# A Novel ARX-Based Approach for the Steady-State Identification Analysis of Industrial Depropanizer Column Datasets

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## Abstract

**:**

## 1. Introduction

## 2. Steady-State Identification Techniques

#### 2.1. F-Like Test

- A filtering value is estimated from the data:$${X}_{f,1}={\lambda}_{1}{X}_{i}+(1-{\lambda}_{1}){X}_{f,i-1}$$
_{f}is the filtered value of X, λ_{1}is a filter factor and i is the time sampling index. - The initial variances are calculated by the following exponentially weighted moving average:$${v}_{f,i}^{2}={\lambda}_{2}{\left({X}_{i}-{X}_{f,i-1}\right)}^{2}+(1-{\lambda}_{2}){v}_{f,i-1}^{2}$$
_{2}is a filter factor for the variance. - The second filtered variance estimate is computed as:$${\delta}_{f,i}^{2}={\lambda}_{3}{\left({X}_{i}-{X}_{i-1}\right)}^{2}+(1-{\lambda}_{3}){\delta}_{f,i-1}^{2}$$
_{3}is another filter factor for the variance. - The ratio based on the two variances corresponds to the SSI index and is obtained by the following equation:$$R=\frac{(2-{\lambda}_{1}){v}_{f,i}^{2}}{({\delta}_{f,i}^{2})}$$

_{1}= 0.2, λ

_{2}= λ

_{3}= 0.1. The work in [4] also identified possible process issues that can affect R.

#### 2.2. Adaptive Polynomial

_{1}is used as an indicator of the steady-state condition. The only parameter to be tuned in this technique is the size of the window.

#### 2.3. Wavelet-Based Method

_{1}and d

_{2}, respectively, are calculated. In this work, we employ the “diff” function of MATLAB for that purpose. The thresholds T

_{s}, T

_{w}and T

_{u}are computed by:

_{1}and $\overline{{d}_{1}}$ is the median of d

_{1}.

_{T}and R

_{i}, can take values between zero and one. In particular, one indicates a steady-state condition and zero an unstable/transient condition. The values between zero and one also indicate an unstable condition.

#### 2.4. Proposed ARX-Based Approach

_{i}and b

_{i}, are identified by least-squares as follows:

^{T}X, which carries the model information. This matrix, which is comprised of the measured and the input variables, can be analyzed by means of a singular value decomposition (SVD) given by:

^{T}X from which the eigenvalues, diag(λ

_{1}, λ

_{2}, …, λ

_{n}), can be obtained, and U and V are unitary matrices. For different process systems applications, SVD is also used in principal component analysis (PCA) for fault and diagnosis detection by combining observations of measured variables.

^{T}X, as follows:

^{T}X is less than a certain specified threshold, the system is at steady state. This can be explained by the fact that the ARX model matrix, X

^{T}X, presents a singularity (represented by the eigenvalue close to zero) in the steady-state condition where the ARX model is not identifiable. In the absence of noise, the smallest eigenvalue of the model matrix would be zero at steady state. However, in the transient state, where there is enough system excitation, the eigenvalues will be well conditioned. In other words, this SSI technique is based on the continuous analysis of the matrix X

^{T}X. Note that in this proposed approach, it is necessary to define the size of the ARX window and the threshold ( ) for Equation (20). In this work, we use a value of ϵ = 10

^{−2}and n = 1 for a

_{n}and b

_{n}for all SSI simulations. The selected ARX model order could be changed if performance improvement of the method was necessary. Furthermore, for performance improvement, although here we consider only one threshold value for all variables, the user could choose to define one of such values for each variable if needed.

## 3. Description of Case Studies

#### 3.1. Simulated Dataset

#### 3.2. CSTR System

_{r}], the concentration of A and reactor temperature, respectively; and two manipulated inputs, u = T

_{c}, coolant liquid temperature, and u = F

_{0}, the inlet flow rate. The optimization problem solved by the controller at each time step is given by:

_{sp}is the output setpoint.

#### 3.3. Industrial Depropanizer Column

## 4. Steady-State Identification Results

#### 4.1. Simulated Data Example

_{1}= 0.0587, λ

_{2}= 0.3, λ

_{3}= 0.02, were taken from Table 2 of [13], because these factors improved the performance of the method, and a λ = 1.2 was employed for the wavelet approach.

#### 4.2. CSTR System

^{−2}, i.e., below this value, the process is considered at steady state. To build the ARX model matrix, information is taken from the manipulated input u = T

_{c}and the controlled output y = T

_{r}. The other techniques only used information of the controlled output y = T

_{r}. Finally, the F-like test considered the following tuning parameters, λ

_{1}= 0.2, λ

_{2}= λ

_{3}= 0.1.

_{1}= 0.0587, λ

_{2}= 0.3, λ

_{3}= 0.02. Figure 10 presents the results for these new conditions. Note in this figure that the polynomial technique improved its performance when compared to the previous case. However, the technique still fails to indicate the steady-state condition at some points, for example during the time interval between 15 and 25. For guaranteed performance, the polynomial technique may need further development, including further studies on its index definition. The F-like test with the new tuning is now able to identify the transients at Time 70 and at the first instants. Finally, the ARX-based technique considerably improved its performance after re-tuning, showing comparable results to the F-like test, with less Type-I errors. From the number of tuning parameters point of view, the ARX-based approach requires the size of the window and the threshold (ϵ), which has an intuitive definition, as it depends on the singularity of the model matrix. The F-like test is based on three λ’s. The above results indicate that the ARX-based technique offers an alternative with comparable performance to the F-like test, but with less tuning parameters.

#### 4.3. Industrial Depropanizer Column

#### 4.3.1. Analysis of the Reflux Drum (TA1)

_{1}= 0.0587, λ

_{2}= 0.3, λ

_{3}= 0.02 for the F-like test and a λ = 1.2 for the wavelet method.

#### 4.3.2. Loop Data from the Heat Exchanger (HX4)

_{1}= 0.0587, λ

_{2}= 0.3, λ

_{3}= 0.02 for the F-like test and a λ = 0.06 for the wavelet approach. In this figure, the top two parts depict the data associated with the output and input variables. These variables show a transient behavior in the first 410 min, followed by a steady-state period. The results of the F-like test in this figure show that this approach is able to correctly identify the steady-state part after 400 time units. However, the initial period was not identified properly. The wavelet method performs in a similar manner by missing most of the transient periods between 0 and 400 min, but detecting the steady-state period afterwards. The ARX-based approach has a slightly better performance. It identified correctly the entire steady-state period after 400 time units, and the transient parts between 60–160 and 300–340 min.

^{−3}. The steady-state period was once again correctly identified by both the ARX-based and the wavelet methods, in which the ARX-based one only missed some stable points around 410–430 min. Thus, both methods demonstrated a satisfactory level of accuracy for this application. Table 2 summarizes the SSI results for different frequencies in terms of their success rate in identifying steady-state/transient points. Therefore, this case study shows that the performances of the methods improve when reducing the measurement sampling. This conclusion suggests that, for really fast sampling, the SSI analysis is giving a higher weight to the faster dynamics (including local noises and oscillations), rather than the overall dynamic characteristics of the system that are relevant to the steady-state detection. It is also important to note that depending on the noisy characteristics of the dataset, the wavelet method may require a data pre-filtering step (for the correct calculation of the first and second derivatives), which would preclude its online implementation. That is not the case of our proposed ARX-based approach that uses a model that is computationally tractable for online application.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Case 1. Simulated data results with 3% of noise, in which a window of 10 was used for the polynomial technique.

**Figure 2.**Case 2. Simulated data results with 5% of noise with gross error, in which a window of 10 was used for the polynomial technique.

**Figure 4.**CSTR data results with 3% of noise, in which a window of 70 was used for the polynomial and the ARX-based techniques.

**Figure 10.**CSTR data results with 3% of noise, in which a window of 30 was used for the polynomial and the ARX-based techniques.

**Figure 11.**Depropanizer reflux drum case temperature experimental data and SSI results for the F-like test and the polynomial methods.

**Figure 12.**Depropanizer reflux drum case temperature experimental data and SSI results for the wavelet method.

**Figure 13.**Depropanizer reflux drum case temperature experimental data, in which 5% of Gaussian noise is added to the data, and the SSI results for the F-like test and the polynomial methods.

**Figure 14.**Depropanizer reflux drum case temperature experimental data, in which 5% of Gaussian noise is added to the data, and the SSI results for the wavelet method.

**Figure 15.**Depropanizer reflux drum case temperature experimental data and SSI results for the F-like test and the polynomial methods.

**Figure 16.**Depropanizer reflux drum case temperature experimental data and SSI results for the wavelet method.

Parameters | Name | Value |
---|---|---|

k_{0} | frequency factor | 7.210 × 10^{10} min^{−1} |

F_{0} | inlet flow rate | 0.1 m^{3}/min |

C_{0} | concentration of A in the inlet flow | 1 mol/m^{3} |

r | radius of the tank | 0.219 m |

T_{c} | coolant liquid temperature | 300 K |

U | heat transfer coefficient | 54,936 J/(min m^{2} K) |

h | level of the tank | 0.659 m |

E/R | activation energy and gas constant | 8.75 × 10^{3} |

T_{0} | temperature of the inlet flow | 350 K |

−∆H | heat of reaction | 5.7 J/mol |

ρ | density | 1 × 10^{3} kg/K |

C_{p} | specific heat capacity | 239 J/(kg K) |

**Table 2.**Success rate of steady-state identification (SSI) methods for different frequencies for the HX4 case.

Frequency | F-Like Test | Wavelet | ARX-Based |
---|---|---|---|

1 | 37% | 30% | 46% |

3 | 50% | 60% | 83% |

5 | 56% | 88% | 92% |

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**MDPI and ACS Style**

Rincón, F.D.; Roux, G.A.C.L.; Lima, F.V.
A Novel ARX-Based Approach for the Steady-State Identification Analysis of Industrial Depropanizer Column Datasets. *Processes* **2015**, *3*, 257-285.
https://doi.org/10.3390/pr3020257

**AMA Style**

Rincón FD, Roux GACL, Lima FV.
A Novel ARX-Based Approach for the Steady-State Identification Analysis of Industrial Depropanizer Column Datasets. *Processes*. 2015; 3(2):257-285.
https://doi.org/10.3390/pr3020257

**Chicago/Turabian Style**

Rincón, Franklin D., Galo A. C. Le Roux, and Fernando V. Lima.
2015. "A Novel ARX-Based Approach for the Steady-State Identification Analysis of Industrial Depropanizer Column Datasets" *Processes* 3, no. 2: 257-285.
https://doi.org/10.3390/pr3020257