# Delay-compensated controllers for two-inputtwo-output (TITO) multivariable processes.

код для вставкиСкачатьASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING Asia-Pac. J. Chem. Eng. 2007; 2: 510?516 Published online 12 October 2007 in Wiley InterScience (www.interscience.wiley.com) DOI:10.1002/apj.095 Research Article Delay-compensated controllers for two-input/two-output (TITO) multivariable processes A. Seshagiri Rao, V. S. R. Rao and M. Chidambaram* Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India Received 19 March 2007; Revised 24 July 2007; Accepted 13 August 2007 ABSTRACT: A decentralized controller design based on the direct synthesis method for two-input/two-output (TITO) processes in the Smith predictor configuration is proposed. The decentralized controllers are designed individually on the basis of the diagonal individual transfer functions of the process. Using the direct synthesis principle, the two decentralized controllers are designed using the individual diagonal transfer functions of the process matrix and the individual desired diagonal closed-loop transfer functions. Further, these direct synthesis controllers are implemented in the Smith predictor configuration for each output. While converting from the synthesis controller to the Smith predictor configuration, the decentralized controller parameters are directly obtained in terms of the individual closedloop desired tuning parameters. The method is compared with the recently reported methods. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. KEYWORDS: Smith predictor; time delay; TITO process; direct synthesis method INTRODUCTION In process industries, two-input/two-output (TITO) processes are the most commonly encountered multivariable processes. Moreover, many processes with inputs/outputs beyond two can be treated as several two-by-two subsystems in practice.[1] Since different time delays are present in different control loops, and owing to the interactions, the difficulty further increases in multivariable systems. Alevisakis and Seborg[2] have extended the original Smith delay compensator to multiinput/multi-output (MIMO) processes for a single time delay. Later, many authors have proposed modified design methods for the Smith predictor.[3 ? 6] Decentralized controllers are mostly used for multivariable processes because it is simple, easy to implement and also easy to tune. In view of the fact that a conventional proportional-integral-derivative (PID) controller often results in excessive oscillation of the set point response in contrast with a proportional-integral (PI) controller, Chien et al .[7] have proposed a modified implementation form of the PID controllers and a tuning rule for multiloop control systems. Improved control design methods have also been proposed by many authors.[8 ? 11] Huang et al .[8] proposed multiloop *Correspondence to: M. Chidambaram, Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India. E-mail: chidam@iitm.ac.in ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. PI/PID controller design formulas by decomposing a multiloop system into a number of single loop systems and formulating an effective open loop process. Liu et al .[11] proposed TITO PI/PID controller design formulas by using the Maclaurin series for approximating the obtained controller expression for each loop. Lee et al .[9] proposed analytical PI/PID controller expressions based on IMC principles. However, the methods proposed involve several rigorous mathematical calculations and the methods are complicated. In the present work, a simple design method based on the direct synthesis method is proposed. The method of designing controllers by Lee et al .[9] is modified without approximating the time delay. Direct synthesis controllers are implemented in the Smith predictor configuration. THEORETICAL DEVELOPMENTS The general block diagram of the feedback control for TITO processes with a decentralized controller is shown in Fig. 1 where Gp (s) is the transfer function matrix of the process and Gc is the transfer function matrix of the decentralized controller. The closed-loop transfer function matrix for set-point changes is given by y(s) = H(s) = [I + Gp (s)Gc (s)]?1 Gp (s)Gc (s) yr (s) (1) Asia-Pacific Journal of Chemical Engineering DELAY-COMPENSATED CONTROLLERS FOR TITO MULTIVARIABLE PROCESSES d Gc e yr - Gc11 0 0 u Gc22 action. A time delay compensator enhances the closedloop performance by compensating for the time delay. After a time delay compensator is used, the main controller should be designed on the basis of the process without considering the time delay. To develop a compensator and to design a compensated controller, in the present work, the same direct synthesis controller, Eqn (5), is used. Gp G11 G21 G12 G22 y Figure 1. Simple decentralized feedback for TITO processes. According to the direct synthesis method, the controller matrix is given by Gc (s) = Gp ?1 (s)R(s)[1 ? R(s)]?1 (2) where R(s) [= (y(s)/yr (s))d ] is the desired closedloop transfer function matrix for TITO processes with time delay. The interaction is a major problem in multivariable processes. To get decoupled responses, the off-diagonal elements are dropped to zero i.e. Gp = diag (G11 , G22 ). With that, the closed loop responses are decoupled and the controller for i th loop can be written as Gci (s) = Gii ?1 (s)Ri (s)[1 ? Ri (s)]?1 DELAY-COMPENSATED CONTROLLER DESIGN Without approximating the time delay in Eqn (5) for controller design, Eqn (5) can be structurally implemented as shown in Fig. 2. Further, Fig. 2 can be written in an alternative way as shown in Fig. 3. Figure 4 is the Smith predictor structure for process Gii (s), where Gcii is the predictor controller. Now the controller is designed for a first order plus time delay (FOPTD) process. For a FOPTD process, Gii ? = kii /(?ii s + 1). (3) ?is+1 e Where Ri (s) is the desired closed loop transfer function for i th loop. Note: The off-diagonal processes are neglected just for simplicity. It will be shown later that even by neglecting the off-diagonal elements, the obtained closed-loop responses are satisfactory. Let us consider the process as Gii (s) = Gii + (s) Gii ? (s) where Gii + (s) is non-minimum phase (NMP) elements of the process i.e. the time delay and right half plane zero. Gii ? (s) is the process without the NMP elements. According to the direct synthesis method, the desired closed-loop transfer function should contain the process NMP elements. Hence, in the present work, the desired closed-loop transfer function is considered as Ri (s) = (?i s + 1) (?s + 1)2 Gii + (s) Figure 2. Eqn (5). [(?i s + 1)2 ? (?i s + 1)Gii + (s)] (5) As Gii + (s) contains the time delay term, this time delay is approximated either using a Taylor series expansion or a Pade approximation. After approximating the time delay and simplifying, Eqn (4) results in a PID controller according to the direct synthesis method. However, because of the time delay approximation, there will be a limit on the controller gain and therefore there will be a limit on the possible amount of control ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Structural implementation of Gcii , (?is+1)G?1 ii?(s) e + + u (?is+1)2 Gii+(s)Gii?(s) Figure 3. Fig. 2. (4) (?i s + 1) u Gii+(s) Upon substitution in Eqn (3), the controller is obtained as Gci (s) = Gii ? ?1 (s) ?1(s) Gii? (?is+1)2 + + Alternative structure for Gcii ?1(s) (?is+1)Gii? e + + +- u [?i(s+1)2?(?is+1)] Gii?(s) Gii+(s)Gii?(s) Figure 4. Delay-compensated controller in the Smith predictor form. Asia-Pac. J. Chem. Eng. 2007; 2: 510?516 DOI: 10.1002/apj 511 512 A. SESHAGIRI RAO, V. S. R. RAO AND M. CHIDAMBARAM Substituting in Gcii results in Gcii = (?i s + 1)(?ii s + 1) (6) kii [(?i s + 1)2 ? (?i s + 1)] Expanding and simplifying yields Gcii = (?i s + 1)(?ii s + 1) kii (2?i ? ?i )s[(?i 2 /(2?i ? ?i ))s + 1)] (7) Equation (7) can be further simplified to a PI controller form by considering ?ii = ?i 2 /(2?i ? ?i ) as: ?i 1 (?i s + 1) Gcii = = 1+ (8) kii (2?i ? ?i )s kii (2?i ? ?i ) ?i s where ?i = 2?i ? (?i 2 /?ii ). In a conventional PI controller form [kci (1 + (1/?Ii s))], the controller parameters are obtained as: ?i and ?Ii = ?i (9) kci = kii (2?i ? ?i ) Hence for a given TITO process with time delay, the two decentralized controllers can be calculated from Eqn (9) on the basis of the processes G11 and G22 with the assumption that there is no plant model mismatch. But, in practice, the effect of the other manipulated variables affect the process output and hence the predictor structure should be implemented by considering the other processes also, as shown in Fig. 5 for TITO processes. Here G ij (i = 1, 2; j = 1, 2) is the process model with time delay and Giim is the process model without time delay. SELECTION OF THE TUNING PARAMETER (?I ) The tuning parameter for the individual closed-loop responses should be chosen such that the resulting controller should be able to provide both nominal and [G11m G12m] + G11 G12 yr1 + Gc11 + - 0 - + 0 u yr2 + y1 G11 G12 G21 G22 Gc22 + - G21 G22 [G21m G22m] y2 + + Figure 5. Decentralized Smith predictor configuration. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering robust control performances. Many simulation results are carried out on different TITO time delay processes and it is observed that the initial value of the tuning parameter can be considered as half of the corresponding time constant of the process. So, the selection of the tuning parameter is purely based on the simulation results. The proposed method of selection of the tuning parameter is simple. If both the nominal and robust control performances are achieved with these values, these values can be taken as the final tuning parameters; else, the tuning parameters should be tuned around this value till robust control performances are achieved. In practice, the selection of the tuning parameter is based on the complete knowledge of the plant and its dynamics. In general, the tuning parameters selected on the basis of the diagonal processes are able to provide robust control performances. However, when there is large process model mismatches in all the processes, the tuning parameter selected on the basis of the diagonal processes may not give robust control performances. In that case, only the tuning parameters need to be detuned to get robust performances. In the present work, no detuning is done for uncertainties. The final values for the tuning parameters are calculated for the nominal processes only after doing many simulation studies to get improved performances. SIMULATION STUDY For the purpose of simulation studies, different TITO time delay processes are considered and the closed-loop performance is compared with the recently reported methods in the literature. Example 1: Consider the widely studied Wood and Berry distillation process[12] ? ? 12.8e?s ?18.9e?3s +1 21s + 1 ? G(s) = ? 16.7s ?7s 6.6e ?19.4e?3s 10.9s + 1 14.4s + 1 Using Eqn (9), the decentralized controllers parameters are calculated on the basis of the principal diagonal process transfer functions as kc1 = 0.248, ?I 1 = 12.157, kc2 = ?0.134, ?I 2 = 11.5517. The tuning parameters are selected as ?1 = ?2 = 8. With these controller parameters, the proposed method is simulated after giving a unit step change in set points for the two controlled variables and a negative step input of magnitude 0.1 in the load at t = 200 s for two manipulated variables, respectively. For comparison, the methods proposed by Lee et al .,[9] Liu et al .[11] and Lee and Edgar[10] are considered. The corresponding responses are shown in Fig. 6. From the responses, it can be seen that the proposed method gives better control performances. The Asia-Pac. J. Chem. Eng. 2007; 2: 510?516 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DELAY-COMPENSATED CONTROLLERS FOR TITO MULTIVARIABLE PROCESSES corresponding control action responses are shown in Fig. 7. It can be seen that the proposed method gives a smoother control action compared to the other methods. It should be noted that Refs [9?11] have already shown that the performance of their methods are better than those of many previous methods. In practice, there always exists a model plant mismatch, and therefore the closed-loop performance should be evaluated by considering the process uncertainties. The uncertainty may exist as an input uncertainty, an output uncertainty or a process uncertainty. Process uncertainty is considered in the present work. Usually the delay compensators are sensitive to the time delay mismatches in a process if the ratio between time delay and time constant is large, but if the controller design is robust, the delay-compensated controller is able to give robust control performances.[13] In the present work, +30% perturbation in each process time delay is considered, and the corresponding closed-loop responses are shown in Fig. 8. It is clear from the figure that the proposed method performs better. Unlike in other methods, the responses for the proposed method are smooth, even though there exist delay perturbations. Figure 9 shows the corresponding control action responses. From the figure, it can be seen that the control action responses of the proposed method are smooth compared to the other methods. 1.5 1.5 1 y1 y1 1 0.5 0.5 0 0 0 50 100 150 200 250 300 100 150 200 250 300 0 50 100 150 Time 200 250 300 1.5 y2 1.5 y2 50 2 2 1 1 0.5 0.5 0 0 ?0.5 ?0.5 0 0 50 100 150 Time 200 250 300 Figure 8. Responses for the Wood and Berry distillation Responses for the Wood and Berry distillation column for perfect parameters: solid line ? proposed; dash ? Ref. [11]; dot ? Ref. [9]; dash dot ? Ref. [10]. This figure is available in colour online at www.apjChemEng.com. Figure 6. column for a perturbation of +30% in each process time delay: solid ? proposed; dash ? Ref. [11]; dot ? Ref. [9]; dash dot ? Ref. [10]. This figure is available in colour online at www.apjChemEng.com. 1 0.5 0.5 U1 U1 1 0 0 ?0.5 0 50 100 150 200 250 ?0.5 300 100 150 200 250 300 0 50 100 150 Time 200 250 300 0.1 0.05 0.05 0 U2 U2 50 0.15 0.1 ?0.05 ?0.1 0 0 ?0.05 0 50 100 150 Time 200 250 300 Figure 7. Control action responses for the Wood and Berry distillation column for perfect parameters: solid ? proposed; dash ? Ref. [11]; dot ? Ref. [9]; dash dot ? Ref. [10]. This figure is available in colour online at www.apjChemEng.com. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. ?0.1 Figure 9. Control action responses for the Wood and Berry distillation column for a perturbation of +30% in each process time delay: solid ? proposed; dash ? Ref. [11]; dot ? Ref. [9]; dash dot ? Ref. [10]. This figure is available in colour online at www.apjChemEng.com. Asia-Pac. J. Chem. Eng. 2007; 2: 510?516 DOI: 10.1002/apj 513 A. SESHAGIRI RAO, V. S. R. RAO AND M. CHIDAMBARAM y1 The decentralized controller parameters are calculated on the basis of the principal diagonal transfer functions of the process as kc1 = 34.4, ?I 1 = 36.56, kc2 = ?24.07, ?I 2 = 26.74. The tuning parameters are selected as ?1 = 22.5 and ?2 = 18 respectively. With these controller parameters, the performance of the closed-loop system is evaluated after giving unit step input to the set points of the two controlled variables simultaneously and a negative step input of magnitude 10 at t = 1000 s in the load for the two manipulated variables. The responses are shown in Fig. 10. For comparison, the methods of Huang et al .[8] and Lee and Edgar[10] are considered. From the figure, it can be seen that the proposed method gives improved performance compared to the other two methods. A perturbation of +30% in each process time delay is considered, and the performance is evaluated for all the methods, and corresponding responses are shown in Fig. 11. It is clear from the figure that the proposed method performs better. The methods of Huang et al .[8] and Lee and Edgar[10] result in more overshoot and settling time under nominal and mismatch conditions, whereas the proposed method results in significantly less overshoot and settling time under both nominal and mismatch conditions. A similar improvement in closed-loop performance is observed when the mismatch is considered in all the process gains and time constants. Also, it is observed that the control action responses (not shown) for the 1.2 1 0.8 0.6 0.4 0.2 0 0 500 0 500 1000 1500 1000 1500 1.5 1 0.5 0 ?0.5 Time Figure 10. Responses for the Wardle and Wood column for perfect parameters: solid ? proposed; dash ? Ref. [10] ; dot ? Ref. [8]. This figure is available in colour online at www.apjChemEng.com. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. Asia-Pacific Journal of Chemical Engineering 1.5 y1 1 0.5 0 0 500 0 500 1000 1500 1000 1500 1.5 1 y2 Example 2: Consider the Wardle and Wood distillation column[14] ? ? ?0.101e?12s 0.126e?6s 1 (48s + 1)(45s + 1) ? G(s) = ? 60s +?8s ?0.12e?8s 0.094e 38s + 1 35s + 1 y2 514 0.5 0 ?0.5 Time Figure 11. Responses for the Wardle and Wood column for a perturbation of +30% in each process time delay: solid ? proposed; dash ? Ref. [10]; dot ? Ref. [8]. This figure is available in colour online at www.apjChemEng.com. proposed method are smooth compared to that of the other two methods. Example 3: To show the advantage of the proposed method, the Wood and Berry[12] distillation process is modified in which the time delay of each process is considered as 3 times the corresponding time constant. The modified process matrix is given by: ? 12.8e?50.1s +1 G(s) = ? 16.7s?32.7s 6.6e 10.9s + 1 ? ?18.9e?63s 21s + 1 ? ?19.4e?43.2s 14.4s + 1 For the proposed method, using Eqn (9), the decentralized controller parameters are calculated on the basis of the principal diagonal process transfer functions as kc1 = 0.14, ?I 1 = 15.37, kc2 = ?0.072, ?I 2 = 14. The tuning parameters are selected as ?1 = ?2 = 12. It is to be noted that the tuning parameters are increased in this case compared to those in Example 1. The reason is that with the same controller settings as in Example 1, the proposed method is able to provide good control responses at the cost of more control action. To minimize the control action, in this case, the tuning parameters are slightly increased. One can see that there is an almost 50% increase in the tuning parameters. But this increase in tuning parameters is to reduce the control action only. In the Lee et al .[9] method, the tuning parameters are selected as 80 and 80, which provided the best performance for their method. With these controller parameters, the proposed method is simulated after giving a unit step change at t = 0 s in set point r1 and at t = 300 s in set point r2 for the two controlled variables and a negative step input of magnitude 0.1 in the load at t = 700 s for Asia-Pac. J. Chem. Eng. 2007; 2: 510?516 DOI: 10.1002/apj Asia-Pacific Journal of Chemical Engineering DELAY-COMPENSATED CONTROLLERS FOR TITO MULTIVARIABLE PROCESSES Y2 Y1 the two manipulated variables respectively. For comparison, the method of Lee et al .[9] is considered. The corresponding responses are shown in Fig. 12. From the responses, it can be seen that the proposed method gives significantly better control performances at higher time delays. The corresponding control action responses are shown in Fig. 13. A similar improvement in control action responses is obtained compared to the other method. When the closed-loop performance is evaluated by considering perturbations in the process parameters, the proposed method is able to track the set points and reject the disturbances whereas the method of Lee et al .[9] is not able to track the set point; in other words, their responses are completely oscillatory but will be stable if enough time is given. 2.5 2 1.5 1 0.5 0 ?0.5 3 2.5 2 1.5 1 0.5 0 ?0.5 0 200 400 600 800 1000 1200 0 200 400 600 Time 800 1000 1200 Figure 12. Responses for Example 3 for the perfect model: solid ? proposed method; dash ? Lee et al.[9] This figure is available in colour online at www.apjChemEng.com DISCUSSION The design of controllers in the proposed method is straightforward and hence the proposed method can be implemented practically easily. With the proposed method of design of controllers, both nominal and robust control performances are achieved. Also, the compensated structure and the controller?s parameters are obtained simultaneously unlike in any other compensated controller, in which the controllers are designed after designing the compensator.[15] The method can be extended to MIMO systems, which needs further studies. Also, the advantage of the present method depends on the diagonal dominance of the process transfer function matrix. The performance of the controllers is clearly given in the graphs. The same difference can be given by calculating the integral of absolute error (IAE) or integral of squared error (ISE) values. In the present work, the set point changes and load disturbances are given simultaneously and therefore the error values are calculated as a sum of both servo and regulatory problems. ISE values are calculated for all the examples for perfect parameters and are shown in Table 1. From the ISE values one can see that the proposed method shows improved control action responses for all the examples. It can be observed that for Example 2 the time constants of the diagonal processes are 60 and 35, respectively. Thus the tuning parameters are considered differently. However, in Examples 1 and 3, the time constants are very close. Therefore, same value for the tuning parameters (?1 and ?2 ) is chosen. In the literature, the closed-loop response is usually chosen as a FOPTD model. However, it is shown here that by selecting the closed-loop specification as a second-order model, the closed-loop performance 0.2 U1 0.15 Table 1. ISE values for the control action of all methods for perfect parameters. 0.1 0.05 Example 1 0 ?0.05 0 200 400 600 800 1000 1200 0.1 U2 0.05 Method U1 U2 Total Proposed Liu et al .[11] Lee et al .[9] Lee and Edgar[10] 9.24 9.90 9.7 9.95 1.01 2.10 1.8 3 10.26 12.00 11.5 12.95 2.4 4.5 6.0 7.3 11.2 13.0 2.61 4.01 27.2 42.98 0 Example 2 ?0.05 ?0.1 0 200 400 600 Time 800 1000 1200 Control action responses for Example 3 for the perfect model: solid ? proposed method; dash ? Lee et al.[9] This figure is available in colour online at www.apjChemEng.com Proposed �?5 Lee and Edgar[10] �?5 Huang et al .[8] �?5 Example 3 Figure 13. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. 4.9 6.7 7.0 Proposed Lee et al .[9] 24.58 38.97 Asia-Pac. J. Chem. Eng. 2007; 2: 510?516 DOI: 10.1002/apj 515 Y1 A. SESHAGIRI RAO, V. S. R. RAO AND M. CHIDAMBARAM 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 ?0.2 Asia-Pacific Journal of Chemical Engineering CONCLUSIONS 0 50 100 150 200 250 300 200 250 300 Time Y2 516 0 50 100 150 Time Figure 14. Responses for Example 1 for the perfect model: solid ? second order; dash ? first order desired closed-loop transfer functions. This figure is available in colour online at www.apjChemEng.com. can be improved both for perfect model conditions and for perturbations. Hence in the present work, the closed-loop trajectory is chosen as a second-order model instead of the traditional FOPTD. The controller parameters obtained by considering FOPTD model (exp(?? s)/?s + 1) as the desired closed transfer function is kci = ?ii /kii ?i , ?Ii = ?ii . The controller parameters calculated for Example 1 are kc1 = 0.1631, ?i 1 = 16.7, kc2 = ?0.0928 and ?i1 = ?14.4. The performance comparisons between the two for Example 1 are shown in Fig. 14 after giving a unit step change in set points for the two controlled variables and a negative step input of magnitude 0.1 in the load at t = 200 s for the two manipulated variables, respectively. It can be seen from the responses that the second-order desired closed-loop model gives better performances. A similar improvement is obtained when there are perturbations in the process parameters. ? 2007 Curtin University of Technology and John Wiley & Sons, Ltd. A simple method of designing the controllers for TITO time delay processes in the Smith predictor configuration is proposed with direct design of the predictor controllers. The method has two tuning parameters and these two can be selected to get robust control performances. Significant improvement in control performances is obtained when compared to recently reported methods, particularly when the individual time delays of the processes are large compared to the corresponding time constants. REFERENCES [1] F.G. Shinskey, Process Control Systems, 4th ed., McGrawHill, New York, 1996; p.153. [2] G. Alevisakis, D.E. Seborg, Int. J. Control, 1973; 3(17), 541?551. [3] B.A. Ogunnaike, W.H. Ray, AIChE J., 1979; 25(6), p.1043. [4] N.F. Jerome, W.H. Ray, AIChE J., 1986; 32(6), 914?931. [5] Q.G. Wang, B. Zou, Y. Zhang, Chem. Eng. Res. Design, 2000; 78, Part A, 565?572. [6] Q.G. Wang, Y. Zhang, M.S. Chiu, Chem. Eng. Sci., 2002; 57, 115?124. [7] I.L. Chien, H.P. Huang, J.C. Yang, Ind. Eng. Chem. Res., 1999; 38, 1456?1468. [8] H.P. Huang, J.C. Jeng, C.H. Chiang, W. Pan, J. Process Control, 2003; 13, 769?786. [9] M. Lee, K. Lee, C. Kim, J. Lee, AIChE J., 2004; 50(7), 1631?1635. [10] J. Lee, T.F. Edgar, Ind. Eng. Chem. Res., 2005; 44, 7428?7434. [11] T. Liu, W. 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