Control Engineering Practice 11 (2003) 733–764
A survey of industrial model predictive control technology S. Joe Qina,*, Thomas A. Badgwell b,1 a
Department of Chemical Engineering, The University of Texas at Austin, 1 Texas Lonhorns, C0400, Austin, TX 78712, USA b Aspen Technology, Inc., 1293 Eldridge Parkway, Houston, TX 77077, USA Received 8 November 2001; accepted 31 August 2002
Abstract
This paper provides provides an overview overview of commerci commercially ally available available model model predicti predictive ve control control (MPC) (MPC) technolo technology, gy, both linear and nonlinear, based primarily on data provided by MPC vendors. A brief history of industrial MPC technology is presented first, followed by results of our vendor survey of MPC control and identification technology. A general MPC control algorithm is presented, and approaches taken by each vendor for the different aspects of the calculation are described. Identification technology is reviewed to determine similarities and differences between the various approaches. MPC applications performed by each vendor are summarized by application area. The final section presents a vision of the next generation of MPC technology, with an emphasis on potential business and research opportunities. r 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Introduction
Model predictive control (MPC) refers to a class of comput computer er contro controll alg algori orithm thmss that that utiliz utilizee an explic explicit it process model to predict the future response of a plant. At each control interval an MPC algorithm attempts to optimize future plant behavior by computing a sequence of future manipulated manipulated variable adjustments. adjustments. The first input in the optimal sequence is then sent into the plant, and the entire entire calcul calculati ation on is repeat repeated ed at subseq subsequent uent contro controll interv intervals als.. Ori Origin ginall ally y develo developed ped to meet meet the specialized control needs of power plants and petroleum refineries, MPC technology can now be found in a wide variety variety of applicati application on areas including including chemicals, chemicals, food processing, automotive, and aerospace applications. Several recent publications provide a good introduction to theoretical and practical issues associated with MPC technology. Rawlings (2000) provides an excellent introd introduct uctory ory tutori tutorial al aimed aimed at contro controll practi practitio tioner ners. s. Allgower, Badgwell, Qin, Rawlings, and Wright (1999) presen presentt a more more compre comprehen hensiv sivee overvi overview ew of nonlin nonlinear ear MPC MPC and and movi moving ng hori horizo zon n esti estima mati tion on,, incl includ udin ing g a summ summar ary y of recen recentt theo theore reti tica call devel develop opme ment ntss and and *Corresponding *Corresponding author. Tel.: +1-512-471-4417; fax: +1-512-471+1-512-4717060. E-mail address:
[email protected] [email protected] u (S.J. Qin). 1 At the time of the survey TAB was with Rice University.
numerical solution techniques. Mayne, Rawlings, Rao, and Scokaert (2000 (2000)) provide a comprehensive review of theoretical results on the closed-loop behavior of MPC algorithms. Notable past reviews of MPC theory include those of Garc of Garcıa, Prett, and Morari (1989); (1989); Ricker (1991); (1991); Morar Mo rarii and Lee (19 (1991) 91);; Musk Muskee and Rawlings (1993), (1993), Rawlings, Meadows, and Muske (1994); (1994); Mayne (1997), (1997), and Lee and Cooley (1997). (1997). Several books on MPC have rece recent ntly ly been been publ publis ished hed (Allgower & Zhe Zheng, ng, 200 2000 0; Kouvaritakis & Cannon, 2001; 2001; Maciejowski, 2002). 2002). The The author authorss presen presented ted a survey survey of indust industria riall MPC technology based on linear models at the 1996 Chemical Process Control V Conference (Qin (Qin & Badgwell, 1997), 1997), summarizing applications through 1995. We presented a review of industrial MPC applications using nonlinear models at the 1998 Nonlinear Model Predictive Control work worksh shop op held held in Asco Ascona na,, Swit Switze zerl rlan and d (Qi Qin n an and d Badgwe Bad gwell, ll, 200 2000 0). Fro Froisy isy (19 (1994) 94) and Kul Kulhav havy, y, Lu, and Sam Samad ad (20 (2001) 01) descri describe be industr industrial ial MPC practi practice ce and future developments from the vendor’s viewpoint. Young, Bartusiak, and Fontaine (2001), (2001), Downs (2001), (2001), and Hillestad and Andersen (1994) report development of MPC MPC techno technolog logy y withi within n operat operating ing compan companies ies.. A survey of MPC technology in Japan provides a wealth of inform informati ation on on applic applicati ation on issues issues from from the point point of view view of MPC MPC user userss (Ohsh Ohshima, ima, Ohno Ohno,, & Hashimoto, 1995). 1995 ).
0967-066 0967 -0661/02 1/02/$ /$ - see front front matter matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 0 2 ) 0 0 1 8 6 - 7
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S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
In rece recent nt year yearss the the MPC MPC land landsc scap apee has has chan change ged d dras drasti tica call lly, y, with with a larg largee incr increa ease se in the the numb number er of reporte reported d applic applicati ations ons,, signifi significan cantt improv improveme ements nts in technic technical al capabi capabili lity, ty, and merger mergerss betwee between n severa severall of the vendor vendor compan companies ies.. The primar primary y purpos purposee of this this paper is to present an updated, representative snapshot of commer commercia cially lly ava availa ilable ble MPC techno technolog logy. y. The informat formation ion reporte reported d here here was collec collected ted from from vendor vendorss starti starting ng in mid-1 mid-199 999, 9, refle reflecti cting ng the the stat status us of MPC MPC practi practice ce just just prior prior to the new millenni millennium, um, roughly roughly 25 years after the first applications. A brief brief histor history y of MPC technolog technology y develo developme pment nt is presented first, followed by the results of our industrial survey. Significant features of each offering are outlined and discussed. MPC applications to date by each vendor are then then summar summarize ized d by applic applicati ation on area. area. The final final sect sectio ion n pres presen ents ts a view view of next next-g -gen ener erat atio ion n MPC technology, emphasizing potential business and research opportunities.
the process considered by Kalman and co-workers can be described by a discrete-time, linear state-space model: xk þ1 ¼ Axk þ Buk þ Gwk ;
ð1aÞ 1aÞ
yk ¼ Cxk þ nk :
ð1bÞ 1bÞ
The vector u represents process inputs, or manipulated variab var iables les,, and vector vector y descri describes bes measur measured ed proces processs outputs. outputs. The vector x represe represents nts proces processs states states to be controlled. The state disturbance wk and measurement noise nk are indepe independe ndent nt Gaussi Gaussian an noise noise with with zero zero mean. mean. The initial state x0 is assumed to be Gaussian with non-zero mean. The The obje object ctiv ivee func functi tion on F to be minimized penali penalizes zes expecte expected d val values ues of square squared d input input and state state deviations from the origin and includes separate state and input weight matrices Q and R to allow for tuning trade-offs: N
F¼
EðJ Þ;
J ¼
X
2 ðjjxk þ j jjQ þ jjuk þ j jj2R Þ:
ð2Þ
j ¼1
2. A brief history history of industrial MPC
This This sect sectio ion n pres presen ents ts an abbr abbrev evia iate ted d hist histor ory y of industrial MPC technology. Fig. 1 shows an evolutionary ary tree tree for for the the most most si sign gnifi ifica cant nt indu indust stri rial al MPC algorithms algorithms,, illustrati illustrating ng their connections connections in a concise concise way. Control algorithms are emphasized here because relatively little information is available on the development of industrial identification technology. The following sub-sec sub-sectio tions ns descri describe be key alg algori orithm thmss on the MPC MPC evolutionary tree.
The norm terms in the objective function are defined as follows: 2 jjxjjQ ¼ xT Qx:
ð3Þ
Implicit in this formulation is the assumption that all vari variab able less are are writ writte ten n in term termss of devi deviati ation onss from from a desired steady state. It was found that the solution to this this proble problem, m, known known as the linear linear quadratic quadratic Gaussian Gaussian (LQG) controller, involves two separate steps. At time interval k ; the output measurement yk is first used to obtain an optimal state estimate xk jk : #
2.1. LQG
xk jk À1 ¼ Axk À1jk À1 þ Buk À1 ;
ð4aÞ 4aÞ
The development of modern control concepts can be traced to the work of Kalman et al. in the early 1960s (Kalman, 1960a, b). b). A greatly simplified description of their results will be presented here as a reference point for the discussion to come. In the discrete-time context,
xk jk ¼ xk jk À1 þ K f ðyk À Cxk jk À1 Þ:
ð4bÞ 4bÞ
#
#
#
#
#
Then the optimal input uk is computed using an optimal proportional state controller: uk ¼ ÀKc xk jk :
ð5Þ
#
2000 DMC+
RMPCT SMCA
1990
SMOC
Connoisseur
I DC DC O OM M- M
RMPC HI EC EC O ON N
PFC
PCT
4th generation MPC 3rd generation MPC 2nd generation MPC
QDMC
1980 DMC
IDCOM
1st generation MPC
1970
LQG
1960
Fig. 1. Approximate Approximate genealogy of linear MPC algorithms.
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
Here, the notation xi j j j refers to the state estimate at time i giv given en inform informati ation on up to and includ including ing time time j : The Kalman filter gain K f is computed from the solution of a matrix Ricatti equation. The controller gain Kc can be found by constructing a dual Ricatti equation, so that the same numerical techniques and software can be used for both calculations. The infinite prediction horizon of the LQG algorithm endows the algorithm with powerful stabilizing properties. For the case of a perfect model, it was shown to be stabilizing for any reasonable linear plant (stabilizable and the states states are detecta detectable ble throug through h the quadra quadratic tic criterion) as long as Q is positive semidefinite and R is positive definite. Extensions to handle practical issues such as controlling outputs, achieving offset-free control, and computcomputing ing the the stea stead dy-st y-stat atee targ target etss fol follow lowed rapi rapidl dly y (Kwakernaak & Sivan, 1972). 1972). However, constraints on the process inputs, states and outputs were generally not addressed in the development of LQG theory. LQG LQG theory theory soon soon became became a standa standard rd approa approach ch to solve control problems in a wide range of application areas. Goodwin, Graebe, and Salgado (2001) estimate that there may be thousands of real-world applications of LQG with roughly 400 patents per year based on the Kalm Kalman an filte filter. r. Howe Howeve ver, r, it has has had had li litt ttle le impa impact ct on control control technology technology development development in the process process indusindustries. The most significant of the reasons cited for this failure include (Richalet, (Richalet, Rault, Testud, & Papon, 1976; 1976; Garcıa, Prett, & Morari, 1989): 1989): #
!
* * * * *
constraints; process nonlinearities; model uncertainty (robustness); unique performance criteria; cultural reasons (people, education, etc.).
It is well known that the economic operating point of a typical process unit often lies at the intersection of constra constraint intss (Prett & Gi Gille llette, tte, 198 1980 0). A succ succes essf sful ul industr industrial ial contro controll ller er for the proces processs industr industries ies must must theref therefore ore maint maintain ain the sys system tem as close close as possib possible le to constraints without violating them. In addition, process units units are typica typically lly comple complex, x, nonlin nonlinear, ear, constra constraine ined d multivariable systems whose dynamic behavior changes with time due to such effects as changes in operating condit condition ionss and cataly catalyst st agi aging. ng. Process Process units units are also also quite individual so that development of process models from fundamental physics and chemistry is difficult to justi justify fy econom economica ically lly.. Indeed Indeed,, the applic applicati ation on areas areas where LQG theory had a more immediate impact, such as the aerospace industry, are characterized by physical system sys temss for which which it is technic technicall ally y and econom economica ically lly feasi feasibl blee to deve develo lop p accu accura rate te fund fundam amen enta tall mode models ls.. Process units may also have unique performance criteria that that are difficu difficult lt to express express in the LQG LQG framew framework ork,,
735
requiring time-dependent output weights or additional logic to delineate different operating modes. However, the most significant reasons that LQG theory failed to have have a stro strong ng impa impact ct may may have have been been rela relate ted d to the the culture of the industrial process control community at the time, in which instrument technicians technicians and control control engineers either had no exposure to LQG concepts or regarded them as impractical. This environment led to the development, in industry, industry, of a more general model based control methodology in which the dynamic optimization problem is solved onli line ne at each each cont contro roll execu executi tion on.. Proc Proces esss inpu inputs ts are are computed so as to optimize future plant behavior over horizon. In the a time interval known as the prediction horizon. general case any desired objective function can be used. Plant Pla nt dynami dynamics cs are descri described bed by an explic explicit it proces processs model whic which h can can take take,, in prin princi cipl ple, e, any any requ requir ired ed mathem mathemati atical cal form. form. Proces Processs input input and output output conconstraints are included directly in the problem formulation so that that future future constr constrain aintt violat violation ionss are antici anticipat pated ed and prevented ted. The first input of the optimal input sequence is injected into the plant and the problem is sol solved ved agai again n at the the nex next tim time inter nterva vall usi using upda update ted d pro process cess measu easure rem ments ents.. In add additio ition n to deve develo lopi ping ng more more flexi flexibl blee cont contro roll tech techno nolo logy gy,, new new process identification technology was developed to allow quick estimation of empirical dynamic models from test data, substantially reducing the cost of model development. ment. This This new method methodolo ology gy for indust industria riall proces processs modeling and control is what we now refer to as MPC technology. In modern modern proces processin sing g plants plants the MPC MPC contro controll ller er is part of a multi-level hierarchy of control functions. This is illust illustrate rated d in Fig. 2, 2, whic which h show showss a conv conven enti tion onal al cont contro roll stru struct ctur uree on the the left left for for Unit Unit 1 and and a MPC MPC structure on the right for Unit 2. Similar hierarchical stru structu cture ress have have been been desc descri ribe bed d by Rich Richalet, alet, Raul Rault, t, Test Te stud ud,, an and d Pa Papo pon n (1 (197 978) 8) and Pr Pret ettt an and d Ga Garc rcıa (1988).. At the (1988) the top top of the the stru struct ctur uree a plan plantt-wi wide de optimizer optimizer determines determines optimal optimal steady-state steady-state settings for each each unit unit in the the plan plant. t. Thes Thesee may may be sent sent to loca locall optimizers at each unit which run more frequently or consider a more detailed unit model than is possible at the plantplant-wid widee level. level. The unit unit optimi optimizer zer comput computes es an optima optimall econom economic ic steady steady state state and passes this to the dynamic constraint control system for implementation. The dynam dynamic ic constra constraint int contro controll must must move move the plant plant from from one one cons constr trai aine ned d stea steady dy stat statee to anoth another er whil whilee minimizing constraint violations along the way. In the conventional structure this is accomplished by using a combination of PID algorithms, lead-lag (L/L) blocks and high/low select logic. It is often difficult to translate the control requirements at this level into an appropriate conventional control structure. In the MPC methodology this combination of blocks is replaced by a single MPC controller. !
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S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764 Unit 1 - Conventional Control Structure
Unit 2 - Model Predictive Control Structure Global Economic Optimization (every day)
Plant-Wide Optimization
Unit 2 Local Optimizer
Unit 1 Local Optimizer
Local Economic Optimization (every hour)
High/Low Select Logic
PID
L/L
SUM
PID
Model Predictive Control (MPC)
SUM
Unit 1 DCS- PID Controls
FC
Dynamic Constraint Control (every minute)
PC
TC
LC
Unit 2 DCS-PID Controls
FC
PC
TC
Basic Dynamic Control (every second)
LC
Fig. 2. Hierarchy Hierarchy of control system functions in a typical processing plant. Conventional Conventional structure is shown at the left; MPC structure is shown at the right.
Although the development and application of MPC technology was driven by industry, it should be noted that that the the idea idea of cont contro roll llin ing g a syst system em by solv solvin ing g a sequence of open-loop dynamic optimization problems was not new. Propoi (1963), (1963), for example, described a moving moving horizon horizon controller controller.. Le Leee an and d Ma Mark rkus us (1 (196 967) 7) anticipated current MPC practice in their 1967 optimal control text: One techni technique que for obtain obtaining ing a feedba feedback ck contro controlle llerr synthesis from knowledge of open-loop controllers is to meas measure ure the the curr curren entt cont contro roll proc proces esss stat statee and and then then comp comput utee very very rapi rapidl dly y for for the the open open-l -loo oop p cont contro roll func functi tion on.. The The first first port portio ion n of this this func func-tion is then used during a short time interval, after which hich a new measu easure rem ment ent of the the fun functio ction n is computed for this new measurement. The procedure is then repeated.
2.2. IDCOM IDCOM The first description of MPC control applications was presented by Richalet et al. in 1976 Conference (Richalet (Richalet et al., 1976) 1976) and later summarized in 1978 Automatica pape paperr (Ri Rich chal alet et et al al., ., 19 1978 78). ). They They desc descri ribe bed d thei theirr approach as model predictive heuristic control (MPHC). The solution software was referred to as IDCOM, an acronym for Identification and Command. The distinguishing features of the IDCOM approach are: *
*
*
*
*
Ther Theree is is,, howe howeve ver, r, a wide wide gap gap betw betwee een n theo theory ry and practi practice. ce. The essent essential ial contri contribut bution ion of indust industry ry was was to put put thes thesee idea ideass into into prac practi tice ce on oper operat atin ing g unit units. s. Out Out of this this expe experi rien ence ce came came a fres fresh h set set of problems that has kept theoreticians busy ever since.
impulse response model for the plant, linear in inputs or internal variables; quadratic performance objective over a finite prediction horizon; future plant output behavior specified by a reference trajectory; input input and output output constra constraint intss includ included ed in the forformulation; optimal optimal inputs inputs computed computed using a heuristic heuristic iterative iterative algorithm, interpreted as the dual of identification.
Richalet et al. chose an input–output representation of the process in which the process inputs influence the process process outputs outputs directly. directly. Process inputs are divided divided into
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
mani manipu pula late ted d vari variab able less (MVs (MVs)) whic which h the the cont contro roll ller er adjust adjusts, s, and distur disturban bance ce var variab iables les (DVS) (DVS) which which are not available for control. Process outputs are referred to as controlled variables (CVs). They chose to describe the relationship between process inputs and outputs using a discrete-time finite impulse response (FIR) model. For the the si sing ngle le inpu input, t, si sing ngle le outp output ut (SIS (SISO) O) case case the the FIR FIR model looks like: yk þ j ¼
N X
hi uk þ j Ài :
ð6Þ
i ¼1
This This mode modell predi predict ctss that that the the outp output ut at a give given n time time depends on a linear combination of past input values; the summat summation ion weight weightss hi are the impuls impulsee respon response se coeffic coefficien ients. ts. The sum is trunca truncated ted at the point point where where past past inpu inputs ts no long longer er influ influen ence ce the the outp output ut;; this this repr represe esent ntat atio ion n is ther therefo efore re only only poss possib ible le for for stab stable le plants. The finite impulse response was identified from plant test data using an algorithm designed to minimize the distance between the plant and model impulse responses in parame parameter ter space. space. The contro controll proble problem m was solved solved using the same algorithm by noting that control is the mathematical dual of identification. The iterative nature of the the cont contro roll algo algori rith thm m allo allows ws inpu inputt and and outp output ut constraints to be checked as the algorithm proceeds to a soluti solution. on. Because Because the control control law is not linear linear and could not be expressed as a transfer function, Richalet et al. al. refe referr to it as heuristic. heuristic. In toda today’ y’ss cont contex extt the the algo algori rith thm m woul would d be refe referr rred ed to as a li line near ar MPC controller. The MPHC MPHC alg algori orithm thm drives drives the predic predicted ted future future output output trajec trajectory tory as closel closely y as possib possible le to a refere reference nce trajectory, defined as a first order path from the current output value to the desired setpoint. The speed of the desired closed-loop response is set by the time constant of the reference trajectory. This is important in practice beca becaus usee it prov provid ides es a natu natura rall way way to cont contro roll the the aggres agg ressiv sivene eness ss of the alg algori orithm thm;; increa increasin sing g the time time constant leads to a slower but more robust controller. Richalet et al. make the important point that dynamic cont contro roll must must be embe embedd dded ed in a hier hierar arch chy y of plant plant control functions in order to be effective. They describe four four leve levels ls of cont contro rol, l, very very si simi mila larr to the the stru struct ctur uree shown in Fig. 2: * *
* *
Level 3—Time and space scheduling of production. Level 2—Optimization of setpoints to minimize costs and ensure quality and quantity of production. Level 1—Dynamic multivariable control of the plant. Level 0—Control of ancillary systems; PID control of valves.
real real econ econom omic ic bene benefit fitss come come at leve levell 2 wher wheree bette betterr dynamic control allows the controlled variable setpoint to be moved closer to a constraint without violating it. This argument provides the basic economic motivation for using MPC technology. This concept of a hierarchy of control functions is fundamental to advanced control applications and seems to have been followed by many practitioners. Pr Pret ettt an and d Ga Garc rcıa (1 (198 988) 8),, for for exam exampl ple, e, describe a very similar hierarchy. Richa Richalet let et al. describe describe applic applicati ations ons of the MPHC MPHC algori alg orithm thm to a fluid fluid cataly catalytic tic cracki cracking ng unit unit (FCCU) (FCCU) main main fracti fractiona onator tor column column,, a power power plant plant steam steam gengenerator erator and a poly-v poly-viny inyll chlori chloride de (PVC) (PVC) plant. plant. All of thes thesee exam exampl ples es are are cons constra train ined ed mult multiv ivar aria iabl blee proprocess cesses es.. The The main main frac fracti tion onat ator or exam exampl plee invo involv lved ed cont contro roll llin ing g key key tray tray tempe tempera ratu ture ress to stab stabil iliz izee the the compos compositi ition on of heavy heavy and light light produc productt stream streams. s. The contro controlle llerr adjust adjusted ed produc productt flowrat flowrates es to compen compensat satee for inlet inlet tempera temperature ture disturb disturbanc ances es and to maint maintain ain the the leve levell of a key key inte intern rnal al tray tray.. The The powe powerr plan plantt stea steam m gene genera rator tor prob proble lem m invo involv lved ed cont contro roll llin ing g the the temp temper erat ature ure and and pres pressu sure re of steam steam deli deliver vered ed to the the turb turbin ine. e. This This appl applic icat atio ion n is inte intere rest stin ing g beca becaus usee the process response time varied inversely with load on the the syst system em.. This This nonl nonlin inea eari rity ty was was over overco come me by execut executing ing the contro controlle llerr with with a var variab iable le sample sample time. time. Benefit Benefitss for the main main fracti fractiona onator tor applic applicati ation on were were reported as $150; 000=yr; due to increasing the flowrate of the light product stream. Combined energy savings from two columns in the PVC plant were reported as $220; 000=yr: !
2.3. DMC Engineers at Shell Oil developed their own independent MPC technology in the early 1970s, with an initial applic applicati ation on in 197 1973. 3. Cutler Cutler and Ramake Ramakerr presen presented ted details of an unconstrained multivariable control algorithm which they named dynamic matrix control (DMC) at the 1979 Nation National al AIChE AIChE meetin meeting g (Cutler & Ramaker, 1979) 1979) and at the 1980 Joint Automatic Control Conference (Cutler (Cutler & Ramaker, 1980). 1980). In a companion paper paper at the the 1980 1980 meeti meeting ng Pre Prett tt and Gi Gille llette tte (19 (1980) 80) descri described bed an applic applicati ation on of DMC DMC techno technolog logy y to an FCCU reactor/regenerator in which the algorithm was modi modifie fied d to hand handle le nonl nonlin inea earit ritie iess and and cons constr trai aint nts. s. Neither Neither paper paper discus discussed sed their their proces processs identi identifica ficatio tion n technology. Key features of the DMC control algorithm include: * *
They They point point out that signifi significan cantt benefit benefitss do not come from from simply simply reducin reducing g the var variat iation ionss of a contro controll lled ed variable through better dynamic control at level 1. The
737
*
linear step response model for the plant; quadratic performance objective over a finite prediction horizon; future future plant plant output output behavi behavior or specifi specified ed by trying trying to follow the setpoint as closely as possible;
738 *
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
optimal inputs computed as the solution to a leastsquares problem.
The linear linear step step respons responsee model model used used by the DMC DMC algo algori rith thm m rela relate tess chang changes es in a proc proces esss outp output ut to a weighted sum of past input changes, referred to as input moves. For the SISO case the step response model looks like: N N À1
yk þ j ¼
X
si Duk þ j Ài þ sN uk þ j ÀN :
ð7Þ
i ¼1
The move weights si are the step response coefficients. Mathematically the step response can be defined as the integral of the impulse response; given one model form the other can be easily obtained. Multiple outputs were handle handled d by superp superposi ositio tion. n. By using using the step step respon response se model one can write predicted future output changes as a linear combination of future input moves. The matrix that that ties ties the the two two toge togethe therr is the the so-c so-cal alle led d Dynamic Matrix. Matrix. Using Using this this represe representa ntatio tion n allows allows the optima optimall move vector to be computed analytically as the solution to a least-s least-squa quares res problem problem.. Feedfo Feedforwa rward rd contro controll is readily readily included included in this formulation formulation by modifying the predicted future outputs. In practice the required matrix inverse can be computed off-line to save computation. Only Only the first row of the final contro controlle llerr gai gain n matrix matrix needs to be stored because only the first move needs to be computed. The objecti objective ve of a DMC DMC contro controlle llerr is to drive drive the output as close to the setpoint as possible in a leastsquares sense with a penalty term on the MV moves. This results in smaller computed input moves and a less aggr aggres essi sive ve outpu outputt resp respon onse se.. As with with the the IDCO IDCOM M referen reference ce trajec trajectory tory,, this this techni technique que provid provides es a degree degree of robustness to model error. Move suppression factors also provide an important numerical benefit in that they can be used to directly improve the conditioning of the numerical solution. Cutler and Ramaker showed results from a furnace temper temperatu ature re contro controll appli applicat cation ion to demons demonstra trate te imimprov proved ed cont contro roll qual qualit ity y usin using g the the DMC DMC algo algori rith thm. m. Feedforward response of the DMC algorithm to inlet temperature changes was superior to that of a conventional PID lead/lag compensator. In their paper Prett and Gillette (1980) described an appli applicati cation on of DMC DMC techno technolog logy y to FCCU FCCU reacto reactor/ r/ rege regene nera rato torr cont contro rol. l. Four Four such such appl applic icat atio ions ns were were already completed and two additional applications were underway at the time the paper was written. Prett and Gillette described additional modifications to the DMC algo algori rith thm m to prev preven entt viol violat atio ion n of abso absolu lute te inpu inputt cons constr trai aint nts. s. When When a pred predic icte ted d futu future re inpu inputt came came suffici sufficientl ently y close close to an absolu absolute te constra constraint int,, an extra extra equation was added to the process model that would drive the input back into the feasible region. These were referre referred d to as time time var varian iantt constra constraint ints. s. Becaus Becausee the
decision to add the equation had to be made on-line, the matrix inverse solution had to be recomputed at each control execution. Prett and Gillette developed a matrix tearin tearing g soluti solution on in which which the origin original al matrix matrix invers inversee could be computed off-line, requiring only the matrix inverse corresponding to active time variant constraints to be computed on-line. The initial IDCOM and DMC algorithms represent the first generation generation of MPC MPC techno technolog logy; y; they they had an enormo enormous us impac impactt on industr industrial ial process process contro controll and served to define the industrial MPC paradigm. 2.4. QDMC QDMC The original IDCOM and DMC algorithms provided excellent excellent control control of unconstrain unconstrained ed multivari multivariable able processes. Constraint Constraint handling, handling, however, was still somewhat what ad hoc. hoc. Engi Engine neers ers at Shel Shelll Oil Oil addr addres esse sed d this this weakness by posing the DMC algorithm as a quadratic progra program m (QP) (QP) in which which input input and output output constra constraint intss appear explicitly. Cutler et al. first described the QDMC algorithm in a 1983 AIChE conference paper (Cutler, (Cutler, Morshedi, & Hay Haydel del,, 198 1983 3). Garcıa an and d Mo Mors rshe hedi di (1986) publis published hed a more more compre comprehen hensiv sivee descri descripti ption on several years later. Key features of the QDMC algorithm include: !
* *
*
*
linear step response model for the plant; quadratic performance objective over a finite prediction horizon; future future plant plant output output behavi behavior or specifi specified ed by trying trying to follow the setpoint as closely as possible subject to a move suppression term; opti optima mall inpu inputs ts comp comput uted ed as the the solu soluti tion on to a quadratic program.
Garcıa and Morshedi show how the DMC objective function can be re-written in the form of a standard QP. Future projected outputs can be related directly back to the input move vector through the dynamic matrix; this allows all input and output constraints to be collected into a matrix inequality involving the input move vector. Alth Althou ough gh the the QDMC QDMC algo algori rith thm m is a some somewh what at adadvanced vanced contro controll alg algori orithm thm,, the QP itself itself is one of the simplest possible optimization problems that one could pose. The Hessian of the QP is positive definite for linear plants plants and so the result resulting ing optimi optimizat zation ion proble problem m is convex. This means that a solution can be found readily using standard commercial optimization codes. Garcıa and and Mors Morshe hedi di wrap wrappe ped d up thei theirr pape paperr by presenting results from a pyrolysis furnace application. The QDMC QDMC contro controlle llerr adjust adjusted ed fuel fuel gas pressu pressure re in three burners in order to control stream temperature at thre threee loca locati tion onss in the the furn furnac ace. e. Thei Theirr test test resu result ltss demonstrated dynamic enforcement of input constraints and decoup decouplin ling g of the temper temperatu ature re dynami dynamics. cs. They They !
!
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
reported good results on many applications within Shell on problems as large as 12 Â 12 (12 process outputs and 12 proc proces esss inpu inputs) ts).. They They stat stated ed that that abov abovee all, all, the the QDMC algorithm had proven particularly profitable in an on-l on-lin inee opti optimi miza zati tion on envi enviro ronm nmen ent, t, prov provid idin ing g a smooth transition from one constrained operating point to another. The QDMC algorithm can be regarded as representing a second generation of MPC technology, comprised of algo algori rith thms ms whic which h prov provid idee a syst system emat atic ic way way to impl implem emen entt inpu inputt and and outp output ut cons constra train ints ts.. This This was was accomp accomplis lished hed by posing posing the MPC MPC proble problem m as a QP, with the solution provided by standard QP codes. 2.5. IDCOM-M IDCOM-M,, HIECON, HIECON, SMCA, and SMOC SMOC As MPC MPC techno technolog logy y gai gained ned wider wider accepta acceptance nce,, and problems tackled by MPC technology grew larger and more complex, control engineers implementing second genera generatio tion n MPC MPC techno technolog logy y ran into into other other practi practical cal problems. The QDMC algorithm provided a systematic approa approach ch to incorp incorpora orate te hard hard input input and output output conconstra strain ints ts,, but but ther theree was was no clea clearr way way to hand handle le an infea infeasi sibl blee solu solutio tion. n. For For exam exampl plee it is possi possibl blee for for a feedforw feedforward ard distur disturban bance ce to lead lead to an infeas infeasibl iblee QP; what should the control do to recover from infeasibility? The The soft soft cons constr trai aint nt form formul ulati ation on is not not comp comple letel tely y satisf satisfacto actory ry because because it means means that that all constra constraint intss will will be violated to some extent, as determined by the relative relative weig weight hts. s. Clea Clearl rly y some some outp output ut cons constr trai aint ntss are are more more impor importan tantt than than others others,, howev however, er, and shoul should d never never be viol violat ated ed.. Woul Would d not not it make make sens sensee then then to shed shed low low priority constraints in order to satisfy higher priority ones? In practice, process inputs and outputs can be lost in real time due to signal hardware failure, valve saturation or direct operator intervention. They can just as easily come come back back into into the the cont contro roll prob proble lem m at any any samp sample le interval. This means that the structure of the problem and the degrees of freedom available to the control can change dynamically. This is illustrated in Fig. 3, 3, which illust illustrat rates es the shape shape of the proces processs transf transfer er functi function on
matrix matrix for three three genera generall cases. cases. The square plant case, which occurs when the plant has just as many MVs as CVs, leads to a control problem with a unique solution. In the real world, square is rare. More common is the fat plant case, in which there are more MVs available than there are CVs to control. The extra degrees of freedom available in this case can be put to use for additional objectives, such as moving the plant closer to an optimal operating point. When valves become saturated or lower leve levell cont contro roll acti action on is lost lost,, the the plan plantt may may reac reach h a condition in which there are more CVs than MVs; this is the thin plant case. In this situation it will not be possible to meet meet all all of the the cont contro roll obje object ctiv ives es;; the the cont contro roll specifi specificat cation ionss must must be relaxe relaxed d someho somehow, w, for examp example le by minimizing CV violations in a least-squared sense. Fault tolerance is also an important practical issue. Rather than simply turning itself off as signals are lost, a practical MPC controller should remain online and try to make the best of the sub-plant under its control. A majo majorr barr barrie ierr to achi achiev evin ing g this this goal goal is that that a well well conditioned multivariable plant may contain a number of poorly conditioned sub-plants. In practice an MPC controller must recognize and screen out poorly conditioned tioned sub-pl sub-plant antss before before they they result result in erratic erratic contro controll action. It also also beca became me incre increas asin ingl gly y diffi difficu cult lt to tran transl slat atee control control requiremen requirements ts into relative weights for a single single objective function. Including all the required trade-offs in a single objective function means that relative weights have have to be assi assign gned ed to the the valu valuee of outp output ut setp setpoi oint nt violat violation ions, s, output output soft soft constra constraint int violat violation ions, s, inputs inputs moves, moves, and optimal input target violations. violations. For large problems it is not easy to translate control specifications into a consistent set of relative weights. In some cases it does not make sense to include these variables in the same same objecti objective ve functi function; on; drivin driving g the inputs inputs to their their optimal optimal targets may lead to larger violation of output soft constraints, for example. Even when a consistent set of relative relative weights can be found, care must be taken to avoid scaling problems that lead to an ill-conditioned solution. Prett and Garcıa (1988) commented on this problem: !
MV's
MV's
CV's
Thin Plant
Over-determined degrees of freedom < 0
CV's
739
Square Plant
Unique solution degrees of freedom = 0
MV's CV's
Fat Plant
Under-determined degrees of freedom > 0
Fig. 3. Process structure determines determines the degrees of freedom available available to the controller. Adapted from Froisy (1994). (1994).
740
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
The combin combinati ation on of multip multiple le object objective ivess into into one objecti objective ve (functi (function) on) does does not allow allow the designe designerr to reflect the true performance requirements. These issues motivated motivated engineers at Adersa, Adersa, Setpoint, Setpoint, Inc., and Shell (France) to develop new versions of MPC algorithms. The version marketed by Setpoint was called IDCOM-M (the M was to distinguish this from a single input/single output version called IDCOM-S), while the near nearly ly iden identic tical al Ader Adersa sa vers versio ion n was was refe referre rred d to as hierar hierarchi chical cal constra constraint int contro controll (HIECO (HIECON). N). The IDCOM-M COM-M contro controll ller er was first descri described bed in a paper paper by Grosdidier Grosd idier,, Froi Froisy, sy, and Hamm Hammann ann (1988 (1988)). A seco second nd paper presented at the 1990 AIChE conference describes an application of IDCOM-M to the Shell Fundamental Control Problem (Froisy (Froisy & Matsko, 1990) 1990) and provides additional additional details concerning concerning the constraint constraint methodomethodology. Distinguishin Distinguishing g features features of the IDCOM-M IDCOM-M algorithm include: * *
*
*
* *
linear impulse response model of plant; contro controlla llabil bility ity superv superviso isorr to screen screen out ill ill-co -condi ndi-tioned plant subsets; multi-objective function formulation; quadratic output objective followed by a quadratic input objective; controls a subset of future points in time for each output, called the coincidence points, chosen from a reference trajectory; a single move is computed for each input; constraints can be hard or soft, with hard constraints ranked in order of priority.
An important distinction of the IDCOM-M algorithm is that it uses two separate objective functions, functions, one for the outputs and then, if there are extra degrees of freedom, one one for for the the inpu inputs ts.. A quad quadra rati ticc outp output ut obje object ctiv ivee func functi tion on is mini minimi mize zed d first first subj subjec ectt to hard hard inpu inputt constraints. Each output is driven as closely as possible to a desired value at a single point in time known as the coincidence point. point. The name comes from the fact that this this is where where the desire desired d and predicte predicted d val values ues should should coincide. The desired output value comes from a first orde orderr refe referen rence ce traj trajec ecto tory ry that that starts starts at the the curre current nt measure measured d val value ue and leads leads smooth smoothly ly to the setpoi setpoint. nt. Each Each outp output ut has has two two basi basicc tuni tuning ng para parame mete ters rs;; a coincidence point and a closed-loop response time, used to define the reference trajectory. Grosdidier et al. (1988) provide simulation results for a representativ representativee FCCU regenerator regenerator control control problem. problem. The problem involves controlling flue gas composition, flue gas temperature, and regenerator bed temperature by manipulating feed oil flow, recycle oil flow and air to the regenerator regenerator.. The first simulation simulation example example demondemonstrates how using multiple inputs can improve dynamic performance performance while while reaching reaching a pre-determi pre-determined ned optimal optimal steady-state condition. A second example demonstrates how the controller switches from controlling one output
to controll controlling ing another another when when a measure measured d disturb disturbance ance causes causes a constrai constraint nt violati violation. on. A third third example example demondemonstrates the need for the controllability supervisor. When an oxygen analyzer fails, the controllability supervisor is left with only flue gas temperature and regenerator bed temperature to consider. It correctly detects that controlling both would lead to an ill-condition ill-conditioned ed problem; this is because these outputs respond in a very similar way to the inputs. Based on a pre-set priority it elects to control only the flue gas temperature. When the controllability supervi supervisor sor is turned turned off the same same simulati simulation on scenario scenario leads to erratic and unacceptable input adjustments. Setpoint engineers continued to improve the IDCOMM technology, and eventually combined their identification, simulation, simulation, configuratio configuration, n, and control control products products into into a sing single le inte integr grat ated ed offe offeri ring ng call called ed SMCA SMCA,, for for Setpoint Setpoint Multivari Multivariable able Control Control Architectur Architecture. e. An improved numerical solution engine allowed them to solve a sequence of separate steady-state target optimizations, providing providing a natural natural way to incorporate incorporate multiple multiple ranked ranked control objectives and constraints. In the the late late 1980 1980’s ’s engi engine neers ers at Shel Shelll Resea Researc rch h in France France developed developed the Shell Multivariable Multivariable Optimizin Optimizing g Controller (SMOC) (Marquis (Marquis & Broustail, 1998; 1998; Yousfi & Tourn Tournier, ier, 1991 1991)) whic which h they they desc descri ribed bed as a brid bridge ge between state-space and MPC algorithms. They sought to combin combinee the constra constraint int handli handling ng featur features es of MPC with the richer framework for feedback offered by statespace methods. To motivate this effort they discussed the control of a hydrotreater unit with four reactor beds in series. The control control system must maintain maintain average average bed temperature at a desired setpoint and hold temperature differences between the beds close to a desired profile, while while preven preventin ting g violat violation ion of maxim maximum um temper temperatu ature re li limi mits ts with within in each each reac reacto tor. r. Ma Mani nipu pula late ted d vari variab able less include include the first reactor inlet temperature temperature and quench flows flows betwee between n the beds. beds. A typica typicall MPC MPC input/ input/outp output ut model would view the manipulated variables as inputs and the bed temperatures as independent outputs, and a constant output disturbance would be assigned to each bed temperature. But it is clear that the bed temperatures are not independent, in that a disturbance in the first first bed will will ultima ultimatel tely y affect affect all three three downstr downstream eam reactors. In addition, the controlled variables are not the process outputs but rather linear combinations thereof. Stat Statee-sp spac acee cont contro roll desi design gn meth methods ods offer offer a natu natura rall solution to these problems but they do not provide an optima optimall way to enforce enforce maximu maximum m bed temper temperatu ature re constraints. The SMOC algorithm includes several features that are are now now cons consid idere ered d esse essent ntia iall to a ‘‘mo ‘‘mode dern rn’’ ’’ MPC MPC formulation: *
State-space models are used so that the full range of linear dynamics can be represented (stable, unstable, and integrating).
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764 *
*
*
*
An explicit disturbance model describes the effect of unmeasured unmeasured disturbanc disturbances; es; the constant constant output output disturbance is simply a special case. A Kalman filter is used to estimate the plant states and unmeasured disturbances from output measurements. A distinction is introduced between controlled controlled variables appearing appearing in the control objective and feedback variables that are used for state estimation. Input and output constraints are enforced via a QP formulation.
The SMOC algorithm is nearly equivalent to solving the LQR LQR proble problem m with with input input and output output constra constraint ints, s, except that it is still formulated on a finite horizon. As such, it does not inherit the strong stabilizing properties of the LQR algorithm. algorithm. A stabilizin stabilizing, g, infinite-hor infinite-horizon izon formulation of the constrained LQR algorithm would come only after academics began to embrace the MPC para paradi digm gm in the the 1990 1990ss (Rawlings & Mu Muske ske,, 199 1993 3; Scokaert & Rawlings, 1998). 1998). The The IDCO IDCOMM-M, M, HIEC HIECON ON,, SMCA SMCA,, and and SMOC SMOC algorithms represent a third generation of MPC technology; others include the PCT algorithm sold by Profimatics, and the RMPC algorithm sold by Honeywell. This generation distinguishes between several levels of constraints (hard, soft, ranked), provides some mechanism to recover from an infeasible solution, addresses the issues resulting from a control structure that changes in real real time time,, prov provid ides es a ri rich cher er set set of opti option onss for for feedback, and allows for a wider range of process dynamics (stable, integrating integrating and unstable) and controller controller specifications. 2.6. DMC-plus DMC-plus and RMPCT In the last 5 years, years, increase increased d compet competiti ition on and the mergers of several MPC vendors have led to significant changes in the industrial MPC landscape. In late 1995 Honeyw Honeywell ell purcha purchased sed Profima Profimatic tics, s, Inc. Inc. and formed formed Honeywell Honeywell Hi-Spec Hi-Spec Solutions. Solutions. The RMPC RMPC algorithm algorithm offered by Honeywell was merged with the Profimatics PCT contro controlle llerr to create create their their curren currentt offeri offering ng called called RMPC RM PCT. T. In earl early y 1996 1996,, Aspe Aspen n Tech Techno nolo logy gy Inc. Inc. purchased both Setpoint, Inc. and DMC Corporation. This was followed by acquisition of Treiber Controls in 1998. The SMCA and DMC technologies were subsequently merged to create Aspen Technology’s current DMCDMC-pl plus us prod produc uct. t. DMCDMC-pl plus us and and RM RMPC PCT T are are representati representative ve of the fourth generation MPC technology sold today, with features such as: * *
Windows-based graphical user interfaces. Multiple Multiple optimizati optimization on levels levels to address address prioritized prioritized control objectives.
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Additional flexibility in the steady-state target optimization, including QP and economic objectives. Direc Directt consid considera eratio tion n of model model uncert uncertain ainty ty (robus (robustt control design). Improved identification technology based on prediction error method and sub-space ID methods.
*
*
*
These and other MPC algorithms currently available in the market marketpla place ce are descri described bed in greate greaterr detail detail in the next section.
3. Survey of MPC technology technology products products
The The industr industrial ial MPC MPC techno technolog logy y has change changed d conconsiderably since the publication of our first survey 5 years ago (Qin (Qin & Badgwell, 1996), 1996), which included data from five vendors: Adersa, DMC, Honeywell, Setpoint, and Treibe Treiberr Contro Controls. ls. In late late 199 1995 5 Honeyw Honeywell ell purcha purchased sed Profima Profimatic ticss and formed formed Honeyw Honeywell ell Hi-Spe Hi-Spec. c. In early early 1996, Setpoint and DMC were both acquired by Aspen Technology. Two years later Aspen purchased Treiber Controls so that three of the companies in our original survey had merged into one. These mergers, continued prod produc uctt deve develo lopm pmen ent, t, and and the the emer emergen gence ce of viab viable le nonl nonlin inea earr MPC MPC prod produc ucts ts (Qin & Badgw Badgwell, ell, 1998 1998)) changed the MPC market enough for us to believe that an updated survey would be worthwhile. We began collecting data for the present survey in mid-19 mid-1999, 99, when when we solici solicited ted inform informati ation on from from eight eight ven vendors dors in orde orderr to asse assess ss the the curr curren entt stat statu us of commercial MPC technology. The companies surveyed and their product names and descriptions are listed in Tabl Ta bles es 1 an and d 2. In this this surv survey ey we adde added d two two new new companies offering linear MPC products: Shell Global Solutions (SGS) and Invensys Systems, Inc. (ISI) (Lewis, (Lewis, Evans, & Sand Sandoz, oz, 1991 1991)) in the UK. UK. Three Three nonlin nonlinear ear MPC vendors were also included: Continental Controls,
Table 1 Companies and products included in Linear MPC technology survey Comp Compan any y
Prod Produc uctt name name
Desc Descri ript ptio ion n
Ader Adersa sa
HIECO IECON N PFC GLIDE LIDE DMC-p DMC-plu luss DMC-plus model RMPC RM PCT T
Hiera ierarc rchi hica call cons constr trai aint nt cont contro roll Predi edicti ctive fun functi ctional contr ontrol ol Iden Identi tific fica ation tion pack packag agee Dynam Dynamic ic matri matrix x contro controll packag packagee Identification Identification package
Aspen Aspen Tech Tech
Hone Honeyw ywel elll Hi-Spec Shell Global Solutions Inven Invensys sys a
SMOC-IIa
Robu Robust st mode modell pred predic icti tive ve cont contro roll technology Shell multivariable optimizing control
Connoi Connoiss sseu eurr
Contr Control ol and identi identifica ficatio tion n packag packagee
SMOC-I was licensed to MDC Technology and Yokogawa in the past. Shell global solutions is the organization that markets the current SMOC technology. technology.
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S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
DOT Products, and Pavilion Technologies. We believe that that the the tech techno nolo logy gy sol sold by thes thesee com compani panies es is representative of the industrial state of the art; we fully recogn recognize ize that that we have have omitte omitted d some some MPC MPC vendor vendorss from our survey, especially those who just entered the market market (e.g., (e.g., Fisher-Rose Fisher-Rosemoun mount, t, ABB). ABB). Some compacompanies were not asked to participate, some chose not to participate, and some responded too late to be included in the paper. Only companies which have documented successful MPC applications were asked to participate. It should be noted that several companies make use of MPC MPC tech techno nolo logy gy deve develo lope ped d in-h in-hou ouse se but but were were not not included in the survey because they do not offer their technology externally. These MPC packages are either well known to academic researchers or not known at all for proprietary reasons. The SMOC algorithm originally develo developed ped at Shell Shell France France is includ included ed in this this survey survey because it is now commercially available through SGS. MDC MDC Techno Technolog logy, y, Inc. Inc. and Yokoga Yokogawa wa had li licen cense se agreements with Shell. Init Initia iall data data in this this surv survey ey were were coll collec ecte ted d from from industrial MPC vendors using a written survey. Blank copies copies of the survey survey form are availabl availablee upon upon reques requestt from the authors. Survey information was supplemented by published published papers, product literature (DMC (DMC Corp Corp., ., 1994;; Setpo 1994 Setpoint int Inc., 1993; 1993; Honey Honeywell well Inc., 1995), 1995), and pers person onal al comm commun unic icat atio ion n betw betwee een n the the auth author orss and and
vend vendor or repr represe esent ntat ativ ives. es. Resu Result ltss of the the line linear ar MPC MPC survey are summarized in Tables 3, 4 and 6. 6. Nonlinear MPC survey results are summarized separately in Tables 5 and 7. 7. While the data are provided by the vendors, the analysis is that of the authors. In presenting the survey resu result ltss our our inte intent ntio ion n is to high highli ligh ghtt the the impo import rtan antt feat featur ures es of each each algo algori rith thm; m; it is not not our our inte intent nt to determine the superiority of one product versus another. The purpose of showing the application numbers is to givee a relati giv relative ve magni magnitud tudee on how MPC is applie applied d to diff differ eren entt area areas. s. The The abso absolu lute te numb number erss are are not not very very important as they are changing fast. The numbers are not not exac exactl tly y comp compar arab able le as the the size size of each each MPC MPC appl applic icat atio ion n can can be very very diffe differe rent nt.. Wi With th this this unde underrstanding in mind, we first discuss the overall procedure for contro controll design design and tuning tuning.. Then Then we descri describe be the vari variou ouss mode modell form formss used used for for both both the the line linear ar and and nonlinear technology. The last two sections summarize the the main main feat featur ures es of the the iden identi tific ficati ation on and and cont contro roll products sold by each vendor. 3.1. Control Control design and tuning The MPC contro controll design design and tuning tuning procedur proceduree is gene genera rall lly y desc descri ribe bed d as foll follow owss (DM DMC C Co Corp rp., ., 19 1994 94;; Setpoint Inc., 1993; 1993; Honeywell Inc., 1995): 1995): *
Table 2 Companie Companiess and products products included included in Nonline Nonlinear ar MPC technolo technology gy survey Company
Product name
Description
Adersa
PFC
Aspen Tech Continental Controls, Inc. DOT Products
Aspen Target MVC
Predictive functional control Nonlinear MPC package Multivariable control
Pavilion Pavilion Technolo Technologie giess
Process Process Perfecter Perfecter
NOVA-NLC
NOVA nonlinear controller Nonline Nonlinear ar control control
*
*
*
*
From the stated control objectives, define the size of the problem, and determine the relevant CVs, MVs, and DVs. Test Test the plant plant sys system temati atical cally ly by var varyin ying g MVs MVs and DVs; capture and store the real-time data showing how the CVs respond. Derive a dynamic model either from first-principles or from from the plant test data using using an identi identifica ficatio tion n package. Configure the MPC controller and enter initial tuning parameters. Test the controller off-line using closed-loop simulation to verify the controller performance.
Table 3 Comparison of linear MPC identification technology Product
Test protocol
Model forma
Est. methodb
Uncert. bound
DMC-plus RMPCTc AIDAd Glide Connoisseur
step, PRBS PRBS, step PRBS, step non-PRBS PRBS, step
VFIR, LSS FIR, ARX, BJ LSS, FIR, TF, MM TF FIR, ARX, MM
MLS LS, GN, PEM PEM-LS, GN GD, GN, GM RLS, PEM
Yes Yes Yes Yes Yes
a
Model Form: finite impulse response (FIR), velocity FIR (VFIR), Laplace transfer function (TF), linear state-space (LSS), auto-regressive with exogenous input (ARX), Box–Jenkins (BJ), multi-model (MM). b Est. method: least-squares (LS), modified LS (MLS), recursive LS (RLS), subspace ID (SMI), Gauss–Newton (GN), prediction error method (PEM), gradient descent (GD), global method (GM). c The commercial name for RMPCT is profit-controller. d AIDA: advanced identification data analysis.
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
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Table 4 Comparison of linear MPC control technology Company
Aspen Tech
Honeywell Hi-Spec
Adersa
Adersa
Invensys
SGS
Product Linear Model Formsa Feedback b Rem Ill-condc SS Opt Objd SS Opt Conste Dyn Opt Objf Dyn Opt Constg Output Trajh Output Horizi Input Param j Sol. Methodk Refe Referrenc ences
DMC-plus FSR L,S,I,U CD, ID IMS L=Q½I; O; y; R IH,OS,R Q[I,O,M],S IH S,Z FH MMB SLS Cutle tler and and Ram Ramaker (19 (1979) and DMC Corp., (1994)
RMPCT ARX, TF L,S,I,U CD, ID SVT Q[I,O] IH,OH Q[I,O] IH,OS S,Z,F FH MM ASQP Honeywell Inc., (1995)
HIECON FIR L,S,I CD, ID — — — Q[O],Q[I] IH,OH,OS,R S,Z,RT FH SM ASQP Richalet (1993)
PFC LSS,TF,ARX L,N,S,I,U CD, ID — Q[I,O] IH,OH Q[I,O],S IA,OH,OS,R S,Z,RT CP BF LS Richalet (1993)
Connois. ARX,FIR L,S,I,U CD, ID IMS L[I,O] IH,OH Q[I,O,M] IH,OS,R S,Z FH MMB ASQP
SMOC LSS L,S,I,U KF IMS Q[I,O],R IH IH,OS Q[I,O] Q[ IH,OS IH S,Z,RTB,F FH MMB ASQP Marquis and Broustail (1998)
a
Model form: Finite impulse response (FIR), finite step response (FSR), Laplace transfer function (TF), linear state-space (LSS), auto-regressive with exogenous input (ARX), linear (L), nonlinear (N), stable (S), integrating (I), unstable (U). b Feedback: Constant output disturbance (CD), integrating output disturbance (ID), Kalman filter (KF). c Removal of Ill-conditioning: Singular value thresholding (SVT), input move suppression (IMS). d Steady-state optimization objective: linear (L), quadratic (Q), inputs (I), outputs (O), multiple sequential objectives ðyÞ; outputs ranked in order of priority (R). e Steady-state optimization optimization constraints: Input hard maximum, minimum, minimum, and rate of change constraints (IH), output hard maximum maximum and minimum constraints (OH), constraints ranked in order of priority (R). f Dynamic optimization objective: Quadratic (Q), inputs (I), Outputs (O), input moves (M), sub-optimal solution (S). g Dynamic optimization constraints: Input hard maximum, minimum and rate of change constraints, IH with input acceleration constraints (IA), output hard maximum and minimum constraints (OH), output soft maximum maximum and minimum minimum constraints (OS), constraints ranked in order of priority (R). h Output trajectory: Setpoint (S), zone (Z), reference trajectory (RT), RT bounds (RTB), funnel (F). i Output horizon: Finite horizon (FH), coincidence points (CP). j Input parameterization: Single move (SM), multiple move (MM), MM with blocking (MMB), basis functions (BF). k Solution method: Least squares (LS), sequential LS (SLS), active set quadratic program (ASQP).
*
*
Download the configured controller to the destination machine and test the model predictions in openloop mode. Close the loop and refine the tuning as needed.
All All of the the MPC MPC pack packag ages es surv survey eyed ed here here prov provid idee software tools to help with the control design, model deve develo lopm pmen ent, t, and and clos closed ed-l -loo oop p si simu mula lati tion on steps steps.. A si sign gnifi ifican cantt amou amount nt of time time is curr curren entl tly y spen spentt at the the closed-loop simulation step to verify acceptable performance and robustness of the control system. Typically, tests are performed to check the regulatory and servo response of each CV, and system response to violations of major constraints is verified. The final tuning is then tested for sensitivity to model mismatch by varying the gain gai n and dynami dynamics cs of key proces processs models models.. Howev However, er, even even the the most most thor thorou ough gh si simu mula lati tion on test testin ing g usua usuall lly y cannot exhaust all possible scenarios. Of the the prod produc ucts ts surv survey eyed ed here here,, only only the the RM RMPC PCT T package provides robust tuning in an automatic way. This is accomplished using a min–max design procedure in which the user enters estimates of model uncertainty
directly. Tuning parameters are computed to optimize perf perfor orma manc ncee for for the the worst worst case case mode modell mism mismat atch ch.. Robustn Robustness ess checks checks for the other other MPC MPC contro controlle llers rs are performed by closed-loop simulation. 3.2. Process Process models The The techni technical cal scope of an MPC MPC produc productt is largel largely y defined by the form of process model that it uses. Tables 3 and 5 show that a wide variety of linear and nonlinear model forms are used in industrial MPC algorithms. It is helpful to visualize these models in a two-dimensional space, as illustrated in Fig. 4. 4. The horizontal axis refers to the source of information used for model development, and the vertical axis designates whether the model is linear or nonlinear. The far left side of the diagram represents empirical models that are derived exclusively from process test data. Because empirical models mainly perform fitting between the points of a data set, they gene genera rall lly y canno cannott be expe expect cted ed to accu accura rate tely ly pred predic ictt process behavior beyond the range of the test data used to develop them. At the far right side of the diagram lie
744
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
Table 5 Comparison of nonlinear MPC control technology Company
Adersa
Aspen Technology
Continental Controls
DOT Products
Pavilion Technologies
Product
PFC
MVC
Nonlinear model Formsa Feedback b Rem Ill-condc SS Opt Objd SS Opt Conste Dyn Opt Objf Dyn Opt Constg Output Trajh Output Horizi Input Param j Sol. Methodk Refe Refere renc nces es
NSS-FP
Aspen Target NS NSS-NNN
SNP-ARX
NOVA NLC NSS-FP
Process Perfecter NN NNN-ARX
S,I,U CD,ID — Q[I,O] IH,OH Q[I,O],S IA,OH,OS,R
S,I,U CD,ID,EKF IMS IM Q[I,O] IH,OH Q[ Q[I,O,M] IH,OS-l1
S CD
S,I CD
S,I,U CD,ID
Q[I,O] IH,OS Q[I,O,M] IH,OS
— — (Q,A)[I,O,M] IH,OH,OS
Q[I,O] IH,OH,OS Q[I,O] IH,OS
S,Z,RT CP BF NLS Rich Richal alet et (1993)
S,Z,RT CP MM QPKWIK De Oliveira and Biegler (1994, 1995), Sentoni et al. (1998), Zhao et al. (1998), Zhao, Guiver, Neelakantan, and Biegler (1999) and Turner and Guiver (2000)
S,Z,RT FH SM GRG2 Berkowitz and Papadopoulos Papadopoulos (1995), MVC 3.0 User Manual (1995), Berkowitz, Berkowitz, Papadopoulos, Papadopoulos, Colwell, and Moran (1996), Poe and Munsif (1998)
S,Z,RTUL FH MM NOVA Bartusiak and Fontaine (1997) and Young et al. (2001)
S,Z,TW FH MM GRG2 Demoro, Demoro, Axelrud, Johnston, and Martin, 1997, Keeler, Martin, Boe, Piche, Mathur, and Johnston, (1996), Martin et al. (1998); Martin and Johnston (1998) and Piche et al. (2000)
IMS
IMS
—
a
Model form: Input–output (IO), first-principles (FP), nonlinear state-space (NSS), nonlinear neural net (NNN), static nonlinear polynomial (SNP), stable (S), integrating (I), unstable (U). b Feedback: Constant output disturbance (CD), integrating output disturbance (ID), extended Kalman filter (EKF). c Removal of Ill-conditioning: Input move suppression (IMS). d Steady-state optimization objective: Quadratic (Q), inputs (I), outputs (O). e Steady-state optimization optimization constraints: Input hard maximum, minimum, minimum, and rate of change constraints (IH), output hard maximum maximum and minimum constraints (OH). f Dynamic optimization objective: Quadratic (Q), one norm (A), inputs (I), outputs (O), input moves (M). g Dynamic optimization constraints: input hard maximum, minimum and rate of change constraints (IH), IH with input acceleration constraints (IA), output hard maximum and minimum constraints (OH), output soft maximum and minimum constraints (OS), output soft constraints with l 1 exact penalty treatment (OS-l1) (De Oliveira and Biegler, 1994). h Output trajectory: Setpoint (S), Zone (Z), reference trajectory (RT), upper and lower reference trajectories (RTUL), trajectory weighting (TW). i Output horizon: finite horizon (FH), coincidence points (CP). j Input parameterization: Single move (SM), multiple move (MM), basis functions (BF). k Solution Solution method: method: Nonlinea Nonlinearr least least squares squares (NLS), (NLS), multi-st multi-step ep Newton Newton method method (QPKWIK (QPKWIK)) generali generalized zed reduced reduced gradient gradient (GRG), (GRG), mixed mixed complementarity complementarity nonlinear nonlinear program (NOVA).
models models derived derived purely purely from theoretical theoretical considerati considerations ons such as mass and energy balances. These first-principles models are typically more expensive to develop, but are able able to predic predictt proces processs behavi behavior or over over a much much wider wider range of operating operating conditions. conditions. In reality reality process process models models used used in MPC MPC tech techno nolo logy gy are are base based d on an effec effecti tive ve combination of process data and theory. First principles models, models, for example, example, are typically typically calibrated calibrated by using process test data to estimate key parameters. Likewise, empi empiri rica call mode models ls are are ofte often n adju adjuste sted d to acco accoun untt for for known known proces processs physic physics; s; for exampl examplee in some some cases cases a mode modell gain gain may be know known n to have have a cert certai ain n si sign gn or value. The MPC products surveyed here use time-invariant models that fill three quadrants of Fig. of Fig. 4; nonlinear first-
princi principle pless models models,, nonlin nonlinear ear empiri empirical cal models models,, and linear empirical models. The various model forms can be derived as special cases of a general continuous-time nonlinear state-space model: x ¼ f ðx; u; v; wÞ;
ð8aÞ 8aÞ
y ¼ gðx; uÞ þ n;
ð8bÞ 8bÞ
’
%
%
where uARmu is a vector of MVs, yARm y is a vector of CVs, xARn is a vector of state variables, vARmv is a vect vector or of meas measure ured d DVs, DVs, wARmw is a vector of unmeas unmeasured ured DVs DVs or proces processs noise, noise, and nARmx is a vector vector of measure measuremen mentt noise. noise. The follow following ing section sectionss describe each model type in more detail.
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
745
Table 6 Summary of linear MPC applications by areas (estimates based on vendor survey; estimates do not include applications by companies who have licensed vendor technology) a Area
Aspen Technology
Honeywell Hi-Spec
Adersa b
Invensys
Refining Petrochemicals Chemicals Pulp and paper Air & Gas Utility Mining/Metallurgy Food Processing Polymer Furnaces Aerospace/Defense Automotive Unclassified Total First App.
1200 45 4 50 100 18 — — 8 — 17 — — — 40 1833 DMC:1985 IDCOM-M:1987 OPC:1987 603 Â 283
480 80 20 50 10 10 6 — — — — — 40 696 PCT:1984 RMPCT:1991
280 — 3 — — — 7 41 — 42 13 7 1045 1438 IDCOM:1973 HIECON:1986
225 Â 85
—
Largest App.
SGSc
Total
25 20 21 — — 4 16 10 — 3 — — 26 125
450 450
1985 550 144 68 10 14 37 51 17 45 13 7 1601 4542 45
1984
1985
31 Â 12
—
a
The numbers reflect a snapshot survey conducted in mid-1999 and should not be read as static. A recent update by one vendor showed 80% increase in the number of applications. b Adersa applications through January 1, 1996 are reported here. Since there are many embedded Adersa applications, it is difficult to accurately report their number or distribution. Adersa’s product literature indicates over 1000 applications of PFC alone by January 1, 1996. c The number of applications of SMOC includes in-house applications by Shell, which are unclassified. Therefore, only a total number is estimated here.
The PFC algorithm can be used with several different model types. The most general of these is a discrete-time first-p first-prin rincip ciples les model model that that can be derive derived d from from 8 by integrating across the sample time:
Nonlinear
Aspen Target MVC Process Perfecter
PFC NOVA-NLC First Principles
Empirical DMCplus HIECON RMPCT PFC Connoisseur SMOC
xk þ1 ¼ f ðxk ; uk ; vk ; wk Þ;
ð9aÞ 9aÞ
yk ¼ gðxk ; uk Þ þ nk ;
ð9bÞ 9bÞ
although special care should be taken for stiff systems.
Linear
Fig. Fig. 4. C Cla lass ssifi ifica cati tion on of mode modell type typess used used in indu indust stri rial al MPC MPC algorithms.
3.2.1. Nonlinear first-principles models Nonlinear first-principles models used by the NOVANLC NLC algo algori rith thm m are are deri derived ved from from mass mass and and energ energy y balances, and take exactly the form shown above in 8. Unkn Unknow own n mode modell para parame mete ters rs such such as heat heat tran transf sfer er coeffic coefficien ients ts and reacti reaction on kineti kineticc consta constants nts are either either estimat estimated ed off-li off-line ne from from test test data data or on-line on-line using an extended Kalman filter (EKF). In a typical application the process model has between 10 and 100 differential algebraic equations.
3.2.2. Linear 3.2.2. Linear empirical empirical models Line Linear ar empi empiri rica call mode models ls have have been been used used in the the majo majori rity ty of MPC MPC appl applic icat atio ions ns to date date,, so it is no surpri surprise se that that most most of the current current MPC MPC produc products ts are based on this model type. A wide variety of model forms are are used used,, but but they they can can all be deri deriv ved fro from 9 by linearizing about an operating point to get: xk þ1 ¼ Axk þ Bu uk þ Bv vk þ Bw wk ;
ð10a 10aÞÞ
yk ¼ Cxk þ Duk þ nk :
ð10bÞ 10bÞ
The SMOC and PFC algorith algorithms ms can use this this model model form. form. An equiva equivalen lentt discre discrete-t te-tim imee transf transfer er functi function on model can be written in the form of a matrix fraction description (Kailath, (Kailath, 1980): 1980): yk ¼ ½I À U y ðqÀ1 ÞÀ1 ½Uu ðqÀ1 Þuk þ Uv ðqÀ1 Þvk
þ
À1 Uw ðq Þwk þ
nk ;
ð11Þ 11Þ
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S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
where qÀ1 is a backward shift operator. The output error identificati identification on approach approach (Lju Ljung, ng, 199 1999 9) mini minimi mizes zes the the measuremen measurementt error nk ; whic which h resu result ltss in nonl nonlin inea earr À1 parameter estimation. Multiplying ½I À U y ðq Þ on both sides of the above equation results in an autoregressive model with exogenous inputs (ARX), yk ¼ U y ðqÀ1 Þyk þ Uu ðqÀ1 Þuk þ Uv ðqÀ1 Þvk
þ Uw ðqÀ1 Þwk þ fk ;
ð12a 12aÞÞ
where Þnk : fk ¼ ½I À U y ðqÀ1 Þn
ð12bÞ 12bÞ
This This mode modell form form is used used by the the RM RMPC PCT, T, PFC, PFC, and and Connoisseur algorithms. The equation error identification approach minimizes fk ; which is colored colored noise even though the measurement noise nk is white. The RMPCT identification algorithm also provides an option for the Box–Jenkins model, that lumps the error terms in to one term ek : yk ¼ ½I À U y ðqÀ1 ÞÀ1 ½Uu ðqÀ1 Þuk þ Uv ðqÀ1 Þvk
þ ½H ½ He ðqÀ1 ÞÀ1 Ue ðqÀ1 Þek :
ð13Þ 13Þ
For a stable system, a FIR model can be derived as an approx approxima imatio tion n to the discre discrete-t te-time ime transfe transferr functi function on model 11: N X u
yk ¼
N X v
Hi u uk Ài þ
i ¼1
N X w
Hvi vk Ài þ
i ¼1
14Þ Hw i wk Ài þ nk : ð14Þ
i ¼1
This This model del form form is used used by the the DMC-pl -plus and and HIEC HIECON ON algo algori rith thms ms.. Typi Typica call lly y the the samp sample le time time is chosen so that from 30 to 120 coefficients are required to describe the full open-loop response. An equivalent velocity form is useful in identification: N X u
Dyk ¼
N X v
Hi u Duk Ài þ
i ¼1
Hvi Dvk Ài
i ¼1 N w
X
þ
Hw i Dwk Ài þ Dnk :
ð15Þ 15Þ
i ¼1
An altern alternati ative ve model model form form is the finite step step respon response se model (FSR) (Cutler, 1983); given by: N u À1
yk ¼
X
u Si u Duk Ài þ SN u u k ÀN u
i ¼1 N Nv À1
þ
X
Svi Dvk Ài þ SvN v vk ÀN v
i ¼1 N w À1
þ
X
v Sw i Dwk Ài þ SN w wk ÀN w þ nk ;
ð16Þ 16Þ
i ¼1
P j
where S j ¼ i ¼1 Hi and Hi ¼ Si À Si À1 : The The FSR FSR mode modell is used used by the the DMCDMC-pl plus us and and RM RMPC PCT T algo algo-rithms. The RMPCT, Connoisseur, and PFC algorithms also also prov provid idee the the opti option on to ente enterr a Lapl Laplac acee tran transf sfer er function model. This model form is then automatically
converted to a discrete-time model form for use in the control calculations. 3.2.3. Nonlinear empirical models Two Two basic basic types types of nonlin nonlinear ear empiri empirical cal models models are used used in the the prod produc ucts ts that that we surv survey eyed ed.. The The Aspe Aspen n Target product uses a discrete-time linear model for the state dynamics, with an output equation that includes a linear term summed with a nonlinear term: xk þ1 ¼ Axk þ Bu uk þ Bv vk þ Bw wk ;
ð17a 17aÞÞ
yk ¼ Cxk þ Du uk þ N ðxk ; uk Þ þ nk :
ð17bÞ 17bÞ
Only stable processes can be controlled by the Aspen Target product, so the eigenvalues of A of A must lie strictly within within the unit unit circle circle.. The nonli nonlinea nearr functio function n N is obtained from a neural network. Since the state vector x is not necess necessari arily ly limite limited d to physic physical al var variab iables les,, this this nonl nonlin inea earr mode modell appe appear arss to be more more gene genera rall than than meas measur urem emen entt nonl nonlin inea earit rity. y. For For exam exampl ple, e, a Wi Wien ener er model with a dynamic linear model followed by a static nonlinear mapping can be represented in this form. It is claimed that this type of nonlinear model can approximate any discrete time nonlinear processes with fading memory (Sentoni, (Sentoni, Biegler, Guiver, & Zhao, 1998). 1998). It is wel elll kn know own n th tha at ne neu ura rall ne netw twor ork ks ca can n be unreliable when used to extrapolate beyond the range of the training training data. The main problem problem is tha thatt for a sigmoidal neural network, the model derivatives fall to zero as the network extrapolates beyond the range of its training data set. The Aspen Target product deals with thiss pro thi proble blem m by cal calcul culati ating ng a mode modell confid confidence ence index (MCI) (MC I) onon-li line. ne. If the MC MCII ind indica icates tes tha thatt the neu neural ral networ net work k pre predic dictio tion n is unr unreli eliabl able, e, the neu neural ral net non non-line li near ar ma map p is gr grad adua uall lly y tu turn rned ed of offf an and d th thee mo mode dell calculation relies on the linear portion fA; B; Cg only. Another feature of this modeling algorithm is the use of EKF to correct for model-plant mismatch and unmeasured disturbances (Zhao, Guiver, & Sentoni, 1998). 1998). The EKF provides a bias and gain correction to the model on-line. This function replaces the constant output error feedback scheme typically employed in MPC practice. The MVC MVC alg algori orithm thm and the Proces Processs Perfect Perfecter er use nonlinear input–output models. To simplify the system identification task, both products use a static nonlinear model superimposed upon a linear dynamic model. Marti Ma rtin, n, Boe, Boe, Keeler Keeler,, Timmer Timmer,, and Havene Havenerr (1998) (1998) and later Piche, Sayyar-Rodsari, Johnson, and Gerules (2000) desc descri ribe be the the deta detail ilss of the the Proc Proces esss Perf Perfec ecter ter modeli modeling ng approa approach. ch. Their Their presen presentat tation ion is in single single-input–single-output form, but the concept is applicable to multi-input–multi-output models. It is assumed that the process input and output can be decomposed into a steady steady-st -state ate portio portion n which which obeys obeys a nonli nonlinea nearr static static model and a deviation portion that follows a dynamic model. For any input uk and output yk ; the deviation
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
variables are calculated as follows: duk ¼ uk À us ;
ð18a 18aÞÞ
d yk ¼ yk À ys ;
ð18bÞ 18bÞ
where us and ys are the steady-state values for the input and output, output, respec respectiv tively ely,, and follow follow a rather rather genera generall nonlinear relation: ys ¼ hs ðus Þ:
ð19Þ 19Þ
The deviat deviation ion var variab iables les follow follow a second second-or -order der linear linear dynamic relation: d yk ¼
2 X
ai d yk Ài þ bi duk Ài :
ð20Þ 20Þ
i ¼1
The identification of the linear dynamic model is based on plant test data from pulse tests, while the nonlinear static model is a neural network built from historical data. It is believed that the historical data contain rich steady-state information and plant testing is needed only for the dynamic sub-model. Bounds are enforced on the model gains in order to improve the quality of the neural network for control applications. The use of the composite model in the control step can be described as follows. Based on the desired output target yd s ; a nonlinear optimization program calculates the best input and output values u f s and y f s using the nonlinear static model. During the dynamic controller calculation, the nonlinear static gain is approximated by a linear interpolation of the initial and final steady-state gains, K s ðuk Þ ¼
K si þ
K f s À K si f
us À ui s
duk ;
ð21Þ 21Þ
where ui s and u f s are the current and the next steady-state values for the input, respectively, and
d ys ; ¼ dus ui
ð22a 22aÞÞ
d ys f ; K s ¼ dus u
ð22bÞ 22bÞ
K si
s
f s
which are evaluated evaluated using the static static nonlinear nonlinear model. Bounds on K si and K f s can be applied. Substituting the approx approxima imate te gai gain n Eq. (21) (21) into into the li linea nearr sub-mo sub-model del yields, d yk ¼
2 X
2 ai d yk Ài þ bi duk Ài þ gi duk Ài ; %
ð23a 23aÞÞ
i ¼1
where bi ¼ %
gi ¼
bi K si ð1 À
Pn
Pn
j ¼1 a j Þ
j ¼1 b j
bi ð1 À
Pn
;
f j ¼1 a j ÞK s f us j ¼1 b j
Pn
ð23bÞ 23bÞ À K si À ui s
:
ð23cÞ 23cÞ
747
The The purp purpos osee of thi this app approxi roxim matio ation n is to redu reduce ce computational computational complexity during the control calculation. It can be seen that the steady-state target values are calculated from a nonlinear static model, whereas the dynami dynamicc contro controll moves moves are calcul calculate ated d based based on the quadratic quadratic model in Eq. (23a). However, the quadratic quadratic model coefficients (i.e., the local gain) change from one control execution to the next, simply because they are rescaled to match the local gain of the static nonlinear model. This approximation strategy can be interpreted as a successive linearization at the initial and final states followed by a linear interpolation of the linearized gains. The interpolati interpolation on strategy strategy resembles resembles gain-schedu gain-scheduling, ling, but the overall model is different from gain scheduling becaus becausee of the gai gain n re-scal re-scaling ing.. This This model model makes makes the assump ass umptio tion n that that the proces processs dynami dynamics cs remain remain linear linear over over the the enti entire re rang rangee of oper operat atio ion. n. Asym Asymme metr tric ic dynami dynamics cs (e.g., (e.g., differ different ent local local time time consta constants nts), ), as a result, cannot be represented by this model. 3.3. MPC modeling modeling and identification identification technology technology Table 3 summarizes essential details of the modeling and and iden identi tific ficat atio ion n tech techno nolo logy gy sold sold by each each vend vendor or.. Model Modelss are usuall usually y develo developed ped using using proces processs respon response se data, obtained by stimulating the process inputs with a carefully designed test sequence. A few vendors such as Ader Adersa sa and and DOT DOT Prod Produc ucts ts advo advoca cate te the the use use of first first principles models. 3.3.1. Test protocols 3.3.1. protocols Test signals are required to excite both steady-state (low (low freq freque uenc ncy) y) and and dyna dynami micc (med (mediu ium m to high high frefrequency) dynamics of a process. A process model is then identi identified fied from from the proces processs input– input–outp output ut data. data. Ma Many ny vendor vendorss believ believee that that the plant test is the single single most most important important phase in the implementation implementation of DMC-plus DMC-plus controllers. To prepare for a formal plant test, a pre-test is usually necessary for three reasons: (i) to step each MV and adjust existing instruments and PID controllers; (ii) to obtain the time to steady state for each CV; and (iii) to obtain data for initial identification. Most Most identi identifica ficatio tion n packag packages es test test one (or at most most several) manipulated variables at a time and fix other variables at their steady state. This approach is valid as long as the process is assumed linear and superposition works. works. A few packag packages es allow allow severa severall MVs MVs to change change simultaneou simultaneously sly with uncorrelated uncorrelated signals for different different MVs. The plant test is run 24 hours a day with engineers monitoring the plant. Each MV is stepped 8 to 15 times, with the output (CV) signal to noise ratio at least six. The plant test may take up to 5–15 days, depending on the time to steady state and number of variables of the unit. Two requirements are imposed during the test: (i) no PID configu configurati ration on or tuning tuning change changess are allowe allowed; d;
748
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
and (ii) operators may intervene during the test to avoid critical critical situations, situations, but no synchronizi synchronizing ng or correlated correlated moves are allowed. One may merge data from multiple test periods, which allows the user to cut out a period of data which may be corrupted with disturbances. If the the lowe lowerr leve levell PID PID cont contro roll tuni tuning ng chan change gess signifi significant cantly ly then then it may may be necess necessary ary to constru construct ct a new process model. A model is identified between the inpu inputt and and outp output ut,, and and this this is comb combin ined ed by disc discre rete te convolution with the new input setpoint to input model. It appears that PRBS or PRBS-like stepping signals are the primary test signals used by the identification packages. The GLIDE package uses a binary signal in which the step lengths are optimized in a dedicated way. Others use a step test with random magnitude or more random signals like the PRBS (e.g., DMC-plus-Model, Connoisseur, and RMPCT). 3.3.2 3.3.2.. Linear Linear model identificatio identification n The model parameter parameter estimation estimation approaches approaches in the MPC products products are mainly mainly based based on minimi minimizin zing g the following least-squares criterion, J ¼
L X
m 2 jjyk À yk jj ;
ð24Þ 24Þ
k ¼1
using either an equation error approach or an output error error approa approach ch (Lju Ljung, ng, 198 1987 7). The major major differ differenc encee betwee between n the equati equation on error error approa approach ch and the output output error approach appears in identifying ARX or transfer func functi tion on model models. s. In the the equa equati tion on erro errorr appr approa oach ch,, past output measurements are fed back to the model in Eqn. (12a), m þ Uu ðqÀ1 Þuk þ Uv ðqÀ1 Þvk ; yk ¼ U y ðqÀ1 Þyk
ð25Þ 25Þ
whil whilee in the the outp output ut erro errorr appr approa oach ch,, the the past past mode modell output estimates are fed back to the model, yk ¼ U y ðqÀ1 Þyk þ Uu ðqÀ1 Þuk þ Uv ðqÀ1 Þvk :
ð26Þ 26Þ
The equation error approach produces a linear leastsqua square ress prob proble lem, m, but but the the esti estima mates tes are are bias biased ed even even though the measurement noise n in Eqn. (11) is white. The The outpu outputt erro errorr appr approa oach ch is unbi unbias ased ed give given n whit whitee measure measuremen mentt noise. noise. Howeve However, r, the ARX model model paraparameters appear nonlinearly in the model, which requires nonlinear parameter estimation. One may also see that the the equa equati tion on erro errorr appr approa oach ch is a oneone-st step ep ahea ahead d m predic predictio tion n approa approach ch with with referen reference ce to yk ; while while the outp output ut erro errorr appr approa oach ch is a long long rang rangee pred predic icti tion on m : approach since it does not use yk Using Using FIR models models result resultss in a linearlinear-inin-par parame ameter ter model and an output error approach, but the estimation variance may be inflated due to possible overparametrization. In DMC-plus-Model, a least-squares method is used to estimate FIR model parameters in velocity form (Eqn. (15)). The advantage of using the velocity form is to reduce the effect of a step-like unmeasured
disturbance disturbance (Cutler & Yocu Yocum, m, 1991 1991). ). Howe Howeve ver, r, the the velo veloci city ty form form is sens sensit itiv ivee to high high freq freque uenc ncy y nois noise. e. There Therefo fore re,, DMCDMC-pl plus us-M -Mod odel el allo allows ws the the data data to be smoothed prior to fitting a model. The FIR coefficients are then then conver converted ted into into FSR coeffic coefficien ients ts for contro control. l. Connoisseur uses recursive least squares in a prediction error formulation formulation to implement implement adaptive features. features. It is worth noting that subspace model identification (SMI (SMI)) (Lari Larimore, more, 1990 1990;; Ljung Ljung,, 1999 1999)) alg algori orithm thmss are now implemented in several MPC modeling algorithms. SMOC uses canonical variate analysis (CVA) to identify a state-space model which is also the model form used in the the SMOC SMOC cont contro roll ller er.. Seve Severa rall othe otherr vend vendor orss are are deve develo lopi ping ng and and test testin ing g thei theirr own own vers versio ions ns of SMI SMI algorithms. RMPCT RM PCT adopts adopts a three-s three-step tep approa approach: ch: (i) identi identify fy eith either er a Box– Box–Jen Jenki kins ns mode modell usin using g PEM PEM or an FIR FIR mode modell usin using g Chol Choles esky ky deco decomp mpos osit itio ion; n; (ii) (ii) fit the the identified model to a low-order ARX model to smooth out large variance due to possible overparametrization in the FIR model. The output error approach is used to fit the ARX model via a Gauss–Newton method; and (iii) (iii) conver convertt the AR ARX X models models into into Laplac Laplacee transfe transferr functions. As an alternative to (ii) and (iii), RMPCT has the option to fit the identified model directly to a fixed structure Laplace transfer function. When the model is used used in cont contro rol, l, the the tran transf sfer er func functi tion on mode models ls are are discretized into FSR models based on a given sampling interval. The advantage of this approach is that one has the flexibility to choose different sampling intervals than that used in data collection. Model Model uncerta uncertaint inty y bounds bounds are provid provided ed in several several prod produc ucts ts such such as RM RMPC PCT. T. In GLID GLIDE, E, cont contin inuo uous us transfer function models are identified directly by using gradient gradient descent descent or Gauss–New Gauss–Newton ton approaches. approaches. Then model model uncerta uncertaint inty y is identi identified fied by a global global method method,, which finds a region in the parameter space where the fitting fitting criterio criterion n is less less than than a giv given en val value. ue. This giv given en value must be larger than the minimum of the criterion in order to find a feasible region. Most linear MPC products allow the user to apply nonlin nonlinear ear transfo transforma rmatio tions ns to var variab iables les that that exhibi exhibitt sign signifi ifica cant nt nonl nonlin inea eari rity ty.. For For exam exampl ple, e, a loga logari rith thm m transf transform ormati ation on is often often perfor performed med on compos compositi ition on variables for distillation column control.
3.3.3. Nonlinear model identification The The most most diffi difficu cult lt issu issuee in nonl nonlin inea earr empi empiri rica call modeling is not the selection of a nonlinear form, be it polynomial or neural network, but rather the selection of a robust robust and reliab reliable le identi identifica ficatio tion n alg algori orithm thm.. For exampl example, e, the Aspen Aspen Target Target identi identifica ficatio tion n alg algori orithm thm discussed discussed in Zha Zhao o et al. (1998) (1998) builds builds one model model for each output separately. For a process having m y output variables, overall m y MISO sub-models are built. The
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following procedure is employed to identify each submodel from process data. 1. Spec Specif ify y a roug rough h time time cons consta tant nt for for each each inpu input– t– outp output ut pair pair,, then then a seri series es of first first orde orderr filte filters rs or a Lague aguerr rree model del is cons constr truc ucte ted d for for each each inpu inputt (Zh Zhao ao et al al., ., 19 1998 98;; Se Sent nton onii et al al., ., 19 1998 98). ). The The filte filterr stat states es for for all all inpu inputs ts compr compris isee the the stat statee vector x: 2. A static linear model is built for each output f y j ; j ¼ 1; 2; y; m y g using the state vector x as inputs using partial least squares (PLS). 3. Model Model reducti reduction on is then then perfor performed med on the inputinputstate–output model identified in Steps 1 and 2 using principal component analysis and internal balancing to eliminate eliminate highly collinear collinear state variables. 4. The The reduce reduced d model model is rearra rearrange nged d in a statestate-spa space ce model ðA; BÞ; whic which h is used used to gene genera rate te the the stat statee sequence fxk ; k ¼ 1; 2; y; K g: If the the model odel conconverges, i.e., no further reduction in model order, go to the next step; otherwise, return to step 2. 5. A PLS model is built between the state vector x and the output y j : The PLS model coefficients form the C matrix. 6. A neural neural network network model model is built built betwee between n the PLS late latent nt facto factors rs in the the prev previo ious us step step and and the the PLS PLS residu residual al of the output output y j : This This step generates generates the nonlinear static map g j ðxÞ: The use of the PLS latent factors instead of the state vectors is to improve the robustness robustness of the neural network training and reduce the size of the neural network. A nove novell feat featur uree of the the iden identi tific ficat atio ion n algo algori rith thm m is that the dynamic model is built with filters and and the the filte filterr stat states es are are used used to pred predic ictt the the outp output ut variab var iables les.. Due Due to the simpli simplisti sticc filter filter struct structure ure,, each each input variable has its own set of state variables, making the A matrix matrix block-diag block-diagonal. onal. This treatment treatment assumes assumes that each state variable is only affected by one input variable, i.e., the inputs are decoupled. For the typical case where input variables variables are coupled, the algorithm algorithm coul could d gene genera rate te stat statee vari variab able less that that are are li line near arly ly depend dependent ent or collin collinear ear.. In other other words, words, the result resulting ing stat statee vect vector or woul would d not not be a mini minima mall real realiz izat atio ion. n. Neverth Neverthele eless, ss, the use of a PLS algorith algorithm m makes makes the esti estima mati tion on of the the C matrix matrix well-c well-cond onditi itione oned. d. The iteration iteration between the estimation estimation of A; B and C matrices will will likel likely y elimi eliminate nate the initia initiall error error in estima estimatin ting g the process process time constants. Process nonlineari ritty is added to the the model with with conc concer ern n for for mode modell vali validi dity ty usin using g the the mode modell confide confidence nce index. index. When When the model is used used for extrapolation, only the linear portion of the model is used. The use of EKF for output error feedback in Aspen Target is interesting; the benefit of this treatment is yet to be demonstrated.
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3.4. MPC control control technology technology MPC controllers are designed to drive the process from one constrained steady state to another. They hey may rece receiv ivee optim ptimal al stea steady dy-s -sta tate te targ target etss from an overlying optimizer, as shown in Fig. 2, 2, or they may compute an economically optimal operating point using using an interna internall steady steady-st -state ate optimi optimizer zer.. The genera generall objectives of an MPC controller, in order of importance, are: 1. prevent violation of input and output constraints; 2. drive drive the CVs to their their steady steady-sta -state te optima optimall val values ues (dynamic output optimization); 3. drive drive the MVs MVs to their their steady steady-sta -state te optima optimall val values ues using remaining degrees of freedom (dynamic input optimization); 4. prevent excessive movement of MVs; 5. when signals and actuators fail, control as much of the plant as possible. The translation of these objectiv tives into a mathem mathemati atical cal proble problem m statem statement ent involv involves es a number number of appr approx oxim imat atio ions ns and and trad tradee-of offs fs that that defin definee the the basi basicc char charac acter ter of the the cont contro roll ller er.. Like Like any any desi design gn prob proble lem m ther theree are are many many possi possibl blee solut solutio ions ns;; it is no surp surpris risee that that ther theree are are a numb number er of diff differ erent ent MPC MPC control formulations. Tables 4 and 5 summarize summarize how each each of the the MPC vend vendor orss has has acco accomp mpli lish shed ed this this translation. Fig. 5 illust illustrate ratess the flow of a represe representa ntativ tivee MPC calculation at each control execution. The first step is to read read the the curr curren entt valu values es of proce process ss inpu inputs ts (DVs (DVs and and MVs) and process outputs (CVs). In addition to their numeric numerical al val values ues,, each each measur measureme ement nt carrie carriess with with it a sensor status to indicate whether the sensor is functioning properly or not. Each MV will also carry information on the status of the associated lower level control func functi tion on or valv valve; e; if satu satura rate ted d then then the the MV will will be perm permit itted ted to move move in one one dire direct ctio ion n only only.. If the the MV controller is disabled then the MV cannot be used for for cont contro roll but but can can be cons consiidere dered d a measu easure red d disturbance (DV). The remain remaining ing steps steps of the calcul calculati ation on essent essential ially ly answer answer three questions: questions: *
*
*
where is the process now, and where is it heading? (output feedback); where should the process go to at steady state? (local steady-state optimization); what is the best way to drive the process to where it needs to go? (dynamic optimization).
The fol following sec sections describe these and othe otherr aspe aspect ctss of the the MPC MPC calc calcul ulat atio ion n in grea greate terr detail.
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Read MV, DV, CV values from process
Output feedback (state estimation)
Determine controlled process subset
Remove ill-conditioning ill-conditioning
Curre Current ntly ly,, ther theree are are only only two two MPC MPC prod produc ucts ts that that exploit Kalman filter technology for output feedback. As disc discus usse sed d earl earlie ier, r, the the limi limitat tatio ions ns list listed ed abov abovee moti motiva vated ted the the orig origin inal al deve develo lopm pmen entt of the the SMOC SMOC algorithm algorithm (Marquis & Broustail, 1998). 1998). The currently currently sold SMOC product still retains a Kalman filter. The Aspen Target product includes an EKF for feedback, based based on a straig straightf htforw orward ard extens extension ion for nonlin nonlinear ear systems (Kailath, (Kailath, Sayed, & Hassibi, 2000). 2000). For stable processes, the remaining MPC algorithms included in our survey use the same form of feedback, base based d on comp compar arin ing g the the curre current nt meas measure ured d proc proces esss m output yk to the current predicted output yk : m bk ¼ yk À yk :
ð27Þ 27Þ
The bias bias bk term term is adde added d to the the mode modell for for use use in subsequent predictions: Local Steady-State Optimization
Dynamic Optimization
yk þ j ¼ gðxk þ j ; uk þ j Þ þ bk :
ð28Þ 28Þ
This form of feedback is equivalent to assuming that a step step dist distur urba bance nce ente enters rs at the the outp output ut and and remai remains ns constant for all future time (Morari (Morari & Lee, 1991; 1991; Lee, Morari, & Garcıa, 1994). 1994). Muske and Rawlings (1993) analyzed this assumption in the context of the Kalman filter 4; the corresponding filter gain is !
K f ¼ ½0 IT
Output MV's to process Fig. 5. 5. Flow of MPC calculation at each control execution.
3.4.1 3.4.1.. Output Output feedback feedback (state estimation estimation) ) In the output feedback step, the controller makes use of avai availa labl blee meas measur urem emen ents ts in orde orderr to esti estima mate te the the dynamic state of the system. It is at this point in the calculation where the failure to embrace LQG concepts has had the most detrimental impact on industrial MPC technology. technology. Instead of separating separating the overall overall problem problem into its two natural components, state estimation and state state contro control, l, most most industr industrial ial MPC products products do not incorporate the concept of a process state at all, and rely instea instead d upon upon ad-hoc ad-hoc biasin biasing g scheme schemess to incorp incorpora orate te feedback. This has several implications: *
*
*
*
*
Additional process output measurements (non-CVs) that could improve the accuracy of state estimation are not easily incorporated into the control structure; Addi Additi tion onal al effo effort rt is requ requir ired ed to cont contro roll li line near ar combinations of process states (average bed temperature, for example), or to control unmeasured outputs; Unmeasured disturbance model options are severely limited; Ad hoc fixes must be introduced to remove steadystate offset for integrating and unstable systems; Meas Measur urem emen entt nois noisee must must be addr addres essed sed in a subsuboptimal manner.
which means no feedback for the process state estimates and identity feedback for the output disturbance. Muske and Rawlings (1993) show that a wide variety of other other distur disturban bance ce models models is possib possible. le. In partic particula ularr they show that the constant output disturbance model leads to sluggish rejection of disturbances entering at the proces processs input, input, a point point also also made made by Shinsk Shinskey ey in his criticism of MPC (Shinskey, (Shinskey, 1994). 1994). This problem can be addres addressed sed direct directly ly by buildi building ng an input input distur disturban bance ce model, similar to the internal model principle. In the context of the discrete-time linear model 10, a general disturbance model can be written as: xk þ1 ¼ Axk þ Bu uk þ Bv vk þ Bw wk þ Bs sk ;
ð29a 29aÞÞ
sk þ1 ¼ sk ;
ð29bÞ 29bÞ
pk þ1 ¼ pk ;
ð29cÞ 29cÞ
yk ¼ Cxk þ Duk þ nk þ C p pk ;
ð29dÞ 29dÞ
Here sk ARms represents represents a constant constant state disturbance, disturbance, m p and pk AR represents a constant output disturbance. The control designer does not have complete freedom in specif specifyin ying g the number number and locati location on of distur disturban bances ces,, howe howeve ver, r, beca becaus usee the the dist distur urba banc ncee stat states es must must be observable. observable. In practice practice this means that the total number number of disturbances cannot exceed the number of measured outputs ðms þ m p pm y Þ; and and that that for for a give given n proc proces esss model the disturbance is not allowed to enter in certain state, state, or output output directi directions ons (Musk Muske, e, 1995 1995). ). Desig Design n of
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sensible disturbance models for MPC applications is a ripe ripe area area for future future academ academic ic resear research, ch, especi especiall ally y for nonlinear systems. For stable processes, processes, the constant constant output output disturbance disturbance mode modell prov provid ides es inte integr gral al acti action on to the the cont contro roll ller er,, effectively effectively removing steady-state steady-state offset due to disturdisturbances and model mismatch (Rawlings (Rawlings et al., 1994). 1994). For integrating integrating and unstable unstable processes, processes, however, the constant stant outp output ut dist distur urba banc ncee mode modell fail failss beca becaus usee the the estimat estimator or contai contains ns the unstab unstable le poles poles of the proces processs (Muske & Rawlings, 1993). 1993). This can be addressed easily through the use of input or state disturbance models, but most industrial MPC are forced to address the issue usin using g ad hoc hoc fixes, fixes, si sinc ncee they they do not not inco incorp rpor orat atee a Kalma Kalman n filter. filter. The approa approach ch used used in the DMC-p DMC-plus lus product is typical, where it is assumed that a constant integrating disturbance has entered the process, and a rotation rotation factor is used used to desc descri ribe be the the frac fracti tion on of predic predictio tion n error error due to the consta constant nt integr integrati ating ng disdisturbance, turbance, relative relative to the constant constant output output disturbance disturbance.. For a specifi specificc output output var variab iable le that that is an integra integratin ting g mode, mode, the corres correspon pondin ding g distur disturban bance ce model model can be represented as: xk þ1 ¼ xk þ bu uk þ bv vk þ bw wk þ a pk ;
ð30a 30aÞÞ
pk þ1 ¼ pk ;
ð30bÞ 30bÞ
yk ¼ cxk þ duk þ xk þ pk ;
ð30cÞ 30cÞ
where a is the the rota rotati tion on fact factor or.. In this this case case it can be shown that the disturbance pk is observable only if the rotation factor a is nonzero. While this scheme provides offset-free control for integrating systems, choice of an appropriate rotation factor is difficult in practice, and control of noisy integrating systems is often poor. The The MPC MPC prod produc ucts ts incl includ udee feat featur ures es to addr addres esss a variety variety of feedback feedback scenarios encountered encountered in applications. In some cases the CV measurement may not be available at each control execution; this may happen, for example, when the CV measurement is provided by an analyzer. In this case a typical solution is to skip the bias upda update te for for the the affe affect cted ed CV for for a numb number er of cont contro roll intervals. A counter is provided to disable control of the CV if too many executions go by without feedback. 3.4.2 3.4.2.. Determinin Determining g the controlled sub-process sub-process Afte Afterr the the proc proces esss stat statee has has been been esti estima mate ted, d, the the controller must determine which MVs can be manipulated and which CVs should be controlled. In general, if the the meas measur urem emen entt status status for a CV is good good,, and and the the operator has enabled control of the CV, then it should be controlled. An MV must meet the same criteria to be used for control; in addition, however, the lower level control functions must also be available for manipulation. If the lower level controller is saturated high or low, one can add a temporary hard MV constraint to the
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prob proble lem m to prev preven entt movi moving ng the the MV in the the wron wrong g direction. If the lower level control function is disabled, the MV cannot be used for control. In this case it should be treated as a DV. From these decisions a controlled subp subpro roce cess ss is defin defined ed at each each contr control ol execu executi tion on.. In general the shape of the subprocess changes in real-time as illustrated in Fig. 3. 3. In applications, it is typical to include the low level cont contro roll outp output utss (e.g (e.g., ., valv valvee posi positi tion ons) s) in the the MPC MPC control formulation as additional CVs. These CVs are then forced to stay within high and low saturation limits by treating them as range or zone control variables. This serves serves to keep keep the lower lower level level contro controlle llers rs away away from from valve constraints that would otherwise compromise their performance. In most MPC products, sensor faults are limited to complete failure that goes beyond pre-specified control limits. limits. Sensor faults faults such as significant significant bias and drifting drifting that are within normal limits are generally not detected or identified in these products. The DMC-plus, Connoisseur, and RMPCT algorithms distingu distinguish ish between between a critical CV fail failur uree and and a noncritical CV failure. If a noncritical CV fails, the RMPCT and DMC-pl DMC-plus us algorith algorithms ms continu continuee control control action action by setting the failed CV measurement measurement to the model predicted value, which means there is no feedback for the failed CV. If the noncritical CV fails for a specified period of time, RMPCT drops this CV from the control objective func functi tion on.. If a crit critic ical al CV fail fails, s, the the DMCDMC-pl plus us and and RMPCT RMPCT controllers controllers turn-off immediately. immediately.
3.4.3. Removal 3.4.3. Removal of ill-condit ill-conditionin ioning g At any any part partic icul ular ar cont contro roll exec execut utio ion, n, the the proc proces esss encoun encounter tered ed by the contro controll ller er may requir requiree excessi excessive ve inpu inputt move moveme ment nt in orde orderr to cont contro roll the the outp output utss independently. This problem may arise, for example, if two controlled outputs respond in an almost identical way to the ava availa ilable ble inputs. inputs. Consider Consider how difficu difficult lt it woul would d be to inde indepe pend nden entl tly y cont contro roll adja adjace cent nt tray tray temper temperatu atures res in a distil distillat lation ion colum column, n, or to contro controll both regenerator and cyclone temperature in an FCCU. It is important to note that this is a feature of theprocess the process to be cont contro roll lled ed;; any any algo algori rith thm m whic which h atte attemp mpts ts to contro controll an ill-c ill-cond onditi itione oned d proces processs must must addres addresss this this problem. A high condition number in the process gain matrix means that small changes in controller error will lead to large MV moves. Although the conditioning of the full control problem will almost certainly be checked at the design phase, it is nearly nearly imposs impossibl iblee to check check all possib possible le sub-pr sub-proces ocesses ses which may be encountered during future operation. It is therefore important to examine the condition number of the controlled sub-process at each control execution and to remove remove ill-co ill-condi nditio tionin ning g in the model model if necess necessary. ary. Two strategies are currently used by MPC controllers to
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accomplish accomplish this task; singular value thresholdin thresholding, g, and input move suppression. The singular value thresholding (SVT) method used by the RM RMPCT PCT contro controlle llerr involv involves es decom decompos posing ing the proces processs model model based based on singul singular ar val values ues using using URV URV deco decomp mpos osit itio ion. n. Sing Singul ular ar valu values es belo below w a thre thresho shold ld magnitude are discarded, and a process model with a lower condition number is then reassembled and used for contro control. l. The neglec neglected ted singul singular ar val values ues repres represent ent directions along which the process hardly moves even if a large MV change is applied; the SVT method gives up thes thesee dire directi ction onss to avoi avoid d erra errati ticc MV chan change ges. s. This This meth method od solv solves es the the illill-co cond ndit itio ioni ning ng prob proble lem m at the the expens expensee of neglec neglectin ting g the smalle smallest st singul singular ar val values. ues. If the magnitude of the neglected singular values is small compared to the model uncertainty, it may be better to neglect them anyway. After SVT, the collinear CVs are approximated with the principal singular direction. In the case of two collinear CVs, for example, this principal direction is a weighted average of the two CVs. Note that the SVT approach is sensitive to output weighting. If one CV is weighted much more heavily than another, this CV will represent the principal singular direction. Controllers that use input move suppression (IMS), such as the DMC-plus algorithm, provide an alternative strategy strategy for dealing dealing with ill-conditioni ill-conditioning. ng. Input move supp suppre ress ssio ion n fact factor orss incr increa ease se the the magn magnit itud udee of the the diagonal elements of the matrix to be inverted in the least-square least-squaress solution, solution, directly directly lowering lowering the condition condition number. The move suppression values can be adjusted to the point that erratic input movement is avoided for the commonly encountered sub-processes. In the case of two collinear CVs, the move suppression approach gives up a little bit on moving each CV towards its target. The move suppression solution is similar to that of the SVT solution in the sense that it tends to minimize the norm of the the MV moves oves.. In the the limit imit of infin nfinite ite move suppression the condition number becomes one for all sub-processes. It would be useful to find a set of finite move suppression factors which guarantee that all subproc proces esses ses have have a cond condit itio ion n numb number er grea greater ter than than a desired threshold value; the authors are not aware of any academic work that addresses this question. 3.4.4 3.4.4.. Local steady-state steady-state optimizati optimization on Almost all of the MPC products we surveyed perform a sepa separa rate te loca locall stea steady dy-s -sta tate te opti optimi miza zati tion on at each each contro controll cycle cycle to comput computee steady-s steady-stat tatee input, input, state, state, or output output targets targets.. This This is necessa necessary ry becaus becausee the optima optimall targets may change at any time step due to disturbances entering the loop or operator inputs that redefine the control problem. The optimization problem is typically form formul ulat ated ed so as to driv drivee stea steady dy-s -stat tatee inpu inputs ts and and outputs as closely as possible to targets determined by the local economic optimization (just above the MPC algorithm in Fig. 2), 2), without violating input and output
constra constraint ints. s. Consta Constant nt disturb disturbanc ances es estim estimated ated in the output output feedbac feedback k step step appear appear explic explicitl itly y in the steady steady-state optimization so that they can be removed. Rao and Rawlings Rawlings describe a representativ representativee formulatio formulation n of the problem (Rao (Rao & Rawlings, 1999). 1999). The local steady-state optimization uses a steady-state model which may come from linearizing linearizing a comprehensi comprehensive ve nonlinear nonlinear model at each control execution or may simply be the steady-state vers versio ion n of the the dyna dynami micc mode modell used used in the the dyna dynami micc optimization. The Connoi Connoisse sseur ur contro controlle llerr uses uses a Linear Linear Progra Program m (LP) (LP) to do the local local steady steady-sta -state te optim optimiza izatio tion. n. The dist distin ingui guish shin ing g feat featur uree of an LP solu soluti tion on is that that the the opti optima mall stead steady-s y-sta tate te targ target etss will will lie lie at the the verte vertex x of the constr constrain aintt bounda boundary. ry. If the constra constraint int bounda boundary ry changes changes frequently due to model mismatch or noise, noise, the opti optima mall stea steady dy-s -stat tatee solu soluti tion on may may boun bounce ce arou around nd unnecessari unnecessarily, ly, leading to poor overall control perforperformance. Typical solutions to this problem include heavily filtering the output signals and detuning the optimizer by addi adding ng a term term that that mini minimi mize zess move moveme ment nt of the the solution. These solutions slow down the movement of the steady steady-st -state ate target target but also also hamper hamper reject rejection ion of process disturbances. An alternative solution based on direct direct incorp incorpora oratio tion n of model model uncert uncertain ainty ty has been been proposed, but has not yet been implemented commercially (Kassmann, (Kassmann, Badgwell, & Hawkins, 2000). 2000). The RMPCT, PFC, Aspen Target, MVC, and process perfecter algorithms use Quadratic Programs (QP) for the steady-state target calculation. The QP solution does not necess necessari arily ly lie at the constra constraint int boundary boundary,, so the optimal steady-state targets tend not to bounce around as much as for an LP solution. The DMC-plus algorithm solves the local steady-state optimization problem using a sequence of LPs and/or QPs. QP s. CVs CVs are are rank ranked ed by prio priori rity ty such such that that cont contro roll performance of a given CV will never be sacrificed in order to improve performance of a lower priority CV. The prediction error can be spread across a set of CVs by grouping them together at the same priority level. The calcul calculati ation on procee proceeds ds by optimi optimizin zing g the highes highestt prio priori rity ty CVs CVs first first,, subj subjec ectt to hard hard and and soft soft outp output ut cons constr trai aint ntss on the the same same CVs CVs and and all all inpu inputt hard hard constra constraint ints. s. Subseq Subsequen uentt optimi optimizat zation ionss preserv preservee the future trajectory of high priority CVs through the use of equality constraints. Likewise inputs can be ranked in priori priority ty order order so that that inputs inputs are moved moved sequen sequentia tially lly toward towardss their their optima optimall val values ues when when extra extra degree degreess of freedom permit. All of the products we surveyed enforce hard input constra constraint intss in the steady steady-st -state ate optimi optimizat zation ion.. One can always always find steady steady-sta -state te target targetss that that are feasib feasible le with with resp respec ectt to the the inpu inputt cons constr trai aint nts, s, prov provid ided ed they they are are defined in a sensible way. The same is not true, however, for for outp output ut cons constr trai aint nts. s. This This is beca becaus usee if a larg largee disturbance enters the process, it may not be possible,
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giv given the the avai availlabl able input nput spac space, e, to com complet pletel ely y remove the disturbance at steady-state. For this reason, the DMC-p DMC-plus lus,, SMOC, SMOC, MVC, MVC, and Process Process Perfec Perfecter ter prod produc ucts ts prov provid idee for for soft soft outp output ut cons constr trai aint ntss in the the steady-state optimization. These allow for some violation of an output constraint, but the magnitude of the violat violation ion is minim minimize ized d in the object objective ive functi function. on. The rema remain inin ing g prod produc ucts ts may may be subj subjec ectt to stea steady dy-s -sta tate te infeas infeasibi ibili lity ty proble problems ms unless unless other other precau precautio tions ns are taken. Similar infeasibility problems may arise due to failure to zero out integrating modes of the process at steady state. The DMC-plus algorithm, for example, includes an explicit constraint that forces integrating (ramp) CV variables to line out at steady state. If this constraint is violated, due perhaps to a large disturbance entering the loop loop,, the the DMCDMC-pl plus us prod produc uctt will will shut shut off off with with a warning warning message. message. Other products most likely likely address address this issue in a similar way. An alternative solution is to penalize the violation using a soft constraint, so that the controller will try its best to line out the integrating CVs at steady state without shutting off. If the disturbance diminishes, or the operator provides more room with the input constraints, it is possible that the next steady-state calculatio calculation n will be feasible. feasible. The The NOVA NOVA-N -NLC LC contr control olle lerr does does not not perf perfor orm m a separate separate steady-state steady-state optimizati optimization; on; the steady-state steady-state and dynamic dynamic optimizati optimizations ons are performed performed simultaneou simultaneously. sly. This This may may impro improve ve cont contro roll ller er perf perform orman ance ce as long long as the the stea steady dy-s -sta tate te and and dyna dynami micc obje objecti ctive vess do not not confl conflic ict. t. For For exam exampl ple, e, with with sepa separa rate te obje object ctiv ives es it is possible that a steady-state target may be computed that cannot be achieved, due to constraints that appear only only in the the dyna dynami micc opti optimi miza zati tion on.. Comb Combin inin ing g the the objectives leads to a more complex numerical problem, however. 3.4.5. Dynamic optimization optimization At the dynamic optimization level, an MPC controller must compute a set of MV adjustments that will drive the the proc proces esss to the the desi desire red d stea steady dy-s -sta tate te oper operat atin ing g poi point witho ithout ut viol violat atiing cons constr tra aints ints.. All of the the MPC products we surveyed can be described (approximately) as minimizing the following dynamic objective function: M
J ðu Þ ¼
P X
y
q
q
fjjek þ j jjQ j þ jjs j jjT g
j ¼1 M M À1
þ
X
q
q
u fjjek þ j jjR j þ jj Duk þ j jjS j g
ð31a 31aÞÞ
j ¼0
xk þ j ¼ f ðxk þ j À1 ; uk þ j À1 Þ yk þ j ¼ gðxk þ j ; uk þ j Þ
8 j ¼ 1; P ;
8 j ¼ 1; P
and subject to inequality constraints: y À s j pyk þ j py þ s j %
s j X0
8 j ¼ 1; P ;
upuk þ j pu %
8 j ¼ 0; M À 1;
%
DupDuk þ j pDu %
ð31cÞ 31cÞ
8 j ¼ 0; M À 1:
%
The objecti objective ve functi function on in Eq. (31a) (31a) involv involves es four four conflic conflictin ting g contri contribut bution ions. s. Future Future output output behavi behavior or is contro controlle lled d by penali penalizin zing g deviat deviation ionss from from the desire desired d y r r ; y e y y output output trajectory trajectory k þ j defined defined as k þ j k þ j À k þ j ; over a prediction horizon of length P : Output constraint violations are penalized by minimizing the size of output constraint constraint slack varia variables bles s j : Future input deviations deviations from from the desire desired d steadysteady-stat statee input input us are controlled controlled u u us ; over a using input penalties defined as ek À k þ j þ j control horizon of length M : Rapid input changes are penali penalized zed with with a separa separate te term term involv involving ing the moves moves Duk þ j : The size of the deviations is measured by a vector norm, usually either an L1 or L2 norm ðq ¼ 1; 2Þ: The relative importance of the objective function contributions is controlled by setting the time dependent weight matrices Q j ; T j ; S j ; and R j ; thes thesee are are chos chosen en to be positive semi-definite. The solution is a set of M input adjustments: uM ¼ ðuk ; uk þ1 ; y; uk þM À1 Þ:
ð32Þ 32Þ
Most of the MPC controllers use a quadratic objective func functi tion on simi simila larr to 31a 31a ðq ¼ 2Þ for for the the dyna dynami micc optimizati optimization. on. For this case the dynamic dynamic optimizati optimization on takes the form of a QP, and can be solved reliably using standard software. However, for very large problems, or very very fast fast proces processe ses, s, there there may may not not be suffic sufficie ient nt time time available to solve a QP. For these reasons the DMC-plus and PFC algorithms use fast sub-optimal algorithms to generate approximate solutions to the dynamic optimization. In the DMC-plus algorithm, when an input is predicted to violate a maximum or minimum limit it is set equal to the limit and the calculation is repeated with the MV remove removed. d. The PFC algorith algorithm m perfor performs ms the calculation without constraints and then clips the input values val ues if they they exceed exceed hard hard constr constrain aints. ts. Both Both of these these techni technique quess will will preven preventt violat violation ion of hard hard input input conconstrain straints ts but will, will, in genera general, l, involv involvee a loss loss of perfor perfor-mance mance that is difficult difficult to predict. predict. Typically Typically the resulting perfor performan mance ce is accept acceptabl able, e, but the soluti solutions ons do not satisfy satisfy the Karush–Ku Karush–Kuhn–Tu hn–Tucker cker necessary necessary conditions conditions for optimality. The DMC-plus and SMOC algorithms penalize only the last input input deviat deviation ion in order order to drive drive the system towards the optimal steady state: RM À1 b0:
ð31bÞ 31bÞ
8 j ¼ 1; P ;
%
R j ¼ 0
subject to a model constraint:
753
8 j oM À 1; ð33Þ 33Þ
If the final input weight is large enough and the process is stable, this is approximately equivalent to having a
754
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terminal state constraint. If the dynamic solution at the end of the horizo horizon n is signifi significan cantly tly differ different ent from from the steady-state targets, which means the terminal states are not effectively effectively constrained, constrained, the DMC-plus DMC-plus controller controller will will autom automati atical cally ly turn turn off. off. This This settin setting g effecti effectivel vely y provid provides es nomina nominall stabil stability ity for DMC-p DMC-plus lus contro controlle ller. r. The final input weights are also applicable to integrating processes where the derivative of the integrator is driven to zero. The The RM RMPC PCT, T, HIEC HIECON ON,, PFC, PFC, and and SMOC SMOC algo algo-rithms do not penalize input moves directly (S j =0). The HIECON and PFC algorithms use a predefined output reference reference trajectory trajectory to avoid aggressive aggressive MV moves. The RMPC RM PCT T contro controll ller er defin defines es a funn funnel el,, whic which h will will be desc descri ribe bed d late laterr in the the pape paper, r, and and finds finds the the opti optima mall trajectory and optimal MV moves by minimizing: ðuM ; yr Þ ¼ arg min
P X
2 r jjyk þ j À yk þ j jjQ
j ¼1
þ jj uk þM À1 À us jj2R
ð34Þ 34Þ
subject to the funnel constraints. The relative priority of the two terms is set by the two weighting matrices. In the case that the first term is completely satisfied, which is typical due to the funnel formulation, the CV error will vanish and the minimization is in fact performed on the second term only. In this case the results will be similar to having two separate objectives on CVs and MVs. In the case of an infinite number of solutions, which is also typica typicall due to ‘‘rel ‘‘relaxi axing’ ng’’’ the trajec trajectory tory,, a minim minimum um norm solution to the MVs is obtained due to the use of singular value thresholding. Usin Using g a si sing ngle le obje objecti ctive ve func functio tion n in the the dyna dynami micc optim optimiza izatio tion n means means that that tradetrade-off offss betwee between n the four four contribution contributionss must be resolved resolved using the relative relative weight weight matrices Q j ; T j ; S j ; and R j : The HIECON HIECON algorithm algorithm reso resolv lves es confl conflic icti ting ng dyna dynami micc cont contro roll obje objecti ctive vess by solving solving separate separate CV and MV optimization optimization problems. problems. The decision is made, a priori , that CV errors are more important than MV errors. A quadratic output optimization problem is solved first, similar to 31 but including y only the ek þ j terms. For the thin and square plant cases this will provide a unique solution and the calculation terminates. For the fat plant case there are remaining degree degreess of freedo freedom m that that can be used to optimi optimize ze the inpu inputt sett settin ings gs.. For For this this case case the the cont contro roll ller er solv solves es a separate quadratic input optimization problem, similar u to 31 but but incl includ udin ing g only only the the ek terms. The input input þ j terms. optimization includes a set of equality constraints that pres preserv ervee the the futu future re outp output ut traje traject ctor orie iess foun found d in the the output output optimi optimizat zation ion.. This This elimin eliminate atess the need need to set weights to determine the trade-off between output and input errors, at the cost of additional computation. The PFC controller includes only the process input and output output terms terms in the dynami dynamicc objecti objective, ve, and uses uses constant weight matrices ðQ j ¼ Q; T j ¼ 0; R j ¼ R; S j ¼
0; q ¼ 2). The Aspen Target and MVC products include all four terms with constant weights (Q j ¼ Q; T j ¼ T; R j ¼ R; S j ¼ S; q ¼ 2). The NOVA-NLC product adds to this the option of one norms (Q j ¼ Q; T j ¼ T; R j ¼ R; S j ¼ S; q ¼ 1; 2). Instead Instead of using using a referen reference ce trajec trajector tory, y, the Proces Processs Perfecter product (T j ¼ T; R j ¼ 0; S j ¼ 0; q ¼ 2) uses a dynamic objective with trajectory weighting that makes Q j gradually increase over the horizon P : With this type of weig weight htin ing, g, cont contro roll erro errors rs at the the begi beginn nnin ing g of the the horizon are less important than those towards the end of the horizon, thus allowing a smoother control action.
3.4.6. Constraint formulations formulations There are three types of constraints commonly used in indust industria riall MPC MPC techno technolog logy; y; hard, hard, soft, soft, and setpoi setpoint nt approx approxima imatio tion. n. These These are illustr illustrate ated d in Fig. 6. 6. Hard Hard constraints, shown in the top of Fig. of Fig. 6, 6, are those which should never be violated. Soft constraints, shown in the middle of Fig. of Fig. 6 are those for which some violation may be allowed; the violation is typically minimized using a quadratic penalty in the objective function.
Hard Constraint past
future
quadratic penalty
Soft Constraint past
future
quadratic penalty Setpoint approximation of Soft Constraint past
future
Fig. 6. The three basic basic types types of constrai constraint; nt; hard, hard, soft and setpoint setpoint approximation. Hard constraints (top) should not be violated in the future. Soft constraints (middle) may be violated in the future, but the violation is penalized in the objective function. Setpoint approximation of constrai constraint nt (bottom) (bottom) penalize penalizess deviatio deviations ns above above and below below the constrai constraint. nt. Shades Shades areas areas show violati violations ons penalize penalized d in the dynamic dynamic optimization. Adapted from Froisy (1994). (1994).
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
Another way to handle soft constraints is to use a setpoint approximation, as illustrated at the bottom of Fig. 6. 6. Setpoi Setpoints nts are defined defined for each each soft soft constra constraint int,, resulting in objective function penalties on both sides of the constraint. The output weight is adjusted dynamically, however, so that the weight becomes significant only when the CV comes close to the constraint. When a violation is predicted the weight is increased to a large value so that the control can bring the CV back to its cons constr trai aint nt limi limit. t. As soon soon as the the CV is with within in the the constra constraint int limit, limit, the steady steady-st -state ate target target is used used as the setpoint instead. All of the MPC algorithms allow hard MV maximum, minimum, and rate of change constraints to be defined. These are generally defined so as to keep the lower level MV controllers in a controllable range, and to prevent viol violen entt move moveme ment nt of the the MVs MVs at any any si sing ngle le cont contro roll exec execut utiion. on. The PFC PFC alg algorit orithm hm also also acco accom mmodates dates maximu maximum m and minimu minimum m MV accele accelerat ration ion conconstraint straintss which which are useful useful in mechan mechanica icall servo servo contro controll applications. With the exception of the DMC-plus algorithm, all of the MPC products enforce soft output constraints in the dynam dynamic ic optimi optimizat zation ion.. Hard Hard output output constra constraint intss are prov provid ided ed as opti option onss in the the PFC, PFC, HIEC HIECON ON,, and and NOVA-NLC algorithms, but these must be used carefully, fully, because because enforcement enforcement of hard output constraints constraints can lead lead to closed closed-lo -loop op instab instabili ility ty for nonmin nonminim imum um phase phase proces processes ses (Zafiri Zafiriou, ou, 1990 1990;; Muske & Rawlings, 1993). 1993 ). Hard output constraints can also cause feasibility proble problems, ms, especi especiall ally y if a large large disturb disturbanc ancee enters enters the proces process. s. In the PFC and HIECON HIECON algorith algorithms, ms, hard hard output constraints are ranked in order of priority so that low low prio priori rity ty cons constr trai aint ntss can can be drop droppe ped d when when the the problem becomes infeasible. The Connoisseur algorithm provides constraint optimization to match the number of acti active ve cons constra train ints ts with with the the numb number er of degr degree eess of freedom available to the controller, i.e., the number of unconstrained MVs. 3.4.7 3.4.7.. Output Output and input trajectories trajectories Industrial MPC controllers use four basic options to specify future CV behavior; a setpoint, zone, reference trajectory or funnel. These are illustrated in Fig. 7. 7. The y u shaded areas correspond to the ek þ j and ek terms in þ j 31a. All of the controllers provide the option to drive the CVs to a fixed setpoint, with deviations on both sides penalized in the objective function. In practice this type of specification is very aggressive and may lead to very large input adjustments, unless the controller is detuned in some fashion. This is particularly important when the intern internal al model model differ differss signifi significan cantly tly from from the proces process. s. Several of the MPC algorithms use move suppression factors for this purpose. All of the controllers also provide a CV zone control option, designed to keep the CV within a zone defined
755
quadratic penalty
past
Setpoint
future
quadratic penalty
past
Zone
future
Reference Trajectory quadratic penalty past
future
Funnel quadratic penalty past
future
Fig. 7. Four options for specifying future future CV behavior; setpoint, zone, refer referenc encee trajec trajector tory y and funnel funnel.. Shaded Shaded areas areas show show viola violatio tions ns penalized in the dynamic optimization.
by upper and lower boundaries. One way to implement zon zone con contro trol is to defi define uppe upperr and and lower ower soft soft constraints constraints.. Other implement implementations ations are possible, possible, however. The DMC-plus algorithm, for example, uses the setpoint approximation of soft constraints to implement the upper and lower zone boundaries. The HIECON, PFC, SMOC, Aspen Target, Target, MVC, MVC, and and NOVA NOVA-N -NLC LC algo algori rith thms ms prov provid idee an opti option on to spec specif ify y a desi desired red futu future re path path for for each each CV call called ed a reference reference trajectory trajectory.. A first first or seco second nd orde orderr curv curvee is drawn from the current CV value to the setpoint, with the speed of the response determined by one or more trajectory trajectory time constants. constants. Future CV deviations deviations from the reference trajectory are penalized. In the limit of zero time constants the reference trajectory reverts back to a pure pure setpoi setpoint; nt; for this this case, case, however however,, the contro controlle llerr would be sensitive to model mismatch unless some other strategy such as move suppression is also being used. A drawback drawback of the reference trajectory formulation formulation is that
756
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
it penal penaliz izes es the the outp output ut when when it happ happen enss to drif driftt too too quic quickl kly y towa toward rdss the the setp setpoi oint nt,, as migh mightt happ happen en in resp respon onse se to an unme unmeas asur ured ed dist distur urba bance nce.. If the the CV moves too quickly due to model mismatch, however, the reference trajectory is beneficial in that it will slow down the CV and minimize overshoot. The reference trajectory can be interpreted mathematically as a filter in the feedbac feedback k path, path, simila similarr to the robust robustnes nesss filter filter recomrecommended by IMC theory (Morari (Morari & Zafiriou, 1989). 1989). In gene genera ral, l, as the the refe refere renc ncee traj trajec ecto tory ry time time consta constant ntss increase, the controller is able to tolerate larger model mismatch. The NOVA-NLC reference trajectory is slightly more general in that it allows separate specification of upper and and lowe lowerr traje trajecto ctori ries es with with diff differe erent nt dyna dynami mics cs and and setpoints. setpoints. In the controller controller objective, only deviations deviations above above the upper upper trajec trajectory tory and deviat deviation ionss below below the lower trajectory are penalized. penalized. This provides provides additional additional freedom freedom during during the transi transient ent that that the contro controlle llerr can utilize for other tasks. The RMPCT algorith algorithm m attemp attempts ts to keep keep each each CV within within a user user defined defined zone, zone, with with setpoi setpoints nts defined by setting the maximum and minimum zone limits equal to each each other. other. When the CV goes outside outside the zone, the RMPCT algorithm defines a CV funnel, shown at the bottom of Fig. of Fig. 7, 7, to bring the CV back within its range. The slope of the funnel is determined by a user defined performance ratio, defined as the desired time to return to the zone divided by the open-loop response time. A weighted average open-loop response time is used for multivariable systems. The SMOC algorithm uses a variation of funnel when a zone CV falls out of its range. In this case a zone with reference trajectory bounds is defined to drive the CV back into the control zone. The The NOVA NOVA-N -NLC LC uppe upperr and and lowe lowerr traje traject ctor orie ies, s, RMPCT funnel, and SMOC reference trajectory bounds are advant advantage ageous ous in that that if a disturb disturbanc ancee causes causes the predicted future CV to reach the setpoint more quickly
DV
than than a referen reference ce trajec trajectory tory allows allows,, the contro controll ller er will will take take no actio action. n. This This effec effecti tive vely ly frees frees up degr degree eess of freedom for the controller to achieve other objectives. In the same same situat situation ion,, other other contro controlle llers rs would would move move the inputs to bring the CV back onto the defined trajectory. This is illustrated in Fig. 8 for the case of the RMPCT funnel. All of the MPC algorithms surveyed here provide MV setp setpoi oint ntss to driv drivee the the inpu inputs ts towa toward rdss thei theirr opti optima mall values when there are sufficient degrees of freedom. 3.4.8. Output 3.4.8. Output horizon horizon and input parameterization parameterization Industrial MPC controllers generally evaluate future CV behavior over a finite set of future time intervals called the prediction horizon. horizon. This is illustrated at the top of Fig. 9. 9. The finite output horizon formulation is used
Finite Horizon
past
future
Coincidence Points past
future
Fig. 9. Output horizon options. options. Finite horizon (top) includes P future points. points. A subset subset of the predicti prediction on horizon, horizon, called the coincide coincidence nce points points (bottom) (bottom) may also be used. used. Shaded Shaded areas areas show violations violations penalized in the dynamic optimization.
DV
SP
SP B
B
A
A
CV
(a)
prediction horizon P
Funnel
CV
(b)
Fig. 8. 8. MPC based on a funnel allows a CV to move back to the setpoint faster than a trajectory would require if a pulse disturbance releases. releases. A trajectory based MPC would try to move away from the setpoint to follow the trajectory.
S.J. Qin, T.A. Badgwell / Control Engineering Practice 11 (2003) 733–764
by all of the algorith algorithms ms discus discussed sed in this this paper. paper. The length of the horizon P is a basic tuning parameter for these these contro controlle llers, rs, and is genera generally lly set long long enough enough to capture the steady-state effects of all computed future MV moves. Most Most of the MPC contro controlle llers rs use a multi multiple ple point output output horizo horizon; n; this this means means that that predic predicted ted object objective ive function contributions are evaluated at each point in the future. The PFC, Process Perfecter, and Aspen Target controllers allow the option to simplify the calculation by considering only a subset of points in the prediction horizo horizon n called called coincidence coincidence points. points. This his con concept cept is illust illustrat rated ed at the bottom bottom of Fig. 9. 9. A separate set of coincidence points can be defined for each output, which is useful when one output responds quickly relative to another. The full finite output horizon can be selected as a special case. Industrial MPC controllers use three different methods to parameterize parameterize the MV profile; these are illustrate illustrated d in Fig. 10. 10. Most Most of the the MPC MPC algo algori rith thms ms comp compute ute a sequ sequen ence ce of futu future re move movess to be spre spread ad over over a finite finite control control horizon horizon,, as shown at the top of Fig. 10. 10. The length of the control horizon M is another basic tuning parame parameter ter for these these contro controlle llers. rs. The most most commo common n
u Multiple Moves
past
future
757
parameterization, referred to as multiple moves, means that a separate input adjustment is computed for each time time poin pointt on the the cont contro roll hori horizo zon. n. Perf Perfor orma manc ncee improves as M increases, at the expense of additional comput computati ation. on. To simpli simplify fy the calcul calculati ation, on, the Aspen Aspen Target, Connoisseur, and RMPCT algorithms provide a move move bloc blocki king ng opti option on,, allo allowi wing ng the the user user to spec specify ify points on the control horizon where moves will not be computed. This reduces the dimension of the resulting optimi optimizat zation ion proble problem m at the possib possible le cost cost of contro controll performance. The HIECON and MVC algorithms compute a single future input move, as shown in the middle of Fig. 10. 10. This This greatl greatly y simpli simplifies fies the calcul calculati ation on for these these alg algoorithms rithms,, which which is helpfu helpfull for the HIECON HIECON algorith algorithm m where two separate dynamic optimizations are solved at each control execution. However, using a single move involves a sacrifice of closed-loop performance that may be difficult to quantify. The The PFC contro controll ller er parame parameter terize izess the future future input input profi profile le usin using g a set set of poly polyno nomi mial al basi basiss func functio tions ns.. A possible solution is illustrated at the bottom of Fig. of Fig. 10. 10. This This allows allows a relati relativel vely y comple complex x input input profile profile to be spec specifi ified ed over over a larg largee (pot (poten enti tial ally ly infini infinite te)) cont contro roll horizon, using a small number of unknown parameters. This This may may prov provid idee an adva advant ntag agee when when cont contro roll llin ing g nonlinear systems. Choosing the family of basis functions establishes many of the features of the computed input profile; this is one way to ensure a smooth input signal, for example. If a polynomial basis is chosen then the order can be selected so as to follow a polynomial setp setpoi oint nt sign signal al with with no lag. lag. This This feat featur uree is ofte often n important for mechanical servo control applications.
control horizon M
u Single Move
past
future
u Basis Function
past
future
Fig. 10. Input parameterization parameterization options. Multiple move option (top), single move option (middle), basis function parameterization (bottom).
3.4.9. Numerical 3.4.9. Numerical solution solution methods methods It is intere interesti sting ng to consid consider er the var variou iouss numeri numerical cal solution methods used in the MPC control calculations. Here we focus on the dynamic optimization step, which takes the form of the nonlinear program 31. For the linear MPC technology, this reduces to a QP. The PFC algo algorit rithm hm with with line linear ar mode models ls solv solves es this this QP in the the simple simplest st possib possible le way, way, using using a leastleast-squa squares res soluti solution on followed by clipping to enforce input constraints. While this is clearly sub-optimal, only a single linear system must be solved, so the PFC controller can be applied to very fast processes. The DMC-plus algorithm is slightly more more comp comple lex x in that that it solv solves es a sequ sequen ence ce of leas leasttsquares problems in order to more accurately approximate the QP solution. The remaining linear products use various active set QP method methodss to solve solve for the input input profile profile.. Unfort Unfortuunately nately,, these these alg algori orithm thmss do not exploi exploitt the proble problem m structu structure re efficie efficientl ntly, y, and the soluti solution on time time genera generally lly grows grows in propor proportio tion n to M 3 ; where M is the contro controll horizo horizon. n. More More efficie efficient nt soluti solutions ons are possib possible le if the model is retained as an equality constraint, yielding a
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solution time that grows linearly in proportion to M : Rao, Wright, and Rawlings describe one such method based based on interi interior or point point techni technique quess (Rao, Wrigh Wright, t, & Rawlings, 1998). 1998). The The nonl nonlin inea earr MPC MPC prod produc ucts ts must must of cour course se use use more more powerf powerful ul soluti solution on techni technique ques; s; these these are usuall usually y based upon a sequence of iterations in which a linearized versio version n of the proble problem m is solved solved.. The Aspen Target Target prod produc uctt uses uses a mult multi-s i-ste tep p Newt Newton on-t -typ ypee algo algori rith thm m developed developed by De Oli Olivei veira ra and Bie Biegle glerr (19 (1995, 95, 199 1994) 4),, name named d QP QPKW KWIK IK,, whic which h has has the the adva advant ntag agee that that inter interme medi diat atee solu solutio tions ns,, alth althou ough gh not not opti optima mal, l, are are guaranteed feasible. This permits early termination of the optimization algorithm if necessary which guarantees a feasib feasible le soluti solution. on. Aspen Target Target uses uses the same QPKWIK engine for local steady-state optimization and the dynamic MV calculation. The The MVC MVC and and proc proces esss perf perfec ecte terr prod produc ucts ts use use a generalized reduced gradient code called GRG2 developed by Lasdon and Warren (1986). (1986). The NOVA-NLC prod produc uctt uses uses the the NOVA NOVA opti optimi miza zati tion on pack packag age, e, a proprietary mixed complementarity nonlinear programming code developed by DOT Products. The The PFC PFC cont contro roll ller er,, when when used used with with a nonl nonlin inea earr model, performs an unconstrained optimization using a nonlinear least-squares algorithm. The solution can be comput computed ed very very rapidl rapidly, y, allow allowing ing the contro controlle llerr to be used for short sample time applications such as missile track trackin ing. g. Some Some perf perform orman ance ce loss loss may may be expe expect cted ed,, howe howeve ver, r, si sinc ncee inpu inputt cons constra train ints ts are are enfo enforce rced d by clipping.
For example, in some cases a single application may be defined as a large MPC controller that encompasses an enti entire re chem chemic ical al plan plant. t. In othe otherr case cases, s, such such as an automo automobil bilee engine engine contro control, l, the vendor vendor may may report report a single single applicatio application n even when several several thousand thousand copies copies of the completed controller are sold. Note also that this is a count of MPC applications performed by the vendors themselves; this does not include a significant number of in-house applications performed by licensees of vendor technology, such as ExxonMobil and DuPont. Nor does it incl includ udee any any cons consid ider erat atio ion n of MPC MPC tech techno nolo logy gy developed completely in-house by operating companies such as Eastman Chemical. More than 4600 total MPC applications are reported in Tables 6 and 7, 7, over twice the number in our previous survey from 5 years earlier (Qin (Qin & Badgwell, 1997). 1997). It is therefore safe to conclude that overall usage of MPC technology is still growing rapidly. All of the vendors report a significant number of applications in progress so it is likely that this number will continue to increase in the coming years. It should be noted that the size of thes thesee appl applic icat atio ions ns rang ranges es from from a sing single le vari variab able le to several hundreds of variables. Therefore, one should not use use the the numb number er of appl applic icat atio ions ns as an indi indica cati tion on of market share. Tables 6 and 7 show that MPC technology can now be found found in a wide wide var variet iety y of appli applicat cation ion areas. areas. The largest single block of applications is in refining, which amounts to 67% of all classified applications. This is also one of the original application areas where MPC tech techno nolo logy gy has has a soli solid d trac track k reco record rd of succ succes ess. s. A significant number of applications can also be found in petroc petrochem hemica icals ls and chemic chemicals als,, althou although gh it has taken taken longer for MPC technology to break into these areas. Significant growth areas include the chemicals, pulp and paper paper,, food food proc proces essi sing ng,, aero aerosp space ace and and auto automo moti tive ve industries. Table 6 shows that AspenTech AspenTech and Honeywell Honeywell HiSpec are highly focused in refining and petrochemicals, with a handful of applications in other areas. Adersa
4. Applications Applications summary summary
Tables 6 and 7 summarize the reported applications of linear and nonlinear MPC technology through 1999. It is important to note that the vendors were free to define application; for this reason one must what constitutes an application; be careful when drawing conclusions from these data.
Table 7 Summary of nonlinear MPC applications by areas (estimates based on vendor survey; estimates do not include applications by companies who have licensed vendor technology) Area
Air and Gas Chemicals Food Processing Polymers Pulp & Paper Refining Utilities Unclassified Total
Adersa
Aspen Technology
Continental Controls
Pavilion Technologies
5
5 9 15 1 13
5
43
18 15
2 1
5 1 3
DOT Products
6
2 1 36
Total
18 22 9 21 1 13 7 2 93
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and and Inve Invens nsys ys appa appare rent ntly ly have have a broa broade derr rang rangee of experi experienc encee with with applic applicati ations ons in the food food process processing ing,, mining/me mining/metallu tallurgy, rgy, aerospace aerospace and automotive automotive areas, among among others others.. The appli applicat cation ionss report reported ed by Adersa Adersa include a number of embedded PFC applications, so it is difficult difficult to report their number number or distributi distribution. on. While only a total number was reported by SGS, this includes a number of in-house SMOC applications by Shell, so the distribution is likely to be shifted towards refining and petrochemical applications. The bottom of Table 6 lists the largest linear MPC appl applic icat atio ions ns to date date by each each vend vendor or,, in the the form form of (outputs)Â (outputs)Â(inputs). The numbers show a difference in phil philos osop ophy hy that that is a matt matter er of some some cont contro rove vers rsy. y. AspenT AspenTech ech prefers prefers to solve solve a large large contro controll proble problem m with a single controller application whenever possible; they report an olefins application with 603 outputs and 283 inputs. Other vendors prefer to break the problem up into meaningful sub-processes. The nonlinear MPC applications reported in Table 7 are spread more evenly among a number of application areas. Areas with the largest number of reported NMPC application applicationss include include chemicals, chemicals, polymers, and air and gas processing. It has been observed that the size and scope of NMPC applications are typically much smaller than than that that of line linear ar MPC MPC appl applic icat atio ions ns (Martin & Johnston, 1998). 1998). This is likely due to the computational complexity of NMPC algorithms.
5. Limitations Limitations of existing technology technology
Many of the Many the curr curren entl tly y avai availa labl blee indu indust stri rial al MPC MPC algori alg orithm thmss suffer suffer from from limita limitatio tions ns inheri inherited ted from from the original DMC and IDCOM technology. Problems with the control technology, which have been discussed by others (Morari (Morari & Lee, 1991; 1991; Muske & Rawlings, 1993), 1993), include: * * * *
limited model choices; sub-optimal feedback; lack of nominal stability; sub-op sub-optim timal al or ineffici inefficient ent soluti solution on of the dynami dynamicc optimization.
Two of the linear linear MPC algorith algorithms, ms, DMC-plus DMC-plus and HIECON, rely on convolution models (impulse and step response). This can be problematic when controlling a process with widely varying time constants; for this case it is typi typica call to sacri sacrific ficee dyna dynami micc cont contro roll of the the fast fast proc proces esss mode modess in orde orderr to keep keep the the mode modell leng length th reasonable. A potentially more significant problem with the impulse and step response models is that they are limited to strictly stable processes. While it is certainly possib possible le to modify modify the alg algori orithm thmss to accomm accommoda odate te a pure integrator, these modifications may lead to other prob proble lems ms,, such such as addi adding ng the the deri deriva vati tive ve of a nois noisy y
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output signal into the feedback path. It is not possible, in genera general, l, to represe represent nt an unstab unstable le proces processs using using an impulse response model. All of these problems can be overcome by using an auto-regressive parametric model form such as a state-space or ARX model. The bias update feedback technique used by industrial MPC controllers is probably the best assumption that can be used for stable plants in the total absence of disturbance disturbance and measuremen measurementt informatio information, n, but better feedback is possible if the distribution of disturbances can be charac character terize ized d more more careful carefully. ly. The bias bias update update method implicitly assumes that there are no stochastic disturb disturbanc ances es affect affecting ing the sys system tem state, state, and the meameasurement is perfect. Measurement noise is typically dealt with with separa separatel tely y in applic applicati ations ons using using ad hoc filteri filtering ng techniques. The bias update fails for purely integrating syst system ems, s, so a numb number er of ad hoc hoc meth methods ods have been been develo developed ped by practi practitio tioner nerss to handle handle this this case. case. These These meth method odss tend tend to work work poor poorly ly in the the pres presen ence ce of significant significant measurement measurement noise. noise. Industrial Industrial practitioner practitionerss need to recognize that the Kalman filter provides a rich fram framew ewor ork k to addr addres esss thes thesee prob proble lems ms.. Musk Muskee and and Rawlings Rawlings have demonstrate demonstrated d how better performance performance can be achieved for MPC applications by using a statespace model and an optimal state observer (Muske (Muske & Rawlings, 1993). 1993). Of course, these problems motivated the the orig origin inal al deve develo lopm pmen entt of the the SMOC SMOC algo algori rith thm m (Marquis & Broustail, 1998), 1998), which uses a Kalman filter for feedback. Tuning MPC controllers for stable operation in the presence of constraints may be difficult, even when the process model is perfect. Currently much effort is spent on closed closed-lo -loop op simula simulatio tion n prior prior to commi commissi ssioni oning ng a cont contro roll ller er.. It must must be reco recogn gniz ized ed,, howe howeve ver, r, that that simula simulatin ting g all possib possible le combin combinati ations ons of active active conconstraints is impossible in practice. It seems far better to build an algorithm that is inherently stable in all such cases; cases; in contro controll theory theory this is known known as a nominally stabilizing algorithm. Sco Scokae kaert rt and Raw Rawli lings ngs (19 (1998) 98) show how this can be done for the linear case by using infin infinite ite pred predic ictio tion n and and cont contro roll hori horizo zons ns.. Chen Chen and and Allgower ower presen presentt a quasiquasi-infi infinit nitee horizo horizon n method method for nonlinear systems (Chen (Chen & Allgower, 1995). 1995). Research results results for the nonlinear nonlinear case are covered thoroughly thoroughly by Mayne Ma yne et al. (2000) (2000). None None of the curren currently tly ava availa ilable ble industrial MPC algorithms makes full use of these ideas. Sub-optimal solutions to the dynamic optimization 31 are used in several of the packages, presumably in order to speed up the solution time. This seems difficult to jus justi tify fy for for the the case case of refin refinin ing g and and petr petroc oche hemi mica call applications, where the controllers run on the order of once once each each minu minute. te. Go Good odwi win n et al al.. (2 (200 001) 1) give give an example where unconstrained LQR solution with antiwindup has significant performance deterioration compared pared to an MPC solution solution under under severe severe constr constrain aints. ts. Howe Howeve ver, r, for for high high spee speed d appl applic icat atio ions ns wher wheree the the .
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controller must execute in a few milliseconds, such as tracking the position of a missile, it may not be feasible to solve a QP at every control execution. For this case a good sub-optimal solution may be the only option. None one of the the line linear ar MPC pro product ductss expl exploi oitt the the struc structu ture re of QP in the the dyna dynami micc opti optimi miza zati tion on.. The The singular value technique in RMPCT is one of the few effo effort rtss in usin using g robu robust st nume numeri rica call tech techni niqu ques es.. By proper properly ly exploi exploitin ting g the struct structure ure of the proble problem, m, it should be possible to handle significantly larger and/or faster processes (Rao (Rao et al., 1998). 1998). Model Model uncerta uncertaint inty y is not addres addressed sed adequa adequatel tely y by current MPC technology. While most of the identification packages provide estimates of model uncertainty, only one vendor (Honeywell) provides a way to use this info inform rmati ation on in the the cont contro roll desi design gn.. All All of the the MPC MPC algo algori rith thms ms prov provid idee a way way to detu detune ne the the cont contro roll to improve improve robustness, robustness, but the trade-off trade-off between between performance and robustness is generally not very clear. Until this this conn connec ecti tion on is made made,, it will will not not be poss possib ible le to dete determ rmin inee when when a mode modell is accu accura rate te enou enough gh for for a particular particular control application. application. There are many robust MPC result resultss ava avail ilabl ablee in the academ academic ic litera literature ture,, but most focus only the dynamic optimization and will not work work for for some someth thin ing g as si simp mple le as a setp setpoi oint nt chan change ge.. Promi Promisin sing g result resultss that that may eventu eventuall ally y impac impactt MPC MPC practice include the robust stability conditions presented by Vuthandam et al. for a modified QDMC algorithm (Vuthandam, Genceli, & Niko Nikolaou, laou, 1995 1995); ); the robust robust MPC algorithm algorithm presented presented by Kothare, Balakrishnan, and Mora Morari ri (1996) and the robust robust steady steady-st -state ate target target calcul calculati ation on descri described bed by Kas Kassma smann, nn, Bad Badgwe gwell, ll, and Hawkins (2000). (2000). More research is needed in this area. Current model identification technology suffers from a number of limitations: *
*
*
*
*
*
plant test signals are typically step-like and require close attention attention of experienced experienced engineers engineers during during the test; there is no tool to determine whether the collected data are adequate to represent the process dynamics for MPC design; in practice the plant is generally over tested, increasing the implementation cost during the test period; period; identi identifica ficatio tion n alg algori orithm thmss are mainly mainly of the leastleastsquares type; there are only a few that use modern subspace identification and prediction error methods; statistical efficiency and consistency of identification algorithms are not addressed; there is a lack of model validation methods to verify whether the identified model is adequate for control or whet whethe herr si sign gnifi ifica cant nt model del dete deteri rior ora ation tion has has occu occurr rred ed afte afterr the the MPC MPC cont contro roll ller er has has been been commissioned; ther theree is litt little le prac practi tice ce exce except pt in Conn Connoi oiss sseu eurr in multivariable closed-loop identification which would
*
be very useful for on-line model updating or adaptive MPC implementation; there there is no sys system temati aticc approa approach ch for buildi building ng nonnonlinea linearr dynami dynamicc model modelss for NMPC. NMPC. Guide Guidelin lines es for plant tests are needed to build a reliable nonlinear model. This is important because even more test data will will be requir required ed to develo develop p an empiri empirical cal nonlin nonlinear ear model than an empirical linear model.
6. Next-generati Next-generation on MPC technology technology
MPC vendors were asked to describe their vision of next-generation MPC technology. Their responses were combined with our own views and the earlier analysis of Froisy Froi sy (1994 (1994)) to come come up with with a comp compos osite ite view view of future MPC technology. 6.1. Basic Basic controller controller formulation formulation Because it is so difficult to express all of the relevant control control objectives objectives in a single single objective objective function, nextgeneration MPC technology will utilize multiple objective tive funct functio ions ns.. The The infin infinit itee pred predic icti tion on hori horizo zon n has has bene benefic ficia iall theo theore reti tica call prop proper erti ties es and and will will prob probab ably ly become become a standard standard feature. Output and input trajectory trajectory options options will include setpoints, setpoints, zones, trajectories trajectories and funnels. funnels. Input parameteriza parameterization tion using basis functions functions may may beco become me more more wide widesp spre read ad,, and and infin infinit itee cont contro roll horizons with moves computed at each control interval may also become possible. 6.2. Adaptive Adaptive MPC A few adaptive adaptive MPC alg algori orithm thmss such such as the GPC algo algori rith thm m intr introd oduc uced ed by Clar Clarke ke et al. al. have have been been proposed proposed (Clark Clarke, e, Mohta Mohtadi, di, & Tuffs Tuffs,, 1987 1987)) but only two adaptive MPC algorithms have reached the marketplace (Connoisseur (Connoisseur from Invensys Invensys and STAR from Dot Prod Produc ucts ts (Dol Dollar lar,, Mel Melton ton,, Mor Morshe shedi, di, Gl Glasg asgow, ow, & Repsher, Repsh er, 1993 1993)) )).. This This is desp despit itee the the stro strong ng mark market et incentive for a self-tuning MPC controller. This reflects the difficulty of doing adaptive control in the real world. Barring Barring a theoretical theoretical breakthrough, breakthrough, this situation situation is not likely to change in the near future. On the other hand, adaptive and on-demand tuning PID PID cont contro roll ller erss have have been been very very succ succes essf sful ul in the the market marketpla place. ce. This This sugges suggests ts that that adapti adaptive ve MPC MPC concontrolle trollers rs may emerge emerge for SISO SISO loops loops as adapti adaptive ve PID tech techno nolo logy gy is gene genera rali lize zed d to hand handle le more more diffi difficu cult lt dynamics. 6.3. Robust Robust MPC With one excepti With exception on (Honey (Honeywel well), l), indust industria riall MPC contro controlle llers rs rely rely solely solely on brute-f brute-forc orcee simula simulatio tion n to
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evaluate the effects of model mismatch. Robust stability guarantees would significantly reduce the time required to tune and test industrial MPC algorithms. It is likely that that recent recent robust robust researc research h results results (Kas Kassma smann nn et al. al.,, 2000;; Kothare et al., 1996; 2000 1996; Scokaert & Mayne, 1998) 1998) will eventually make their way into MPC products. Robust stabil stability ity guaran guarantee teess will will then then be combin combined ed with with uncerta certain inty ty estim estimate atess from from iden identi tific ficat atio ion n softw softwar aree to greatly simplify design and tuning of MPC controllers. 6.4. Nonl Nonlinear inear MPC Next-ge Next-gener nerati ation on MPC MPC techno technolog logy y will will allow allow nonnonlinear models to be developed by seamlessly combining process knowledge with operating data. A continuoustime fundamental model will be defined from a graphical representation of the plant. Process test signals will be desi design gned ed auto automa mati tica call lly y so as to expl explor oree impo import rtan antt regi region onss of the the oper operat atin ing g spac spacee wher wheree the the mode modell is inadequate for control. Closed-loop process tests will be conduc conducted ted using using PRBS PRBS signal signals, s, with with minim minimal al on-site on-site time requirements. Owing to the difficulty in data based modeling of nonlinear processes, first principles models and other other altern alternati ative ve modeli modeling ng method methodss (Foss, Lohmann, & Marq Marquardt, uardt, 1998 1998)) will will become become necessary necessary in practice.
7. Discussion Discussion and conclusions conclusions
MPC technology has progressed steadily since the first IDCOM and DMC applications over 25 years ago. Our survey data show that the number of MPC applications has has appr approx oxim imat atel ely y doub doubled led in 4 year yearss from from 1995 1995 to 1999. 199 9. MPC MPC applic applicati ations ons show show a solid solid founda foundatio tion n in refining and petrochemicals, and significant penetration into a wide range of application areas from chemicals to food processing. processing. The MPC technology technology landscape landscape has changed dramatically in recent years, making it difficult for us to keep track of the swift progress in academic resea researc rch h and and indu indust stri rial al appl applic icat atio ions. ns. Ma Majo jorr rece recent nt developments include: *
*
*
Techno Technolog logica icall change changess due to the develo developme pment nt of new algorithms algorithms by academia academia and industry, computer hardware and software advances, and the breadth of industrial applications. Orga Organi niza zati tion onal al chang changes es due due to acqu acquis isit itio ions ns and and mergers, such as the acquisition by Aspen Technology ogy of DMCC DMCC,, Setp Setpoi oint nt,, Trei Treibe berr Contr Control ols, s, and and Neuralware. Integr Integrati ation on of MPC MPC technol technology ogy into into Distri Distribut buted ed Control System (DCS) platforms, illustrated by the acquisitio acquisition n of Predictive Predictive Control Ltd. by Invensys, Invensys, the parent company of Foxboro.
*
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Increased applications by end-users who license MPC software from vendors.
Curren Currentt genera generatio tion n linear linear MPC MPC techno technolog logy y offers offers signifi significan cantt new capabi capabilit lities ies,, but severa severall unneces unnecessar sary y limita limitatio tions ns still still remain remain.. Some Some produc products ts do not allow allow the use of parame parametri tricc models models.. Feedba Feedback ck option optionss are seve severe rely ly limi limited ted,, ofte often n lead leadin ing g to poor poor contr control ol of integrating integrating systems and slow rejection of unmeasured unmeasured disturbances. The algorithms are not nominally stabilizing, ing, so that that tuning tuning choices choices must must be tested tested extens extensive ively ly throug through h closed closed-lo -loop op simula simulatio tion. n. None None of the linear linear MPC MPC algo algori rith thms ms make makess use use of mode modern rn nume numeri rica call solu soluti tion on meth method ods, s, limi limiti ting ng the the size size and and spee speed d of processes that they can be applied to. Appli Applicat cation ionss of nonlin nonlinear ear MPC produc products ts will will concontinu tinuee to grow row in thos thosee area areass where ere linear near MPC technology has proven difficult to apply, such as control of polymerization processes. The nonlinear MPC algorithms reported here differ in the simplifications used to generate generate a tractable tractable control calculatio calculation. n. The models are usuall usually y specifi specified ed as linear linear dynami dynamicc subsys subsystem temss with with added nonlinearity. The NOVA-NLC controller is the only current offering that allows the use of a rigorous nonlinear model derived from first principles. An impor importan tantt observ observati ation on is that that the majori majority ty of indu industr stria iall MPC MPC contr control olle lers rs use use empi empiri rica call dyna dynami micc mode models ls iden identi tifie fied d from from test test data data.. Impr Improv ovem emen ents ts in identificati identification on technology technology that result in shorter shorter overall overall process testing times will have a large positive impact on the bottom line for MPC technology users. Choosing an MPC technology for a given application is a complex question involving issues not addressed in this this paper. paper. Here, Here, we have have emphas emphasized ized the technic technical al capabilities of MPC technology. However, if a vendor is to be selecte selected d to design design and imple implemen mentt an advanc advanced ed control system, it would be wise to weigh heavily their experience with the particular process in question. Research needs as perceived by industry are mostly control engineering issues, not algorithm issues. Industrial practitioners do not perceive closed-loop stability, for example, to be a serious problem. Their questions are are more more like like:: Whic Which h vari variab able less shoul should d be used used for for cont contro rol? l? When When is a mode modell good good enou enough gh to stop stop the the identi identifica ficatio tion n plant plant test? test? How do you determ determine ine the source source of a proble problem m when when a contro controlle llerr is perfor performi ming ng poorly poorly?? When When can the added added expens expensee of a nonlin nonlinear ear MPC MPC contro controll ller er be justi justifie fied? d? How How do you you desig design n a control system for an entire plant? How do you estimate the the benefi benefits ts of a cont contro roll syst system em?? Answ Answeri ering ng thes thesee questions will provide control practitioners and theoreticians with plenty of work in the foreseeable future. Just as academicians do not agree on which control algorithms algorithms are best, industrial industrial practitioner practitionerss also have very different views on the future trends of MPC. Just as PID controllers controllers have many different different implementa implementations tions
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(series (series form, parallel parallel form, etc.), MPC controller controllerss are not not anti antici cipa pated ted to conv conver erge ge to a si sing ngle le form form in the the foreseeable foreseeable future. While While academic academic research research will continue to develop new algorithms and prove new facts in the MPC domain, industrial development will consider its its own own prio priori riti ties es with with a heal health thy y inte interac racti tion on with with acad academ emia ia.. Alth Althou ough gh ther theree is stil stilll plen plenty ty of room room to refine refine or develo develop p new MPC algorith algorithms, ms, pushin pushing g the technology to new application areas could be at least equally equally important important and challengi challenging. ng. Given the uneven distribution in the MPC applications today, it is safe to anticipate that much more development in both theory and practice is still ahead of us.
Acknowledgements
The authors wish to thank the vendor representativ representatives es who responded to our MPC survey for their time and effort. We also wish to thank the following people for their constructive criticism of this paper: Pierre Carrette, Jim Downs, Harry Forbes, Brian Froisy, Pierre Grosdidier, didier, John Guiver, Rob Hawkins, Hawkins, Dean Kassmann, Kassmann, Geoff Lewis, Joseph Lu, Greg Martin, Steve Piche, Jim Rawli Rawlings, ngs, Jaques Jaques Richal Richalet, et, Andy Andy Spezia Speziale, le, Dougla Douglass White, Robert Young, and Hong Zhao.
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