and tructure. Thi i due to the following reaon: - If the tranfer function template to fit P ˆ( ) i exactly equal to the actual proce to identify, the identification procedure will provide an exact reult. Thi i due to the fact that the template i fitted with the ame degree of freedom than the true proce. A reult, the behaviour of G ˆ( ) will be equal to G ( in all the frequencie range. - If the proce ha a higher order than the template or a different tructure, thi will produce the reult to be exact at the range of frequencie between _ and _ but with dicrepancie at other frequencie. Thi i a conequence of fitting the template with leer degree of freedom than the true proce. The effect i that the behavior of the model at frequencie out of the range Modofy can become very inaccurate. Such fact will be epecially notoriou and viible at frequencie below _ or at the teady tate when the fitted model i a FOPTD or a SOPTD, that i, when the current proce doe not have integral dynamic. The olution propoed in thi work conit of forcing the ytem to illate during the econd tet at a very low frequency a cloe to zero a poible. To reduce the frequency of the limit cycle below _ an additional delay will be added to the ytem during the econd tet. Next ome example are given in order to explain better the problem and preent the olution. 3.. Identification of IFOPD procee Example : To tart illutrating the event-baed identification procedure, let conidering the proce [8], 0. e (5) Initially, the proce i controlled by a PI tuned to force the ytem to illate. The controller parameter elected for uch a goal are [ K p, Ti 0]. In all the imulation, meaurement noie wa not conidered and wa et to. The data obtained in the firt tet were G( j ) ( j) at frequency For the econd tet, K p wa increaed to. to obtain a limit cycle at a higher frequency and T i wa not changed. Now, the econd tet data were and _ G( j _ ) ( j). The model parameter were obtained by applying (5) and (6) to the previou data to get j _ ) and j _ ), and after that, uing (9), (0) and (). The reulting model and reult obtained from other relay-baed identification method are preented in Table I. It can be appreciated that the event-baed procedure give reult of the ame quality a more elaborated method baed on tate-pace [] and curve-fitting [8] approache. Table I: Model and error for (5) where pc. 6. Event-baed.0000 e 0.05 procedure.0000 By [8] By [].0000 e e The accuracy of the etimated proce model i computed uing the frequency domain etimation error index ( E ~ ) for each of the proce model i found by applying integral of abolute error (IAE) criterion a pc ~ j) j) E d, j) 0 where pc i the phae cro over frequency of the actual proce P (, that i, the frequency where phae hift i equal to -80º. Example : Let now conidering the identification of the higher-order proce preented in [8], ( ) e 5 5 (6) After two conecutive tet with the two et of control parameter [ K 0., T 50] and p [ K p 0., Ti 50], the model obtained i hown and compared in Table II. The obtained data were and j ) ( j) i 439

5 for the firt tet, and and _ j ) ( j) for the econd one. _ found to be [ K p 5, Ti 0] and [ K p 6, Ti 0]. The reult of the fitting can be found in Table III. It mut be noticed that the reult of the event-baed procedure i very accurate becaue the actual proce ha the ame order that the template to fit. Example 4: Now the following high-order proce i going to be identified a a FOPTD model 4 P ( ( ) (8) Figure 3: Plot of (6) and the identified model. In the Example, the Nyquit plot of the model and the proce are apparently imilar in the third quadrant (ee Figure 3). In particular, the model and the proce behave in a imilar way between _ and _. However, there are dicrepancie at lower frequencie. Indeed, at the frequencie 0.0, 0.00 and 0.000, the difference between the true proce and model, that i, j) j), are 0.088, 0.87, and 8.73, repectively (ee detail in Figure 3 of the Nyquit point at 0. 00). Table II: Model and error for (6) where pc By the eventbaed e.9670 procedure By [8] By [3] e e Identification of FOPTD procee Example 3: Conider the following proce e (7) that i being well controlled by a SSOD-PI tuned with [ K p, Ti 0]. By increaing the proportional gain, the et of parameter ued to enter the ytem into two different table limit cycle are where pc. Applying the procedure a before, that i, with two et of control parameter that force the ytem to illate, for example, [ K p.5, Ti 3] and [ K p.6, Ti 3], the etimated model i.75.93e Table III: Model and error for (7) where pc By the event-baed e procedure By [] By [] e e Obviouly, uch reult i not acceptable a the teady gain i far from the correct value of one producing an ~ etimation error index very high ( E ). With the identification procedure a originally defined, the fitting of the model i good around the two Nyquit point defined by the frequencie of the two limit cycle but not at 0. In thi example, uch illation frequencie are and , and the difference are mall, P P but not for 0, ) j ( j ) j ( j P ( 0) 0).93 ) ) It can be oberved in Figure 4 that the identified model fit correctly around the illation frequencie meaured in the two tet. In particular, the fitting i exact for _ a it i the frequency 440

6 elected for getting L with (8) once K and T are known by (6) and (7). theoretical ection i located between arg( / N (, )) and arg( / N (, )) 0. 75, that i, between -80º and -35º. Figure 4: Nyquit plot of (8) and the fitted model. If a fitting of a SOPTD model i tried, the new reult improve with repect to the previou FOPTD model, 0.94 ˆ.0 e (.70 ) Figure 5: Step to modify the econd tet to get illation at frequencie near zero. A way to get that i by rolling G( around the center of Nyquit map. A the rotation can be done by adding a delay L ad to G (, that i, L ad G' ( G( e, bound for L ad to aure that P will be rotated inide the grey area are given by, ~ with E , but there i till a 0% of dicrepancy at 0, arg P 0.75 arg P Lad (9) P ( 0) 0) Modifying the procedure A practical olution to make a correct identification i to generate in the econd tet a limit cycle at a frequency _ a near zero a poible. A point of G ( with a frequency are, in general, far from the interection with the DF of the even-baed ampler and, alo, due to the integral action of the PI controller, the point will be located along the negative real axi of the Nyquit map. The olution i to add new dynamic to G ( to allow that the very low frequency range of the new ytem G' ( interect in ome point with N( ). To undertand how to modify the etimation procedure to make the econd tet with a low frequency limit cycle, ee the tep depicted in Figure 5. Step conit in rotating a unknown Nyquit point P G( j _ ), where _ i a very low frequency, to the grey area depicted in Figure 5. That area repreent the theoretical ection of the Nyquit map where the interection of G '( with the negative reciprocal of N ( ) can be produced after a radial movement of the point P (Step ). Thi Auming that the frequency _ elected for the econd tet i low enough (e.g., 0. _ ), and becaue of the integral action of the PI controller, we can conider arg P 0.5 (that i, -90º) at very low frequencie. Thu, from (9) practical bound for L ad could be, Lad (30) _ and fixing _ 0. _, 5.5 Lad (3) _ Now, a the unknown point j _ L P G( j ad _ ) e i located in the grey area but far from the interection with / N( ), it i neceary to give a econd tep. Thi tep conit Lad in a radial tranlation of the new ytem G( e looking for an interection with / N( ). That mut be done by reducing the proportional gain a it can be appreciated in Figure 5. Unfortunately, the calculation of thi gain i not intuitive and mut be done by trial and error. 44

7 Example 5: We identify the previou proce 4 P ( ( ) by applying the modified procedure. The firt tet i run with the ame parameter a in the previou example, that i, [ K.5, T 3], and the reult i p i _ and arg G( j _ ) 46º. By fixing _ 0. _ and according to (30), we obtained 8.5 L ad 4.. The econd tet i run with the following et of parameter [ K p 0.5, Ti 3, Lad.4] and the frequency meaured i , that i cloe to 0.. In Table IV and Figure 6, the new etimation i preented and compared with the model obtained by a more elaborated method. Table IV: FOPTD model of (8) where pc By the event-baed.006 e procedure.503 By [3] e Example 6: Table V how the fitting of 4 P ( ( ) to a SOPTD model uing the reult of the two tet of the Example 5. A aid before, if the tructure of the actual proce and the model to fit are the ame, the original method i valid for any model and it i not neceary in the econd tet to force the ytem to illate at a very low frequency. But if the tructure of the actual proce i higher than the model template it will be neceary to modify the method a explained before. However, the original method i valid for FOPTDI fitting of high-order procee with one pole at the origin (ee Example 7). It i due to the double integral action introduced by the proce and the controller. Forcing the econd limit cycle at a very low frequency can be done by reducing the proportional gain ued in the firt tet. The effect of thi action produce two conequence in the Nyquit plot of G ( : (a) to be moved radially toward the origin, and (b) the reduction of the phae margin a conequence of a lower integral gain ( K p / Ti ). The radial movement produced an approach of the low frequencie to the origin, and the reduction of the phae margin reaure the interection of G ( with the reciprocal of the decribing function of the eventbaed block. 4 Figure 6: Nyquit plot of ( ) and the identified model with the modified procedure. Table V: SOPTD model of (8) where pc..7 By the event-baed.0007 e procedure (.589 ) e By [3] (.76 ) Example 7: To produce a new limit cycle at a very low frequency uing the proce of the Example, a new imulation i run with the control parameter [ K p 0.00, Ti 50]. It mut be noticed that the proportional gain ha been ignificantly reduced with repect to the parameter applied in the econd tet in Example (that are [ K p 0., Ti 50] ). Now, the frequency of thi new limit cycle i new It can be oberved in Figure 7 the difference in the frequencie of the limit cycle depending on the elected et of controller parameter. The identified FOPTDI model of (6) uing data from the limit cycle at and i _.000e _ (3) With thi new model, the dicrepancie at lower frequencie with repect to (6) have been reduced. For the frequencie 0. 0, 0.00 and 0.000, the difference j) j) are 0.07, 0.67, and.66, repectively (compare thee value with thoe preented at the end of Example ). 44

8 Figure 7: Plot of G ( where 5 ( ) e ( ) parameter. Concluion 5 and C ( change it In thi paper, an autotuning method completely deigned for event-baed PI control loop ha been preented. The identification approach i baed on the information obtained from two limit cycle produced by the SSOD ampler and the PI controller. Simulation example have proven the effectivene of the method. However, there are ome iue that need to be improved. For example, regarding the identification of FOPTD and SOPTD model, it i neceary to improve the procedure to determine the econd tet, epecially the etimation of the new proportional gain to apply in the Step. Thi will be part of future invetigation. Acknowledgement Work funded by Spanih Minitry of Economy and Competitivene under contract DPI C--R and DPI C--R. Reference [] Bajarangbali, S., Mahji, J. (05) Identification of integrating and critically damped ytem with time delay, Control Theory and Technology 3(), pp [] Bajarangbali R., Majhi S., Pandey S. (04) Identification of FOPDT and SOPDT proce dynamic uing cloed loop tet, ISA Tranaction 53(4), pp [3] Berner J., Hägglund T., Åtröm K.J. (06) Aymmetric relay autotuning - Practical feature for indutrial ue, Control Engineering Practice 54, pp [4] Bechi, M., Dormido, S., Sánchez, J, Viioli, A., Yebra, L.J. (04) Event-baed PI plu feedforward control trategie for a ditributed olar collector field, IEEE Tranaction on Control Sytem Technology (4), pp [5] Bechi, M., Dormido, S., Sánchez, J, Viioli, A. (05) Cloed-loop automatic tuning technique for an event-baed PI controller, Ind. Eng. Chem. Re. 54(4), pp [6] Bechi, M., Dormido, S., Sánchez, J, Viioli, A. (05) An Event-baed PI Controller Autotuning Technique for Integral Procee. t IEEE International Conference on Event-Baed Control, Communication, and Signal Proceing (EBCCSP- 05)., Krakow (Poland). [7] Gelb, A; Van der Velde, WW. (968) Multiple- Input Decribing Function and Nonlinear Sytem Deign. New York, USA: McGraw-Hill. [8] Panda, C., Vijayan, V. Sujatha, V. (0) Parameter etimation of integrating and time delay procee uing ingle relay feedback tet, ISA Tran 50(4), pp [9] Romero J.A., Sanchí, R. (06) A new method for tuning PI controller with ymmetric end-ondelta ampling trategy, ISA Tran 64, pp [0] Sánchez, J., Guinaldo, M., Dormido, S., Viioli, A. (08) Identification of proce tranfer function parameter in event-baed PI control loop, ISA Tran 75, pp [] Sánchez, J., Guinaldo, M., Dormido, S., Viioli, A. (08) Enhanced event baed identification procedure for proce control. Ind. Eng. Chem. Re. DOI: 0.0/ac.iecr.7b0539. [] Srinivaan, K., Chidambaram, M. (003) Modified relay feedback method for improved ytem identification, Comput. Chem. Eng. 7(5), pp [3] Tao, L., Furong, G. (008) Identification of integrating and untable procee from relay feedback, Comput. Chem. Eng. 3(), pp [4] Vivek, S., Chidambaram, M. (005) Identification uing ingle ymmetrical relay feedback tet. Computer & Chemical Engineering 9(7), pp by the author. Submitted for poible open acce publication under the term and condition of the Creative Common Attribution CC-BY-NC 3.0 licene ( 443