**4. Conclusions**

In this paper, an LSQ and PLS combined hybrid inverse problem approach has been proposed to realize model-based diagnosis for the distillation process. LSQ is used to identify parameters that best-represent an abnormal state of distillation on the basis of a nonlinear dynamic model. PLS regression is then used to fit these parameters with input/output signals and forecast their developing trajectories. The correction interval of PLS significantly a ffects the speed and accuracy of the fault diagnosis process. The approach has been carried out to successfully identify stripper-related faults in the TEP benchmark process. For fault 7, QFE decreases by 81.60% and the running speed increases about 1.7 times compared to the base case. For fault 8, QFE decreases by 92.31% and the running speed increases about 13 times compared to the base case. Therefore, it has been proven to be a computationally e fficient scheme for model-based diagnosis. In conclusion, compared with a single nonlinear LSQ-based approach, the presented hybrid inverse problem approach enables a trade-o ff between accurate LSQ and fast PLS and is more suitable for real-time fault diagnosis.

In the future, it would be helpful to combine this approach with some process history-based approaches, like a bond graph [25], to enhance its vital ability to locate fault-specific sections prior to fault diagnosis.

**Author Contributions:** All authors participated to the elaboration of the manuscript. Investigation, methodology and writing-original draft, S.S.; investigation and software, Z.C.; data curation, X.Z.; methodology and supervision, W.T. All the authors discussed the results. All authors have read and agreed to the published version of the manuscript.

**Funding:** Financial support for carrying out this work was provided by the National Natural Science Foundation of China (Grant No. 21576143).

**Conflicts of Interest:** The authors declare no conflicts of interest.
