A Parametric Dynamic Matrix Controller Approach for ...A Parametric Dynamic Matrix Controller Approach for Nonlinear Chemical Processes EDINZO IGLESIAS1, OSCAR CAMACHO2;5, MARCO SANJUAN3,

A Parametric Dynamic Matrix Controller Approach for NonlinearChemical Processes

EDINZO IGLESIAS1, OSCAR CAMACHO2,5, MARCO SANJUAN3, CARLOS SMITH4

SILVIA M. CALDERON1, ANDRES ROSALES5

1 Escuela de Ingenierıa Quımica2 Escuela de Ingenierıa Electrica

Universidad de Los AndesNucleo La Hechicera, Merida, 5101

VENEZUELA3 Departamento de Ingenierıa Mecanica

Universidad del NorteBarranquilla, Atlantico

COLOMBIA4Chemical Engineering Department

University of South FloridaTampa, FL 33620

USA5Facultad de Ingenierıa Electrica y Electronica

Escuela Politecnica NacionalQuito

[email protected], [email protected],[email protected], [email protected], [email protected],

[email protected]

Abstract: This study presents a Parametric Dynamic Matrix Controller (PDMCr) to handle inde-pendently the process gain, time constant and dead time effects over the DMC performance. Thecontroller is complemented with a fuzzy supervisory module that monitors changes on the processnonlinear behavior and keeps updated the embedded process model, saving time by avoiding repeatingthe open-loop step response identification. The new PDMCr is able to improve regulation and controltasks, even in those cases when the operation point is far from the initial identification point. Threeexamples are used to assess the PDMCr performance, a linear FOPDT and two nonlinear chemicalprocesses: a mixing tank, and a neutralization reactor. The PDMCr improves the DMC performance,tracking set point and rejecting disturbances with shorter settling times and less overshoot, it remainsstable when DMCr oscillates; and show more tolerance to noise than the DMCr, keeping a stablebehavior for longer times under noisy input signals. When both performances are comparable, the IAEvalues for the PDMCr are until 23% lower compared to the DMCr.

Key-Words: Nonlinear DMC, Nonlinear Chemical Processes, Parametric DMC, Tuning parame-ters, Fuzzy Logic, Supervisor Module

1 Introduction

Dynamic Matrix Control (DMC) was originally de-veloped by Cuttler and Ramaker in 1979 [1]. It wasfirst intended to fulfill requirements of petrochemical

and power plants; more than three decades later, it hasbeen successfully applied in chemical, food process-ing, pulp and paper and aerospace industries [2] [3].DMC is a Model Predictive Control technique, whichdecisions are driven by real data of the process dy-

WSEAS TRANSACTIONS on SYSTEMS and CONTROLEdinzo Iglesias, Oscar Camacho, Marco Sanjuan, Carlos Smith, Silvia M. Calderon, Andres Rosales

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namics over a future horizon time, this is its key issue[4]. The essence of DMC is to find the optimal con-trol output vector using a least squares solution for theresiduals of non-compensated errors coming from theDynamic Matrix, a linear combination of future out-puts and future inputs of the process [2]. It is a ModelBased Control (MBC) which success is due to its abil-ity to anticipate and eliminate feed forward and feed-back disturbances [4] with a less aggressive outputand a more robust response compared to other tech-niques such as the traditional and universally imple-mented PID control [5]. As a MBC technique, DMCallows intrinsic dead time compensation [2], and dueto its Dynamic Matrix can be easily extended to mul-tivariable cases [4]. Industrial processes are nonlin-ear by nature and DMC approaches them using a lin-ear step response model. This may result in nega-tive consequences such as very slow to oscillatory re-sponses [6] [7] [8] and in some cases, large IntegralSquared Error (ISE) values for set-point tracking dueto steady state error [5]. Ill-conditioned matrices arealso a problem and lead to poor DMC performances[9].Currently, many of the DMC improvement alterna-tives are focused on two points, improvements on howto deal with nonlinearities in MIMO systems withstrong variable interactions [10], and improvementson the control parameter tuning for horizon and movesuppression factor [11]in order to avoid fluctuationand decrease execution frequency [5].Proposals to solve the DMC weaknesses can begrouped in two sets: one set is focused on the DMCalgorithm reformulation, and the other one is focusedon the control parameter tuning. Examples in the firstset are McDonald and McAvoy [8], Brengel and Sei-der [12], Peterson, et. al. [13], Aufderheide and Be-quette [14]. Some of these proposals suggest the addi-tion of a disturbance vector to take into account the ef-fect of nonlinearities in the prediction horizon [13], ora multiple model structure: standard DMC plus FirstOrder Plus Dead Time (FOPDT) [14]. McDonaldand McAvoy [8] proposed a gain and time schedulingtechnique to update the DMC algorithm and enhanceits control performance. The second set uses the con-trol parameter tuning to improve DMC performance.Recently Jeronimo and Coelho [11] used SISO pro-cesses represented by a FOPDT model to proposeauto tuning and self-tuning methods based on DMCminimum realization (optimal move suppression fac-tor and optimal horizon) and online minimization ofthe objective function to reduce set-point tracking er-

ror and control ringing.The work presented here proposes a hybrid approachthat includes changes in the standard DMC’s con-trol law to isolate the process gain, time constantand dead time effects; and an auto-tuning module.The traditional Dynamic Matrix Controller (DMCr) istransformed into a Parametric Dynamic Matrix Con-troller (PDMCr). Also an auto tuning technique issuggested to calculate the optimal move suppressionfactor based on regression analysis using a FOPDTmodel. The input parameters for tuning come from afuzzy supervisor module that monitors changes on theprocess nonlinear behavior, detects them and sendsthe information to update the PDMCr. The newPDMCr is able to improve regulation and controltasks, even in those cases when the operation pointis far from the initial identification point.The paper is organized as follows: section two showshow to implement a DMCr for a SISO system. Sec-tion three describes the proposed PDMCr, and sectionfour presents the fuzzy supervisor model. Section fivepresents the PDMCr performance in a mixing processand a neutralization reactor compared to the conven-tional DMCr. The final section shows the noise effecton the PDMCr, and finally conclusions are presented.

2 Conventional DMC Implementa-tion

There are many ways to describe the DMC algorithm;for the purpose of this research, the form suggestedby Sanjuan [15] is used. The following discussiondescribes the DMC implementation for Single-Input-Single-Output (SISO) systems. The process is identi-fied first using a step change in the signal to the valveinput (∆m), sampling the sensor signal provides theprocess response. This data is collected in a samplingvector (Sv) until a new steady state is reached. A sam-pling time between one tenth and one fifth of the pro-cess time constant is usually recommended, resultingin a Sampling Size (SS) between 25 and 50 samples.The length of the sample vector is directly related tothe Prediction Horizon (PH), which is the number offuture steps where the process output variable will bepredicted. Each component of the sampling vectoris transformed by subtracting the final steady state.Each element of this new vector is then divided by∆m. The final component of this new vector (Av) isthe process gain KP. For the DMC algorithm the vec-tor Av is used to build the dynamic matrix process A


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as follows:

A =

Av 0 . . . . . . 0KP Av 0 . . . 0KP KP Av 0 0. . . . . . . . . . . . . . .KP KP . . . . . . Av

PH×CH

(1)

The number of columns of A in Eq.(1) correspondsto the Control Horizon (CH). For industrial appli-cations only 5 or 10 step moves are investigated bythe optimizer [13] [16]. The lower CH is, the slowerthe controller (less aggressive) will be and vice versa.Once CH is chosen, the Prediction Horizon (PH) isdefined as:

PH = SS+CH−1 (2)

The dynamic matrix control A Eq.(1) is now used tofind the control output vector ∆M. This task is doneby the minimization of the residuals R from the pre-dicted error vector E using a least square technique.

A∆M = E +R (3)

The controller output ∆M can be calculated from Eq.(4)

∆M =(AT A

)−1AT E (4)

Usually a suppression factor λ is used as a tuning pa-rameter to change the aggressiveness of the controller,in that case the control move is expressed as:

∆M =(AT A+λ

2I)−1

AT E (5)

Equation (5) is the DMC standard control law pro-posed by Cutler and Ramaker [1]. This study pro-poses to isolate the dynamics process characteristics(process gain KP, time constant τ, delay time to) fromthe A matrix, in order to take direct actions over theirspecific changes.

3 A New Approach for DMC Struc-ture

As it was previously mentioned, the DMC control lawin Eq.(5) is built upon the assumption of a fixed linearprocess model. This becomes a limitation in somecases where process are highly nonlinear and there-fore very sensitive to disturbances. For highly non-linear processes, a more convenient way to expressthe DMC control algorithm could be in a parametric

form, as a function of the process parameters: gain,time constant and dead time, and the suppression fac-tor λ [17]:

∆M = f (KP,τ, to,λ) (6)

Once the effect of each process parameter is isolatedfrom the control law, this algorithm can adapt itselfto specific changes in the process gain, and/or pro-cess time constant, and/or dead time without the needof additional identification steps. All of this keepingthe smoothing role of the suppression factor, and theDMC’s original advantages such as its ability to han-dle multivariable cases and processes with high delaytimes.The proposal for a new controller algorithm beginswith Eq. (1), where the process gain KP can be fac-tored out as a common term. The Av can then be ex-pressed as a function of a vector V whose elementsvaries from 0 to 1 and contains information about theprocess time constant and the dead time. The dy-namic process matrix becomes equal to:

A = KP

V 0 . . . . . . 01 V 0 . . . 01 1 V 0 0. . . . . . . . . . . . . . .1 1 . . . . . . V

PH×CH

= KPU

(7)

The DMC control law in Eq.(5) is transformed usingthis new process dynamic matrix Eq.(7) into a newform called Parametric Dynamic Matrix Control law(PDMC).

∆M =1

KP

(UTU +λ

2I)−1

UT E (8)

Similarly, when the process dead time changes, thosechanges can be included directly into the matrix Uusing the vector V .

V =

00...1

SS×1

=

[Zun×1

Du(SS−n)×1

]SS×1

(9)

In Eq.(9) Zu is a a vector with zeros which dimensionsn× 1 are related to the process dead time, because nrepresents how many sampling periods Ts correspondto it. The vector Du is composed by the remaining nonull elements in vector V . A graphical representationis shown in figure 1. If Eq.(9) is used to build the


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Fig. 1: Discrete Data Contended Inside Vector Av

process dynamic matrix, this becomes equal to:

A = KP

Zu 0 0 0 . . . 0Du Zu 0 0 . . . 01 Du Zu 0 . . . 01 1 Du Zu . . . 0. . . . . . . . . . . . . . . . . .1 1 1 . . . 1 Zu1 1 1 . . . 1 Du

PH×CH

(10)

Equation (11) shows that the U matriz is composed oftwo matrices, the Z and D matrices. The Z matrix iscomposed by Zu vectors, one in each column of the Umatrix; their values are defined by the designer. TheD matrix is composed for the remaining elements inthe U matrix.

A = KP

[Zn×CH

D(PH−n)×CH

]PH×CH

(11)

To illustrate how these changes will affect the con-trol law, we show here how the term

(UTU

)−1UT

changes if Eq.(11) is included into in the PDMC lawEq.(8) (without the suppression factor to keep it sim-ple):

(UTU

)−1UT =[ZT

CH×n... DT

CH×(PH−n)

] Zn×CH

. . .DCH×(PH−n)

−1

×

[ZT

CH×n... DT

CH×(PH−n)

](12)

(UTU

)−1UT =

(ZT Z +DT D

)−1[ZT ... DT

]=[(

DT D)−1 ZT ...

(DT D

)−1 DT]

=[Z

...(DT D

)−1 DT]

(13)

After some mathematical manipulations and the ad-dition of the suppression factor λP the proposed formfor the PDMC law Eq.(8) can be written as:

∆M =1

KP

([Z

...(DT D

)−1DT]+λ

2PI[

ZT ...DT])

E

(14)

The suppresion factor inside Eq.(14) regulates thecontroller aggresiveness, and it is calculated as inequation (15).

λP =λ

KP(15)

Standard DMC tuning equations for λ [16][3][18]were tested on nonlinear processes in a previous workand gave very aggressive controller responses [19].For this reason we used the equation proposed byIglesias [17]:

λ = 1.631KP

( toτ

)0.4094(16)

Equation (16) is the final result of a two-stage proce-dure (factorial experiment plus nonlinear regressionanalysis) designed to measure the effect of changes inthe process variables due to the controller aggresive-ness. To measure the controller agresiveness, Iglesias[17] defined a cost function, called Performance Pa-rameter (PP). The cost function combines the Inte-gral of the Absolute Value of the Error (IAE) and theIntegral of the Absolute Value of the Change in Ma-nipulated Valve signal (IMV). This cost function isexpressed as:

PP =∫

∞

0|e(t) |dt +Γ

∫∞

0|mss−m(t) |dt (17)

The PP depends on process parameters such as KP, τ,toτ

, Tsτ

and Γ (a weighted parameter in the cost func-tion). A 35 factorial experiment was performed tostudy their effect over PP. Only main effects andsecond order interactions were considered. The ef-fect of each variable and their possible interactionswere tested using the optimal λ values (those thatminimize the PP) for each one of the experiments.All experiments were performed on a FOPDT sys-tem in SISO control loop under set point changes


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(±10%TransmitterOut put) at different times.Significant factors (p < 0.05) were the primary vari-ables and their interactions KPτ, KP

toτ

, τtoτ

and toτ

Tsτ

.The optimal suppression factors (from each one ofthe signicant variables and their combinations) werefit to different nonlinear tuning equations and the bestequation found is presented in Eq.(16) [17] [19].Once the PDMCr agressiveness was handled, the nextstep was to find how the changes in the process timeconstant could be compensated in the PDMC algo-rithm. Due to the complex dynamic behavior of realprocesses, there is not an easy way to isolate the infor-mation concerning to the process time constant, as itwas previously done for the process gain and the deadtime. The matrix D in Eq.(14) contains the processdynamic information after the dead time. Therefore,the process time constant is inside this matrix. A non-linear correction to adjust changes in the process timeconstant is proposed:

Dad j = D

1− exp−kiTs

τnew

1− exp−kiTs

τprev

(18)

The terms inside Eq.(18) are ki, the ith term in matrixD row; Ts, the sampling time used to record processdata; τnew is the new process time constant; τprev is thenew process time constant. The correction factor is aratio of two exponential terms with the typical FirstOrder Plus Dead Time (FOPDT) form. The numer-ator is a function of the new process time constant,whereas the denominator is a function of the previousprocess time constant value. The exponential form ofthe time constant correction was chosen based on thegood agreement between empirical models and thedynamic behavior of real processes. The effective-ness of our PDMCr is restricted to those processesable to be modeled as FOPDT processes. The ideawas to adjust the ith term in the D matrix using its cor-responding correction factor as expressed in Eq.(18).The process time constant correction completes thePDMCr, and its control law is then expressed as:

∆M =1

KP

([Z

...(DT

ad jDad j +λ2PI)−1

DTad j

])CH×PH

×E

(19)

Eq.(19) is the parametric law of the Dynamic MatrixController. Its mathematical form offers some advan-tages when it is neccesary to update the process modelafter disturbances. If the process gain changes, thechange is included into the PDMC model using the

factors 1KP

and λP. If the process dead time changes,the change is included into the PDMC model, Eq.(14) changing the number of columns n of matrix Zinside the matrix D. If the process time constant τ

changes, the Dad j matrix is adjusted using the cor-rection factor. All the adjustments can be done withno need to recalculate the Process Dynamic Matrix A.This could save time because the controller algorithmdoes not need a new identification procedure after dis-turbances.

4 Definitions of Function Spaces andNotation

To use the PDMCr in an efficient way (adjust thecontrol system to compensate for nonlinearities), it isnecessary to detect and quantify changes in processparameters on-line, as soon as they happen, and trans-fer these changes to the PDMCr and tuning equation.There are several ways to do this, but one of the mostreliable procedures is based on the concept of mod-eling error [20][21]. Modeling error, em, is definedas the difference between the actual process output ofc(t), and the controller predictive value cP (t) fromthe embedded model inside the controller. This con-cept applies for all MBC controllers.In a multivariable model there are several ways to de-termine the modeling errors. For example, em can becalculated using the time to reach the steady-state, theratio of maximum peak to minimum peak, the damp-ing ratio, the decay ratio, etc. In this work 19 differentindicators were studied and they are presented in Ta-ble 1. These indicators, defined as Modeling Error In-dexes (MEI), were statistically related to the changesin the process parameters reported in Table 2. Valuesfor the Modeling Error Indexes were measured fromthe process response after each one of the changes inTable 2 was applied.A total of 61236 simulations were performed to eval-uate the 19 indexes in Table 1. Just a set of 15 MEIsshowed statistically significant correlations with thechanges in the process parameters (p < 0.05). Multi-ple linear regression analyses were applied to these15 indicators using Equation (20) [17]. Just threeof the MEIs, Time for Maximum Peak/τ, Time forMinimum Peak/τ and 10th Correlation Coefficient ofmodeling error, achieved statistically significant re-sults and showed the maximum changes in their val-ues. The multilinear model coefficients for Eq.(20)


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are shown in Table 3.

MEI = β1 +β2τ+β3toτ+β4∆KP +β5∆τ+β6∆to+

(20)

β7τtoτ+β8τ∆KP +β9τ∆τ+β10τto

The optimal MEI were included in a Fuzzy Infer-ence System (FIS) to create a Supervisor Module forpredicting on-line changes in process parameters andreadjusting the process model, as well as the suppres-sion factor in the PDMCr. The Supervisor Modulewas designed to work only when a set point changeis detected. This design could seem peculiar becauseset points are commonly constant. However the ideabehind this design is to use the set point change asa tool to test the system for changes in process pa-rameters that could cause controller aging. From apractical point of view this mean that when a plantoperator detects an inadequate control loop responserejecting disturbances, the operator would perform aset point change in one direction and later the samechange in the opposite direction to return back to theinitial conditions. In this way the supervisor has twoopportunities to evaluate the process parameter andadjust the PDMCr to the new conditions.

The supervisor records the em values until the sys-

Fig. 2: Nonlinear relationship among two of the MEIsand γ∆K p

tem reaches the new set point. The data is used todetermine the Time for Maximum Peak/τ (MEI1) ,Time for Minimum Peak/τ (MEI2) and 10th Corre-lation Coefficient of modeling error (MEI3). Thesevalues are then used to determine the changes in theprocess parameters ∆KP, ∆τ and ∆to by the optimiza-tion of the cost function CF shown in Equation (21),

Table 1: Modeling Error Indexes used to predict Pro-cess Parameters Changes


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Table 2: Factors and Levels Used to Record ModelingError and Develop the Regression Equations

∆Cset (%TO) KP(%TO

%CO

)τ

toτ

∆KP ∆τ ∆to-15 0.5 1 0.2 -40 -40 -15-7.5 1.5 3 0.6 -30 -30 -107.5 2.5 5 1 -20 -20 -515 -10 -10 0

0 0 510 10 1020 20 1530 3040 40

Table 3: Parameters for Regression Equations Usedin Supervisor Module

β Time for Maximum Peak/ τ Time for Minimum Peak/ τ 10th Corr. Coeff. of emβ1 12.5494 7.96350 -0.64086β2 -2.3297 -1.20200 -0.00542β3 0.789060 5.31050 0.26446β4 0.003125 0.19760 0.01052β5 -0.003750 -0.03070 0.00154β6 -0.033750 0.14062 0.05056β7 0.210940 0.18164 0.08960β8 0.210940 0.00546 0.000924β9 -0.0003125 -0.04484 0.000354β10 0.037500 -0.00312 -0.00409

where F1, F2 and F3 are the appropriate regressionequations corresponding to each MEI. All of themhave the same form that Equation (20) with the pa-rameters defined on Table 3. The results for ∆KP, ∆τ

and ∆to are expressed as a percentage of change in therespective parameter. Then, the new process param-eters are estimated using the following expressions,where γ is a correction factor with value between 0and 1.

CF = |MEI1calc−F1 (β,∆KP,∆τ∆to) | (21)

+|MEI2calc−F2 (β,∆KP,∆τ∆to) |+|MEI3calc−F3 (β,∆KP,∆τ∆to) |

KPad j = KP

(1+

γ∆K p∆KP

100

)(22)

τad j = τ

(1+

γ∆τ∆τ

100

)(23)

toad j = to

(1+

γ∆to∆to100

)(24)

There is an uncertainty in the β values due to the in-evitable failures in the goodness of fit. The idea be-hind the γ factor is improving the process parameterprediction using Fuzzy Rules. These rules take ad-vantage of the experience gained by the authors ascontrol engineers, after running more than 60000 sim-ulations for different process changes. Every MEIvalue is fuzzified using three membership functions:Zero (Z), Small (S) and Large (L). Once the fuzzyrules are evaluated for each input, the Fuzzy inferencesystem gives a set of fuzzy values as a result. For ex-ample, if the Time for Maximum Peak/τ is Large, theTime for Minimum Peak/τ is Large and the 10th Cor-relation Coefficient of modeling error is small, thenthe γ∆K p is small. The final defuzzification operationto transform fuzzy values of γ into crisp values is per-formed using the centroid method [17].Figure 2 shows the kind of nonlinear relationshipamong MEIs and γ∆K p that can be obtained evaluatingthe fuzzy rules. It would be very complex to expressmathematically the relationship among the variables,but using 12 Fuzzy Logic rules is easy to do so. Oncethe initial process parameters are corrected based onthe information provided by the modeling error, theadjusted process parameters are sent to the PDMCr toadjust the control law and determine the best tuningparameter. Figure 3 shows a schematic representationof the proposed approach.

Fig. 3: Schematic representation of the ParametricDMC

5 Simulation Results

The process model uncertainty is one the most impor-tant limitations of all MPCrs, including those basedon the DMC algorithm. Controllers work as wellas the model fits the real process behavior. If non-representative data from the process dynamic is fedto the controller at the identification stage, there


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will surely be time, resources and money losses in-volved on its industrial implementation [2]. Oncethe controllers have been implemented, their embed-ded model cannot be adjusted and validated to vary-ing process conditions, and therefore suffer signifi-cant deterioration, leading controllers to unsuccessfulperformances [2].A set of tests were performed to evaluate the PDMCr.The first test was designed to use a known FOPDTsystem as the process in the control loop. A sequen-tial set of set point changes were induced to the con-trol loop at different times. Simultaneously, the pro-cess parameters were modified by +25%, in orderto emulate the nonlinear behavior of the process. Adisturbance was fed to the process at 400s. Figure 4shows the results when standard DMCr and PDMCrwere used. The standard DMCr could not compen-sate for the changes in FOPDT parameters and be-came oscillatory. The PDMCr using the Fuzzy Su-pervisor Module detected and estimated the processparameters changes,then compensated for them, al-lowing a stable process control. Figure 5 shows

Fig. 4: Comparison of standard DMCr and PDMCrperformances applied to a FOPDT process model

the comparison of process parameters estimation per-formed by the Fuzzy Supervisor Module during the

Fig. 5: Comparison among actual process parame-ters and estimated parameters by the Fuzzy Supervi-sor Module applied to a FOPDT process model

test presented in Figure 4. The figure 5 shows that theparameters are updated after every set point changeis detected.The time to update the process parametervaries and it depends how long takes to the model-ing error to settle down. The process parameters pre-dicted by the Fuzzy Supervisor are accurate enough toguide the PDMCr and adapt its response to the chang-ing process condition.As a second test, the PDMCr and standard DMCrwere incorporated to control the mixing process de-scribed by Iglesias et. al.[19][22][23] (see Figure 6).Figure 7 shows the comparison when facing consec-utive set point changes. Figure 7 shows that everytime the set point was decreased by 5%, the stan-dard DMCr became more and more oscillatory un-til finally reached a completely oscillatory behavior.This was a consequence of the nonlinear characteris-tic in the mixing process. On the contrary, the PDMCrshowed a smooth response and it was able to trackthe set point changes during the test. It also rejecteda cold temperature increment used as disturbance attime 700s. Every time a set point change was detectedby the Fuzzy Supervisor, the model parameters wereestimated and adjusted to adapt the controller to thenew process conditions. When the disturbances af-fected the process, the PDMC tracked the set point


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Fig. 6: Schematic representation of a mixing tank

avoiding large deviation. The two controllers were

Fig. 7: Performance comparison between the DMCrand the PDMCr for the mixing tank process

also tested using the neutralization reactor describedby Iglesias et. al.[19] (Fig. 8). A series of consecutiveset point changes were induced, and a reduction of15% in acid stream concentration was used as distur-bance; figure 9 shows the results. PDMCr tracked theset point with less overshoot that the standard DMCr,for both set point changes and disturbance rejection.The IAE values were 4284 for standard DMCr and

Fig. 8: Schematic Representation for a neutralizationreactor

3214 for PDMCr, a difference of about 23%. Figure

Fig. 9: Performance comparison between standardDMCr and PDMCr for the neutralization reactor

10 shows another test performed using the neutral-ization reactor. Initially a reduction on acid streamaffects the process as disturbance; later, two consecu-tive set point changes in opposite directions (reachingthe initial value again), are induced into the controlloop, to allow the PDMCr estimate and update theprocess parameters (Fig. 10 a.). Later two consec-utives changes in acid stream affect again to the pro-cess. Figure 10 shows that PDMCr tracked set pointwith less deviation than the standard DMCr. The totalIAE values were 8080 for standard DMCr and 6717


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for PDMCr; a reduction about 17%. The last two dis-turbances were compensated in less time and with lessovershoot when PDMCr was used. This test couldrepresent the way the control engineer should use theconsecutive set point changes to allow the PDMCradapt to varying operating conditions.

Fig. 10: Neutralization reactor test for disturbancesRejection: a.Changes induced on the acid streamb.DMCr response c.PDMCr response

6 Effect of noise

To test the effect of noise on the DMCr and PDMCrperformance, the mixing process describe in Fig.6was modified to include a noisy signal in cold streamF2. The noise used had an ARMA(1,1) structure:Zn = φ1Zn−1+an−θ1an−1, (φ1 = 0.6,θ1 = 0.3).Two different values of variance were used to gener-ate the noise: σ2a = 0.02 y σ2a = 0.03. Figure11shows cold stream signal affected by the noise. Fig-ure 12 shows the effect of noise on the DMCr andPDMCr performances. The same test shown in Figure7 is now used with the noisy cold water flow shownin Figure 11. Figures 12(a) and (b) compare stan-dard DMCr and PDMCr when noise has a varianceequal to 0.02, whereas Figures 12(c) and (d) show thesame test but this time with variance 0.03. The testshows that the PDMCr was sensitive to the presenceof noise as it was expected for controllers of a discretenature. A slight variation on noise variance couldcause oscillatory behavior on the PDMCr, see Figure12(d). However, compared with the standard DMCr,

Fig. 11: Cold Stream F2 Affected by Noise WithStructure ARMA(1,1)

the PDMCr showed more tolerance to the noise pres-ence, and kept a stable behavior for longer times.

Fig. 12: Noise Effect on DMCr and PDMCr perfor-mance when applied to the mixing tank

7 ConclusionThe proposed PDMCr was tested under noisy inputsignals and varying process dynamics, common char-acteristics of real industrial processes. The PDMCrwas able to overcome problems with a better perfor-mance that the standard DMCr, controlling nonlinearprocesses. The PDMCr also showed more toleranceto continuous noisy input signals, keeping a stable be-havior for longer times as compared to the DMCr.The PDMCr tracked set point changes with less de-viation compared to the standard DMCr, and com-pensated disturbances in less time and with less over-


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shoot. The PDMCr kept its stability under conditionswhere the DMCr was completely unstable. Whenboth performances were similar to each other, the IAEvalues for PDMCr were up to 23% lower than for theDMCr.The PDMCr-Fuzzy Supervisor Module set presentedin this study exerted a continuous supervisory role onthe process model in order to avoid its aging. The su-pervisory module kept the model updated using onlythree MEI values coming from closed-loop set pointchange responses. This was an advantage comparedto the conventional procedures, because it was notnecessary to do a new open-loop step-response modelidentification. This could save time and effort duringits implementation.Future studies should be focused on the adaptationand implementation of the PDMCr to control MIMOsystems.

Acknowledgements: Oscar Camacho thanks thePROMETEO Project of SENESCYT, Republic ofEcuador, for its sponsorship for the realization of thiswork.

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[17] E. Iglesias, Using Fuzzy Logic to Enhance Con-trol Performance of Sliding Mode Control andDynamic Matrix Control, Doctoral Dissertation,University of South Florida, Tampa, FL, USA,2006.

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A Parametric Dynamic Matrix Controller Approach for ...A Parametric Dynamic Matrix Controller Approach for Nonlinear Chemical Processes EDINZO IGLESIAS1, OSCAR CAMACHO2;5, MARCO SANJUAN3,