2 model predictive control tuning methods a review

7/29/2019 2 Model Predictive Control Tuning Methods a Review

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Model Predictive Control Tuning Methods: A Review

Jorge L. Garriga and Masoud Soroush*

Department of Chemical and Biological Engineering, Drexel UniVersity, Philadelphia, PennsylVania 19104

This paper provides a review of the available tuning guidelines for model predictive control, from theoretical

and practical perspectives. It covers both popular dynamic matrix control and generalized predictive controlimplementations, along with the more general state-space representation of model predictive control and othermore specialized types, such as max-plus-linear model predictive control. Additionally, a section on stateestimation and Kalman filtering is included along with auto (self) tuning. Tuning methods covered rangefrom equations derived from simulation/approximation of the process dynamics to bounds on the region ofacceptable tuning parameter values.

1. Introduction

Model predictive control (MPC) has been implementedwidely in the process industries since its introduction in the early1960s.1-8 Zadeh and Whalen9 first proposed the connectionbetween the minimum time optimal control problem and linearprogramming, and Propoi10 introduced the idea of a moving

horizon approach to the optimization problem, which has sincebecome the foundation of all MPC implementations. Earlyimplementations of MPC, such as the linear quadratic Gaussian(LQG) controller, did not have much impact in the processindustry due its inability to handle constraints, process nonlin-earities, and uncertainty, and its unique performance criteria.Later implementations of MPC, such as dynamic matrix control(DMC)11 and model algorithmic control (MAC),12 addressedsome of the shortcomings of LQG by introducing constraints,both input and output, into the formulation in a systematicmanner. Further additions included the set point trajectory toincrease robustness and move suppression coefficients to limitthe size of the manipulated variables action.7 While theseadditions provided MPC with enormous flexibility, they com-plicated the task of controller tuning. The controller designerhad to set the prediction horizons (P0 and P), control horizon(M), model horizon (N), weights on the outputs (Q), weightson the rate of change of inputs (), weights on the magnitudeof the inputs (), the reference trajectory parameter (), andconstraint parameters. Efficient setting of this large set of tunableparameters often requires a systematic tuning guideline for MPC.

The tuning methods herein can be classified into two types:the methods are either explicit formulas for the variousparameters based on approximation/simulation of the processdynamics or bounds on where the tuning parameters lie basedon parameters of the process dynamics. For instance, the tuningguidelines proposed by Shridhar and Cooper,13 assume that the

actual process is well-represented by a first-order-plus-dead-time approximation. Similarly, the bounds proposed by McIn-tosh et al.14,15 are based on the simulated process delay andrise time. In recent years, advances in subspace identification16,17

and automated testing have allowed practitioners to dramaticallyreduce the cost of tuning MPC applications.18 The autocova-riance least-squares technology developed by Rawlings group19,20

is expected to make Kalman filtering much more accessible byautomatically identifying the main tuning parameters.

Yamuna Rani and Unbehauen21 reviewed various methodsto tune DMC and GPC controllers. The authors work spans

from 1985 to 1994 and does not include any other formulationsof MPC controllers, such as state-space representations and max-plus-linear (MPL) systems. Similarly, it does not include real-world applications of the tuning strategies presented or methodsdeveloped by industrial applications. Though there are a limitednumber of review papers on tuning explicitly, various other

review papers cover sparsely theorems regarding minimumvalues of the tuning parameters for robust stability.1-3,7

Although a typical MPC law has many tunable parameters,according to Badgwell,22 who has tuned MPC for a great numberof industrial applications, the controller tuning is not as difficultas it looks because of the MPC approach of formulating thecontrol problem as an open-loop optimization in the timedomain. In this formulation, the translation between controlspecifications and tuning factors is more transparent, whichplaces the control problem within the grasp of a process operator.The details of the MPC formulation matter when it comes totuning.22 For example, if a state-space model is used then tuningfor integrators will be transparent. If not, one should expect to

find some ad-hoc tuning parameters to deal with integrators.Furthermore, an infinite horizon objective function eliminatestuning factors and improves performance. Two fundamentalnotions that one should consider in tuning an MPC applicationare:22

The first act of tuning is to develop an appropriate processmodel. If the model is accurate enough, then the rest ofthe tuning is straightforward. And if the controller exhibitspoor performance, then one should consider the model poor(inaccurate) unless proven otherwise. This review paper isnot concerned with proper model development or selectionfor MPC applications.

The trade-off between performance and robustness is thesecond fundamental notion. Fast closed-loop performancerequires a high precision model. Most MPC users in theprocess industries are willing to detune the controllersomewhat so that the controller will remain stable as theprocess operating point varies.

This study gives a survey of the current scope of tuningstrategies, both from a theoretical perspective and from real-world applications of MPC. We provide a broad treatment ofthe available methods to tune not only the popular DMC andGPC controllers, but also include sections on new developmentssuch as MPL systems and methods to calculate the covariancematrices and Kalman filter for state estimation. Finally, a section

* To whom correspondence should be addressed. Phone. (215) 895-1710. Fax: (215) 895-5837. E-mail: [email protected].

Ind. Eng. Chem. Res. 2010, 49, 35053515 3505

10.1021/ie900323c 2010 American Chemical SocietyPublished on Web 03/19/2010


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on autotuning gives, according to the authors current knowl-edge, the most comprehensive treatment of MPC tuning in theliterature.

This paper is organized as follows. An overview of themathematical preliminaries dealing with the formulation of thecontrol law is given first. Following this are successive sectionsfor offline tuning; each dealing individually with a tunableparameter. A section of autotuning procedures follows the offlinetuning sections. A case study section is included to compare

the effectiveness of the various tuning guidelines. Finally,concluding remarks are presented.

2. Mathematical Preliminaries

Consider a linear, time-invariant, discrete-time plant modelof the form:

where kis the sampling instant, x is the vector of state variables,u is the vector of manipulated inputs, and y is the vector ofmodel outputs. A Rnn, B Rnm, G Rng, and C Rpn

are constant matrices. {wk}k)0Nd

and {Vk}k)0Nd

are the vectors ofplant disturbances and output measurement disturbances, re-spectively, modeled as zero-mean Gaussian noise sequences withcovariance Qw and Rv, respectively. Estimates of the plant statevariables can be obtained from the Kalman filter

where the Kalman gain is given by

P(k|k - 1) represents the covariance matrix of the estimationerror, (k) x(k) - x(k|k- 1) and is the solution to the Ricattiequation

Consider a general moving-horizon minimization problem ofthe form:

subject to

where rj is the reference trajectory for the controlled output yj,m is the number of inputs, and p is the number of outputs. Eachreference trajectory, rj, is defined in this manner:

where yspj is the set point for the output yj, which can be constantor time-varying, and j is a tunable parameter that takes a valuebetween 0 and 1 and sets the time constant of the referencetrajectory rj. j is the relative order (degree) of the output yjwith respect to the input vector, u.

3. Offline Tuning Methods

3.1. Prediction Horizons. In this section, various techniquesto tune the prediction horizons will be discussed. Intuitively,the prediction horizons describe the lower and upper limits ofthe time horizon (window) in the future over which the controllertries to induce a desired response to the plant outputs. The finalprediction horizon (upper limit of the horizon) is set to be infiniteto ensure stability23 or finite, in which case the final horizonshould be tuned properly based on tuning rules to ensure theclosed-loop stability of the control system. Since DMC and GPCare the most popular implementations, theoretically and industri-ally, the tuning rules for these are given first. Newer imple-mentations, such as state-space representations and max-plus-linear (MPL) systems, are given toward the end of the section.A list of the tuning guidelines for setting the prediction horizonsis given in Table 1.

For DMC, two heuristics and a theorem are given followedby formulas to determine the final prediction horizon, P: As adefault setting, one can set P equal to 10.21 Similarly, as a rule

of thumb, Wojsznis et al.24

suggested setting the predictionhorizon so large that further increment has no significant effecton control performance. In an article by Nagrath et al.,25 theauthors used the default DMC value of the prediction horizon(i.e., 10) to tune a state-space representation of a continuous-stirred tank reactor (CSTR) with excellent results. Garcia andMorari1 presented a theorem to ensure closed loop stability basedon the final prediction horizon. The theorem states that ifQ )I and ) 0, then there is a sufficiently small M (controlhorizon) and sufficiently large P > N + M - 1 to ensure theclosed-loop system is stable. Maurath et al.26 suggests that Pbe chosen so that between 80 and 90% of the process steady-state is included in the calculation of P, with 85% being an

average value. Similarly, in ref 27, the authors proposed aminimum value for the prediction horizon with respect to thecontrol horizon and the process delay. This ensures that thedynamic matrix is of full column rank and thus invertible.Cutler28 proposed setting the value of the prediction horizonaccording to the sum of the control horizon and model horizon.The author suggested increasing Muntil there is no further effecton the first move of the controller to a set point change.

Georgiou et al.29 suggested adding the rise times at 60 and95% of process steady-state divided by the sampling time(period) minus one to determine P and applied this method tothree distillation column examples.29 The control response ofthe first column, column A, was quite excellent. There was nooffset and only very slight oscillation in the output variables.

The response of the second column, column B, yielded poorerperformance than the first column due to large load disturbance.

x(k + 1) ) Ax(k) + Bu(k) + Gw(k)y(k) ) Cx(k) + v(k)

x(k + 1|k) ) Ax(k|k - 1) + Bu(k) +AL(k)(y(k) - Cx(k|k - 1))

L(k) ) P(k|k - 1)CT[CP(k|k - 1)CT + Rv]-1

P(k + 1|k) ) AP(k|k - 1)AT + GQwGT

-

AP(k|k - 1)CT[CP(k|k - 1)CT + Rv]-1

CP(k|k - 1)AT

minu(k),...,u(k+M-1)

{j)1

p

l)P0

P

qjl(rj(k + l) - yj(k + l))2

+

i)1

m

l)0

P-1

il(ui(k + l))2+i)1

m

l)0

M-1

il(ui(k + l) -

ui(k + l - 1))2}

x(k + 1) ) Ax(k) + Bu(k) + Gw(k)y(k) ) Cx(k) + v(k)u(k + l) ) u(k + l - 1), l ) M,...Du(k) e d, k ) 0,1,...Hx(k) e h, k ) 1,2,...

rj(k) ) yj(k)rj(k + 1) ) yj(k + 1)

lrj(k + j - 1) ) yj(k + j - 1)

rj(k + j) ) (1 - j)yspj + jyj(k + j - 1)

rj(k + j + 1) ) (1 - j)yspj + jyj(k + j)

lrj(k + P) ) (1 - j)yspj + jyj(k + P - 1)

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This illustrated that DMC is more sensitive to processnonlinearities than traditional proportional-integral (PI) con-trollers.29 The controller for column C was not designed due tothe high computational effort necessary because of a very largetime constant. Due to the large time constant and use ofreasonable sampling times (periods), P would have been verylarge. This, according to Ogunnaike,30 can lead to (a) controlmoves of very large sizes; i.e. excessive movement of thevariable which often results in an unacceptably oscillatorysystem response, and (b) bad conditioning of ATA that canresult in poor control. However, using two nonlinear transfor-mations of the two output variables, the control performanceof both columns B and C were improved dramatically. Adisadvantage of these two methods is that one needs to simulatethe process offline beforehand in order to determine the varioustime constants.

Hinde and Cooper31 proposed converting the model intoFOPDT (first-order-plus-dead-time) form and using five timesthe overall process time constant plus the dead time divided by

the sampling time to determine the value of P. This methodallows for tuning of the prediction horizon without simulatingthe system beforehand. Unfortunately, it is only applicable tosingle-input single-output (SISO), open-loop stable, minimumphase, nonintegrating processes. Due to these constraints on thesystem, this tuning procedure is very specialized, applying tovery few systems of practical importance.

Shridhar and Cooper13 proposed a tuning strategy for SISOand multiple-input multiple-output (MIMO) unconstrained DMCcontrollers that are open-loop stable, including nonsquaresystems. The basis of their tuning method is the conditionnumber (c) of matrix A of the process. Their choice of conditionnumber, 500, was based on the rule of thumb that the

manipulated variable move sizes for a change in set point shouldnot exceed 2-3 times the final change in manipulated vari-able.13 Additionally, to derive their equation, they approximatedthe dynamics of the process via an FOPDT model of the processwith zero-order hold. From this, they arrived at an equation thatdetermines P from the maximum of the various combinationsof process inputs Sand outputs R. All five of the above methodsassume a starting value for the prediction horizon of 1.

Genceli and Nikolau32 formulated robust stability conditionsbased on the tuning parameters for a linear DMC with endconditions. They proposed a bound on the prediction horizonto guarantee stability of the closed loop system with constraintson the inputs and outputs. It is based on the maximum betweenthe difference of the minimum and maximum inputs divided

by the maximum change in input or 1. This allows for robuststability of the closed-loop system in the presence of active

constraints. This condition implies that a prediction horizon ofa minimum length of P ) M + 1 guarantees closed-loop

stability. In particular, ifumax g umax - umin, then a shortprediction horizon of length P ) 2 with a control horizon ofone (M ) 1) guarantees closed-loop stability.32 For the longhorizon case, where P g M + N - 2, the move suppressioncoefficients can all have equal values and guarantee closed-loop stability. To find the optimal value for the predictionhorizon, the authors proposed minimizing the initial onlinecalculated optimal objective function value subject to con-straints on the set point and disturbance.32

For GPC, various methods to tune the prediction horizonsexist. Typically, they fall into a class of ranges on the finalprediction horizon or formulations based on assumptions of theprocess model. All, except two, assume the starting value ofthe prediction horizon to be 1.14,15,33,34 Clarke and Mohtadi35

propose setting the starting value (P0) to the order of the Apolynomial in the GPC model. Clarke et al.36 proposed settingthe value different than one only if it is known a priori thatthe dead-time of the plant is at least k sample-intervals then[P0] can be chosen as k or more to minimize computations.McIntosh et al.14,15 proposed four methods for the initial andfinal value of the prediction horizons, here named as GPC1-GPC4 (Table 2). In their second method (GPC2), they proposedsetting the starting value of the prediction horizon to the orderof the B polynomial in the GPC model plus 1, i.e., nb + 1.

For the value of the final prediction horizon, Clarke et al.36

proposed using a default value of 10 for a large class of plants.However, they proposed that P should be given a value between

the order of the B polynomial of the process model and theoutput-response rise-time in samples. Yoshitani and Hasegawa37

applied this tuning strategy to a continuous annealing processwith good results. The authors applied the strategy to theautotuning formulation of GPC to smoothly track the transitionfrom strip temperature control to furnace temperature control.Yamuna Rani and Unbehauen21 proposed a method to tune theprediction horizon based on the maximum dead time and settlingtime of the system. Banerjee and Shah,33 for systems that arestable, proposed using an equation with the rise time for 95%of the process steady-state. Karacan et al.38 applied GPC tunedusing this method to a packed distillation column and obtainedgood results. They used the default value of 10 for the prediction

horizon, but stated that the prediction horizon should be themaximum of the default value and the process-output rise-timein steps. Edouard et al.,39 for a catalytic reverse flow reactor,proposed the largest of the two controlled variable activeconstraint lengths.

Clarke and Mohtadi35 proposed a bound on the predictionhorizon based on the order of the A polynomial and the risetime of 95% of the process steady-state. As given in Table 2,McIntosh et al.14,15 proposed three different bounds on theprediction horizon depending upon how other parameters in theGPC scheme are tuned. The first bound (GPC1) is based onthe default tuning settings of their four methods. The secondbound developed by McIntosh et al.14,15 results from deviatingfrom the default method by either proposing a strategy for the

move suppression weights (GPC2), the initial value of P(GPC2), the value for the control horizon (GPC2 and GPC3),

Table 1. Guidelines for Tuning the Prediction Horizon

MPC formulation value or bound ref

DMC P ) [(t80 + t90)/2]/Ts 26DMC P > M + td/Ts 27DMC P ) M + N 28DMC P ) t60/Ts + t95/Ts - 1 29DMC P ) (5p + td)/Ts 31DMC P ) max(5rs/Ts + krs) 13

krs ) d,rs/Ts + 1DMC P - 1 g max((umax - umin)/umax, 1) 32

GPC nb < P e tr/Ts 34GPC P ) dmax + (ts/Ts)/3.5 21GPC P ) t95/Ts 33GPC P ) max(10, tr/Ts) 38GPC P ) max (P1f, P2f) 39GPC 2na - 1 < P < t95/Ts 35DMC, GPC low RPN: P ) t80/Ts, P0 ) 1 41

high RPN: P ) t90/Ts, P0 ) t10/Ts

Table 2. GPC Tuning Guidelines

method P0 P M ref

GPC1 1 dmax + 1 < P < t95/Ts 1 trial and error 14, 15GPC2 nb + 1 M + P0 + 1 < P < t95/Ts na + 1 rel[B(1)]2 14, 15GPC3 1 M + P0 + 1 < P < t95/Ts na + 1 trial and error 14, 15GPC4 1 M + dmax + 1 < P < t95/Ts na + 1 trial and error 14, 15

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or the value for the weights on the outputs (GPC3). The thirdbound developed by McIntosh et al., GPC4, varies from GPC3only in its output weights. McIntosh et al.40 suggests using asupervisory level method to periodically update the tuningparameters based on a performance criterion. One proposed byYamuna Rani and Unbehauen21 is the following:

The performance is quantified in terms of some criteria chosenby the control practitioner in order to judge the quality of thecontrol system. One possibility is to minimize the integral-squareerror (ISE). Similarly, McIntosh et al.40 proposed using a servoperformance index with a specified overshoot of 5% and aspecified amount of samples after each set point change in theGPC objective function to tune the final prediction horizon. Thiscan be periodically updated to improve controller performanceby resolving the optimization problem. The advantage of allthese strategies of GPC is that they are easy to implement dueto their straightforward approaches with minimal offline simula-tion and yield robust performance.

Trierweiler and Farina41 proposed a tuning strategy based onthe robust performance number (RPN) of the system. Thestrategy applies to both DMC- and GPC-type MPC controllers.Before tuning, the transfer functions of the system should bescaled based on a procedure outlined in the paper and theattainable performance of the system should be determined.Once that is complete, the RPN number can be calculated todetermine how attainable the performance is. If the RPN is low,they propose setting the initial prediction horizon to 1 and thefinal prediction horizon to the rise time of 80% of the processsteady-state. However, if the RPN is high, they propose settingthe initial prediction horizon to the rise time of 10% of theprocess steady-state and the final prediction horizon to the risetime of 90% of the process steady-state.

Lee and Yu42

proposed a tuning strategy for a state-estimation-based controller, which is applicable to systems witha state-space model. They recommended first tuning for nominalstability and then detuning the system for robust performance.For systems with stable inverses, the prediction horizon ischosen equal to the control horizon. If the system has unstableinverses, one can either choose an infinite horizon formulation,solve a Lyapunov equation to determine the appropriate terminalweights and keep the prediction horizon equal to the controlhorizon, or make P larger than Mby the systems settling timein sampling units and constrain the output error at zero bychoosing a large terminal weight. These settings are kept whenthe system is detuned for robust performance. Zhang and Li16

proposed tuning the final prediction horizon by choosing initialvalues of the input weights, initial prediction horizon (typicallyset to the process delay time), and the rate of change weights.Once these are chosen, one plots the achievable performancebenchmark for different values of P against the installedcontroller and a benchmark controller, which in the article wasan LQG controller indentified via subspace identification. Fromthis, one sets the value of P by considering the systemspecification and the variance trade-off in the benchmarkplots.16 This method is applicable to state-space representationswithout active constraints.

Max-plus-linear (MPL) models are a new class of modelsused for discrete event systems, which are dynamical systemsthat often arise in the context of manufacturing systems,

telecommunication networks, railway networks, parallel com-puting, etc.43 In an article by van den Boom and De Schutter,44

the authors give a strategy to tune the initial and final predictionhorizon. The authors proposed setting the initial predictionhorizon to 1 and the final prediction horizon to the length ofthe impulse response of the system.44 van den Boom et al.45

proposed setting the final prediction horizon according to acomplex rule that requires knowledge of the spectral radius ofA and the shifted state, input, and output matrices by formulasusing the spectral radius. Both of these studies are forconstrained SISO MPL MPC systems.

In summary, there are a multitude of strategies for tuning P.Typically P0 can be set to 1 or the amount of time delay in theoutput response. DMC tuning strategies tend to use either thevarious time constants from the open-loop simulations or anFOPDT representation of the process and formulas derived fromthe time constants of the FOPDT representation. GPC tuningstrategies almost universally are based both on the structure ofthe model and the various time constants obtained from open-loop simulations, similar to some DMC strategies.

3.2. Control Horizon. This section reviews guidelines fortuning the control horizon. The control horizon affects howaggressive or conservative the control action is. This leads to atrade-off: increasing the control horizon from 1 creates a morerobust, but more aggressive controller with increased compu-tational load; however, keeping it at 1 increases the conserva-tiveness and decreases the robustness of the controller but at asavings of computation. As in the previous section, this sectionwill first start out with the tuning guidelines for DMC and GPC

and then go into more current control formulations. Similarly,the guidelines to tune the control horizon are listed in Table 3.

For DMC controllers, Yamuna Rani and Unbehauen21 sug-gested a default value of 1 for the control horizon. Maurath etal.26 proposed tuning the control horizon via trial and error:they stated that the horizon should be large enough to extendover all significant adjustments in the manipulated variableneeded to implement a set-point change. Georgiou et al.29

proposed setting the control horizon based on the time it takesfor the output response to reach 60% of the process steady-state. Hinde and Cooper,31 using an FOPDT approximation ofthe process dynamics, proposed setting the control horizon basedon the time constant of the approximated system. As stated

before, using this formulation allows for tuning without simulat-ing the process offline and calculating the steady-state timeconstants. It is only applicable to SISO, open-loop stable,minimum phase, nonintegrating processes, which makes it verylimiting. Shridhar and Cooper13 also proposed a method fortuning based on FOPDT approximation of the process, similarto that of Hinde and Cooper,31 for open-loop stable, uncon-strained MIMO systems. It is similar to the equation for theprediction horizon, though the process time constant is notmultiplied by five. Genceli and Nikolaou32 proposed setting thecontrol horizon greater than or equal to the maximum of thedifference between the maximum input and the minimum inputdivided by the maximum change in input or 1. This ensuresrobust stability of the closed-loop system with active constraints.

In GPC formulations, as in DMC formulations, the defaulttuning value for the control horizon is 1.14,15,21,33-35 Clarke et

tp(t) ) tp(t - 1) (k(Pm(t) - Pd)tp(t - 1)

Pd

Table 3. Guidelines for Tuning the Control Horizon

MPC formulation value or bound ref

DMC M ) t60/Ts 29DMC M ) p/Ts 31DMC M ) max(rs/Ts + krs) 13

krs ) d,rs/Ts + 1DMC Mg max((umax - umin)/umax, 1) 32GPC Me d/Ts 35DMC, GPC M ) int(P/4) 41



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al.36 proposed setting the control horizon to 1 for typical plantsor equal to the number of poles near the stability boundary foropen-loop unstable plants.36 Yoshitani and Hasegawa37 appliedthis tuning strategy successfully to the autotuning formulationof GPC implemented on a continuous annealing process. Thecontrol problem was to smoothly track the transition from striptemperature control to furnace temperature control. Similarly,Clarke and Mohtadi35 proposed setting M equal to 1 for mostplants or to less than or equal to the plant dead-time. Karacan

et al.38

used the default value of 1 for a packed distillation towerwith good results. However, in the work of McIntosh et al.,14,15

for GPC2-GPC4, the authors set the control horizon to the valueof the A polynomial of the process model plus 1.

Trierweiler and Farina41 proposed a strategy for DMC andGPC controllers based on the robust performance number (RPN)of the system. After the proper scaling is done, as outlined inthe paper, and the attainable performance is computed, theauthors proposed setting the control horizon based on the finalprediction horizon divided by 4. Rawlings and Muske,46

assuming an infinite horizon formulation of the control law,proposed tuning the control horizon based on the stability ofAor on the ability to stabilize {A, B}. IfA is stable, they proposed

setting the control horizon Mg 1. If {A, B} are stabilizable,they propose setting Mg r, where r is the number of unstablepoles. In GPC formulations, the tuning of M can typically bedone solely on the basis of the structure of the system withoutthe need for offline simulation.

For state-space representations, Lee and Yu42 proposed tuningthe system first for nominal stability, then detuning for robustperformance. They suggested setting the control horizon aslarge as possible within the computational limits.42 Edouardet al.39 proposed setting the control horizon to 1 when tuningan MPC controller with a state-space representation to decreasethe computational burden of the constrained optimizationproblem. They applied this strategy to a catalytic reverse flowreactor. Nagrath et al.25 suggested a tuning strategy for a state-space representation of a CSTR; as in ref 46, they proposedsetting the control horizon to greater than or equal to the numberof unstable poles of the system.25 Zhang and Li16 proposedtuning the control horizon by choosing initial values of the inputweights, initial prediction horizon (typically set to the processdelay time), and the rate of change weights. Once those arechosen, one plots the achievable performance benchmark fordifferent values of P against the installed controller and abenchmark controller, which in the article was an LQGcontroller indentified via subspace identification. From this, onesets the value ofMby considering the system specification andthe variance trade-off in the benchmark plots described in ref16. This method is applicable to state-space representations

without active constraints.For linear quadratic regulator (LQR) controllers, Scokaert and

Rawlings47 presented a strategy to tune the control horizon onan infinite horizon LQR formulation. Their strategy is to setthe value of the control horizon, solve the related MPCoptimization as formulated within the study, and check whetherthe terminal output state satisfies the constraints. If the terminaloutput state satisfies, then one should terminate the algorithmand use that control horizon; if it does not, then one shouldincrease the control horizon and reinitialize the algorithm untilthe terminal output state satisfies. This allows for the weightsin the control, both the input and output weights, to be set tounity. State-space representations typically require offline

simulation and some trial-and-error tuning because of thestructure of the representation. For MPL systems, van den Boom

and De Schutter44 proposed setting the control horizon equalto the upper bound of the minimal system order, which is inthe range 1 e Me P.44 This is applicable to unconstrainedSISO MPL MPC only.

In summary, for DMC, GPC, and state-space representations,most strategies propose setting M ) 1 or to the number ofunstable poles. A few, such as those in refs 14 and 15 for GPC,suggest setting to a different value (see Table 2). This saves incomputational time and leads to a less aggressive control action,at the cost of robustness to process-model mismatch andunmeasured disturbances.

3.3. Model Horizon. This section reviews guidelines pro-posed for setting the model horizon in a finite-impulse or a finite-step response model for DMC. The model horizon affects thecondition of the matrix A. As the model horizon increases,matrix A becomes more ill-conditioned.29 Guidelines for tuningthe model horizon are given in Table 4. Georgiou et al.29

proposed setting the model horizon larger than the time required

for the slowest open-loop process output response to reach 95%of the steady state. Shridhar and Cooper13 and Wojsznis et al.24

proposed setting the model horizon equal to the final predictionhorizon.

3.4. Weights on the Outputs. This section presents a reviewof recommendations made on tuning the output weights. Weightson the output are used to scale the control variables and directmore control efforts toward more important controlled outputsto achieve tighter control of these controlled outputs. The tuningguidelines for the weights on the outputs are listed in Table 5.

Rowe and Maciejowski48 used an LQG/LTR approach to tunean infinite horizon state-space MPC controller. Using thattechnique, the authors derived an expression for the output

weights for minimum phase systems as the product of thetranspose of the output matrix C times the output matrix. Fornonminimum phase systems, the technique only works wherethe closed loop bandwidth is made small enough.48 Rowe andMaciejowski49 provided another formula based on H loopshaping of the process model. This applies for nonstrictly propercases. For strictly proper cases, the expression is the same asthat in ref 48. Both studies assume that the weights on the rateof change of inputs is 1, i.e., ) I.

Yoshitani and Hasegawa37 used a linearly decreasing weighton the furnace temperature and a linear increasing weight onthe upper limit of the furnace temperature to achieve smoothtemperature transition from the initial reference temperature tothe upper allowable limit. This was applied to a state-space

model of a continuous annealing process. Nagrath et al.25increased the output weight on the reactor temperature of a

Table 4. Guidelines for Tuning the Model Horizon

MPC formulation formula or bound ref

DMC N> (max(t95,i))/Ts 29DMC P ) N ) max(5rs/Ts + krs) 13

krs ) d,rs/Ts + 1

Table 5. Guidelines for Tuning the Weights on the Outputs


state-space Q ) CTC 48

state-space nonstrictly proper case: 49Q ) X - A

T(I + XBBT)-1XAX )

2XW-1

W ) (2 - 1)I - ZX > min ) (1 + F(XZ))1/2

X ) ATXA - ATXB(I + BTXB)-1BTXA + CTCZ ) AZAT - AZCT(I + CZCT)-1CZAT + BBT

state-space strictly proper case: Q ) CTC 49DMC, GPC Qs ) 1/(1 + yZ,s)1/2 41



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nonlinear state-space model of a CSTR over the concentrationin order to make the input action fast so that the output tracksthe set point aggressively.

Trierweiler and Farina41 presented an approach for settingthe input weights on the basis of the output zero directions ofthe RHP-zero closest to the origin calculated for the scaledsystem GS(s). These directions give an idea how the RHP-zeroeffect is distributed on the plant output. This approach can beused for DMC and GPC controllers. Wojsznis et al.24 suggested

that Q should have higher coefficients applied to the areas ofthe step response with higher process gain and lower values orzero to areas with smaller process gain. They also suggestedusing linearly increasing coefficients with some initial coef-ficients set to zero to improve robustness. The number of initialvalues set to zero is called the patience factor. This method iseffective when the initial part of a step response is uncertainbut the steady-state gain is reasonably well-defined.24

The tuning ofQ can be typically left to the control engineerwho is familiar with the process requirements. If one or moreprocess outputs are more important than others or they decreasein importance over time while others increase, such as in ref37, one can set Q such that the control action accordingly

matches the requirements.3.5. Weights on the Rate of Change of Inputs. This section

reviews existing approaches to tuning the weights on the rateof change of the inputs. Penalizing the rate of change yields amore robust controller but at the cost of the controller beingmore sluggish. Setting a small penalty or none whatsoever givesa more aggressive controller that is less robust. Table 6 liststhe tuning guidelines herein discussed regarding the weightson the rate of change of inputs.

For DMC controllers, Garcia and Morari1 presented a theoremregarding a finite value of the rate-of-change-of-inputs weightsthat stabilizes the closed-loop. It states the following: there existsa * > 0 such that for all > * the closed loop is stable for

all P, Mg

1 and Q>

0. Georgiou et al.

29

used a constrainedlinear minimization procedure to determine the correct sup-pression factors. They used in the inequality constraint theminimum integral absolute error of the outputs such that thechanges of the manipulated variables were similar in maximummagnitude to those of the proportional-integral (PI) controllerthat the MPC formulation was compared to. Maurath et al.27

proposed a trial-and-error tuning strategy for the weights onthe rate of change of inputs. They proposed setting the rate-of-change weights to an initial value and then simulating theclosed-loop system to determine whether the controller is tooaggressive or too sluggish.27 To determine the next weight touse, the authors suggested increasing the value of the currentweight by 10. Hinde and Cooper31 used an approach to set the

rate of change of inputs (suppression) weights based on thedesired controller performance defined as a short rise time with

10-15% overshoot. Genceli and Nikolaou32 provided twoequations for the suppression weights: one is for the suppressionweight at the terminal state, and the other is a recurrence relationfor the other suppression weights. Using this method, the robuststability of the closed-loop system is guaranteed. Sarimveis etal.50 introduced a strategy to tune nonsquare DMC systemsbased on ref 32. The authors derived an optimization algorithmto determine the suppression weight. The algorithm is basedon modifying the end conditions in ref 32 such that there is a

pseudoinverse of the nonsquare dynamic matrix. Next, theobjective is used as a Lyapunov candidate function, and theoptimization is conducted to see if it is monotonically decreasingwith a suboptimal solution. If this is the case, then you cansolve the nonconvex optimization for the suppression coef-ficients with respect to the pseudoinverse of the dynamic matrix.Shridhar and Cooper13 derived a formula for the suppressionweight by approximating the dynamics of the process using anFOPDT model. From this approximation, they derived anequation that relates the suppression weights to the various othertuning parameters. This method is applicable to open-loop stableprocesses including nonsquare systems. Wojsznis et al.24

proposed using decreasing values of the suppression weights

on moves greater than M ) 1. According to the authors, thismakes the first move less aggressive since the next moves arerelaxed and can be used with lower penalties for error correction.Although the second and next moves may violate moveconstraints, this is not a practical concern since the controllermoves are recalculated.24 As a result of this strategy, thecontroller moves are smaller and the controller is more robust.

For GPC controllers, Clarke and Mohtadi35 proposed settingthe value of the suppression weights to zero. However, ifnumerical stability is required, then they propose setting thesuppression weight equal to some small value, . Banerjee andShah33 proposed using a suppression weight greater than 1 butless than 2 (i.e., 1 < < 2), even in open-loop stable systems,

to increase the robustness and detune the performance. Karacanet al.38 suggested tuning the suppression weight by a trial-and-error process. They first chose the other parameters, namely theinitial and final prediction horizons and the control horizon, andthen varied until they achieved the best result. For their system,which was a packed distillation column, their value was )1.2. Yamuna Rani and Unbehauen21 proposed a method to tunethe suppression weights based on optimally tuning the finalprediction horizon in ref 40. The affine constants are determinedusing the previously tuned values.21 This allows for immediateupdating of the suppression weights when the final predictionhorizon is retuned in ref 40. McIntosh et al.14,15 proposed astrategy for initially setting and then periodically updating theweights on the rate of change of the inputs. In their strategy,which is used in the GPC2 formulation, the rate of change ofinputs weights is initially set to a value. From these initial values,the weights can be updated using a second formula. Yoshitaniand Hasegawa37 used a suppression weight of 0.6 for acontinuous annealing process to increase the robustness of thecontroller, although the authors noted that it was not critical solong as the final prediction horizon was properly tuned.

Trierweiler and Farina41 proposed a formula for calculatingthe suppression weight based on a rescaling of the system. It isbased on using a frequency representation of the model todetermine the frequency where the RPN is at its maximum. Thisapplies to both DMC and GPC controllers with inactiveconstraints. Rowe and Maciejowski48 used an LQG/LTR ap-

proach to tune an infinite horizon state-space MPC controller.For minimum phase systems, they set the suppression weight

Table 6. Guidelines for Tuning the Weights on the Rate of Changeof Inputs


DMC ) Kp2, is typically set to 10 31DMC P ) [j)-N+1P j + (b + wb) i)1N Ei + j)-N+1P

(aj + waj)]/[1 - (i)1N Ei)/|G|]32

j-1 ) j - aj - waj - j, 1 e j e Pvarious terms are defined in the article

DMC s2 ) (M/500) r)1R {Qr2Krs2[P - krs - (3/2)(rs/Ts) + 2 - ((M - 1)/2)]}, s ) 1, 2, ..., S

13

krs ) d,rs/Ts + 1GPC ) mP + c 21GPC start ) m|tr(GTG)|/M 14, 15

) rel[B(1)]2

DMC, GPC s ) [(1 + uZ,s)log10(RPN + 1)]1/2 mean|gsi,j(sup)| 41



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equal to zero since recovery is always guaranteed from the LQGformulation. For nonminimum phase systems, they set thesuppression weights equal to a small value to help ensurerecovery from the LQG formulation. Nagrath et al.25 set thesuppression weight to 1 for a CSTR with a nonlinear state-space plant model with good results. Zhang and Li16 proposeda tuning strategy for the rate of change of outputs weights. Theauthors suggested setting the values of the initial and finalprediction horizons, the control horizon, and the input weights

ad hoc. Then, one should plot the MPC-achievable curveplotted with the MPC-achievable benchmark variance . . . asordinate and the input variance . . . as the abscissa by varyingparameter [] along with the installed controller curve on thesame plot and varying until the installed controller curveapproaches that of the achievable benchmark curve.16

For MPL MPC strategies, van den Boom and De Schutter44

proposed a bound on the suppression weight: 0 < < 1. vanden Boom et al.45 further refined this by proposing a boundbased on the prediction horizon: < 1/P, where P is the finalprediction horizon. Both of these bounds apply for constrainedSISO MPL MPC. For unconstrained MIMO systems, Necoaraet al.43 suggested a slight adjustment of the approach proposed

in ref 45 < 1/mP, where m is the number of manipulated inputsand P is the final prediction horizon.The rate-of-change-of-inputs weights can be used as system-

atic constraint to minimize controller action. From the theoremin ref 1, there is always a weight that guarantees closed-loopstability for DMC controllers. Knowing this, as long as onesimulates the system and confirms that the closed-loop systemis stable with a particular value of rate-of-change-of-inputweight, increasing above this value is at the discretion of thecontrol engineer based on the knowledge of the process tofurther suppress controller action. However, increasing it toomuch creates a sluggish controller, which may be unwanted ina process with small time constants.

3.6. Weights on the Magnitude of the Inputs. This sectionreviews the known strategies for tuning the weights on themagnitude of the input variables. This parameter penalizesexcess controller action. It is a way to remove a constraint fromthe optimization problem and thus make it more computationallyattractive. This particular penalty allows one to remove theconstraints on the minimum and maximum sizes of the inputs.

Nagrath et al.25 used a value of the weights on the magnitudeof the inputs of zero. The authors reasoning was that it wasunnecessary since there were no magnitude constraints for thesystem they studied. This tuning strategy was applied to anonlinear state-space model of a CSTR. Garriga and Soroush51

quantitatively described the effect of increasing the size of theweights on the magnitude of the inputs. In the limiting case, asf , the Jacobian of the closed loop approaches the Jacobianof the open loop and thus the controller is operating in openloop mode.

3.7. Reference Trajectory Parameters. This section reviewsthe purpose and tuning of the reference trajectory parameters,0 e j < 1, j ) 1, ..., p. The reference trajectories are used toensure a smooth transition from the current output values tothe desired set point values.36

Clarke et al.36 suggested using a value of the referencetrajectory parameter around 1 to ensure a slow transition fromthe current measured variable to the real set point. However,Garriga and Soroush51 noted that the reference trajectoryparameter loses its effect when a long final prediction horizon

is implemented. Seborg et al.52 proposed a way to set thereference trajectory parameter. The desired time constant for

each output is specified indirectly by a performance ratio for

the output, where the performance ratio is the ratio of the desiredclosed-loop settling time to the open-loop settling time.52 Inthis way, a small performance ratio correlates with a smallreference trajectory parameter.

3.8. Constraint Parameters. This section provides tuningguidelines regarding when the constraints on the system areactive or inactive. Typically, the objective to tuning theconstraints involves knowing the window when the constraintsare active or inactive in order to make the optimization problemfeasible. Table 7 lists the tuning guidelines for the constraintparameters discussed in this section.

Sarimveis et al.50 provided a method to determine the windowof active output soft constraints for DMC-type controllersapplied to nonsquare systems. Rawlings and Muske46 proposedtwo methods to tune the output constraints based on the stabilityof the open-loop system. Using an infinite horizon controlmethod, the authors derived two bounds on when the constraintsare active in a stable plant. Agarwal et al.53 used an algorithmbased on Bayesian inference networks and statistics to determinewhich input and output constraints can be changed to optimizethe process. The inference network was used to determine whichvariable should be used as evidence in the statistical calculation,and based on a set target, the probabilities of the other variableswere calculated and determined whether or not to change theconstraint. This method applies to all MPC formulations.

3.9. Covariance Matrix and Kalman Filter Gain. Thissection presents a review of several methods of determining

the covariance matrices in a Kalman filter when it is necessaryto estimate a state in the absence of a measurement of the stateneeded to calculate MPC control moves. Since the covariancematrices are not known a priori, they must be estimated usingone of four known methods: Bayesian approaches,54,55 maxi-mum likelihood,56,57 covariance matching,58 and correlationtechniques.59-62 According to Odelson et al.,19 Bayesian andmaximum likelihood methods have not been used widelybecause their computation time is often excessive. Covariancematching is the calculation of the covariance matrices from theresiduals of the state estimation, but it has shown to give biasedestimates of the true values.19

Odelson et al.19,20 introduced a new method for estimating

the covariance matrices based on correlation techniques andgives necessary and sufficient conditions for the estimate toapproach the true value as the sample size increases. Odelsonet al.19 used this method to solve an optimization problem forthe optimum gain (Kalman) filter, even though there may notbe enough information to find the covariance uniquely. Sincethe optimization is nonlinear, the authors suggested initiallysetting the filter L to some value, process the data and computeA1 and R 1, and then solve the linear least-squares minimizationover only Rv and PCT. Substituting these values into theconstraint then gives a new value for L. This procedure shouldbe repeated until convergence.19 The authors tested the methodon a laboratory-sized reactor and in an industrial-sized facilityat the Eastman Chemical Company. kesson et al.63 proposed

a predictor-corrector algorithm, which can be viewed as anextension of an interior-point method for solving quadratic

Table 7. Guidelines for Tuning the Constraint Parameters


DMC k1 - 1 g Mg max((ujmax - ujmin)/ujmax) - 1,1 e j e ni

50

state-space k1 ) max{[ln(hmin/[|H|K(T)|x0|])]/ln(max), 1} 46k2 ) M + max{[ln(hmin/[|H|K(T)|xM|])]/ln(max),

0}



8/11

programming (QP) problems, to solve the optimization problemand determine the optimum Kalman filter. Valappil and Geor-gakis64 used a Taylor series expansion around the nominalparameter values, on the assumption that the dependence waslinear. This gave a closed-form solution for the covariancematrix. To account for nonlinearities, the authors also proposedusing Monte Carlo simulations to account for the dependenceon the fitted parameters.

4. Auto (Self) Tuning Methods

This section provides a review of several autotuning proce-dures available in the literature for MPC. The obvious advantageto using an autotuning method is that the control engineer isnot required to have a great amount of systems knowledge toinitialize the tuning procedure. Similarly, the tuning parametersare updated along with the optimization algorithm; thus, theyare always set to the optimal values. However, this comes at acomputational price since the engineer is now required tocalculate two optimization procedures per time step, rather thanthe one in an offline tuning strategy.

Han et al.65 proposed an autotuning strategy for unconstrainedDMC using particle swarm optimization (PSO). As a startingpoint for tuning values within the optimization, the authorssuggested using the method proposed in ref 13. Their procedureto tune the parameters via PSO is outlined in ref 65. Once thecriteria are met in the sixth step, the output of the optimizationis a solution to the min-max problem posed within paper:minimize the tuning parameters while maximizing the use ofthe process model parameters. This strategy allows for rapidautotuning each time the optimization problem is calculated.Suzuki et al.66 used PSO to tune MPC. Instead of tuning all ofthe parameters, the authors allow the user to set the predictionand control horizons offline and then use PSO to determineautomatically the weights on the magnitude of inputs, outputs,and rate of change of inputs. This saves computational time

since the optimization to determine the tuning parameters issmaller than in ref 65. van der Lee et al.67 used geneticalgorithms (GA) and multiobjective fuzzy decision making(MOFDM) to autotune unconstrained DMC. The processinvolves: (1) setting the tuning parameters to an initial value;(2) running simulations for a set of initial parameters; (3)calculating the objective for each tuning parameter set; (4)performing MOFDM; (5) discarding the worst set of tuningparameters and replace with the best; (6) performing GA if lessthan the number of predetermined generations; otherwise, exitwith the optimal tuning parameters. This was applied to a hotwater mixing tank with four different definitions of optimality.

Al-Ghazzawi et al.68 proposed an autotuning strategy for

constrained DMC that involves the exploitation of thesensitivity of the closed-loop response to the tuning param-eters. The online tuning strategy involves a linear approxima-tion of the relationship between the process output and theMPC tuning parameters.68 This allows for an automaticadjustment in the tuning parameters so that the control actiondoes not violate the constraints. Baric et al.69 have proposeda redefinition of the objective function to include theoptimization of the weights on the magnitude of the inputs.It involves changing the traditional terminal state costfunction into a new linear programming cost function withthe vector of optimizers, which represent the upper boundson the components of vectors x(k) and u(k), respectively.69

Once the new optimization problem is solved, the control

engineer not only obtains the solution for all feasible statevariables, but also determines the input weights for the

specified range. This strategy is applicable to constrainedDMC. Sanchez et al.17 proposed an algorithm to tunerestricted structure control systems using subspace identifica-tion of the process model. Once the first N terms of theimpulse response model are known, the multivariable pa-rameters can be calculated by minimizing a finite horizonLQG optimization problem subject to nonlinear constraints.

Liu and Wang70 proposed two algorithms for the self-tuningof GPC-type controllers. Their recursive algorithms ensure boththe satisfaction that the objective function reaches a minimumand that there is convergence upon optimal tuning parameters.It applies to unconstrained GPC only since it requires solving

a closed form control increment. Similarly, Tsai et al.71,72proposed a self-tuning strategy for a GPC controller. Theyapplied the strategy to an oil-cooling machine used for coolinghigh speed machine tools71 and a plastic injection moldingprocess.72 The algorithm is based on using a pseudo randombinary signal (PRBS) as a test signal for the initial parameterestimation via recursive least-squares estimation (RLSE). Oncethis is done, the algorithm automatically calculates the optimalvalues of the tuning parameters.

5. Case Study

This section presents a comparison of closed-loop perfor-mances obtained using several tuning guidelines. The processis a continuous-stirred tank reactor (CSTR), in which a first-order reaction takes place. A model predictive controller isimplemented to control the temperature of the reactor. Theprocess has a model of the form:

The model parameter values are given in Table 8.The chemical reactor has three equilibrium points (steady

states) corresponding to Q ) 3 kJ/min: a high A-concentration,low temperature stable node (9.969 kmol/m3, 323.26 K), a lowA-concentration, high temperature stable node (0.403 kmol/m3,564.83 K), and a mid A-concentration, mid temperature saddlepoint (5.190 kmol/m3, 443.94 K). Using a first-order truncatedTaylors series expansion of the model around the highA-concentration, low temperature stable equilibrium point, thelinear approximation of the nonlinear model around the steadystate is described by

Table 8. Parameter Values of the CSTR Model

R ) 8.314 100 kJ/kmol KZ ) 2.000 104 1/minEa ) 5.600 104 kJ/kmol-H ) 5.000 104 kJ/kmolFs ) 9.000 102 kg/m3

c ) 2.200 100 kJ/kg KTi ) 2.952 102 K ) 1.800 102 minV ) 1.000 10-2 m3

CAi ) 1.000 101 kmol/m3

dCAdt

) -Zexp(-EaRT )CA +CAi

- CA

dTdt

)(-H)Z

Fscexp(-Ea

RT)CA + Ti - T +

Q

FscV

A ) [-0.00557 -1.15 10-5

0.000451 0.00527 ]b )

[

0

0.0505 ]c ) [ 0 1 ]



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where the manipulated input u ) Q and the controlled outputy ) T. The control objective is to operate the reactor at the lowconcentration, high temperature, stable equilibrium point, whileinitially (at time t ) 0) the reactor is at the high concentration,low temperature stable steady state. Using the tuning strategiesoutlined in refs 13, 29, and 31, a closed-loop simulation of theCSTR is carried out using the model predictive control toolboxof MATLAB. The values within the estimation tab in the modelpredictive control toolbox remained at their default settings. Onthe basis of the strategy in ref 13, P ) 7, M ) 4, and )0.0342; on the basis of the strategy in ref 29, P ) 38, M ) 9,and ) 0.4; and on the basis of the strategy in ref 31, P ) 6,M ) 1, and ) 0.4293.

The responses of the reactor temperature are depicted inFigure 1. One can see that the first strategy yields a fasterresponse, but at the premium of a more aggressive manipulatedinput profile, rate of energy supplied to the reactor, as shownin Figure 2. Since this plant has a large apparent time constant(275 min), the relatively small gain in response time of theoutput is not justified by the much higher input magnituderequired to achieve it. Moreover, the third tuning strategy hadno overshoot, while the first had the largest overshoot of allthree strategies. On the basis of this, the third tuning strategyyields better output response than the first one. Using the second

strategy, both the final prediction horizon and the control horizonare higher than the values that resulted using the FOPDT tuning

strategies. From Figure 1, one can see that the response time ofthe process lies somewhere in between the two FOPDT tuningstrategies. Figure 2 shows that the input necessary for thisstrategy is overall about the same as the one needed in ref 31.However, increased cost in computation due to the higher Pand M may not be justified using this strategy.

From this example, there is a trade-off between highercomputational cost and robustness on one end and ease of tuningwithout being able to operate at an unstable equilibrium point.

If the control engineer wishes to operate within the stableequilibrium zone or from one stable equilibrium point to another,then the strategy in ref 31 yields good results. However, if oneneeds to operate in an unstable equilibrium zone, then refs 13and 31 cannot be used since they are for stable plants only andthe control engineer must use the tuning guidelines proposedin ref 29.

6. Concluding Remarks

This paper provided a broad review of many tuning methodsfor several classes of MPC formulations. The review coversboth theoretically based strategies and heuristic/industrialmethods to tune not only DMC and GPC MPC, but also general

state-space representations and other formulations such as MPLMPC. The study also touched on methods of calculating thecovariance matrices for state estimation and the Kalman filtergain along with auto (self) tuning procedures.

As stated before, the first act of tuning in MPC is to developan appropriate process model. If the model is accurate enough,then the rest of the tuning will be straightforward. And if thecontroller exhibits poor performance, then one should considerthe model poor (inaccurate) until proven otherwise. In thisreview paper, methods for proper model development orselection for MPC applications were not reviewed. In recentyears, advances in subspace identification and automated testinghave allowed practitioners to dramatically reduce the cost of

tuning MPC applications.22

Advances in the autocovarianceleast-squares technology are expected to make Kalman filteringmuch more accessible by automatically identifying the maintuning parameters.

Acknowledgment

This study was supported in part by the National ScienceFoundation (NSF) through the grant CBET-0651706. Theauthors would like to thank Thomas A. Badgwell for hisinsightful comments on MPC tuning methods.

Notation

c ) heat capacity of the reacting solutiondmax ) maximum dead timegi ) estimated coefficients of the step response modelhmin ) smallest output constraint boundk ) adjustable gain factor in ref 21, typically set to 1k1 ) time (in samples) when the constraint becomes active in ref

50k1 and k2 ) lower and upper bound of the output constraint window

in ref 46m ) detuning factor in refs 14 and 15na ) order ofA polynomial in GPC formulationnb ) order ofB polynomial in GPC formulationni ) number of inputs in ref 50nw ) output constraint window lengtht10 ) rise time for 10% of process steady-statet60 ) rise time for 60% of process steady-state

Figure 1. Temperature responses for the three tuning strategies.

Figure 2. Manipulated input responses for the three tuning strategies.



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t80 ) rise time for 80% of process steady-statet90 ) rise time for 90% of process steady-statet95 ) rise time for 95% of process steady-statet95,i ) rise time for 95% of yi process steady-statetd ) process delay timetp(t) and tp(t - 1) ) current and previous tuning parameter value,

respectivelytr ) process rise timets ) process settling time

umax ) maximum inputujmax ) maximum value for input uj in ref 50umin ) minimum inputujmin ) minimum value for input uj in ref 50uZ,s ) input zero direction for each respective inputx0 and xM ) vectors of initial and final state valuesymax ) maximum outputymin ) minimum outputyZ,s ) output zero direction for each respective outputA ) shaped plants state matrix in ref 49B ) shaped plants input matrix in ref 49B(1) ) B polynomial of the GPC process model evaluated at 1CAi ) concentration of reactant A in the inlet stream of the CSTR

CA ) concentration of reactant A in the outlet stream of the CSTREa ) activation energyEi ) maximum additive errors in the step response modelG ) lower triangular matrix of estimated model coefficientsH ) output constraint matrixK(T) ) condition number of the Jordan decomposition of the state

matrix AKp ) PID controller gain from ref 31Krs ) FOPDT model gain for R outputs and S inputsM ) control horizonN ) model horizonP0 ) initial prediction horizonP1

f ) final prediction horizon for first constraint in ref 39

P2f

) final prediction horizon for second constraint in ref 39P ) final prediction horizonPd ) desired performancePm(t) ) current measurement of the performanceQ ) weights on the outputsQr ) output weight for R outputsQ ) rate of thermal energy supplied to the CSTRR ) universal gas constantRPN ) robust performance numberT ) temperature of the outlet stream of the CSTRTi ) temperature of the inlet stream of the CSTRTs ) sampling time (period)V ) volume of reacting mixture in the CSTRX ) stabilizing solution of LQR problem in ref 49Z ) stabilizing solution of Kalman filter in ref 49Z ) pre-exponential factor of the reaction rate equationmax ) absolute value of the maximum eigenvalue of the state matrix Arel ) constant computed from the initial values of the weights in

refs 14 and 15F ) spectral radiusFs ) solution density) CSTR residence timep ) FOPDT process time constantd,rs ) FOPDT process delay constant(s) for R outputs and Sinputsrs ) FOPDT process time constant(s) for R outputs and S inputssup ) frequency where RPN reaches maximal value ) tuning factor in ref 31 based on the desired performance of

systemumax ) maximum change in input

ujmax ) maximum change in value for each input in ref 50(-H) ) heat of reaction ) weights on the rate of change of inputs

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ReceiVed for reView February 25, 2009ReVised manuscript receiVed September 19, 2009

Accepted February 18, 2010

IE900323C


2 model predictive control tuning methods a review

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