**3. Case Study**

The stripper of the TEP simulator [22] is used as an example to test the feasibility of the proposed hybrid inverse problem approach to fault diagnosis. The fault set related to this stripper includes two types, that is, a step and a random type. This work chooses fault 7 and 8 as typical examples for these two types. The advantages and disadvantages of this approach are illustrated in comparison with the base approach with LSQ only.

#### *3.1. Solving the LSQ Inverse Problem with Di*ff*erent Initial Values*

LSQ is greatly affected by the initial values of iterated variables, so the first step is to discuss different setting methods for initial values of the LSQ algorithm. The following test is firstly based on the fault diagnosis process under fault 7. Fault 7 occurs at 8 h when the header pressure loss of the stripping stream entering the stripper bottom decreases abruptly. The fault diagnosis result using LSQ only is given in Figures 3 and 4, which has been published in our early work [16], henceforth referred to as the base case for comparison. In this base case, the overall running time and quantity of function evaluations (QFE) are 539 seconds and 1826, respectively. In the fault diagnosis algorithm, the boundary state at *t* = 0 is considered a normal state, this is, without anomaly. Therefore, the pressure loss coefficient is set as 1 uniformly in Figure 3. When *t* > 0, the fault diagnosis algorithm is activated to obtain the pressure loss coefficient in a timely manner based on the measurements of a plant.

**Figure 3.** Fault parameter in the base case of fault 7.

**Figure 4.** Function evaluation in the base case of fault 7.

(1) Set the initial value with the previous time point

The simplest value-setting method for the initial value is to directly use one from a previous time point. If there are only small changes in the fault parameter between two consecutive time points, the initial value given can thus be accepted. This is also the initial value-setting method adopted by the base case.

(2) Set the initial value with the linear fitting method

Figure 5 shows the diagnosis result with initial values given by the linear fitting method. It indicates that QFE decreases greatly and becomes stochastically stable before the fitted point number equals 30. After that, a rising function evaluation number curve is seen because of the growing time lag of the fitted line. For this reason, 5 or 8 is chosen as the candidate for the optimal fitting point number. Because more fitting points will definitely lead to a heavier computational load for the fitting operation, 5 is finally chosen as the optimal fitting point number. Despite this, QFE is only cut down by 2%, contributing little to the computational efficiency of the diagnosis algorithm.

**Figure 5.** Fault diagnosis of fault 7 with linear fitted initial values.

(3) Set the initial value with the parabolic fitting method

Following the linear fitting method presented above, the parabolic fitting method, the simplest nonlinear fitting method, is utilized here to predict the initial values. Figure 6 shows the diagnosis result with this method, revealing a larger QFE than the base case. This reflects the essentially linear change of the fault parameter between neighboring time points. Therefore, the nonlinear fitting method does not achieve the satisfactory goal of reducing QFE.

**Figure 6.** Fault diagnosis of fault 7 with parabola fitted initial values.

#### (4) Set the initial value with the grey model

Different from the above regression methods, grey system theory uses partial description information of a system to generate a grey sequence revealing the potential rule of data with the goal of system whitening [23]. It has advantages such as a small amount of data, fast operation, easy iteration, and high accuracy [24]. Figure 7 gives the diagnosis result with the grey model-predicted initial values. It shows that QFE decreases by 4.7% at most when adopting 15 as the number of sampling points. So far, the highest reduction ratio of function evaluations to the base case has been obtained by the grey model, which is, therefore, the best method for estimating initial values. At the same time, the limited improvement given by the grey model shows the need for further improvement of the diagnosis algorithm based on other factors.

**Figure 7.** Fault diagnosis of fault 7 with grey model-predicted initial values.

#### *3.2. Solving the LSQ Inverse Problem with Di*ff*erent Numbers of Iterations*

As one popular optimization algorithm, LSQ requires a large number of iterations to obtain accurate fault parameters at each sampling time, so its high computational cost is its main disadvantage. In fact, the aim of fault diagnosis is to find the abnormal trend of fault parameters in a timely manner during a given time interval. In this process, completely converged calculation at each time point is not necessary. Based on the idea of tracking approximation, the proposal of the present paper distributes the inner iterative computation into an outer integration progress to decrease the maximum number of iterations at each sampling point. Figure 8 shows the diagnosis result with different maximum numbers of iterations. It presents a grea<sup>t</sup> decrease of QFE when reducing the maximum number of iterations. In particular, when the iteration number equals 1, QFE decreases by 55% compared to the base case, being far greater than the value obtained by the grey model.

**Figure 8.** Fault diagnosis of fault 7 with different numbers of iterations.

The fault diagnosis result obtained by this strategy is shown in Figure 9, in contrast with that obtained by the base case. The exact coherence of fault parameters between these two cases evidences no loss of accuracy with this fast algorithm. Meanwhile, minor parameter fluctuation is observed due to the insufficient iterative computation of this algorithm.

**Figure 9.** Fault diagnosis of fault 7 with one iteration only.

#### *3.3. Hybrid Inverse Problem-Solving Strategy*

In the above sections, two kinds of improvements—increasing the prediction accuracy of initial values and decreasing the number of iterations—were conducted for the least squares algorithm. The computational results show that the latter has a significant effect on the fault diagnosis speed. Generally, these algorithms use the passive trial and error method to solve the inverse problem of fault diagnosis. Fault parameters are defined as input variables for the system model used by LSQ, different from their output variable role defined in the inverse problem. In the following, an alternative inverse problem model using a direct mapping of fault parameters from measurements will be considered to avoid the time-consuming model solving process.

In an information view of fault diagnosis, the inverse problem defined herein is a typical multiple input-multiple output (MIMO) system in which measurable/controllable variables and fault parameters constitute input and output parts, respectively. The linear MIMO model is given by the PLS method in this work owing to the small data change for both input and output variables in a short sampling interval. Furthermore, periodic correction for this linear model by LSQ is necessary to preserve its accuracy.

Figure 10 shows the comparison of fault diagnosis for the base case and the case using a hybrid strategy. It proves the feasibility and accuracy of this strategy, but indicates larger fluctuations of the fault parameter with the hybrid algorithm. Therefore, PLS is suitable for replacing LSQ, but its application should be controlled properly. Factors affecting the efficiency of this strategy will be discussed hereafter.

**Figure 10.** Fault diagnosis of fault 7 with the hybrid algorithm.

<sup>(1)</sup> Number of PLS components

PLS is something of a cross between multiple linear regression and principal component analysis. It constructs components as linear combinations of the original variables, while allowing for correlation between independent and dependent variables. The number of components is therefore of primary importance to the accuracy of the PLS model. Figure 11 depicts the percent of variance explained in the dependent variable as a function of the component number. A maximum of 16 components is assumed in Figure 11 because the independent variables consist of a total of 16 variables for the stripper in TEP. It can be seen that more than 95 percent variance was explained by the first three components, which were, accordingly, chosen as the principle components in the following PLS modeling process.

**Figure 11.** Number analysis of PLS components under fault 7.

(2) Sampling data set for PLS modeling

The training data sets for PLS modeling were composed of 12 measured variables, 4 manipulated variables, and 1 fault parameter. As shown in Figure 1, the fault parameter may be obtained from LSQ or PLS, so the PLS model can be built on LSQ-generated fault parameter sets (Vector I) or mixed sets (Vector II).

Figure 12a,b show the fault diagnosis results with Vector II as training data sets for PLS, whereas Figure 12c,d give results with Vector I as training data sets. The root mean square error (RMSE) of fault parameters between the base case and the proposed approach was also calculated and adhered to Figure 12a–d. As illustrated in Figure 1, a second PLS (PLS II) was performed with the aim of keeping the continuity of fault parameters when no abnormal signals were detected. Figure 12 shows the fault parameters obtained with (b) and without (a) in this second PLS based on Vector II. The worse result obtained with PLS than without PLS to predict the fault parameter in normal states evidences an adverse propagation effect of PLS prediction error on the PLS model itself. The fact that diagnosis results with Vector I as training data sets for PLS coincide exactly with the base case no matter whether they are with (d) or without (c) the second PLS further proves this conclusion. Consequently, PLS modeling should be conducted based on the fault parameters generated by the LSQ algorithm to preserve its accuracy.

Generally, the time-saving prediction of fault parameters with the PLS method may be equal to several sampling periods before each time-consuming LSQ in Figure 1. Although this scheme can cut down the running time of fault diagnosis greatly, the using frequency of PLS should be limited to an allowable range since the PLS accuracy strongly depends on new fault parameters generated by LSQ. In other words, it is crucial to correct the PLS model with LSQ-generated data after consecutive calls of PLS.

**Figure 12.** *Cont*.

**Figure 12.** Fault diagnosis under fault 7 (**a**) based on Vector II data, but without second PLS; (**b**) based on Vector II data and with second PLS; (**c**) based on Vector I data, but without second PLS; (**d**) based on Vector I data and with second PLS.

Figure 13 shows the effect of the correction interval on QFE, the running time, and RMSE, respectively. We can see from Figure 13a,b that QFE decreases, as does the running time of the fault diagnosis process, when increasing the correction interval. In particular, their decreasing magnitude becomes small when the correction interval exceeds 5. However, there appears to be a rapid growth of calculation error, as can be seen from Figure 13c. Although the effect of correction interval 4 or 5 on RMSE is not significant, both QFE and the running time will increase for the correction interval of 4 compared with 5. Therefore, 5 is the appropriate calling number of PLS in a correction interval. With this calling number, QFE decreases by 81.60% and the running speed increases about 1.7 times compared to the base case with this calling number.

Table 1 summarizes the approaches that can effectively reduce QFE in fault diagnosis. The best results obtained are indicated in boldface. It leads to the conclusion that the approach proposed in this paper evaluates the fault parameter markedly faster than the pure LSQ-based algorithm used in [16].


**Table 1.** Algorithm improvement concerning function evaluations.

The above feasibility test was implemented based on a step-type fault, with fault 7 as an example. Next, another type of fault, a random type with fault 8 as an example, will be tested with the aforementioned hybrid structure. In the case of fault 8, the composition of the feed stream (containing component A, B, and C only) changes randomly from the 8 h point. The essentially random sampling values form this fault are fed to the hybrid fault diagnosis algorithm as a preset input from an outside battery. It is satisfactory to discriminate the fault type from fault diagnosis results, with no need for a statistics analysis of random sampling values. Its fault diagnosis result with LSQ only is shown in Figures 14 and 15, also published in our early work [16]. The overall running time and QFE for this diagnosis process are 5103 seconds and 19,256, respectively. This case is referred to as the base case for fault 8 in the following discussions.

**Figure 13.** Fault diagnosis with different correction intervals of PLS under fault 7 with respect to (**a**) quantity of function evaluations (QFE); (**b**) the running time; (**c**) the root mean square error (RMSE).

**Figure 14.** Fault parameter in the base case of fault 8.

**Figure 15.** Function evaluation in the base case of fault 8.

Similar to Figure 11, Figure 16 shows the percent of variance explained by independent components. The first two components make more than 80% contributions and were thus selected as the components in PLS modeling under fault 8.

**Figure 16.** Number analysis of PLS components under fault 8.

Figure 17 shows the effect of the correction interval on QFE (a), the running time (b), and RMSE (c) under fault 8. We can see that QFE and the running time decrease, but RMSE increases, when increasing the correction interval. 5 is chosen as the optimal correction interval since the former two indices do not decrease significantly, while RMSE remains small under this choice. Besides, larger values of the former two indices than fault 7 are observed in Figure 17, indicating that a random-type fault consumes more time than a step-type fault due to its stochastic computing load.

**Figure 17.** Fault diagnosis with different correction intervals of PLS under fault 8 with respect to (**a**) QFE; (**b**) the running time; (**c**) RMSE.

With 5 as the correction interval, the diagnosis result of three compositions in the bottom feed of the stripper is shown in Figure 18. It indicates nearly the same composition trajectories for the hybrid approach and base case, and proves the feasibility and accuracy of our proposed approach. Finally,

**Figure 18.** Fault diagnosis with a hybrid scheme under fault 8 for the (**a**) composition of A; (**b**) composition of B; (**c**) composition of C.
