Non-Gaussian Assessment of the Benefits from Improved Control

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World The International International Federation of Congress Automatic Control Control Toulouse, France,Federation July 9-14, 2017 The of Automatic Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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Non-Gaussian Assessment of the Benefits Non-Gaussian Assessment of Non-Gaussian Assessment of the the Benefits Benefits from Improved Control from Improved Control from Improved Control Pawel D. Doma´ nski Pawe ll D. Doma´ n ski Pawe D. Doma´ n Pawel D. Doma´ nski ski Institute of Control and Computation Engineering, Institute of of Control and Computation Engineering, Institute and Engineering, University of Technology, InstituteWarsaw of Control Control and Computation Computation Engineering, Warsaw University of Technology, Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665 Warsaw, Poland Warsaw University of Technology, ul. Nowowiejska 15/19, 00-665 00-665 Warsaw, Poland Poland ul. 15/19, Warsaw, (e-mail: [email protected]) ul. Nowowiejska Nowowiejska 15/19, 00-665 Warsaw, Poland (e-mail: [email protected]) (e-mail: (e-mail: [email protected]) [email protected])

Abstract Control quality significantly contributes to the process technical and financial perforAbstractProductivity, Control quality significantly contributes contributes to the process technical and financial perforAbstract Control significantly to process and performance. environmental and energy management push systems towards their Abstract Control quality quality significantlyissues contributes to the the process technical technical and financial financial performance. Productivity, environmental issues and energy management push systems towards their mance. Productivity, environmental issues and energy management push systems towards their technological constraints calling for better regulation closer to process limitations. Any control mance. Productivity, environmental issues and energycloser management push systems towards their technological constraints calling for better regulation to process limitations. Any control technological constraints calling for better regulation closer to process limitations. Any control improvement constraints initiative should befor predated with the estimation of the potential benefits associtechnological calling better regulation closer to process limitations. Any control improvement initiative should beproject. predated with the estimation estimation of on thethe potential benefits associimprovement initiative should be predated with the of the potential benefits associated with control rehabilitation The assessment is based appropriate measures. improvement initiative should be predated with the estimation of the potential benefits associated with control rehabilitation project. The assessment is based on the appropriate measures. ated with control The assessment is on measures. Classical are based on project. Gaussian approach. However, investigation of industrial data ated withmethods control rehabilitation rehabilitation The assessment is based based on the the appropriate appropriate measures. Classical methods are based based with on project. Gaussian approach. However, investigation of industrial data Classical methods are on Gaussian approach. However, investigation of industrial data frequently is not compliant normal assumption on control signals. This paper proposes Classical are based with on Gaussian approach. However, investigation of paper industrial data frequentlymethods is notnon-Gaussian compliant normal assumption on control control signals. This proposes frequently is compliant normal on signals. This approach using probabilistic distributions Cauchy, Laplace and L´eproposes vy. The frequently is not notnon-Gaussian compliant with with normal assumption assumption on like control signals. This paper paper proposes approach using probabilistic distributions like Cauchy, Laplace and L´ e vy. The approach using non-Gaussian probabilistic like methodology is illustrated on the exemplarydistributions industrial data. approach using non-Gaussian probabilistic distributions like Cauchy, Cauchy, Laplace Laplace and and L´ L´eevy. vy. The The methodology is illustrated on the exemplary industrial data. methodology is on industrial data. methodology is illustrated illustratedFederation on the the exemplary exemplary industrial © 2017, IFAC (International of Automatic Control) data. Hosting by Elsevier Ltd. All rights reserved. Keywords: monitoring and performance assessment, statistical data analysis, non-Gaussian Keywords: monitoring and performance assessment, statistical Keywords: monitoring monitoring and performance assessment, statistical data data analysis, analysis, non-Gaussian non-Gaussian distributions, same limit method, controlassessment, benefits statistical Keywords: and performance data analysis, non-Gaussian distributions, same limit method, control benefits distributions, same limit method, control benefits distributions, same limit method, control benefits 1. 1. 1. 1.

INTRODUCTION INTRODUCTION INTRODUCTION INTRODUCTION

Industrial processes are often non-stationary time-varying Industrial processes are often non-stationary non-stationary time-varying Industrial processes are time-varying systems with many correlations impacted by disturbances. Industrial processes are often often non-stationary time-varying systems with many correlations impacted by disturbances. systems with disturbances. They cause lot correlations of challengesimpacted for the by control system systems with aamany many correlations impacted by disturbances. They cause cause lotdesign of challenges challenges for the the control system They a lot of for control system implementation, and tuning. Dynamic goals are They cause a lotdesign of challenges for the controlgoals system implementation, and tuning. tuning. Dynamic are implementation, design and Dynamic goals are addressed by the base control with single point or cascade implementation, design and tuning. Dynamic goals are addressed by the the base control with single point or cascade addressed single point cascade PID loops.by arecontrol many with reports showing that base addressed byThere the base base control with single point or or cascade PID loops. There are many reports showing that base PID loops. There are many reports showing that base control tuning brings significant financial benefits (Marlin PID loops. There aresignificant many reports showing that base control tuning brings financial benefits (Marlin control tuning brings significant financial benefits (Marlin et al. (1991); ski et al. (2016)). control tuning Doma´ bringsn significant financialFurther benefitsimprove(Marlin et al. al. is (1991); Doma´ nski ski et al. al.supervisory (2016)). Further improveet (1991); n et (2016)). Further improvement often Doma´ obtained with applications of et al. (1991); Doma´ n ski et al. (2016)). Further improvement is often obtained with supervisory applications of ment is often obtained with supervisory applications of Advanced process Control (APC) and Process Optimizament is often obtained with supervisory applications of Advanced process Control (APC) and Process OptimizaAdvanced process Control (APC) and Process Optimization (PO) (Gabor al. (2000); Laing al. (2001)). Advanced process et Control (APC) and et Process Optimization (PO) (PO) (Gabor (Gabor et al. (2000); (2000); Laing et al. (2001)). (2001)). tion et al. Laing et al. tion al. for (2000); Laing et al. Thus(PO) there(Gabor arises a et need methodologies to (2001)). compare conThusrehabilitation there arises aacost need for for methodologies to compare conThus there need to control expected economic benefits. Thus there arises arises acost needagainst for methodologies methodologies to compare compare control rehabilitation against expected economic benefits. trol rehabilitation cost against expected economic benefits. Suchrehabilitation decisions arecost mostly based on the financial basis. trol against expected economic benefits. Such decisions decisions are mostly mostly based on the the financial basis. Such are based on financial basis. Estimation techniques allowing calculation of the benefits Such decisions are mostly based on the financial basis. Estimation techniques allowing calculation of the benefits Estimation techniques allowing calculation of benefits resulting from the control system improvement have been Estimation techniques allowing calculation of the the benefits resulting from from the control control system improvement have been resulting the system improvement have been proposed by Bauer et al. (2007); Wei and Craig (2009). resulting from the control system have been proposed by Bauer Bauer etthe al.decision (2007); improvement Wei and Craig Craig (2009). proposed by et al. (2007); Wei and (2009). The cost element of is simple as it may be proposed by Bauerofetthe al.decision (2007); is Wei and Craig (2009). The cost cost element simple as it may may be The element of the decision is simple as it be easily derived from past projects or obtained from control The cost element ofpast theprojects decisionorisobtained simple as it may be easily derived from from control easily projects control systemderived vendor.from Thepast benefit part or is obtained evaluatedfrom specifically easily derived from projects or from control system vendor. Thepast benefit part is obtained evaluated specifically system vendor. The benefit part is evaluated specifically for each case. The algorithm is based on the mitigation system vendor. Thealgorithm benefit part is evaluated specifically for each case. The is based on the mitigation for each The is on of process variability, leading towards results for each case. case. The algorithm algorithm is based basedquantitative on the the mitigation mitigation of process process variability, leading towards quantitative results of variability, leading towards quantitative results (Ali (2002)). Frequently, one may assume upper or lower of process variability, leading towards quantitative results (Ali (2002)). Frequently, one may assume upper or lower (Ali (2002)). Frequently, one may assume upper or lower limitation for the variable. Reduction of its variability (Ali (2002)). Frequently, one Reduction may assume upper or lower limitation for the variable. of its variability limitation for the variable. Reduction of its variability through better control enables to shift ofit its closer to the limitation for the variable. Reduction variability through better better control enables to shift it it closer to the the through control enables to shift to constraint and thus to generate benefit. Ascloser the variable through better control enables to shift it closer to the constraint and thus to generate benefit. As the variable constraint thus to generate benefit. As the is explicitlyand linked with performance, may constraint and thus to the generate benefit. the As benefits the variable variable is explicitly explicitly linked with the performance, the benefits may is linked with the performance, the benefits may be calculated. The method assumes that shape of the is explicitly linked with the performance, the benefits may be calculated. The method assumes that the shape of the be calculated. The method assumes that the shape of the variable histogram is Gaussian and standard deviation is be calculated. The method assumes that the shape of the variable histogram is Gaussian Gaussian and standard deviation is variable histogram is and standard deviation is used as the variability measure. Apart from that, other variable histogram is Gaussian and standard deviation is used as the variability measure. Apart from that, other used as variability measure. Apart that, methods were proposed, like probabilistic used as the the variability measure. Apart from fromoptimization that, other other methods were proposed, like probabilistic probabilistic optimization methods were proposed, like optimization approach Zhao et al. (2011). methods were proposed, like probabilistic optimization approach Zhao et et al. (2011). (2011). approach approach Zhao Zhao et al. al. (2011).

The rationale for this work originates from observations The rationale rationale for this this work originates originates from observations The for from gained in several Control Performance AssessThe rationale forcommercial this work work originates from observations observations gained in several commercial Control Performance Assessgained in several commercial Control Performance Assessment (CPA) projects. It appears that the properties of gained in several commercial Control Performance Assessment (CPA) projects. It appears that the properties of ment (CPA) projects. It appears that the properties of industrial data do not follow expected Gaussian assumpment (CPA) projects. It appears that Gaussian the properties of industrial data do not not Fat follow expected assumpindustrial data do follow expected Gaussian assumption. They are unlike. tail distributions, like Cauchy industrial data do not Fat follow Gaussian assumption. They are unlike. unlike. tailexpected distributions, likeshown Cauchy tion. are Fat tail distributions, Cauchy or L´eThey vy α-stable enable better fitting. It islike in tion. They are unlike. Fat tail distributions, like Cauchy or L´ e vy α-stable enable better fitting. It is shown in or L´ e vy α-stable enable better fitting. It is shown in Doma´ n ski (2016b) that statistical factors of these funcor L´evy α-stable enable better fitting. Itofisthese shown in Doma´ n ski (2016b) that statistical factors funcDoma´ n ski (2016b) that statistical factors of these functions, nlike or stability α factors are more appropriate Doma´ ski scaling (2016b) γ statistical of appropriate these functions,robust like scaling γthat or stability α are are more tions, like scaling γ or stability α more appropriate and measures of control loop variability. To follow tions, like scaling γ or stability α are more appropriate and robust measures of control loop variability. Tomethod follow and robust measures of control loop variability. To follow this observation, non-Gaussian versions of the and robust measures of control loop variability. Tomethod follow this observation, non-Gaussian versions of the this observation, versions of should be applied. non-Gaussian This work follows that direction. Three this observation, non-Gaussian versions of the the method method should be applied. applied. This work follows that direction. Three should be This work follows that direction. Three Probabilistic Density Functions (PDF) are tested: Cauchy, should be applied. This work follows that Three Probabilistic Density Functions (PDF) aredirection. tested: Cauchy, Probabilistic are Cauchy, α-stable and Density LaplaceFunctions to extend(PDF) standard the same limit Probabilistic Density Functions (PDF) are tested: tested: Cauchy, α-stable and Laplace to extend standard the same limit α-stable and Laplace to extend standard the same limit method for the general non-Gaussian situation. α-stable andthe Laplace tonon-Gaussian extend standard the same limit method for general situation. method for non-Gaussian situation. method for the the general general non-Gaussian situation. The manuscript starts with the presentation of considered The manuscript starts with the presentation of considered The manuscript starts with the presentation of statistical functions. It is followed by the extension of the The manuscript startsItwith the presentation of considered considered statistical functions. is followed by the extension of the the statistical functions. It is followed by the extension of benefit estimation with these PDFs. The methodology is statistical functions. It isthese followed by The the extension of the benefit estimation with PDFs. methodology is benefit estimation with these PDFs. The methodology is verified estimation on the industrial dataPDFs. and the paper concludes benefit with these The methodology is verified on the the industrial data and paper concludes verified on data the paper concludes with final and directions for the further research. verified onremarks the industrial industrial data and and the paper concludes with final remarks and directions for further research. with final remarks and directions for further research. with final remarks and directions for further research. 1.1 Statistical measures 1.1 Statistical measures measures 1.1 1.1 Statistical Statistical measures The control quality measures are widely used in industry Themeasure control quality quality measures are widely used in industry The control measures widely used to assess controlare loops (Gao et in al.industry (2016); The control and quality measures are widely used in industry to measure and assess control loops (Gao et al. (2016); to measure and assess control loops (Gao et al. (2016); Bauer et al. (2016)). Normal distribution delivers the most to measure and assess control loops (Gao et al.the (2016); Bauer et al. (2016)). Normal distribution delivers most Bauer et al. (2016)). Normal distribution delivers the most popular performance indicators. Mean value and standard Bauer etperformance al. (2016)). Normal distribution delivers the most popular indicators. Mean value and standard popular performance indicators. Mean value and standard deviationperformance are commonly used. Importance these meapopular indicators. Mean value of and standard deviation are commonly used. Importance of these meadeviation commonly used. of these measures and are their acceptance is Importance unquestionable. Standard deviation are commonly used. Importance of these measures and their acceptance is unquestionable. Standard sures and their is Standard deviation about signal variability. Higher value sures and informs their acceptance acceptance is unquestionable. unquestionable. Standard deviation informs about and signal variability. Higher value deviation informs about signal variability. Higher value means larger variations poorer control, while small deviation informs about and signal variability. Higher value means reflect larger variations poorer control, while small means larger variations and values situation. But control, they arewhile valid,small once means larger opposite variationssituation. and poorer poorer control, while small values reflect opposite But they are valid, once values opposite situation. But they are valid, signal reflect properties are Gaussian. Normality may be once valivalues reflect opposite situation. But they are valid, once signal graphically properties are are Gaussian. Normality may be valivalisignal properties Gaussian. Normality may be dated through visual inspection of histogram signal properties are Gaussian. Normality may be validated graphically through visual inspection of histogram dated visual or withgraphically normality through tests. dated visual inspection inspection of of histogram histogram or with withgraphically normality through tests. or normality tests. or with normality tests.

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Proceedings of the 20th IFAC World Congress 4942 Pawel D. Domański et al. / IFAC PapersOnLine 50-1 (2017) 4941–4946 Toulouse, France, July 9-14, 2017

Review of control loops from different process industries Doma´ nski (2015) shows that only minority (≈ 6%) has normal properties. Majority witnesses fat tails well fitted with α-stable (> 60%) or Cauchy PDF (≈ 30%). This can be explained by process complexity, correlations, time varying delays and human impact.

Bauer and Craig (2008). All of them are using assumption about Gaussian shapes of the controlled variable. Normal approach is followed by extensions with other PDFs.

Cauchy PDF Cauchy PDF is an example of the fat-tail distribution (1). The shape for values further from mean does not decay so fast as it is with normal PDF. It is symmetric function. Its parameters have meaning similar to normal PDF. Location factor δ ∈ R informs about distribution position, while scale γ > 0 reflects variability.   γ2 1 P DFδ,γ (t) = (1) πγ (t − δ)2 + γ 2

The method is based on the evaluation of normal distribution for some variable informing about economic performance. Thus the method assumes Gaussian properties of the process behavior. Improvement potential is evaluated on the basis of the algorithm presented below Ali (2002):

2.1 Standard Gaussian approach

(1) Evaluate histogram of the selected variable or the performance index. (2) Fit normal distribution to the obtained histogram which is described by two parameters: mean value and standard deviation σ. (3) It is assumed that mean value (Mimprov for the improved system and Mnow for the original one) is kept within the same distance from potential limitation. The idea is to shift the mean value towards the respective constraint. For the confidence level of 95% it is equal to a = 1.65. Such a value is used in the calculations. The mean value for the improved operation is estimated. Standard deviation σ0 relates to the original system and σ1 to the improved one.

Laplace PDF Laplace distribution is called double exponential. It forms a function of differences between two independent variables with identical exponential distributions. Probability density function is given by formula (2). |t−µ| 1 P DFµ,b (t) = e− b , (2) 2b where µ ∈ R is a location factor and b > 0 is a scale parameter. The shape decays exponentially and is characterized by factor b. L´evy α-stable distributions This density function belongs to the family of stable distributions. It has more degrees of freedom (3) as it uses four parameters. α (3) P DFα,β,δ,γ (t) = exp {iδt − |γt| (1 − iβl (t))} ,  πα    sgn (t) tg for α = 1 2 l (t) = , 2  −sgn (t) ln |t| for α = 1 π 0 < α ≤ 2 is called stability index, |β| ≤ 1 is a skewness factor, δ ∈ R is location and γ > 0 is scale parameter.

Stability parameter α is responsible for shape. Location δ keeps information about function position. Additionally we have two more shaping parameters. β informs about distribution skewness, while scaling γ has the meaning very similar to γ parameter of Cauchy PDF. There may be different combinations of them. α = 2 reflects independent stochastic process realizations, especially α = 2, β = 0, γ = 1 and δ = 1 we get exact normal distribution equation.

Mimprov = Mnow · a · (σ0 − σ1 ) (4) (4) Finally percentage improvement is calculated on the basis of the following equation: Mimprov − Mnow ∆M = 100 · (5) Mimprov Let assume that we fit PDF to the histogram with parameters (x0 , σ0 ). We need to keep the same limit at point of x∗ = x0 + k · σ0 , where k determines the point shift. We assume that better control improves variability by factor c, i.e. new σ1 = c · σ0 . Thus we may maintain the same limit with density function shifted by benefit (6).  k M = k · σ0 − c · σ0 2( − ln c) (6) 2

1.2 PDF fitting to data There are various methods for PDF to histogram fitting. Maximum Likelihood Estimation using Octave successive quadratic programming solver (Axensten (2006)) is applied for Cauchy and Laplace function. α-stable fitting uses regression approach (Koutrouvelis (1980)). 2. BENEFIT ESTIMATION The task to predict possible improvements associated with upgrade of a control system exists in literature for a long time Tolfo (1983). From the early days it was mostly associated with the implementation of APC. There are three well established approaches called: same limit, same percentage and final percentage rules Bauer et al. (2007);

Figure 1. Gauss same limit rule graphical example

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Let us select k = 2.0, c = 0.75, u0 = 1.0, γ0 = 0.5 (solid black line). Limit point is x = 2.0 with function value 0.106. We obtain improved variability with γ1 = 0.38 (blue dashed line). Thus we may shift the function by benefit factor M = 0.198 towards the same limit (green dotted line). Figure 1 shows example graphical visualization.

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2.0 with function value of 0.14. We obtain new improved variability of b1 = 0.38 (blue dashed line). Thus we may shift distribution by benefit factor M = 0.142 towards the same limit (green dotted line). Figure 3 presents graphical visualization of the Laplace example.

This approach is very popular despite some deficiencies. But practice shows frequent situations with non-Gaussian histograms. Thus other density functions are considered. 2.2 Algorithm for Cauchy distribution The idea of the algorithm is similar and control system improvement is measured through PDF broadness (σ). The distribution function is presented in (1). Assume we have fitted PDF to the histogram with parameters (x0 , γ0 ). We also have to maintain the same limit at point of x0 = x0 + k · γ0 . We assume that better control improves variability by factor c, i.e. new γ1 = c · γ0 . We keep the same limit with density function shifted by benefit (7).  M = k · γ0 − γ0 c(1 − c)(1 + k 2 ) (7)

Let us assume that k = 2.0, c = 0.75, u0 = 1.0, γ0 = 0.5 (solid black line). Resulting limiting value is at point x = 2.0 with function value of 0.127. We obtain new improved variability of γ1 = 0.38 (blue dashed line). Thus we may shift distribution by benefit factor M = 0.107 towards the same limit (green dotted line). Figure 2 presents graphical visualization of the Cauchy example.

Figure 3. Laplace same limit rule graphical example

2.4 Algorithm for L´evy distribution Control improvement benefit estimation using L´evy density function is not as straightforward as for Laplace or Cauchy PDF. In previous cases we had one, single factor responsible for variability, i.e. σ (Gauss), γ (Cauchy) and b (Laplace). Now we have three different parameters addressing variability. Scaling parameter γ is responsible for distribution broadness. Stability factor α reflects long tails and in such a way it also affects the same limit rule. Skewness β may also impact variability limit. Thus we should consider different combinations of these parameters in solving the same limit rule. It is infeasible analytically. Considering analogy of the stable distribution with the nature of the control system performance each of these parameters has different meaning. By analogy, distribution broadness should be a measure of control quality. Recent works confirm that hypothesis. Scaling factor of α-stable density function can play the role of robust control quality measure. In fact it works almost identically as Cauchy factor γ (Doma´ nski (2016a)).

Figure 2. Cauchy same limit rule graphical example 2.3 Algorithm for Laplace distribution The basic function is presented in (2). Let us assume that we have identified histogram fitting distribution with parameters (x0 , b0 ). One wants to maintain the same limit at point of x0 = x0 + k · b0 . Additionally we estimate that improvement in control diminishes variability by factor c, i.e. new b1 = c · b0 . Thus we may maintain the same limit with density function shifted by benefit (8). M = k · b0 − (c − b0 ) · (k − ln c) (8) Let us assume that k = 2.0, c = 0.75, u0 = 1.0, b0 = 0.5 (solid black line). Resulting limiting value is at point x =

Unfortunately, stability (α) is also impacted by control quality. It is connected with persistence properties of the variable time series and the control system. Skewness manipulation may be also used to improve regulation, as we may allow variations in the direction opposite to the limitation reducing more dangerous ones close to the constraint. Such behaviour is frequently observed in the systems with cooling and water spraying, like in steam temperature control in the desuperheater systems. Thus α-stable PDF is not considered further whereas Cauchy scaling plays very similar role and is analytically available. Nonetheless it remains the open issue.

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2.5 Comparison Three approaches for benefit estimation are evaluated. In all three cases the same example is considered, assuming the same values for the PDF’s scaling factors and the same relative limitation (measured by the density function broadness). In all the cases we have obtained different estimation for the benefit (Table 1). distribution

PDF limit

value at limit

benefit - M

Gauss Cauchy Laplace

0.75 · σ 0.75 · γ 0.75 · b

0.106 0.127 0.135

0.198 0.107 0.142

Table 1. Comparison of benefit estimates Gaussian approach is the most optimistic, i.e. it predicts the highest benefits with the same assumptions. In contrary Cauchy approach is the most conservative. One may call it the realistic solution from the engineering perspective. The Laplace lies between. The differences are caused by the heaviness of the density function tails. One may notice potential high risk to overestimate benefits (in our case almost twice higher – 85%) of Gauss approach in case of fat-tail histogram.

Figure 5. Histogram & PDF fitting for Var2 (before)

3. INDUSTRIAL VALIDATION Industrial validation is performed on the anonymous control data from gas processing industry. The process has undertaken major control rehabilitation. Thus we may compare data before and after tuning. Four exemplary variables are selected, called Var1, Var2, Var3 and Var4. Plots of their histograms with fitted PDFs before the tuning are presented in Figures 4. . . 7. The histograms with fitted PDFs for variables after the tuning are presented in Figures 8. . . 11. Figure 6. Histogram & PDF fitting for Var3 (before)

Figure 4. Histogram & PDF fitting for Var1 (before) Comparison of presented data consists of three elements. Table 2 shows mean square error representing fitting quality of each PDF to the variable histogram. Variables Var1 and Var 4 are of Gaussian character, which is also visible in histogram graphs. In contrary two other variables have fat

Figure 7. Histogram & PDF fitting for Var4 (before) tail properties. It is difficult to assign unambiguously single appropriate density function. It seems that tuning changes their fat tail properties. Var2 changed its character from

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Figure 8. Histogram & PDF fitting for Var1 (after)

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Figure 11. Histogram & PDF fitting for Var4 (after) Gauss

Cauchy

Laplace

before

Var1 Var2 Var3 Var4

10.54 62.61 28.47 11.84

15.40 12.75 18.75 20.05

13.34 28.31 15.14 17.58

after

Var1 Var2 Var3 Var4

5.84 29.09 27.53 9.80

11.30 11.13 15.56 14.25

9.22 9.22 15.61 13.09

Table 2. Histograms PDF fit mean square error Table 3 shows changes in variability measured with PDF factors σ, γ and b. Their mitigation is significant. Reductions for Gaussian variables (Var1 and Var4) are similar, while differences for fat tail variables are prominent. Gauss - σ before after

Figure 9. Histogram & PDF fitting for Var2 (after)

0.43 2.30 84.3%

Cauchy - γ before after

Var1 change reduction

2.73

1.52

2.10

Var2 change reduction

1.66

0.42

0.90

Var3 change reduction

0.72

0.33

0.51

Var4 change reduction

0.64

0.35

0.47

0.39 1.27 76.6% 0.17 0.55 76.8% 0.47 0.17 26.5%

0.24 1.28 84.5%

Laplace - b before after

0.13 0.29 69.6% 0.06 0.26 81.3% 0.27 0.08 22.5%

0.33 1.78 84.5% 0.22 0.68 75.5% 0.10 0.40 79.5% 0.37 0.11 22.4%

Table 3. Improvement in variability measured according to Gauss, Cauchy and Laplace

Figure 10. Histogram & PDF fitting for Var3 (after) Cauchy to Laplace, while Var3 in an opposite direction. Thus we should be cautious with PDF selection.

Table 4 shows possible benefits for the variables obtained with shifting towards limitations. It includes absolute shifts and percentage reduction. It should be noted that changes in the ranges of 1.5÷4.5 are visible and financially measurable. Unfortunately, it is difficult to directly match shifts in benefits with improvements. It is due to the fact that practically nobody can guarantee that setpoint values are shifted exactly according to the same limit rule.

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Final operating decisions are taken with accordance to various factors: safety, controllability, operating practices and habits. The rule only suggests expected potential. MEAN VALUE before after Var1 change reduction %

514.64 530.00 -15.36 -3.0%

Var2 change reduction %

438.99 450.01 -11.02 -2.5%

Var3 change reduction %

71.14

67.99 3.15 4.4%

Var4 change reduction %

66.00

67.00 -1.00 -1.5%

Table 4. Realized variable benefit (mean value)

4. CONCLUSION The paper deals with the the potential benefits estimation that result from control system rehabilitation, like structure upgrade, tuning or APC implementation. Such an analysis is performed at the very initial stage of the process rehabilitation, i.e. during feasibility study and as such does not determines any methods and means how the estimated benefit can be achieved. There are established methods supporting that task. They are based on the same limit rule idea and address reductions of process fluctuations resulting from control improvements. Variability is measured with the normal standard deviation. The method assumes that the variables properties are Gaussian. Actually, there are frequent situations when the normality assumption does not hold, with evident fat tail properties. Following that observation the same rule method is extended towards fat-tail distributions of Cauchy and Laplace to capture industrial reality. Respective relations are evaluated. It is shown with the analytical calculations and on real data examples. The main observation is that Gaussian approach tends to overestimate potential benefits. Fat tail Cauchy density function is the most conservative. This conservatism enables realistic predictions about potential benefits of the control upgrade. Application of the fat tail versions of the same limit method may significantly increase method industrial applicability. The method is general and does not consider any assumptions on the variable properties, like non-stationarity or nonlinearity. That subject needs further attention and may be addressed with persistence or fractal measures. More comprehensive results can be obtained with functions having more degrees of freedom, like α-stable distribution. This direction remains open for further research. REFERENCES Ali, M.K. (2002). Assessing economic benefits of advanced control. In Process Control in the Chemical Industries, 146–159. Chemical Engineering Department, King Saud University: Riyadh, Kingdom of Saudi Arabia.

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