An Algorithm for Tuning NMPC Controllers with Application to Chemical Processes

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1 Ths s an open access artcle publshed under an ACS AuthorChoce Lcense, whch permts copyng and redstrbuton of the artcle or any adaptatons for non-commercal purposes. Artcle pubs.acs.org/iecr An Algorthm for Tunng NMPC Controllers wth Applcaton to Chemcal Processes Federco Lozano Santamaría and Jorge M. Goḿez* Grupo de Dsenõ de Productos y Procesos, Departamento de Ingenería Químca, Unversdad de los Andes. Carrera 1 No. 18A-1, Bogota, Colomba Downloaded va on September 3, 18 at :4:3 (UTC). See for optons on how to legtmately share publshed artcles. ABSTRACT: The advantages of lnear/nonlnear model predctve control (N)MPC for dealng wth the multple nput multple output problem, for performng optmzaton and for handlng constrants are well-known and because of that t has been appled wdely n the chemcal ndustry. However, there s a recurrent problem for ths knd of controllers and t s how to defne the best tunng parameters to acheve a good closed-loop response. Ths s an open queston for research even when the common practce s to defne the (N)MPC tunng parameters by tral and error or by the expertse of the control engneer. There are some works that propose a systematc defnton of the (N)MPC tunng parameters, but most of them are restrcted to certan predcton models or do not defne all the tunng parameters. Ths papers presents a general tunng algorthm for (N)MPC controllers that allows the systematc defnton of all the tunng parameter and t s not restrcted to a specfc predcton model or to a problem formulaton. It s based on multobjectve optmzaton for the defnton of the objectve functon weghts and on the optmzaton of a closed loop performance ndex for the defnton of the horzon lengths. Three cases of chemcal processes are consdered n detal to llustrate the applcaton and advantages of the proposed (N)MPC tunng algorthm. 1. INTRODUCTION Nonlnear model predctve control (NMPC) s an advanced control strategy wdely used n dfferent ndustres because of ts capablty to handle constrants, to perform optmzaton and to handle the multvarable problem drectly. The essence of NMPC s to solve an optmal control problem (OCP) n a recedng horzon scheme to construct the control law, n other words, at every tme perod t mnmzes certan performance ndex subject to a (non)lnear model (equalty constrants) and operatonal lmts (nequalty constrants). Even though (N)MPC controllers have been used n many applcatons t s clear that ther good performance depends on two mportant factors: the use of an accurate enough model to represent the plant dynamc and the selecton of tunng parameters to obtan a good closed loop response. 1 If the controller exhbts poor performance, even when ts tunng parameters have been mproved, t s possble that the model used for predcton s not approprate and t should be refned. 1 Ths paper s not focused on the selecton of predcton models for (N)MPC, rather t assumes that a nonlnear model can descrbe the process dynamc precsely. On the other hand, f the model s adequate, the closed loop performance of the system can be mproved selectng approprate tunng parameters for the controller. Usually, the (N)MPC tunng has been done va tral and error or based on the desgner expertse, but t s possble to tune the controller systematcally and there are many studes that try to solve the tunng problem from dfferent perspectves. 1 In addton, the ncreasng number of (N)MPC applcatons stpulates a systematc tunng procedure. The tradtonal tral and error methodology for tunng (N)MPC controllers s an neffcent strategy because t requres the evaluaton of a large number of combnatons. Also, many of the parameters have coupled or overlappng effects on the performance of the controller; thus, t s clear the need for a more precse and consstent method for tunng these controllers. The man dea of a systematc tunng procedure for (N)MPC controllers s to defne the controller parameters (e.g., predcton horzon, control horzon, weghts of the dfferent objectves) based on performance ndexes of the closed loop behavor of the system. It mght seem easy to defne a performance ndex for the closed loop response as a smlar ndex s defned for the OCP that s solved perodcally, but there could be more aspects that are mportant for the control engneer that are not represented n the objectve functon of the OCP. Moreover, the OCP s usually nonlnear and s subject to nequalty constrants, whch make t mpossble to fnd an analytcal expresson of the control law. 3 In consequence, the performance ndex defned for the closed loop behavor of the system cannot be related explctly wth the tunng parameters of the controller. In the lterature there are many works dealng wth the problem of tunng (N)MPC controllers from dfferent perspectves, but none of them, as far as we know, s capable to defne all the Receved: March 1, 16 Revsed: July 6, 16 Accepted: August 5, 16 Publshed: August 5, Amercan Chemcal Socety 915 DOI: 1.11/acs.ecr.6b111

2 Industral & Engneerng Chemstry Research tunng parameters for any knd of (N)MPC controller and the development of a more general methodology s stll a challenge and an open area for research. 4 For example, Shrdhar and Cooper 5 present a tunng strategy based on a complaton of heurstcs and on the defnton of a well-condtoned optmzaton problem. Ths strategy begns wth the approxmaton of the predcton model to a frst order plus dead tme (FOPDT) model and then t uses some smple equatons to calculate all the tunng parameters of the controller, ncludng the length of the horzons. Nevertheless, the approxmaton to a FOPDT model and the use of heurstcs mght not be accurate enough to guarantee a good closed loop performance or a robust controller. It s mportant to note that ths approach can also be appled to nonlnear MPC, but frst the model must be approxmate to a FOPDT model n order to dentfy some characterstcs of the process such as tme constants and dead tmes whch are requred n the tunng expressons. A more rgorous approach proposed by D Carno and Bemporad 6 s a tunng strategy based on matchng the MPC closed loop performance to the closed loop performance of a favorte controller. To match the two controllers they propose a mnmzaton of the dfference of the control law predcted by both controllers. However, t only defnes the objectve functon weghts and t s lmted to lnear models wthout nequalty constrants n whch t s possble to fnd an explct soluton of the feedback control sequence. In another work by Francsco and Vega 7 a multobjectve optmzaton strategy, specfcally goal programmng, s used to solve the conflctng objectves of the controller and to defne the penalzaton weghts of the rate of change of the manpulated varables. Then, a random search s used to defne the length of the horzons based on the same performance ndex of the multobjectve optmzaton problem. Ths strategy s lmted to lnear models wthout constrants and for nonlnear models t s necessary to lnearze them around the operaton pont whch reduces the effcacy of the fnal tunng parameters on the closed-loop performance. One fnal example s the tunng procedure presented by Vallero et al. 8 n whch a multobjectve optmzaton procedure s used to defne the weghts of the conflctng objectves assumng a fx control and predcton horzon. The multobjectve OCP s solved usng the enhanced normalzed normal constrant or normal boundary ntersecton method and the Pareto front s defned. Once the analyst has chosen a compromsed soluton on the Pareto front, the result of the multobjectve optmzaton can be translated nto a weghted sum whch s a smpler and more common approach for NMPC controllers. In ths Artcle, we present a tunng algorthm for NMPC controllers that allows the defnton of the tunng parameters (predcton horzon, control horzon, and the objectve functon weghts) and that s not restrcted to a specfc type of predcton model or problem formulaton. The proposed tunng algorthm s based on the constrant optmzaton of a performance ndex of the closed loop behavor of the controller. It uses a multobjectve optmzaton technque, specfcally a utopa pont trackng approach, to defne the weghts of the dfferent objectves and a nonlnear ft to approxmate the behavor of the closed-loop performance ndexes wth respect to the length of the horzons. Ths fnal step allows the formulaton and soluton of a new optmzaton problem n whch the tunng parameters of the controller are the optmzaton varables. The rest of ths Artcle s structured as follows: secton presents a bref descrpton of NMPC and dynamc optmzaton. The tunng strategy, whch s the man contrbuton of ths work, 916 Artcle s descrbed n detal n secton 3. It presents each step of the algorthm and the dfferent optmzaton technques used wthn t such as multobjectve optmzaton usng the utopa trackng approach and the approxmaton of the performance ndexes to quadratc functons of the predcton and control horzons. Secton 4, descrbes the advanced step NMPC (asnmpc) controller. Secton 5 llustrates the applcaton of the NMPC tunng algorthm wth two case studes of chemcal processes, a CSTR reactor, and an extractve dstllaton column. For the frst case, two possble scenaros are consdered whch result n a fnal repertory of three examples that apply the proposed tunng algorthm. The fnal conclusons and consderatons of ths Artcle are presented n secton 6.. BACKGROUND ON NMPC AND DYNAMIC OPTIMIZATION Nonlnear model predctve control (NMPC) s an advanced control strategy that solves an OCP at every tme perod. The soluton to ths problem s a control sequence over a future horzon, but from ths sequence only the frst element s mplemented n the plant to construct the control law. Fgure 1 presents a smple example of how model predctve control works. At the current tme (t k ), an OCP subject to a Fgure 1. Representaton of the recedng horzon strategy n the (N)MPC formulaton. model of the process s defned and solved onlne n a future predcton horzon (N p ) where only a number of control actons are allowed (N u ). The soluton of the optmzaton problem s the predcton of the future behavor of the process and the necessary control actons to acheve the desred operaton. However, only the frst value of the manpulated varables s feedback to the process and the rest future actons predcted are dscarded. The defnton of the OCP s updated based on new measurements of the state varables and t s solved agan. The crtcal step n the mplementaton of NMPC controllers s the soluton of the OCP because t has to be solved perodcally and n less tme than a samplng tme nterval. A general formulaton of the OCP s gven by eq 1, where x d are the dfferental states, x a the algebrac states, y the measured outputs, w I the dsturbances n the states, w O the dsturbances n the outputs, and u the manpulated varables. The objectve functon, eq 1a, s composed by two terms: a runnng cost (ϕ), whch s a weghted sum of dfferent objectves lke a quadratc trackng functon, a quadratc penalzaton of the rate of change of the manpulated varables or an energy or economc functon, and a termnal cost (ϕ f ), whch s only a functon of the states at the fnal tme of the predcton horzon and s normally used to DOI: 1.11/acs.ecr.6b111

3 Industral & Engneerng Chemstry Research approxmate the fnte horzon problem to the nfnte horzon problem. 9 The optmzaton problem s subject to a set of dfferental and algebrac eqs eqs 1b 1d that can be lnear defnng a quadratc programmng problem (QP) or nonlnear defnng a nonlnear programmng problem (NLP). Equaton 1f s also an equalty constrant, but t s presented separately from the others because t represents the measured outputs for the control problem. The nequaltes of the problem, eq 1e, are normally assocated wth varables bounds or bounds on ther rate of change, but they can also be nonlnear expressons. tk mn OBJ = ϕ( t, y, u) d t + ϕ( y, u ) f x, x, u f f d a tk+τnp (1a) x s.t. d dt d = f(, t x, x, u, w) d a I (1b) x d ( t k ) = x d (1c) ct (, xd, xa, u, wi ) = (1d) ht (, xd, xa, u, wi ) (1e) y = gt (, xd, xa, wo ) (1f) Note that eq 1c represents the ntal condton of the dfferental states whch s updated at each samplng nstant. In the presence of model plant msmatch the smple approach of updatng the ntal condtons wth the value of the states observed, f all the dfferental states can be measured, can produce a steady-state offset that deterorates the performance of the controller. Thus, an output feedback method s requred to obtan an offset free response. 1 Usng the formulaton presented n eq 1, the output feedback can be done usng an observer such as an extended Kalman flter (EKF) or a movng horzon estmator (MHE) whch soluton provdes the ntal condtons of the problem (x d ) and the value of the dsturbances (w I,w O ). There are other smpler approaches for handlng the feedback, when all the dfferental states are measured, n order to elmnate the offset and they are the conventonal feedback and the steady-state target settng. 1 In the frst one, a constant bas, calculated as the dfference between the output predcted and the output measured, s added to the rght-hand sde of eq 1f. 4 In the second one, the set-ponts of the outputs are replaced by a reference value, whch s obtanng by solvng a steady-state optmzaton problem that mnmzes the dfference between the reference and the set-pont subject to the model that consders the bas. 1 If the optmzaton problem gven by eq 1 s of large-scale or has complex nonlnear terms ts soluton mght take an mportant amount of tme ntroducng delays and even nstabltes n the closed loop control. In addton, the effects of the tunng parameter of the controller on the closed loop performance are not easy to predct and they mght not be monotonc.,11 3. NMPC TUNING ALGORITHM The optmal tunng problem wth respect to the controller parameters can be formulated as follows: 11 mn Jx ( d, xa) p u (a) N, N, W, τ s.t. f ( xd, xa, u, wi ) = (b) 917 u = gx (, x, N, N, W, τ) Artcle d a p u (c) hx ( a, xd ) (d) Where the objectve functon, eq a, s a performance ndex for the closed loop behavor of the controller. Equaton b s a general set of dfferental algebrac equatons that represent the real plant, and t s a functon of the states, the control law, and the dsturbances. Ths allows to tune the controller for a specfc type of antcpated or predefned dsturbances. Equaton d are nequalty constrant for the closed loop behavor (e.g., the maxmum overshoot allowed). Note that the objectve functon of the problem defned by eq s an ndcator of the whole closed loop performance of the process (e.g., the ntegral absolute error), whch s dfferent from the objectve of eq 1 that represents the OCP objectve (e.g., a quadratc functon of trackng wth respect to a specfc set-pont). Also, the OCP defned n eq 1 s solved recurrently at every samplng nterval, whle the problem of eq s only solved once for the defnton of the controller tunng parameters. In most cases t s mpossble to defne exactly the eq b. Moreover, eq c s the explct feedback control law expressed as a functon of the dfferental states, the algebrac states and the controller tunng parameters (e.g., predcton horzon N p, control horzon N u, objectve functon weghts W, samplng tme τ); and t can only be defned n ths way for lnear model predctve controllers wthout nequalty constrants. If t were possble to formulate explctly the optmzaton problem of eq ts soluton would be the tunng parameters of the NMPC controller that mnmze the specfed closed loop performance ndex. Therefore, there s a need to formulate the optmzaton problem of eq n a dfferent way to fnd an alternatve soluton to ths problem consderng ts lmtatons. The algorthm presented hereafter s one alternatve for solvng ths optmzaton problem for tunng NMPC controllers, but before ntroducng t n detal t s mportant to analyze the tunng parameters of the controller and ther general effect on the closed loop performance. Samplng Tme. It s the tme nterval n whch a new control sgnal s feedback to the system. The samplng nterval should be small enough to capture adequately the dynamc of the process, but the smaller t s the optmzaton problem becomes larger. 1 It s not adequate as a tunng parameter because t should be fxed based on the equpment nstalled and the lmtatons of sensors and actuators. It also mpacts drectly those parameters that are specfed n samplng ntervals lke the predcton and control horzons. 5 It s mportant to consder the Shannon s samplng theorem when selectng the samplng tme because t defnes an upper bound for ths parameter to avod loss of nformaton durng the samplng of a sgnal, n other words, t states that the samplng frequency should be greater than two tmes the maxmum frequency of the sgnal. 13 Note that the frequency of nterest here can be referred to the manpulated varables or to the dsturbances, beng the frst one more mportant because t ndcates how fast the actuators can response. Also, there are some emprcal gudelnes that suggest a samplng tme between 1% and 3% of the perod of oscllaton for oscllatory systems and of the rse tme for nonoscllatory systems. 14 Predcton Horzon. It defnes the upper lmt of the tme horzon for the predcton of the process behavor. If the process model s lnear, ncreasng the predcton horzon has a monotonc effect on the closed loop performance mprovng ts DOI: 1.11/acs.ecr.6b111

4 Industral & Engneerng Chemstry Research Artcle stablty. It should be selected as large as possble but fnte. As the predcton horzon s ncreased the sze and complexty of the dynamc optmzaton problem does also ncrease. 1 Control Horzon. Ths s the number of control actons that are allowed durng the OCP and t s lower or equal than the predcton horzon. It affects how aggressve the control acton s and defnes a trade-off between robustness of the controller for large control horzons and easy of computaton of the OCP soluton for small control horzons. 1,1 The control nputs can also be parametrzed n a blockng scheme n whch the user specfes ponts on the control horzon that are not computed and are set equal to the prevous control acton. 4 Before ntroducng the rest of the NMPC tunng parameters t s mportant to consder n detal the objectve functon of eq 1. Usually ths objectve functon s composed of dfferent objectves (e.g., quadratc trackng of the set-pont, penalzaton of the rate of change of the manpulated varables) and the compromse among these s expressed n a sngle objectve functon constructed as a weghted sum, eq 3. Also, eq 4 shows the most common objectve functon used n (N)MPC where W are weghts for every ndvdual objectve (ϕ ). In eq 4, the frst term s a quadratc trackng of the outputs wth respect to a set-pont, the frst summaton s done over the number of outputs n y and the second one over the dscrete tmes untl the predcton horzon N p, and the second term s a penalzaton of the rate of change of the manpulated varables, the frst summaton s done over the number of manpulated varables n u and the second one over the dscrete tmes untl the control horzon N u. The second term can also be expressed as the change of the manpulated varables wth respect to a reference. Note that ths s a dscrete tme representaton = 1 mn OBJ = Wϕ( t, y, u) xd, xa,u n o (3) = 1 k= 1 j= 1 k= sp mn OBJ = W ( y y ) + W ( Δu ) xd, xa,u ny k, j Np nu Nu 1 jk, Weghts on the Outputs. Tths weghts are used n the objectve functon to scale the varables and to prortze the dfferent objectves accordng to ther relatve mportance. Increasng a specfc weght, whle keepng the rest constant, drves the control effort toward a more tghter control n that varable. 1 Weghts on the Change or Rate of Change of the Inputs. They suppress the aggressve changes of the manpulated varables makng a more robust controller, but they also make the response on the controlled varables slower. 5 Settng a small penalty on ths part of the objectve functon produces a more aggressve response and decreases the ablty of the controller to handle drastc changes or dsturbances. Algorthm for Tunng NMPC Controllers. An algorthm for tunng NMPC controllers s developed n whch the fnal result s the value of all the tunng parameters wth the excepton of the samplng tme that s consdered as an nput for the algorthm. Fgures and 3 present every step of the algorthm and the shadowed blocks are the stages n whch the tunng parameter values are obtaned. The algorthm conssts of three phases. The frst one s the defnton of the model, the plant and the ntalzaton of some parameters and sets. The second one s (4) Fgure. Phase I of the algorthm for tunng NMPC controllers. the defnton of the OCP objectve functon weghts usng multobjectve optmzaton and the computaton of key performance ndexes (KPI) for each confguraton. Note that a KPI can be any type of metrc used to quantfy the closedloop performance of the process and some examples are the ntegral square error (ISE), the overshoot percentage, the maxmum soluton tme of the OCP and the economc proft. Fnally, the thrd phase s the soluton of an optmzaton problem formulated based on the closed loop performance and the KPIs obtaned. Each step of the algorthm s descrbe next. In Phase I of the algorthm the model, the real plant and the ntalzaton parameters are defned. The algorthm starts by defnng the model of the process. For example, t can be a set of dfferental equatons that descrbes the dynamc behavor of the system ncludng nequalty constrants. These two elements correspond to eq 1b 1e of the OCP formulaton. Wthn the tunng algorthm t s mportant to consder the model plant msmatch because every practcal mplementaton of NMPC exhbts a msmatch and the way ths s handled has a sgnfcant effect on the tunng parameters of the controller. 15 Therefore, t s necessary to defne a smulaton model for the real plant f the tunng cannot be done usng the plant tself. Some alternatves for modelng the real plant behavor are usng a more detaled model than the one used for predcton, ntroducng whte nose n the measurements or modfyng the parameters of the predcton model. Once ths two models, or the plant measurements and the predcton model, are defned the next step s to 918 DOI: 1.11/acs.ecr.6b111

5 Industral & Engneerng Chemstry Research Artcle Fgure 3. Phase II and III of the algorthm for tunng NMPC controllers. consder the state estmaton problem and an alternatve to handle model plant msmatch. In ths paper we consder, wthout loss of generalty, that all the states can be measured, thus we do not solve a state estmaton problem; but estmators lke the Kalman flter, extended Kalman flter or MHE (movng horzon estmaton) can be used n ths step of the tunng algorthm. 16 For handlng the model plant msmatch and to avod computatonal delays we used the asnmpc strategy coupled wth NLP senstvty whch s a fast alternatve to update the value of the manpulated varables once the real measurements of the process are avalable. 17 Ths s descrbed n more detal n secton 4 because t s appled for the case studes presented and the algorthm s not restrcted to usng t. Also, other alternatves can be used here lke a proportonal correcton of the manpulated varables based on the error n the measured varables. 18 The rest of phase I ncludes defnng the type of control problem for whch the controller s beng desgned (e.g., dsturbance rejecton or set-pont trackng) and ntalzng some parameters and sets. The parameters that must be ntalzed are the samplng tme (τ), that should be selected based on the system lmtatons and on the Shannon s samplng theorem; the set of control objectves (ϕ), whch corresponds to each element presented n the weghted sum objectve functon, eq 3; a set of performance ndexes (e.g., ntegral square error, overshoot percentage, economc proft) (KPI_Set) and a set of dfferent confguratons of predcton horzon and control horzon (H_Set). Note that the H_Set elements are only an ntal confguratons of N p and N u and later n the algorthm more elements wll be added to the set. However, the ponts of ths ntal confguraton must have a good dsperson allowng a good samplng of the whole regon defned by N p and N u to ft response surfaces n ths space. 19 One alternatve to ntalze the H_Set s presented n Fgure 4, ths confguraton s based on a central composed desgn of experments for a nonunform regon and allows a good samplng of the space defned by N p and N u. 19 Note that the samplng space shown n Fgure 4 depends on the value of the parameter N p max whch s an upper bound for the predcton horzon and t s defned by the analyst. In ths step of the tunng algorthm addtonal parameters mght be ncluded such as an nput blockng scheme or nternal model control (IMC) flter constants, but t wll ncrease the number of cases that must be evaluated n the phase II. In other words, the H_Set wll be composed not just by the predcton and control horzon, but t wll have addtonal elements correspondng to the addtonal aspects of the controller. For example, f an nput blockng s ncluded n the controller each element of the H_Set wll have the form of (N p, N u,bs),wherebs s the blockng scheme. Also, addng more tems to each element of the H_Set ncreases the complexty of the phase III, but t s not a restrcton for applyng the tunng algorthm proposed. In phase II of the algorthm for each element of the set H_Set, whch means for each confguraton of predcton horzon 919 DOI: 1.11/acs.ecr.6b111

6 Industral & Engneerng Chemstry Research Artcle Fgure 4. Intal confguraton of H_Set. For the examples presented later n ths paper N p max takes values of 1 and 5, whch means that the samplng s done at values of N p equals to (, 5, 8, 1) and (, 5, 38, 5), respectvely. and control horzon, the followng s done: () Solve the NMPC problem for each ndvdual control objectve (e.g., quadratc trackng of one state varable wth respect to ts set-pont, quadratc penalzaton of the rate of change of a manpulated varable). At every perod of the NMPC, when an OCP s solved, the values of all the objectves are computed and stored (ϕ,j,k * ). Also, the notaton here ndcates that the objectve s computed for every tme perod when the objectve j s mnmzed and the horzon confguraton k s used. It s mportant to menton that for a specfc set of ndexes (e.g., =1,j =1,k = 1) the varable ϕ,j,k * s actually a vector ndexed n the dscrete tme t because the objectve functon value s computed at every samplng tme. The astersk (*) symbol ndcates the value of the varable/expresson at the optmal soluton. () Once the NMPC has been solve for every ndvdual objectve (ϕ j ) an upper bound (ϕ U,k ) and a lower bound (ϕ L,k ) for each objectve are computed from the values stored prevously. Note that the upper and lower bounds are computed usng the maxmum and mnmum operators over the j ndex and the dscrete tme (t) on the varable ϕ,j,k *. Then, usng these bounds the range of all objectves s computed. () The OCP objectve functon weghts are calculated based on the range of each objectve. (v) Another NMPC problem s solved usng a composed objectve functon whch s defned as the weghted sum of all the objectves and for ths fnal result the KPIs are computed and saved. The las three steps of phase II of the proposed algorthm are better explaned n the followng paragraphs and n the U Appendx A whch shows n great detal how to compute ϕ,k and ϕ L,k for the case study 1 (secton 5.1). At the end of Phase II a set of weghts and a set of KPIs must have been defned for each confguraton of predcton and control horzon. The dea behnd ths way of calculatng the objectve functon weghts s a multobjecve optmzaton technque called utopa trackng approach whch mnmzes the dstance between the Pareto front and the utopa pont. Fgure 5 llustrates the utopa pont, the nadr pont, the Pareto font and the dstance that s mnmzed usng ths approach for a case consstng only of two objectves. Zavala and Flores use ths utopa trackng approach to solve onlne the multple conflctng objectves n a NMPC controller. They solve n parallel as many optmzaton problems as objectves are defned 9 Fgure 5. Illustraton of the utopa trackng approach for two objectves. to fnd the utopa pont and then the OCP that s solved to defne the feedback value of the manpulated varables has the followng objectve functon: 1 = 1 L ϕ ϕ mn U L xd, xa,u ϕ ϕ no (5) Note that n eq 5 s expressed n terms of the elements of the followng vectors: ϕ s a vector of objectves, ϕ L s a vector of lower bounds for the objectves (utopa pont), and ϕ U s a vector of upper bounds for the objectves (nadr pont). Also, each dstance from the objectve to the utopa pont s standardzed dvdng t by the range of the objectve because the order of magntude of the objectves mght dffer sgnfcantly and produces an ll-condtoned optmzaton problem. It s possble to avod the onlne soluton of more than one optmzaton problem wthn every tme nterval f the global coordnates of the utopa and the nadr pont are defned prevously. To compute the utopa and the nadr pont t s necessary to solve a NMPC problem for each objectve and, once ths s done, t s possble to use eq 5 as the objectve functon of the OCP. However, consderng that the utopa pont s an unreachable pont because t mnmzes all the objectves smultaneously, the absolute value operator of eq 5 can be omtted and the objectve functon can be rearranged as follows: = 1 L = 1 L ϕ ϕ ϕ ϕ mn = mn U L U L xd, xa,u ϕ ϕ xd, xa,u ϕ ϕ n n = 1 = 1 L ϕ ϕ = mn U L U L xd, xa,u ϕ ϕ ϕ ϕ (6) n n Fnally, takng nto account that the argument of mnmzng a functon plus a constant s the same as mnmzng only the functon (t s also vald f the problem s constraned) 3 the OCP objectve functon can be expressed as a weghted sum of the ndvdual objectves where each weght s the nverse of the objectve range, eq 7. DOI: 1.11/acs.ecr.6b111

7 Industral & Engneerng Chemstry Research arg mn ( x* * d, xa, u* ) = ( xd, xa, u) n arg mn 1 ϕ = U ( x, x, u) ϕ ϕ d a = 1 = 1 L ϕ ϕ U L U ϕ ϕ n = = 1 n arg mn + C = ( x, x, u) L d a n ϕ ϕ L Wϕ When the utopa trackng approach s used for defnng the OCP weghts all the objectves receve the same relatve mportance. 4 The phase III of the algorthm s composed of two steps. The frst one s the computaton of a response surface that s a functon of the predcton horzon (N p ) and the control horzon (N u ) for every KPI n the KPI_Set. Note n Fgure 4 that only specfc measurements of the KPI are avalable after phase II of the proposed tunng algorthm and there s not an explct functon of the KPI n the whole space defned by N p and N u. Because of ths t s necessary to use these measured ponts, whch are computed after each teraton of phase II, to approxmate a functon or response surface that can predct accurately the behavor of the KPIs. To compute the response surface t s necessary to solve the problem RS-KPI (response surface for KPI), stated n eq 8. Ths problem mnmzes the devaton between the measured ponts (obtaned n Phase II) and a specfc functon; n other words, t s a lnear or nonlnear ft. In ths paper we use a quadratc approxmaton as t s shown n eq 9 to descrbe how each KPI depends on N p and N u, but other types of response surfaces can be used for ths approxmaton. mn ( K K) a1, a,..., am HSet _ s.t. K = f( a, a,..., a ), H_ Set 1 m f ( a, a,..., a ) = a N + a N + a N + a N 1 m p 1 p u 1 u + a N N + a (7) (8) 11 p u (9) The second step of the phase III s to solve the horzons problem (HP) whch s smlar to the problem stated by eq, but the only decson varables are N p and N u because the objectve functon weghts have been defned prevously. The HP mnmzes a KPI of the closed loop behavor of the system subject to the nature of the control and predcton horzon, to a constrant n the soluton tme of the OCP, and to other nequalty constrants based on the performance ndexes (e.g., maxmum allowable overshoot percentage, mnmum rse tme). eq 1 presents a general formulaton of the HP, where the functon F s a KPI functon obtaned from solvng the RS-KPI problem, eq 8, for a specfc performance ndex. G s a vector of KPI functons and each one corresponds to the soluton of a RS-KPI problem. mn FN (, N) Np, Nu s.t. t sol τ GN (, N) N N N ( N, N ) p u p u max u p p u p + (1) The fnal step of the algorthm s to check f the soluton of the HP s already n the H_Set, f so the algorthm fnsh and the arguments of the soluton of the HP are the predcton and Artcle control horzon for the NMPC controller. Note that ths par of ponts have a vector of objectve functon weghts assocated, whch were found n phase II. If the soluton of HP s not n the H_Set ths new pont s added to the set, and the phase II s repeated for ths pont to defne the correspondng objectve functon weghts. 4. NLP SENSITIVITY AND ASNMPC CONTROLLER The deal NMPC assumes that the OCP can be solved nstantaneously, that all the states are measured and that there s no model plant msmatch. However, f all these assumptons are consdered vald durng the tunng of the controller the fnal result mght not be robust and the controller could fal even under small dsturbances. Thus, t s necessary to consder an addtonal level of complexty. For dong so, n ths paper we consder the advanced step NMPC (asnmpc) controller that avods computatonal delays and t s capable of handlng model plant msmatch. The asnmpc solves the OCP n advanced, whch means that at the current tme t k and wth the measurements of the current states and manpulated varables, the predcton model s used to predct the states at the next tme perod t k+1 and the OCP for ths last perod s solved between t k and t k+1. Usng ths strategy the delays generated due to the computatonal tme are avoded, but there can be stll a msmatch between the states predcted and the measured values at t k+1, thus a rapd correcton usng NLP senstvty has to be done. If the parameters (p) of an optmzaton problem, whch n the case of the OCP correspond to the ntal condtons and dsturbances, are assocated wth artfcal constrants of the form x p p =, where x p s a varable, the NLP senstvty calculatons can be reduced to solve the lnear system gven by eq Note that ths s a lnear equaton n whch H L s the Hessan matrx of the Lagrange functon (L), A s the Jacoban matrx of the equalty constrants, V s a dagonal matrx, whch contans the dual varables (v), L s the Lagrange functon, c s a vector of equalty constrants, Z s a dagonal matrx whch contans all the varables of the problem, and λ are the Lagrange multplers for the equalty constrants. Equaton 11 s obtaned after applyng the chan rule over the frst order optmalty condtons of a constraned NLP problem when t s solved usng an nteror pont algorthm, such as the one mplemented n the solver IPOPT. These dervatves are taken wth respect to the parameters to obtan the senstvty of the varables (z,x p ). Also, the left-hand sde matrx corresponds to the KKT matrx at the optmal soluton whch s avalable after solvng the optmzaton problem. For a more detaled dscusson about asnmpc controller, ts propertes and NLP senstvty calculatons the reader s referred to the followng references. 5 7 H λ L zx L( z, x,, v) A I p p Δz xzlz (, x, λ, v) Lz (, x, λ, v) cz (, x) I Δ x p p xx p p p xp p p T T A c z x Δ λ x (, ) p p Δv V Z Δλ I = Δp (11) 91 DOI: 1.11/acs.ecr.6b111

8 Industral & Engneerng Chemstry Research Artcle Table 1. Model Parameters for Cases 1 and parameter value parameter value V 1.9 m 3 k h 1 v 1.5 m 3 /h E kj/kmol F {A,B,C,M} {36.3, 453.6,, 45.36} kmol/h C p,w 75.3 kj/kmol K C p, {A,B,C,M} {146.5, 75., 19.6, 81.7} kj/kmol K UA 3386 kj/h K T 4 C T w,n 15.5 C C (t =){A,B,C,M} {, 4.64,, } kmol m 3 ΔH rxn kj/kmol Along ths manuscrpt, we focus on NMPC and thus the OCP, after dscretzaton of the dfferental equatons, s defned as a large-scale NLP. The solver IPOPT s chosen for solvng ths NLP problem because t has been proven effectve for the type of formulatons that arse n chemcal processes 3,8 and t has been apply n NMPC controllers wth good results. 9 The IPOPT solver mplements a determnstc optmzaton algorthm based on the Newton s method for solvng the KKT condtons of the problem. Moreover, t uses a prmal-dual nteror pont algorthm, n whch the nequalty constrants are handled n the objectve functon ntroducng a logarthm barrer and a barrer parameter, and a flter lne search method for solvng the frst order optmalty condtons. sipopt s a toolbox that can be found as an extenson of IPOPT solver and that performs the NLP senstvty calculatons automatcally, eq 11. The only requrement for the user s to defne the artfcal constrants wthn the problem formulaton and a seres of suffxes used to communcate the problem wth the solver ndcatng whch of the varables are artfcal. For further nformaton about the solvers IPOPT and ts extenson sipopt the reader s referred to the references. 5,3 5. EXAMPLES Ths secton presents three examples that show the applcaton of the tunng algorthm descrbed n secton 3. All the optmzaton problems were formulated n Pyomo, 31 an optmzaton software mbedded n Python, and solved wth the nonlnear solver IPOPT, the NLP senstvty calculaton was performed wth the extenson of ths solver sipopt. The whole tunng algorthm was programed n Python Case 1: CSTR wth Two Control Objectves. Ths s an example adapted from ref 3, n whch a CSTR s used for the producton of propylene glycol, and t s desred to control the reactor temperature durng the start-up operaton and n the presence of dsturbances. The hydrolyss reacton of propylene oxde to produce propylene glycol s carred out wth excess of water, n the presence of methanol and usng a strong acd catalyst. The reacton s frst-order n propylene oxde concentraton. The dynamc model and operatonal constrants for ths example are gven by eq 1, where A s propylene oxde, B s water, C s propylene glycol, and M s methanol. Ths model s based on molar balances eq 1a and energy balances eq 1b assumng that the hold-up of coolng flud n the jacket s neglgble; also the parameters of the model are presented n Table 1. Here C s the molar concentraton of speces, r s the reacton rate, V s the reactor volume, ϑ s the stochometrc coeffcent of speces, v s the volumetrc feed flow rate, T s the reactor temperature, C p s the heat capacty, F s the molar flow rate, ΔH rxn s the reacton enthalpy, T w s the nlet temperature of the coolant flud, U s the global heat transfer coeffcent, and A s the heat transfer area. The objectve of ths control problem s to mantan an operatonal temperature of 63 C manpulatng the coolant flud feed 9 flow rate (m ẇ ). Note that the reactor do not start from ts steady state operaton pont, thus t s a start-up operaton and after h a dsturbance s ntroduced; more specfcally the nlet temperature drops lnearly from 4 to 1 C n an 1 h nterval. dc =ϑ r + C C v ( ), { A, B, C, M} dt V (1a) dt Q F C T T ΔH rv A B C M p ( ) ( ) {,,, }, rxn = dt CVC r = k exp E RT C A { A, B, C, M} p, (1b) (1c) = UA Q m wcp,w( Tw,n T) 1 exp m wcp,w (1d) T 8 C (1e) The real plant s consdered to be the same model descrbed by eq 1 but decreasng the frequency factor of the reacton rate expresson by %. Addtonal model plant msmatch s ntroduced addng whte nose of ampltude 1% to the measurements of the concentraton and temperature at each samplng perod. Accordng to the phase I of the algorthm proposed n secton 3 after defnng the type of control problem, the real plant model for smulaton and the predcton model t s necessary to defne the samplng tme, the control objectves and the KPIs that evaluate the closed-loop performance of the controller. For ths example a samplng tme of.1 h s selected whch s small enough consderng that the tme constant of ths process for changes around ts fnal steady state s approxmately.9 h. Two control objectves are defned for ths problem: a quadratc trackng of the temperature set-pont wthn the predcton horzon, eq 13, and a penalzaton of the drastc changes of the coolant feed flow rate wthn the control horzon, eq 14. Fnally, three KPIs are selected to evaluate the closed-loop behavor of the process: the ntegral square error (ISE) of the temperature trackng, the maxmum soluton tme of the OCP wthn the NMPC scheme and the average concentraton of propylene glycol (C) durng the tme wndow of the operaton. ϕ = 1 ϕ = = N ( T T ) sp p (13) = N 1 w ( Δm ) u (14) Usng these defntons the phase I of the algorthm s over, and one can proceed to execute phase II and III. Nevertheless, DOI: 1.11/acs.ecr.6b111

9 Industral & Engneerng Chemstry Research before dong so t s necessary to defne the horzons problem (HP) whch n ths case s the mnmzaton of the ISE of the temperature trackng, eq 15a, subject to a lower lmt n the average concentraton of propylene glycol ( CC ), eq 15b, toa upper lmt n the tme allowed for solvng (t sol ) the OCP, eq 15c, and to the nature of the predcton and control horzons, eq 15d. Note that every element of the optmzaton problem defned n eq 15 are KPIs of the NMPC closed-loop performance and a response surface s defned for each one after solvng the RS-KPI problem, n ths case we use a quadratc approxmaton for each functon. Examples of these response surface functons are presented n Fgure 6, whch are constructed for the fnal teraton of the algorthm and from top to bottom s the OCP maxmum soluton tme, the average concentraton of propylene glycol (C) and the ISE of the Fgure 6. Response surface for the KPIs n the case 1. temperature trackng. In addton, the quadratc fttng of the response surfaces s good enough for these cases producng a R statstc greater than.9 and, furthermore, ths functons capture the behavor of the KPIs as a functon of the horzons whch s not lnear nor monotonc. t= tf mn ( T T ) dt s.t. C t sol t= C τ = sp.35 kmol/m.1 h 3 Artcle (15a) (15b) (15c) Nu Np Np max = 5 (15d) After takng all these defnton nto account and executng phases II and III of the algorthm the fnal tunng parameters are N p = N u = 11, W 1 = , and W = For a step by step computaton of the objectve functon weghts, the reader s referred to the Appendx A. Frst of all, note that the fnal confguraton of predcton horzon and control horzon s not ncluded n the ntalzaton of H_Set, ths s a result of solvng the HP problem every tme the phase III of the algorthm s executed. In addton, wth every teraton of phase III the response surfaces of the KPIs are mproved by addng a new pont to the samplng space, and ths new pont s closer to the optmal soluton of the HP. Second of all, the weghts for the composed objectve functon are the result of phase II that executes the utopa trackng multobjectve approach for the confguraton N p = N u = 11 and scale all the objectves to have the same level of mportance. 5.. Case : CSTR wth Three Control Objectves. Ths example uses the same model and consderatons of the case 1 wth the only excepton that an addtonal objectve for the OCP s ntroduced and t s the maxmzaton of the propylene glycol concentraton at the fnal tme of the predcton, eq 16. As the utopa trackng approach consders all the objectves n the mnmzaton sense t s necessary to mnmze the negatve of the concentraton n order to maxmze t. Introducng ths addtonal objectve ncreases the complexty of the tunng algorthm as an addtonal weght has to be determned and, durng Phase II, an extra NMPC problem has to be solved. However, ths shows that the proposed tunng algorthm s capable of handlng as many objectves as the analyst decde to nclude even f t s a nonstandard objectve, such as eq 16 that represents a fnal cost. ϕ = C ( t = τn ) 3 C p (16) The results of executng the tunng algorthm wth ths change are clearly dfferent from those of the case 1 and are the followng: N p = 19, N u = 14, W 1 = , W = ,andW 3 =.376. Addng one extra objectve to the problem formulaton changes the predcton horzon, the control horzon and the weghts of the common objectves snce the prortes n the objectve functon and the velocty wth whch the controller response to dsturbances change. Comparng case 1 and case the value of W 1 does not change sgnfcantly but the value of W s reduced allowng more drastc and fast changes of the coolng flud feed flow rate to acheve a tghter temperature control and thus maxmze the concentraton of the desred product at the end of the predcton horzon, n fact the ISE of the temperature trackng s mproved by a.56% n case. Moreover, the horzon lengths for case are larger than for case 1 whch ntroduces more control actons durng the predcton horzon 93 DOI: 1.11/acs.ecr.6b111

10 Industral & Engneerng Chemstry Research Artcle The phase I of the algorthm uses the same ntalzaton for the H_Set as the prevous two cases and the controller s desgned for dsturbance rejecton usng as benchmark dsturbance a snusodal dsturbance n the ethanol feed composton of ampltude 5% and perod of 1 mn. Addtonal model plant msmatch s ntroduced by addng a random nose of ampltude 5% n the lqud hold-up values at each samplng perod. The samplng tme s chosen as.5 mn whch s close to the tme constant of the process for changes around ts steady state ( mn), but lower than the rse tme ( 6 mn). Four objectves are consdered for the OCP formulaton eq 17 among them s one economc objectve of proft maxmzaton (t ncludes the proft from dstllate D and the cost from the reboler duty Q R ), one trackng objectve of ethanol mole fracton n dstllate (x D EtOH ), and two penalzatons of the changes of the manpulated varables, the reflux rato (R) and the reboler duty (Q R ), wth respect to a reference value. Fnally, three KPIs are selected to evaluate the closedloop behavor of the process: the total economc proft, the maxmum soluton tme of the OCP wthn the NMPC scheme and the average ethanol mole fracton n dstllate. ϕ = 1 Np = CD C Q D Q R, (17) ϕ = Np = EtOH D, ( x x ) D sp (18) Fgure 7. Profles for the controlled varables (a) and the manpulated varable (b) for case 1 and case. ncreasng the robustness of the controller. The profles of the fnal tunng confguraton for the asnmpc controllers are shown n Fgure 7 and t also compares the results of the case 1 and case. In addton, Fgure 7 shows a tghter temperature control for case than for case 1 that s acheved wth faster and more drastc changes of the manpulated varable. Usng the tunng parameters obtaned from the algorthm proposed n ths paper t s possble to mantan the reactor temperature around ts setpont even n the presence of nose, dsturbances and model plant msmatch wthout drastc changes n the coolng flud flow rate Case 3: Extractve Dstllaton Column. Ths fnal example s a larger and more complex problem than the prevous two whch serves to show that the NMPC tunng algorthm proposed s capable of handlng hghly nonlnear models and actve nequalty constrants that produces complex nteractons of the tunng parameters of the controller. Ths case s the desgn of a NMPC controller wth economc orentaton for an extractve dstllaton column used n the separaton of the azeotropc mxture of water and ethanol usng glycerol as an entraner. More detals about ths case and a detal descrpton of the dynamc model used for predcton can be found n a prevous work of the authors. 33 In summary, the model s composed by a set of dfferental algebrac equatons that represent the mass balances, energy balances, phase equlbrum equatons, and hydraulc relatons for each stage of the column. Also, the system s subject to a mnmum purty constrant of the dstllate, whch s a molar fracton of ethanol greater or equal than 99.5%. 94 ϕ = 3 ϕ = 4 Nu = Nu = R, R,ref ( Q Q ) ( R R ) ref (19) () For executng phase II and III of the algorthm t s necessary to formulate the horzons problem (HP), whch s the maxmzaton of the economc proft subject to the ethanol purty constrant n dstllate and to the nature of the predcton and control horzon as t s shown n eq 1. Every element n the HP s a KPI of the closed loop behavor and thus t s approxmated wth a response surface solvng the RS-KPI problem. Examples of these response surfaces are presented n Fgure 8 whch are constructed for the fnal teraton of the algorthm and from top to bottom s the OCP maxmum soluton tme, the average ethanol percentage n dstllate and the economc proft. t= tf mn ( CD CQ)dt t= D s.t. x 99.5% t sol D EtOH τ =.5 mn max u p p N N N = 1 Q R (1a) (1b) (1c) (1d) After takng all these defntons nto account and executng phase II and III of the algorthm the fnal tunng parameters of the asnmpc controller are N p =8,N u =5,W 1 =.7, W = 5.3, W 3 =.1, and W 4 = Note that usng ths tunng algorthm the order of magntude of the objectve weghts can vary sgnfcantly due to the dfferences n scales of every objectve. For example, the economc objectve, eq 17, s on the order of 1 3, the mole fracton s expressed as a percentage DOI: 1.11/acs.ecr.6b111

11 Industral & Engneerng Chemstry Research Artcle Fgure 8. Response surface for the KPIs n the case 3. so ts order s of 1, the reboler duty s on the order of 1 and the reflux rato s on the order of 1 1. However, ths dfferences n the weghts allows a better prortzaton of the conflctng objectves presented n the composed objectve functon of the OCP. Fgure 9 shows a comparson of the closed loop profles usng the tunng parameters obtaned from the proposed algorthm and a set of arbtrary parameters defned by tral and error, as those defned n a prevous work. 33 Usng a tral and error approach s dffcult to defne the penalzaton weghts for the manpulated varables n order to use all of them to control a specfc varable and n ths case Fgure 9a and Fgure 9b show how poorly tuned penalzaton weghts drve all the control effort toward only one manpulated varable, the reflux rato, even when two varables are avalable. The proposed tunng Fgure 9. Profles for the case 3 and comparson of the asnmpc controller performance usng the tunng algorthm versus a tral and error alternatve. (a) Reboler duty, (b) reflux rato, and (c) ethanol mole fracton n dstllate. algorthm avods ths knd of behavor and defnes the composed objectve functon weghts such as all the parameters receve the same level of mportance and all the manpulated varables can be used to control the process. Also, the tunng algorthm for ths case ams to maxmze the economc proft of the process whle satsfyng the constrants and an ncrease of the 1.3% s obtaned. Note that ths mprovement n economc proft s obtaned even when the predcton and control horzons are smaller than those defned n the tral and error approach. 6. CONCLUSIONS AND PERSPECTIVES Ths Artcle presents a general tunng algorthm for NMPC controllers that conssts of three man phases. The frst one s 95 DOI: 1.11/acs.ecr.6b111