Improving the Regulatory Response of PID Controller Using Internal Model Control Principles

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1 International Journal of Control Science and Engineering 9, 9(): 9-4 DOI:.59/j.control.99. Improving the Regulatory Repone of PID Controller Uing Internal Model Control Principle Arun R. Pathiran Dept. of Electrical and Electronic Technology, Federal TVET Intitute, Addi Ababa, Ethiopia Abtract The Internal Model Control () Scheme i the model baed control tructure, while the conventional feedback cheme with Proportional plu Integral (PI) and Proportional plu Integral plu Derivative (PID) controller i the mot widely ued control tructure in indutry. The propoed cheme combine thee two cheme to attain an improved input diturbance rejection repone. The regulatory repone to the change in the input diturbance can be improved via tuning the controller where the repone to the etpoint change i unditurbed. Comparion and imulation reult are preented to illutrate the effectivene of the propoed two degree of the freedom control cheme. Keyword, PID, Regulatory repone. Introduction The proportional integral (PI) and proportional integral derivative (PID) controller are the widely ued controller in the proce indutrie due to their implicity, robutne and wide range of applicability in the regulatory control layer []. A urvey of more than, controller in the proce indutrie report that more than 97% of the regulatory controller utilize the PID algorithm []. The PID tuning rule propoed by variou author have been compiled and reported in [, 5]. Large number of tuning procedure have been propoed in the literature to tune PID controller for different objective and pecification. Thee tuning procedure can be claified a model baed approache and non-model baed approache wherea, in the former proce model are ued to tune PID controller and in the later proce information i ued to tune PID controller [4, 5]. The direct ynthei method [6] and -PID tuning method [7, 8] are the popular model baed PID tuning method for achieving deired etpoint tacking repone. Thee method do not necearily reult in PI/PID controller; however, by approximating the proce model into Firt Order Plu Dead Time (FOPDT) or Second Order Plu Dead Time (SOPDT) model, the controller form can be reduced to that of PI/PID controller. In thee method, cloed loop time contant, i.e. the filter time contant i a ingle convenient tuning parameter to adjut the cloed loop * Correponding author: arun.pathiran@gmail.com (Arun R. Pathiran) Publihed online at Copyright 9 The Author(). Publihed by Scientific & Academic Publihing Thi work i licened under the Creative Common Attribution International Licene (CC BY). repone and robutne [9]. The direct ynthei approache and -PID approache yield very good performance for etpoint change, but the load diturbance repone for lag-dominant (including integrating) procee with a mall time-delay/ time-contant ratio i found to be unatifactory [, ]. The well-known PID controller tuning rule propoed by ZN how better diturbance rejection performance than -PID deign method for lag dominant procee []. In order to improve the regulatory repone, a new type of filter wa choen and -PID tuning rule wa derived []. A direct ynthei deign baed PID controller tuning relation were derived for cloed loop diturbance model []. To avoid exceive overhoot in the et-point repone, a et-point weighting factor i utilized. Conidering the filter tructure of [] -PID, tuning relation are derived by approximating the controller uing Maclaurin erie [9]. A model reduction technique to reduce the high-order proce model into a low-order model wa propoed and alo the implified -PID tuning rule were propoed by conidering the lag-time dominant procee [4]. A new filter for diturbance rejection i taken in [5] and the correponding analytical -PID tuning relation were derived. The modified integral time contant of PID controller offer improved regulatory repone. On the other hand, it alo give undeired overhoot in the ervo repone. Thi overhoot can be minimized by mean of a etpoint filter. Feedforward compenation plu feedback controller cheme provide two degree of freedom control and it allow two different deign objective to be atified the feed forward compenation i mainly ued for dominant

2 Arun R. Pathiran: Improving the Regulatory Repone of PID Controller Uing Internal Model Control Principle diturbance rejection and it need tranfer function model of the proce [6]. It may be noted that the feedback controller will provide compenation for the poible model inaccuracie. The diturbance oberver (DOB) can be ued to etimate the diturbance, and then the etimated value can be ued in the feed forward compenation cheme [7]. In thi work, the feedback PID controller and cheme are combined to achieve atifactory ervo a well a regulatory performance. The controller i deigned to achieve improved diturbance rejection performance, without acrificing the ervo performance and alo it provide compenation for the external diturbance and proce parameter variation. The feedback controller will provide compenation for the external diturbance and proce parameter variation. The detailed deign procedure of the propoed cheme i preented in ection and the imulation example are reported in ection followed by concluding remark in ection 4.. Combined Feedback plu Control Scheme A. Feedback plu control cheme The combined feedback plu control cheme i hown in Figure. The ignal r, d and y are reference, diturbance and proce output repectively. Gp () and Gm () are the proce and it model repectively. Cp () and CI () are the PI/PID and controller repectively. The proce and model repone i compared to eparate the diturbance repone. From the repone the diturbance i etimated uing the controller which contain an invere of the proce model. The etimated value i being ued to compenate for the unmeaured diturbance. It hould be noted that the enable the deigner to improve the diturbance repone without altering the ervo repone. The nominal cloed loop tranfer function relating proce output to the etpoint and diturbance are a follow: y () C () () () P GP Hr = = () r () + mc () () + C () G () I P P ( ) y () GP() Gm() CI() Hd () = = () d () + mc () I() + CP() GP() In the above equation m () i the model error i.e. m () = Gp() Gm(). Aume that the proce model Gm () i perfect and exactly matche with the proce behavior i.e. Gp() = Gm() then the cloed loop tranfer function are a follow: y () CP() Gm() Hr () = = () r () + CP() Gm() ( ) y () Gm() Gm() CI() Hd () = = d () + CP() Gm() From equation, it can be inferred that the propoed cheme proce repone to the etpoint change i independent of the controller and i dependent on the PID controller. Equation 4 how that the regulatory repone i a function of both PID and controller. The controller take the following form r C () p CI() Gm () f() (4) = (5) Gp () Gm () () Figure. Feedback plu Control Scheme Scheme + Where Gm () i the invertible portion of the proce model and f() i the filter. Uually the filter ha the following form: f() = ( λ + ) n (6) In the above filter n i the order of the filter and it i elected baed on the proce model and λ i the only tuning parameter. It could be noted that the lower value of λ improve the diturbance prediction capabilitie []. B. Tuning of the propoed control cheme A hown in equation and 4 the tability of the propoed control ytem i function of only PID controller. The regulatory repone can be modified independently by tuning the controller. In thi work, the Simplified tuning rule propoed by Skogetad i conidered to tune the primary PI/PID controller. Then, the controller i tuned to improve the regulatory performance. Step involved in the deign of the propoed control cheme Tune the PID controller uing Simplified PI/PID tuning rule and then Tune the filter parameter λ to achieve the deired regulatory performance. From equation and it can be inferred that the proce characteritic equation i function of Model Plant Mimatch u d CI y

3 International Journal of Control Science and Engineering 9, 9(): 9-4 (MPM) ( m () = G () G () ). The model inaccuracy p m affect the cloed loop performance and robutne. Lower value of λ improve the diturbance repone. On the other hand, it affect the robutne of the cloed loop ytem in cae of MPM. While deigning the controller, the proce uncertaintie ha to be conidered to yield a table cloed loop performance. In the propoed cheme, the number of tuning parameter ha increaed by one compared to the conventional PI/PID controller.. Simulation Example The following erie form of PID controller i ued in the combined and S-PID control cheme in the all imulation example: τ () I+ τ () D+ u= Kc r y () τi τf+ With τ F = ατ D. where K c i the controller gain, τ I the integral time, τ D the derivative time, τ F filter time contant and α the filter time contant factor. In order to avoid derivative kick, derivative on meaurement i implemented. The value of α =. ha been choen in the imulation tudy. Thi value wa choen in order to not bia the reult, but in practice (and epecially for noiy procee) a larger value of α in the range.. i normally ued [4]. The performance of the propoed control cheme i validated through the imulation example. To evaluate the controlled ytem performance, a unit tep etpoint change (r=) and a unit tep input (load) diturbance (G d =G P and d=) ha been conidered. The output performance metric namely Integral abolute error (IAE) and the input performance metric namely total variation (TV) of the manipulated input u are ued a the evaluation criteria for the comparion. The IAE and TV value are defined a follow: IAE = r() t y() t dt (7) (8) TV= uk ( + ) uk ( ) k= (9) Maximum enitivity M i a claical meaure of cloed-loop ytem robutne. The reciprocal of M i the hortet ditance between the Nyquit curve of the loop tranfer function and the critical point. The typical value of M hould be in range.-.. The maximum enitivity i then given by, M = max ω + Cp ( jω ) Gm ( jω ) () The range of M over.-. correpond to a gain margin of 6.-. and a phae margin of []. Moreover, the performance i compared with the Simplified and ZN PI/PID control cheme. B. Comparion with other tuning method Simplified -PID etting The proce model are uually repreented in the following form k θ Gm () = e ( τ+ )( τ+ ) () where k i the proce gain, τ, τ are the proce time contant, and θ the dead time. In [4] PI and PID etting are derived for FOPDT and SOPDT proce model. The S-PID etting for erie form of PID controller i a follow:.5τ Kc = kθ τi = min ( τ,8θ) () τ = τ ZN-PID etting [9] The Z-N PID tuning rule i a follow: Kc =.6 Ku = Pu / τd = Pu /8 D () Where K u and Pu are the proce ultimate gain and ultimate period. Example Conider a lag dominated FOPDT proce model Gm () = e, the propoed combined control + cheme PI controller i deigned uing S tuning relation. To improve the regulatory repone the controller filter contant i choen a.. The, S-PI and ZN-PI control cheme are alo tuned and the controller parameter are reported in Table. The etpoint-diturbance repone of the control cheme are imulated and hown in Figure. The controller repone are alo hown in Figure. The performance meaure are computed for the deigned control cheme and reported in Table. The roboutne meaure M i computed and alo reported in Table. The performance meaure and repone how that the propoed control cheme give improved regulatory repone compared to, Simplified -PI and ZN-PI control cheme. The control cheme how good ervo repone at the cot high controller output variation and the regulatory performance i found to be not atifactory. The Simplified -PI control cheme provide comparable ervo and regulatory repone. The propoed cheme robutne i depend on the primary PI/PID controller for the cae of no plant model mimatch. The propoed cheme robutne i better than the and ZN-PI cheme.

4 Arun R. Pathiran: Improving the Regulatory Repone of PID Controller Uing Internal Model Control Principle Table. FOPDT proce: comparion of ervo-regulatory performance of propoed control cheme with other control cheme Proce Outpout Method Propoed Controller parameter K c =.5 λ =. M Setpoint Diturbance IAE TV IAE TV λ = Simplifed -PI ZN-PI k c =.5 k c =.7 = Figure. Servo-regulatory repone of the propoed control cheme,, Simplified -PI and ZN-PI control cheme for FOPDT proce Controller Output Propoed S - PI Time [ec] Propoed S - PI Time [Sec] Figure. Controller repone of the propoed control cheme,, S-PI and ZN-PI control cheme for FOPDT proce It hould be noted that the ervo repone of the propoed and S-PI control cheme are identical. The additional controller performe only during the preence of diturbance and plant model mimatch. Hence the ervo repone i identical to S-PI cheme. The combined cheme give improved ervo and regulatory performance than the ZN-PI control cheme with minimum controller output variation. The combined cheme effectively reduce the undeired effect of unknown input diturbance in the proce repone compared to the conventional feedback control cheme. Example Conider a lag dominated SOPDT proce model Gm () = e, the combined control cheme ( + )( 5+ ) PID controller i deigned uing S PID tuning relation and the controller filter contant i choen a. The, S-PID and ZN-PID control cheme are alo tuned and the controller parameter are reported in Table. The etpoint-diturbance repone of the control cheme are hown in Figure 4. The controller repone are hown in Figure 5. The performance and robutne meaure are computed for the deigned control cheme and reported in Table. From the performance meaure and repone it can be inferred that the cheme give very good ervo repone with very high controller output variation. Alo, it regulatory repone i not atifactory a compared with other cheme. When the cheme i combined with the feedback control cheme the regulatory performance ha remarkably improved without altering the ervo repone of feedback controller. Table. SOPDT proce: comparion of ervo-regulatory performance of propoed control cheme with other control cheme Method Controller Setpoint Diturbance M parameter IAE TV IAE TV Propoed k c =.5 τd = λ = λ = Simplified -PID ZN-PID k c =.5 τ = 5 d k c = 4.7 = 5.8 τ =.46 d From the performance meaure and repone it can be inferred that the combined control cheme give improved regulatory repone than the S-PID cheme. The propoed cheme give improved ervo and regulatory performancez than the ZN-PID control cheme with minimum controller output variation. Moreover the ZN-PID cheme give more overhoot for etpoint change with high controller output variation. The imulation example preent the effectivene of propoed two-degree

5 International Journal of Control Science and Engineering 9, 9(): 9-4 of freedom controller tructure. The addition of controller improve the regulatory repone without diturbing the ervo repone. Proce Output Propoed S - PID D Time [ec] Table. Uncertainty analyi: FOPDT proce with +% change in proce gain, time contant, and dead-time Method Setpoint Diturbance IAE TV IAE TV Propoed S-PI ZN-PI From the Table and ervo-regulatory repone, it can be inferred that the propoed control cheme performance deviation from the nominal performance i le than the other control cheme. The controller how more deviation from the nominal performance. Thi analyi clearly how that the propoed combined control cheme offer improved performance than other control cheme in the preence of uncertainty in the proce parameter. Figure 4. Servo-regulatory repone of the propoed control cheme,, S-PID and ZN-PID control cheme for SOPDT proce Controller Output Propoed S - PID D Proce Output Propoed S - PI Time [ec] Time [ec] Figure 6. Servo-regulatory repone of propoed cheme,, S-PI and ZN-PI control cheme for FOPDT proce with +% uncertainty Figure 5. Controller repone of the propoed control cheme,, S-PID and ZN-PID control cheme for SOPDT proce C. Uncertainty analyi The propoed control cheme, controller i a model baed controller. The performance of controller depend on the proce model; the uncertaintie in proce parameter will affect the performance of the deigned controller. In order to ae the performance of the control cheme, an uncertainty of +% in the proce gain, time contant and dead-time of FOPDT proce ha been aumed and imulation tudie have been performed. The performance meaure uch a IAE and TV value of all the control cheme are computed and reported in Table. It may be noted that the controller etting of propoed and other control cheme were determined uing the nominal proce model. The ervo-regulatory repone and controller output of the control cheme in the preence of model-plant-mimatch i hown in Figure 6 and 7 repectively. Controller Output Propoed S - PI Time [ec] Figure 7. Controller repone of propoed cheme,, S-PI and ZN-PI control cheme for FOPDT proce with +% uncertainty

6 4 Arun R. Pathiran: Improving the Regulatory Repone of PID Controller Uing Internal Model Control Principle 4. Concluion Two degree of freedom control cheme ha been propoed by combining the feedback and controller. The ervo repone depend upon the primary feedback controller and the regulatory repone can be tuned independently via controller. The propoed cheme ue the principle to achieve an improved performance for the input diturbance. controller filter parameter λ i the only additional tuning parameter and by tuning thi parameter, regulatory performance can be improved without acrificing the ervo performance. The imulation reult how that the propoed cheme give improved regulatory repone than the S-PI/PID controller and give improved ervo-regulatory performance than the ZN-PI/PID controller. REFERENCES [] K.J. Atrom, T. Hagglund, The Future of PID Control, Control Engineering Practice, pp [] L.D. Deborough, and R.M. Miller, Increaing cutomer value of indutrial control performance monitoring Honeywell experience, Chemical Proce Control-VI (Tucon, Arizona, Jan. ), AIChE Sympoium Serie No. 6. vol. 98, USA,. [] A. O Dwyer, Handbook of PI and PID controller tuning rule. Imperial College Pre, London, 6. [4] K.J. Atrom, and T. Hagglund, Advanced PID control. ISA: Reearch Triangle Park, NC 779, 6. [5] Pathiran AR, Prakah J. Deign and Implementation of a model-baed PI-like control cheme in a reet configuration for table ingle-loop ytem. Can. J. Chem. Eng. 4; 9(9), pp [6] D.E. Seborg, T.F. Edgar, and D.A. Mellichamp, Proce dynamic and control. John Wiley & Son, New York, 4. [7] D.E. Rivera, M. Morari, and S. Skogetad, Internal model control: PID controller deign Ind. Eng. Chem. Proce De., 5, 986, pp [8] M. Morari, and E. Zafiriou, Robut Proce Control. Prentice-Hall, New Jerey, 989. [9] Y. Lee, S. Park, M. Lee, and C. Broilow, PID controller tuning for deired cloed-loop repone for SI/SO ytem, AIChE Journal, 44, 998, pp.6-. [] D. Chen, and D.E. Seborg, PI/PID controller deign baed on direct ynthei and diturbance rejection, Ind. Eng. Chem. Re., 5,, pp [] D.B. Santoh Kumar, R. Padma Sree, Tuning of baed PID controller for integrating ytem with time delay, ISA Tranaction, 6, 6, pp.4-5. [] J.G. Ziegler, and N.B. Nichol, Optimum etting for automatic controller, Tran. ASME, 64, 94, pp [] I. G. Horn, J. R. Arulandu, J. G. Chritopher, J. G. VanAntwerp, and R. D. Braatz, Improved filter deign in internal model control, Ind. Eng. Chem. Re. 5, 996, pp [4] S. Skogetad, Simple analytic rule for model reduction and PID controller tuning, Journal of Proce Control,,, pp [5] M. Shamuzzoha, and S. Skogetad, -PID controller for improved diturbance rejection of time-delayed procee, Ind. Eng. Chem. Proce De., 46, 7, pp [6] ZhenpengYu, JiandongWang, Performance aement of tatic lead-lag feedforward controller for diturbance rejection in PID control loop, ISA Tranaction, 64, 6, pp [7] R. Vilanova, O. Arrieta, and P. Pona, baed feedforward controller framework for diturbance attenuation on uncertain ytem, ISA Tran. 48, 9, pp [8] L. Juan, W. Chao, L. Shihua, Y. Jun, and L. Shengquan, Optimal diturbance rejection control approach baed on a compound neural network prediction method, Journal of Proce Control, 4,4, pp [9] J.G. Ziegler and N.B. Nichol, Optimum etting for automatic controller, Tran. ASME, vol. 64, no. 8, pp , 94.

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