Diagnosability-optimization design of fixed-wing unmanned aerial vehicle based on causal matching and tree seed algorithm

Abstract

With mission intensity and difficulty escalating, the future market demand for UAV reliability and safety grows dramatically. So, this paper proposes an optimal design strategy based on causal matching and the Tree Seed Algorithm (TSA) based on structural analysis. First, a novel algorithm is designed for finding the minimum set of consistency relations for diagnosability analysis. The complexity of this algorithm is at a polynomial level, which is a significant improvement over previous algorithms with exponential complexity. A causal consistency search algorithm is also innovatively proposed for causal diagnosability analysis considering the causal constraints of the dynamic variables. Secondly, a diagnosability optimization strategy based on TSA is designed to balance the diagnosability requirements and the design cost of consistency relations. This strategy can satisfy the system’s diagnosability demand under different causal constraints with minimum consistency relations. Finally, a fixed-wing UAV model is established to analyze the diagnosability under different causal constraints qualitatively. Based on the TSA, consistency relations with the minimum integrated diagnosis cost and the best diagnosis performance are preferred.

Keywords

Fixed-wing UAV structural analysis causality tree seed algorithm diagnosability

Introduction

Due to the advantages in manufacturing cost, mobility, safety, stealth performance, etc., UAVs have been developed rapidly in the military, police, and civil fields.¹ In the military field, UAVs are mainly used in military exercises, weapon trials, enemy reconnaissance, fire strikes, etc.;² in the civil field, UAVs are primarily used in environmental monitoring, crop plant protection,³ communication networking,⁴ engineering quality testing,⁵ logistics, and transportation, etc.; in the police field: UAVs are mainly used in firefighting and rescue, disaster monitoring, personnel search and rescue, emergency lighting, area inspection, aerial photography modeling, etc.⁶ Fixed-wing UAVs play an essential role in logistic supply and fire strikes because of the advantages of extended flight time, high speed, and large payload. With the escalation of combat intensity and the increasing difficulty of combat, the future battlefield has an ever-increasing demand for UAVs’ flexibility, safety, and reliability. When performing missions, UAVs may fail due to strikes or interference by enemy forces, which in turn leads to the failure of combat missions.⁷ In addition, when a UAV fails in the air, its flight trajectory can change uncontrollably, which damages itself or surrounding structures and may pose a risk to human life and health when it crashes to the ground. As UAVs are used in broader applications, ensuring their reliability and safety has become essential.

At the present stage, improving the reliability of UAVs starts from two aspects: increasing the reliability of critical components. However, increasing the reliability of key components cannot only avoid the occurrence of faults but also increase the system design cost. Secondly, it improves the performance of the fault diagnosis system. Fault diagnosis techniques for UAVs can be broadly classified into three categories: model-based approaches, data-driven approaches, and hybrid approaches. Model-based approaches are methods that analyze the redundancy relationships of a system based on its mathematical model. They can be further classified in detail into parameter estimation methods,⁸ state estimation methods,⁹ parity space methods,⁹ and structural analysis methods.¹⁰ Data-driven methods use large amounts of data from the system to analyze redundancy. Specifically, they can be subdivided into statistical methods,¹¹ neural networks,¹² expert systems,¹³ fuzzy logic,¹⁴ time-frequency analysis,¹⁵ pattern recognition,¹⁶ and other methods. Hybrid methods can take into account the advantages of the first two fault diagnosis methods and have been developed to a certain extent in the field of UAV fault diagnosis.¹⁷ However, the performance of a fault diagnosis system is often limited by the fault information provided by the system. If the system’s fault information is insufficient or missing, even the most accurate fault diagnosis algorithms will not help. Therefore, it is necessary to start from the source of fault diagnosis to improve the fault diagnosis performance of UAVs. Diagnosability is a design property that represents the fault diagnosis performance. On the one hand, diagnosability ensures that the system structure provides sufficient fault information;¹⁸ on the other hand, it also ensures that the system completes fault diagnosis with low cost and high efficiency and guarantees the fault diagnosis performance of the system throughout its life cycle.¹⁹ Therefore, in this paper, the reliability of UAVs is improved from the perspective of optimal design for diagnosability.

The structural analysis method differs from previous model-based fault diagnosis methods that rely on constructing, analyzing, and deriving an accurate analytical model of the system.²⁰ Structural analysis focuses on obtaining relevant information for fault diagnosis from the structural model of the system. In the design stage, structural analysis is used for diagnosability design: first, by analyzing the structural diagnosability of the system, diagnosability optimization design is carried out; second, it does not require the establishment of an accurate mathematical model, which is advantageous for the diagnosability design of large and complex control systems; and third, it avoids complex numerical operations, which is more efficient than using analytical methods.²¹ The redundancy sub-model is the basis for diagnosability studies based on structural analysis. Redundancy sub-models are also known as analytic redundancy relations, minimal structural overdetermination, test subsystems, minimal evaluation chains, etc.²² In this paper, we collectively refer to them as minimum consistency relations. The computational complexity of standard consistency relation search algorithms is exponential, especially when computing large and complex systems. To improve the performance of the algorithms, Chen proposed the idea of structural equivalence classes to simplify the number of algorithm iterations.²³ Based on the greedy algorithm, Schmid prefers the minimum set of consistency relations based on the number of system faults.²⁴ Therefore, in this paper, we propose a matching-based minimum consistency relation search algorithm, which is polynomial in complexity and improves over previous algorithms. This algorithm dramatically reduces the complexity of the consistency relation search algorithm.

The consistency relation is limited by the causality of the dynamic variables.²⁵ Differential causality: firstly, differential causality will limit the consistency relation, which will affect the diagnosability of the system to a certain extent; secondly, if the differential operation is performed on the dynamic variables in a noisy environment, the higher the order of differentiation, the greater the effect of noise on the system. Integral causation: firstly, if noise is present, the stability of the consistency relation cannot be guaranteed when performing the integral operation; secondly, the initial values of the dynamic variables, the system modeling uncertainty, and the system bias all impact the integral causation. However, all these limitations of integral causation can be avoided to some extent. Mixed causation implies no causal restrictions exist in the design of the consistency relation. Their effects differ if integral and differential causality appear in a linear algebra ring. Differential rings can be broken using integral causality; however, they cannot be broken by relying solely on differential causality. Therefore, differential causality should be avoided as much as possible while calculating consistency relations. Mixed causality can take care of both the system noise and the stability of the consistency relation. Using lower-order differential operations in mixed causality is more advantageous than using differential causality alone in dealing with system noise. To fully consider the impact of such causality on system diagnosability, this paper proposes a diagnosability analysis method based on causal matching to qualitatively analysis the diagnosability of the system under different causal constraints.

In addition, conformance relationships are somewhat limited by system cost constraints. Not all consistency relations are required to participate in the system diagnosability analysis if the diagnosability requirement and the design cost of consistency relations are to be considered simultaneously. Therefore, finding the most miniature set of consistency relations that satisfy the system diagnosability requirements is necessary, which is a combinatorial optimization problem. Bionic intelligent optimization algorithms play a significant advantage in solving optimization problems. Standard bionic intelligent optimization algorithms include artificial bee colony algorithm,²⁶ ant colony algorithm,²⁷ cuckoo algorithm,²⁸ bat algorithm,²⁹ firefly algorithm,³⁰ chimpanzee algorithm,³¹ squirrel searching algorithm,³² whale optimization algorithm³³ and so on. TSA is derived from the natural tree growth and dispersal.³⁴ The main reason for adopting this algorithm in this paper lies in the unique search mechanism of the algorithm, which can effectively prevent the algorithm from falling into local optimal solutions with the help of the randomness of seed propagation. In recent years, the TSA has been widely used and has been developed to a certain extent in parameter estimation,³⁵ controller design,³⁶ neural network training,³⁷ and data prediction.³⁸ Therefore, this paper proposes an optimal design strategy for diagnosability based on the tree species optimization algorithm to find the minimum set of consistency relations that satisfy the diagnosability requirements of the system.

To meet the increasing demand for the reliability and safety of UAVs, this paper proposes an optimal design strategy for diagnosability based on the structural analysis method. The influence of the causality of dynamic variables on diagnosability is fully considered. The main contributions of this paper are as follows:

(1) A new minimum consistency relation search algorithm is proposed for finding the minimum set of consistency relations;

(2) A consistency relation search algorithm considering causal constraints is proposed for the diagnosability analysis of a system under different causal constraints;

(3) A diagnosability optimization strategy based on the TSA is proposed to find the minimum set of consistency relations that satisfy the system’s diagnosability requirements.

The rest of the paper is organized as follows. Section “Structural modeling of causal diagnosability” establishes a system causal diagnosability model to lay the foundation for system causal diagnosability analysis. In Section “Diagnosability evaluation based on structural analysis,” a consistent relationship search algorithm based on matching is proposed; secondly, a causal consistency search algorithm is proposed based on the causal constraints of dynamic variables. Section “Diagnosability optimization design” establishes a diagnosability optimization model and proposes a diagnosability optimization strategy based on TSA. In Section “Fixed-wing UAV diagnosability analysis,” based on the fixed-wing UAV model, its causal diagnosability is simulated, analyzed and optimally designed. Section “Conclusion” concludes the main work of this paper.

Structural modeling of causal diagnosability

Structural modeling

Let $E = {e_{1}, e_{2}, \dots}$ be the set of constraint equations and $V = {v_{1}, v_{2}, \dots}$ be the set of internal variables. Then, the system model $M$ can be represented by a tuple $(E, V)$ . The variables in $V$ can be categorized into known variables $Z$ and unknown variables $X$ , that is, $V = Z \cup X$ . The structural model is an abstract representation of the analytical model.

The structural model can be described by a bipartite graph $G (E, V, \bar{A})$ .²⁵ $\bar{A}$ is the set of edges:

\bar{A} = {(e_{i}, v_{j}) | e_{i} \in E, v_{j} \in V, v_{j} i s i n e_{i}}

(1)

It can also be described by the correlation matrix $A = [a_{ij}]$ of $G (E, V, \bar{A})$ :

a_{ij} = {\begin{matrix} 1, (e_{i}, v_{j}) \in \bar{A} : \\ 0, otherwise \end{matrix}

(2)

A specific pattern of $M$ does not need to be described by all the equations in $E$ . Let $Q$ be a mapping from $E$ to $V$ :

\begin{matrix} Q : 2^{E} \to 2^{V} \\ E' \mapsto Q (E') = {v \in V | \exists e \in E' s . t . (e, v) \in \bar{A}} \end{matrix}

(3)

Let $R$ be the mapping from $V$ to $E$ :

\begin{matrix} R : 2^{V} \to 2^{E} \\ V' \mapsto R (V') = {e \in E | \exists v \in V' s . t . (e, v) \in \bar{A}} \end{matrix}

(4)

Due to $V = Z \cup X$ , it can be further defined that $Q_{X}$ is the mapping from $E$ to $X$ , and $Q_{Z}$ is the mapping from $E$ to $Z$ .

The model in a specific behavioral pattern can be represented as $(E', Q (E'))$ and the corresponding structural model as $G (E', Q (E'), \bar{A}')$ . Structural analysis is concerned with the model associated with the unknown variable $X$ . For descriptive purposes, the structural model related to $X$ is defined as $G (E, X, \bar{A})$ .

Structural model of causal diagnosability

The basic principle of consistency-based fault diagnosis is to design consistency relations associated with faults.³⁹ Assume that the system model $M_{1}$ is

\begin{matrix} \begin{matrix} e_{1} : d x_{1} = - x_{1} + x_{2} + x_{5} e_{9} : d x_{1} = \frac{d}{dt} x_{1} \\ e_{2} : d x_{2} = - 2 x_{2} + x_{3} + x_{4} \\ e_{3} : d x_{3} = - 3 x_{3} + x_{5} + f_{1} + f_{2} e_{10} : d x_{2} = \frac{d}{dt} x_{2} \\ e_{4} : d x_{4} = - 4 x_{4} + x_{5} + f_{3} e_{11} : d x_{3} = \frac{d}{dt} x_{3} \\ e_{5} : d x_{5} = - 5 x_{5} + u + f_{4} \\ e_{6} : y_{1} = x_{1} e_{12} : d x_{4} = \frac{d}{dt} x_{4} \\ e_{7} : y_{2} = x_{2} \\ e_{8} : y_{3} = x_{3} e_{13} : d x_{5} = \frac{d}{dt} x_{5} \end{matrix} \end{matrix}

(5)

Then $X_{1} = {d x_{1}, d x_{2}, d x_{3}, d x_{4}, d x_{5}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, f_{1}, f_{2}, f_{3}, f_{4}}$ , $Z_{1} = {u, y_{1}, y_{2}, y_{3}}$ . $d x_{i}$ represents the derivative of the dynamic variable $x_{i}$ . $F_{1} = {f_{1}, f_{2}, f_{3}, f_{4}}$ are system faults.

Causality²⁵ exists in differential equations

d x_{i} = \frac{d}{dt} x_{i}

(6)

(1) Derivative causality: $d x_{i}$ can be obtained by differentiating of $x_{i}$ ;

(2) Integral causality: $x_{i}$ can be obtained by integration of $d x_{i}$ ;

The consistency relationship is calculated as follows: based on the redundant sub-model, a part of the equations is preferred to solve for the unknown variables; secondly, based on the redundant equations, the consistency test is performed. As shown in Figures 1 and 2, the flow chart of the calculation of the consistency relationship is given. The circles in the figure represent system variables, the black bars represent static constraint equations, and the red bars represent dynamic constraint equations. As shown in Figure 1, $x_{3}$ and $x_{5}$ are obtained by $e_{8}$ and $e_{3}$ ; dynamic variables $d x_{3}$ and $d x_{5}$ are obtained by $e_{11}$ and $e_{13}$ . Finally, the consistency relation $r_{1}$ is obtained through $e_{5}$ :

r_{1} : = dd y_{3} + 8 d y_{3} - 15 y_{3} - u - d f_{1} - d f_{2} - 5 f_{1} - 5 f_{2} - f_{4}

(7)

Since $f_{1}$ and $f_{2}$ appear in $e_{3}$ , and $f_{4}$ appears in $e_{5}$ . When either fault in ${f_{1}, f_{2}, f_{4}}$ occurs, $r_{1} ≄ 0$ ; otherwise, $r_{1} \approx 0$ . From the computational procedure of $r_{1}$ , it can be seen that the solution of dynamic variables ( $d x_{3}$ , $d x_{5}$ ) requires only differential operations. Therefore, $r_{1}$ can also be called derivative causal consistency.

Figure 1.

Derivative causal consistency.

Figure 2.

Integral causal consistency.

As shown in Figure 2, the consistency relations $r_{2}$ and $r_{1}$ involve the same equations, but the order of calculation is different. $x_{3}$ is computed by $e_{8}$ ; $d x_{5}$ , $x_{5}$ , and $d x_{3}$ are obtained by $e_{5}$ , $e_{13}$ , and $e_{3}$ in sequence. $e_{13}$ and $e_{5}$ form a differential loop and can be broken by an integrator, that is, $x_{5} = \frac{u + f_{4}}{5} (1 - e^{- 5 x_{5}}) + C e^{- 5 x_{5}}$ , let $x_{5} = h (u, f_{4})$ . Finally, the consistency relation $r_{2}$ is obtained based on $e_{11}$ :

r_{2} = \frac{d}{dt} y_{3} + 3 y_{3} - h (u, f_{4}) - f_{1} - f_{2}

(8)

As can be seen from the computational flow of $r_{2}$ , the solution of the unknown variables involves only integral causality. Therefore, $r_{2}$ is called integral causal consistency.

Suppose differential and integral causation are involved in the computation of the consistency relation. In that case, it is a mixed causal consistency, and its computation process is similar to that of Figures 1 and 2, which will not be repeated here.

To facilitate the characterization of the causality effect on the diagnosability, let

\bar{A} = {\bar{A}}_{\cdot} \cup {\bar{A}}_{D} \cup {\bar{A}}_{I}

(9)

(1) ${\bar{A}}_{D}$ denotes the edge in the differential equation associated with $d x_{i}$ ;

(2) ${\bar{A}}_{I}$ denotes the edge in the differential equation associated with $x_{i}$ ;

(3) ${\bar{A}}_{\cdot}$ denotes the remaining edge in $\bar{A}$ .

The causal structural model can be expressed as $G (E, V, {\bar{A}}_{\cdot} \cup {\bar{A}}_{D} \cup {\bar{A}}_{I})$ . For example, $M_{1}$ has the structural model shown in Figure 3. The rows represent the equations; the columns represent the internal variables; the dots represent the corresponding edges, which indicate that the variables corresponding to the columns are in the equations corresponding to the rows. The blue dots represent the edges related to the unknown variables. “D” represents the edge of the differential equation associated with $d x_{i}$ , and “I” represents the edge associated with $x_{i}$ . Red dots represent edges related to faults. Black dots represent edges associated with known variables. The consistency relation is mainly concerned with the unknown variables. Therefore, the diagnosability analysis based on structural analysis is primarily oriented to $G (E, X, \bar{A})$ , as shown in the blue part of Figure 3.

Figure 3.

Causal structural model of $M_{1}$ .

Diagnosability evaluation based on structural analysis

The tools for structural analysis are matching and structural decomposition. The fault diagnosis principle is based on structural analysis, which analyzes the structural model to identify the subsystems that imply consistent relationships.⁴⁰ Diagnosability evaluation is performed based on the sensitivity between the consistency relations and the faults.

Given a structural model of the system $M$ as $G (E, X, \bar{A})$ , let $φ : 2^{M} \to Z$ be a mapping:

φ (M) = φ (G) = max_{M' = (E', X') \subseteq M} (| E' | - | X' |)

(10)

$φ (M)$ is the structural redundancy of $M$ . || represents the set’s base (the number of elements inside).²² If $φ (M) > 0$ , then there is a consistency relation in $M$ . The key objective of structural analysis is to identify the subsystem $M'$ that contains consistency relations in $M$ .

Fault diagnosis principles based on structural analysis

Dulmage-Mendelsohn decomposition

The Dulmage-Mendelsohn (DM) decomposition⁴¹ is a commonly used tool for structural analysis. It is transformed into an upper triangular form by ordering $G (E, X, \bar{A})$ appropriately, as shown in Figure 4.

Figure 4.

DM decomposition.

In Figure 4, the shaded part is “0 or 1,” and the blank part is “0.” Based on the DM decomposition, $M$ is divided into three subsystems: structural underdetermined subsystem $M^{-} = (E^{-}, X^{-})$ , which $| E^{-} | < | X^{-} |$ and $φ (M^{-}) < 0$ , structural exactly determined subsystem $M^{0} = (E^{0}, X^{0})$ , which $| E^{0} | = | X^{0} |$ and $φ (M^{0}) = 0$ ; and structural overdetermined subsystem $M^{+} = (E^{+}, X^{+})$ , which $| E^{+} | > | X^{+} |$ and $φ (M^{+}) > 0$ . Therefore, the consistency relation exists in the structural overdetermined subsystem $M^{+}$ . Thus, the structural redundancy can also be expressed as

φ (M) = φ (G) = | E^{+} | - | X^{+} |

(11)

$G (E, X, \bar{A})$ is decomposed into three subgraphs: structural underdetermined subgraph $G^{-} (E^{-}, X^{-}, {\bar{A}}^{-})$ , structural exactly determined subgraph $G^{0} (E^{0}, X^{0}, {\bar{A}}^{0})$ , and structural overdetermined subgraph $G^{+} (E^{+}, X^{+}, {\bar{A}}^{+})$ . A set of Hall components $G_{1}, G_{2}, \dots, G_{n}$ ⁴² exists in the exactly determined subgraph $G^{0}$ . And each Hall component $G_{i}$ is structurally exactly determined.

To simplify the definition, + and 0 are used here as the structural decomposition operations to obtain structural overdetermined and exactly determined subsystem. For example, $M_{1}$ undergoes DM decomposition to get $M_{1} = M_{1}^{+}$ .

Fault diagnosis principles based on structural analysis

Definition 1 (Consistency Relation) Given a system model $M$ , the set of equations $E^{r}$ constitutes a Consistency Relation ( $CR$ ) if $E^{r} \subseteq E^{+}$ and $φ ((E^{r}, Q (E^{r}))) > 0$ .

The consistency relation between $r_{1}$ and $r_{2}$ can be expressed as $C R_{1} = C R_{2} = {e_{3}, e_{5}, e_{8}, e_{11}, e_{13}}$ . Assume that the same fault does not appear in different equations. If there is a fault $f$ in equation $e$ , define it as $e_{f}$ . For example, the fault equation for $M_{1}$ is $e_{f_{1}} = e_{f_{2}} = e_{3}$ , $e_{f_{3}} = e_{4}$ , $e_{f_{4}} = e_{5}$ . Faults in $M^{+}$ are detectable. If $e_{f} \in E^{+}$ , then $f$ is detectable. A more minor consistency relation is contained within $E^{+}$ : $\exists CR' \subseteq E^{+}$ $s . t . 0 < φ ((CR', Q (CR'))) \leq φ (M^{+})$ .

Definition 2 (Minimum Consistency Relation) $CR$ is a Minimum Consistency Relation ( $MCR$ ) if and only if $φ ((CR, Q (CR))) = 1$ .

It can be seen that $C R_{1} = C R_{2} = {e_{3}, e_{5}, e_{8}, e_{11}, e_{13}}$ is $MCR$ . The fault diagnosis principle based on structural decomposition converts the sensitivity relation between the consistency relation and fault $f$ into the sensitivity relation between $MCR$ and $f$ . A fault $f_{i}$ is detectable if and only if

\exists C R_{j}, s . t . e_{f_{i}} \in C R_{j}

(12)

Fault $f_{i}$ can be isolated from $f_{j}$ if and only if

\exists C R_{k}, s . t . e_{f_{i}} \in C R_{k} \land e_{f_{j}} \notin C R_{k}

(13)

So, $MCR$ is advantageous in analyzing fault isolability.

Consistency relation based on matching

Matching is mainly used in structural analysis to determine the order in which unknown variables are solved. The principle of matching-based fault diagnosis suggests that the consistency relation is not obtained by algebraic elimination between the equations but by determining the matching. The matching equations are solved for the matching unknown variables and brought into the non-matching equations, yielding the consistency relation.

Matching

Given the system $G (E, X, \bar{A})$ , where $a = (e, x) \in \bar{A}$ . Define the mapping $p_{E}$ and $p_{Z}$ :

\begin{matrix} p_{E} : \bar{A} \to E \\ p_{E} (a) = e \end{matrix}

(14)

\begin{matrix} p_{X} : \bar{A} \to X \\ p_{V} (a) = x \end{matrix}

(15)

A matching $N$ is a subset of $\bar{A}$ :

\begin{matrix} \forall a_{1}, a_{2} \in N, s . t . \\ a_{1} \neq a_{2} \Rightarrow p_{E} (a_{1}) \neq p_{E} (a_{2}) \land p_{X} (a_{1}) \neq p_{X} (a_{2}) \end{matrix}

(16)

$N$ is a maximum matching if $\forall N_{i} \in 2^{\bar{A}}$ , $N \subset N_{i}$ , such that $N_{i}$ is not a matching. Matching $N$ is complete if $| N | = | E |$ or $| N | = | X |$ . A matching in $| N | = | E |$ is row complete matching, and a matching in $| N | = | X |$ is column complete matching. A matching in $| N | = | E | = | X |$ is perfect matching.²⁴ The matching set $N$ is only partially ordered, and system maximum matching is not unique—extensions to mappings $p_{E}$ and $p_{Z}$ :

\begin{matrix} π_{E} : N \to 2^{E} \\ π_{E} (N) = {e \in E | \exists a \in N, e = p_{E} (a)} \end{matrix}

(17)

\begin{matrix} π_{X} : N \to 2^{V} \\ π_{X} (N) = {x \in X | \exists a \in N, x = p_{X} (a)} \end{matrix}

(18)

The matching $N$ satisfies:

\forall N \in N, π_{E} (N) \subseteq E \land π_{X} (N) \subseteq X

(19)

| N | \leq min {| E |, | X |}

(20)

| π_{E} (N) | = | π_{X} (N) | = | N |

(21)

Complete matching search algorithms

For the convenience of description, the related operation of graph theory is given here. Let $G$ represent the structural graph, ${\bar{A}}_{2} \subseteq \bar{A}$ be the subset of edges, $E_{2} \subseteq E$ be the subset of equations, $X_{2} \subseteq X$ be the subset of variables, and $N$ be the matching:

(1) $G (E, X, \bar{A}) - {\bar{A}}_{2}$ represents the structural subgraph $G (E, X, \bar{A} \ {\bar{A}}_{2})$ .

(2) $G - X_{2}$ and $G - E_{2}$ are the structural subgraphs after $X_{2}$ and $E_{2}$ , and their corresponding edges are sequentially deleted from $G$ .

(3) $N_{E} (X_{2})$ and $N_{X} (E_{2})$ represent the sets of equations and unknown variables in $N$ corresponding to $X_{2}$ and $E_{2}$ , respectively.

(4) $G (E_{2})$ and $G (X_{2})$ represent the structural subgraphs of $E_{2}$ and $X_{2}$ , respectively.

Algorithm 1 is designed to solve for the maximum matching set. Lines 4–6 of Algorithm 1 shows that once a matching relation is determined, the structural model’s equation endpoints and variable endpoints will be removed accordingly, and the algorithm cuts off when no edges can be matched. Taking the structural subgraph $G_{1} ({e_{1}, e_{2}, e_{3}})$ of $M_{1}$ and applying Algorithm 1, the maximum matching set of $G_{1}$ can be obtained as ${(e_{1}, x_{1}), (e_{2}, x_{3})}$ , ${(e_{1}, x_{1}), (e_{2}, x_{2}), (e_{3}, x_{3})}$ , ${(e_{1}, x_{2}), (e_{2}, x_{3})}$ , ${(e_{1}, x_{2}), (e_{3}, x_{3})}$ . From the return result of Algorithm 1, it can be seen that the maximum matching is not unique.

Algorithm 1:

Match

≔FindMatch(

G (E, X, \bar{A})

)

Match

=Ø;

i = 1

;
for each edge

(e, x)

\bar{A}

Match (i) = Match (i) \cup (e, x)

;

G : = G - E ({e}) - X ({x})

\bar{A} (G) \neq Ø

\begin{matrix} Match (i) = Match (i) \cup \\ FindMath (G (E, X, \bar{A})) \end{matrix}

else

i = i + 1

;
end if
end for
Delete the same matching in

Match

Return

Match

The consistency relation is closely related to column complete matching. Based on Algorithms 1 and 2 is designed to solve the column complete matching. Algorithm 2 has a return value if and only if $φ (G) \geq 0$ and $G^{-} = Ø$ ; otherwise, the return value of Algorithm 2 is the empty set. From Algorithm 2, it can be seen that the column complete matching is not unique, which is determined by the matching characteristics. If required to fetch a row complete or perfect matching, change the judgment condition in line 7 of Algorithm 2 to $| N | = | E |$ or $| N | = | E | = | X |$ accordingly. Applying Algorithm 2 for the structural subgraph $G_{1}$ yields the column complete matching as ${(e_{1}, x_{1}), (e_{2}, x_{2}), (e_{3}, x_{3})}$ .

Algorithm 2:

Xmatch

≔ FindXMatch
(

G (E, X, \bar{A})

)

Match

:=FindMatch

(G (E, X, \bar{A}))

;

Xmatch = Ø

;

i = 1

;
for each match

N

in Mach do
if

| N | = | X |

Xmatch (i) = N

;

i = i + 1

;
end if
end for
Return

Xmatch

Minimum consistency relation search algorithm

Lemma 1 Given the system $G (E, X, \bar{A})$ , $φ (G) \geq 1$ , $N \subseteq \bar{A}$ is the column matching of $G^{+}$ . If $\exists e_{i} \in E \ π_{E} (N)$ , $s . t . Q_{X} (e_{i} \cup π_{E} (N)) \subseteq N_{X} (e_{i} \cup π_{E} (N))$ , then the set of equations ${e_{i} \cup π_{E} (N)}$ constitutes a consistency relation.

Proof: since $Q_{X} (e_{i} \cup π_{E} (N)) \subseteq N_{X} (e_{i} \cup π_{E} (N))$ then $\forall x \in Q_{X} (e_{i} \cup π_{E} (N))$ , $\exists e \in π_{E} (N)$ , $s . t . (e, x) \subseteq N$ , that is, $| Q_{X} (e_{i} \cup π_{E} (N)) | = | N |$ . Since $| {e_{i} \cup π_{E} (N)} | = | N | + 1$ , so $φ ({e_{i} \cup π_{E} (N)}, Q_{X} ({e_{i} \cup π_{E} (N)})) \geq 1$ . Hence, ${e_{i} \cup π_{E} (N)}$ forms a consistency relation.

Proof End. Then, the consistency relation of $G (E, X, \bar{A})$ can be expressed as $C R_{i} = {e_{i} \cup π_{E} (N)}$ . The structural redundancy of $M_{i} = (C R_{i}, Q (C R_{i}))$ is $φ (M_{i}) \geq 1$ . However, not all the equations in $π_{E} (N)$ are needed to compute the consistency relation. Therefore

\exists E_{min} \subseteq π_{E} (N), s . t . φ (((e_{i} \cup E_{min}), Q ((e_{i} \cup E_{min})))) = 1

(22)

Theorem 1 $E'$ is a minimal consistency relation if and only if a column complete matching $N$ is in the system $(E', Q_{X} (E'))$ , $E' = {(E')}^{+}$ , and $| E' | - | π_{X} (N) | = 1$ .

Proof: $N$ in $(E', Q_{X} (E'))$ , so $| π_{X} (N) | = | Q_{X} (E') |$ . Since $E' = {(E')}^{+}$ , so

\begin{matrix} φ (E', Q_{X} (E')) = | E' | - | Q_{X} (E') | \\ = | E' | - | π_{X} (N) | \\ = 1 \end{matrix}

(23)

Based on Definitions 4 and 5, it can be shown that this is $MCR$ .

Necessity. If $E'$ is $MCR$ , then $E' = {(E')}^{+}$ and $φ (E', Q_{X} (E')) = | E' | - | Q_{X} (E') | = 1$ . So $(E', Q_{X} (E'))$ is minimal structural overdetermined, and the number of equations in the system is one more than the number of unknown variables. So for $\forall x_{i}, x_{j} \in Q_{X} (E')$ , there are $e_{i} = R_{X} (x_{i}), e_{j} = R_{X} (x_{j})$ corresponding to them, and $e_{i} \neq e_{j}$ . For $\forall x_{i} \in Q_{X} (M')$ , there exists an ordered pair $(e_{i}, x_{i})$ . The ordered pairs form the set $N = {(e_{i}, x_{i})}$ . And $\forall a_{1}, a_{2} \in N$ , $p_{E} (a_{1}) \neq p_{E} (a_{2}) \land p_{E} (a_{1}) \neq p_{X} (a_{2})$ . So, $| N | = | Q_{X} (E') |$ , $N$ is the column complete matching. Since $| N | = | π_{X} (N) | = | π_{E} (N) |$ , so

\begin{matrix} | E' | - | π_{E} (N) | \\ = | Q_{X} (E') | + 1 - | N | \\ = 1 \end{matrix}

(24)

Proof End.

Lemma 2 $N$ is the column complete matching of $MCR$ . So

(1) The redundancy equation $e = MCR \ π_{E} (N)$ satisfies

Q_{X} (e) \subseteq π_{X} (N)

(25)

(2) The unknown variables in $N$ satisfies

Q_{X} (MCR) = π_{X} (N)

(26)

Based on Theorem 1, Lemma 2 is valid.

It can be seen that $MCR$ facilitates fault isolability analysis. However, there exist some $MCR$ that do not contain fault information. So, the operator $F$ is constructed to determine the availability of $MCR$ :

F (MCR) = {\begin{matrix} 1, \exists e_{f} \in MCR \\ 0, otherwise \end{matrix}

(27)

Design Algorithm 3 to solve the minimal consistency relation set ( $MCRs$ ) based on Theorem 1. $CR$ exists in the structural overdetermined part, so line 2 of Algorithm 3 obtains the structural overdetermined part by structural decomposition: $G = G^{+}$ . Based on Algorithm 2, line 3 of Algorithm 3 gets the column complete matching $N$ . Lines 5 and 6, based on condition (1) of Lemma 2, use each redundant equation $e$ in $E \ π_{E} (N)$ as a starting point to find the equation $N_{E} (Q_{X} (e))$ in $π_{E} (N)$ that matches the unknown variable $Q_{X} (e)$ in $e$ . In lines 7–9, based on condition (2) of Lemma 2, find the unknown variable $Q_{X} (E') \ N_{X} (E' \ e)$ that has not been matched in $E' = {e \cup N_{E} (Q_{X} (e))}$ and the equation $N_{E} (Q_{X} (E') \ N_{X} (E' \ e))$ that matches it. The final $MCR$ obtained is $MCR' = E' \cup N_{E} (Q_{X} (E') \ N_{X} (E' \ e))$ . It follows from the construction process of the $MCR'$ that $MCR' = {(MCR')}^{+}$ and ${MCR' \ e}$ are its column complete matching equations, so $| MCR' | = | Q_{X} (MCR') | + 1$ . For $\forall e' \in MCR'$ , ${MCR' \ e'}$ cannot form a consistency relation. Based on Theorem 1, $MCR'$ is the minimum consistency relation. To ensure the availability of $MCR$ , lines 10–12 of Algorithm 3 use the availability judgment criterion of equation (27) to obtain the $MCRs$ available to the system. Applying Algorithm 3, the $MCRs$ of $M_{1}$ can be obtained as ${e_{3}, e_{5}, e_{8}, e_{11}, e_{13}}$ , ${e_{2}, e_{4}, e_{5}, e_{7}, e_{8}, e_{10}, e_{12}, e_{13}}$ , ${e_{2}, e_{3}, e_{4}, e_{7}, e_{8}, e_{10}, e_{11}, e_{12}}$ , ${e_{2}, e_{3}, e_{4}, e_{5}, e_{7}, e_{10}, e_{11}, e_{12}, e_{13}}$ , ${e_{1}, e_{5}, e_{6}, e_{7},$ . $e_{9}, e_{13}}$ , ${e_{1}, e_{3}, e_{6}, e_{7}, e_{8}, e_{9}, e_{11}}$ , ${e_{1}, e_{2}, e_{4}, e_{6}, e_{7}, e_{8}, e_{9}, e_{10}, e_{12}}$ , ${e_{1}, e_{2}, e_{4}, e_{5}, e_{6}, e_{8}, e_{9}, e_{10}, e_{12}, e_{13}}$ , ${e_{1}, e_{2}, e_{3}, e_{4}, e_{6}, e_{9}, e_{10}, e_{11}, e_{12}}$ , ${e_{1}, e_{2}, e_{3}, e_{4}, e_{6}, e_{7}, e_{9}, e_{10}, e_{11}, e_{12}}$ , ${e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{9}, e_{10}, e_{11}, e_{12}, e_{13}}$ .

Algorithm 3:

MCRs

= FindMCRs

(G (E, X, \bar{A}))

G = G^{+}

;

MCRs : = Ø

;

j = 1

;

Xmatch

=FindXMatch(

G

);
for each matching

N

Xmatch

do
for each equation

e

E \ π_{E} (N)

MCRs (j) = {e \cup N_{E} (Q_{X} (e))}

;
While

Q_{X} (MCRs (j)) \neq N_{X} (MCRs (j) \ e)

\begin{matrix} MCRs (j) = MCRs (j) \cup \\ N_{E} (Q_{X} (MCRs (j)) \ N_{X} (MCRs (j) \ e)) \end{matrix}

end while
if

F (MCRs (j)) = 0

MCRs (j) = Ø

;
else

j = j + 1

;
end if
end for
end for
Delete the same elements in

MCRs

Return

MCRs

Computational complexity

This section focuses on analyzing the computational complexity of Algorithm 5. The number of $MCR$ increases exponentially with the structural redundancy. Structural redundancy tends to be related to the number of sensors configured in the system. It is generally low due to the price constraints of sensor configurations.

From Algorithm 3, it is clear that the $MCR$ is in the structural overdetermined part, and there is a direct link between the column complete of the structural overdetermined part and the structural redundancy $\bar{φ}$ . Assuming the structurally overdetermined part is $M^{+} = (E^{+}, X^{+})$ , the maximum number of column complete matching is

(\begin{matrix} | E^{+} | \\ | X^{+} | \end{matrix}) = \frac{(| E^{+} |)!}{(| X^{+} |)! \cdot (| E^{+} | - | X^{+} |)!} = \frac{(| E^{+} |)!}{(| X^{+} |)! \cdot \bar{φ}!}

(28)

From Lemma 1, a redundant equation plus a column complete matching is an $CR$ . The number of redundant equations is $\bar{φ}$ , so the maximum number of $MCR$ that can be constituted by each column complete matching is $\bar{φ}$ . The complexity of Algorithm 5 is

(\begin{matrix} | E^{+} | \\ | X^{+} | \end{matrix}) \cdot \bar{φ} = \frac{(| E^{+} |)!}{(| X^{+} |)! \cdot (\bar{φ} - 1)!} \leq \frac{(| E^{+} |)!}{(| X^{+} |)!} \leq {| E^{+} |}^{\bar{φ}}

(29)

Thus, for a given structural redundancy $\bar{φ}$ , the computational complexity of Algorithm 5 is of polynomial order, whereas the complexity of previous algorithms is at least exponential.

Consistency relations based on causal matching

Causal matching of hall components

$r_{1}$ and $r_{2}$ belong to the same $MCR = {e_{3}, e_{5}, e_{8}, e_{11}, e_{13}}$ of $M_{1}$ , but the causal relationship is different. The causal difference between $MCR$ in the consistency relationship calculation process depends on the following:

(1) The redundant equation for the final calculation of the consistency relation.

(2) The column complete matching in $MCR$ is utilized to solve for the unknown variables.

$e_{5}$ is used as the redundant equation for $r_{1}$ ; $e_{11}$ is used as the redundant equation for $r_{2}$ . The results of the DM decomposition of the matching relation used for $r_{1}$ and $r_{2}$ are shown in Figure 5(a) and (b), and the edges circled with black borders in the figure are the Hall components shown in Figure 4, that is, the matching relations used by the $MCR$ under the current causal constraints.

Figure 5.

Causal matching relationship of $M_{1}$ : (a) derivative causality, (b) integral causality, (c) mixed causality.

“int,”“der,” and “mix” denote integral, derivative, and mixed causality. After $MCR$ removes any equation $e \in MCR$ , $(MCR \ e, Q (MCR \ e))$ is exactly determined. The Hall component $G_{1}, G_{2}, \dots, G_{n}$ obtained after DM decomposition of the exactly determined system is unique. The perfect matching relation exists in the Hall component. For example, the exactly determined system shown in Figure 5(a) has the Hall components after DM decomposition as $G_{1} = {(e_{8}, x_{3})}$ , $G_{2} = {(e_{11}, d x_{3})}$ , $G_{3} = {(e_{3}, x_{5})}$ , $G_{4} = {(e_{13}, d x_{5})}$ . Its Hall component corresponds exactly to the column complete matching ${(e_{8}, x_{3}), (e_{11}, d x_{3}), (e_{3}, x_{5}), (e_{13}, d x_{5})}$ of $r_{1}$ . This also interprets the $r_{1}$ computation order in Figure 1. From Figure 5(a), it can be seen that $r_{1}$ contains only derivative causal matchings, so $r_{1}$ is derivative causality. The Hall components shown in Figure 5(b) are $G_{1} = {(e_{8}, x_{3})}$ , $G_{2} = {(e_{13}, x_{5}), (e_{13}, d x_{5}), (e_{5}, x_{5}), (e_{5}, d x_{5})}$ , $G_{3} = {(e_{3}, d x_{3})}$ . $| G_{2} | > 1$ illustrates that a differential loop, as shown in Figure 2, occurs when solving for $x_{5}$ and $d x_{5}$ . Integral causality can break the differential loop, so in the calculation of $r_{2}$ , the integral causality is adopted, that is, the column-matching relation adopted is $(e_{8}, x_{3}), (e_{5}, d x_{5}), (e_{13}, x_{5}), (e_{3}, d x_{3})$ . That is, $(e_{13}, x_{5}), (e_{5}, d x_{5})$ is adopted as the perfect matching relation for $G_{2}$ , that is, integral causality is used to compute the unknown variables, so $r_{2}$ is integral causality.

The causality of a consistency relation depends on which columns complete matching are used for its internal Hall components. The perfect matching $N_{int}$ of an integral causal Hall component satisfies $N_{int} \subseteq {\bar{A}}_{\cdot} \cup {\bar{A}}_{I}$ ; the perfect matching $N_{der}$ of a derivative causal Hall component satisfies $N_{der} \subseteq {\bar{A}}_{D}$ and $| N_{der} | = 1$ .

$G^{0}$ is integral causal if there exists a perfect matching $N_{int}$ for its Hall component $G_{i}$ :

N_{int} \subseteq {\bar{A}}_{\cdot} \cup {\bar{A}}_{I}

(30)

$G^{0}$ is derivative causal if there exists a perfect matching $N_{der}$ for its Hall component $G_{i}$ :

N_{der} \subseteq {\bar{A}}_{\cdot} \cup {\bar{A}}_{D}

(31)

if N_{der} \cap {\bar{A}}_{D} \neq 0, then | N_{der} | = 1

(32)

$G^{0}$ is mixed causal if a perfect matching $N_{mix}$ exists such that the Hall component of $G^{0} (N_{mix}) = G^{0} - E (G^{0}, N_{mix} \cap {\bar{A}}_{I}) - X (G^{0}, N_{mix} \cap {\bar{A}}_{I})$ is derivative causal.

For $MCR = {e_{3}, e_{5}, e_{8}, e_{11}, e_{13}}$ of $M_{1}$ , if $e_{3}$ is used as the redundancy equation, the matching relationship of Hall components is shown in Figure 5(c), which is mixed causality. Algorithm 4 is designed to determine the causality. Relying on derivative causality alone cannot break the differential loop, so only integral causality and mixed causality matching relationships can be used when $| G_{i} | > 1$ .

Algorithm 4:

Casual

≔ FindCasual(

MCR

)

HallCasual

= Ø;

Casual

= Ø;

i = j = 1

;

G = G (MCR)

for each equation

e

MCR

do
for each Hall component

G_{i}

MCR \ e

do
if

| G_{i} | > 1 \land \bar{A} (G_{i}) \cap ({\bar{A}}_{I} \cup {\bar{A}}_{D}) \neq Ø

HallCasual (i, j)

=‘mix’;
% ‘mix’ is der or int casual. So ‘mix’== ‘mix’ or ‘int’.
else if

| G_{i} | > 1 \land \bar{A} (G_{i}) \subseteq {\bar{A}}_{\cdot}

HallCasual (i, j)

=‘none’;
%‘none’ is no casual.
else if

| G_{i} | = 1 \land \bar{A} (G_{i}) \cap {\bar{A}}_{D} \neq Ø

HallCasual (i, j)

=‘der’;
else if

| G_{i} | = 1 \land \bar{A} (G_{i}) \cap {\bar{A}}_{I} \neq Ø

HallCasual (i, j)

=‘int’;
else if

| G_{i} | = 1 \land \bar{A} (G_{i}) \cap {\bar{A}}_{\cdot} \neq Ø

HallCasual (i, j)

=‘none’;
end if

j = j + 1

;
end for
if

HallCasual (i, :)

\‘none’==‘int’;

Casual (i)

=‘int’;
else if

HallCasual (i, :)

\‘none’==‘der’;

Casual (i)

=‘der’;
else

Casual (i)

=‘mix’;
end if

i = i + 1

;
end for
Return

Casual

$HallCasual (i, j)$ represents the causal matching of the jth Hall component $G_{j}$ of an exactly determined system $((MCR \ e_{i}), Q (MCR \ e_{i}))$ . And $Casual (i)$ represents the causality of the exactly determined system $((MCR \ e_{i}), Q (MCR \ e_{i}))$ . For example, applying Algorithm 4 yields the causality of $MC R_{1} = {e_{3}, e_{5}, e_{8}, e_{11}, e_{13}}$ as: {“mix,”“der,”“int,”“int,”“mix”}.

Causal consistency relational search algorithm

$MCR$ has different causality of consistency relations depending on the redundant equations chosen or the column complete matching. Based on Algorithms 3 and 4, Algorithm 5 is designed to find the $MCRs$ that imply causality in the system.

Algorithm 5 [

intMCRs

derMCRs

mixMCRs

]≔
FindMCRCasual

(G (E, X, {\bar{A}}_{\cdot} \cup {\bar{A}}_{I} \cup {\bar{A}}_{D}))

MCRs

=FindMCRs(

G

);

CasualMCRs

=Ø;

i = j = 1

;
for each

MCR

MCRs

CasualMCR (i, :)

=FindCasual(

MCR

);
if‘int’∈

CasualMCR (i, :)

intMCRs (j)

MCR

;
else if‘der’∈

CasualMCR (i, :)

derMCRs (j)

MCR

;
else if‘mix’∈

CasualMCR (i, :)

mixMCRs (j)

MCR

;
end if

i = i + 1

;

j = j + 1

;
end for
Return

intMCRs

derMCRs

mixMCRs

For $M_{1}$ , the causal $MCRs$ is obtained by applying Algorithm 5. For ease of description, the representation is based on the $MCRs$ number of $M_{1}$ obtained by applying Algorithm 3 in Section “Minimum consistency relation search algorithm”: $intMCRs$ =[1,2,4,5,8,11]; $derMCRs$ =[1,2,3,5,6,7,9]; $mixMCRs$ = $MCRs$ .

Diagnosability evaluation

Based on Algorithm 5, $MCRs = {MC R_{1}, MC R_{2}, \dots, MC R_{m}}$ under any causal constraints can be found. The system faults are $F = {f_{1}, f_{2}, \dots, f_{n}}$ , and based on Definition 6, the fault signature matrix $FSM = {[f m_{ij}]}_{n \times m}$ is established.

f m_{ij} = {\begin{matrix} 1, if e_{f_{i}} \in MC R_{j} \\ 0, otherwise \end{matrix}

(33)

A fault isolation matrix $FIM = {[f_{ij}]}_{n \times n}$ is built based on Definition 7

f_{ij} = {\begin{matrix} 1, if \exists MC R_{k}, s . t . e_{f_{i}} \in MC R_{k} \land e_{f_{j}} \notin MC R_{k} \\ 0, otherwise \end{matrix}

(34)

Based on FSM and FIM, the fault detection rate (FDR) can be obtained as

FDR = \frac{\sum_{i = 1}^{n} or (f m_{i \cdot})}{n}

(35)

Where $or (f m_{i \cdot})$ represents the logical or operation between the ith row of $FSM$ .

The fault isolation rate (FIR) is

FIR = \frac{\sum_{i = 1}^{n} f k_{i}}{\sum_{i = 1}^{n} or (f m_{i \cdot})}

(36)

Where $f k_{i}$ is

f k_{i} = {\begin{matrix} 1, if f_{ij} = 0 \land f_{ii} = 1 \\ 0, otherwise \end{matrix}

(37)

Therefore, a comprehensive diagnosability analysis can be performed based on structural analysis. According to equation (33), Algorithm 6 is designed to find the FSM. Based on equation (34), Algorithm 7 is designed to find the FIM.

Algorithm 6

FSM : =

FindFSM (

CRs

F

)
%

F

is the system faults set;

FSM = Ø

;

i = 1

;

j = 1

;
for each fault

f_{i}

F

do
for each

C R_{j}

CRs

do
if

e_{f_{i}} \in C R_{j}

FSM (i, j) = 1

;
else

FSM (i, j) = 0

;
end if

j = j + 1

;
end for

i = i + 1

;
end for
Return

FSM

Algorithm 7

IsoM : =

FindIsoM (

CRs

F

)

FSM

==FindFSM (

CRs

F

)
[m, n]=size(

FSM

);

IsoM

=ones(m, m); %
for each fault

f_{i}

f_{j}

F

do
for k=1:n do
if

FSM (i, k) = = 1 \land FSM (j, k) = = 0

IsoM (i, j) = 0

;
end if
end for
end for
Return

IsoM

Applying Algorithms 6 and 7, the fault signature matrix and fault isolation matrix of $r_{1}$ and $r_{2}$ are obtained as shown in Figure 6. The dots in the figure represent the “1” element of the corresponding matrix. From Figure 6(a), it can be seen that the $f_{1}$ , $f_{2}$ , $f_{4}$ are detectable and the fault $f_{3}$ is not detectable. So, $F D R = 75 %$ . From Figure 6(b), it can be seen that $f_{1}$ , $f_{2}$ , $f_{3}$ , $f_{4}$ cannot be isolated from each other. So, $FIR = 0$ .

Figure 6.

The diagnosability analysis of $r_{1}$ and $r_{2}$ : (a) fault signature matrix, (b) fault isolation matrix.

Diagnosability optimization design

Diagnosability optimization model

In the consistency relationship design, not all $MCR$ in the $MCRs$ are needed. So, minor integrated diagnosis costs must be preferred to meet the diagnosability requirements. The diagnosability optimization model is as follows:

Step 1: Given the system $G (E, X, {\bar{A}}_{\cdot} \cup {\bar{A}}_{I} \cup {\bar{A}}_{D})$ , find the $MCR s_{i},$ $i = int, der, mix$ under different causality based on Algorithm 5;

Step 2: Obtain fault signature matrix $FS M_{i}$ under different causality based on Algorithm 6.

Step 3: Obtain fault isolation matrix $Iso M_{i}$ under different causality based on Algorithm 7.

Step 4: Based on the dimension $n_{i}$ of $intMCRs$ , $derMCRs$ , $mixMCRs$ , build the corresponding optimization vector $x_{i} = (x_{i 1}, x_{i 2}, \dots, x_{i n_{i}})$ :

x_{ij} = {\begin{matrix} 1, if MC R_{j} is selected . \\ 0, otherwise \end{matrix}

(38)

Step 5: Based on Algorithm 8, establish the optimization matrix $X_{i}$ .

Step 6: The diagnosability optimization model is:

\begin{matrix} min & g (x_{i}) \end{matrix}

(39)

s . t . FindIsoM (FS M_{i} \cdot X_{i}, F) = Iso M_{i}

(40)

Algorithm 8:

X_{i} : =

FindX (

x_{i}

)
Let

E = [e_{1}, e_{2}, \dots, e_{n_{i}}]

be a

n_{i}

-dimensional unit matrix;

k = 1

;
for each

x_{ij}

x_{i}

do
if

x_{ij} = 1

y_{k} = e_{j}

;

k = k + 1

;
end if
end for

X_{i} = [y_{k}]

;
Return

X_{i}

In the above equation, $g (x_{i})$ is the cost of the preferred $MCRs$ , and if it wants to minimize the size of the preferred $MCRs$ , then $g (x_{i}) = | x_{i} |$ .

TSA-based diagnosability optimization

The diagnosability optimization problem is a complex NP problem. TSA is a heuristic algorithm miming the propagation between trees and seeds.

The population size of a tree population $T = (T_{1}, T_{2}, \dots, T_{N})$ is $N$ . Where the position $(T_{i 1}, T_{i 2}, \dots, T_{iD})$ of $T_{i}$ in $D$ -dimensional space is defined as

T_{ij} = L_{j, min} + r_{ij} (H_{j, max} - L_{j, min})

(41)

In the above equation, $T_{ij}$ is the position of $T_{i}$ in the $j$ -dimensional space; $r_{ij}$ is a random number in the range $[0, 1]$ ; $L_{j, min}$ is the lower bound of the $j$ -dimensional space; and $H_{j, max}$ is the upper bound of the $j$ -dimensional space. In TSA, the seed $S_{i}$ is generated from the tree $T_{i}$ , where the position update formula of the seed $S_{i}$ is

S_{ij} = T_{ij} + α_{ij} \times (Bes t_{j} - T_{rj})

(42)

S_{ij} = T_{ij} + α_{ij} \times (T_{ij} - T_{rj})

(43)

In the above formula, $α_{ij}$ is a random number in the range of $[- 1, 1]$ , which controls the size and direction of the seed position update step; $Bes t_{j}$ is the optimal position of $T$ in the $j$ -dimensional space; $T_{rj}$ is the random position of $T$ in the $j$ -dimensional space. According to the algorithm characteristics, the maximum number of tree seeds is 25% of the population, and the minimum is 10% of the population. Then, the steps of the $MCR$ optimization algorithm based on TSA are:

Step 1: Parameter initialization: number of populations $N = 500$ ; parameter $ST = 0.1$ ; spatial dimension $D = n_{i}$ ; maximum number of iterations $MaxFEs = 20$ ; $FEs = 1$ ;

Step 2: Based on the defined range of the $MCR$ optimization vector $x_{i} = (x_{i 1}, x_{i 2}, \dots, x_{i n_{i}})$ , use equation (41) to initialize the tree’s position in the search space randomly.

Step 3: Generate seeds for each tree $T_{i}$ in the population $T$ :

3.1 Initialize the seed tree $ns$ ;

3.2 Update the seed position based on equation (42) production when $rand < ST$ ; otherwise, update the seed position based on equation (43);

3.3 Determine whether the seed satisfies the constraints based on equation (40). If it meets, calculate the seed fitness value based on equation (39); otherwise, its fitness value is infinite;

3.4 $FEs = FEs + 1$ ;

Step 4: Prefer $N$ individuals with optimal fitness from the current population as a new population;

Step 5: If $FEs < MaxFEs$ , repeat Step 3 and Step 4; otherwise, end the algorithm and return the optimal tree.

Fixed-wing UAV diagnosability analysis

Fixed-wing UAV structural model

The fixed-wing UAV model can be decomposed into a longitudinal motion that includes airspeed, pitch angle, and altitude and a lateral motion that provides for roll angle and heading angle. Although the longitudinal and lateral motions are coupled, for most airframes, the effects of this kinetic coupling are minimal, and unwanted effects can be mitigated by control algorithms that suppress perturbations.⁴³ The concerns for the lateral dynamics model are roll angle, roll rate, heading angle, and yaw rate. The control surfaces for the lateral dynamics are the ailerons and the directional wing. The ailerons are mainly used to control the roll rate, while the directional wings are used primarily to control the yaw angle.⁴⁴ A fixed linearized model is established by considering small changes in the selected state trajectories of the nonlinear system. The linear model $M_{uav}$ of the lateral system⁴⁵ is

\begin{matrix} dx = A x + B u + F \\ y = C x \end{matrix}

(44)

$x = {[β, p, r, ϕ, ψ, δ_{a}, δ_{r}]}^{T}$ is the system state: $β$ is the sideslip angle, $p$ is the roll speed, $r$ is the yaw speed, $ϕ$ is the roll angle, $ψ$ is the yaw angle, $δ_{a}$ is the rudder angle, and $δ_{r}$ is the aileron angle. $u = {[u_{1}, u_{2}]}^{T}$ is the system input: $u_{1}$ is the rudder angle input, $u_{2}$ is the aileron input. $F = {[f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}]}^{T}$ is system faults.

A = [\begin{matrix} - 0.277 & 0 & - 32.9 & 9.81 & 0 & - 5.432 & 0 \\ - 0.103 & - 8.532 & - 0.102 & 0 & 0 & 0 & - 28.64 \\ 0.365 & 0 & - 0.639 & 0 & 0 & - 9.490 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - 10 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 5 \end{matrix}]

(45)

B = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 20 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 10 \end{matrix}]}^{T}

(46)

C = [\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]

(47)

According to the parameter matrix of the system, the trim value of $M_{uav}$ can be obtained as $x_{trim} = {[0, 0, 0, 0, 0, 0, 0]}^{T}$ , $u_{trim} = {[0, 0]}^{T}$ , $y_{trim} = {[0, 0, 0]}^{T}$ , $d x_{trim} = {[0, 0, 0, 0, 0, 0, 0]}^{T}$ ; its eigenvalue is $λ = [0, - 8.5, 0, - 0.4 + 3.5 i, - 0.4 + 3.5 i, - 10, - 5]$ . The structural model of $M_{uav}$ and the result of DM decomposition are shown in Figure 7.

Figure 7.

Structural analysis of $M_{uav}$ : (a) structural model, (b) DM decomposition.

From Figure 7(a), the constraint equation $d x_{i} = \frac{d}{dt} x_{i}, i = 1, 2, 3, 4, 5, 6, 7$ has been added to facilitate the causal analysis. From Figure 7(b), it can be seen that $M_{uav} = M_{uav}^{+}$ . It can be seen that the system faults $f_{i}, i = 1, 2, 3, 4, 5, 6, 7$ are all detectable. However, the diagnosability of $M_{uav}$ under different causal conditions needs to be further analyzed.

Causal diagnosability analysis

Algorithm 3 is applied to obtain the $MCRs$ of $M_{uav}$ as:

\begin{matrix} MC R_{1} = {e_{3}, e_{5}, e_{6}, e_{8}, e_{10}, e_{13}, e_{15}, e_{16}} \\ MC R_{2} = {e_{2}, e_{4}, e_{5}, e_{7}, e_{8}, e_{9}, e_{10}, e_{12}, e_{14}, e_{15}, e_{17}} \\ MC R_{3} = {e_{2}, e_{3}, e_{4}, e_{7}, e_{8}, e_{9}, e_{10}, e_{12}, e_{13}, e_{14}, e_{16}, e_{17}} \\ MC R_{4} = {e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{9}, e_{10}, e_{12}, e_{13}, e_{14}, e_{15}, e_{16}, e_{17}} \\ MC R_{5} = {e_{1}, e_{5}, e_{6}, e_{8}, e_{9}, e_{10}, e_{11}, e_{15}, e_{16}} \\ MC R_{6} = {e_{1}, e_{3}, e_{6}, e_{8}, e_{9}, e_{11}, e_{13}, e_{16}} \\ MC R_{7} = {e_{1}, e_{3}, e_{5}, e_{8}, e_{9}, e_{10}, e_{11}, e_{13}, e_{15}} \\ MC R_{8} = {e_{1}, e_{3}, e_{5}, e_{6}, e_{9}, e_{10}, e_{11}, e_{13}, e_{15}, e_{16}} \\ MC R_{9} = {e_{1}, e_{2}, e_{4}, e_{6}, e_{7}, e_{8}, e_{9}, e_{11}, e_{12}, e_{14}, e_{16}, e_{17}} \\ MC R_{10} = {e_{1}, e_{2}, e_{4}, e_{5}, e_{6}, e_{7}, e_{9}, e_{10}, e_{11}, e_{12}, e_{14}, e_{15}, e_{16}, e_{17}} \\ MC R_{11} = {e_{1}, e_{2}, e_{4}, e_{5}, e_{6}, e_{7}, e_{8}, e_{10}, e_{11}, e_{12}, e_{14}, e_{15}, e_{16}, e_{17}} \\ MC R_{12} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{7}, e_{8}, e_{9}, e_{11}, e_{12}, e_{13}, e_{14}, e_{17}} \\ MC R_{13} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{6}, e_{7}, e_{9}, e_{11}, e_{12}, e_{13}, e_{14}, e_{16}, e_{17}} \\ MC R_{14} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{6}, e_{7}, e_{8}, e_{11}, e_{12}, e_{13}, e_{14}, e_{16}, e_{17}} \\ MC R_{15} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{7}, e_{9}, e_{10}, e_{11}, e_{12}, e_{13}, e_{14}, e_{15}, e_{17}} \\ MC R_{16} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{7}, e_{8}, e_{10}, e_{11}, e_{12}, e_{13}, e_{14}, e_{15}, e_{17}} \\ MC R_{17} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}, e_{10}, e_{11}, e_{12}, e_{13}, e_{14}, e_{15}, e_{16}, e_{17}} \end{matrix}

(48)

To verify the performance of Algorithm 3, it is simulated and compared with the minimum consistency relation search algorithms of literature⁴⁶ and literature⁴⁷ mentioned in the introduction, and the simulation results are shown in Table 1, taking the simulation time of the algorithms as an index. It can be seen that Algorithm 3 is 1400 times faster compared to the algorithm shown in the literature⁴⁶ and 1500 times faster compared to the algorithm shown in the literature.⁴⁷

Table 1.

Performance comparison of minimum consistency relationship algorithms.

Algorithm	Execution time (s)
Algorithm 3	0.01
Algorithm of literature⁴⁵	142.21
Algorithm of literature⁴⁶	159.46

The $MCRs$ of $M_{uav}$ under different causality is obtained by applying Algorithm 5, as shown in the second column of Table 2. The fault signature matrix under different causality is analyzed using Algorithm 6, as shown in Figure 8. The fault isolation matrix under different causalities using Algorithm 7 is analyzed in Figure 9. The results of the diagnosability evaluation of UAVs under different causal constraints are shown in columns 3 and 4 of Table 2. Figure 9 shows that the fault isolation characteristics under mixed causality are the best: the faults $f_{1}, {f_{2}, f_{4}, f_{7}}, f_{3}, f_{5}, f_{6}$ are isolated from each other, but the faults in ${f_{2}, f_{4}, f_{7}}$ are not isolated from each other. Secondly, the isolability of derivative causation is stronger than integral causation. Under the integral causality, no fault can be isolated from $f_{3}$ and $f_{6}$ . Under derivative causality, more than mixed causality, no fault can be isolated from $f_{5}$ .

Table 2.

Casual $MCRs$ .

Causality	$MCRs$	FDR (%)	FIR (%)
Integral	$MC R_{1}$ $MC R_{3}$ $MC R_{6}$ $MC R_{8}$ $MC R_{13}$ $MC R_{14}$ $MC R_{17}$	100	28.57
Derivative	$MC R_{1}$ $MC R_{2}$ $MC R_{5}$ $MC R_{7}$ $MC R_{16}$	100	42.86
Mixed	$MC R_{1}$ , $MC R_{2}$ , $MC R_{3}$ , $MC R_{4}$ , $MC R_{5}$ , $MC R_{6}$ , $MC R_{7}$ , $MC R_{8}$ , $MC R_{9}$ , $MC R_{10}$ , $MC R_{11}$ , $MC R_{12}$ , $MC R_{13}$ , $MC R_{14}$ , $MC R_{15}$ , $MC R_{16}$ , $MC R_{17}$ ,	100	57.14

Figure 8.

Fault signature matrix: (a) integral causality, (b) derivative causality, (c) mixed causality.

Figure 9.

Fault isolation matrix: (a) integral causality, (b) derivative causality, (c) mixed causality.

TSA-based diagnosability optimization design

To verify the performance of TSA, the sine cosine algorithm,⁴⁸ the genetic algorithm,⁴⁹ and the simulated annealing algorithm⁵⁰ are selected for comparative verification. The population size of the algorithm is 500, the number of iterations is 100, and the rest of the simulation parameters are consistent with the corresponding references. The test functions are

F_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}

(49)

F_{2} (x) = \sum_{i = 1}^{n} i \cdot x_{i}^{4} + rand

(50)

$F_{1} (x)$ is a single-peak function, mainly to verify the optimization speed of the algorithm; $F_{2} (x)$ is a multi-peak function, primarily to confirm the convergence accuracy of the algorithm as well as the global convergence ability. The extreme values of both $F_{1} (x)$ and $F_{2} (x)$ are zero. Figure 10(a) and (c) show the functions’ images in the two-dimensional space. The dimensions of $F_{1} (x)$ and $F_{2} (x)$ in the simulation process are 30, and the range of independent variables of $F_{1} (x)$ is [−100, 100], and that of $F_{2} (x)$ is [−1.28, 1.28]. The fitness curves of $F_{1} (x)$ and $F_{2} (x)$ are shown in Figure 10(b) and (d). The simulation results show that TSA is better than other algorithms in terms of optimization search speed and global convergence ability.

Figure 10.

The performance validation of TSA: (a) $F_{1} (x)$ , (b) the fitness curves of $F_{1} (x)$ , (c) $F_{2} (x)$ , (d) the fitness curves of $F_{2} (x)$ .

The minimum $MCRs$ under mixed causality is obtained based on TSA. The fitness function is $g (x_{i}) = | x_{i} |$ . The fitness curve is shown in Figure 11(a), the fault signature matrix of the preferred $MCRs$ is shown in Figure 11(b), and the fault isolation matrix is shown in Figure 11(c).

Figure 11.

TSA-based diagnosability optimization results: (a) fitness curve, (b) fault signature matrix, (c) fault isolation matrix.

Figure 11(a) shows that the algorithm’s ability to find the solution is significantly more substantial than the other three algorithms, and the optimal solution is found at the beginning of the algorithm iteration. The TSA algorithm is optimized to obtain the number of $MCR$ as 5. Thus, it can be seen that the cost of designing the system’s coherent relationship has been reduced by 71%. Figure 11(b) shows that the preferred $MCRs$ is ${MC R_{1}, MC R_{2}, MC R_{3}, MC R_{7}, MC R_{9}}$ . Figure 11(c) shows that the fault isolation characteristics of the preferred $MCR$ are in line with the fault isolation characteristics shown in Figure 9(c). Therefore, the TSA-based diagnosability optimization results satisfy the system’s fault isolation requirements and ensure optimal diagnosability with minimal MCRs. However, the TSA-preferred MCRs are not unique; the set of MCRs can also be:

\begin{matrix} {MC R_{2}, MC R_{3}, MC R_{5}, MC R_{7}, MC R_{9}} \\ {MC R_{4}, MC R_{8}, MC R_{10}, MC R_{13}, MC R_{15}} \\ {MC R_{2}, MC R_{3}, MC R_{6}, MC R_{7}, MC R_{10}} \\ {MC R_{1}, MC R_{2}, MC R_{5}, MC R_{7}, MC R_{14}} \\ {MC R_{4}, MC R_{5}, MC R_{7}, MC R_{9}, MC R_{12}} \\ ⋮ \end{matrix}

(51)

The minimum $MCRs$ is not unique and can be further optimized based on the diagnosability requirements. Now, take ${MC R_{1}, MC R_{2}, MC R_{3}, MC R_{7}, MC R_{9}}$ as an example to design the consistency relation ${r_{1}, r_{2}, r_{3}, r_{4}, r_{5}}$ . Since the matching relation implies the computation order of the unknown variables, computing the consistency relation can be seen from the matching relation. The mixed causal matching relation of ${r_{1}, r_{2}, r_{3}, r_{4}, r_{5}}$ is shown in Figure 12.

Figure 12.

Mixed causal matching: (a) $r_{1}$ , (b) $r_{2}$ , (c) $r_{3}$ , (d) $r_{4}$ , (e) $r_{5}$ .

The simulation results of $M_{uav}$ are shown in Figure 13 when there is no fault.⁵¹ And ref1 and ref2 are the reference input signals. The LQ controller is developed to maintain the stability of the system’s sideslip angle. It is assumed that only one fault occurs at a time. When $f_{1}$ occurs, the simulation results are shown in Figure 14.

Figure 13.

Fault-free simulation results.

Figure 14.

Simulation results when $f_{1}$ occurs.

When each fault occurs, the simulation results of the consistency relationship are shown in Figure 15. It can be seen that $r_{1}$ is sensitive to the faults ${f_{3}, f_{5}, f_{6}}$ ; $r_{2}$ is sensitive to the faults ${f_{2}, f_{4}, f_{5}, f_{7}}$ ; $r_{3}$ is sensitive to the faults ${f_{2}, f_{3}, f_{4}, f_{6}, f_{7}}$ ; $r_{4}$ is sensitive to the faults ${f_{1}, f_{5}}$ ; $r_{5}$ is sensitive to the faults ${f_{1}, f_{2}, f_{4}, f_{7}}$ . It can be seen that the degree of sensitivity of ${r_{1}, r_{2}, r_{3}, r_{4}, r_{5}}$ to faults is approximately the same as that of the fault signature matrix shown in Figure 11(b). The difference is that the sensitivity of $r_{4}$ to $f_{3}$ and the sensitivity of $r_{5}$ to $f_{6}$ are lower. This is related to the unknown inputs, the noise, the faults’ magnitude, and the dynamic variables’ causality.⁵²

Figure 15.

Consistency simulation results.

Effect of causality on diagnosability

Based on Table 2, it can be seen that $MC R_{1}$ contains both integral causality, differential causality, and mixed causality. Applying Algorithm 4 yields the causality of $MC R_{1}$ as {“mix,”“mix,”“der,”“mix,”“int,”“mix,”“int,”“der”}. When $e_{3}$ is selected as the redundancy equation, the matching relationship is shown in Figure 12(a), which can be seen as a mixed causal relationship. Here, $e_{6}$ and $e_{10}$ are adopted sequentially as redundant equations, and their matching relations are shown in Figure 16. Figure 16(a) shows that $MC R_{1}$ with $e_{6}$ as the redundant equation is differential causal; Figure 16(b) indicates that $MC R_{1}$ with $e_{10}$ as the redundant equation is integral. The simulations are performed separately, and the simulation results are shown in Figure 17.

Figure 16.

Matching relations of $MC R_{1}$ : (a) differential causal, (b) integral causal.

Figure 17.

Impact of causality on $r_{1}$ : (a) simulation results of $r_{1}$ under different causal relationships, (b) sensitivity of $r_{1}$ to $f_{5}$ for different causality.

For ease of description, $mr 1$ , $dr 1$ , and $ir 1$ are used to represent the mixed, differential, and integral causal consistency relation in turn. As shown in Figure 17(a), $mr 1$ , $dr 1$ , and $ir 1$ simulation results under different fault modes are displayed from top to bottom. It is obvious that $MC R_{1}$ under different causal relationships is still guaranteed to be sensitive to the faults ${f_{3}, f_{5}, f_{6}}$ , but its fault sensitivity is different. Under differential and mixed causality, when a fault occurs, the jumps of $mr 1$ and $dr 1$ are similar but in opposite directions. Under integral causation, the jump direction of $ir 1$ is identical to that of $mr 1$ , but the jump degree is much larger than that of the other two causality. When $f_{5}$ occurs, the simulation results of $mr 1$ , $dr 1$ , and $ir 1$ are shown in Figure 17(b), which shows that the jumps of $mr 1$ , $dr 1$ , and $ir 1$ are pretty different, and the sensitivity of $ir 1$ to $f_{5}$ is the largest. Therefore, causality has an impact on the design of consistency relationships. This further validates the rationality and validity of the study in this paper.

Conclusion

To address the growing need for reliability, this paper proposes a fixed-wing UAV diagnosability optimization strategy based on causal matching and TSA to improve reliability and diagnosability. First, a series of algorithms are designed to analyze the diagnosability under different causal constraints qualitatively. The algorithm design process is as follows: in the first step, design Algorithm 1 to find the maximum matching of the system; in the second step, design Algorithm 2 to find the complete matching based on Algorithm 1; in the third step, design Algorithm 3 to find the minimum consistency relation based on Algorithm 2, with due consideration to the MCR’s availability; The computational complexity of Algorithm 3 is polynomial level, which is a significant improvement over the previous exponential-level algorithms. It is more than 1400 times faster than other algorithms in terms of optimization speed. In the fourth step, Algorithm 4 is designed to determine the causality implied by the MCR when the MCR selects any equation as a redundant equation. In the fifth step, Algorithm 5 is designed based on Algorithms 3 and 4 to find the MCRs under a particular causality. Based on the MCRs obtained by Algorithms 5 –7 are designed to get the fault characterization matrix and fault isolation matrix and analyze the performance qualitatively under different causality constraints. Detectability and isolation under different casualties make up for the lack of current research on causal diagnosability.

Secondly, based on the TSA, the diagnosability optimization design model under different causal constraints is established to consider the cost of consistent design and the demand for diagnosability. Finally, for the fixed-wing UAV model, based on Algorithm 5, the MCR under different causal constraints is obtained, and the diagnosability of the UAV under different causal conditions is qualitatively analyzed. Based on the fixed-wing UAV diagnosability optimization model, the TSA algorithm is used to prefer the set of MCRs with the minimum integrated diagnosis cost. From the simulation results, it can be seen that five MCR are optimized from the original set of MCRs, and the preferred five MCR can achieve the diagnosability performance achieved by the original 17 MCR, which dramatically reduces the design cost of the consistency relation. In addition, to further verify the reasonableness of the diagnosability optimization results, the system residuals are designed based on the MCRs. The rationality of the diagnosability analysis based on causal matching is also further verified by the sensitivity relationship between residuals and faults.

In this paper, the diagnosability of the system is analyzed and optimally designed from a qualitative perspective only. Therefore, in the future, the diagnosability of the system can be analyzed and optimally designed from a quantitative perspective. Secondly, based on the fault diagnosis principle of structural analysis, the structural redundancy of the system can be increased by sensor configuration to achieve diagnosability analysis of the faults in the exactly determined part. Exploring the diagnosability in the structurally underdetermined part of the system is also crucial.

Footnotes

Author contributions

Concept and design: Xian-Jun SHI and Xu-ping GU; data collection and analysis: Xu-ping GU; drafting of the article: Xu-ping GU; critical revision of the article for important intellectual content: Xu-ping GU and Xian-Jun SHI; study supervision: Xian-Jun SHI. All the authors approved the final article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

XuPing Gu

Data availability statement

All data that support the findings of this study are included in this manuscript and its supplementary information files.

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