PREDICTIVE CONTROL OF A PROCESS WITH VARIABLE DEAD-TIME. Smaranda Cristea*, César de Prada*, Robin de Keyser**

Transcription

1 PREDICIVE CONROL OF A PROCESS WIH VARIABLE DEAD-IME Smaranda Cristea, César de Prada, Robin de Keyser Department of Systems Engineering and Atomatic Control Faclty of Sciences, c/ Real de Brgos, s/n, University of Valladolid, Spain fax: ; tel: prada@atom.va.es, smaranda@atom.va.es Department of Control Engineering & Atomation University of Gent, ecnologiepark-zwijnaarde 9, B-905 GHEN, Belgim rdk@atoctrl.rg.ac.be Abstract: is paper deals wit te control of variable-delay processes, were te delay depends on te vale of te maniplated variable, wic reslts in a non-linear system difficlt to control. As a reference process, te case of a eated tank were te controlled variable is te liqid temperatre and te placement of te sensor introdce a transport delay in te control loop, as been considered. is callenging problem is approaced from te perspective of predictive control, sing te non-linear EPSAC controller. Copyrigt 005 IFAC Keywords: Variable dead-time, Predictive control, Nonlinear control, Delay systems. INRODUCION ime delays in te feedback loop appear freqently in indstrial processes, bot from te process itself and cased by te controllers. e presence of deadtime in a feedback control loop as a significant impact on te performance of te control loop. Depending on te magnitde of te delay, performance of te feedback loop can be severely limited. A difficlt and callenging problem is to control a process witot instantaneos measrement of te state variables, or via delayed actators (Ricard, 003). Dead-time compensating controllers (DC) ave received attention in te literatre. e most widely sed metod in indstry is te Smit predictor, (Smit, 957). In Hägglnd (996), a PI controller wit model-d prediction (PPI) was presented. Anoter dead-time compensation in te controller design is proposed in Normey-Rico et al. (997). A simple modified Smit predictor controller was also related in Cien et al. (00). Dead-time compensators seems to be a good option for improving te system response witot increasing too mc te complexity of te controllers. Bt all of tese dead-time compensators consider stable or integrating processes wit a constant dead-time, identified like: K n ds K n ds Pn = e or Pn = e s + s n In te process indstry is not ncommon to find transport delays de to te placement of measring elements far away from te process being controlled. ese transport delays introdce a pre significant dead time tat canges wit te vale of te flow, wic, in trn, is often a maniplated variable. is means tat te difficlty of te control problem increases significantly as tere is a relationsip between te dead-time and te maniplated variable. e model ten becomes: d ( ) s Pn = G( s) e wic is non-linear. e paper tries to contribte to te soltion of tis problem tat is still an open one in spite of te many efforts developed in te past. (Ricard, 003). e approac follows te NMPC ideas, taking advantage of te non-linear EPSAC (De Keyser, 998) implementation to gain in adeqacy to te problem as well as in comptation time. In tis sense, it does not present a new approac, bt sows

2 te advantages of te proposed metod wen dealing wit variable time delays de to its dependence of te maniplated variable. Oter NMPC linearization metods as been proposed in te literatre (Kovaritakis, 999), (Lee, 00), (Alvarez, 000) bt none is focsed to te variable delay problem, nor offers clear advantages over te non-linear EPSAC. e paper content is split in five sections. After te introdction, section describes te process to be controlled. In section 3 a brief review of te basic ideas of te non linear predictive control is presented followed by te nonlinear EPSAC algoritm. Section 4 is devoted to experimental reslts concerning te controlled process and finally, some conclding remarks are provided.. PROCESS DESCRIPION Wit te prpose of illstrating te ideas beind te proposed controller, a simple, bt callenging problem as been cosen. e process considered in or case stdy consists of a stirred tank (Fig. ) wit inlet volmetric flow q. e liqid is eated by a constant amont of eat Q spplied by a sbmerged electrical eating element and flows ot of te tank by overflow. e objective is to keep te tank temperatre at a desired vale maniplating te inlet flow q wose temperatre is i. e temperatre is measred, not in te tank, bt in te efflent pipe in a place located at a distance L from te tank. is distance L introdces a transportation delay to te control loop tat depends on te flow, wic is te maniplated variable. e matematical model tat describes ow te otlet temperatre ( responds to canges in inlet flow q( is given by an energy balance: d( Q A = q( ( i ( ) + () dt ρc p were te constant parameters ρ, A, c p, density, cross section and specific eat, are caracteristic of te tank and te eat losses to te srrondings are assmed to be negligible. e measred variable is denoted as d and its response will be te same as ( except tat it will be delayed by a transportation time d(: ( = ( t d( ) () q, i Q d Fig. L q, d e time delay d( is inversely proportional to maniplated variable: distance L LS d ( = = = (3) velocity q( q( S were S is te cross section of te efflent pipe. e model () is non-linear, bt if te measred variable were, it cold be fairly well controlled sing standard controllers. Neverteless, te strong relationsip between te dead-time and te otpt of te controller, provides a variable delay tat increases te difficlty of te problem and eiter force to detne te controller in order to gain stability margin or oblige s to consider explicitly te dynamic non linear model of te process for te development and implementation of te control system. 3. NONLINEAR PREDICIVE CONROLLER Nonlinear predictive control (NMPC) is a natral extension of te linear MPC tecniqe. Most of te algoritms are d on te se of an internal nonlinear plant model, wic captres te main process caracteristics, followed by a dynamic optimization tat provides te optimal maniplated variables. In or case, te pysical model ()-(3) was sed trying to reflect te dynamics of te process. In tis section, te NMPC and EPSAC approaces will be presented briefly. 3. NMPC Controller e objective of te non-linear model predictive control (NMPC) is finding te ftre optimal maniplated variable seqence in order to imize a fnction d on a desired otpt trajectory over a prediction orizon. e cost fnction is te integral over te sqares of te residals between te model predicted otpts y pred and te set point vales r over te prediction interval N τ (were N is te prediction orizon and τ is te sampling time). A typical formlation is ( k / k ),..., ( k + N / k ) N j= 0 β J = [ ( k + j tk + N τ tk γ [( y ( r( ) pred dt + (4) e cange in te maniplated variable is also inclded in te imization. Oter formlations inclde penalty terms or additional constraints in order to garantee certain stability properties. In order to perform te optimization, it is necessary to parameterise te maniplated variable; oterwise an infinite nmber of decision variables wold appear in te problem. An sal possibility is te discretization of te along te control orizon (N ) were te inpt remains constant over te sampling period τ:

3 ( = (k), kτ t < (k+)τ (k) = (N -) k > N - y optimize y y e imization (4) is done nder te constraints of te continos model eqations and te typical ones applied on te maniplated and controlled variables: ( k) y y( k) y ( k) (5) δ(t/ time Only te first component of te N moves optimal control seqence, is implemented, and te wole procedre is repeated every sampling period. e controller law soltion leads to a dynamic nonlinear programg problem, wic cold be formlated generically as a real time imization of a non-linear fnction wit constraints, were te index J can be compted by simlation eac time te optimization algoritm needs it. 3. Nonlinear iterative EPSAC formlation e key idea of tis formlation is to approximate te non-linear predictions by iterative linearizations arond ftre trajectories, so tat tey converge to te tre non-linear optimal soltion. For tis prpose, te ftre seqence of maniplated variables is considered as te sm of a basic ftre control scenario, called ( t + k /, k 0 and optimizing ftre control actions δ ( t + k /, 0 k N : ( t + k / = ( t + k / + δ ( t + k / (6) In tis way, see Fig., te otpt predictions can be considered in a first approximation as being te cmlative reslts of two effects: y( t + k / y ( t + k / + yoptimize( t + k / (7) k=,..n e component y ( t + k / is calclated as te otpt of te non-linear model sing te known (postlated) seqence ( t + k / as te model inpt. e oter component, y optimize ( t + k /, is te response to te δ(t+k / component of te inpt. Notice tat (7) is only an approximation d on te se of te sperposition principle. We will come back on tis point later on. Assg tat te pertrbations δ(t+k / are small enog, its is possible to derive an expression for tis term as te one tat corresponds to a linearised model along ( t + k /. As te terms δ(t+j / are implses at times t+k, except te last one at time N -, tat is a step, and te effect at time t+k of an implse at time t+j is k-j, ten te term y optimize ( t + k / is te acmlative effect of a series δ(t+/ Fig. time of implse inpts and a step inpt (De Keyser, 998): yoptimize ( t + k / = kδ( t / + k δ( t + / + (8)... + g k N ( / + δ were te parameters,,.., k,... N are te coefficients of te nit implse response of te system at te crrent operating point, wereas te vales g k refer to te nit step response coefficients. Notice tat te vale of te parameters reflects te effect of te variable delay arond tis specific control actions. Using matrix notation, te prediction eqation ten becomes Y = Y + GU (9) were Y = [ y ( /... y ( / U = [ δ( t /... δ( / (0) N + N N... g N N = N + N N... g N N + G N N N... g N N + A simple relationsip exists between te control actions and δ: ( t / δ( t / ( t + / δ( t + / = A + b () ( / δ( / wit te matrix A and te vector b given by: A =, () ( t / ( t ) ( + / ) ( / ) = t t t t b... ( / ( /

4 In tis description, te coefficients of te matrix G are compted sing te linearized model abot te crrent trajectory at eac sampling instant. In case of a complex dynamic model, obtaining a linearized model is not an easy task. o avoid it, a possible alternative is to se te non-linear model to calclate te coefficients k and g k from pertrbations in te model sing a simlation procedre. aking into accont te relationsip (), and recalling tat k = gk gk, a new formla is obtained for y optimize ( t + k / : y + ( t + k / = g [ ( t ) + ( ( optimize k N g k i + i= [ ( t + i) + ( t + i ) ( t i) (3) e step response coefficients can be calclated eac sampling instant simlating te nonlinear model of te system wit a particlar ftre seqence ( t + k / taking as initial conditions te crrent process state and evalating te predictions y ( t + k /. A simple coice for ( t + k / cold be ( t ) + (. Wit tis option and considering tat y ( t + k / = y ( t + k / + yoptimize( t + k / (4) te coefficients g k verify te formla c0g k = y ( t + k / y ( t + k / (5) g k c g k c... g k N + cn were g = 0 0 c = ( t ) + ( 0 c = j ( t + j ) ( (6) ( t + j), j Based on (9)...(), te cost fnction is a qadratic form in U N N [ y( t + k / r( t + k / + β [ ( t + k / J = γ k = N k = 0 ( R Y GU) ( R Y GU) + β ( AU + b) ( AU + b) = γ (7) and te optimisation problem, te imization of J sbject to te constraints (5), can be solved wit simple qadratic programg tecniqes (QP). As mentioned before, as (7) is only an approximation becase te sperposition principle does not old, te controls ( t + k / = ( t + k / + δ ( t + k /, d on te soltion δ ( t + k / of (7), are sboptimal. However if te approac is repeated iteratively in te same sampling instant by redefining ( t + k / ( t + k / and recalclating δ ( t + k / and ( t + k / ntil δ ( t + k / 0, ten, as δ ( t + k / 0, te term y optimize ( t + k / is practically zero and ten (7) olds. So, te soltion obtained in tis iteration, converge to te optimal = non-linear soltion. In tis way, te non-linear optimization problem is replaced by several QP problems. o redce te nmber of iterations, wic is crcial for te efficiency of te algoritm, te initial vale of ( t + k / is important. A simple and effective coice, (De Keyser, 998), is to start wit te optimal control policy derived at te previos sample ( t + k / ( t + k / t ). In tis paper, tis strategy as been sed, leading to a very small nmber of iterations per sampling time, sally tree or for. 3.3 State estimation On te oter and, te crrent state is needed as an initial condition at eac iteration to predict te ftre beavior and select an optimal control seqence. Since te state of te nonlinear system is not directly measrable at time t, bt wit a delay, an estimation tecniqe was implemented in order to reconstrct te crrent state of te system. e tecniqe cosen was a receding orizon one. e model state is compted d on te process model and past vales of maniplated variable. e present measred otpt d ( is sed as initial condition to simlate te process model starting at past time t d, and applying te known past discrete control seqence (t-ne), (t-ne+),,(t-). e receding orizon NE depends on te dead time: NE=d/τ. So, te vale ( is te one obtained integrating te model from IME=t-NEτ to IME=t wen te past controls are applied. (Fig.3) 4. SIMULAION RESULS Several experiments ave been carried ot in a simlated environment in order to test te nonlinear EPSAC controller on te process previosly described and to compare it wit oter approaces. In all of tem te sampling period is 0 seconds Measred otpt: (t-ne)= d ( past past controls ftre State estimation: ( Receding Horizon: NE time Present moment t Model otpt Recorded otpt Fig.3 State estimation

5 wereas te oter parameters, prediction orizon, control orizon and control weigt, are N =5, N =, γ=, β=0 respectively. For te maniplated variables, te constraints were fixed to = 5, = 00 and teir canges were limited to = 5 ; = 5.e controlled variables are constrained by y 5 and y 30. = = e simlations ave been performed sing te simlation langage EcosimPro wit te process being represented by eqations ()..(3), were te parameters ad te following vales: te area of te tank was A=500 cm, te lengt of te pipe L=50 m, te initial level =5 cm, te inpt temperatre of te liqid i =5 ºC and te initial vale of te inlet flow q=50 cm 3 /s. 4. Setpoint tracking Fig. 4 sows te evoltion of te tank temperatre wit te proposed non-linear EPSAC controller dring 0000 seconds were several set point step canges were performed, sown in dotted line. e estimation of te temperatre in te tank is also sown. e controller was started wit a reference vale of.8 ºC (te steady-state vale). e temperatre follows fairly well te set point canges. Notice tat no delay appears in te grap becase we ave represented te temperatre in te tank and not te measred temperatre d. Also notice tat te graps of and its estimation are sperposed, so tat no clear difference can be seen between tem. e maniplated variable is represented in Fig. 5 and te time varying delay can be seen in Fig.6. In order to compare tese reslts wit te ones obtained wit a linear predictive controller wit fixed delay time and to keep te same metodology, te linear EPSAC was applied wit te following CARIMA model: C( q ) A( q ) y( = B( q ) ( + e( (8) q Fig. 5 e maniplated variable (non linear controller) were A( q ) = 0.875q B( q C( q ) = ( q ) = ) q k b A e open-loop time constant is τ = = 50 q seconds. Using te same parameters and constraints as te non linear controller and taking into accont an identical simlation seqence, tree cases was considered, wit different vales for te pre time delay k b taking into accont te range represented in Fig. 6. In te first case, te constant model delay was nderestimated. Fig. 7 illstrates te evoltion of te temperatre considering a constant small vale for te dead-time: d=00 seconds < τ (k b =5). e response as a very big oversoot and it is too oscillatory. In te second experiment, sown in Fig. 8, an overestimated fix vale of: d= 300=τ (k b =5) is considered. e response improves, bt is oscillatory and worst tan te one of Fig.4. Notice tat overstating a delay is not a soltion for te stabilisation of te system as it is sown in Fig.9, wic corresponds to te same linear controller operating wit a model wit a fixed dead-time of d=440 3τ (k b =). As we can see, te response is too oscillatory and clearly nacceptable. Fig.4 Setpoint tracking of te controlled variable Fig. 6 e dead-time

6 Fig.7 e controlled variable (sing linear controller wit d=00 seconds) Fig.9 e controlled variable (sing linear controller wit dead-time d=440 seconds) As a final test, te fll NMPC controller was implemented and te experiments repeated. Reslts were identical to Fig. 4 bt te comptation time was twice te one needed for te non-linear EPSAC to obtain te same reslt. 6. AKNOWLEDGMENS e researc was spported by te Spanis Department of Science and ecnology, trog project DPI CONCLUSIONS In tis paper an alternative approac to te soltion of NMPC problems ave been presented, applied to a difficlt control problem de to te variable delay involved. One important point to mention is tat te way in wic te internal model is sed in nonlinear EPSAC, gives trog te te gross effect of te variable delay, particlarly wen N >, and provides an atomatic fitting to te cange of tis effect via te step and implse coefficients, improving its response. Anoter advantage is te redction in comptation time over a pre NMPC approac. Notice tat cases of variable, bt comptable, delay, like te one considered ere, in wic te main nonlinearity is te variable delay, can be represented by te model d ( ) s Pn = G( s) e for wic, te proposed non-linear EPSAC approac provides an attractive and simple soltion. Fig.8 e controlled variable (sing linear controller wit dead-time d=300 seconds) REFERENCES Alvarez,., adeo, F.,Mejias. D., El Gomari M.Y., Villanova, R., Prada C. de. (000). Constrained Predictive Control of Robotic maniplators sing on-line linearized models. Worksop on Systems wit time-domain constraints, Eindoven, Holland. Cien, I.L., Peng, S.C. and Li, J.H. (00). Simple control metod for integrating processes wit long deadtime. Jornal of Process Control,, De Keyser, R.M.C. (998) A gentle introdction to model d predictive control. EC-PADI International Conference on Control Engineering and Signal Processing. Lima, Perú, Plenary Paper. Hägglnd,. (996). An indstrial dead-time compensating PI controller. Control Engineering Practice, 4, Kovaritakis, B., Cannon, M., Rossiter, J.A. (999). Non linear model predictive control. International Jornal of Control, 3(), Lee, Y.I., Kovaritakis, B., Cannon, (00). Constrained receding orizon predictive control for non-linear systems. Atomatica, 38, Normey-Rico, J., Bordons, C. and Camaco, E. (997). Improving te robstness of dead-time compensating PI controllers. Control Engineering Practice, 5, Ricard, J.P. (003). ime-delay systems: an overview of some recent advances and open problems. Atomatica, 39, Smit, O.J.M (957). Closer control of loops wit dead time. Cemical Engineering. Progress, 53, 7-9