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Diagnosis of poor performance in model predictive controllers: Unmeasured Disturbance versus Model-Plant Mismatch Viviane Rodrigues Botelho, Jorge Otávio Trierweiler, and Marcelo Farenzena Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b00907 • Publication Date (Web): 12 Oct 2016 Downloaded from http://pubs.acs.org on October 18, 2016

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Industrial & Engineering Chemistry Research

Diagnosis of poor performance in model predictive controllers: Unmeasured Disturbance versus ModelPlant Mismatch Viviane Botelho, Jorge Otávio Trierweiler*, Marcelo Farenzena Group of Intensification, Modeling, Simulation, Control, and Optimization of Processes. Chemical Engineering Department. Federal University of Rio Grande do Sul (UFRGS) , R. Eng. Luiz Englert, s/n. Campus Central - Porto Alegre - RS – BRAZIL.

ABSTRACT: Poor model quality in model predictive controller (MPC) is often an important source of performance degradation. A key issue in MPC model assessment is to identify whether the bad performance comes from model-plant mismatches (MPM) or unmeasured disturbances (UD). This paper proposes a method for distinguishing between such degradation sources, where the main idea is to compare the statistical distribution of the estimated nominal outputs with the actual modeling error. The proposed approach relies on the assessment of three case studies: a simple SISO Linear MPC and two multivariable cases, where the linear controller is subject to a linear and nonlinear plant, respectively. Results show that the proposed method provides a good indicator of the model degradation source, even when both effects are present but one of them is dominant.

KEYWORDS: model predictive control, model-plant mismatch, unmeasured disturbance, model quality assessment

ACS Paragon Plus Environment

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INTRODUCTION Model predictive controllers (MPCs) are the standard solution of the supervisory control layer,

since they can work with multivariable complex dynamic systems. The MPC uses a dynamic process model to predict the behavior of controlled variables (CVs) along the future horizon, based on past control actions and disturbances. From this result, an optimization algorithm calculates the control actions that lead the process toward its optimal trajectory, respecting the constraints. The maintenance of MPC is an important and challenging problem, since performance degradation may stem from many different sources, such as: bad tuning (control and prediction horizon, weighting matrices, sampling time, etc.), poor model quality, poor disturbance rejection, and inappropriate constraint setup1. Among all these sources, poor model quality is the most frequent and impactful. Considering that a model is an abstraction of the real system behavior, modeling inconsistences will always be present. However, sometimes these inconsistences are so strong that the closed loop cannot achieve good performance. Therefore, it is necessary to quantify the modeling error, which cannot be compensated by feedback controller and, therefore, will deteriorate the corresponding closed loop behavior2. Several methods are focused on model quality investigation. Some of them are based on model validation metrics and investigate the need for a model re-identification3-5. Other approaches6-8 are focused on identifying which portion of the model (i.e. controlled variable or pair-controlled versus manipulated variable) is degraded. A key issue of MPC model assessment is to identify the source of a modeling inconsistency, which could be a model-plant mismatch (MPM) or an unmeasured disturbance (UD). The first occurs when the model cannot adequately describe the relations between its input and output variables and a re-identification is required. An UD is characterized by the absence of an input


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variable (signals) in the process model. Both cause similar effects in the process output and isolating each effect is not trivial. For example, a sensor fault could be understood as a UD, since the measurement errors are independent of the controller actuation (i.e., come from an external source). An MPM occurs, for example, due to equipment fouling, such as the relation between its inputs and outputs variables are no longer adequately represented by the model identified at the beginning of its operation. Several methods for disturbance detection were proposed. Thornhill & Horch provide an overview of the most important. According to their work, different approaches are needed depending on whether the disturbance is oscillating or non-oscillating. For the first case, the methods fall into three main classes, namely those which use the time domain, those using autocovariance functions, and spectral peak detection. Most of the methods are off-line and exploit these advantages, such as the use of the entire data history. In the case of non-oscillatory, spectral decomposition methods as principal component analysis (PCA), independent component analysis (ICA), and non-negative matrix factorization (NNMF) have been used to find significant spectral features. The knowledge of the UD effects is essential for the controller model assessment. In case of classical control (PID), the process model is intrinsic in the PID tuning. So, the model assessment is indirect, and the objective is determining how the current controller tuning is attenuating (or propagating) the UD effect and which is the limit of performance improvement through re-tuning (minimal attainable variance). In case of MPC, the process model is directly used by the controller, so, in this case, the main objective of the model assessment is to evaluate the predictive capacity of the controller to detect inconsistencies in the process model (MPM) or absence of variables in the model (UD).


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This paper proposed a new data-based approach for MPC model assessment. The method complements an extensive class and techniques available in literature (e.g., Schafer & Cinar10; Badwe et al.11, Sun et al.1; Botelho et al.12; Botelho et al.13), which detect the lack of quality in the prediction model, but are incapable of detecting the source of modeling problems. Once an unconformity between the model prediction and the actual outputs is detected, it is necessary to diagnose its causes, which can be related to MPM and/or UD. Here, we propose a statistical approach to identify whether the performance degradation is related to MPM or UD.

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PROPOSED METHOD The proposed method is based on the statistical comparison of the system outputs in the

absence of MPM and UD (called nominal output) with the difference between this value and actual outputs (called nominal error). The next sections describe the fundamental concepts of the proposed approach. 2.1

Estimation of nominal outputs

The first step in our approach consists of the nominal output estimation following the method proposed by Botelho et al.12-13. We assume the control loops illustrated in Figure 1, with a MPC controller and nominal model , which is used in the MPC to describe the real plant . The model-plant mismatch (MPM) magnitude is ∆. The theoretical system without mismatch is shown in Figure 1a, for which nominal closed-loop outputs are . is the nominal complementary sensitivity function. The real system, in a scenario subject to MPM, is shown in Figure 1b, where corresponds to the setpoints, are the manipulated variables, are the

measured outputs, are the simulated outputs of the nominal model perturbed by the actual

control actions , and is the actual complementary sensitivity function. Figure 1c illustrates the


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case with an unmeasured disturbance (UD), where is a sequence of independent random variables, is the unknown disturbance model, and are the disturbance signals.

Figure 1: Schematic diagram of closed-loop (a) nominal case, (b) system with model-plant mismatch (MPM), and (c) system with unmeasured disturbance (UD). Botelho et al.12 define the nominal output as the output of the system in the absence of MPM or UD. This can be estimated according to:

(1)

is the nominal sensitivity funcion, given by:

(2)

The nominal sensitivity function ( ) is a square transfer matrix that characterizes the system response in closed loop. Its dimensions are equal to the number of outputs. The diagonal elements ( ) give the closed-loop behavior of the outputs when their references (setpoints or soft constraints) are changed. The remaining elements provide the impact of these reference variations on the other outputs. Thus, Botelho et al.13 suggest the estimation of the nominal


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output considering only the diagonal sensitivity function ( ) to locate the controlled variable (CV) with model errors. For this case, equation (1) can be rewritten as:

(3)

produces a softening effect on the simulation residuals ( ) and retains only the

part not removed by the controller feedback that is impacting the performance of the corresponding output. Thus, CVs without significant MPM or UD will have ≅ , because

their simulation errors are close to zero. provides the complete diagnosis of the model, showing the effect of the MPMs or UDs in the corresponding output as well as how it is propagating onto the others. Thus ≠ even for variables without significant MPM or UD, considering the existence of MPM or UD in another CV model. The stronger the coupling between the channels, the larger the difference between . 2.2

Relation between nominal outputs and modeling errors

According to Botelho et al.12 is the estimated output free from modeling errors, which

includes model-plant mismatches and unmeasured disturbances. Defining the nominal error as the effect of the modeling problems in the loop, we have:

(4)

The outputs of a system with MPM and UD are14:

(5)

where is the disturbance signal entering in the loop. First, let us consider a system under a

model-plant mismatch only (Figure 1b). In this case, 0 and equation 5 is reduced to:

(6)

Analogously, the nominal output is (see Figure 1a):


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(7)

and equation 4 becomes: ∆

(8)

From (7) and (8) we can observe that the relation between the nominal error ( ) and nominal output ( ) in the presence of model-plant mismatch are a function of the same input signal (i.e.,

) passing through two different functions (∆ and ). Although these functions are different,

and are functions of , making them dependent. Therefore, the statistical behavior of and

will be correlated.

Now, let us consider a system under unmeasured disturbance only (Figure 1c). In this case, , so and . Then the measured output could be written as:

(9)

And the nominal error is: - ( ) -

(10)

From (7) and (10) we can observe that the relation between the nominal error ( ) and nominal output ( ) in the presence of unmeasured disturbance is given by two independent signals ( and ) passing through a “single function” (and its complementary, i.e., and - ), which

means that the variations of and will also be independent.

In summary, when a process output is under a MPM, the control objectives ( ) are the only

signals showing that and are dependent, which explains why their variation have

similarities. However, when a process output is under UD, will be dependent solely on the

external signal , while is dependent solely on the control objectives. This means that the

variation of and do not bear resemblance. Therefore, considering that the process is


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sufficient excited, the similarity between the variations of and are indicative of MPM presence. Remark 1: Although the premise of the method is the evaluation of similarity between and , the direct application of Pearson correlation is not ideal to compare them because both signals are estimated through (see equations 7, 8, and 10). This means that the correlation tends to be high in the presence of any MPM, even when its impact is negligible compared to an UD present in the same output. Since most real processes have MPM, its influence will mask the effect of UD, even when the importance is high. Then, we proposed the comparison of the statistical distribution along a moving window, which will be described below. Remark 2: Taking into account that the objective is to know the MPMs and UDs contained in each output, it is more appropriate to compare (equation 3) and , since they capture the isolated effect of each de modeling error, disregarding the effects of interaction between the CVs (see Botelho et al.13). The is defined by:

–

(11)

Thus, using and , it is possible to locate the modeling problem in each output. 2.3

Statistical Distribution in a Moving Window

Our proposed diagnosis procedure to distinguish between model-plant mismatches and unmeasured disturbances consists in a comparative analysis of the statistical distributions. A moving window ( ) is defined and the statistical distributions of and are

determined for each subset. Figure 2 illustrates the procedure, where ! is the sampling time.


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Figure 2: Illustration of a moving window evaluation procedure. The statistical distribution is evaluated by the skewness (!"#) and kurtosis ("$!) coefficients. These indexes show how far the signal is from a normal distribution. It is important to note that the data is seldom normally distributed, so we consider these indicators merely as a reference. The kurtosis coefficient ("$!) provides the shape of the probability density function. A high value of kurtosis means that the data has a large number of observations far from the mean, when compared with a normal distribution. The sample skewness (!"#) provides an indicator of how asymmetric the dataset is. For instance, a positive value of skewness means that there is a higher concentration of values smaller than the mean15, whereas a negative value corresponds to a higher concentration of values above the mean value. These coefficients for and in each moving window ( ) are: "$! &' %

∑&'

-. ) * +

&' /∑ -. )

,

0 0

* + 1

(12)


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"$!2&' %

!"# &' √ %

!"#2&' √ %

∑&' +

-. ) 333

&' /∑ -. )

,

0 0

(13)

333 + 1

∑&'

-. ) * +

67∑&'

-. )

5

67∑&'

-. )

0

5

(14)

5

(15)

* + 8

∑&' +

-. ) 333

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5 0

333 + 8

where "$! &' and "$!2&' are respectively the kurtosis coefficients of and at the % % moving window , !"# &' and !"#2&' are the skewness coefficients of and at % %

the moving window , 333 and * are the mean value of each corresponding dataset.

Figures 3 and 4 illustrate the qualitative difference between MPM and UD. When a MPM is present, the variation of statistical distributions of and shows the major variation at

the same time (see the peaks), which does not occur when a UD is present.


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(a)

(b)

(c)

Figure 3: Hypothetical case with MPM: (a) measured , estimated , and ; (b) kurtosis coefficients along a moving window; and (c) skewness coefficients along a moving window.

(a)

(b)

(c)

Figure 4: Hypothetical case with UD: (a) measured , estimated and ; (b) kurtosis coefficients along a moving window; and (c) skewness coefficients along a moving window. 2.4

Model Diagnosis

As described in Section 2.2, the diagnosis of MPM and UD is grounded on the comparison of the nominal output and corresponding nominal error. Two approaches based on the data scatter are considered for quantifying the relation between the variations of the statistical distributions. For both approaches, a scan is performed varying

size in the neighborhood of the prediction

horizon. This scan is necessary to ensure that all dynamic behaviors that impact the controller performance will be captured. It is important to take care with the size of the moving windows,


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since a small

is strongly impacted by the noise as well as a large

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is strongly impacted

by the sampling (i.e., effect of large amount of past data) and, in both cases the diagnosis can be misleading. Thus, we suggest vary

from 0.5 :ℎ to 2 :ℎ, where :ℎ is the MPC’s prediction

horizon, to capture only the inconsistencies in the work frequency of MPC. The first approach considers the Pearson correlation. For each evaluated

the correlation

between the statistical distributions of and are evaluated: =>?&'

=>@A &' , A2&' C % %

7DE@A &' C. DE@A2&' C % %

(16)

where => and DE represents, respectively, the covariance and variance of the corresponding signals, A is the evaluated statistical distribution coefficient ("$! or !"#, from equations 12 to

15) and =>?&' is the correlation indicator referent to the statistical distribution A at the moving

window . A high correlation denotes the existence of similarity between the statistical

distributions of and , indicating the presence of a model-plant mismatch. For the

proposed indicators, the threshold between the MPM or UD is defined according the confidence interval of correlation (), given by:

FGH/0 J#

(17)

where FGH/0 is the critical value of a standard normal variate at the confidence level K and # is

the number of sampled data in . Case at least one of the distributions (kurtosis or &' &' skewness) present the values of =>L or => LM higher than , a significant correlation between

the statistical distribution of and exist, indicating the presence of a MPM

disrupting the controller performance. Otherwise, the problem is due to a UD. Since several

are evaluated, a graphical evaluation of the results is suggested to provide the complete


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diagnosis. Alternatively, according to our experience, for a reasonable dataset size (# between

&' &' 200 and 5000) and for a confidence level of 95%, when the mean of absolute =>L or => LM

along the moving windows are higher than 0.1, a MPM is dominant. The mean absolute values (=> 333L D#N => 333 LM ) are defined as: 333? =>

&' ∑0PQ &'-.RPQ|=>? |

#&'

(18)

where #&' are the number of evaluated moving windows.

The second approach is based on confidence ellipse scatter. For each evaluated

the ellipses

CS@A2&' C, where A is the statistical are constructed considering the covariance matrix of @A &' % %

distribution coefficient ("$! or !"#, from equations 12 to 15). For each case, the angle of the

largest eigenvalue corresponds to the ellipse inclination (T?&' ). The ellipse’s major axis (D?&' ) and minor axis (U?&' ) are given, respectively, by the square root of the largest and the lowest

eigenvalues multiplied by the critical chi-square value (χWXYZ 0) associated with a given probability level16. Figure 5 illustrates the expected behavior for each scenario: the confidence ellipse is less circular (i.e., the greater are the ratios between the major and minor axis) and more diagonal (the nearest to []4 are the inclination) when the similarity between statistical distributions is more significant, indicating the presence of a model-plant mismatch.


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(a) (b) Figure 5: Expected response of linear and elliptical approximation (a) under UD and (b) under MPM. &' &' The confidence ellipse indicators (=L and = LM ) take into account the shape and the

inclination of the ellipse. They are defined as: =?&'

bc ,^_`a

d

J1 fg?&'

(19)

where fg?&' is the ratio between the largest and smallest ellipse diameters (fg?&' U?&' ⁄D?&' ). For the calculation of the confidence ellipse indicators the angles must be reduced

&' &' to half of the first quadrant (i.e., 0 ? &' were calculated with moving windows ( ) varying from 10 to 40. Here, only the and =?

indices based on the derivatives were used, once they have shown a better result, as presented in Section 3.1. Figure 20 shows the results for all

and Table 6 summarizes this analysis

through the mean values. The evaluation was performed only for . since the MPMs and UDs were only added in this CV.

Figure 19: Shell heavy oil fractionator case: Perturbations in the setpoints.


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UD1

UD2

UD3

UD4

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0 -0.2

0 -0.2

Index

1

Index

1

Index

Index

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0 -0.2

0 -0.2

-0.4

-0.4

-0.4

-0.4

-0.6

-0.6

-0.6

-0.6

-0.8

-0.8

-0.8

-1

10

20

30

40

50

60

70

-1

10

20

30

MW

40

50

60

70

-1

-0.8 10

MW

co

dkts

20

30

40

50

60

-1

70

MW

co

dskn

ce

dkts

ce

10

20

30

40

50

60

70

MW

dskn

CI

Figure 20: Shell heavy oil fractionator case: Correlation and confidence ellipse indicators for scenarios with MPM or UD. Table 6: Shell heavy oil fractionator case: Mean Indicators for Scenario containing MPM or

333 vwrnop

UD in . . 333 vwrpnq

333 vxrnop

333 vxrpnq

MPM1

0.03

0.10

0.11

0.22

MPM2

0.05

0.13

0.08

0.21

MPM3

0.21

0.43

0.36

0.52

MPM4

0.02

0.11

0.03

0.18

UD1

0.016

0.024

0.031

0.036

UD2

0.018

0.044

0.025

0.036

UD3

0.015

0.031

0.021

0.039

UD4

0.10

0.11

0.038

0.039

The results in Figure 20 show that, for all evaluated cases, at least one of the statistical distributions tends to exceed the when a MPM is present. The mean values presented in Table

6 provide a similar diagnosis. The highest 333 =? for each scenario is superior to 0.1 when a MPM

is present and does not exceed 0.05 when an unmeasured disturbance occurs, except for UD4,


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where 333 = LM 0.11. The => 333? works in all scenarios, indicating at least one => 333? higher than 0.18 when a MPM is present and not exceeding 0.04 in cases with UD. Based on this evaluation, we can conclude that the => 333? is more reliable than = 333? . Furthermore, the difference in the => 333? between the cases with MPM and UD is higher, allowing an easier and more conclusive the interpretation. Moreover, to guarantee the reliability of results, we recommend to evaluate both kurtosis and skewness. Considering that the method is based on a statistical evaluation, not always the similarities between the statistical distributions are evident for both coefficients. Thus, when more coefficients are evaluated, greater the likelihood of finding the MPM effects. We also simulated and applied the method to scenarios containing MPM and UD at the same time. To prove the method efficiency, the following procedure is adopted: first, data were generated only with the MPMs. Then data are generated only with the UDs. With these datasets, the real dominant effect (MPM or UD) could be truly determined and the result compared with the diagnosis of the proposed method when the system is susceptible to MPM and UD at the same time. The real dominant effect is determined by comparison of the Variance Index12 of the datasets, which is defined as: DE

DE DE

(26)

Thus, the DE was calculated for the data generated only with MPM (DE&& ) and only with UD (DE ). The results were compared with the Variance Index for the case with MPM+UD

(DE&& ). The closer the DE&& or DE of DE&& , the higher the dominance of

the corresponding effect. Table 7 shows the results. The proposed method was applied to the data containing both effects and the results are presented in Figure 21 and Table 8. Table 7: Shell heavy oil fractionator case: Determination of the real dominant effect using the DE for . .


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MPM1+UD1

MPM2+UD2

MPM3+UD3

MPM4+UD4

3.21

1.24

9.72

1.48

1.19

2.67

1.25

3.02

t

3.07

3.46

9.78

3.09

Dominant

MPM

UD

MPM

UD

t

MPM1+UD1

MPM2+UD2

MPM3+UD3

MPM4+UD4

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0 -0.2

0 -0.2

Index

1

Index

1

Index

Index

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0 -0.2

0 -0.2

-0.4

-0.4

-0.4

-0.4

-0.6

-0.6

-0.6

-0.6

-0.8

-0.8

-0.8

-1

10

20

30

40

50

60

70

-1

10

20

30

MW

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-1

70

-0.8 10

MW

co

dkts

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-1

70

10

20

30

MW

co

dskn

ce

dkts

ce

dskn

40

50

60

70

MW

CI

Figure 21: Shell heavy oil fractionator case: Correlation and confidence ellipse indicators for scenarios with MPM and UD. Table 8: Shell heavy oil fractionator case: Mean Indicators for Scenario containing MPM and UD in . . 333rnop vw

333rpnq vw 333 vxrnop

333 vxrpnq

MPM1+UD1

MPM2+UD2

MPM3+UD3

MPM4+UD4

0.09

0.03

0.46

0.06

0.31

0.04

0.35

0.08

0.16

0.04

0.58

0.03

0.35

0.03

0.43

0.04


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The comparison of Table 7 with Figure 21 and Table 8 shows that the method is capable to detect the dominant effect when both model-plant mismatches and unmeasured disturbances are occurring at the same time. For the case MPM1+UD1, the DE&& is nearest of DE&& ,

indicating that the MPM is dominant (Table 7). The = 333 LM and => 333 LM tends to exceed the

(mean values are equal to 0.31 and 0.35, respectively). For the case MPM4+UD4, the DE&& is nearest to DE , indicating that the UD is dominant. The 333 = LM and 333 => LM

remains inside the (mean values equal to 0.08 and 0.04, respectively). Similar results are obtained for MPM2+UD, when an unmeasured disturbance is dominant as well in MPM3+UD3, when the dominant effect is a model-plant mismatch. 3.3

The quadruple-tank process

This case study aims to illustrate the application of the method to a nonlinear multivariable plant. The system is composed of four cylindrical tanks connected according to Figure 22. Water split up using the valves, with openings equal to S. and S0 , respectively. More details can be found in Johanson21.

Figure 22: Diagram of the quadruple-tank process case study. The Mass balances around each tank are:


31


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nℎ. =N. =N5 S. ". ℎ. P. ℎ5 P5 N$ . . . .

Page 32 of 41

(27)

Nℎ0 =N0 =N, S0 "0. ℎ0 P0 ℎ, P, N$ 0 0 0 0

(28)

Nℎ5 =N5 1 S0 "0 ℎ5 P5 0 N$ 5 5

(29)

Nℎ, =N, 1 S. ". ℎ, P, . N$ , ,

(30)

Where ℎ is the level of each tank, ". . and "0 0 are the pump output flows, is the cross-

section area of each tank, =N is the discharge coefficient of each tank, and S: is the discharge exponent. Table 9 provides the model parameters. Table 9: The quadruple-tank process: model parameter values.

28 =0

vr

2.525 =0.R /!

32 =0

n

3.35 =5 ⁄!

vr vr vr

28 =0 32 =0

3.145 =0.R /! 2.525 =0.R /! 3.145 =0.R /!

n

w

3.33 =5 ⁄!

w w w

0.5 0.5 0.5 0.5

To illustrate the proposed approach, a MPC controller was simulated in Matlab/Simulink, whose controlled variables are the four levels (ℎ. , ℎ0 , ℎ5 and ℎ, ) and the manipulated variables

are pump voltages (. and 0 ) and valve openings (S. and S0 ).


32

Page 33 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The linear model, used by the MPC, was obtained from the linearization of the nonlinear model at the operating point defined by the manipulated variables . 3.2, 0 3.15, S.

0.43 , and S0 0.34, given by: 0.048 { z! 0.016 z 0.0009 z! 0.016 ! z z 0 z z z 0.045 y! 0.018

!0

0.0025 0.028! 0.0002

0.35 ! 0.015

0.035 ! 0.011

0.0055 ! 0 0.024! 0.0002

0

0.31 ! 0.018

0.078 0.028! 0.25

!0

0

0.0096 0.41! 0.0004 0.323 ! 0.011 0.37 ! 0.026 0

(31)

The controller was tuned using the RPN methodology22, whose values are shown in Table 10. Table 11 shows the different scenarios evaluated in this case study. Scenario 0 is the nominal case, without MPM and unmeasured disturbances. Scenarios 1 to 4 and 7 to 10 show a MPM whereas the others have unmeasured disturbances, which are included in the output (Scenarios 4,5, 11 and 12) as well in the input signals (Scenarios 13 and 14). The data used in this study was simulated using a sequence of step setpoint changes (Figure 23) and including a white noise with magnitude 2% of the variables range added on the output measurements. The plant is simulated considering the nonlinear model of equations 27 to 30. Table 10: The quadruple-tank process: MPC tuning parameters. Sample Time

10s

Prediction Horizon

48

Control Horizon

12

CVs Weights

Q. = Q0 Q5 Q, = 10

Move Suppression MVs lower limits MVs upper limits

. = 0 . 0 = 50

M M M 1 M Q = 2Q 0.1, S1Q S2Q 0.05 1 Q = 2Q 10, S1Q S2Q 0.95


33


M M M ℎ1 M Q ℎ2Q ℎ3Q ℎ4Q 0

CVs lowe limits

ℎ1 Q ℎ2Q ℎ3Q ℎ4Q 20

CVs upper limits

20 h1set

h2set

h3set

h4set

15

yset

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 41

10

5

0

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Time [s]

Figure 23: The quadruple-tank process: Perturbations in the setpoints. Table 11: The quadruple-tank process: Scenarios Scenario

Parameter

Value

0

--

--

1

=N.

7.86 =0.R /!

0

256 =0

2 3

=N5

4

,

5

Unmeasured disturbance in ℎ4

6


7

S:3

8 9

S:2 =N0

6.29 =0.R /! 7.44 =0 0.8 70! 1 0.25 5! 1 1

0.9

10.1 =0.R /!


34

Page 35 of 41

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

.

11


12


13

Unmeasured disturbance: 1

14

Unmeasured disturbance: S2

7 =0

1.4! 0.105 10! 5 500! 1

0.2! 0.45 80! 0.15 150! 1

The method was applied in the most affected controlled variable for each scenario. This selection was made considering the corresponding CV with the highest variance of DE (equation 26). For each selected CV, the confidence ellipses considering

48 are

&' &' presented in Figures 25 and 26. Figure 24 shows the =? and =>? for a moving windows size

( ) varying from 24 to 96. Table 12 summarizes the results, presenting the highest variances 333? . indexes of each case as well the mean values of 333 =? and => . Table 12: The quadruple-tank process: =>? and =? for each scenario. Scenario

Evaluated CV

Ivar

vwrnop

vxrpnq

vxrnop

1

ℎ1

vwrpnq

1.45

0.359

0.418

0.457

0.596

2.58

0.189

0.129

0.182

0.149

4.54

0.308

0.343

0.554

0.695

2.85

0.357

0.408

0.530

0.710

1.63

0.006

0.002

0.052

0.041

1.89

0.006

0.001

0.045

0.046

2.38

0.310

0.270

0.306

0.568

2 3 4 5 6 7

ℎ1 ℎ2 ℎ4 ℎ4 ℎ2 ℎ3


35


ℎ2

8

ℎ1

10

ℎ1

11

ℎ3

12

ℎ1

13

ℎ1

14

0.408

0.491

0.565

0.706

1.86

0.499

0.524

0.603

0.709

1.60

0.28

0.26

0.49

0.55

2.31

0.006

0.002

0.001

0.008

1.89

0.007

0.002

0.04

0.04

2.92

0.01

0.004

1.73

0.01

0.008

Scenario2

Scenario3

0.05

0.04

0.05

0.08

Scenario4

Scenario5

1

1

0.5

0.5

0.5

0.5

0.5

0

-0.5

-1

40

60

0

-0.5

-1

80

40

60

MW

0

-0.5

-1

80

40

MW

Scenario6

60

0

-0.5

-1

80

Index for h4

1

Index for h4

1

Index for h2

1

Index for h1

40

MW

Scenario7

60

0

-0.5

-1

80

Scenario8

Scenario9

0.5

0.5

0.5

0.5

-0.5

-1

40

60

-0.5

-1

80

40

60

MW

-0.5

-1

80

40

MW

Scenario11

60

0

-0.5

-1

80

Scenario12

Scenario13

0.5

0.5

0.5

-1

40

60

80

Index for h1

0.5

Index for h3

1

-0.5

-1

40

60

MW

0

-0.5

-1

80

MW

co

dkts

40

60

80

dskn

-0.5

-1

40

60

80

MW

0

-0.5

-1

40

MW

co

80

0

Scenario14

1

0

60

MW

1

-0.5

40

MW

1

0

Index for h1

0.5

Index for h2

1

Index for h2

1

Index for h3

1

0

80

Scenario10

1

0

60

MW

1

0

40

MW

Index for h1

Index for h1

Scenario1

Index for h2

1.93

ℎ2

9

Index for h1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 41

60

80

MW

ce

dkts

ce

dskn

CI

Figure 24: The quadruple-tank process: Correlation and confidence ellipse indices for all scenarios


36

dkts y0/dt

Scen8

-10 -10

10

dktse0/dt

dktse0/dt 0

-10 -10

10

0

0

-10 -10

10

dktsy0/dt

0

0

Scen11

-10 -10

10

0

0

Scen12

-10 -10

10

-10 -10

10

0

0

Scen13

Scen14 10

0

-10 -10

10

dkts y0/dt

0

0

-10 -10

10

dktsy0/dt

0

dskny0/dt

Scen8

2

0 -2 0

dskny0/dt

-2

2

0

0

-2

2

0

0

-2

2

-2

2

0 -2 0

dskny0/dt

0 -2

2

-2

2

0 -2 0

2

Scen14

2

-2

0

dskny0/dt

Scen13

2

-2

0

2

dskny0/dt

Scen12

-2

dskny0/dt

0

dskny0/dt

2

-2

0

2

dskne0/dt

-2

2

Scen11

-2

dskny0/dt

0

dskny0/dt

2

-2

0

2

dskne0/dt

-2

2

Scen10

2

-2

0

dskny0/dt

dskne0/dt

dskne0/dt

0

0

-2

Scen9

-2

dskny0/dt

-2

2

dskny0/dt

2

-2

0

0

Scen7

dskne0/dt

-2

2

dskne0/dt

-2

2

0

48.

Scen6

dskne0/dt

0

2

dskne0/dt

-2

0

Scen5

dskne0/dt

-2

2

Scen4 dskne0/dt

0

Scen3 dskne0/dt

2

dskne0/dt

dskne0/dt

Scen2

10

dktsy0/dt

Figure 25: The quadruple-tank process: Kurtosis derivative confidence ellipse for

Scen1

10

dktsy0/dt

10

0

dkts y0/dt

0

dktsy0/dt

10

-10 -10

10

dktsy0/dt

0

dkts y0/dt

0

Scen7 10

dktse0/dt

dktse0/dt

0

-10 -10

10

10

0

Scen6 10

dkts y0/dt

Scen10

0

dkts y0/dt

-10 -10

10

10

10

0

0

0

dktsy0/dt

Scen9

10

-10 -10

0

dktsy0/dt

Scen5 10

dktse0/dt

0

dktse0/dt

0

-10 -10

10

Scen4 10

dktse0/dt

0

dktse0/dt

dktse0/dt

dktse0/dt

0

-10 -10

dktse0/dt

Scen3 10

dktse0/dt

Scen2 10

dktse0/dt

Scen1 10

dskne0/dt

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dktse0/dt

Page 37 of 41

2 0 -2

2

dskny0/dt

Figure 26: The quadruple-tank process: Skewness derivative confidence ellipse for

-2

0

2

dskny0/dt

48.

Figure 24 shows that the indices based on the derivatives of skewness and kurtosis allow us to distinguish the root cause of the problem and points to the channel with mismatch, in all &' &' scenarios, since when a MPM is present, the =? and =>? tend to violate the (see

Scenarios 1 to 4 and 7 to 10), which do not occur when the system is corrupted by an UD added in the output signals (Scenarios 5,6,10 and 11) as well in the input signals (Sencarios 11 to 14). The ellipses (Figure 25 and 26) tend to be more sloped and less circular when a MPM is present. Table 12 shows that =i is superior to 0.129 when a MPM is present and not exceeding 0.008


37


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 41

when a UD occurs. Similarly, =>i is superior to 0.149 when a MPM is present and does not exceed 0.08 when a UD occurs. In the simulated scenarios, the nonlinearity of the plant does not impact the system behavior, since the diagnosis was compatible with the added MPM/UD. However, when the nonlinearity is significant, the diagnosis will indicate a MPM.

4

CONC LUS IONS This paper proposed an approach to distinguish between model-plant mismatch or unmeasured

disturbance impacting in the performance of model predictive controllers. The idea behind the method is to compare the nominal system outputs with the nominal error. When a MPM is responsible for the performance degradation, these signals have similarities because both are dependent on the control actions. However, when a UD is present, the nominal error depends on the disturbance signal, which comes from an external source. Thus, the nominal error does not have relation with the nominal output. The comparison between the nominal error and nominal output was performed considering the statistical distribution of the signals along a moving window. The statistical distribution is defined from the kurtosis and skewness coefficients. Four indicators were proposed: one was based on the Pearson’s correlation coefficient (=>), another based in the confidence ellipse of the statistical distributions (=) both considering the raw statistical distributions or its derivative. Three case studies are evaluated: a SISO Linear MPC, the Shell fractionator process, and the quadruple-tanks nonlinear process. The results show that the =>i , i.e., the indicator based on the Pearson’s correlation coefficient using the derivatives of the statistical distribution, is the most reliable, since it provides the correct diagnosis in all evaluated scenarios. The other indicators also provide good results, but they are less sensitive to the modeling errors, since in


38

Page 39 of 41

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some scenarios the results are in the neighborhood of the threshold value, leading to inconclusive diagnosis when the difference between MPM and UD is small. When the system contains MPM and UD at the same time, the method is capable of detecting the dominant effect, indicating higher values of =>i and =i , as the effect of the MPM is more evident. For the diagnosis of MPM and UD, two thresholds are proposed, one based on the graphical analysis considering the confidence interval of the correlation and another based in a heuristic rule. Results show that both approaches provide the same diagnosis in all evaluated cases. The results also show that, when a MPM occurs, kurtosis and skewness do not necessarily both have indicator values higher than 0.1.

Thus, the analysis of kurtosis and skewness are

complementary and if just one of them satisfies the criteria, it is enough for indicating MPM.

ACKNOWLEDGMENT The authors received financial support from Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul (FAPERGS) and Petrobras. CORRESPONDING AUTHOR CONTACT INFORMATION *

Phone: +555133084072. E-mail: [email protected]

REFERENCES (1) Sun, Z.; Qin, J.; Singhal, A.; Megan, L. Performance monitoring of model-predictive controllers via model residual assessment. J. Process Control. 2013, 23, 473.


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Page 40 of 41

(2) Whang, H.; Song, Z.; Xie, L. Parametric Mismatch Detection and Isolation in Model Predictive Control System. Proceedings of the 8th IFAC Symposium on Advanced Control of Chemical Processes, Furama Riverfront, Singapore, July 10-13,2012. (3) Huang, B.; Malhotra, A.; Tamayo, E. Model Predictive Control Relevant Identification and Validation. Chem. Eng. Science. 2003, 58, 2389. (4) Conner, J.; Seborg, D. Assessing the Need for Process Re-identification. Ind. Eng. Chem. Res. 2005, 44, 2767. (5) Jiang, H.; Shah, S.; Huang, B.; Wilson, B.; Patwardhan, R.; Szeto, F. Model analysis and performance analysis of two industrial MPCs. Control Eng. Practice. 2012, 20, 219. (6) Badwe, A.; Gudi, R.; Patwarhan, R.; Shah, S.; Patwarhan, S. Detection of model-plant mismatch in MPC applications. J. Process Control. 2009, 19, 1305. (7) Kano, M.; Shigi, Y.; Hasebe, S.; Ooyama, S. Detection of Significant Model-Plant-Mismath from Routine Operation Data of Model Predictive Control System. Proceedings of the 9th International Symposium on Dynamics and Control of Process Systems, Leuven, Belgium, July 5-7,2010. (8) Ji, G.; Zhang, K.; Zhu, Y. A method of MPC model error detection. J. Process Control. 2012, 22, 635. (9) Thornhill, N.; Horch, A. Advances and new directions in plant-wide disturbance detection and diagnosis. Control Eng. Practice. 2007, 15, 1196. (10) Schäfer, J.; Cinar, A. Multivariable MPC system performance assessment, monitoring, and diagnosis. J. Process Control. 2004, 14, 113. (11) Badwe, A.; Patwarhan, R.; Shah, S.; Patwarhan, S.; Gudi, R. Quantifying the impact of model-plant mismatch on controller performance. J. Process Control. 2010, 20, 408. (12) Botelho, V.; Trierweiler, J.; Farenzena, M.; Duraiski, R. A methodology for detecting model-plant mismatches affecting MPC performance. Ind. & Eng. Chem. Research. 2015, 54, 12072. (13) Botelho, V.; Trierweiler, J.; Farenzena. MPC model assessment of highly coupled systems. Industrial & Engineering Chemistry Research (submitted for publication). (14) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control: Analysis and Design. John Wiley & Sons: New York, 1996. (15) Adams, K.; Lawrence, E. Research Methods, Statistics, and Applications. Sage Publications: California, 2015. (16) Santos-Fernández, E. Multivariate Statistical Quality Control Using R. Springer: New York, 2013.


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(17) Trierweiler, J.; Machado, V. Which is the best criterion for identification of dynamic models? Proceedings of 7th IFAC Symposium on Dynamics and Control of Process Systems. Cambridge, Massachusetts, USA, 5-7 July, 2004. (18) Prett, D.; Morari, M. The Shell Process Control Workshop. Butterworths: Boston, 1987. (19) Maciejowski, J. Predictive Control with Constraints. Prentice Hall: New Jersey, 2002. (20) Farenzena, M. Novel methodologies for assessment and diagnostics in control loop management. Ph.D. Thesis, Federal University of Rio Grande do Sul, Brazil, 2008. (21) Johanson, K. The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero. IEEE Transactions on Control Systems Technology. 2000, 8, 456. (22) Trierweiler, J.; Farina, L. RPN tuning strategy for model predictive control. J. Process Control. 2003, 13, 591.

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