Nonlinear Model Predictive Control of a Distillation Column Using Wavenet Based Hammerstein Model

Transcription

1 Engineering Letter, 0:4, EL_0_4_04 Nonlinear Model Predictive Control of a Ditillation Column Uing Wavenet Baed Hammertein Model K. Rameh, A. Hiyam, N. Aziz and S.R. Abd Shukor Abtract - Ditillation column are fairly complex multivariable ytem and need to be controlled cloe to optimum operating condition becaue of economic incentive. Nonlinear Model Predictive Control (NMPC cheme i one of the bet option to be explored for proper control of ditillation column. In the preent work, a new wavenet baed Hammertein model NMPC ha been developed to control ditillation column. An experimentally validated equilibrium model wa ued a plant model in nonlinear ytem identification and in NMPC. wo multiple-input-ingle-output (MISO wavenet baed Hammertein model are developed to model the dynamic of the ditillation column. he nonlinear model parameter were etimated uing iterative predictionerror minimization method. he Uncented Kalman Filter (UKF wa ued to etimate the tate variable in NMPC and the NLP problem wa olved uing equential quadratic programming (SQP method. he cloed loop control tudie have indicated that the performance of developed NMPC cheme wa good in controlling the ditillation column. Key word - Ditillation column; Nonlinear model predictive control; Sequential quadratic programming; Wavenet baed Hammertein model. I. INRODUCION Ditillation column are important proceing unit in petroleum refinerie and other chemical proceing indutrie (CPI for eparating feed tream, and for purification of final and intermediate product tream []. he eparation need relatively large amount of energy. Cloe control of ditillation column improve the product quality, minimize energy uage and maximize the plant throughput and it economy []. Mot of the indutrial ditillation column are currently controlled by multiloop controller baed on linear model. Among the Manucript received June, 0, revied July 7, 0. K. Rameh i with Proce Sytem Engineering (PSE Group in Faculty of Chemical & Natural Reource Engineering, Univeriti Malayia Pahang, 6300 Gambang, Kuantan, Pahang, Malayia. (Phone: , fax: , rameh@ump.edu.my A. Hiyam i with Proce Sytem Engineering (PSE Group in Faculty of Chemical & Natural Reource Engineering, Univeriti Malayia Pahang, 6300 Gambang, Kuantan, Pahang, Malayia. ( ahiyam@ump.edu.my N. Aziz i with School of Chemical Engineering, Engineering Campu, Univeriti Sain Malayia, Seri Ampangan, 4300 Nibong ebal, Pulau Pinang, Malayia. ( chnaziz@eng.um.my S.R. Abd Shukor i with School of Chemical Engineering, Engineering Campu, Univeriti Sain Malayia, Seri Ampangan, 4300 Nibong ebal, Pulau Pinang, Malayia. ( chyamrizal@eng.um.my multivariable controller, Model Predictive Control (MPC i an important advanced control technique which can be ued for difficult multivariable control problem [3]. MPC refer to the cla of control algorithm in which dynamic proce model i ued to predict and optimize the proce performance. he current generation of commercially available MPC technology i baed on linear dynamic model, and i refereed by the general term linear model predictive control (LMPC. Many procee uch a high purity ditillation column, multi-grade polymer reactor are ufficiently nonlinear to preclude the ucceful application of LMPC technology [4]. hi ha led to the development of nonlinear model baed controller uch a nonlinear model predictive control (NMPC in which more accurate nonlinear model i ued for proce prediction and optimization. Many author have tudied the performance of NMPC to control ditillation column uing different nonlinear model namely emi-rigorou reduced order model [5], NARX model [6], Hammertein model [7], Recurrent Dynamic Neuron Network (RDNN model [8] and grouped neural network (GNN model [9]. Fo et al. [0] in their cae tudy on proce modeling in Germany and Norway concluded that depite the commercially available modeling tool, the effort pent for all kind of modeling activitie i the mot time conuming tep in an indutrial project where model baed proce engineering technique are applied. he NMPC problem formulation involve online computation of a equence of manipulated input which optimize an objective function and atify proce contraint. he development of NMPC technique for large cale ytem may require problem formulation which exploit the pecific tructure of the nonlinear model. Finally, NMPC require online olution of a nonlinear program (NLP at each iteration. he olution of uch NLP problem can be very time conuming, epecially for large cale ytem. An additional complication i that the optimization problem generally i nonconvex becaue the nonlinear model equation are poed a contraint []. Conequently, NLP olver deigned for convex problem may converge to local minima or even diverge. So it i neceary to find out an improved olution algorithm for nonconvex NLP problem. he vital part of the preent tudy are to develop uitable nonlinear model for ditillation column, formulate NMPC problem and to find out an efficient optimization algorithm to be ued with NMPC. wo multiple-input-ingle-output (Advance online publication: November 0

2 Engineering Letter, 0:4, EL_0_4_04 (MISO wavenet baed Hammertein model are developed in thi tudy to model the dynamic of the ditillation column. An experimentally validated equilibrium model wa ued a plant model in nonlinear ytem identification and in NMPC. he remainder of the paper i organized a follow. Section wo provide the nonlinear ytem identification of new wavenet baed Hammertein model. he NMPC problem formulation, tate etimation and optimization algorithm are explained in Section hree and the reult of cloed loop control tudie are preented in Section Four. Finally concluding remark are mentioned in Section Five. II. WAVENE BASED HAMMERSEIN MODEL he detail of pilot plant ditillation column ued in thi tudy and the experimental validation of equilibrium model were dicued in []. he Hammertein model conit of a nonlinear tatic element followed in erie by a linear dynamic element. In thi tudy, two eparately parameterized nonlinearity tructured multiple-input-ingleoutput (MISO wavenet baed Hammertein model are developed to model the dynamic of the ditillation column. he reaon for uing two MISO model intead multipleinput-multiple-output (MIMO model i that the MISO model provide better prediction compared to MIMO model [3] he firt MISO model uing reflux flow rate (L and reboiler heat load (Q R a input, and top product compoition (x D a output, while, the econd MISO model uing reflux flow rate (L and reboiler heat load (Q R a input, and bottom product compoition (x B a output. In both the MISO Hammertein model a new wavenet baed nonlinear function i ued to decribe the nonlinear tatic block and Output Error (OE model i ued to decribe the linear dynamic block. he linear block i the Output Error (OE model, given by the following equation. B ( q B ( q y( x( x( ( A ( q A ( q he polynomial B ( q, A ( q, B ( q and A ( q in Equation ( are defined a nk ( nk bq bq ( nknb bnb q a q a q a na na q nk ( nk b q b q ( nknb bnb q a q a q a na na q B ( q ( A ( q (3 B ( q (4 A ( q (5 where, nb i the number of coefficient in B (q - and B (q - na i the number of coefficient in A (q - and A (q - nk i the delay from input to output b, b,, b nb are the coefficient of polynomial B a, a,, a na are the coefficient of polynomial A b, b,, b nb are the coefficient of polynomial B a, a,, a na are the coefficient of polynomial A Wavenet tructure baed nonlinear function x = F(u i ued to repreent the tatic nonlinearity of the Hammertein model. x( F( u ( u r PL af ( b(( u r Q c ak f ( bk(( u r Q c awg( bw(( u r Q cw awk g( bwk (( u r Q cwk d (6 x( F( ( P L a f ( b(( Qc ak f ( bk (( Qck awg( bw(( Q cw awk g( bwk (( Q cwk d (7 Subtituting Equation (6 and (7 in Equation (, the output of the wavenet baed Hammertein model y( i given by B ( q y( A ( q a k aw k k f ( b k g( bw ( u r PL a (( u r Q c k k k B ( q ( P L a f ( b(( Qc A ( q ak f ( bk (( Qck awg( bw(( Q aw g( bw (( u r Q cw d (( u r Q cw f ( b k k d (( u r Q c aw g( bw (( u r Q cw cw (8 he caling function f(u and f(u in Equation (6 & (7 are given by f u exp( 0.5 u u (9 f u exp( 0.5 u (0 ( ( he wavelet function g(u and g(u in Equation (6 & (7 are given by g( u (dim( u u uexp( 0.5 u u ( g u (dim( u u u exp( 0.5 u ( ( where, P and P are nonlinear ubpace parameter Q and Q are linear ubpace parameter r and r are regreor mean L and L are linear term coefficient in wavenet function aw k and aw k are wavelet coefficient bw k and bw k are wavelet dilation coefficient cw k and cw k are wavelet tranlation coefficient a k anda k are caling coefficient b k and b k are caling dilation coefficient c k and c k are caling tranlation coefficient d and d are output offet he experimentally validated equilibrium model wa ued a plant model to generate data required for nonlinear model identification. he ytem identification toolbox verion 7.0 in MALAB wa employed for parameter etimation. he model parameter were etimated uing iterative predictionerror minimization method. Random Gauian input equence i ued to make change in reflux flow rate (L and (Advance online publication: November 0

3 Engineering Letter, 0:4, EL_0_4_04 ABLE I PARAMEERS OF MISO HAMMERSEIN MODELS Model parameter Firt MISO model Second MISO model Linear OE model Parameter Nonlinear ubpace Parameter Linear ubpace parameter a = -.334, a = 0.36 a = -.36, a = b =, b = P = 55.3 P =.99 Q = 55.3 Q =.99 a = -.338, a = a = -.09, a = 0.09 b =, b = P = 55.3 P =.984 Q = 55.3 Q =.984 Linear term coefficient in wavenet function L = L = L = L = Output offet d = d = d = d = Regreor mean r = r = r = r = Wavelet coefficient aw = aw = aw = aw = Wavelet dilation coefficient bw = 5.99 bw = Wavelet tranlation coefficient cw = -.04 cw = bw = 5.53 bw = 5.98 cw = cw = Scaling coefficient a = 0 a = a = a = Scaling dilation coefficient b = b = 6.0 Scaling tranlation coefficient c = c = b = 3.04 b = 6.00 c = c = reboiler heat load (Q B imultaneouly, in order to generate data ued for nonlinear model identification. he model parameter are given in able I. he firt and econd MISO Hammertein model howed 94.64% and 95.% agreement with the equilibrium model repectively. III. NMPC he developed nonlinear wavenet baed Hammertein model i of the following form. X ( k F[ X (, U (3 Y ( h[ X (4 (4 where X ( [ x ( x U ( [ u( Y [ y ( y ( x and x are n-dimenional vector of tate variable, u (reflux flow rate and u (reboiler heat load are m- dimenional vector of manipulated input variable, and y (top product compoition and y (bottom product compoition are p-dimenional vector of controlled output variable. In thi work, two eparate MISO model were developed (one for each output intead of uing a MIMO model. he optimization problem i given by min J [ Y ( k P \ ] U ( k \, U ( k \, U ( k M \ P LY ( k j \ k, U ( k j0 j \, U ( k j \ (5 where U(k + j\ i the input U(k + j calculated from information available at time k, Y(k + j\ i the output Y(k + j calculated from information available at time k, ΔU(k + j\ = U(k + j-\ U(k + j\, M i the control horizon, P i the prediction horizon and φ and L are nonlinear function of their argument. he function φ and L can be choen to atify wide variety of objective and in thi tudy, the quadratic function of the following form i conidered: L [ Y ( k j \ k Y Q[ Y ( k j \ k Y [ U ( k j \ k U R[ U ( k j \ k U U ( k j \ SU ( k j \ [ Y ( k P \ Y Q[ Y ( k P \ Y (7 (6 where U ( and Y ( are teady-tate target for U and Y repectively, and Q, R and S are poitive-definite weighing (Advance online publication: November 0

4 Engineering Letter, 0:4, EL_0_4_04 matrice. he principal controller tuning parameter are M, P, Q, R, S and the ampling period Δt. An important characteritic of proce control problem i the preence of contraint on input and output variable. A major advantage of NMPC, compared to other nonlinear control trategie i that it provide explicit contraint handling capacity. In ditillation control uing NMPC, input contraint take the following form: umin u umax (8a min max (8b umin u umax (8c u u (8d min max where, u min - Minimum value of the reflux flow rate u max - Maximum value of the reflux flow rate u min - Minimum value of the reboiler heat load u max - Maximum value of the reboiler heat load Δu min - Minimum value of rate of change of reflux flow rate Δu max - Maximum value of rate of change of reflux flow rate Δu min - Minimum value of rate of change of reboiler heat load Δu max - Maximum value of rate of change of reboiler heat load he reflux flow rate bound are et to be [0., 0.75] l/min. he lower bound for reflux flow rate 0. l/min wa meant to keep the input phyically meaningful, namely, the reflux flow rate hould be poitive and have ome minimum value. he upper bound of reflux flow rate i approximately 50% of the nominal capacity, which would eldom occur in operation. he reboiler heat load bound are et to be [0, 5] kw. he lower bound meant that the reboiler heat load hould not be negative, wherea upper bound 5 kw wa the maximum heater capacity of the reboiler. Output contraint uually are aociated with operational limitation uch a equipment pecification and afety conideration. he output contraint can be poed a y y min y (9a max y y (9b min ymax where y - Minimum value of the top product compoition min y - Maximum value of the top product compoition max y - Minimum value of the bottom product compoition min y - Maximum value of the bottom product compoition max he top product compoition bound are et to be [0.5, ]. he lower bound for output 0.5 wa meant that the top product purity hould not be le than 50%. he upper bound wa meant that the maximum value of top product purity i 00% and beyond that i practically not meaningful. he bottom product compoition bound are et to be [0, 0.5]. he lower bound for bottom product compoition 0 wa meant that the maximum value of bottom product purity i 00%. he upper bound of bottom product compoition 0.5 wa meant that the bottom product purity hould not be le than 50%. A major advantage of NMPC compared to other nonlinear control trategie i that it provide the contraint handling capability. he deired product purity i achieved by olving the nonlinear optimization problem ubject to the following inequality contraint. U min U ( k j \ U max, 0 j M (0a U min U ( k j \ U max, 0 j M (0b Ymin Y ( k j \ Ymax, j P (0c In addition, the nonlinear model equation are poed a a et of following equality contraint: X ( k j \ F[ X ( k j \, U ( k j \ ], 0 j P ( Y ( k j \ h[ X ( k j \ ], j P ( where X(k\ = X( if the tate variable are meaured. It i important to note that input contraint are hard contraint in the ene that they mut be atified. Converely, output contraint can be viewed a oft contraint becaue their violation i neceary to obtain a feaible optimization problem. NMPC calculation require meaurement or etimate of the tate variable and in the preent work, UKF wa ued to etimate the tate variable in the NMPC problem. In UKF, the tate ditribution i repreented by a Gauian Random Variable (GRV, which i pecified uing a minimal et of carefully choen ample point. hee ample point completely capture the true mean and covariance of the GRV, and when propagated through the true non-linear ytem, capture the poterior mean and covariance accurately to the third order of aylor erie expanion for any nonlinearity [4]. Finally, the NMPC problem wa olved uing fmincon function in MALAB optimization toolbox verion 3..which ue a equential quadratic programming (SQP method. IV. CLOSED LOOP CONROL SUDIES Cloed loop control tudie were carried out to verify the performance of wavenet baed Hammertein model NMPC. hee tudie include diturbance rejection of NMPC for feed flow rate and feed compoition change, and et point tracking of NMPC for change in et point of top and bottom product compoition. he cloed loop imulation tudie were done through MALAB verion (R007a. A ampling interval of min wa choen for cloed loop control tudie. he NMPC parameter M, P, Q, R, S and the ampling period Δt are choen by repeated tuning and the final value are; ampling period Δt = min, prediction horizon P = 30 time tep (30 min, control horizon M = 6 time tep (6 min, Q = (,0.5, R = (, and S = (,. he experimentally validated equilibrium model (Advance online publication: November 0

5 Engineering Letter, 0:4, EL_0_4_04 for ditillation column wa ued a plant model in cloed loop control tudie. he performance of the NMPC wa tudied by making three feed flow rate diturbance: a +0% increae at t = 0 min, again a +0% increae at t = 40 min and a -0% decreae at t = 80 min. he repone of product compoition for thee feed flow rate diturbance are hown in Figure and it can be een from the Figure that the Hammertein NMPC uccefully rejected the feed flow rate diturbance within 0 min. he integral of abolute value of error (IAE wa calculated for each output and the numerical value of IAE performance index were indicated in all the NMPC repone of product compoition. he correponding repone of manipulated variable to feed flow rate diturbance are hown in Figure. he performance of the NMPC in et point tracking wa tudied by making -3% change in x D at t = 0 min followed by -3% change in x B at t = 50 min. he repone of product compoition for thee et point change are hown in Figure 3 along with the correponding numerical value of IAE performance index. It can be een from the Figure 3 that the Hammertein NMPC uccefully tracking the etpoint quickly. he correponding repone of manipulated variable for et point change are hown in Figure 4. x D IAE D = Set point Hammertein NMPC xd IAE D = Set point Hammertein x B IAE B = ime (min IAE B = 0-3 Fig. 3. Hammertein NMPC repone of product compoition to -3% change in x D at t =0 min and -3% change in x B at t = 50 min xb ime (min Fig.. Hammertein NMPC repone of product compoition to a feed flow rate diturbance L (l/min L (l/min 0. Q B (kw ime (min Fig. 4. Hammertein NMPC repone of manipulated variable to -3% change in x D at t =0 min and -3% change in x B at t = 50 min QB (kw ime (min Fig.. Hammertein NMPC repone of manipulated variable to a feed flow rate diturbance he control performance of the Hammertein NMPC for imultaneou change in both et point and load variable wa tudied by making following change: a -3% change in x D along with a +0% change in feed flow rate at t = 0 min and a -3% change in x B along with a -0% change in feed compoition at t = 50 min. he repone of product compoition for thee change along with their actual et point are hown in Figure 5 and the correponding repone of manipulated variable are hown in Figure 6. It wa noted that the change in Q B i cloe to the upper limit of the reboiler heat load. he IAE performance of index for top and bottom product compoition wa found to be and repectively. (Advance online publication: November 0

6 Engineering Letter, 0:4, EL_0_4_04 9 x D 7 6 IAE D = Set point Hammertein NMPC REFERENCES [] J.J. Down and J.E. Do. "Preent tatu and future need-a view from North American indutry". in Proceeeding of the international conference on chemical proce control 99. Padre Iland, X, USA. x B Fig. 5. Hammertein NMPC repone of product compoition for imultaneou change in et point and diturbance L (l/min Q B (kw IAE B = ime (min ime (min Fig. 6. Hammertein NMPC repone of manipulated variable for imultaneou change in et point and diturbance V. CONCLUSION wo wavenet baed MISO Hammertein model were developed to be ued with NMPC to control ditillation column. An experimentally validated equilibrium model wa ued a plant model in nonlinear ytem identification and in NMPC. he model parameter were etimated uing iterative prediction-error minimization method. he Uncented Kalman Filter (UKF wa ued to etimate the tate variable in NMPC and the NLP problem wa olved uing equential quadratic programming (SQP method. he cloed loop control tudie indicated that the developed NMPC technique performed well in controlling the ditillation column by rejecting the diturbance in regulatory control and tracking the et point quickly in ervo control.. [] S. Hurowitz, et al., "Ditillation control configuration election". Journal of Proce Control, (4: pp [3] G.C. Goodwin, S.F. Graebe and M.E. Salgado, Control Sytem Deign. 00, Upper Saddle River, NJ: Prentice Hall. [4] M.A. Henon, "Nonlinear model predictive control: Current tatu and future direction". Computer and Chemical Engineering, (: pp [5] S.N. Maiti, S. Ganguly and D.N. Saraf, "Some new approache for the control of a ditillation column and their experimental evaluation on a pilot plant". Computer & Chemical Engineering, (Supplement : pp [6] G.R. Sriniwa, et al., "Nonlinear identification and control of a highpurity ditillation column: a cae tudy". Journal of Proce Control, : pp [7] K.P. Fruzzetti, A. Palazoglu and K.A. McDonald, "Nonlinear model predictive control uing Hammertein model". Journal of Proce Control, (: pp [8] A.M. Shaw and F.J. Doyle III, "Multivariable nonlinear control application for a high purity ditillation column uing a recurrent dynamic neuron model ". Journal of Proce Control, (4: pp [9] J. Ou and R.R. Rhinehart, "Grouped neural network model-predictive control". Control Engineering Practice, 003. (7: pp [0] F. Michelen and B. Fo, "A comprehenive mechanitic model of a continuou Kamyr digeter". Applied Mathematical Modeling, : pp [] M. Cannon, "Efficient nonlinear model predictive control algorithm". Annual Review in Control, : pp [] K. Rameh, N. Aziz and S.R. Abd Shukor, "Nonlinear model predictive control of a ditillation column uing NARX model". Computer Aided Chemical Engineering, : pp [3] E. Ekinat, S.H. Johnon and W.L. Luyben, " Ue of Hammertein model in identification of nonlinear ytem". A.I.Ch.E. Journal, (: pp [4] E.A. Wan and R.V.D. Merwe. "he Uncented Kalman Filter for Nonlinear Etimation". in In Proceeding of Sympoium 000 on Adaptive Sytem for Signal Proceing, Communication and Control (AS-SPCC Lake Louie, Alberta, Canada.: IEEE. (Advance online publication: November 0