**1. Introduction**

Distillation is a widely used energy consuming unit operation in modern petroleum and chemical industries. Its separation on raw materials, intermediate products, and crude products exerts a strong influence on the energy consumption and product quality of the industrial process involved. The fault diagnosis technique benefits from catching the deterioration symptom of critical parameters in time and predicting their deterioration trend effectively when a distillation column enters an abnormal state. To improve the reliability of control systems, the problem of fault detection and diagnosis has been paid more attention over the past two decades. A robust fault diagnostic method for multiple open-circuit faults and current sensor faults in three-phase permanent magne<sup>t</sup> synchronous motor drives has been proposed by Jlassi [1]. The proposed observer-based algorithm relies on an adaptive threshold for fault diagnosis. A composite fault tolerant control (CFTC) with disturbance observer scheme has been considered for a class of stochastic systems with faults and multiple disturbances [2]. The problem of fault diagnosis for a class of nonlinear systems has been investigated via the hybrid method of an observer-based approach and homogeneous polynomials technique [3]. As one of the quantitative model-based fault diagnosis methods for the distillation process, the parameter estimation method identifies the process parameters of physical meaning, such as the heat transfer coefficient, thermal resistance, etc., and hereby explains abnormal reasons based on the relationship between input and output signals. From the point of view of control theory, the parameter estimation method provides a closed loop structure with a good computational stability and convergence since parameters are input into the reasoning model again after estimation. Based on a nonlinear dynamic model, parameter estimation, however, becomes a nonlinear optimization problem, whose heavy computational load is a serious bottleneck limiting its application. Some improvements have therefore emerged in the

last decade to simplify both the model and algorithm. For example, a hybrid fault detection and diagnosis scheme was implemented in a two-step pattern, that is, neural networks were activated to deduce the root reason for a fault state after the fault-related section of a plant was located by a Petri net [4]. A fault diagnosis technique was proposed based on multiple linear models, in which several linear perturbation models suitable for various operation regimes were identified by a Bayesian approach and then combined with a generalized likelihood ratio method to perform fault identification tasks [5]. A fault detection and diagnosis scheme, which uses one tier of a nonlinear rigorous model and another tier of a linear simplified model to monitor the distillation process and identify abnormal parameters, respectively, was developed to consider the accuracy and speed of nonlinear and linear models simultaneously [6]. A three-layer nonlinear Gaussian belief network was constructed and trained to extract useful features from noisy process data, where the absolute gradient was monitored for fault detection and a multivariate contribution plot was generated for fault diagnosis [7].

The determination of fault parameters on the basis of known input and output signals of distillation can be approached as the solution of an inverse problem [8–10]. The most widely used method to solve such a problem is its least squares (LSQ) formulation as the minimization of an error function between the real measurements and their calculated values, similar to the above improved parameter estimation methods. Meanwhile, meta-heuristics for LSQ optimization are popular due to their inherent advantages, like their global optimum and the few requirements for problem formulation [11,12]. However, the running speed of the LSQ-based method is slow owing to its time-consuming iterative optimization of fault parameters. During iterations, the process of solving the forward problem and then adjusting its parameters is repeated until the di fference between the measured values and the calculated ones reaches a minimum. For this reason, a direct derivation of fault parameters from input and output signals instead of trial and error in forward problems is a very current topic, such as the decomposition of solution space followed by polynomial approximation [13], usage of artificial neural networks (ANN) for the automated reconstruction of an inhomogeneous object as pattern recognition [14], and inverse regression between the disturbance and characteristic distances based on single variable perturbation [15].

Based on the LSQ-based fault diagnosis work [16], a hybrid inverse problem approach that uses partial least squares (PLS) to fit and forecast trajectories of the fault parameters generated by LSQ is proposed in this paper to accelerate the model-based fault diagnosis process. PLS is a popular tool for key performance monitoring, quality control, and fault diagnosis in large-scale chemical industry [17,18]. It has been improved from relative contributions of process variables or blocks on faults [19,20], and the orthogonal decomposition of measurement space before deducing new specific statistics with non-overlapped domains [21].

This work aims to test the feasibility of an LSQ and PLS combined hybrid inverse problem approach for model-based fault diagnosis. In the following sections, the proposed hybrid inverse problem solving approach is explained and its positive action in terms of speeding up the diagnosis process is proved via a case study of a stripper simulator in the Tennessee-Eastman process (TEP) compared to the base approach with LSQ only. The e ffect of initial values, iteration times, calling period, etc. on the performance of the proposed method is also analyzed.

#### **2. Hybrid Fault Diagnosis Structure**

Figure 1 shows the structure of fault diagnosis formulated as a parameter-estimation inverse problem solved by the least squares optimization algorithm. The first step is fault detection, in which the system outputs estimated by dynamic simulation with a nonlinear model are compared with those measured online from a plant to check whether the present state adheres to its theoretical estimation. The di fference is defined by statistic Q. When Q is greater than its threshold Q<sup>α</sup>, the process is considered as deviating from its predefined state and the second step, fault diagnosis, is conducted. Otherwise, the fault detection step continues. Before fault diagnosis, a dynamic simulation based on the process model should be firstly performed to check its coincidence with real measurements from a

plant under normal conditions. In this procedure, the model is manually calibrated to guarantee the later detected anomaly coming from a fault occurring in the plant other than an error of the model [16]. During fault diagnosis, fault parameters are obtained as a solution of an inverse problem with LSQ and PLS. PLS regression of fault parameters generated by LSQ is utilized to predict fault parameters. PLS also runs when the statistic Q lies within its threshold Qα to give continuous fault parameter estimation. Therefore, the aforementioned hybrid inverse problem approach to parameter estimation is composed of one complex optimization part with a nonlinear model and another simple regression part with a linear model.

**Figure 1.** Hybrid inverse problem-solving process for fault diagnosis.

In Figure 1, the nonlinear model is solved once for dynamic simulation at one sampling interval to detect any fault, but is solved many times for fault diagnosis. Therefore, fault diagnosis consumes more computation time than dynamic simulation based on the same nonlinear model. The hybrid inverse problem-solving strategy replaces LSQ with PLS as much as possible since PLS calculates fault parameters directly after fitting the relationship of system outputs and fault parameters. Therefore, such a hybrid inverse problem approach can be expected to reduce the calculation load of fault diagnosis greatly.

#### *2.1. Obtaining Fault Parameters by the LSQ Algorithm*

The nonlinear fault parameters can be obtained rigorously by minimizing the deviation of measurable variables from their unsteady-state simulation values. The deviation **r** is defined in Equation (1) with a normalized version, Equation (2), where **y**meas and **y**sim represent data measured and simulated, respectively. It indicates an anomaly when its aggregated index **Q** exceeds the corresponding threshold Q<sup>α</sup>, as defined in Equations (3) and (4). In this phase, fault parameters θ are solved based on an optimization formulation (LSQ) of fault parameters about the mechanism model of distillation composed of measurable variables **y**, manipulated variables **u**, disturbance ω, and state variables **x**, as shown in Equation (5).

$$\mathbf{r} = \mathbf{y}\_{\text{meas}} - \mathbf{y}\_{\text{sim}} \tag{1}$$

$$\mathbf{r}^\* = \frac{\mathbf{r} - \mathbf{r}\_{\text{mean}}}{\sigma\_r} \tag{2}$$

$$\mathbf{Q} = \mathbf{r}^{\mathsf{T}} \mathbf{r}^\* \tag{3}$$

$$Q\_a = \chi\_a^2(m) \tag{4}$$

$$\begin{aligned} \min\_{\boldsymbol{\Theta}\_{\boldsymbol{f}}} & \mathbf{Q} \\ \text{s.t.} & \mathbf{y} = f(\mathbf{u}, \boldsymbol{\omega}, \mathbf{x}, \boldsymbol{\Theta}) \end{aligned} \tag{5}$$

As LSQ is essentially nonlinear, it is time-consuming and should not be performed frequently in practice. In a small enough range of one time point, fault parameter θ can be considered as a linear function of measurable variable **y** (see Equations (6) and (7)). In the present study, **y** and θ are scalar variables. Therefore, a revised multiple linear regression method (PLS) is used in this paper to obtain the explicit correlation between fault parameters and measurable variables.

$$\mathbf{y} = f(\boldsymbol{\Theta}) \approx f(\boldsymbol{\Theta}\_0) + f'(\boldsymbol{\Theta}\_0)(\boldsymbol{\Theta} - \boldsymbol{\Theta}\_0) \tag{6}$$

$$\boldsymbol{\Theta} = \frac{1}{f'(\boldsymbol{\Theta}\_0)} [\mathbf{y} - f(\boldsymbol{\Theta}\_0)] + \boldsymbol{\Theta}\_0 \tag{7}$$

#### *2.2. Obtaining Fault Parameters by the PLS Algorithm*

Most PLS methods are applied to regression modeling, replacing the general multivariate regression and principal component regression to a large extent. Comparatively, PLS can not only exclude the correlation of original variables, but also filter the noise of both independent variables and dependent variables. Its prediction ability is stronger and more stable because it uses fewer characteristic variables to describe the regression model.

Firstly, the data **X** = **[y u]<sup>T</sup>**∈**R***l*×*<sup>n</sup>* and **Y** = θ∈**R***l*×*<sup>c</sup>* are normalized and decomposed, respectively, where **T**, **P**, and **E** denote the score, load, and residual matrix of X, respectively; U, Q, and F denote the score, load, and residual matrix of Y, respectively; and *a* and *n* denote the number of PLS components and variables, respectively. The external relations are obtained as Equations (8) and (9).

$$\mathbf{X} = \mathbf{T}\_a \mathbf{P}\_a^T + \mathbf{E}, a < n \tag{8}$$

$$\mathbf{Y} = \mathbf{U}\_{\mathbf{d}} \mathbf{Q}\_{\mathbf{a}}^{\mathrm{T}} + \mathbf{F}\_{\mathbf{r}} \mathbf{a} < n \tag{9}$$

Then, their internal relationship is determined as Equation (10).

$$\mathbf{U}\_{a} = \mathbf{T}\_{a}\mathbf{B},\tag{10}$$

where **B** is the internal regression matrix.

> The PLS model is finally obtained as Equation (11).

$$\mathbf{Y} = \mathbf{T}\_a \mathbf{B} \mathbf{Q}\_a^T + \mathbf{F} \tag{11}$$

When the independent variable **X** is known, PLS can be used to predict the dependent variable **Y**. The calculation procedure is given in Equations (12) and (13).

$$\mathbf{f}\_h = \mathbf{E}\_{h-1} \mathbf{w}\_h \tag{12}$$

$$\mathbf{E}\_{\rm li} = \mathbf{E}\_{\rm h-1} - \mathbf{f}\_{\rm h} \mathbf{p}\_{\rm h'}^{\rm T} \tag{13}$$

where **t** and **p** represent the element vector in the score and load matrix, respectively; **w** represents the weight vector; *h* denotes the component index; and **E**0 = **X**.

Therefore, dependent variables can be predicted using Equation (14).

$$\mathbf{Y} = \sum\_{h=1}^{a} \mathbf{v}\_{h} \mathbf{f}\_{h} \mathbf{q}\_{h}^{\mathrm{T}} \tag{14}$$

## *2.3. Correcting PLS by LSQ*

Because PLS extracts linear features of fault parameters, it should be corrected continuously by LSQ. The correction framework is shown in Figure 2, where the red color loop represents the correction process. The rigorous iterative LSQ is performed once to supply one accurate value of fault parameters for the PLS training set when an anomaly is detected, not enough sampling points are collected, and correction is needed. The main contribution of this work lies in the frequent usage of fast PLS prediction of fault parameters instead of slow LSQ in the fault diagnosis algorithm. However, compared to LSQ, PLS' accuracy is limited because of its linear regression essence, so it should be corrected periodically by rigorous LSQ results. Sufficient LSQ results are needed to form the training set of the PLS model before PLS correction. Therefore, if not enough sampling points of fault parameters are given by LSQ for PLS training, LSQ should be activated correspondingly to supply one sampling point of fault parameters into the PLS training set to meet the periodical correction requirements of PLS. In this case, the boundary value θ0 in Equation (7) is kept stable and accurate, and the prediction accuracy of PLS is guaranteed as a consequence.

**Figure 2.** Correction framework of partial least squares (PLS) by least squares (LSQ).
