**1. Introduction**

A complex system, for instance, control system [1], servo system [2], energy storage system [3], is widely used in aviation, aerospace, electronics and other fields. Due to the complex structure and poor working environment, the system performance can be degraded, which affects the operation reliability of the system. Therefore, it is crucial to assess the health status of the complex system to provide decisions for managemen<sup>t</sup> and maintenance [4].

In the current research of health assessment, there are mainly three methods called the data-based method, the qualitative knowledge-based method, and the semi-quantitative information-based method. The data-based methods assess the system performance by fitting the nonlinear relationship between the input and output of the system based on observation data, such as deep learning, neural network [5–7]. Since it is a pure black-box modelling, the assessment results cannot be explained, and there is a problem of overfitting. The qualitative knowledge-based methods provide interpretable assessment progress based on the operation mechanism of the system and expert knowledge, for example, fuzzy reasoning, belief rule base [8,9]. Due to the subjectivity of expert knowledge, the model assessment accuracy is poor. The semi-quantitative information-based methods provide both qualitative knowledge and quantitative data concurrently, providing interpretable and accurate assessment results [10]. Therefore, the health assessment based on semiquantitative information is basically concentrated in this paper.

The evidential reasoning (ER) rule [11], as a representative semi-quantitative informationbased method, originated from the Dempster-Shafer (DS) evidence theory [12], and is regard as a generalized Bayesian inference process [11]. DS evidence theory is regarded as a special case of ER, when the indicator reliability is equal to 1 [13]. In the ER rule, the quantitative data and qualitative knowledge can be effectively integrated by adopting the orthogonal operations.

**Citation:** Li, Z.; Zhou, Z.; Wang, J.; He, W.; Zhou, X. Health Assessment of Complex System Based on Evidential Reasoning Rule with Transformation Matrix. *Machines* **2022**, *10*, 250. https://doi.org/ 10.3390/machines10040250

Academic Editor: Davide Astolfi

Received: 27 January 2022 Accepted: 24 March 2022 Published: 31 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Reference value is introduced to divide the input information status, then the initial evidence can be generated. To deal with the data uncertainty, the evidence weight and reliability are introduced. Particularly, the weight reflects the relative importance of multiple pieces of evidence in the aggregation of evidence. The reliability reflects the ability of the information sources to provide correct information. The reliability is influenced by the performance of the information source and external noise [14]. By clearly differentiating the two concepts, the ER rule is widely used in many fields, such as multi-attribute decision-making [15], fault diagnosis [16], health assessment [17], etc.

When using the ER rule to assess the health status of a complex system, a set of mutually exclusive and collectively exhaustive assessment result grades need to be settled in advance. First, health assessment indicators are selected, and the indicators are equivalent to evidence. Then, input indicator reference grades are introduced to conveniently collect the initial evidence pointed to the assessment grades. Finally, the initial evidence, evidence weight and reliability are integrated based on the ER rule, and the health assessment results of the complex system can be obtained. Therefore, as an important part of the assessment, indicator reference grades determine the belief distribution of initial evidence, which directly affected the assessment results.

The above assessment process determines that the input indicator reference grades and output assessment result grades strictly correspond to each other. However, in practice, the assessment result grades are determined in advance, which leads to the disaccord with input indicator reference grades. For example, the input indicator reference grades can be easily divided into "normal" and "fault" based on industry-standard, but health assessment result grades are predetermined as "health", "subhealth", "fault".

In the process of health assessment, the relationship between input indicator references grades and assessment result grades does not exactly correspond to each other. Therefore, for the sake of dealing with this problem, there are two methods to solve it. First, regarding the relationship as a one-to-one correspondence [18,19], then the input indicator references grades can be matched with assessment result grades. Second, based on expert knowledge, adding the reference grades to realize an input and output in accordance [4,8]. However, the first method neglects the consistent relationship in engineering practice, and the accuracy of the assessment results is influenced. The second method violates the prior mapping relationship, resulting in randomness and no standard of the assessment result.

To deal with the mentioned issues, a new health assessment model for a complex system based on the ER rule is proposed. First, the transformation matrix is determined according to the expert's knowledge. The input information can be converted into the initial belief distribution with regard to assessment result grades by using the rule-based information transformation technique. Thus, a general information transformation framework is constructed. Second, the evidence weight and reliability are determined by expert knowledge and the synthesis of static and dynamic characteristics separately. Then, the health assessment model of a complex system based on ER rule is constituted. Finally, to further enhance the model precision, an objective function is established to optimize the model parameters. In this paper, there are two innovations as follows:

(1) Based on transformation matrix, the mapping relationship between the antecedents and the consequent of the assessment model is established, which solves the inconsistent problem between the indicators reference grades and pre-defined assessment result grades in the engineering practice. Due to the subjectivity and limitations of the expert's knowledge, the initial values of transformation matrix may deviate from real status, hence the need to build a optimization model to further optimize the values of transformation matrix.

(2) On the basis of parameters calculation, the optimization algorithm is employed to enhance the assessment result accuracy. Then, a complete health assessment model for complex system is constituted.

This paper is organized as follows. The framework and related problems of the health assessment model are described in Section 2. In Section 3, the health assessment model based on ER for a complex system is proposed. The optimization of model parameters is presented in Section 4. In Section 5, the bus of control system and the engines are taken as examples to validate the effectiveness of the proposed model. Conclusions and future work are defined in Section 6.
