A Liquid Launch Vehicle Safety Assessment Model Based on Semi-Quantitative Interval Belief Rule Base

Abstract

As the propulsion part of a space launch vehicle and nuclear weapon missile, the health status of the liquid rocket determines whether the space launch vehicle and nuclear weapon missile can function normally. Therefore, it is of great significance to evaluate the health status of the liquid rocket. As the structure of the liquid rocket is becoming increasingly sophisticated, subjective judgment alone can no longer meet the needs of the actual system. As an expert system and a gray-box model, the belief rule base (BRB) can process both qualitative and quantitative information. The expert knowledge base is used in the safety assessment of a liquid rocket. However, in practical applications, the traditional BRB model still has two problems, which are that (1) when there are too many premise attributes, it easily leads to the explosion of combination rules, and (2) the reliability of rules is not considered in the process of model reasoning. Therefore, this paper proposes the BRB model with intervals (intervals-BRB) on the basis of traditional BRB. The interval-BRB retains the advantage of the traditional BRB, which can handle semi-quantitative information. In addition, the proposed model changes the reference point of the prerequisite attribute to the reference interval and changes the rule combination. This solves the problem of the traditional BRB explosive combination rule. The ER-rule (evidential reasoning rule) is introduced into the reasoning procedure, and the weight of the rule and the reliability of the rule are considered at the same time, which solves the shortcoming of the traditional BRB, which does not consider the reliability of the rule in reasoning. Finally, the CMAES optimization algorithm is used to optimize the initial model to obtain better performance. Finally, the model is verified by the actual data set of a liquid rocket, and the experimental results show that the model can achieve good experimental results.

1. Introduction

Liquid launch vehicles are rockets powered by liquid rocket engines. It is generally composed of a power plant, arrow body structure, control system and other components [1]. There are single stage rockets and multiple stage rockets [2]. Liquid rockets are mainly used as space vehicles and the propulsion part of missile nuclear weapons [3]. As the propulsion part of the carrier vehicle, the liquid rocket plays an irreplaceable role, and its health state directly affects whether the carrier vehicle can be successfully launched or whether the missile weapon can reach the established goal [4,5,6]. Therefore, the assessment of the health state of the liquid rocket is of great significance; fortunately, many researchers are attracted to conduct study in this area [7].

At present, many scholars have studied the methods of health status assessment. Zhan et al. [8] proposed an evaluation method based on the Mahalanobis distance and power prediction for the health state evaluation of wind turbines. Based on the edge computing model, Zhang [9] used large-scale multimedia health data to evaluate the remote mobilization health status. Aiming at the health status assessment of wind turbines, Wang et al. [10] calculated the health status deterioration index based on state recognition and adopted a dual threshold alarm scheme. Aiming at the health status assessment of equipment, Zhang et al. [11] proposed a real-time assessment scheme using the health degree to quantitatively express the health status of equipment. The Euclidean distance is introduced, and principal component analysis is used to eliminate the redundancy rule. Li et al. [12] proposed to use the analytic hierarchy process (AHP) to evaluate the health status of the radar system of the space launch site and introduced the analytic hierarchy process (AHP) and fuzzy comprehensive evaluation. Zhou et al. [13] proposed a one-dimensional convolutional neural network to analyze the data in order to evaluate the health state of the aero-engine. Liu et al. [14] proposed an evaluation method of multi-channel information fusion and adaptive noise reduction for the health condition of diesel valve clearance. Based on the convolutional neural network, Li et al. [15] used the data of measurement indicators to evaluate mental health status. Based on the confidence rule base, Cheng et al. [16] proposed multiple discount BRBs to evaluate the health status of complex large-scale electromechanical systems. Jiang et al. [17] applied the semi-supervised learning algorithm and distributed fiber optic sensor technology to the health state assessment of pipeline systems.

Through the comprehensive analysis of the above literature, it can be seen that the health status assessment model can be roughly divided into three categories. (1) The data-driven models, which can be divided into supervised models and unsupervised models. Supervised models include traditional machine learning models decision trees, random forests, support vector machines, etc., and unsupervised models are mainly based on clustering [18] and typical models, such as k-means. Both supervised and unsupervised models are supported by a large amount of data, and the relationship between input and output is found through continuous training. The accuracy depends on the data samples, and the modeling process is not interpretable. (2) The knowledge-driven model. The typical knowledge-driven model is an expert system [19,20], which takes the prior knowledge of experts as the core to construct the model and does not need data training to perform classification prediction or regression prediction. The accuracy of the model depends on the reliability of the expert knowledge. (3) The hybrid driven model [21,22]. Based on the hybrid-driven model, the typical belief rule base (BRB) is modeled by expert knowledge, and then a certain amount of data is used to optimize the parameters, which can ensure accuracy and a certain degree of interpretability.

Data-driven models, such as back propagation neural networks (BPNNs) [23,24], extreme learning machines (ELMs) [25] and radial basis functions (RBFs) [26], are usually black box models, which can obtain high accuracy through a large amount of data [27,28]. However, in the aerospace field, it is difficult to obtain a large amount of sample data, which makes the model difficult to fully train. In addition, due to the complex structure and harsh operating environment of the liquid rocket [29], it is difficult to build a model through expert knowledge. Therefore, a hybrid model driven by data and knowledge, such as a belief rule base, provides an alternative solution for modeling liquid rocket healthy state assessment.

BRB [30,31,32,33,34] is a gray-box model proposed by Yang et al. that can process both quantitative and qualitative information and has been widely used in fault detection, safety assessment, aerospace, medical treatment and other fields. BRB can effectively use engineering experience and expert knowledge, integrate small sample test data, and has strong nonlinear modeling ability, as well as high modeling accuracy and model interpretability. However, due to its modeling mechanism, BRB also has some shortcomings, including (1) too many premise attributes and too many reference points (5 premise attributes, 5 reference points for each attribute, $5^5=3125$ rules) in the application context, which easily leads to combination rule explosion. (2) Due to the ignorance of expert knowledge, it sometimes fails to give excellent values when giving the reference points of premise attributes. (3) The ER approach (ER parsing algorithm) [34] is used in evidence fusion without considering the reliability of rules. (4) Neither qualitative analysis nor quantitative analysis can be taken into account. Because qualitative analysis is limited by subjective judgment, quantitative analysis relies on professional mathematical knowledge and has partial one-sidedness. It is difficult to combine the two in the liquid rocket body structure safety state assessment model, so a safety state assessment method based on semi-quantitative information modeling is needed.

When building a traditional BRB, all prerequisite attributes and the corresponding reference values for each prerequisite attribute are set first. This results in the exponential growth of the number of rules in the belief rule base, which easily leads to the problem of combination rule explosion. In recent years, some scholars have studied the problem of combination rule explosion in traditional BRB. The current solution to this problem is to simplify the rule base by adjusting the number of prerequisite attributes and reference values. It is divided into two cases: (1) The number of prerequisite attributes does not change. Zhou et al. [35,36] evaluated the rules by a statistical method of whether the utility reached a certain threshold to achieve rule reduction. (2) A variable number of prerequisite attributes. Chang et al. [37] introduced multi-dimensional scaling (MDS) [38], principal component analysis (PCA) [39], Isomap [40], gray target (GT) [41] and other traditional dimension reduction methods to the belief rule base. The rule base is reconstructed by processing the expert information to select the main prerequisite attributes.

The above rule reduction methods have been proven to be effective in simplifying belief rule base. However, after comprehensive analysis, it is found that these methods are not enough. For example, methods such as MDS and GT cannot guarantee the high accuracy of the model, even if they can effectively reduce the rules. PCA, Isomap and other methods can effectively reduce the rules while ensuring the high accuracy of the model. However, PCA has some limitations in the screening of important attributes, and Isomap is difficult to implement due to its high algorithmic complexity. In addition, these methods need to build traditional BRBs first and then perform rule reduction on the basis of the traditional rule base. This significantly increases the algorithmic complexity. Therefore, in essence, these methods are still unable to reduce the rules with high precision and efficiency.

Aiming at the shortcomings of rule reduction research, a new interval-based belief rule base (intervals-BRB) is proposed from the perspective of modeling. This semi-quantitative modeling approach of intervals-BRB not only solves the problem of traditional BRB combination rule explosion, but also avoids the inaccuracy of expert knowledge setting reference values. At the same time, it also considers the rule reliability.

Aiming at the shortcomings of the BRB model mentioned above, this paper proposes an interval-based BRB modeling method (intervals-BRB). Compared with the traditional BRB, the intervals-BRB has the following innovations: (1) In the modeling process, the interval is introduced into BRB, the reference point of the traditional BRB prerequisite attribute is changed to the reference interval, and the combination of rules is changed to build a new belief table. It can effectively solve the problem of combination explosion caused by excessive attributes of traditional BRB. The method of constructing a BRB by such intervals is also different from the traditional interval BRB. The traditional interval BRB only changes the reference point into the reference interval but does not change the rule combination method and the model reasoning process [42]. (2) In the reasoning process, the ER-rule is introduced, and the reliability of evidence is taken into account, which makes the results more accurate. (3) Adopting the semi-quantitative modeling method. Both qualitative use of expert knowledge and quantitative data can be used. This enhances the ability to handle different types of data.

The main structure of the paper is as follows. In the first part, the original health state assessment model of a liquid rocket is analyzed and discussed. On the basis of revealing the shortcomings of the original model, the health state assessment model of a liquid rocket based on intervals-BRB is proposed. The second part describes the three problems that need to be solved when building the intervals-BRB model. The third part presents the construction, reasoning and optimization process of the health state assessment model of a liquid rocket engine based on intervals-BRB. In the fourth part, an example is used to verify the intervals-BRB method and analyze the experimental conclusions. Section 5 concludes the paper and gives an outlook on future work.

2. Traditional BRB Model and Problem Description

The model proposed in this paper is an improvement on the basis of the traditional BRB. Therefore, this chapter first introduces the modeling process of the traditional BRB, then, aiming at highlighting the shortcomings of the traditional BRB model, the interval BRB model is proposed, and three problems that need solving in the modeling process of the interval BRB model are introduced.

2.1. Traditional BRB Model

The traditional BRB model does not consider the optimization process, and the modeling process is mainly divided into four steps. The specific process follows Step 1–4.

Step 1: Firstly, the problem is analyzed to determine the main influencing factors of the problem and the possible results of the problem. In order to facilitate understanding, we assume that a problem has two premise attributes and four possible results.

Step 2: Determine the reference point and reference value of the premise attribute. The reference points represent the points with typical significance of a certain attribute, which are usually the locations where most of the points are distributed and contain the upper and lower bounds of the value. The number of the reference points is determined by experts according to the problem. If premise attribute 1 has upper and lower bounds of $[0,100]$ and is densely distributed at 35 and 50, and premise attribute 2 has upper and lower bounds of $[0,60]$ and is densely distributed at 25 and 45, we can set their reference points and reference values as in

Table 1. Reference point and reference values of premise attribute 1.

Reference Point	Reference Values
A	0
B	35
C	50
D	100

and

Table 2. Reference point and reference values of premise attribute 2.

Reference Point	Reference Values
I	0
J	25
K	45
L	60

.

Step 3: Set the reference value corresponding to the result. The number of reference values of the result is determined according to the problem and practical needs. If our problem has four result references, we can set the result reference table as in

Table 3. Reference value of the result.

Result	Reference Value
D1	2
D2	4
D3	6
D4	8

.

Step 4: The confidence rule of the traditional BRB is described in Formula (1). In Formula (1), the initial parameters of each rule include the rule weight, the attribute weight and the belief degree of the result, and the initial values have been given by experts. The reference point and reference value of the premise attribute are defined, and the reference point and reference value of the result are defined. We can generate the belief rule base as in

Table 4. Corresponding belief tables.

Index of rule	Attribute 1	Attribute 2	Result
1	0	0	(D1,0.9), (D2,0.1), (D3,0), (D4,0)
2	0	25	(D1,0.8), (D2,0.2), (D3,0), (D4,0)
3	0	45	(D1,0.5), (D2,0.3), (D3,0.2), (D4,0)
4	0	60	……
5	35	0	……
6	35	25	……
7	35	45	……
8	35	60	……
9	50	0	……
10	50	25	……
11	50	45	……
12	50	60	……
13	100	0	……
14	100	25	……
15	100	45	……
16	100	60	……

.

$$\begin{array}{l}{R}_k:If x_1 is A_1\wedge x_2 is A_2,\dots ,\wedge x_M is A_M\\ Then \{(O_1,{\beta }_{1,k}),(O_2,{\beta }_{2,k}),\dots ,(O_N,{\beta }_{N,k})\}\\ with rule weight {\theta }_k\\ and attribute weight {\delta }_1,\dots ,{\delta }_M\end{array}$$

(1)

where $x_i(i=1,\dots ,M)$ refers to the premise attributes for modeling objects; $A_i(i=1,\dots ,M)$ represents the set of reference values; $O_i(i=1,\dots ,N)$ is the n evaluation results for the BRB; ${\beta }_{i,k}(i=1,\dots ,N)$ denotes the belief degree of each result under the $k_{th}$ belief rule; let ${\theta }_k$ denote the rule weight of the $k_{th}$ belief rule, and ${\delta }_k$ is the attribute weight of the $k_{th}$ belief rules.

Table 1, Table 2, Table 3 and Table 4 show the process of constructing the belief rule base with traditional BRB, where Table 1 shows the four reference values corresponding to premise attribute 1, Table 2 shows the four reference values corresponding to premise attribute 2, and Table 3 shows the reference value corresponding to the result. According to Table 1 and Table 2, it can be determined to generate $4\times 4=16$ rules, and each rule is the corresponding belief distribution of the result in Table 4.

The inference process of traditional BRB uses the ER parsing algorithm and the rule reliability is not considered. In order to make the results more reasonable, the interval BRB model proposed in this paper considers the rule reliability; thus, the ER parsing algorithm is no longer used but the new inference algorithm is.

2.2. Problem Description

When the traditional BRB has too many premise attributes or too many reference points of premise attributes, it is easy to produce combination rule explosion, and the rule reliability is not considered in the reasoning process. Therefore, this paper proposes the intervals-BRB model, but there are some problems when constructing the intervals-BRB model, which are described as follows.

Problem 1. How to model. To solve the traditional BRB on the premise that attributing too many reference points can cause the rule of the combination explosion problem, and the reasoning rules do not consider the reliability problem, this paper puts forward the intervals-BRB model with traditional BRB modeling-based property reference points into the reference range. The reasoning machine adopts the ER rule and takes into account the rules of reliability. The solution to this problem corresponds toSection 3.1.

Problem 2. How to reason. The belief rule base of a traditional BRB is obtained by the Cartesian product of reference points. The input information needs to be transformed first, and then the matching degree and activation weight are calculated. Finally, a belief distribution is obtained by using the ER parsing algorithm [20], and the final value is obtained by utility calculation. However, the modeling process of the intervals-BRB is different from that of the traditional BRB, so the whole reasoning process also changes. This process can be described as Formula (2).

$$u=g(x,a)$$

(2)

where $x$ represents the evaluation index set of the liquid rocket health state evaluation model, $a$ is the parameter set of the model inference process, $u$ is the liquid rocket health state output, namely, the final belief distribution, and $g(\cdot )$ is the inference function. The solution to this problem corresponds to Section 3.2.

Problem 3. How to optimize. BRB, as a hybrid model of knowledge and data driven by the expert knowledge structure again after the initial model, needs to use training data to optimize the parameters. In the traditional BRB model, the weight of the rules, the weight of the attribute and the belief distribution of the results need to be optimized. In the intervals-BRB model, it is also necessary to optimize the relevant parameters in the model. This process can be described as in Formula (3).

$$a_{best}=optimize(x,u,\gamma )$$

(3)

where $x$ and $u$ are the inputs and outputs, $\gamma$ is the set of parameters used in the optimization process, $a_{best}$ is the set of optimal model parameters and $optimize(\cdot )$ is the optimization function. For the intervals-BRB model, the set of optimized parameters includes the rule weight, the rule reliability, the belief distribution of the result and so on. Since the parameters in these models are subject to certain constraints, the constrained global optimization algorithm can be considered to optimize the model parameters. The solution to this problem corresponds to Section 3.3.

3. The Intervals-BRB Model

The normal operation of liquid rocket is of great significance for space missions. A health state assessment model of a liquid rocket system with high accuracy and interpretability can effectively find the influencing factors of the health state to avoid the occurrence of potential dangers. Therefore, to solve the problem that traditional BRBs easily lead to combination rule explosion and that there is uncertainty in expert knowledge, this paper proposes the intervals-BRB model based on intervals. Section 3.1 defines the intervals-BRB model, Section 3.2 defines the inference procedure of the intervals-BRB model and Section 3.3 defines the optimization procedure of the intervals-BRB model. Figure 1 shows the overall structure diagram of the liquid rocket health state assessment model based on intervals-BRB.

3.1. The Modeling Process of the Intervals-BRB

The modeling process of the intervals-BRB can be divided into three parts: (1) problem mechanism analysis, (2) interval construction and (3) belief table construction. These three parts are outlined in the following:

3.1.1. Problem Mechanism Analysis

The problem mechanism analysis mainly determines two things: (1) the main influencing factors of the problem, that is, the premise attributes and (2) the possible results of the problem.

Taking the health state evaluation of a liquid rocket in the background of this paper as an example, the health state of a liquid rocket is mainly judged by the tilt and sway degree of the liquid rocket during operation, and the health state is defined as very poor, medium, good or excellent. Then, the premise attributes and possible results can be obtained.

• Premise attributes: Tilt degree and shaking degree (two prerequisite attributes)
• Possible results: Very poor, medium, good, excellent (four results)

3.1.2. Building Intervals

For traditional BRB, it is necessary to determine the reference point and reference value of the premise attribute and set the reference value corresponding to the result. For the intervals-BRB model, the reference point is replaced by the reference interval, and the result reference value does not change.

The reference interval means dividing the interval into the upper and lower bounds of the attribute; it represents the different ranges that the attribute may fall into. In general, the range that the attribute value may fall into is determined according to expert knowledge, and the interval is divided according to the range. For example, the upper and lower bounds of the attribute are from 0–100, and the attribute is most likely to fall between 30 and 40, so it can be divided into three intervals from 0–30, 30–40 and 40–100. If the attribute is likely to fall into other intervals, the corresponding division can be used. The number of reference intervals is defined according to the problem (more reference intervals will improve the accuracy, but the problem complexity will also increase.

For example, if premise attribute 1 is divided into 10 intervals and premise attribute 2 is divided into 20 intervals, then $10+20=30$ belief rules need to be established. Here, innovatively, each interval is used as a rule, and for the time being, each rule only considers the impact of a single attribute. However, the traditional BRB requires the establishment of $10\times 20=200$ rules, resulting in an exponential increase in the number of rules. By combining rules with interval addition, the number of rules can be greatly reduced and redundant rules can be removed. This advantage becomes even more apparent when the number of attributes and reference values is greatly increased. The construction process of the belief rule base for the traditional BRB model is described in Section 2.1, and for the intervals-BRB model in Section 3.1.3.

Reference interval setting: (Upper bound, typical meaning interval (can be multiple), lower bound).

Example: The value of premise attribute 1 is between 0 and 100, and the sample distribution is dense from 30 to 40 and from 50 to 60. In this case, five reference intervals are defined, which are defined as [L, M1, M2, M3, H] (symbol). The specific reference points and reference intervals of premise attribute 1 are shown in

Table 5. Reference point and reference intervals of premise attribute 1.

Reference Point	Reference Interval
L	[0, 30]
M1	[30, 40]
M2	[40, 50]
M3	[50, 60]
H	[60, 100]

.

The result (very poor, medium, good, excellent) is defined as [O1, O2, O3, O4]. Based on the utility representation, the representation of the result can be defined as in

Table 6. Result and reference value.

Result	Reference Value
O1	0
O2	2
O3	4
O4	6

.

For interval BRB, one of the most significant changes is changing the reference point to the reference interval. When the reference interval is reasonably set, the scale of the belief rule base can be greatly reduced while ensuring the high accuracy of the model.

3.1.3. Building Belief Rule Base

The intervals-BRB model is a modeling method based on IF-THEN rules, which is composed of a series of belief rules. Assume that the different prerequisite properties are independent of each other, the relationship between evaluation indicators and results can be described by Formula (4).

$$\begin{array}{l}BeliefRul{e}_k:\\ If x_1\in [a_1,b_1]\vee x_2\in [a_2,b_2]\vee \dots \vee x_M\in [a_M,b_M]\\ Then result is\{(O_1,{\beta }_{1,k}),(O_2,{\beta }_{2,k}),\dots ,(O_N,{\beta }_{N,k})\}\\ with rule weight {\omega }_k\\ and rule reliability{r}_k\\ {\displaystyle \sum _{i=1}^N{\beta }_{i,k}\le 1},k\in \{1,2,\dots ,L\}\end{array}$$

(4)

where $x_i(i=1,\dots ,M)$ refers to the premise attributes for modeling objects; $[a_i,b_i](i=1,\dots ,M)$ represents the reference interval set of premise attributes; $O_i(i=1,\dots ,N)$ is the n evaluation results for the intervals-BRB; ${\beta }_{i,k}(i=1,\dots ,N)$ denotes the belief degree of each result under the $k_{th}$ belief rule; let ${\omega }_k$ denote the rule weight of the $k_{th}$ belief rule, and $r_k$ is the rule reliability of the $k_{th}$ belief rules.

The disjunctive notation used in Formula (4) is based on the characteristics of the proposed model. The use of symbols here is mainly based on how the model generates the rule base. There are two reasons for the notation here. First, “$\wedge$“ and” $\vee$“reflect the combination of belief rules. Yang [43] proposed the BRB modeling method and this model was used for oil spill detection by Xu [44]. Formula (1) is the representation of the belief rule of the traditional BRB. It can be seen that the traditional belief rule base combines rules in the form of Cartesian products, so it uses “$\wedge$“. However, the intervals-BRB are combined using interval addition, so “$\vee$“ is used here. Second, “$\vee$“ can reflect the independence of the premise attributes. The properties are assumed to be independent of each other, so the symbol “$\vee$“ is used.

From the above, it can be seen that the use of “$\vee$“ symbols is mainly designed based on the rule generation process. However, when rules are fused, the model takes the form of symbols “$\wedge$“. At this point, the model considers all attributes. Each attribute sample falls into a different interval and activates a different rule.

In order to understand the process of intervals-BRB constructing the belief rule base more specifically, intervals-BRB and traditional BRB are, respectively, used to construct the belief rule base under the same conditions.

Table 7. Reference point and reference intervals of premise attribute 1.

Reference Point	Reference Intervals
A	[0, 30]
B	[30, 40]
C	[40, 60]
D	[60, 100]

,

Table 8. Reference point and reference interval of premise attribute 2.

Reference Point	Reference Intervals
I	[0, 10]
J	[10, 40]
K	[40, 50]
L	[50, 60]

,

Table 9. Reference value of the result.

Result	Reference Value
D1	2
D2	4
D3	6
D4	8

and

Table 10. Corresponding belief table.

Index of Rule	Interval	Result
1	Attribute 1: [0, 30]	(D1,0.9), (D2,0.1), (D3,0), (D4,0)
2	Attribute 1: [30, 40]	(D1,0.8), (D2,0.2), (D3,0), (D4,0)
3	Attribute 1: [40, 60]	(D1,0.5), (D2,0.3), (D3,0.2), (D4,0)
4	Attribute 1: [60, 100]	……
5	Attribute 2: [0, 10]	……
6	Attribute 2: [10, 40]	……
7	Attribute 2: [40, 50]	……
8	Attribute 2: [50, 60]	……

show the process of intervals-BRB constructing the belief rule base, where Table 7 is the four reference intervals corresponding to premise attribute 1, Table 8 is the four reference intervals corresponding to premise attribute 2 and Table 9 is the reference value corresponding to the result. According to Table 7 and Table 8, it can be determined that $4+4=8$ rules are generated, and each rule is the corresponding confidence distribution of the result. Table 10 is the built belief base.

As shown in Table 10, the setting of each rule in the belief rule base only considers the effect of one attribute on the result. Therefore, Formula (4) adopts or joins the method.

For example, when attribute 1 of the data sample is 35 and attribute 2 is 15, the two attribute data are in the interval [30,40] and [10,40], respectively. Then, rule 2 and rule 6 are activated, respectively. Finally, rule 2 and rule 6 are fused using the ER-rule algorithm. It can be seen that the rules corresponding to all attributes are considered when the rules are fused. This represents the symbol “$\wedge$“.

The biggest difference between intervals-BRB and traditional BRB is that the reference point of the premise attribute is changed into the reference interval, which greatly reduces the number of rules and makes up for the defect that experts cannot perfectly determine the reference point to some extent. Therefore, for intervals-BRB, the setting of the reference interval is particularly important. For the setting of the reference interval, the setting of the traditional BRB reference point can be referred to, to a certain extent. Within the upper and lower bounds of premise attributes, the range of the premise attribute is most likely to be determined according to expert experience to set the interval. If the interval division is smaller, the precision of the model will be higher, but the number of rules and model complexity will increase at the same time. If the interval division is larger, the number of rules will shrink, and the corresponding model accuracy will also decrease, so it must be divided into the appropriate range to meet the requirements of the accuracy and complexity of the model.

3.2. The Inference Process of the Intervals-BRB

Intervals-BRB acts as a gray-box model and is driven by both data and expert knowledge. The main components of the intervals-BRB are the knowledge base, inference engine and optimization process. The reasoning process of the intervals-BRB model mainly has three steps. After judging the activated rules according to the input information, ER-rule is used for rule fusion and, finally, the expected utility is obtained through utility calculation to obtain the results of the modeling object. The reasoning procedure for ER rules is detailed below.

Step 1: Suppose that each rule activated by the intervals-BRB model is independent evidence and there are $K$ independent evidence $e_k(k=1,\dots ,K)$. The belief distribution of evidence can be directly obtained from the activated rule and the $k_{th}$ evidence can be expressed as Formula (5).

$$e_k=\{({\theta }_n,{\beta }_{n,k}),n=1,\dots ,N;(\Theta ,{\beta }_{\Theta ,k})\}$$

(5)

where ${\theta }_n(n=1,\dots ,N)$ is the evaluation level. $P_{n,k}$ represents the belief degree that this evaluation scheme is evaluated as ${\theta }_n$ under evidence $e_k$, where $N$ is the number of evaluation levels. $\Theta =\{{\theta }_1,{\theta }_2,\dots ,{\theta }_n)$ is the discernment frame and ${\beta }_{\Theta ,k}$ represents global ignorance.

Step 2: The weight ${\omega }_k$ and the reliability $r_k$ of the independent evidence $e_k$ are defined in advance by the model. The weighted belief distribution of the $K_{th}$ evidence with reliability is shown in Formulas (6)–(8).

$$m_k=\{({\theta }_n,{\tilde{m}}_{\theta ,k}),\forall \theta \subseteq \Theta ;(\beta (\Theta ),{\tilde{m}}_{\beta (\Theta ),k})\}$$

(6)

where $\beta (\Theta )$ is a power set and ${\tilde{m}}_{\theta ,k}$ is the mixed probability quality of indicator $k$ under lever ${\theta }_n$ and satisfies:

$${\tilde{m}}_{\theta ,k}=\{\begin{array}{l}0, \theta =\varnothing \\ C_{rw,k}{m}_{\theta ,k}, \theta \subseteq \Theta ,\theta \ne \varnothing \\ C_{rw,k}(1-r_k), \theta =\beta (\Theta )\end{array}$$

(7)

$${\displaystyle {\sum }_{\theta \in \Theta }{\tilde{m}}_{\theta ,k}+{\tilde{m}}_{\beta (\Theta ),k}=1}$$

(8)

where $C_{rw,k}=1/(1+{\omega }_k-r_k)$ is the regularization coefficient, $m_{\theta ,k}$ is the basic probability quality of indicator $k$ under level ${\theta }_n$ and $\varnothing$ is an empty set and $m_{\theta ,k}={\omega }_k{\beta }_{\theta ,k}$.

Step 3: K independent evidence $e_k(k=1,\dots ,K)$, ${\beta }_{\theta ,e(b)}$ for proposition $\theta$ can be obtained by iterating the following Formulas (9)–(12).

$${\widehat{m}}_{n,e(b)}=[(1-r_b)m_{\theta ,e(b-1)}+m_{p(\Theta ),e(b-1)}{m}_{\theta ,b}]+{\displaystyle {\sum }_{A\cap B=\theta }{M}_{A,e(b-1)}{m}_{B,b},\forall \theta \in \Theta }$$

(9)

$${\widehat{m}}_{P(\Theta ),e(b)}=(1-r_b)m_{\beta (\Theta ),e(b-1)}$$

(10)

$$m_{n,e(b)}=\{\begin{array}{l}0, \theta =\varnothing \\ \frac{{\widehat{m}}_{n,e(b)}}{{\displaystyle \sum _{A\subseteq \Theta }{\widehat{m}}_{A,e(b)}+{\widehat{m}}_{\beta (\Theta ),e(b)}}}, \theta \in \Theta ,\theta \ne \varnothing \end{array}$$

(11)

$${\beta }_{n,e(b)}=\{\begin{array}{l}0, \theta =\varnothing \\ \frac{{\widehat{m}}_{n,e(b)}}{{\displaystyle \sum _{A\subseteq \Theta }{\widehat{m}}_{A,e(B)}}}, \theta \subseteq \Theta ,\theta \ne \varnothing \end{array}$$

(12)

where $k=1,\dots ,L,m_{\theta ,e(1)}=m_{\theta ,1},m_{\beta (\Theta ),e(1)}=m_{\beta (\Theta ),1}$, $m_{\theta ,e(b)}$ is the joint probability density of the $b$ pieces of evidence, and ${\beta }_{\theta ,e(b)}$ is the joint belief degree of the $b$ pieces of evidence.

Step 4: Let the utility of evaluating grade ${\theta }_n$ be $u({\theta }_n)$. Then, the expected utility can be calculated by Formula (13), where $u$ is the output in Formula (2).

$$u={\displaystyle \sum _{n=1}^{N}u({\theta }_n){\beta }_{\theta ,e(b)}}$$

(13)

The overall flow chart of ER-rule is shown in Figure 2.

3.3. The Optimization Process of the Intervals-BRB

Considering limited expert knowledge, the model has difficulty accurately describing the real condition. Therefore, it is necessary to optimize the intervals-BRB model. The objective optimization function of the intervals-BRB model is described as Formula (14).

$$\begin{array}{l}\min MSE(a),a=\{r,\omega ,\beta \}\\ MSE(a)=\frac{1}{N}{\displaystyle \sum _{n=1}^N{({\widehat{Y}}_n(t)-{\widehat{\varphi }}_n(t))}^2}\end{array}$$

(14)

where $MSE(\cdot )$ denotes the mean square error (MSE) function, $N$ is the number of test samples, ${\widehat{Y}}_n(t)$ is the actual predicted value for the $n_{th}$ sample, ${\widehat{\varphi }}_n(t)$ represents the expected predicted value of the $n_{th}$ sample and $a$ is the set of parameters in the inference process. Before optimization, it is first necessary to determine the constraints of rule weight $\omega$, rule reliability $r$ and belief degree $\beta$, which are described as Formulas (15)–(18).

• Rule weight. During the construction of the belief rule base model, the constraint conditions that rule weight ${\omega }_k$ corresponding to $k_{th}$ belief rule $BeliefRul{e}_k$ needs to satisfy are Formula (15).

  $$0\le {\omega }_k\le 1,k=1,2,\dots ,L$$

  (15)

• Rule reliability. When building the belief rule base model, it is necessary to determine the rule reliability $r_k$ corresponding to $k_{th}$ belief rule $BeliefRul{e}_k$, and the corresponding constraints are represented as Formula (16).

  $$0\le r_k\le 1,k=1,2,\dots ,L$$

  (16)

• Belief degree. For the $k_{th}$ rule, the probability of occurrence of the corresponding $n_{th}$ result $O_n$ can be expressed by the belief degree ${\beta }_{n,k}$, and the constraint condition of the belief degree ${\beta }_{n,k}$ can be described as Formula (17).

  $$0\le {\beta }_{n,k}\le 1,n=1,2,\dots ,N;k=1,2,\dots ,L$$

  (17)

• Meanwhile, the sum of the belief values of all belief rules can be expressed as Formula (18).

  $${\displaystyle \sum _{n=1}^N{\beta }_{n,k}\le 1,n=1,2,\dots ,N;k=1,2,\dots ,L}$$

  (18)

The Covariance Matrix Adaptive Evolution Strategy (CMA-ES) [45] algorithm is an improved algorithm based on the evolutionary algorithm, which achieves the optimal purpose by simulating the biological evolution process. It is mainly used to solve nonlinear and non-convex optimization problems. The method starts from a random initial search point, generates the first population A according to a certain probability distribution and evaluates the fitness of all individuals in the population. Then, according to the fitness of individuals in population A, the better individual is selected to update the evolution strategy so as to adjust the evolution direction of the next population, that is, control the generation of the next population. After each mutation, the optimal solution in the current population and convergence conditions must be compared. If the optimal solution is met, the optimal solution is found and the loop is exited, otherwise the iteration continues. This method has the characteristics of good global performance and high optimization efficiency. This model uses the revised CMAES optimization algorithm [46] for optimization, and the specific optimization process of the model is shown in Figure 3.

3.4. The Overall Flow of the Interval BRB Model

From Section 3.1, Section 3.2, Section 3.3 and Section 3.4 we can conclude the following pseudocode for the intervals-BRB model.

Pseudocode for the Intervals-BRB Model

Step 1: According to Formula (4) and Table 7, Table 8, Table 9 and Table 10, the expert divides the interval for the premise attribute and gives the initial parameters of the model, including the rule weight, the rule reliability and the result belief distribution. Step 2: The model parameters given by experts are taken as the initial parameters, Formula (14) is taken as the optimization objective function, and the iteration rounds are set for optimization. The optimized model is obtained through the revised optimization algorithm CMAES. Step 3: The optimized model is used to test on the test set, assuming that there are m test data and each piece of data has two premise attributes.         For $i=1:m$ 1.Two rules are activated based on the two premise attributes. The belief degree distribution of each rule is obtained according to the Formula (5). 2.Obtain the weighted confidence distribution of the two pieces of evidence through Formulas (6)–(8). 3. Obtain the joint confidence of two pieces of evidence through Formulas (9)–(12). 4. The final result is obtained by utility calculation through Formula (13)         End         Then we get m final results. Step 4: The final MSE of the model is obtained by comparing the predicted value with the actual value.

4. Case Study

In order to verify the proposed liquid rocket safety evaluation method based on semi-quantitative interval BRB, this section is based on the wireless sensor network test platform built in the laboratory for verification. Wireless sensors were used to monitor the vibration frequency, tilt angle, temperature, humidity and other attributes of the rocket model. Since the temperature and humidity of the experimental environment remained basically unchanged during the experiment, only the influence of vibration frequency and tilt angle on the evaluation model was considered. The unit of vibration amplitude is $\mu M$, which represents vibration displacement. The angle of motion is measured in degrees.

The main purpose of this section is to construct the intervals-BRB model for the health state assessment of liquid rockets. In this paper, the specific data of a liquid rocket operation is used as the data set, and there are 515 data in total. Each data takes tilt degree and shaking degree as two input attributes, and the actual running results as the corresponding output results. Figure 4 shows the two input attributes.

In this paper, 70 data are randomly selected from 515 data for testing, and the remaining 445 data are used for training. The running environment of this experiment is Windows 11 and 12th Gen Intel(R) Core(TM) i5-12500H, and the code is run in MATLAB R2021a without using other class libraries.

4.1. Initial Definition

The initial intervals-BRB needs to establish the belief rule, which is constructed as Formula (18).

$$\begin{array}{l}BeliefRul{e}_k\\ \mathrm{If}{x}_1\in [a_1,b_1]\vee x_2\in [a_2,b_2]\\ Thenresultif\{(O_1,{\beta }_{1,k}),(O_2,{\beta }_{2,k}),(O_3,{\beta }_{3,k})\}\\ withruleweight{\omega }_k\\ andrulereliability{r}_k\\ {\displaystyle \sum _{i=1}^N{\beta }_{i,k}}\le 1,k\in \{1,2,\dots ,L\}\end{array}$$

(19)

where $x_1$ represents the degree of tilt, $x_2$ represents the degree of shaking, $Oi(i=1,2,3,\dots )$ represents the health state of the liquid rocket, and the health state grade is divided into three grades in this experiment, poor ($O_1$), good ($O_2$) and excellent ($O_3$).

4.2. Analysis of Case Experimental Results

By analyzing the data characteristics of the indicators, 20 reference intervals are set for each indicator and 40 belief rules are generated accordingly. First, the rule weight and rule reliability are all set to 1; the setting of the initial belief is shown in

Table A1. The initial rule weight, rule reliability and belief degree of the proposed model.

Index	Intervals	Rule Weight	Rule Reliability	Initial Belief Degree
1	X1 (3.1,3.5)	1	1	(0,0.10,0.90)
2	X1 (3.5,4.0)	1	1	(0,0.12,0.88)
3	X1 (4.0,5.0)	1	1	(0,0.15,0.85)
4	X1 (5.0,5.5)	1	1	(0,0.17,0.83)
5	X1 (5.5,6.0)	1	1	(0,0.2,0.8)
6	X1 (6.0,6.5)	1	1	(0,0.22,0.78)
7	X1 (6.5,7.0)	1	1	(0,0.25,0.75)
8	X1 (7.0,7.5)	1	1	(0,0.27,0.73)
9	X1 (7.5,8.0)	1	1	(0.05,0.25,0.70)
10	X1 (8.0,8.5)	1	1	(0.05,0.27,0.68)
11	X1 (8.5,9.0)	1	1	(0.05,0.30,0.65)
12	X1 (9.0,9.5)	1	1	(0.02,0.35,0.63)
13	X1 (9.5,50)	1	1	(0.05,0.45,0.5)
14	X1 (50,55)	1	1	(0,0.6,0.4)
15	X1 (55,56)	1	1	(0,0.65,0.35)
16	X1 (56,58)	1	1	(0.05,0.7,0.25)
17	X1 (58,60)	1	1	(0.25,0.6,0.15)
18	X1 (60,62)	1	1	(0.20,0.65,0.15)
19	X1 (62,64)	1	1	(0.30,0.7,0.)
20	X1 (64,66)	1	1	(0.4,0.6,0.)
21	X2 (0.020,0.025)	1	1	(0,0.1,0.9)
22	X2 (0.025,0.028)	1	1	(0,0.12,0.88)
23	X2 (0.028,0.030)	1	1	(0,0.15,0.85)
24	X2 (0.030,0.035)	1	1	(0,0.17,0.83)
25	X2 (0.035,0.038)	1	1	(0,0.2,0.8)
26	X2 (0.038,0.040)	1	1	(0,0.22,0.78)
27	X2 (0.040,0.045)	1	1	(0,0.25,0.75)
28	X2 (0.045,0.048)	1	1	(0.,0.27,0.73)
29	X2 (0.048,0.050)	1	1	(0.05,0.25,0.70)
30	X2 (0.050,0.053)	1	1	(0.05,0.27,0.68)
31	X2 (0.053,0.055)	1	1	(0.05,0.30,0.65)
32	X2 (0.055,0.060)	1	1	(0.02,0.35,0.63)
33	X2 (0.060,0.063)	1	1	(0.05,0.45,0.50)
34	X2 (0.063,0.065)	1	1	(0,0.60,0.40)
35	X2 (0.065,0.070)	1	1	(0,0.65,0.35)
36	X2 (0.070,0.075)	1	1	(0.05,0.7,0.25)
37	X2 (0.075,0.078)	1	1	(0.22,0.60,0.15)
38	X2 (0.078,0.080)	1	1	(0.20,0.65,0.15)
39	X2 (0.080,0.085)	1	1	(0.30,0.7,0)
40	X2 (0.085,0.090)	1	1	(0.4,0.6,0)

and the optimized rule weight, rule reliability and belief are shown in

Table A2. The rule weight, rule reliability and belief degree of the proposed model after training.

Index	Intervals	Rule Weight	Rule Reliability	Initial Belief Degree
1	X1 (3.1,3.5)	1	0.7867	(0.0026,0.0102,0.9872)
2	X1 (3.5,4.0)	0.9078	0.8851	(0.1686,0.5893,0.2421)
3	X1 (4.0,5.0)	0.7301	0.2220	(0.0378,0.8166,0.1456)
4	X1 (5.0,5.5)	0.2102	0.8172	(0.6683,0.2168,0.1149)
5	X1 (5.5,6.0)	0.7840	0.0083	(0.4943,0.3227,0.1829)
6	X1 (6.0,6.5)	0.9624	0.8145	(0.0000,0.0388,0.9627)
7	X1 (6.5,7.0)	0.2669	0.4096	(0.8316,0.1579,0.0105)
8	X1 (7.0,7.5)	0.5875	0.8089	(0.2892,0.2287,0.4821)
9	X1 (7.5,8.0)	0.3177	0.4827	(0.2032,0.0648,0.7320)
10	X1 (8.0,8.5)	0.5942	0.4554	(0.0539,0.6376,0.3084)
11	X1 (8.5,9.0)	0.6505	0.3649	(0.3222,0.5768,0.1010)
12	X1 (9.0,9.5)	0.9796	0.5075	(0.0063,0.0146,0.9791)
13	X1 (9.5,50)	0.9486	0.8049	(0.9692,0.0154,0.0154)
14	X1 (50,55)	0.2840	0.0454	(0.1347,0.8490,0.0162)
15	X1 (55,56)	0.8845	0.2220	(0.0694,0.1034,0.8272)
16	X1 (56,58)	0.5764	0.8309	(0.9765,0.0077,0.0157)
17	X1 (58,60)	1	0.3629	(0.9693,0.0145,0.0162)
18	X1 (60,62)	0.4915	0.3228	(0.0249,0.2961,0.6789)
19	X1 (62,64)	0.9575	0.7452	(0.9394,0.0519,0.0086)
20	X1 (64,66)	0.8834	0.5223	(0.8345,0.1025,0.0629)
21	X2 (0.020,0.025)	0.7927	0.1554	(0.1621,0.5703,0.2676)
22	X2 (0.025,0.028)	0.2730	0.6253	(0.5733,0.1749,0.2518)
23	X2 (0.028,0.030)	0.2837	0.3165	(0.5019,0.0126,0.4855)
24	X2 (0.030,0.035)	0.5109	0.2151	(0.3179,0.4214,0.2607)
25	X2 (0.035,0.038)	0.3638	0.0533	(0.4531,0.3446,0.2023)
26	X2 (0.038,0.040)	0.4091	0.0056	(0.6193,0.3221,0.0586)
27	X2 (0.040,0.045)	0.0970	0.0478	(0.3255,0.3281,0.3464)
28	X2 (0.045,0.048)	0.3024	0.2062	(0.4142,0.4288,0.1570)
29	X2 (0.048,0.050)	0.5590	0.8618	(0.0231,0.4743,0.5026)
30	X2 (0.050,0.053)	0.1739	0.0594	(0.5092,0.3808,0.1100)
31	X2 (0.053,0.055)	0.1804	0.1872	(0.3137,0.5713,0.1150)
32	X2 (0.055,0.060)	0.1184	0.9862	(0.2392,0.6325,0.1283)
33	X2 (0.060,0.063)	0.0549	0.6556	(0.3431,0.2018,0.4551)
34	X2 (0.063,0.065)	0.6217	0.6834	(0.4593,0.5057,0.0350)
35	X2 (0.065,0.070)	0.5137	0.8129	(0.6539,0.1379,0.2082)
36	X2 (0.070,0.075)	0.0579	0.4612	(0.0970,0.2875,0.6155)
37	X2 (0.075,0.078)	0.8762	0.2636	(0.1264,0.2237,0.6499)
38	X2 (0.078,0.080)	0.7430	0.2221	(0.3961,0.3917,0.2122)
39	X2 (0.080,0.085)	0.4638	0.1083	(0.0466,0.2641,0.6893)
40	X2 (0.085,0.090)	0.1739	0.7077	(0.1216,0.5776,0.3008)

. The upper and lower bounds of the input indicators are shown in

Table 11. The lower bound and upper bound of two attributes.

Attribute	Lower Bound	Upper Bound
$x1$$(\mu M$)	3.1	66
$x2$(degree)	0.02	0.09

, and the reference values of the output results are shown in

Table 12. The result and value of the healthy state assessment.

Result	Value
$O1$	0
$O2$	0.5
$O3$	1

. For the output, the value ranges from 0 to 1, with higher values indicating worse health status assessments.

Through expert knowledge, 20 intervals are divided for each of the two indicators within the upper and lower bounds,

Table 13. Shows 40 intervals corresponding to attribute x1 and attribute x2.

Index	$\mathbf{Intervals} \mathbf{of} \mathbf{X}1 \left(\mathit{\mu }\mathit{M}\right)$	Intervals of X2 (Degree)
1	(3.1,3.5)	(0.020,0.025)
2	(3.5,4.0)	(0.025,0.028)
3	(4.0,5.0)	(0.028,0.030)
4	(5.0,5.5)	(0.030,0.035)
5	(5.5,6.0)	(0.035,0.038)
6	(6.0,6.5)	(0.038,0.040)
7	(6.5,7.0)	(0.040,0.045)
8	(7.0,7.5)	(0.045,0.048)
9	(7.5,8.0)	(0.048,0.050)
10	(8.0,8.5)	(0.050,0.053)
11	(8.5,9.0)	(0.053,0.055)
12	(9.0,9.5)	(0.055,0.060)
13	(9.5,50)	(0.060,0.063)
14	(50,55)	(0.063,0.065)
15	(55,56)	(0.065,0.070)
16	(56,58)	(0.070,0.075)
17	(58,60)	(0.075,0.078)
18	(60,62)	(0.078,0.080)
19	(62,64)	(0.080,0.085)
20	(64,66)	(0.085,0.090)

shows the 40 intervals specific to the 2 premise attributes.

When the two attribute values $x_1$ and $x_2$ are entered, we first determine which intervals $x_1$ and $x_2$ are in and then activate the rules corresponding to the two intervals because every time we enter two attribute values, each input will activate two rules. Since the rule in intervals-BRB is a belief distribution itself, it can be directly used as evidence. The two rules, the rule weight and the rule reliability of the two rules are used as the input of the ER-rule for evidence fusion to obtain the final belief distribution, and then the health status evaluation level is obtained through utility calculation. For example, if the input attribute $x1$ is 4.1 and the input attribute $x2$ is 0.041, then the two activated rules are the rule corresponding to (4.0, 5.0) in the attribute $x1$ and the rule corresponding to (0.040, 0.045) in the attribute $x2$. These two rules are fused by the ER rule to perform utility calculations, and the final health status assessment grade is obtained. Although the extension of the reference point for the reference range reduced the uncertainty of expert knowledge, the result of belief, rules of weight and attribute weight still depends on the expert knowledge; therefore, the intervals-BRB model still needs to be the optimization algorithm adopted to optimize the parameters, using the optimized model to forecast the results compared with the actual value.

For the intervals-BRB model, we set a total of three result reference values, which are 0, 0.5 and 1 numerically. When we use the intervals-BRB model for prediction, the result is usually from 0.9–1 or from 0–0.1. In order to make the result more obvious and easier to understand, we set the threshold value as 0.1. When the predicted value and the actual value are less than the threshold, we make the predicted value equal to the actual value, so as to obtain the final prediction result. We then plot the results using the same threshold of 0.1 and the same comparison method for the traditional BRB and the other prediction models. From Figure 5, we can see that the trained intervals-BRB model has a high accuracy, with an accuracy of 0.97 in the case of small samples.

4.3. Comparison of the Model before and after Improvement

Compared with the traditional BRB model, the intervals-BRB model proposed in this paper mainly improves the traditional BRB model, which easily leads to combination rule explosion (or too many parameters leading to a training time that is too long), and the inference process does not consider the reliability of the rules, which affects the final accuracy. Therefore, intervals-BRB and traditional BRB were used for modeling, the same training set was used for training and the same test set was used for testing. The classification fitting diagram of the intervals-BRB is shown in Figure 5, and the classification fitting diagram of the traditional BRB model is shown in Figure 6.

The traditional BRB model and intervals-BRB model use the same data set and, each time, 70% are randomly selected for training and 30% for testing. The accuracy after 10 rounds of trials is shown in Figure 7 below. The average accuracy of the traditional BRB model is approximately 92%, while the accuracy of the intervals-BRB model is approximately 97%. It can be seen that intervals-BRB has higher accuracy and is robust.

Table 14. Related metrics of the traditional BRB model and intervals-BRB model.

Methods	Parameters	Rules	Accuracy
Intervals-BRB	200	40	97%
Traditional BRB	1766	441	90%

shows the comparison of related metrics between intervals-BRB and traditional BRB. It can be clearly seen from Table 14 that the parameters are reduced from 1766 parameters to 200 parameters, a decrease of approximately 88.7%. The number of rules is reduced by 90% and the accuracy is increased by 7%. The intervals-BRB model proposed in this paper intersects with the traditional BRB model, which can effectively reduce the number of rule bases and the number of parameters. Additionally, the accuracy of the proposed model is not lower than that of the traditional BRB model.

4.4. Other Comparative Experiments

Comparing the intervals-BRB model with the traditional BRB model, BPNN, ELM and RBF. The initial parameters of the BPNN and ELM experiment are set as follows: the number of iterations is 200, the learning rate is 0.00000001 and the target value is 0. The initial parameters of the RBF experiment were set as follows: the expansion speed of the radial basis function was 1, the target value was 0, the maximum number of neurons was 10 and the number of neurons added between two displays was 25. The fitting plots of the intervals-BRB model and traditional BRB model are shown in Figure 5 and Figure 6, and the fitting plots of the other three models are shown in Figure 8, Figure 9 and Figure 10.

The accuracy comparison of different methods is shown in

Table 15. The accuracy of different methods.

Methods	Accuracy (%)
Intervals-BRB	97
BRB	90
BPNN	81.4
ELM	90
RBF	95.71

. It can be seen that the intervals-BRB model has obvious advantages in terms of accuracy and is much faster than the traditional BRB in terms of time consumption. Therefore, the intervals-BRB model proposed in this paper has high practical value.

4.5. Discussion of Experiments

The purpose of this part of the experiment is to verify that the proposed intervals-BRB model can effectively perform safety state assessment through a certain liquid rocket data set. According to the overall framework of the model in Figure 1, the initial parameters are determined by experts, and then the training set and CMAES algorithm are used to optimize the parameters to obtain the optimized model. Then, the optimized model is tested on the test set.

According to Section 4.2 and Section 4.3, it can be known that the improved intervals-BRB model based on the traditional BRB can achieve no less accuracy than the traditional BRB under the condition of fewer parameters. According to Section 4.4, intervals-BRB also has advantages in accuracy when compared with other models. In addition, since intervals-BRB is still an expert system, driven by data and expert knowledge, it is more interpretable than other models.

In the intervals-BRB model, the initial parameters are given by experts, so the initial parameters are reasonable, but in order to improve the accuracy of the model, the model needs to be optimized, but the parameters often become unreasonable in the optimization process. In order to maintain the advantage of interpretability of the intervals-BRB model, interpretability constraints can be added to the optimization process in subsequent experiments for optimization.

5. Conclusions

The intervals-BRB model can deal with semiquantitative information and solve the problem of the traditional BRB, which easily causes combination rule explosion when there are too many premise attributes, and the expert knowledge is easy to consider incompletely when setting the reference point of the premise attributes. The ER-rule is used in the reasoning process, and the reliability of the rule is taken into account to make the result more accurate. Through experimental verification, it can be seen that the intervals-BRB has fewer parameters and a simpler structure; additionally, it has higher accuracy, which provides an effective modeling scheme for the health state assessment of liquid rockets.

If there are fewer premise attributes and fewer reference values, the advantage of the intervals-BRB is not obvious. However, when there are 5 premise attributes and each has 10 reference points, the traditional BRB may not be able to model at all, while the intervals-BRB can still model well, which fully reflects the advantage of intervals-BRB in effectively solving the problem of combination rule explosion. In future research, according to the operational characteristics of liquid rockets, reasonable interpretability constraints can be set in the optimization process to increase the interpretability of the model while ensuring accuracy.