Design of Centralized PID Controllers for TITO Processes*

Transcription

1 6th International Sympoium on Advanced Control of Indutrial Procee (AdCONIP) May 8-3, 07. Taipei, Taiwan Deign of Centralized PID Controller for TITO Procee* Byeong Eon Park, Su Whan Sung, In-Beum Lee Abtract A new method for deigning the centralized proportional-integral-derivative (PID) controller in two-input two-output (TITO) procee i propoed. The propoed method ha two diagonal part PID controller and two off-diagonal part PID controller. The diagonal part PID controller are to attenuate the interaction of the TITO procee and the off-diagonal part PID controller are to track the etpoint. The diagonal part PID controller are directly tuned by the conventional ingle-input ingle-output (SISO) PID tuning method on the bai of the diagonal part of a proce model matrix. And the tuning parameter of the off-diagonal part PID controller are calculated by minimizing the effect of the off-diagonal component of the open-loop tranfer function matrix in frequency domain. The propoed control method how better decoupling and etpoint tracking performance than previou approache. I. INTRODUCTION TITO procee are frequently encountered in chemical and petroleum indutry. Unfortunately, it i not eay to achieve high control performance for the TITO procee if the loop interaction and time delay are ignificant. Hence, a plenty of control cheme have been introduced to control the TITO procee with high control performance by attenuating the interaction in a ytemic way in everal previou decade. One of them i a model predictive control (MPC) that calculate a preent controller output by minimizing the predicted future control error baed on the proce model by olving a contrained optimization problem in each ampling time []. Even though the MPC theoretically provide an optimal control performance and manage a proce contraint, PID controller are till the mot popular proce control cheme for indutrial field becaue of their implicity, robutne and eay maintenance. One of the commonly ued PID control cheme for the TITO procee i a multi-loop (decentralized) PID control hown in Figure. In order to pair proce input and output, the relative gain array (RA) analyi or the ingular value analyi (SVA) [] have been ued. Thee method can provide mot effective proce input * Thi work wa upported by Human Reource Program in Energy Technology of the Korea Intitute of Energy Technology Evaluation and Planning (KETEP), granted financial reource from the Minitry of Trade, Indutry & Energy, Republic of Korea. (No ) Byeong Eon Park i with the Department of Chemical Engineering, Pohang Univerity of Science and Technology, Pohang, yeongbuk, Korea ( bepark@potech.ac.kr) Su Whan Sung i with the Department of Chemical Engineering, Kyungpook National Univerity, Daegu, Korea 7070 ( uwhanung@knu.ac.kr) In-Beum Lee i with the Department of Chemical Engineering, Pohang Univerity of Science and Technology, Pohang, yeongbuk, Korea (correponding author to provide phone: ; fax: ; iblee@potech.ac.kr) Y Y Y Y Figure. Block diagram of the multi-loop (decentralized) control. Figure. Block diagram of the multi-loop (decentralized) control with decoupler. Y Y E E E E E E c c c c c c c c u u u u d d d d U U p p p p Figure 3. Block diagram of the centralized control. and output pairing to achieve high control performance attenuating the proce interaction for the multi-loop PID control cheme. And the tuning method for the multi-loop PID control cheme have been propoed in everal paper [3][5]. Neverthele, if the loop interaction are ignificant, the multi-loop control PID control cheme are tructurally not recommended to attenuate the loop interaction. So, decoupler have been developed to overcome the limitation of the multi-loop control cheme. It i hown in Figure. Typically, mot decoupler are deigned on the bai of the teady-tate model or dynamic model of the procee [6][9]. The tatic decoupler are baed on the teady-tate model o that they cannot guarantee acceptable tranient decoupling performance. While the dynamic decoupler can provide good tranient decoupling performance theoretically, however, it i U U U U p p p p p p p p y y y y y y y y y y y y Y Y Y Y Y Y /7/$ IEEE 53

2 Y Y c c c c Y Y Y Y (3) Y Y c c c c c c c c Y Y Y Y (4) ( c( ( c( 0 c( c( ) (5) ( c( ( c( 0 c( c( ) (6) c c θ A( e A( ( θθ) θ c( c( e (7) A e A θ A( e A( ( θθ) ( θ c( c( e (8) A e A c c kc ( ) ( kc kc ie for i kc ( ) ( kc kc ie for i kc ( kc kc i ) for c( i kc ( kc kc i ) for c( i (9) (0) () () not eay to implement them in real indutry attributable to their complex tructure. An alternative methodology for decoupler i an inverted decoupling which manipulate a proce input a a combination of a controller output and the other proce input, and it how good etpoint tracking and decoupling performance [0]. However, when the order of the numerator i higher than the order of the denominator in the feedback decoupler, the inverted dcoupling i o enitive to noie that it cannot guarantee acceptable control performance. Another control cheme to control TITO procee i a centralized PID control hown in Figure 3. When the proce interaction are highly interactive, the centralized PID control i uperior to the multi-loop PID control cheme ince the centralized PID control can attenuate the proce interaction from the tructural point of view. Lielehto propoed a n n centralized control deign method on the bai of SISO PID deign method []. Wang et al. propoed an automatic tuning method to deign the centralized PID controller uing the equential relay feedback method []. Recently, a tuning method for the centralized PI controller baed on a ynthei method wa introduced [3]. It ue the RA analyi to approximate the invere of the proce tranfer function matrix. The above-mentioned deign method for the centralized PID controller cannot guarantee acceptable decoupling performance in a time delay proce. To conider the ignificant time delay effect of the TITO proce more ytematically, Morilla et al. ued a control tructure of the ideal PID control with a time delay for the centralized control [4]. The Morilla method how good control performance for the TITO procee with ignificant time delay. However, the Morilla tuning algorithm i too much complex to apply in indutry, and it require uer-defined parameter; deired control trajectory. In thi reearch, a new deign method to tune the centralized PID controller i propoed in a conceptually traightforward way. It goe through the following tep. Firt, the effective proce input and output are paired uing the RA analyi. Second, when the RA analyi indicate the proce input paired with the proce output and the proce input paired with the proce output are the bet pairing, the diagonal-part PID controller are tuned by the conventional SISO PID tuning method [4-6]. Third, the off-diagonal part PID controller are tuned to remove the interaction of the proce. The off-diagonal part PID controller have a time delay term to compenate the time delay in a tructural point of view. The propoed method calculate the tuning parameter in an analytic way without olving an iterative optimization problem and requiring additional uer-defined parameter uch a gain margin or phae margin. 54

3 c ( e 4 ( e (8) c ( e (0) c ( e.43 ( e () c ( e.45 ( ) (4) II. PROPOSED DESIN METHOD FOR CENTRALIZED PID CONTROLLER Conider the following tranfer function matrix of TITO proce. Y Y U( U where ij(, i=,; j=, repreent the tranfer function of each ub-proce. Y ( and Y ( are the proce output. U ( and U ( are the proce input. The centralized PID controller can be repreented by (), where, () cij i=,; j=, are the PID controller. Y ( and Y ( are the controller etpoint. U( U c c c c Y Y Y Y The cloed-loop repone equation in (3) and (4) are obtained by combining () and (). Here, without lo of generality, it i aumed that the RA analyi provide the bet paring of Y paired with U and Y paired with U. Then, the tranfer function of (4) hould be diagonal to remove the loop interaction by atifying (5) and (6). Letting ij ij ( () ij A e, i=,; j=,, (5) and (6) can be rewritten a (7) and (8). Note that the controller tranfer function of (7) and (8) have the time delay term ( ) and e. It i not eay to approximate the frequency repone of the time delay term for the ideal PID tructure becaue they how totally different frequency repone, while it i relatively eay to approximate the time delay free term for the ideal PID tructure. So, the ideal PID tructure plu time delay term for a poitive time delay and the ideal PID e ( ) tructure for a negative time delay are choen to remove proce interaction a (9) - (). Time delay-free term of (9), (0) and (), () are linear combination of the tuning parameter and the Laplace variable. Therefore, we can apply linear leat quare method to obtain the tuning parameter of the off-diagonal part PID controller by approximating the frequency repone of ( A( / A( ) c( and ( A( / A( ) c(. For convenience, let Q ( A( / A( ) c(. And, the Laplace variable i replaced to i k, k=,,,n to obtain the frequency repone data where ω denote frequency. Then, the following linear equation can be traightforwardly derived by follow: B ΦΡ. (3) Thu, the matrix B, Φ and P are Q( i0 ) B Q( in ), Φ / i0 i0, / in in kc P kc / i. (4) kc d Since (3) and (4) have imaginary term, the olution of the linear leat quare method hould be calculated by T T Φ' Φ' Φ' B' P (5) where, Re(B) Re Φ B', Φ' Im(B). (6) Im Φ 55

4 Figure 4. Repone of the etpoint change in Wood and Berry column. Figure 6. Repone of the etpoint change in Tyreu tabilizer. Figure 5. Repone of the etpoint change in Wardle and Wood column Figure 7. Repone of the etpoint change in Vinante and Luyben plant. 56

5 TABLE I. CONTROL PERFORMANCE EVALUATION USIN RMSE Wood and Berry column Wardle and Wood column Tyreu tabilizer Vinante and Luyben plant Tuning method Y Y Y Y Y Y Y Y Propoed BLT Wang It mean that the PID tuning parameter can be analytically obtained from (5) without olving iterative nonlinear optimization problem. The PID tuning parameter of can be determined in the ame way with the cae of. III. CASE STUDY c c Four kind of typical TITO procee are controlled by the propoed method and Luyben bigget log-modulu (BLT) method [3] and Wang method [] are imulated to demontrate that the propoed method how almot equivalent or better control performance compared with previou approache a well a it doe not require the uer-defined parameter. A hown in Table, the root-mean-quare error (RMSE) of the propoed method and the other tuning method are preented to evaluate the control performance of the propoed method. The BLT tuning method i applied to the four kind of typical TITO proce, and the Wang method i only applied to Wood and Berry column and Vinante and Luyben plant ince the tuning parameter for the other procee are not included in Wang paper. A. Wood and Berry Column. 8 e e e 9. 4 e 4. 4 ( 3 (7) Wood and Berry column i preented in (7) [5]. Before deigning the centralized PID controller uing the propoed method, the RA analyi hould be carried out to find the bet pairing. In thi cae, the RA analyi how that Y paired with U and Y paired with U are the bet. So, the diagonal part PID controller are imply tuned by the ITAE- tuning method [6]. And the parameter of the off-diagonal part PID controller are traightforwardly calculated by olving the leat quare method. The time delay of the off-diagonal part PID controller are and 4, repectively. Then, the centralized PID controller of (8) i obtained and the control performance i hown in Figure 4, confirming that the propoed method provide almot complete decoupling performance and imilar etpoint tracking performance compared with the BLT method and Wang method. B. Wardle and Wood Column e ( e e (48 )(45 ) 8 0. e 35 (9) Wardle and Wood Column can be repreented by (9) [7]. The RA analyi indicate that the Y paired with U and Y paired with U are the bet. The diagonal part PID controller are traightforwardly tuned by the SISO IMC tuning method [8]. The tuning parameter of the off-diagonal part PID controller are calculated by the linear leat quare method of (5) a hown in (0). Figure 5 how that the propoed method provide acceptable tracking performance of the etpoint change and excellent decoupling performance. C. Tyreu Stabilizer ( 0.53(0 )e (4 ) e (43 )( ) e (33 ) () e (44 )(0 ) Equation () i the Tyreu tabilizer [9]. The RA analyi recommend the Y paired with U and Y paired with U. The diagonal part PID controller are tuned by the SISO ITAE- tuning method [0]. The ITAE- combined with model reduction i ued to tune the c ( becaue the i 3rd order proce [0], []. The calculated centralized PID controller are preented in (). Figure 6 how that the propoed method can control Tyreu tabilizer proce in an efficient way. The reult how the propoed method give remarkable improvement in tracking etpoint change and decoupling performance. D. Vinante and Luyben Plant.e 7.8.8e e e 9. ( 35 (3) Vinante and Luyben plant i given in (3) [3]. The RA analyi how the Y paired with U and Y paired with U are the bet. The diagonal part PID controller are tuned by SISO ITAE- tuning rule [6]. The time delay of ( i larger than the time delay of (. Therefore, the tructure of the off-diagonal part PID controller c ( ) hould be (). The 57

6 off-diagonal part PID parameter can be obtained by olving (3) with linear leat quare method, reulting in the centralized PID controller in (4). The reult how that the propoed method guarantee imilar tracking performance and better decoupling performance for Y when Y etpoint i changed without requiring uer-defined parameter. It i hown in Figure 7. IV. CONCLUSION In thi reearch, a new deigning method for the centralized PID controller i propoed to improve the cloed-loop control performance by removing off-diagonal part component of the open-loop tranfer function matrix in frequency domain. The propoed method provide the centralized PID controller tuning parameter in a traightforward way without uer-defined parameter uch a phae margin, gain margin, overhoot and o on. The cae tudie how that the control performance of the propoed method i almot equivalent or better than previou approache even it i conceptually traightforward and imple. The propoed method can be implemented by adding jut two off-diagonal PID controller to the already exiting multi-loop PID control ytem without breaking conventional control loop becaue it ue conventional tuning parameter to tune the diagonal-part PID controller. Therefore, the propoed method can be applied to indutrial field more eaily. [] Q.-. Wang, C.-C. Hang, and B. Zou, A Frequency Repone Approach to Autotuning of Multivariable Controller, Chem. Eng. Re. De., vol. 75, no. 8, pp , 997. [3] V. Vijay Kumar, V. S. R. Rao, and M. Chidambaram, Centralized PI controller for interacting multivariable procee by ynthei method, ISA Tran., vol. 5, no. 3, pp , 0. [4] F. Morilla, F. Vázquez, and J. arrido, Centralized PID Control by Decoupling for TITO Procee, Proceeding 3th IEEE Int. Conf. Emerg. Technol. Fact. Autom., pp. 3835, 008. [5] R. K. Wood and M. W. Berry, Terminal compoition control of a binary ditillation column, Chem. Eng. Sci., vol. 8, no. 9, pp , 973. [6] A. M. Lopez, C. L. Miller, C. L. Smith, and P. W. Murrill, Controller Tuning Relationhip Baed on Integral Performance Criteria, IntTrum. Technol., vol. 4, pp. 576, 967. [7] A. P. Wardle and R. M. Wood, Problem of application of theoretical feedforward control model to indutrial cale fractionating plant, IChemE Symp. Ser., pp. 688, 969. [8] C. E. arcia and M. Morari, Internal Model Control.. A Unifying Review and Some New Reult, Ind. Eng. Chem. De. Dev., vol., pp , 98. [9] B. D. Tyreu, Multivariable Control Sytem Deign for an Indutrial Ditillation Column relation are obtained directly, Ind. Eng. Chem. De. Dev., vol. 8, no., pp. 778, 979. [0] S. W. Sung, J. O, I.-B. Lee, J. Lee, and S.-H. Yi, Automatic Tuning of PID Controller uing Second Order Plu Time Delay Model, J. Chem. Eng. Japan, vol. 9, no. 6, pp , 996. [] S. W. Sung, J. Lee, and I.-B. Lee, Proce identification and PID control REFERENCES [] M. Morari and J. H. Lee, Model predictive control: pat, preent and future, Comput. Chem. Eng., vol. 3, no. 45, pp , 999. [] E. Britol, On a new meaure of interaction for multivariable proce control, IEEE Tran. Automat. Contr., vol., no., pp. 3334, 966. [3] W. L. Luyben, Simple Method for Tuning SISO Controller in Multivariable Sytem, Ind. Eng. Chem. Proce De. Dev., vol. 5, pp , 986. [4] S. Skogetad and M. Morari, Robut Performance of Decentralized Control Sytem by Independent Deign, 987 Am. Control Conf., vol. 5, no., pp. 95, 987. [5] T. N. L. Vu, J. Lee, and M. Lee, Deign of Multi-loop PID Controller Baed on the eneralized IMC-PID Method with Mp Criterion, International J. Control. Autom. Syt., vol. 5, no., pp. 7, 007. [6] M. Waller, J. B. Waller, and K. V Waller, Decoupling Reviited, Ind. Eng. Chem. Re., vol. 4, no. 0, pp , 003. [7] M. H. Perng and J. S. Ju, Optimally decoupled robut control of MIMO plant with multiple delay, IEEE Proc.-Control Theory Appl., vol. 4, no. I, pp. 53, 994. [8] T. J. McAvoy, Steady-State Decoupling of Ditillation Column, Ind. Eng. Chem. Fundam., vol. 8, no. 3, pp. 6973, 979. [9] J. Lee, D. H. Kim, and T. F. Edgar, Static decoupler for control of multivariable procee, AIChE J., vol. 5, no. 0, pp. 770, 005. [0] J. arrido, F. Vázquez, and F. Morilla, An extended approach of inverted decoupling, J. Proce Control, vol., no., pp. 5568, 0. [] J. Lielehto, MIMO controller deign uing SISO controller deign method, Proceeding 3th IFAC world Congr., pp. 6973,