Consistent and Effective Nonlinearity Index and its Application on

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Consistent and Effective Nonlinearity Index and its Application on Model Predictive Controller Performance Deterioration Fahim Uddin, Lemma Dendena Tufa, and Abdulhalim Shah Maulud Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b01984 • Publication Date (Web): 06 Oct 2018 Downloaded from http://pubs.acs.org on October 8, 2018

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Industrial & Engineering Chemistry Research

1

Consistent and Effective Nonlinearity Index and its

2

Application on Model Predictive Controller

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Performance Deterioration Fahim Uddin1, Lemma Dendena Tufa1,* Abdulhalim Shah Maulud1

4 1

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Chemical Engineering Department, Universiti Teknologi PETRONAS, Seri Iskandar - 32610,

6

Perak Darul Ridzuan, Malaysia

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Keywords: Nonlinearity, Nonlinearity Index, Model Predictive Control, Aspen Plus.

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Abstract

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Many control-relevant systems present the challenges of nonlinearity, directionality and ill-

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conditioning for the control systems, and exhibit poor controller performance. This study

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proposes a Nonlinearity Index to quantify the extent of nonlinearity of such systems. A dynamic

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nonlinear model of a pilot-scale distillation column operating near the azeotropic region was

*

Corresponding author

Lemma Dendena Tufa, E-mail: [email protected], Contact # 060-12-2696 816

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Industrial & Engineering Chemistry Research 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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simulated using Aspen Plus Dynamics. A comparison of the results is made with the

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Nonlinearity Measure proposed by Du et al. Results show that a significant increase is exhibited

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in the proposed nonlinearity index values of the system as the system moves towards the

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azeotropic region. Prediction errors of the linear models are also shown to be correlated to the

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proposed index. Therefore, the proposed Nonlinearity Index is consistent with the existing

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indicators of nonlinearity, and thus its measurements of system nonlinearity are reliable.

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Controller performance for the system at higher values of the proposed index further presents its

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efficacy.

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1. Introduction

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Nonlinearity of the system has always been a concern for system identification, analysis and

23

control 1-2. Mildly nonlinear systems can be approximated as linear systems and thus can

24

effectively be controlled using quasi-linear filtering methods. Whereas, a highly nonlinear

25

system can only be represented effectively by a nonlinear model and thus requires nonlinear

26

filtering. Linear controllers of such plants will always perform poorly and high model-plant

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mismatches (MPMs) will be detected by the MPM algorithms. Moreover, such algorithms are

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not sufficient for the performance improvement of highly nonlinear systems 3-5. As nonlinear

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filtering results in high computation and complexity, it is desirable to avoid nonlinear filtering as

30

much as feasible. This can be achieved by determining the degree of system nonlinearity, which

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can categorise whether the system has sufficient nonlinearity to require a nonlinear control

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structure. However, the quantification of the extent of nonlinearity has not been studied


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extensively, and there is a lot of scope for determining parameters which can evaluate the

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nonlinearity of the system effectively.

35

Nonlinearity is not only displayed by the real systems but also can be found in the first-

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principle models of these systems. Such systems, although conveniently simulated by

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commercially available simulators such as Aspen HYSYS®, Aspen Plus® and MATLAB®, are

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found to be challenging for linear control 6-8. Since these simulations are able to demonstrate the

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dynamic behaviour of the plant, they are widely used in chemical engineering applications for

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conceptualisation, analysis, identification and control 9-11.

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A general classification of the existing nonlinearity indices (NLIs) is as follows:

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1.

Distance from the closest linear model

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2.

Function of Nonlinear curvature around any operating point.

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3.

Nonlinearity tests (only determine if a system is nonlinear, not its degree)

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The idea to measure nonlinearity based on its distance from a linear approximation was

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introduced by Beale in 1960, who used regression analysis for the purpose 12. Later on, an NLI

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for systems involving inverse scattering was proposed as the absolute magnitude of the product

48

of diagonal operators and internal radiation terms of the Neumann series expansion 13.

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NLIs based on the distance between a nonlinear system and its best linear approximation were

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presented by 14-18. A normalised version of 14 was proposed by 19, however, its computation

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required a numerical solution, and thus the authors of 20 proposed to calculate a similar NLI

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using functional expansions. Controllability and observability gramians of the linearised system

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were also proposed to determine nonlinearity of the system 21.

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Use of two linear systems for nonlinearity measurement was proposed 18. In this method, NLI

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is the larger of the distances from N to its greatest-lower and the smallest-upper linear boundary 3 ACS Paragon Plus Environment


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function. However, the extension of this method to multiple-input-multiple-output (MIMO)

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systems is challenging, and determination of the linear boundary functions can prove to be

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difficult 22. Liu et al. 23 proposes an NLI as the minimum distance from the set of all linear

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models. This NLI is able to deal with additive and non-additive noise.

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Gap metric was introduced, which is the distance between the linear approximation of the

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nonlinear system and another linear system 24-26. Applications of this metric for nonlinearity

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estimation and multimodel control have been studied 27-28 via weight-based control 29, the H∞

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loop-shaping technique 30, stability margin 31-32 and integration with the neighbourhood

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estimation algorithm33. However, linearization may lose an important part of the nonlinearity

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which is detrimental for the appropriate nonlinearity measurement.

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Instead of the distance of the system from its linear approximation, some researchers prefer to

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use the decline of control performance as a direct measure of the nonlinearity. This performance

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decline can also be considered as the distance from the linear model in terms of control

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performance. A ‘Performance Sensitivity Measure’ was introduced which basically is an index to

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measure the control-relevant nonlinearity by quantifying the performance loss of linear quadratic

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gaussian (LQG) controller 34. This work was further extended in 35. This measure uses the first

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and second order sensitivity coefficients, which are obtained using the original nonlinear first-

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principle models. However, this methodology has not been extended to the nonlinear systems

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whose first-principle models are not present or difficult to obtain. Two local nonlinearity indices

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were proposed which measure the departure of LQG controller performance and stability from

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optimality 36. These indices were able to be integrated with the control design. Another index

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used the variance in the disturbance rejection performance of a linear controller to measure the

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nonlinearity in an operating range 37. This index was shown to be better than the gap metric for 4 ACS Paragon Plus Environment

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determination of nonlinearity. The lower bound ratio of the minimum variance was also

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proposed as a measure of control-relevant nonlinearity 38. The method is data-driven and is

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applicable to Wiener, Hammerstein and Wiener-Hammerstein structures.

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The advantage of NLIs in this class is that they usually do not require the derivative of the function. However, they have several drawbacks: 23

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1.

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They usually present the worst-case scenario, which may be very different from the normal operation.

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2.

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They are not suited for estimation as they do not account for the randomness of the estimated quantity.

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3.

They usually require high computation and are thus inappropriate for MIMO systems.

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Another way to calculate NLI is to utilise the curvature from differential geometry. Refs. 39-40

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defined an NLI based on curvature, which uses its first and second derivatives. Both of these NLI

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curvatures were calculated 41 and applied to various real-world problems 42-45, and performance

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evaluation based on NLI curvatures was also carried out 46. Another index measured the steady-

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state nonlinearity using partial differentiation 47. The tangential and normal components were

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used to determine the nonlinearity of the process.

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The advantages of curvature-based NLIs are 23:

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1. They are relatively easier to compute in case of easy/defined derivatives.

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2.

They are easy to visualise geometrically.

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3.

Intrinsic curvature is independent of parametrisation.

However, they also incur various disadvantages, such as 23:

99 100

1.

The derivatives must be determined analytically.



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2.

They provide a pessimistic measure of nonlinearity since they usually calculate the

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maximum nonlinearity present in the system at a particular frequency which might

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not be helpful or relevant to a specific control application.

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3.

They ignore high order terms, which also contribute to system nonlinearity.

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Mixed-filtering based NLI was proposed 48, in which a linear and a nonlinear filter were

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applied in parallel and summed. A minimisation function determines the weights of the sum,

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where the weight of nonlinear filter is referred to as NLI. However, its dependence on the filter

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selection and representation of the performance rather than nonlinearity are not justified.

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One way to determine the nonlinearity is to apply any of the several nonlinearity tests

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proposed in the literature 49-54 to the time series or clinical data. These tests determine whether

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the system is linear or not without providing insights to the degree of nonlinearity. However,

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having an estimate of the extent of nonlinearity is more informative than just knowing whether

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the system is nonlinear. The current study adopts the idea of 50 and transforms it into a

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measurable index of nonlinearity for MIMO systems which can assess whether the linear process

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control of a system will fail to deliver acceptable performance. This index is then compared with

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an existing nonlinearity index in the literature and their relationship with the parameters

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indicating system nonlinearity are observed. The NLI-proposed degree of nonlinearity of the

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operating ranges is also tested by observing the system control behaviours of these ranges.

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The rest of the paper is arranged as follows: Section 2 explains the mathematical derivation

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and input requirements of the proposed nonlinearity index. Section 3 discusses the development

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of the case study, its control structure and data generation for the measurement of nonlinearity.

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An existing nonlinearity measure (NM) from the literature is also briefly introduced. Section 4

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presents the results obtained by the application of proposed NLI. The values of the proposed NLI 6 ACS Paragon Plus Environment

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and NM are compared for various operating ranges, input magnitudes and linear model

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prediction errors. Control performance of the system at different NLI values are also shown.

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Finally, Section 5 presents the conclusion of this study.

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2. Current Work

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2.1. Proposed Nonlinearity Index

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A test statistic was introduced by Billings and Voon50 in order to determine the structure of the

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system. In this statistic, the higher order correlation functions were used and compared against

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the maximum allowable value for 95% confidence limits. The test requires that the input u(t) and

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noise e(t) are independent, with their means and odd-order moments being zero. These properties

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can be ensured by using sine wave, gaussian signal, ternary pseudorandom sequence or

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independent uniformly distributed process as the input signal. The correlation function ϕ at delay

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or lags τ between N number of measured values of output with sample mean as 𝑦 is defined as:

 yy

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 yy

138 139

2

2

( )

( )



 E[( y( k  )  y )( y( k )  y ) 2 ] 1 N

N

 [( y

( k  )

k

 y )( y( k )  y ) 2 ]

(1) (2)

If the system is perfectly linear, the odd order moments of the output signal obtained will also

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be zero. Consequently, it was proposed using 95% confidence limits that any dynamic system

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can be appropriately represented by a linear model if:

 yy

142

2

( )

 yy (0)  y

 2 2

y (0)

1.96 N 

(3)

According to Billings and Voon 50, the left-hand side of the equation needs to be evaluated and

143 144

checked whether it exceeds the threshold set on the right-hand side of the equation. However,

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this does not provide any information on the extent of nonlinearity.

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In order to measure the extent of nonlinearity, this equation can be rearranged for this work as: 8 ACS Paragon Plus Environment

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0.5102 N  

147

148

 yy

2

( )

 yy (0)  y

1 y (0)

Defining the proposed Nonlinearity Index as:

NLI  0.5102 N  

149

150

 yy

2

( )

 yy (0)  y

(5) 2 2

y (0)

where an NLI value of zero represents absolute linearity and one represents sufficient

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nonlinearity to use nonlinear model for identification. Since it is a ratio of odd and even order

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moments, the units cancel each other and the resulting NLI carries no units.

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(4)

2 2

In order to define overall nonlinearity of a MIMO system having j number of outputs, a general expression is defined as: j

1 j

NLI   (1  NLI yi )  1

155

(6)

i 1

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2.2. Plant Excitation for NLI measurements

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The nonlinearity test defined in section 2.1 requires that the input signals must have zero odd-

158

order moments 50. Therefore, sine wave signals can be used for the excitation of the system.

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Consequently, the amplitudes of inputs sine waves are varied systematically using appropriate

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step size and introduced to the selected operating regimes. It is better to keep the inputs out of

161

phase and to have different frequencies in order to explore the interaction of the system.

162



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3. Case Study: Nonlinearity Measurement near Azeotropic Region

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In order to demonstrate the effectiveness of the proposed nonlinearity index (NLI), distillation

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column control near azeotropic region is selected as the case study. It has been observed in our

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previous work 55 that as the system moves towards the azeotropic region, the following

168

observations are made:

169

1.

As the operating range moves near high purity/azeotropic point, the nonlinearity of

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the system increases. Hence it is difficult to present the system behaviour effectively

171

using any linear model.

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2.

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system.

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3.

175 4.

177

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As the nonlinearity increases, the prediction accuracy of any linear model identified for the system decreases.

176

178

The wider the operating range gets, the more the nonlinearity is incurred by the

Since controller performance is adversely affected by nonlinearity, any increase in the nonlinearity results in a decrease in control performance.

The magnitude of the proposed NLI is defined to be proportional to the actual nonlinearity of the system, therefore it should be able to achieve the following results.

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1.

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Proposed nonlinearity index should represent an increase in nonlinearity of the system as the operating range moves towards azeotropic point.

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2.

183

Proposed nonlinearity index should indicate an increase in nonlinearity as the operating range of the system gets wider.


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3.

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For higher values of the proposed nonlinearity index, the prediction error of the linear model identified for the system should increase.

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4.

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Increase in the value of the proposed nonlinearity index should result in a decrease in control performance.

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In order to analyse the relevance of NLI as the system moves towards the azeotropic region,

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three different operating regimes with different steady states were selected, differing only in the

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steady-state compositions. Consequently, three Aspen Plus® simulations were developed. These

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simulations were then converted to dynamic mode, and Model Predictive Control (MPC) was

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designed for the top and bottom compositions.

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3.1. Aspen Plus® Dynamic Simulation

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This study was conducted by using Aspen Plus® dynamic simulation of the pilot distillation

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column present at Universiti Teknologi PETRONAS as presented in Figure 1. The column has

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15 stages, 0.35 m apart, with the feed entering at the ninth stage. The ethanol-water separation

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using this column was selected for this study. The feed of binary mixture, composed of 25 mol%

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ethanol with a flow rate of 0.6 m3/hr, enters the column T-300 through feed valve F3-V at 77oC.

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The ethanol-enriched stream is collected as distillate after being condensed. The pressure in the

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condenser was maintained at 1.25 bar to reflect the near-atmospheric distillation systems. A

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detailed account of the development of steady-state simulation for this pilot plant has been

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reported in the literature 11, 56. The column results for the selected operating regimes have been

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summarized in Table 1. The major difference among the operating regimes is the ethanol

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composition among the distillate streams of the operating regimes as shown in Table 2. 11 ACS Paragon Plus Environment


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Figure 1: Process Schematic for the Aspen Plus® Dynamic Simulation of Pilot-Plant Distillation

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Column

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Table 1: Steady-state stream conditions of the simulations Stream conditions

Feed

Distillate

Bottoms

Temperature (C)

77

83.56

91.20

Pressure (kPa)

130.95

124.75

135.78

Flowrate (m3/hr)

0.6

0.13044

0.4806

209 210

Table 2: Ethanol compositions Operating Regimes

SS-82

SS-83

SS-84

Feed (Steady state)

25

25

25

Distillate (Steady state)

82.01

83.02

83.99

Bottoms (Steady state)

17.42

17.35

17.29

Operating Ranges

[80.75 83.25]

[81.75 84.25]

[82.75 85.25]

211 12 ACS Paragon Plus Environment

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The pressure-driven dynamic simulation had a pressure controller as a default in the

213

simulations. This proportional-integral (PI) controller maintained the pressure of the condenser

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by manipulating condenser duty. A PI feed flowrate controller and two proportional (P) level

215

controllers for the condenser and reboiler level were installed in each simulation. Details

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regarding the general setup procedure for dynamic simulations can be found in the literature 8, 57,

217

whereas the configurations of P and PI controllers can be found in 58.

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3.2. Data Generation for NLI Calculation

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In order to generate data for the calculation of proposed NLI, the sine inputs are entered in the

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system for six hours, generating a total number of 600 data samples for analysis, using 0.01 hr as

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sample time. Sine inputs with amplitudes varying from 5% to 20% were used to observe the NLI

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for different widths of operating ranges. Sine inputs with amplitudes of 5% as well as 20% are

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plotted in Figure 2 for illustration.



224 225

Figure 2: Input sequences to the simulation for NLI calculation


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227


3.3. NM Calculation The nonlinearity measure proposed by Du et al. 59 is compared with the proposed NLI. The

228

comparison will be made based on the ability to show a consistent increase in system

229

nonlinearity as the system moves towards the azeotropic region or widens the existing operating

230

range. According to Du et al. 59, the nonlinearity measure can be calculated by the following

231

equation:

NM 

232

 max ( P*)

(7)

bP*, K

233

Where δmax (P*) is the Maximum gap i.e. Gap metric of the system and bP*,K is the gap metric

234

stability margin of the system P* and linearized stable controller K, which is given by the

235

following expression: bP , K 

236 237

I  1  K   I  PK   I  

1

P

 

I  1  P   I  KP   I  

1

K

(8) 

As the gap metric stability margin is a function of the system as well as the controller,

238

appropriate control configuration is required for its calculation. Therefore, proportional-integral-

239

derivative (PID) controller configurations were identified using MATLAB® command ‘pidtune’

240

for PID controller design and used for the calculation. This command selects PID tuning to

241

ensure appropriate open-loop phase margin as well as cross-over frequency for a given system.

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3.4. System Identification and Control

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The plant is excited using Pseudo Random Binary Signal (PRBS) for the development of linear models of the operating regimes for model predictive control, as shown in Figure 3. In order to 15 ACS Paragon Plus Environment


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avoid ill-conditioned modelling of the system, the outputs i.e. top composition yD and bottom

246

composition yB, inputs i.e. reflux flowrate R and reboiler duty QB and disturbance i.e. feed

247

flowrate F were scaled as follows:

a' 

248 249

a  a SS aR

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(9)

Where aSS is the steady-state value of the parameter and aR is its span.

250 251 252 253 254

255

256

Figure 3: Input sequences and output responses for System Identification The 2x3 first order plus time delay (FOPTD) models were identified as shown in the following equations. Note that the delays less than the sampling time Ts=0.01 hr were neglected. For the operating regime SS-82,  4.146  y 'D   0.085s  1  y '    1.179  B   0.198s  1

4.327   1.519   0.084 s  1  R '   0.129 s  1    F' 1.668  Q 'B   0.9723  0.198s  1   0.195s  1 

For the operating regime SS-83, 16 ACS Paragon Plus Environment

(10)

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 2.03  y 'D   0.067 s  1  y '    0.921  B   0.197 s  1

257

258

(11)

0.962   0.3147   R '   0.586 s  1  0.084 s  1    F' 0.907  Q 'B   0.9199  0.207 s  1   0.196 s  1 

(12)

For the operating regime SS-84,  3.891 y '  D   0.276 s  1  y '    3.270  B   0.206 s  1

259

260

2.578   0.1236e 0.08 s  0.069 s  1   R '   0.03s  1   F'  1.419  Q 'B   0.9653   0.189 s  1  0.197 s  1 

Where y’D and y’B are the scaled outputs i.e. top and bottom compositions, R’ and Q’B are the

261

scaled inputs i.e. reflux flowrate and reboiler duty and F’ is the scaled disturbance i.e. feed

262

flowrate.

263

Table 3 shows that the models for the selected regimes fitted more than 95% to estimation data

264

of output yD and 80% to estimation data of output yB, using prediction focus. It can be observed

265

that the identified models for the selected regimes fit the identification data well. The Final

266

Prediction Error (FPE) and Mean Square Error (MSE) were found to be of the order of 10-11 and

267

10-6 respectively for the selected regimes. The Relative Gains (RG) for these models suggest that

268

the top composition (yD) is more influenced by reflux (R) than by the reboiler duty (QB) in all the

269

cases, which is in accordance with the intuition.

270

Table 3: Comparison of the identified models Model Accuracy

SS-82

SS-83

SS-84

Fit% (prediction focus)

[95.74;97.4]

[96.5;97.6]

[95.8;97.2]

FPE × 1011

1.918

2.092

4.790

MSE × 106

12.76

8.646

7.435

Relative Gain

3.81

5.69

0.53



271 272

Using these models, suitable MPCs are designed for composition control as shown in Table 4.

273

Tuning advisor is used from MATLAB MPC toolbox for the controller tuning. Therefore, the

274

system is ready for the evaluation of controller performance. For setpoint tracking performance,

275

the system will incur a step change in the setpoint, the magnitude of which will depend on the

276

operating range. For disturbance rejection, a step change in disturbance will be introduced to the

277

system.

278

Table 4: Tuning parameters selected for MPC using Tuning advisor Tuning parameters

SS-82

SS-83

SS-84

Sampling Time

0.01 hr

0.01 hr

0.01 hr

Prediction Horizon

30

30

30

Control Horizon

10

10

10

Output Weights

[3.75 1.85]

[2.15 1.88]

[2.15 1.88]

Input Weights

[0 0]

[0 0]

[0 0]

Input Rate Weights

[1.5 0.5]

[1.5 0.5]

[1.5 0.5]

279 280

4. Results and Discussions

281

4.1. Perturbation Responses for Determining NLI

282

Figure 4 presents the Aspen Plus simulation responses for the operating regimes SS-82, SS-83

283

and SS-84 for the inputs shown in Figure 2. It is observed that the systems show relatively

284

deterministic linear behaviour when the input amplitudes are low. For higher input values, the


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285

systems show increased nonlinear response. The responses of the system observed in this section

286

are utilised for the calculation of the proposed NLI for the system.

287 288

Figure 4: Output responses of the Aspen Plus® Dynamic Simulation

289

4.2. Variation w.r.t. Input Magnitude and Distance from the Azeotropic Region

290

291

The system is ready to be evaluated by the proposed NLI for its extent of nonlinearity by using

292

the data from Section 4.1. A comparison with a recently proposed nonlinearity measure (NM) by

293

Du et al. (2017) 59 is also made to further demonstrate the efficacy of NLI.



294

The values of the NLI proposed in this study are shown in Table 5. Effects of input magnitudes

295

and distance from the azeotropic point are observed. Since the nonlinearity index has been

296

defined to use the open-loop sinusoidal inputs and its responses, it is relevant to compare the

297

values of nonlinearity index for different operating regimes against same input perturbations in

298

open loop. As required, the NLI values are found to increase with an increase in input

299

magnitude, as well as with the decrement of distance from the azeotropic region.

300

It can be realised from Figure 5 that the values of the nonlinearity index for SS-82 are less than

301

one for all input magnitudes, which means that this operating range can be considered a linear

302

system. A maximum value of 0.90 was observed, which is still below the statistical limit of 1. A

303

linear model is sufficient for acceptable control performance using the model predictive control in

304

such cases. On the other hand, the NLI value of the regime SS-84 for 20% input magnitude is

305

greater than one, which means that a nonlinear model is required for acceptable control

306

performance using the model predictive control in this case.

307

Table 5: Nonlinearity Index (this study) values for the selected system

Operating Ranges

Input Perturbation Magnitudes 0.05

0.10

0.15

0.20

SS-82

0.21

0.44

0.67

0.90

SS-83

0.22

0.48

0.75

1.03

SS-84

0.39

0.77

1.06

1.33

308 309


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310 311

Figure 5: Nonlinearity Index (this study) values for the selected system

312



313

Table 6 and Figure 6 present the NM values for the system as the operating range moves

314

towards the azeotropic region. The NM value for operating range SS-83 with input magnitude

315

0.20 is less than 1, which suggests that the system can be considered linear in this range. This

316

value is also equal to the NM value for input magnitude 0.15, suggesting no increase in the

317

nonlinearity as the operating range is widened. However, the experimental observations in a

318

previous work have shown otherwise 55. Moreover, inconsistencies are observed as the NM for

319

operating range SS-84 with input magnitude 0.10 is less than that of 0.05 and for input

320

magnitude 0.20 is less than that of 0.15. Therefore, it can be concluded that the magnitude of

321

NM is not able to track the actual nonlinearity of the system as the system moves near azeotropic

322

region.

323


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324


Table 6: Nonlinearity Measure (Du, 2017) values for the selected system Operating Ranges (mol%)

Input Perturbation Magnitudes 0.05

0.10

0.15

0.20

SS-82

0.48

0.59

0.77

0.82

SS-83

0.59

0.64

0.84

0.84

SS-84

1.07

0.88

2.67

1.63

325

326 327 328 329

Figure 6: Nonlinearity Measure (Du, 2017) values for the selected system Therefore, the values of the proposed NLI are more consistent and realistic in the prediction of the nonlinearity of the system than the previous observations of NM for the same system.

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332

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4.3. FPE Comparison with NLI and NM To further establish the efficacy of the proposed index, the model residuals can be used as

333

another criterion, since the nonlinearity of the system is also known as its inability to be

334

represented effectively by a single linear model. The results in Table 7 and Figure 7 show that

335

the NLI values are more closely related to the observed logarithmic values of the FPE. The graph

336

shows that the NM values are continuously moving back and forth as the logarithmic values of

337

FPE increase. However, the NLI values are mostly showing a more predictable increase with the

338

increase in the logarithmic values of FPE. The adjusted R2 value for linear regression of NLI

339

against FPE is 0.84 whereas the adjusted R2 value for linear regression of NM against FPE is

340

only 0.40. Therefore, the NLI values are better representative of the nonlinearity of the system

341

than NM.

342

Table 7: Comparison of NLI with NM with Model residuals log10 FPE

-10.7 -9.05 -7.94 -8.03 -8.78 -7.17 -5.95 -7.74 -6.63 -6.13 -5.28 -4.46

NLI

0.21

0.22

0.39

0.44

0.45

0.62

0.75

0.77

0.90

0.02

0.06

0.33

NM

0.48

0.59

1.07

0.59

0.64

0.75

0.84

0.88

0.88

0.88

2.67

1.63

343


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344 345

Figure 7: NLI and NM against FPE for the selected system

346

4.4. Linear MPC Performance for different NLI values

347

In order to show how the system behaviour changes and control performance deteriorates as

348

the Nonlinearity index (NLI) values increase, a comparison is made for the setpoint tracking and

349

disturbance rejection performance of the operating regimes with different NLI values. The

350

system was subjected to linear Model Predictive Control (MPC), which used the identified linear

351

model (Equation 12) for the selected operating ranges. The controller was appropriately tuned.

352

Setpoint changes were introduced as PRBS with magnitudes in accordance with the nonlinearity

353

index values of the system.

354 355

Figure 8(a,b) show that the MPC is well-tuned for the control of the system and is able to track the set points and reject the disturbances in both positive and negative directions. From Figure 25 ACS Paragon Plus Environment


356

8(a), it can be observed that the setpoint for the top composition is achieved within the first five

357

minutes of the change in setpoint, with an overshoot of 40%. Moreover, the system achieves the

358

setpoint for the bottom composition within the first fifteen minutes. In Figure 8(b) the

359

disturbance rejection is efficiently performed. The top and bottom compositions remain close to

360

the tolerance band.

361

These observations show that as the operating regime is mildly nonlinear i.e. has a low value

362

of NLI, the responses of the system are linear, and thus optimal input manipulations are carried

363

out for setpoint tracking and disturbance rejection. It can be concluded that the controller

364

performance of the system is satisfactory.

365 366

(a) Setpoint Tracking – NLI = 0.65 26 ACS Paragon Plus Environment

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367 368 369

(b) Disturbance Rejection – NLI = 0.65 Figure 8: Controller Performance (NLI=0.65)

370

The setpoint tracking and disturbance rejection performance of the operating regime where the

371

system has just enough nonlinearity to be considered as nonlinear is presented in Figure 9(a,b). It

372

is observed that the tracking and rejection performances have just started to deteriorate, and the

373

controller action to achieve or maintain the setpoint has just started to be insufficient. Figure 9(a)

374

reveals that although the setpoints for the top composition are tracked within the first ten minutes

375

with only 35% overshoot, the response for bottom composition shows that a trade-off is being

376

made and the bottom composition is not being brought back to its reference point. The steady27 ACS Paragon Plus Environment


377

state error, however, might be tolerable for some of the control applications where the tolerance

378

bands are wider. A similar behaviour is observed in the disturbance rejection plots i.e. Figure

379

9(b) where the disturbance is not completely rejected but is brought back near the tolerance band.

380

Depending on the product specifications, some plant might tolerate the controller performance

381

of this magnitude. Nevertheless, it can be incurred that further widening of the operating regime

382

will result in poor setpoint tracking and rejection due to the increase in nonlinearity.

383 384

(a) Setpoint Tracking – NLI = 1.00


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385 386 387 388

(b) Disturbance Rejection – NLI = 1.00 Figure 9: Controller Performance (NLI=1.0) Figure 10(a,b) shows the setpoint tracking and disturbance rejection performance of the system

389

when the operating regime is wide enough to be significantly nonlinear (NLI=1.66). The NLI

390

magnitude suggests that the rejection and tracking should be poor, and the results confirm this

391

hypothesis. Large offsets are present, and the controller action is not able to track or maintain the

392

setpoint, thus poor controller performance is achieved.



393 394

(a) Setpoint Tracking – NLI = 1.65


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395 396

(b) Disturbance Rejection – NLI = 1.65

397

Figure 10: Controller Performance (NLI=1.66)

398

4.5. Overall discussion

399

The efficacy of any proposed nonlinearity index depends on its ability to predict the behaviour

400

of the system with sufficient accuracy and consistency. The increased nonlinear behaviour of a

401

system, as well as the decline in the prediction accuracy and linear controller performance, are

402

used in this study as the outcomes of an increase in nonlinearity. The proposed nonlinearity

403

index is evaluated in such conditions and compared with another index proposed in the literature. 31 ACS Paragon Plus Environment


404

These results clearly establish the superiority of the proposed NLI. Therefore, it can be

405

concluded that the system nonlinearity and behaviour is correctly predicted by the proposed NLI.

406

Importantly, the generic nature of this NLI can make it viable for the prediction of nonlinearity

407

for other control-relevant systems.

408

5. Conclusion

409

This study proposes an effective method to determine the extent of nonlinearity in the form of

410

the nonlinearity index (NLI). It is shown that the proposed nonlinearity index is able to estimate

411

the nonlinearity of system and is correlated to the linear model residuals and closed-loop model

412

predictive controller performance. The results are compared with the nonlinearity measure in

413

literature and observed that the proposed index is more consistent and relatable to the existing

414

results of the selected case study. Thus, the proposed NLI is an efficient and computation-

415

friendly index to evaluate the nonlinearity of systems. The index is generic in nature and can be

416

applied to other control-relevant systems.

417

6. Acknowledgements

418

The main author would like to thank Syed A. Taqvi for his valuable insights, comments and

419

suggestions used for the conceptualization and analysis of this study. The authors also recognize

420

the Graduate Assistance scheme provided by Universiti Teknologi PETRONAS for its

421

continuous support throughout the research.

422

7. References 32 ACS Paragon Plus Environment

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Nyström, R. H.; Böling, J. M.; Ramstedt, J. M.; Toivonen, H. T.; Häggblom, K. E.,

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54. Barnett, A. G.; Wolff, R. C., A time-domain test for some types of nonlinearity. IEEE Trans. Signal Process. 2005, 53 (1), 26-33. 55. Uddin, F.; Tufa, L. D.; Shah Maulud, A.; Taqvi, S. A., System Behavior and Predictive

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10 (2).

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59. Du, J.; Johansen, T. A., Control-relevant nonlinearity measure and integrated multimodel control. J. Process Control 2017, 57, 127-139.

554 555

Author Contributions

556 557 558

The manuscript was written through equal contributions of all authors. All authors have given approval to the final version of the manuscript.

559

ABBREVIATIONS

560 561 562

NLI, Nonlinearity Index; MIMO, Multiple-Input-Multiple-Output; NM, Nonlinearity Measure; MPC, Model Predictive Control; RG, Relative Gain; FPE, Final Prediction Error; MSE, Mean Square Error; FOPTD, first order plus time delay; MPM, model-plant mismatch; LQG, linear 39 ACS Paragon Plus Environment


563 564

quadratic gaussian; P, Proportional; PI, Proportional-Integral; PID, Proportional-IntegralDerivative; PRBS, Pseudo Random Binary Signal.

565


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TOC Graphic

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Consistent and Effective Nonlinearity Index and its Application on

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