Design of controller using variable transformations for a nonlinear process with dead time

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Design of controller using variable transformations for a nonlinear
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  • 1. ISA TRANSACTIONS® ISA Transactions 44 ͑2005͒ 81–91Design of controller using variable transformations for a nonlinear process with dead time R. Anandanatarajan,a,* M. Chidambaram,b T. Jayasinghc a Pondicherry Engineering College, Pondicherry, India b IIT, Chennai, India c St. Xaviers College of Engineering, Nagercoil, India ͑Received 30 October 2003; accepted 5 June 2004͒Abstract In this work, a globally linearized controller ͑GLC͒ for a first-order nonlinear system with dead time is proposed.This is similar to the GLC proposed by Ogunnalke ͓Ind. Eng. Chem. Process. Des. Dev. 25, 241–248 ͑1986͔͒ fornonlinear systems without dead time. Two methods are proposed. One is based on the Smith prediction from the modelin the transformed domain and the other is based on Newton’s extrapolation method. The simulation study is made onthe conical tank level process and the results are compared with those obtained using a conventional PI controller andthe Smith PI controller based on the transfer function model about the operating point 39%. Finally, experimentalresults on the laboratory conical tank level process are also given. © 2005 ISA—The Instrumentation, Systems, andAutomation Society.Keywords: Nonlinear control; GLC; Conical tank1. Introduction volves linearizing the modeling equation around a steady state and applying linear control theory re- Time delays occur in many control applications, sults ͓4͔. It is obvious that the controller perfor-typically as a result of transporting material or en- mance in this case will deteriorate as the processergy. When the time delay is significantly greater moves further away from the steady state aroundthan the process dominant time constant, serious which the model was linearized.degradation of the closed-loop performance can be Apart from the ‘‘local linearization’’ approach,expected with traditional controllers. For instance, there are a few other ‘‘special purpose’’ designthe traditional PI controller results in very sluggish procedures ͓5͔ which may be applied directly toclosed-loop response ͓1͔. Faster reference tracking nonlinear systems. However, as noted by Ray ͓5͔,and disturbance rejection can be achieved by the these usually have limited applicability and are of-Smith predictor. However, its main weakness, ten based on accumulated experience with a spe-compared to the PI controller, is a lack of robust- cial type of nonlinear system. A new modifiedness to variations in process parameters ͓2,3͔. Smith predictor is proposed by Vrecko ͓6͔ to im- The traditional and easiest approach to the con- prove the robustness but again it is a linear con-troller design problem for nonlinear systems in- troller. The design of a controller based on the variable *Corresponding author. E-mail address: transformation for a nonlinear system withoutannandaគnatarajan@yahoo.com dead time is presented by Ogunnalke ͓7͔. In this0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.
  • 2. 82 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91present paper, it is shown that how transforma-tions may be found which transform the nonlinearsystem with dead time into one that is ideally lin-ear. The approach is based on the hypothesis that asystem which is nonlinear with dead time in itsoriginal variables is linear in some transformationof the predicted future original variables. Con-structive methods for finding the appropriate trans-formations are presented. The design of a control-ler for the linear transformed system may then becarried out with a great deal of facility. The result- Fig. 1. Conceptual configuration of the GLC controller.ing controller will of course be nonlinear whenrecast in terms of the original system variables. Inthe present work, the specific problem of a conicaltank level process system having dead time is usedto illustrate the potentials of this approach. An ex- where a and b are constants, v is a state variable,ample simulation and real time implementation of and g ( • ) is a function to be determined. Differen-the controller on a conical tank laboratory level tiating Eq. ͑3͒ with respect to t,process are presented. dz dg dx ϭ . ͑5͒ dt dx dt From Eqs. ͑1͒ and ͑2͒2. Design of globally linearized controller„GLC… dx ϭc 1 f 1 ͑ x ͒ ϩc 2 f 2 „x,u ͑ tϪT d͒ …. ͑6͒ dt2.1. Variable transformation Substituting Eq. ͑6͒ in Eq. ͑5͒, A single-input, single-output nonlinear control dz dg dgsystem with dead time can be represented in gen- ϭc 1 f 1 ϩc 2 f 2 . ͑7͒eral by dt dx dx dx Let ϭF„x,u ͑ tϪT d͒ …, ͑1͒ dt dg c 1 f 1͑ x ͒ ϭa ͑8͒where F ( • ) is an arbitrary nonlinear function of x, dxthe system state variable, and u, the control vari- andable, and T d is the dead time in the process. Thefunction F„x,u ( tϪT d) … is split up as dg c 2 f 2 „x,u ͑ tϪT d͒ … ϭb v . ͑9͒ F„x,u ͑ tϪT d͒ …ϭc 1 f 1 ͑ x ͒ ϩc 2 f 2 „x,u ͑ tϪT d͒ …, dx ͑2͒ The nonlinear system is thus mapped towhere f 1 ( x ) is a function of x alone and dzf 2 „x,u ( tϪT d) … is a function of both x and ϭaϩb v . ͑10͒u ( tϪT d) . Both f 1 and f 2 are taken to be nonlin- dtear. c 1 and c 2 are constants. A linear system may be effectively controlled with zϭg ͑ x ͒ ͑3͒ a PI controller as in Fig. 1. From Eq. ͑8͒, the trans- formation g ( x ) which transforms the given non-is a transformation for mapping the nonlinear sys- linear system with dead time into a linear systemtem F ( • ) to a linear system is dz dt ϭaϩb v , ͑4͒ g͑ x ͒ϭ a c1 • ͵ f dxx ͒ . ͑ 1 ͑11͒
  • 3. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 83 2.2. Prediction of process variable and controller output A conventional PI controller gives an output v ( t ) which effectively controls z. The future value of f 2 is predicted either by using Newton’s method or by using the transformed model of the system. Since f 2 is a function of v , once f 2 is predicted, v can be easily obtained. The nonlinear control law can be obtained using Eq. ͑12͒.Fig. 2. Block diagram representation of the nonlinear con-troller with Newton’s predictor method ͑NEM͒. The PI con-troller is designed for the linear system ͓Eq. ͑10͔͒. 2.3. Newton’s extrapolation method Newton’s extrapolation formula, used to predict the future value, isSubstituting the value of dg/dx from Eq. ͑8͒ intoEq. ͑9͒ yields f ͑ pϩh ͒ ϭ f ͑ p ͒ ϩhٌ f ͑ p ͒ ϩhigher-order terms, ͑13͒ c 1 f 1 b v ͑ tϩT d͒ u͑ t ͒ϭ . ͑12͒ c 2 a f 2 „x ͑ tϩT d͒ … where f ( pϩh ) is the future value to be predicted after an interval of h sec and p is the samplingLet time at which the latest value is available. ٌ is the difference operator. ٌ f ( p ) is the difference be- u ͑ t ͒ ϭq ͑ • ͒ , tween the latest sampled value and the previous sampled value. The number of terms that has to bewhere v ( tϩT d) is the predicted manipulated vari- considered in the above formula depends on theable and f 2 „x ( tϩT d) … is the predicted function of number of past data available and the designer’sstate variable. In deriving Eq. ͑12͒, the specific interest. In their work, only the first two terms arecase of f 2 ( x ) •u ( tϪt d) is considered than considered ͓8͔. This method may be called New-f 2 „x,u ( tϪt d) …. In general case u has to be solved ton’s extrapolation method ͑NEM͒.numerically. Thus a nonlinear controller u ( t ) is designedbased on a variable transformation for the first- 2.4. Variable transformation predictororder nonlinear process with dead time. The con-troller performance is tested by simulation of the The nonlinearity in the process and hence in theconical tank level process. The proposed control- model limits the prediction. The model which isler is expected to be highly robust when the oper- used to map the nonlinear process by variableating point of the process is shifted over the entire transformation is linear and this model is usedspan of the tank. q ( • ) is the transformation which here to predict the future variable as shown in Fig.transforms the linear controller output in the trans- 3. The configuration is a combination of Smithformed domain into the nonlinear controller out- predictor and variable transformation and henceput in the original domain. may be called the variable transformation predic- A PI controller is designed and interfaced with tor ͑VTP͒. The transfer function model of thethe pseudolinear system shown in Fig. 2. This is transformed pseudolinear system ͑10͒ ismade possible by the use of the transformations z ( s ) / v ( s ) ϭb/s. The transfer function model b/sg ( x ) and u ( t ) . x * is the set point and z * is the set will give the predicted transformed process vari-point in the transformed domain. Thus the entire able z Ј . The model prediction error e m is the dif-design procedure boils down to the determination ference between the predicted output and theof g ( x ) to compute the corresponding u ( t ) by us- transformed process variable z. e m is added in theing Eq. ͑12͒. feedback path as an additional error.
  • 4. 84 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91Fig. 3. Block diagram representation of the variable trans-formation predictor. The PI controller is designed for thelinear system ͓Eq. ͑10͔͒. Fig. 4. Block diagram of the Smith PI controller. but practically it is found that selection of large3. Application of GLC to conical tank level values of a and b end up with overflow error in theprocess transformed variable. The transfer function of Eq. ͑15͒ is The mathematical model of the conical tank liq-uid level system considered for the study is ex- G ͑ s ͒ ϭb/s. ͑16͒pressed as The forward path transfer function of the closed- dh ͓ u ͑ tϪT d͒ ϩu d͑ tϪT d͒ Ϫc ͱh ͔ loop system with a PI controller is ͫ ͬ ϭ , ͑14͒ dt R2 2 G f͑ s ͒ ϭbK c͑ 1ϩT is ͒ /T is 2 . ͑17͒ ␲ 2h H T i and K c can be obtained by assuming a closed-where H, R, c are total height, top radius, and out- loop time constant and damping factor ͓9͔. Inter-flow valve coefficient of the conical tank, respec- estingly, the tuning parameters are independent oftively. h, u, and u d are liquid level, inflow rate, and local time constant and local gain of the process.disturbance of the conical tank level process, re- The transformation which transforms the presentspectively. The transformation g ( h ) transforms process in to a linear system isthe nonlinear system represented by Eq. ͑14͒ into Ϫ2a⌸R 2a linear system. The transformed system is a g͑ h ͒ϭ h 5/2. ͑18͒pseudolinear system, whose mathematical model 5H 2 cis 4. Tuning of controllers dz ϭaϩb v . ͑15͒ dt The responses are compared with a conventionalThe values of a and b are assumed as Ϫ0.7657 PI controller tuned about nominal operating pointand 0.0177, respectively. Mathematically there is of 39%. The time constant and gain of linearizedno restriction on the selection of values of a and b model are 76 sec and 1.2, respectively. The pro- cess dead time is 32 sec. The Ziegler and NicholsTable 1 Table 2ISE of Regulatory responses for 15% decrease in load. ISE of Regulatory responses for 15% increase in load.Operating ZN Smith Operating ZN Smithpoint PI PI GLC point PI PI GLC39% 1285 3269 928 39% 1236 3367 80724% 2324 2762 1492 24% 1780 2991 216254% 1224 3740 518 54% 1221 3779 448
  • 5. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 85Table 3ISE of servo responses for 20% increase in set point.Operating ZN Smithpoint PI PI GLC39% 23670 35480 1609024% 19260 27530 1578054% 32030 47660 16260͑ZN͒ tuning parameters for the linearized modelare K cϭ0.95 and T iϭ0.020. The tuning param-eters for the Smith PI controller as shown in Fig. 4are based on Haalman’s tuning rule. They are K cϭ0.5747 and T iϭ0.022 ͓10͔. Similarly for bothGLC’s are K cϭ2.87 and T iϭ0.025. The simulation and experimentation are carriedout by taking 39% as nominal level. 24% and 54%are other operating points used to test the robust- Fig. 5. Regulatory responses for a 15% decrease in load atness of the controller tuned at 39% nominal level. 39% nominal operating point.The integral square error ͑ISE͒ values are pre-sented in the Tables 1– 4. Regulatory responses fora 15% decrease in load at nominal operating point a better response ͑26% lesser ISE͒ than the con-39% ͑refer to Fig. 5͒ show that the proposed con- ventional Smith predictor. The ZN-PI controllertroller gives an improved response ͑28% lesser gives an oscillatory response. Regulatory re-ISE͒ while the conventional Smith predictor sponses for a 15% decrease in load at 54% oper-͑150% higher ISE͒ gives a poorer performance ating point but tuned at 39% ͑refer to Fig. 9͒ showthan the ZN-PI controller. that the proposed controller gives an improved Regulatory responses for a 15% increase in load performance ͑58% lesser ISE͒ while the conven-at nominal operating point 39% ͑refer to Fig. 6͒ tional Smith predictor ͑205% higher ISE͒ providesshow that the proposed controller improves the re- a poorer performance than the ZN-PI controller.sponse ͑35% lesser ISE͒ while the conventionalSmith predictor ͑172% higher ISE͒ gives a poorerperformance than the ZN-PI controller. Regulatory responses for a 15% decrease in loadat 24% operating point but tuned at 39% ͑refer toFig. 7͒ show that the proposed controller providesa better response ͑36% lesser ISE͒ than the con-ventional Smith predictor ͑19% higher ISE͒. TheZN-PI controller gives an oscillatory response. Regulatory responses for a 15% increase in loadat 24% operating point but tuned at 39% ͑refer toFig. 8͒ show that the proposed controller providesTable 4ISE of servo responses for 20% decrease in set point.Operating ZN Smithpoint PI PI GLC39% 46920 28900 2003024% 18918 14260 10930 Fig. 6. Regulatory responses for a 15% increase in load at54% 28980 36270 18420 39% nominal operating point.
  • 6. 86 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91Fig. 7. Regulatory responses for a 15% decrease in load at Fig. 9. Regulatory responses for a 15% decrease in load at24% operating point. 54% operating point. Regulatory responses for a 15% increase in load proved response ͑31% lesser ISE͒ than the con-at 54% operating point but tuned at 39% ͑refer to ventional Smith predictor. The ZN-PI controllerFig. 10͒ show that the proposed controller im- provides an oscillatory response. Servo responsesproves the performance ͑63% lesser ISE͒ while for a 20% increase in the set point at nominal op-the conventional Smith predictor ͑209% higher erating point 39% ͑refer to Fig. 12͒ show that theISE͒ gives a poorer response than the ZN-PI con- proposed controller improves the response ͑32%troller. lesser ISE͒ while the conventional Smith predictor Servo responses for a 20% decrease in the set ͑50% higher ISE͒ gives a poorer performance thanpoint at nominal operating point 39% ͑refer to Fig. the ZN-PI controller.11͒ show that the proposed controller gives an im-Fig. 8. Regulatory responses for a 15% increase in load at Fig. 10. Regulatory responses for a 15% increase in load at24% operating point. 54% operating point.
  • 7. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 87Fig. 11. Servo responses for a 20% decrease in set point at39% nominal operating point. Fig. 13. Servo responses for a 20% decrease in set point at 24% operating point. Servo responses for a 14% decrease in the set domain are lesser than the tolerance that can bepoint at 24% operating point but tuned at 39% handled by the computer.͑refer to Fig. 13͒ show that the proposed controller Servo responses for a 20% increase in set pointimproves the response ͑33% lesser ISE͒ more than at 24% operating point but tuned at 39% ͑refer tothe conventional Smith predictor. The ZN-PI con- Fig. 14͒ show that the proposed controller im-troller gives an oscillatory response. At this oper- proves the response ͑18% lesser ISE͒ while theating point it is not possible to decrease the set conventional Smith predictor ͑43% higher ISE͒point greater than 14% because of numerical gives a poorer performance than the ZN-PI con-round-off error. Some values in the transformed troller.Fig. 12. Servo responses for a 20% increase in set point at Fig. 14. Servo responses for a 20% increase in set point at39% nominal operating point. 24% operating point.
  • 8. 88 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91Fig. 15. Servo responses for a 20% decrease in set point at54% operating point. Fig. 17. Servo responses of conical tank level process for Ϫ8% step change at nominal operating point 39% using GLC ͑VTP͒ and GLC ͑NEM͒ with the same PI settings. Servo responses for a 20% decrease in set pointat 54% operating point but tuned at 39% ͑refer toFig. 15͒ show that the proposed controller im- ler and the conventional Smith predictor ͑49%proves the response ͑36% lesser ISE͒ while the higher ISE͒ provides a very poor response.conventional Smith predictor ͑25% higher ISE͒ The servo responses of the conical tank levelgives a poorer response than the ZN-PI controller. process for a 8% decrease in set point at the nomi-Servo responses for a 20% increase in set point at nal operating point of 39% using GLC54% operating point but tuned at 39% as in Fig. controllers—one with NEM prediction and the16 show that the proposed controller improves the other with proposed VTP prediction but both withresponse ͑51% lesser ISE͒ than the ZN-PI control- same tuning parameters—are shown in Fig. 17. The response given by NEM has slight oscilla- tions. The magnitude of oscillations increases as the step magnitude increases. This indicates that the prediction by NEM is very poor compared with the proposed GLC with VTP. Figure 18 shows the experimental servo re- sponses for various set point changes at 39% nominal level. The responses are slightly oscilla- tory. Figures 19 and 20 show the experimental regulatory responses for various magnitudes of load changes at the nominal operating point. The regulatory responses are found to have small os- cillations as compared to the servo responses. Fig- ure 21 shows a block diagram of the experimental setup. 5. ConclusionFig. 16. Servo responses for a 20% increase in set point at A nonlinear controller is designed based on the54% operating point. variable transformation for the first-order nonlin-
  • 9. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 89Fig. 18. Experimental servo responses of conical tank level process for set point changes at nominal operating point 39%using GLC ͑VTP͒.ear process with dead time. The performances of worse when the operating point is shifted to 54%.proposed GLC ͑VTP͒ are tested by simulation and On the other hand, the proposed controller givescompared with ZN PI and Smith PI controllers. better dynamics.The ZN PI controller gives an oscillatory response Consider the regulatory responses at 24% oper-for decrease in set point even at a nominal operat- ating point for decrease in load as shown in Fig. 7.ing point. The situation becomes worse when the The ZN-PI controller gives a highly oscillatory re-operating point is shifted to 24%. For any increase sponse. On the other hand, Smith PI and GLC givein the set point the Smith PI controller gives a an oscillation free response. Similarly, considervery poor response due to a very small dead time the regulatory responses at 24% operating pointto time constant ratio. This situation becomes for an increase in load as shown in Fig. 8. HereFig. 19. Experimental regulatory responses of conical tank level process for increase in load changes at nominal operatingpoint 39% using GLC ͑VTP͒.
  • 10. 90 Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91Fig. 20. Experimental regulatory responses of conical tank level process for decrease in load changes at nominal operatingpoint 39% using GLC ͑VTP͒.also the ZN-PI controller gives a highly oscillatory controller gives oscillatory responses and theresponse but the Smith PI gives an oscillation-free Smith PI gives sluggish responses, GLC gives os-response. GLC gives an oscillatory response but cillation free responses with quick rise time. Thequickly settles unlike that of the ZN-PI controller. simulation results show the robustness of theIn this case the drawback with the GLC is the GLC.large undershoot. But it has less ISE than that of The proposed controller outperforms the SmithSmith PI. Consider the regulatory responses at PI and the ZN-PI controllers when the operating54% operating point ͑refer to Figs. 9 and 10͒. The point of the process is shifted over the entire spanSmith PI controller gives a large undershoot and a of the tank. While comparing VTP and NEM, VTPlarge overshoot compared to that of the ZN-PI gives better dynamics than NEM. As a futurecontroller. But GLC gives a very small overshoot work, the GLC can be tested on other nonlinearand undershoot compared to that of the ZN-PI systems; the work in this direction is ongoing.controller. Consider the servo responses at 39%and 24% ͑refer to Figs. 11 and 13͒. The ZN-PI Acknowledgments We are grateful to Professor K. Ethirajulu for his constant encouragement and providing the facili- ties. We thank Professor P. Dhananjayan for his encouragement. References ͓1͔ Meyer, C., Seborg, D. E., and Wood, R. K., A com- parison of the Smith predictor and conventional feed- back control. Chem. Eng. Sci. 31, 775–778 ͑1976͒. ͓2͔ Hagglund, T., A predictive PI controller for processes with long dead times. IEEE Control Syst. Mag. 12 ͑1͒, 57– 60 ͑1992͒. ͓3͔ Tan, K. K., Lee, T. H., and Leu, F. M., Predictive PI versus Smith control for dead-time compensation. ISA Trans. 40, 17–29 ͑2001͒. ͓4͔ Schneider, D. M., Control of process with time delays. IEEE Trans. Ind. Appl. 24 ͑2͒, 186 –191 ͑1988͒. ͓5͔ Ray, W. H., Advanced Process Control. McGraw-Hill, Fig. 21. Block diagram of experimental setup. New York, 1981.
  • 11. Anandanatarajan, Chidambaram, Jayasingh / ISA Transactions 44 (2005) 81–91 91 ͓6͔ Vrecko, D., Vrancic, D., Juricic, D., and Strmcnik, S., A new modified Smith predictor: The concept, design M. Chidambaram obtained and tuning. ISA Trans. 40, 111–121 ͑2001͒. his B.E. ͑Chemical͒ from An- namalai University, M.E. ͓7͔ Ogunnalke, B. A., Controller design for nonlinear pro- ͑Chemical͒ and Ph.D. from In- cess systems via variable transformations. Ind. Eng. dian Institute of Science in Chem. Process Des. Dev. 25, 241–248 ͑1986͒. 1977 and 1984, respectively. ͓8͔ Chidambaram, M., Anandanatarajan, R., and Jayas- He was faculty member in IIT- ingh, T., Controller design for nonlinear process with Bombay during 1984 to 1991. dead time via variable transformations. Proceedings of Since September 1991 he has been a faculty member in IIT- the International Symposium on Process Systems En- Madras. He had been head of gineering and Control ͑ISPSEC’03͒, IIT, Mumbai, the Department of Chemical 2003, pp. 223–228. Engineering during the period ͓9͔ Chidambaram, M., Applied Process Control. Allied October 2000 to October 2003. Publishers, New Delhi, India, 1998. He has authored three books: Nonlinear Process Control. John Wiley,͓10͔ Tian, Y. C. and Gao, F., Double-controller scheme for 1996; Applied Process Control. Allied Publishers, 1998; Computer Control of Processes. Narosa Publishers, 2002. He has published 135 control of processes with dominant delay. IEE Proc.: research publications in journals and 45 papers in conferences pro- Control Theory Appl. 145 ͑5͒, 479– 484 ͑1998͒. ceedings. His areas of interest in process control are PID control, relay tuning, and nonlinear control. R. Anandanatarajan ob- tained his B.Sc. ͑Mathematics͒ from Madras University in 1984, Bachelor degree in Elec- T. Jayasingh obtained his trical from the Institution of Ph.D. in Control and Instru- Engineers ͑India͒ in 1989, mentation from IIT-Delhi. He M.Sc. ͑Mathematics͒ and M.E. has more than two decades of ͑Process control and Instru- experience in teaching and re- mentation͒ from Annamalai search at Anna University, University in 1994 and 1998, Chennai, India. Presently he is respectively, Ph.D. from Anna a visiting professor at St. University in 2003. Presently Xaviers College of Engineer- he is assistant professor of the ing, Nagercoil, India. His areas Department of Instrumentation of interest include process con-Engineering, Pondicherry Engineering College. He has authored two trol, computer based instru-books titled Signals & Systems and Computer Peripherals and Inter- mentation, and sensor model-facing. ing.