Reconfigurability-optimal design of fixed-wing UAV based on structurally redundant sensor configuration and improved cuckoo search algorithm

Abstract

To enhance the reliability and fault reconfiguration capability of Unmanned Aerial Vehicles (UAV), this study proposes a reconfigurable optimal design method based on a structurally redundant sensor configuration and an improved cuckoo search algorithm (ICSA). First, the precondition for structural reconfigurability is defined, and a fault reconfiguration matrix, along with a reconfigurability evaluation index, is established to assess system reconfigurability qualitatively. Next, a sensor configuration method is developed to improve structural redundancy, and a minimum structural overdetermined search algorithm is designed to achieve a reconfigurable system. Furthermore, the ICSA enhances the convergence speed and optimization accuracy of the conventional cuckoo search algorithm (CSA) by incorporating a genetic algorithm mutation strategy and adaptive parameter tuning. Based on ICSA, a reconfigurable optimal design strategy is proposed to minimize system design costs while meeting reconfigurability requirements. Simulations on a fixed-wing UAV model validate the rationality and effectiveness of the proposed method. The results demonstrate that the structurally redundant sensor configuration achieves a 100% fault detection rate, 97.14% fault isolation rate, and 96.67% fault reconfiguration rate. Additionally, the ICSA-based approach reduces reconfigurable system design costs by 80% overall and by 8% compared to the unimproved algorithm.

Keywords

UAV fault structural analysis reconfigurability cuckoo search algorithm

Introduction

Under the influence of the fresh era of technical innovation and industrial change represented by the Internet, artificial intelligence, new materials, etc., countries all over the world are actively increasing their investment in exploring and applying High-tech innovations, like artificial intelligence, autonomous synergy, and resilient networking, to promote the research and development and innovation of Unmanned Aerial Vehicles (UAV).¹ At present, UAVs have been developed to different degrees in arms,² civil,³ farming,⁴ and police⁵ fields. As the structure and function of UAVs become more and more complex, a local fault often produces a chain reaction, leading to the collapse of the whole system, which causes economic losses and affects mission success. Therefore, reconfiguration control technology must significantly improve the UAVs’ safety and reliability.

Reconfigurability indicates the system’s autonomous capacity to reconfigure itself from faults.⁶ Reconfigurability in the control domain can be subdivided into: first, the ability to restore the system performance through fault-tolerant control after a fault occurs; second, the ability to enable the system performance to be restored after a fault occurs through fault regulation and system reconfiguration operations; and third, if the system stays controllable and observable when a fault happens, it is reconfigurable. Reconfigurability can be divided into reconfigurability evaluation and reconfigurability design. Depending on the evaluation elements, reconfigurability evaluation can be divided into approaches depending on the system’s intrinsic properties, approaches depending on the system performance constraints, approaches depending on the system’s functional requirements, and other approaches. The intrinsic characteristic-based methods measure reconfigurability by establishing the relationship between controllability, observability, stability, or other system characteristics and faults. Yang et al.⁷ analyzed the controllability, observability, stabilizability, and detectability of nonlinear systems based on Lie algebra and established a link between control redundancy and reconfigurability. The approach based on the functional properties measures the reconfigurability by establishing the link between the system’s energy, input-output, time constraints, and other practical performance constraints and the reconfigurability. Steffen⁸ classified the reconfigurability goals into stability, weak, strong, direct, and fault-hiding goals based on the different control goals. He gave sufficient conditions for linear systems under each reconfigurability objective. Yang⁹ proposed a reconfigurability evaluation method for switching systems based on energy constraints. Existing reconfigurable design methods are summarized as pseudo-inverse or control mixers, linear quadratic regulators,¹⁰ gain scheduling or linear parameter variation, adaptive control or model following,¹¹ feature structure assignment, feedback linearization or dynamic inversion, robust control,¹² model predictive control, variable structure or sliding mode control,¹³ intelligent control using expert systems, neural networks, and fuzzy logic for intelligent control.¹⁴ Reconfigurability evaluation and reconfigurable design are complementary. Based on the results of reconfigurability evaluation, for reconfigurable faults, corresponding fault-tolerant control strategies are adopted to reconfigure the faults. The system structure can be further optimized for non-reconfigurable faults to ensure they are reconfigurable. On the one hand, a reconfigurable design strategy with a high degree of complexity can be selected for faults that are more difficult to reconfigure. On the other hand, the system structure can be further optimized. The reconfigurable strategy with low cost is preferred for faults with less difficulty in reconfiguring.

From the current research progress on reconfigurability, it can be seen that, firstly, it’s necessary to create an accurate mathematical model of the system, and the computational amount of both reconfigurability evaluation and reconfigurability design is vast. Secondly, the reconfigurability design is for fault-tolerant control strategies and seldom adjusts the overall optimization of the system reconfigurability. Structural analysis obtains system-related information from the structural model.¹⁵ Structural analysis has been developed to a certain extent regarding controllability, observability, and diagnosability. Schmid et al.¹⁶ established a system structural model for critical defects in battery systems, changed the fault diagnostic issue into a minimal structural overdetermination problem, and offered adequate criteria for fault structural detectability and structural isolation. Frisk et al.¹⁷ proposed a diagnosability design strategy for critical engine faults based on structural analysis. Reissig et al.¹⁸ proposed sufficient conditions for strong structural controllability and observability based on structural analysis. In the framework phase of design, the reconfigurability analysis and design based on structural analysis are vital in enhancing overall dependability and safety. Firstly, it does not require establishing an accurate mathematical model, which is advantageous for researching the reconfigurability of large-scale and complex control systems. Secondly, it adopts the relevant tools based on graph theory, which avoids complex numerical operations. Therefore, this work establishes the system’s structural model based on structural analysis, gives the structural reconfigurability criterion, establishes the sensitive connections between the reconfigurability and the minimum structural overdetermination(MSO), and qualitatively analyses the structural reconfigurability based on the fault reconfiguration matrix.

The MSO is closely related to the structural redundancy and the structural reconfigurability. Therefore, this paper increases the structural redundancy through structurally redundant sensor configurations to improve reconfigurability, increasing the number of MSO. Secondly, the MSO search algorithm is designed for reconfigurable system design. Since the amount of MSO rises dramatically with the system structural redundancy, this work provides a reconfigurable optimal design methodology based on ICSA to consider the reconfigurability requirements, diagnosability requirements, and the reconfigurable system design cost. Finding the reconfigurable system with the best overall performance from the minimum structured overdetermination set(MSOs) of systems is a complex combinatorial optimization issue. Intelligent optimization methods provide substantial advantages in tackling combinatorial optimization issues. The standard bionic intelligent optimization algorithms are ant colony algorithm, hybrid frog hopping algorithm, artificial bee colony algorithm, artificial fish swarming algorithm, firefly algorithm, grey wolf optimization algorithm, fruit fly optimization algorithm, genetic algorithm, and other intelligent optimization algorithms.^19,20 The cuckoo search algorithm (CSA) is a new heuristic brilliant optimization method based on cuckoos’ nest-seeking and egg-laying behavior combined with the Lévy flight of birds.²¹ It has gained wide attention because its algorithm is simple and easy to implement, has fewer parameters, does not easily fall into local optimum, and does not require many parameters to solve complex problems. Currently, CSA is widely used in system reliability optimization,²² neural network training,²³ and 3D path planning.²⁴ To further increase the convergence speed and the accuracy of CSA, this work offers a reconfigurable optimum design method based on the ICSA by optimizing and designing based on CSA.

This work provides a reconfigurability optimization method based on structurally redundant sensor configurations and ICSA to improve the UAVs’s reliability and fault reconfiguration capability. The approach considers the reconfigurability requirements while minimizing the reconfigurable system design cost. The innovations of this work are as follows:

Sufficient conditions for structural reconfigurability are given, and the fault reconfiguration matrix, as well as a reconfigurability evaluation index, are established to analyze the reconfigurability qualitatively;

A structural redundant sensor configuration algorithm is designed to improve the structural redundancy, and a reconfigurable MSO search algorithm is designed;

ICSA and a reconfigurability optimization design strategy based on ICSA are proposed.

The remaining parts are arranged as follows: Section Two introduces the basic principles of structural analysis in detail; Section Three proposes a qualitative evaluation method for the structural reconfigurability; Section Four designs a structural redundant sensor configuration algorithm as well as a reconfigurable MSO search algorithm; Section Five improves CSA and proposes an optimized design strategy for reconfigurability based on ICSA; Section Six, based on the fixed-wing UAV structural model, the reasonableness, and efficacy of the strategy suggested in this research are simulated and verified from the perspective of system optimal design; Section Seven, summarizes the main work of this paper.

Structural analysis

Structural modelling

The system analytical model $M$ can be represented by the constraint equation set $E$ and the variable set $V$ of $M$ , that is, $M = (E, V)$ . The variables $V$ can be divided into known variables $Z$ , unknown variables $X$ , that is, $V = Z \cup X$ . For example, the mathematical model of $M_{1}$ is

\begin{matrix} e_{1} : d x_{1} = - x_{2} + x_{3} + f_{1} \\ e_{2} : d x_{2} = - 2 x_{1} + x_{3} + f_{2} \\ e_{3} : d x_{3} = - 3 x_{1} + u + f_{3} \\ e_{4} : y_{1} = x_{1} + f_{4} \\ e_{5} : y_{2} = x_{2} + f_{5} \end{matrix}

(1)

$d x_{i}$ represents the time differentiation of $x_{i}$ , $E_{1} = {e_{1}, e_{2}, e_{3}, e_{4}, e_{5}}$ represent the constraint equations, $X_{1} = {x_{1}, x_{2}, x_{3}}$ are the state variables, $F_{1} = {f_{1}, f_{2}, f_{3}, f_{4}, f_{5}}$ are the faults, $Y_{1} = {y_{1}, y_{2}}$ are the sensor measurement variables, and $u$ is the control variable.

Structural analysis focuses on obtaining system-relevant information from its structural model. The structural model ignores the algebraic relationships between the system variables and focuses on the logical mapping relationships between the equations and the variables. The structural model can be represented by a bipartite graph $G (E, V, \bar{A})$ , where $\bar{A}$ is the edge set, that is, $\bar{A} = {(e_{i}, v_{j}) | e_{i} \in E, v_{j} \in V, v_{j} is in e_{i}}$ . The structural model can also be represented by the association matrix $A = [a_{ij}]$ of $G (E, V, \bar{A})$ .

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

(2)

The structural model of $M_{1}$ is presented in Figure 1. In Figure 1, the rows of the structural matrix stand for the system equations, the columns stand for the system variables, and the points stand for the column variables associated with the current position in the equations corresponding to the rows; the blue points stand for the edges associated to $X$ , the red points represent the edges associated with $F$ , and the black points represent the edges related to $Z$ .

Figure 1.

The structural model of $M_{1}$ .

When modeling and analyzing a specific behavioral pattern of a system, it is unnecessary to have all the equations in $E$ to describe it. Here, given an equation mapping $Q : 2^{E} \to 2^{V}, Q (E') = {v \in V | \exists e \in E' s . t . (e, v) \in \bar{A}}$ and a variable mapping $R : 2^{V} \to 2^{E},$ $R (V^{'}) = {e \in E | \exists v \in V^{'} s . t . (e, v) \in \bar{A}}$ . Depending on the nature of the variables, the mapping $Q$ can be further defined as $Q_{X} : 2^{E} \to 2^{X}$ , $Q_{Z} : 2^{E} \to 2^{Z}$ , $Q_{F} : 2^{E} \to 2^{F}$ . The mapping $R$ can be further defined as $R_{X} : 2^{X} \to 2^{E}$ , $R_{Z} : 2^{Z} \to 2^{E}$ , $R_{F} : 2^{F} \to 2^{E}$ . Thus, the structural model in a particular mode can be denoted as $G (E^{'}, Q (E^{'}), {\bar{A}}_{E^{'}})$ or $G (R (V'), V', {\bar{A}}_{V'})$ . Structural analysis often focuses on the structural model associated with $X$ , that is, the blue part in Figure 1. For ease of presentation, the structural model related to $X$ is defined as $G (E, X, \bar{A})$ .

Structural decomposition

The key to the structural analysis is finding submodels with structural redundancy. The structural redundancy of $M$ is

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

(3)

$| S |$ represents the base of the set $S$ . The structural analysis mainly uses the Dulmage-Mendelsohn (DM) decomposition²⁵ to find submodels that imply structural redundancy.

The DM decomposition converts the equations and variables of $G (E, X, \bar{A})$ into upper triangular form by ordering them appropriately, as seen in Figure 2(a). The structure matrix is shown in Figure 2(a), where blanks are 0 and shaded parts represent 0 or 1. As shown in Figure 2(a), $G (E, X, \bar{A})$ could be decomposed into a structurally underdetermined subdiagram $G^{-} (E^{-}, X^{-}, {\bar{A}}^{-})$ , a structurally exactly determined subdiagram $G^{0} (E^{0}, X^{0}, {\bar{A}}^{0})$ and a structurally overdetermined subdiagram $G^{+} (E^{+}, X^{+}, {\bar{A}}^{+})$ . $G^{0}$ can be further partitioned into a set of Hall components ${G_{1}, G_{2}, \dots, G_{n}}$ . Each Hall component $G_{i}$ is structurally exactly determined. Further, $M$ may be divided into a structurally underdetermined subsystem $M^{-} = (E^{-}, X^{-})$ , which $| E^{-} | < | X^{-} |$ ; a structurally exactly determined subsystem $M^{0} = (E^{0}, X^{0})$ , which $| E^{0} | = | X^{0} |$ ; and a structurally overdetermined subsystem $M^{+} = (E^{+}, X^{+})$ , which $| E^{+} | > | X^{+} |$ . For ease of description, + and $0$ are the operators for obtaining the structurally overdetermined and structurally exactly determined parts of the system, respectively.

Figure 2.

DM decomposition: (a) DM decomposition schematic and (b) DM decomposition of $M_{1}$ .

From the structural properties of the DM decomposition, it can be seen that the structural redundancy could be further declared as

φ (M) = φ (G) = | E^{+} | - | X^{+} |

(4)

Therefore, the structural redundancy exists mainly in $M^{+}$ . Because of $| E^{+} | > | X^{+} |$ , and there is the possibility of multiple structurally overdetermined subsystems in the $M^{+}$ . For ease of description, the following definitions are given here:

Definition 1 (SO). The equation set $E$ is structurally overdetermined (SO) if and only if

φ (G (E, Q_{X} (E), {\bar{A}}_{E})) > 0

(5)

As shown in Figure 2(a), $E^{+}$ is structural overdetermined. Since $φ (M^{+}) \geq 1$ , $\exists E^{'} \subseteq E^{+}$ , $s . t . φ (G (E^{'}, Q_{X} (E^{'}), {\bar{A}}_{E^{'}})) > 0$ . Further:

Definition 2 (PSO). An SO set $E$ is a proper structurally overdetermined (PSO) set if and only if

E = E^{+}

(6)

Definition 3 (MSO). An SO set $E$ is an MSO set if and only if ${}^{\neg}\exists E^{'} \subseteq E$ , such that $E^{'}$ is structurally overdetermined.

The DM decomposition of $M_{1}$ is shown in Figure 2(b), which indicates that $M_{1} = {(M_{1})}^{+}$ . So $E_{1}$ is a PSO. Applying the MSO-seeking approach in the literature,²⁶ the MSOs of $M_{1}$ can be obtained as $MS O_{1} = {e_{2}, e_{3}, e_{4}, e_{5}}$ , $MS O_{2} = {e_{1}, e_{3}, e_{4}, e_{5}}$ , $MS O_{3} = {e_{1}, e_{2}, e_{4}, e_{5}}$ , $MS O_{4} = {e_{1}, e_{2}, e_{3}, e_{5}}$ , $MS O_{5} = {e_{1}, e_{2}, e_{3}, e_{4}}$ .

Matching

Structural redundancy is a system-structured representation of analytical redundancy. Matching¹⁵ is a key tool for structural analysis to solve structural redundancy. Matching establishes the mapping relationship between system equations and unknown variables, solves for the matched unknown variables using the matching equations, and obtains the analytical redundancy relationship based on the unmatched equations. Unlike the algebraic elimination between equations, matching focuses only on the matching relationship between equations and unknown variables.

$a = (e, x) \in \bar{A}$ is an edge, define the mapping $p_{E} : \bar{A} \to E$ and the mapping $p_{X} : \bar{A} \to X$ , that is, $p_{E} (a) = e$ , $p_{V} (a) = x$ . Call $N \subseteq \bar{A}$ a system matching if and only if $\forall a_{1}, a_{2} \in N,$ $s . t . a_{1} \neq a_{2} \Rightarrow$ $p_{E} (a_{1}) \neq p_{E} (a_{2}) \land p_{X} (a_{1}) \neq p_{X} (a_{2})$ . Call $N \subseteq \bar{A}$ a maximal matching if $\forall N_{i} \in 2^{\bar{A}}$ , $N \subset N_{i}$ , $N_{i}$ is not a matching. Call $N \subseteq \bar{A}$ a complete matching if and only if $| N | = | E |$ or $| N | = | X |$ . When $| N | = | X |$ , $N$ is a column complete matching; when $| N | = | E |$ , $N$ is a row complete matching; when $| N | = | E | = | X |$ , $N$ is a perfect matching.

The mappings $p_{E}$ and $p_{X}$ are extended into the mapping $π_{E} : N \to 2^{E}$ and the mapping $π_{X} : N \to 2^{X}$ based on the matching properties. For $\forall N \in N$ , satisfies

π_{E} (N) = {e | \exists a \in N, e = p_{E} (a)}

(7)

π_{X} (N) = {x | \exists a \in N, x = p_{X} (a)}

(8)

Further, the matching $N$ satisfies

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

(9)

| π_{X} (N) | = | π_{E} (N) | = | N |

(10)

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

(11)

A maximal matching of $M$ is the light dotted line in the DM decomposition schematic shown in Figure 2(a). There exists row complete matching in $G^{-} (E^{-}, X^{-}, {\bar{A}}^{-})$ , perfect matching in $G^{0} (E^{0}, X^{0}, {\bar{A}}^{0})$ , and column complete matching in $G^{+} (E^{+}, X^{+}, {\bar{A}}^{+})$ . Based on equation (4), structural redundancy exists in the structurally overdetermined subsystem. Based on the matching relationship of $G^{+}$ , it can be seen that there are multiple matching equations. However, for MSO, only one non-matching (redundant) equation exists, that is, there are at most $| MSO |$ different column complete matching in MSO. Each column complete matching in MSO corresponds to a structural redundancy relationship. Taking the $MS O_{1} = {e_{2}, e_{3}, e_{4}, e_{5}}$ of $M_{1}$ as an example, its structural redundancy relation is shown in Figure 3.

Figure 3.

Structural redundancy of $MS O_{1}$ .

The circles in Figure 3 represent the variables, the bars represent the equations, and the arrows represent the computational flow. As shown in Figure 3, its column complete matching relationship is ${(e_{4}, x_{1}), (e_{3}, x_{3}), (e_{2}, x_{2})}$ , and its non-matching equation is $e_{5}$ . From the computational flow, it could be noticed that the matching unknown variables $x_{1}$ , $x_{2}$ , $x_{3}$ can all be expressed in the form of known functions:

\begin{matrix} x_{1} = y_{1} = γ_{4} (y_{1}) \\ x_{3} = \int (- 3 y_{1} + u) d x_{3} = γ_{3} (y_{1}, u) \\ x_{2} = \int (- 2 γ_{4} (y_{1}) + γ_{3} (y_{1}, u)) d x_{2} = γ_{2} (γ_{4} (y_{1}), γ_{3} (y_{1}, u)) \end{matrix}

(12)

Solving the analytical redundancy relation based on the redundancy equation (non-matching equation):

r = y_{2} - γ_{2} (γ_{4} (y_{1}), γ_{3} (y_{1}, u)) = 0

(13)

Therefore, all unknown variables in the MSO can demonstrated as functions of known variables, and the analytical expressions shown in equation (13) are different for different column complete matching. If there is a fault in any of the equations in ${e_{2}, e_{3}, e_{4}, e_{5}}$ , based on equation (13), it is known that $r \neq 0$ . Therefore, a fault $f$ in $M$ is said to be structurally detectable²⁷ if and only if

e_{f} \in E^{+}

(14)

where $e_{f}$ denotes the equation impacted by $f$ . Further, the fault $f_{i}$ is said to be isolable from $f_{j}$ if and only if

e_{f_{i}} \in {(E \ e_{f_{j}})}^{+}

(15)

Structural reconfigurability analysis

Structural reconfigurability

Reconfigurability reflects the system’s ability to reconfigure itself from autonomous faults, that is, fault-tolerant control capability. Fault-tolerant control ensures the system can still fulfill its goals and keep running smoothly after a fault occurs. The system goal refers to the control and estimation functions; the control goal ensures that the state is controllable, and the estimation goal ensures that the state is observable. Therefore, reconfigurability reflects the system’s ability to maintain a controllable state and an estimable state after a fault occurs.²⁸ Hence, the reconfigurability is given here based on controllability and observability.

Observability is mainly related to state observability, that is, the potential of reconstructing the state from the system’s outputs. Thus, state observability can be defined at a given time, and state observability is the result of the observability of each state.

Definition 4 (State Observable). For any initial time $t_{1}$ , initial value $x_{1}$ , and termination time $t_{2} \neq t_{1}$ , the state $x$ is said to be observable if the initial value $x_{1}$ can be determined based on the output $y (t)$ and input $u (t)$ of the system at moment $t \in [t_{1}, t_{2}]$ .

Theorem 1 (Structurally Observable). $x$ is structurally observable, $OBS (x)$ , if and only if

\exists MSO, e_{x}, e_{y}, s . t . {e_{x}, e_{y}} \subseteq MSO

(16)

Proof. Based on the structural redundancy relation of MSO in Figure 3, $\forall x \in Q_{X} (MSO)$ , $\exists x = γ (y)$ according to the different column complete matching embedded in the MSO. Therefore, all the unknown variables in the MSO can be represented as a function of the measured variable $y$ , due to $e_{y} \in MSO$ . Proof End.

Analogous to observability, structural controllability is defined here based on the controllability of the state at a given moment.

Definition 5 (State Controllable). $x$ is said to be controllable if the control variable $u (t)$ can convert the initial value $x_{0}$ of $x$ at the moment $t_{0}$ , to the value $x_{f}$ at the moment $t_{f}$ , within the time interval $[t_{0}, t_{f}]$ .

Theorem 2 (Structurally Controllable). $x$ is structurally controllable, $CONT (x)$ , if and only if

\exists MSO, e_{x}, e_{u}, s . t . {e_{x}, e_{u}} \subseteq MSO

(17)

Proof. Based on the structural redundancy relation of MSO shown in Figure 3, according to the complete column matching implied in MSO, since $e_{u} \in MSO$ , so $\forall x \in Q_{X} (MSO)$ , $\exists x = γ (u)$ . Thus, all the unknown variables in the MSO can be represented as functions of the measured variables $u$ . Proof End.

The sufficient conditions for structural reconfigurability are given here based on the relationship between reconfigurability, controllability, and observability.

Theorem 3 (Structural reconfigurability). When a fault $f$ occurs, $x$ is structurally reconfigurable, $RECO {(x)}_{f}$ , if and only if

\exists MSO, s . t . OBS (x) \land CONT (x) \land e_{f} \notin MSO

(18)

Proof. $e_{f} \notin MSO$ shows that the fault $f$ does not affect the solution of the structural redundancy relation of MSO. $OBS (x) \land CONT (x)$ ensures that the structural controllable and observable properties of $x$ still hold after $f$ occurs, that is, $x$ is structurally reconfigurable. Proof End.

Reconfigurability evaluation

Based on Theorem 3, determine whether the MSO is reconfigurable: if ${e_{y}, e_{u}} \subseteq MSO$ , then the MSO is reconfigurable; if ${e_{y}, e_{u}} ⊄ MSO$ , then the MSO is not reconfigurable. Assuming that the system reconfigurable MSOs is ${MS O_{i}}$ .

The state reconfiguration matrix (SRM) $SRM = [M x_{ij}]$ is established based on the affiliation of reconfigurable MSO with each system state.

M x_{ij} = {\begin{matrix} 1, x_{i} \in Q_{X} (MS O_{j}) \\ 0, x_{i} \notin Q_{X} (MS O_{j}) \end{matrix}

(19)

The fault reconfiguration matrix (FRM) $FRM = [f x_{kj}]$ is established based on the system fault set $F = {f_{k}}$ and SRM.

f x_{kj} = {\begin{matrix} 1, \exists i, s . t . M x_{ij} = 1 \land e_{f_{k}} \notin MS O_{i} \\ 0, otherwise \end{matrix}

(20)

Based on FRM, the fault reconfiguration rate (FRR) of the system is

FRR = \frac{\sum_{k} \sum_{j} f x_{kj}}{| F | \cdot | X |}

(21)

Further, the single-fault reconfiguration rate for fault $f_{k}$ is

FR R_{f_{k}} = \frac{\sum_{j} f x_{kj}}{| X |}

(22)

The SRM of $M_{1}$ is displayed in Figure 4(a), and the FRM is displayed in Figure 4(b). As displayed in Figure 4(a), the reconfigurable MSOs of $M_{1}$ are ${MS O_{1}, MS O_{2}, MS O_{4}, MS O_{5}}$ , and each MSO implies the structural redundancy relation of ${x_{1}, x_{2}, x_{3}}$ . As displayed in Figure 4(b), when the fault $f_{3}$ occurs, the system states are all non-reconfigurable. This is because when $f_{3}$ occurs, the system actuator $u$ is affected, resulting in the states being uncontrollable. As a result, the fault reconfiguration rate of $M_{1}$ is 80%; except for $f_{3}$ , which has a reconfiguration rate of 0%, the rest of the faults have a reconfiguration rate of 100%.

Figure 4.

Reconfigurability analysis of $M_{1}$ : (a) SRM and (b) FRM.

Reconfigurability optimization design

Depending on where the fault occurs, system faults can be classified as actuator, sensor, and internal faults. Common actuator faults are jamming, damage, and loose float. In the case of a multi-maneuver UAV, control redundancy exists between the maneuvers. It is assumed that the control effectiveness in jamming or flotation of any maneuvering surface can be allocated to the damaged or normal maneuvering surface through control allocation. Standard control allocation methods include series chaining, generalized inverse, direct allocation, mathematical planning, and allocation methods based on intelligent control theory.²⁹ Therefore, the primary focus here is on the reconfigurability of sensor and internal faults.

Structurally exactly determined sensor configuration

According to Theorem 3, the key to reconfigurability is to construct MSOs that satisfy the reconfigurability requirement. Structural redundancy is a prerequisite and foundation for reconfigurability. Therefore, the structural redundancy needs to be increased by sensor configuration. As shown in Figure 2(a), $G (E, X, \bar{A})$ can be decomposed into $G^{-}$ , $G^{0}$ , and $G^{+}$ , where the structural redundancy relation exists in $G^{+}$ . $G^{-}$ is mainly excluded from most control systems to ensure its diagnosability. Therefore, the MSO design of the structurally underdetermined part is not considered here.

As shown in Figure 2(a), there exists a series of Hall components ${G_{1}, G_{2}, \dots, G_{n}}$ for $G^{0}$ . The Hall components are sorted based on the affiliation between the unknown variables in the Hall components $G_{i} (i = 1, 2, \dots, n)$ :

{\begin{matrix} \begin{matrix} G_{i} < G_{i + 1}, if π_{X} (G_{i}) \subseteq Q_{X} (π_{E} (G_{i + 1})) \\ G_{i} < G_{j}, if G_{i} < G_{k}, G_{k} < G_{j} \end{matrix} \\ \begin{matrix} G_{i} = G_{i + 1}, if π_{X} (G_{i}) ⊄ Q_{X} (π_{E} (G_{i + 1})) \\ G_{i - 1} < G_{i + 1}, if G_{i} = G_{i + 1} (or G_{i} = G_{i - 1}), \\ and π_{X} (G_{i - 1}) \subseteq Q_{X} (π_{E} (G_{i + 1})) \end{matrix} \end{matrix}

(23)

For $G^{0}$ , $φ (G^{0}) = | E^{0} | - | X^{0} | = 0$ ; for Hall components $G_{i} (i = 1, 2, \dots, n)$ , $φ (G_{i}) = 0$ . When adding the measurement equation $e$ , then

\begin{matrix} e : y = x, x \in π_{X} (G^{0}) \\ | E^{0} \cup e | - | X^{0} | = 1 \end{matrix}

(24)

That is, there is a structural redundancy relation in $E^{'} = {e \cup π_{E} (G^{0})}$ . Since $φ ((E^{'}, Q_{X} (E^{'}))) = 1$ , MSO exists in $E^{'}$ . According to equation (23), it is clear that for $E^{″} = {e \cup_{G_{j} \leq G_{i}} π_{E} (G_{j})}$ , then

φ ((E^{″}, Q_{X} (E^{″}))) = | E^{″} | - | Q_{X} (E^{″}) | = 1

(25)

And $\forall e_{i} \in E^{″}$ ,

φ ((E^{″} \ e_{i}, Q_{X} (E^{″} \ e_{i}))) = 0

(26)

So, $E^{″}$ is MSO. Thus, $\forall x \in π_{X} (G_{i})$ , when adding the measure equation $e : y = x$ , $E^{″} = {e \cup_{G_{j} \leq G_{i}} π_{E} (G_{j})}$ is MSO. Therefore, Algorithm 1 is designed to turn the structurally exactly determined part into a structurally overdetermined part by sensor configuration and to obtain the MSOs in it. The pseudo-code of Algorithm 1 is as follows.

Algorithm 1. $[MSOs, S_{D}]$ =GetMSO $(M^{0})$
Find the order of $G_{i} (i = 1, 2, \dots, n)$ in $M^{0}$ ; Find the set of maximal classes $G_{m}$ in ${G_{i}}$ $k = 1$ ; $D_{f} = \emptyset$ ; for every $G_{i}$ in $G_{m}$ do for every $x_{i}$ in $π_{X} (G_{i})$ do $e_{i} : y = x_{i}$ % Adding measurement equations $e_{i}$ $MSOs (k) = {e_{i} \cup_{G_{j} \leq G_{i}} π_{E} (G_{j})}$ $D_{f} (k) = π_{X} (\cup_{G_{j} \geq G_{i}} G_{j})$ $k = k + 1$ ; end for end for $S_{D}$ is the minimal hitting set of $D_{f}$ Return $MSOs$ , $S_{D}$

Algorithm 1.

[MSOs, S_{D}]

=GetMSO

(M^{0})

Find the order of

G_{i} (i = 1, 2, \dots, n)

M^{0}

;
Find the set of maximal classes

G_{m}

{G_{i}}

k = 1

;

D_{f} = \emptyset

;
for every

G_{i}

G_{m}

do
for every

x_{i}

π_{X} (G_{i})

e_{i} : y = x_{i}

% Adding measurement equations

e_{i}

MSOs (k) = {e_{i} \cup_{G_{j} \leq G_{i}} π_{E} (G_{j})}

D_{f} (k) = π_{X} (\cup_{G_{j} \geq G_{i}} G_{j})

k = k + 1

;
end for
end for

S_{D}

is the minimal hitting set of

D_{f}

Return

MSOs

S_{D}

Algorithm 1 returns two parts: one is the minimum sensor configuration set $S_{D}$ that turns the structure structurally exactly determined into the structure overdetermined, and the other is the MSOs implied by the $M^{0}$ after the sensor configurations. Algorithm 1 represents the sensor configurations as unknown variables measured by the measurement equation. Thus, $S_{D}$ that turns $M^{0}$ into $M^{+}$ is the minimal hitting set of $D_{f}$ . For the subsystem $M_{1}^{'} = ({e_{1}, e_{2}, e_{3}}, {x_{1}, x_{2}, x_{3}, f_{1}, f_{2}, f_{3}, u_{1}})$ of $M_{1}$ , applying Algorithm 1, the set of sensor configurations can be obtained as $S_{D} = {{x_{1}}, {x_{2}}, {x_{3}}}$ , and the MSO is ${e_{1}, e_{2}, e_{3}, e_{4}}$ .

Sensor configuration based on diagnosability

The system must diagnose faults before they can be reconfigured. Diagnosability is the prerequisite and basis for reconfigurability. Therefore, algorithms must be designed to ensure the system’s maximum diagnosability. Algorithm 1 ensures that the system is structurally overdetermined through the sensor configurations, which, based on equation (14), also ensures the system’s maximum detectability. Therefore, the algorithm must also be designed to ensure the system’s maximum isolability. The design idea is as follows: firstly, based on Algorithm 1, find out the sensor configuration set $S_{D}$ that satisfies the maximum detectability; secondly, configure $S_{i}$ for each sensor in $S_{D}$ , and when configured as $S_{i}$ , find out the sensor configuration $S_{I}$ that satisfies the system’s maximum isolability; finally, calculate the multi-concatenation set $S$ of each set $S_{j}$ in $S_{I}$ with $S_{i}$ , then $S$ is a sensor configuration set that satisfies the system’s maximum diagnosability.

$M = M^{+}$ can be guaranteed by sensor configuration using Algorithm 1 for $M = (E, V)$ . Based on equation (14), it shows that Algorithm 1 guarantees the system’s maximum detectability. Based on equation (15), it is known that the sufficient condition for the fault $f_{i}$ to be isolated from $f_{j}$ is $e_{f_{i}} \in {(E \ e_{f_{j}})}^{+}$ . Since the structural decomposition of $M^{+}$ after removing any equation may contain only structurally exactly determined and structurally overdetermined parts, that is, the structure of $(E^{+} \ e_{f_{j}})$ contains both ${(E^{+} \ e_{f_{j}})}^{+}$ and ${(E^{+} \ e_{f_{j}})}^{0}$ . Therefore, ensuring that the fault $f_{i}$ can be isolated from $f_{j}$ ensures that ${(E^{+} \ e_{f_{j}})}^{0}$ is structurally overdetermined using Algorithm 1. Further, this also satisfies the $e_{f_{i}} \in {(E \ e_{f_{j}})}^{+}$ . Based on the above analysis, Algorithm 2 is designed to ensure the system’s maximum isolability through sensor configuration.

Algorithm 2. $S_{I}$ = FindIsolability $(M^{+})$
$S_{D_{f}} = \emptyset$ ; $k = 1$ ; for every fault $f_{i}$ in $Q_{F} (E^{+})$ do $M^{0} = {(E^{+} \ e_{f_{i}})}^{0}$ ; $S_{D_{f}} (k)$ =GetMSO $(M^{0})$ $k = k + 1$ end for $S_{I}$ is the minimal hitting set of $S_{D_{f}}$ Return $S_{I}$

Based on Section 4.1, it can be seen that the sensor configuration set obtained by applying Algorithm 1 to the system $M_{1}^{'}$ is $S_{D} = {{x_{1}}, {x_{2}}, {x_{3}}}$ . Therefore, when the sensor configuration is ${x_{1}}$ , Algorithm 2 is utilized to get the sensor configuration that satisfies the maximum isolability as ${x_{2}}$ ; when the sensor configuration is ${x_{2}}$ , Algorithm 2 is utilized to get the sensor configuration set as ${{x_{1}}, {x_{3}}}$ ; when the sensor configuration is ${x_{3}}$ , Algorithm 2 is utilized to get the sensor configuration set as ${x_{2}}$ . Next, only Algorithm 1 and Algorithm 2 must be fused to get the sensor configuration set that meets the system’s maximum diagnosability requirement. Therefore, Algorithm 3 is designed to find the sensor configurations for maximum diagnosability.

Algorithm 3. $S$ = FindMaxDiagnosibility $(M = (E, V))$
$S = \emptyset$ ; $S_{D}$ =GetMSO $(M^{0})$ ; for every sensor set $S_{i}$ in $S_{D}$ do $S_{I}$ = FindIsolability $({(E \cup E_{S_{i}}, X)}^{+})$ % $E_{S_{i}}$ is the measurement equation of $S_{i}$
$S = S \cup {S_{i} \overset{+}{\cup} S_{j} \| S_{j} \in S_{I}}$ ;
end for Delete nonminimal sensor sets in $S$ ; Return $S$

In Algorithm 3, $A \cup^{+} B$ denotes the multi-concatenation set between $A$ and $B$ , that is, ${a_{1}, a_{2}} \cup^{+} {a_{1}, a_{4}} = {a_{1}, a_{1}, a_{2}, a_{4}}$ . For example, by applying Algorithm 3 to $M_{1}^{'}$ , the sensor configuration set that satisfies the maximum diagnosability can be obtained as ${{x_{1}, x_{2}}, {x_{2}, x_{3}}}$ .

Structurally reconfigurable MSO design

Algorithm 1 guarantees that the system is structurally overdetermined, and Algorithm 2 and Algorithm 3 guarantee the structural diagnosability. Therefore, this section first designs Algorithm 4 to find the system’s MSOs based on equation (4); secondly, it designs Algorithm 5 to find the system’s reconfigurable MSOs based on Theorem 3.

Algorithm 4. $MSOs$ =FindMSOs $(M)$
$M = M^{+}$ $MSOs = \emptyset$ if $φ (M) = 1$ do $MSOs = MSOs \cup {E}$ ; else for each equation $e$ in $M$ do $M = {(E \ e, Q_{X} (E \ e))}^{+}$ ; $MSOs = MSOs \cup$ FindMSOs $(M)$ end for end if return $MSOs$

Algorithm 5. $rMSOs$ =FindRMSOs $(M = (E, V))$
$S$ =FindMaxDiagnosibility $(M = (E, V))$ ; select a sensor set $S_{i}$ in $S$ do $M^{'} = (E^{'} = E \cup E_{S_{i}}, V)$ $M^{'} = {(M^{'})}^{+}$ $MSOs$ =FindMSOs $(M^{'})$ ; $rMSOs = \emptyset$ ; for every $MS O_{i}$ in $MSOs$ do if $\exists e_{y}, e_{u}, s . t . e_{y} \in MS O_{i} \land e_{u} \in MS O_{i}$ do $rMSOs = rMSOs \cup MS O_{i}$ ; end if end for return $rMSOs$

Applying Algorithm 4, the MSOs for $M_{1}$ are obtained as $MS O_{1} = {e_{2}, e_{3}, e_{4}, e_{5}}$ , $MS O_{2} = {e_{1}, e_{3}, e_{4}, e_{5}}$ , $MS O_{3} = {e_{1}, e_{2}, e_{4}, e_{5}}$ , $MS O_{4} = {e_{1}, e_{2}, e_{3}, e_{5}}$ , $MS O_{5} = {e_{1}, e_{2}, e_{3}, e_{4}}$ . This is consistent with the conclusion of the MSO search algorithm shown in the literature.²⁶ Based on Theorem 3, to obtain MSOs that satisfy the reconfigurable requirements, Algorithm 5 is designed here to find the reconfigurable MSOs.

Applying Algorithm 5, the reconfigurable MSOs of $M_{1}$ are $MS O_{1}$ , $MS O_{2}$ , $MS O_{4}$ , $MS O_{5}$ . It can be seen that the output of the algorithm is consistent with Figure 4(a).

ICSA-based reconfigurable optimized design

The reconfigurable MSOs can be obtained based on Algorithm 5. On the one hand, to reduce the reconfigurable system design cost, all MSO in MSOs don’t need to undergo reconfigurable analysis and design; on the other hand, it is also vital to take into account the system’s diagnosability and reconfigurability requirements in the process of reconfigurable optimization. Therefore, this paper is based on ICSA to minimize the reconfigurability system design cost while satisfying the system reconfigurability and diagnosability requirements.

CSA

CSA is a novel bionic intelligent optimization algorithm based on the parasitic breeding behavior of cuckoos and adding Lévy flying. The parasitic breeding traits of cuckoos are separated into intraspecific parasitism, cooperative parenting, and nest occupancy. Lévy flight refers to randomized wandering with the truncated-tailed probability spectrum of step length. As a continuous probability distribution, its movement mechanism occasionally moves with extensive step lengths under the premise of many random movements with small step lengths. Based on the movement mechanism of Lévy flight, CSA can avoid falling into local extremes, which increases its global search capability. In addition, the CSA has a simple structure with few parameters, which is advantageous for solving complex and unique problems.

CSA comprises three main elements: selection of the optimum, locally arbitrary flight, and global Lévy flight. For convenience of expression, these three rules are provided here:

(1) Each bird lays one egg at a time, and random selection is followed in selecting nests for incubation;

(2) The ideal nest and optimal solution are kept for the following generation;

(3) The nests’ number is constant, and the likelihood that the nest-owning bird will locate an egg is $p_{a} \in [0, 1]$ . In this situation, the owner bird either tosses the eggs out of the nest or discards the nest to find and establish a new nest elsewhere.

In CSA, the cuckoo, the egg, and the nest are equivalent to the result of the optimization issue. Due to the above rules, the cuckoo’s nesting path and position update formula is:

x_{i}^{t + 1} = x_{i}^{t} + α \oplus L (λ), i = 1, 2, \dots, N

(27)

$x_{i}^{t}$ stands for the position of the $i$ th nest in the $t$ th generation; $α \in [0, 1]$ is the step factor, which is usually taken as 0.01; ⊕ is the dot product symbol; $N$ is the number of nests; and $L (λ)$ obeys the Lévy probability distribution:

L (λ) ~ u = t^{- λ}, 1 < λ < 3

(28)

$λ$ is the power coefficient. In CSA, the following equation is taken to calculate the random step:

L (λ) = \frac{μ}{{| v |}^{1 / β}}

(29)

$β \in [0, 2]$ generally takes the value 1.5, and $β = λ - 1$ ; $μ$ and $v$ obey the normal distribution, $μ ~ N (0, σ^{2})$ , $v ~ N (0, 1)$ , and

σ = {| \frac{Γ (β + 1) \sin (π β / 2)}{Γ [(β + 1) / 2] \times 2^{(β - 1) / 2} \times β} |}^{1 / β}

(30)

$Γ$ is the standard Gamma distribution.

Discard some of the inferior results due to the discard probability $p_{a}$ , and then take a preference random walk to generate the same number of new solutions as the discarded ones:

x_{i}^{t} = x_{i}^{t} + r \times H (u) \times (x_{m}^{t} - x_{n}^{t})

(31)

$r \in (0, 1)$ represents the scaling factor; $H (u)$ is the Hoviside step function; $x_{m}^{t}$ and $x_{n}^{t}$ represent two different nests in generation $t$ , respectively.

In CSA, bird nests represent candidate solutions, so the operation mechanism of CSA is as follows: firstly, initialize the bird nests, generate new individuals based on equation (27), carry out the fitness calculation, and keep the optimal solution. Secondly, compare the random numbers $rand$ and $p_{a}$ , and if $rand > p_{a}$ , the eggs are found by the nest owner bird. Then, discard the poorly adapted individuals based on $p_{a}$ . Finally, adopt the random walk approach based on equation (31) to generate the same number of discarded new individuals as the previous step, compute their fitness, and retain the optimal solution. This approach is taken to iterate until the global optimal solution is found.

ICSA

CSA needs to be upgraded to increase convergence velocity and optimization capabilities. The heart of the enhancement is to modify the impact of parameter choices on the convergence result of the algorithm and to increase the adaptable capacity of CSA.

(1) Mutation Strategy

When solving large-scale NP problems, the increase in population size will increase the algorithm’s complexity. Therefore, the mutation operation of chromosomes based on the genetic algorithm is used to improve the quality of individual populations further:

x_{i}^{t} = x_{best}^{t} + 0.2 \sin (\frac{π}{2} \times \frac{iter}{Max_iter}) \oplus ε

(32)

$x_{best}^{t}$ stands for the optimal solution of the current iteration; $Max_iter$ and $iter$ stand for the maximum number of iterations and the current number of iterations, respectively; $ε \in R^{D}$ obeys a nontrivial distribution; D is the dimension of the problem.

(2) Adaptive Lévy Flight

The Lévy flight parameter $β$ is the core parameter of the CSA position update, and the size of $β$ is proportional to the step size of the position update. In the early stage, to ensure the optimization speed, the step size should be larger; in the late stage, the step size should be smaller to ensure accuracy. Therefore, the adaptive Lévy flight parameters are:

β = (β_{max} - β_{min}) \times \frac{(Max_iter - iter)}{Max_iter} + β_{min}

(33)

$β_{min}$ and $β_{max}$ stand for the maximum and minimum values of $β$ , respectively.

(3) Discard probability $p_{a}$

Literature³⁰ suggests that CSA is advantageous in parameter tuning compared to genetic and particle swarm algorithms. The key lies in the discard probability $p_{a}$ . CSA is insensitive to $p_{a}$ , and $p_{a} = 0.25$ is sufficient to solve most problems. The larger the $p_{a}$ is, the higher the probability that the CSA generates a new solution, and the convergence speed decreases. However, adjusting $p_{a}$ can improve the convergence rate for complex and multimodal problems. Therefore, $p_{a}$ is adjusted according to the dimension of the problem-solving:

p_{a} = \frac{ran d_{0}}{D} + 0.25

(34)

$ran d_{0} \in [0, 1]$ is the random number.

(4) Adaptive Step $α$

A larger $α$ can increase the convergence speed in the early stage of the iteration; in the late stage, a smaller $α$ should be taken to improve the convergence accuracy. Therefore, the adaptive step factor $α$ is:

α = α_{max} \times \exp (\frac{iter}{Max_iter} \times \ln (\frac{α_{min}}{α_{max}}))

(35)

$α_{max}$ and $α_{min}$ represent the maximum and minimum values of the step factor, respectively.

(5) Replacement Strategy

In CSA, the convergence speed is affected when replacing the old and new nests is performed. To better control the step size, the original randomized search mechanism is changed based on the global optimal replacement principle:

x_{i}^{t} = x_{i}^{t} + r \times H (u) \times (x_{best}^{t} - x_{i}^{t})

(36)

Reconfigurability optimization design

Reconfigurability optimization design model

Given the system structural model, the reconfigurable MSOs can be obtained using Algorithm 5. However, to save the reconfigurable system design cost, not all the MSO in the MSOs are needed for reconfigurability analysis and design. Therefore, to balance the reconfigurability requirement and the reconfigurable system design cost, selecting the optimal MSO for reconfigurability analysis and design from the MSOs is necessary. Since diagnosability is the concept and cornerstone of reconfigurability analysis and design, the diagnosability requirement must also be considered here. The model of reconfigurability optimization design is as follows:

Step 1: Build the structural model of $M = (E, X)$ and utilize Algorithm 3 to find the sensor configuration set $S$ that satisfies the maximum diagnosability;

Step 2: The least costly set $S_{i}$ of sensor configurations is preferred from $S$ . Adding the measurement equations $E_{S_{i}}$ , the system $M$ becomes $M^{'} = (E \cup E_{S_{i}}, X)$ ;

Step 3: Using Algorithm 5, find the reconfigurable $MSOs = {MS O_{1}, MS O_{2}, \dots, MS O_{m}}$ of $M^{'}$ ;

Step 4: Calculate the reconfigurable design cost $C = {c_{1}, c_{2}, \dots, c_{m}}$ for each MSO in the MSOs;

Step 5: Create the reconfigurable optimization vector $X = {[x_{i}]}_{m \times 1}$

x_{i} = {\begin{matrix} 1, if MS O_{i} is selected \\ 0, otherwise \end{matrix}

(37)

$x_{i} = 1$ stands for selecting $MS O_{i}$ for reconfigurability design. Then, the cost of reconfigurability design is $C \cdot X$ .

The reconfigurability optimization cost function is

min C \cdot X

(38)

The constraints include:

(1) Diagnosability requirements

Diagnosability requirements need to be balanced with both fault detection rate $FDR$ and fault isolation rate $FIR$ :

FDR \geq FD R_{req}

(39)

FIR \geq FI R_{req}

(40)

$FD R_{req}$ stands for fault detection requirement; $FI R_{req}$ stands for fault isolation requirement.

(2) Reconfigurability requirements

Reconfigurability requirements need to take into account both the global fault reconfigurability rate $FRR$ and the single fault reconfigurability rate $FR R_{f_{i}}$ :

FRR \geq FR R_{req}

(41)

FR R_{f_{i}} \geq {FRR}_{req}^{f_{i}}, f_{i} \in F

(42)

$FR R_{req}$ is the system failure reconfiguration requirement, and ${FRR}_{req}^{f_{i}}$ is the reconfigurable requirement for fault $f_{i}$ .

Reconfigurability optimization design strategy

The steps of ICSA-based reconfigurability optimization design are as follows:

Step 1: Initialize population size $NP$ ; maximum iterations $Max_Iter$ ; mutation probability $p_{m} = 0.25$ ; space dimension $D = | MSOs |$ ; $iter = 1$ ;

Step 2: Based on equation (37), randomly generate NP bird nests. If the constraints equations (39)–(40) are satisfied, then the fitness of the bird’s nest is calculated based on equation (38); otherwise, the fitness of the bird’s nest is infinite. And keep the current optimal bird nest location;

Step 3: Based on equations (33) and (35), calculate the parameter $α$ and $β$ . Based on equation (27), nest locations were updated using Lévy flights and compared with the previous generation of nest locations to retain the better-adapted nests;

Step 4: Determine the size of the random number $ran d_{1}$ and variant probability $p_{m}$ . If $ran d_{1} \geq p_{m}$ , update the nest location based on equation (32) and keep the optimal solution; if $ran d_{1} < p_{a}$ , keep the current nest;

Step 5: Determine the discovery probability $p_{a}$ based on equation (34) and determine the size of random numbers $ran d_{2}$ and $p_{a}$ . If $ran d_{2} \geq p_{a}$ , discard the poorly adapted nest and generate a new nest using equation (36), keeping the optimal solution; if $ran d_{2} < p_{a}$ , maintain the current nest; $iter = iter + 1$ ;

Step 6: If $iter < Max_Iter$ , return to Step 3 and continue iteration; otherwise, output the optimal solution.

Simulation verification

Fixed-wing UAV structural model

The numerical experiments were conducted in MATLAB R2021b on a workstation with an AMD Ryzen 7 6800H CPU and NVIDIA RTX 3050 Ti GPU. GPU-accelerated computations employed CUDA 11.2, and parallel processing was enabled via MATLAB’s Parallel Computing Toolbox (where applicable). In this paper, a small UAV shown in the literature³¹ is taken as the subject of study to validate the efficacy of the suggested scheme in this work from the system design perspective. The lateral kinematics model $M_{uav}$ of this UAV is

dx = A x + B u + F

(43)

A = [\begin{matrix} A_{1} & A_{2} \end{matrix}]

(44)

A_{1} = [\begin{matrix} - 0.0370 & - 5.8003 & 0.3007 & 0 & - 0.3641 & 0 \\ - 0.0034 & - 0.7311 & 0.0113 & 0 & 0.9689 & 0 \\ 0 & 0 & - 0.1702 & 0.1615 & 0 & - 0.9739 \\ 0 & 0 & - 13.0346 & - 1.3769 & 0 & 0.6423 \\ 0.0010 & 3.9620 & - 0.0046 & 0 & - 0.7130 & 0 \\ 0 & 0 & 0.6218 & - 0.1040 & 0 & - 0.3040 \\ 0 & 0 & 0 & 0 & 0 & 1.0129 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0.1613 \\ 1.0000 & - 0.0374 & - 0.0374 & 0 & 0 & 0 \\ 0 & 0 & 74.8478 & 0 & 0 & 0 \\ 0 & 74.8478 & 0 & 0 & 0 & 0 \end{matrix}]

(45)

A_{2} = [\begin{matrix} A_{21} & A_{22} \end{matrix}]

(46)

A_{21} = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 0.0374 & 74.8478 & 0 \\ - 9.8700 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 0.0374 & 0 & - 74.8478 \\ - 7.6533 e - 04 & 0 & 0.1287 & - 0.0081 & 0 & 0.0036 & 0 & 0 & 0 & - 9.4847 e - 04 & - 11.9156 & - 0.0059 \end{matrix}]}^{T}

(47)

A_{22} = {[0]}_{12 \times 3}

(48)

B = [\begin{matrix} B_{1} \\ B_{2} \end{matrix}]

(49)

B_{1} = [\begin{matrix} - 1.1004 & - 1.1004 & - 0.8164 & - 1.2551 & - 1.2551 & - 0.8164 & - 0.2051 \\ 0.0018 & 0.0018 & - 0.0495 & - 0.0786 & - 0.0786 & - 0.0495 & 4.3931 e - 04 \\ - 0.0070 & 0.0070 & 0.0034 & 0.0137 & - 0.0137 & - 0.0034 & 0.0403 \\ 0.7073 & - 0.7.73 & - 3.4956 & - 3.0013 & 3.0013 & 3.4956 & 2.1103 \\ 1.1204 & 1.1204 & - 0.7919 & - 1.2614 & - 1.2614 & - 0.7919 & 0.0035 \\ - 0.3309 & 0.3309 & - 0.1507 & - 0.3088 & - 0.3088 & 0.1509 & - 1.2680 \end{matrix}]

(50)

B_{2} = {[0]}_{6 \times 7}

(51)

$F = {(f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7})}^{T}$ represents the system faults; $u = {(u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7}, u_{8})}^{T}$ is the control input, which represents the left canards, right canards, left outer elevons, right outer elevons, left inner elevons, right inner elevons, and rudder, respectively; $x = {(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}, x_{11}, x_{12})}^{T}$ is the system state, which represents speed, attack angle, sideslip angle, roll rate, pitch rate, yaw rate, heading angle, pitch angle, roll angle, X (Earth coordinate system) axis position, Y (Earth coordinate system) axis position, and Z (Earth coordinate system) axis position, respectively. If the initial state $x_{0} = {(74.8478, 0.1599, 0, 0, 0, 0, 0, 0.1599, 0, 0, 0, - 20)}^{T}$ , the $M_{uav}$ is simulated; the operation results are shown in Figure 5. The plots in Figure 5 represent the simulation results for each state of the system.

Figure 5.

The simulation of $M_{uav}$ .

The structural model and DM decomposition schematic of $M_{uav}$ are displayed in Figure 6. As shown in Figure 6(b), $M_{uav} = {(M_{uav})}^{0}$ . So, $M_{uav}$ is structurally exactly determined with Hall components in order $G_{1} = {{e_{3}, e_{4}, e_{6}, e_{9}}, {x_{3}, x_{4}, x_{6}, x_{9}}}$ , $G_{2} = {{e_{7}}, {x_{7}}}$ , $G_{3} = {{e_{11}}, {x_{11}}}$ , $G_{4} = {{e_{1}, e_{2}, e_{5}, e_{8}}, {x_{1}, x_{2}, x_{5}, e_{8}}}$ , $G_{5} = {{e_{12}}, {x_{12}}}$ , $G_{6} = {{e_{10}}, {x_{10}}}$ . Based on equations (14) and (15), it can be seen that the faults are undetectable and isolated, that is, the fault detection rate $FD R_{1} = 0 %$ and the fault isolation rate $FI R_{1} = 0 %$ .

Figure 6.

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

Reconfigurability optimization design

Diagnosability optimization based on structurally redundant sensor configuration

Diagnosability is the premise and foundation of reconfigurability. Therefore, for $M_{uav}$ , Algorithm 3 is applied to obtain the sensor configuration set that satisfies the maximum diagnosability as ${x_{10}, x_{11}, x_{12}}$ . Thus, the constraint equations of sensor configurations are added:

\begin{matrix} e_{13} : y_{1} = x_{10} + f_{z x_{10}} \\ e_{14} : y_{2} = x_{11} + f_{z x_{11}} \\ e_{15} : y_{3} = x_{12} + f_{z x_{12}} \end{matrix}

(52)

$f_{z x_{10}}$ , $f_{z x_{11}}$ , and $f_{z x_{12}}$ represent the sensor faults of the sensors $y_{1}$ , $y_{2}$ , and $y_{3}$ , respectively. The model of $M_{uav}$ after sensor configuration is defined as $M_{uav}^{'}$ and the structural model of $M_{uav}^{'}$ and the DM decomposition schematic are shown in Figure 7.

Figure 7.

The structural analysis of $M_{uav}^{'}$ : (a) structural model and (b) DM decomposition.

As shown in Figure 7(b), $M_{uav}^{'} = {({M^{'}}_{uav})}^{+}$ . Based on equation (14), it shows that the fault is structurally detectable. Based on Algorithm 5, the reconfigurable MSOs of $M_{uav}^{'}$ can be obtained as displayed in Figure 8(a), its corresponding fault signature matrix $FSM$ as displayed in Figure 8(b), and the fault isolation matrix $FIM$ as displayed in Figure 8(c). The fault signature matrix displays the sensitive link between system faults and MSOs. Based on equation (14) shows that if $FSM (i, j) = 1$ , it means that $MS O_{j}$ can diagnose fault $f_{i}$ . The dots in row i and column j in Figure 8(a) represent $FSM (i, j) = 1$ ; if there are no dots, it means $FSM (i, j) = 0$ . The fault isolation matrix reflects the fault isolation of the system. If $FIM (i, j) = 1$ , fault $f_{i}$ cannot be isolated from fault $f_{j}$ . The dots in the corresponding positions of row i and column j in Figure 8(b) represent $FIM (i, j) = 1$ ; if there is no dot, it means $FIM (i, j) = 0$ .

Figure 8.

Diagnosability analysis of $M_{uav}^{'}$ : (a) MSOs, (b) fault signature matrix, and (c) fault isolation matrix.

For the matrix in Figure 8(a), the rows stand for the equations; the columns stand for the MSOs; the dots indicate that the equations of the rows in which they are located are among the MSO of the column in which they are located. Taking MSO₁ as an example, it can be seen that $MS O_{1} = {e_{3}, e_{4}, e_{6}, e_{7}, e_{9}, e_{11}, e_{14}}$ . The fault signature matrix of the MSOs obtained is displayed in Figure 8(b), and it shows that $\forall f_{i}, \exists MS O_{j}, s . t . FSM (i, j) = 1$ . Thus, the fault detection rate of $M_{uav}^{'}$ is $FD R_{2} = 100 %$ . As shown in the fault isolation matrix in Figure 8(c), $f_{i}$ and $f_{z x_{i}}$ faults are not isolated, where $i = 10, 11, 12$ . Therefore, the fault isolation rate is $FI R_{2} = 97.14 %$ . Figure 7(b) shows that $f_{i}$ and $f_{z x_{i}}$ are in the same black border, where $i = 10, 11, 12$ . Based on the literature,³² it is known that $f_{i}$ and $f_{z x_{i}}$ are a structural equivalence class, so they cannot be isolated from each other, where $i = 10, 11, 12$ . Therefore, the isolability cannot be increased by sensor configuration.

Based on Algorithm 3, the minimum sensor set to satisfy the diagnosability requirements is only one, which is determined by the structural characteristics of complex control systems. For complex control systems, such as UAVs, the correlation between system state variables is strong and belongs to a strongly coupled system, which is manifested in the system structure by larger Hall components that are structurally exactly determined, such as the $G_{1}$ and $G_{4}$ in Figure 6(b). Since the diagnosability in the same Hall component is equivalent, this leads to the equivalence of sensor configurations that satisfy their maximum diagnosability. In particular, when all Hall components contain faults, the number of system minimum sensor configuration sets is typically only one, which is determined by the system’s structural properties.

Reconfigurability optimization based on structurally redundant sensor configurations

Based on Algorithm 5, the reconfigurable MSOs of $M_{uav}^{'}$ can be obtained by sensor configuration, as shown in Figure 8(a), and it can be seen that there are 50 MSO in total. Based on equations (19) and (20), the state reconfiguration matrix $SRM$ and fault reconfiguration matrix $FRM$ are established as shown in Figure 9(a) and (b). The dots in the figure represent matrix elements 1. Based on the fault reconfiguration matrix of Figure 9(b), the fault reconfiguration rate is calculated as shown in Figure 9(c).

Figure 9.

Reconfigurability analysis of $M_{uav}^{'}$ : (a) state reconfiguration matrix, (b) fault reconfiguration matrix, and (c) fault reconfiguration rate.

The state reconstruction capability of MSO can be seen in the SRM shown in Figure 9(a). For example, $MS O_{1}$ can guarantee the reconfigurability of the state ${x_{3}, x_{4}, x_{6}, x_{7}, x_{9}, x_{11}}$ , that is, it can guarantee its structural controllable and observable properties. For $FRM$ in Figure 9(b), it shows that $\forall f_{j} \in {f_{i}, f_{zxi}}$ , $FRM (f_{j}, x_{i}) = 0$ , $i = 10, 11, 12$ . Analogous to Figure 8(c), it shows that the faults that cannot be isolated from each other are not reconfigurable, further validating the reasonableness of the strategy presented in this paper. The reconfiguration rate of faults shown in Figure 9(c) shows that the system’s fault reconfiguration rate is 96.67%, and the reconstruction rate of all faults is 91.67%, except for the reconstruction rate of the fault ${f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}, f_{8}, f_{9}}$ , which is 100%.

ICSA-based reconfigurable optimization

ICSA performance testing

To evaluate the algorithm’s efficiency, four test functions are chosen, and Table 1 displays the parameters of the test functions to confirm ICSA’s performance. Among them, $F_{1} (x)$ and $F_{2} (x)$ are single-peak functions primarily used to confirm the algorithm’s speed at optimization; $F_{3} (x)$ and $F_{4} (x)$ are multi-peak functions used to assess the algorithm’s overall convergence correctness and ability. The genetic algorithm (GA), simulated annealing algorithm (SA), and sine-cosine algorithm (SCA) are selected for comparable testing and validation in this work. Each method has a population of 50 and a maximum iteration number of 100. For each test function, 30 test repetitions are performed to verify the algorithms’ stability. The simulation results are displayed in Figures 10 to 13.

Table 1.

Test function.

Function	Dimension	Range
$F_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	30	[−100,100]
$F_{2} (x) = max \| x \|$	30	[−100,100]
$F_{3} (x) = \sum_{i = 1}^{n} i \cdot x_{i}^{4} + rand$	30	[−5.12,5.12]
$F_{4} (x) = - 20 e^{- 0.2 \sqrt{\frac{\sum_{i = 1}^{n} x_{i}^{2}}{n}}} - e^{\frac{\sum_{i = 1}^{n} \cos (2 π x_{i})}{n}} + 20 + e$	30	[−32,32]

Figure 10.

$F_{1} (x)$ test results: (a) $F_{1} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 11.

$F_{2} (x)$ test results: (a) $F_{2} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 12.

$F_{3} (x)$ test results: (a) $F_{3} (x)$ , (b) fitness curves, and (c) stabilization analysis.

Figure 13.

$F_{4} (x)$ test results. (a) $F_{4} (x)$ , (b) fitness curves, and (c) stabilization analysis.

The images of each function in three dimensions are displayed in Figures 10(a) to 13(a); the fitness curves are displayed in Figures 10(b) to 13(b); and the simulation of each function for 30 repetitions is displayed in Figure 10(c) to 13(c), where Avg represents the mean, Std the variance, Best the best result, and Worst the worst result. Specific numerical expressions are shown in Table 2. It is evident from the (b) graphs of Figures 10 and 11 that ICSA has a much higher convergence speed than other algorithms and from Figures 12 and 13 that ICSA has a significantly higher convergence accuracy than other algorithms. As can be observed from Figures 10(c) to 13(c), ICSA’s stability is markedly more productive than other methods, and its mean and variance are minimal. As a result, ICSA has a more significant edge in terms of algorithm stability, accuracy, and optimization search speed. Therefore, it makes some sense to choose ICSA for reconfigurability optimization design.

Table 2.

Performance comparison (dim = 30).

Function	Indicator	ICSA	CSA	SA	GA	SCA
$F_{1} (x)$	Avg	1.0090e-15	55.9100	1.6396e+03	77.9560	2.9950
	Std	5.5264e-15	3.6.2315	8.9803e+03	426.9837	16.4045
	Best	0	0	0	0	0
	Worst	3.0269e-14	1.6773e+03	4.9187e+04	2.3387e+03	89.8509
$F_{2} (x)$	Avg	1.0691e-04	1.8261	2.3158	24.011	1.8708
	Std	5.8558e-04	10.0020	12.6841	13.1515	10.2466
	Best	0	0	0	0	0
	Worst	0.0032	54.7833	69.4735	72.0337	56.1228
$F_{3} (x)$	Avg	1.5758e-04	0.0323	1.7432	0.1773	0.0028
	Std	8.6309e-04	0.1770	9.5429	0.9711	0.0153
	Best	0	0	0	0	0
	Worst	0.0047	0.9692	52.2688	5.3187	0.0837
$F_{4} (x)$	Avg	8.4724e-10	0.5091	0.6681	0.3945	0.6724
	Std	4.6405e-09	2.7886	3.6594	2.1607	3.6827
	Best	0	0	0	0	0
	Worst	2.5417e-08	15.2737	20.0434	11.8345	20.1710

To evaluate the performance of ICSA, we conducted 30 repeated experiments on a 60-dimensional function. The simulation results are presented in Table 3. A comparative analysis of Tables 2 and 3 demonstrates that ICSA, through its improved variation and adaptive strategy, enhances global search capability, and reduces susceptibility to local optima compared to CSA. Furthermore, ICSA exhibits superior convergence speed, accuracy, and stability relative to SA, SCA, and GA, confirming its overall advantage. Notably, when the variable dimension increases from 30 to 60, ICSA maintains robust performance without significant degradation, outperforming other algorithms across all test functions. This indicates its strong resilience to dimensional changes. Statistical analysis of the 30-run results—based on mean and standard deviation—further confirms that ICSA achieves optimal and stable performance. These findings validate the effectiveness of the proposed ICSA in delivering faster convergence and improved optimization results.

1. Population Size Analysis

Table 3.

Performance comparison (dim = 60).

Function	Indicator	ICSA	CSA	SA	GA	SCA
$F_{1} (x)$	Avg	1.2269e-09	711.9758	4.3438e+03	2.2125e+03	182.6153
	Std	6.7199e-09	3.8997e+03	2.3792e+04	1.2118e+04	1.0002e+03
	Best	0	0	0	0	0
	Worst	3.6860e-08	2.1359e+04	1.3031e+05	6.6374e+04	5.4785e+03
$F_{2} (x)$	Avg	0.0062	2.3446	2.8165	1.8412	2.7533
	Std	0.0341	12.8421	15.4266	10.0847	15.0804
	Best	0	0	0	0	0
	Worst	0.1869	70.3393	84.4592	55.2361	82.5988
$F_{3} (x)$	Avg	2.6615e-04	0.6406	8.2322	2.7574	0.4633
	Std	0.0015	3.5085	45.0895	15.1028	2.5379
	Best	0	0	0	0	0
	Worst	0.0080	19.2166	246.9655	82.7213	13.9004
$F_{4} (x)$	Avg	1.7211e-06	0.6460	0.6839	0.6856	0.6515
	Std	9.4266	3.5382	3.7461	3.7554	3.5683
	Best	0	0	0	0	0
	Worst	5.1632e-05	19.3795	20.5183	20.5693	19.5444

The ICSA was configured with a population size ( $pop$ ) of 50, a mutation parameter ( $k$ ) of 0.5, and a step parameter ( $r$ ) of 0.5. To investigate the impact of these parameters, simulations were conducted on $F_{4} (x)$ .

Tests were performed with population sizes of 20, 40, 60, 80, and 100. Figure 14(a) shows that more significant populations generally yield better convergence. Figure 14(b) and Table 4 indicate that a population size exceeding 80 provides optimal performance for $F_{4} (x)$ .

2. Mutation Parameter ( $k$ ) Analysis

Figure 14.

Results of the impact of population size: (a) fitness curves and (b) stabilization analysis.

Table 4.

Impact factor performance analysis.

Factor	Indicator	Pop = 20	Popp = 40	Popp = 60	Popp = 80	Popp = 100
pop	Avg	2.2832e-07	1.9103e-09	1.8411e-09	6.4027e-11	4.6710e-11
	Std	1.2506e-06	1.0463e-08	1.0084e-08	3.5069e-10	2.5584e-10
	Best	0	0	0	0	0
	Worst	6.8495e-06	5.7309e-08	5.5233e-08	1.9208e-09	1.4013e-09
Factor	Indicator	$k = 0.2$	$k = 0.4$	$k = 0.6$	$k = 0.8$	$k = 1.0$
$k$	Avg	1.4931e-05	3.5367e-07	6.1434e-10	2.4854e-10	3.5948e-08
	Std	8.1781e-05	1.9372e-06	3.3649e-09	1.3613e-09	1.9690e-07
	Best	0	0	0	0	0
	Worst	4.4794e-04	1.0610e-05	1.8430e-08	7.4561e-09	1.0784e-06
Factor	Indicator	$r = 0.2$	$r = 0.4$	$r = 0.6$	$r = 0.8$	$r = 1.0$
$r$	Avg	1.6670e-05	1.7565e-10	4.7760e-11	1.5450e-06	1.0992e-06
	Std	9.1303e-05	9.6208e-10	2.6159e-10	8.4621e-06	6.0204e-06
	Best	0	0	0	0	0
	Worst	5.0009e-04	5.2696e-09	1.4328e-09	4.6349e-05	3.2975e-05

The algorithm was evaluated with $k$ values of 0.2, 0.4, 0.6, 0.8, and 1.0. Figure 15(a) demonstrates that $k = 0.8$ achieves the fastest convergence, while Figure 15(b) and Table 4 reveal that the best accuracy and stability occur within the range of $0.6 \leq k \leq 0.8$ .

3. Step Parameter ( $r$ ) Analysis

Figure 15.

Results of the effect of the variant parameters: (a) fitness curves and (b) stabilization analysis.

Tests were conducted with $r$ values of 0.2, 0.4, 0.6, 0.8, and 1.0. Figure 16(a) indicates that $r = 0.6$ delivers optimal convergence speed, whereas Figure 16(b) and Table 4 show that the highest accuracy and stability are achieved when $0.4 \leq r \leq 0.6$ .

Figure 16.

Results of the effect of the step size parameter: (a) fitness curves and (b) stabilization analysis.

These results provide clear guidelines for parameter selection in ICSA, ensuring balanced convergence speed, optimization accuracy, and stability. In the following simulation, let $pop = 80$ , $k = 0.7$ , and $r = 0.5$ .

ICSA-based reconfigurability optimization

Assuming the $c_{i} = 1$ , to ensure that the reconfigurable system design cost is minimized while taking into account the system diagnosability and reconfigurability requirements, so here, due to the reconfigurable optimization model displayed in equations (38)–(42), using ICSA, the simulation results are generated as seen in Figure 17.

Figure 17.

ICSA-based reconfigurable optimization results: (a) fitness curves, (b) fault signature matrix, (c) fault isolability matrix, and (d) fault reconfiguration matrix.

Figure 17(a) shows that the optimization speed and accuracy of ICSA are more potent than other algorithms, and it can be seen that ICSA optimizes 10 MSO. Compared with before optimization, the reconfigurable cost using ICSA is reduced by 80%; compared with CSA and GA, the reconfigurable cost using ICSA is reduced by 8%; compared with SA, the reconfigurable cost using ICSA is reduced by 10%; the optimization result using SCA is the same as that of ICSA, but the overall convergence pace of ICSA is considerably faster than SCA. ICSA reaches the optimal value in the seventh iteration. In contrast, SCA reaches the optimal value in the 17th iteration. As shown in Figure 17(b), the fault signature matrix of the preferred MSOs shows that the preferred MSOs are the ${M S O_{1}, M S O_{5}, M S O_{6}, M S O_{8}, M S O_{14},$ ${MSO}_{23}, {MSO}_{36}, {MSO}_{37}, {MSO}_{43}, {MSO}_{50}}$ . Figure 8(a) shows that the preferred MSOs contain all the sensor configuration equations, further validating the necessity of sensor configuration. Figure 17(c) shows the fault isolation matrix of the preferred MSOs, and it can be observed that the fault isolation matrix is consistent with Figure 8(c). The fault detection rate is $FDR = 100 %$ , and the fault isolation rate is $FIR = 97.14 %$ . Figure 17(d) shows the fault reconfiguration matrix of the preferred MSOs, and the fault reconfiguration matrix is consistent with Figure 9(b). Therefore, the conclusion of the fault reconfiguration rate remains consistent with Figure 9(c), that is, the total fault reconfiguration rate of the system is 96.67%, and it is 91.67% except for the faults ${f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}, f_{7}, f_{8}, f_{9}}$ which have a reconfiguration rate of 100%.

Fixed-wing UAV reconfigurability Optimization Model

From the system design point of view, based on structurally redundant sensor configurations and reconfigurable optimization design of ICSA, the model of $M_{uav}$ that meets the maximum diagnosability and reconfigurability requirements can be obtained as:

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

(53)

where

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

(54)

F_{y} = {[f_{zx 10}, f_{zx 11}, f_{zx 12}]}^{T}

(55)

The optimal reconfigurable system is ${{MSO}_{1},$ $M S O_{5}, M S O_{6}, M S O_{8}, M S O_{14}, M S O_{23}, M S O_{36}, M S O_{43}, M S O_{50}}$ . Taking $MS O_{1}$ as an example, analogous to equation (13), the residuals $r_{1}$ are designed, and the sensitivity of $r_{1}$ to system faults is shown in the first panel (from top to bottom) of Figure 18. Where the system faults are all slope faults shown in the third panel of Figure 18. From the first panel, it can be seen that $r_{1}$ jumps when the faults ${f_{3}, f_{4}, f_{6}, f_{7}, f_{9}, f_{11}, f_{zx 11}}$ occur, which is consistent with the sensitivity of the $MS O_{1}$ to the fault in the fault signature matrix displayed in Figure 8(b), which deeper verifies the rationality and reliability of the algorithm provided in this article. When the fault $f_{zx 11}$ occurs in the sensor $y_{2}$ , the reconstruction estimation of the state $x_{11}$ based on $MS O_{5}$ is performed, and the simulation outcomes are shown in the second panel of Figure 18. In the second panel, $y_{2}$ represents the output of the sensor under normal conditions; $y_{2 f}$ represents the sensor output under fault condition; $y_{2 R}$ represents the output based on the reconfigurable estimation of $x_{11}$ . It can be seen that even if $f_{zx 11}$ occurs, the control and estimation of $x_{11}$ can still be realized based on $MS O_{5}$ , that is, $x_{11}$ is reconfigurable to the fault $f_{zx 11}$ .

Figure 18.

Reconfigurable performance validation.

Before and after optimization, the changes in the system’s fault detection rate (FDR), fault isolation rate (FIR), fault reconfiguration rate (FRR), and reconfiguration cost (RC) are shown in Table 5. Table 5 represents the order in which the algorithms are run from left to right, “Initial” represents the initial state of the system; “Algorithm 3” represents the analysis state of the system after applying Algorithm 3; “Algorithm 5” represents the analysis state of the system after applying Algorithm 5, “CSA-based,” represents the analysis state of the system after CSA-based reconfigurable optimization, and “ICSA-based,” represents the analysis state of the system after ICSA-based reconfigurable optimization. A blank position in the table means that the current state does not need to take this value into account. From Table 5, it can be seen that the FDR, FIR, FRR, and RC of the system are optimal using the algorithm designed in this paper.

Table 5.

System performance parameter changes.

Parameter	Initial (%)	Algorithm 3 (%)	Algorithm 5 (%)	CSA-based (%)	ICSA-based (%)
FDR	0	100	100	100	100
FIR	0	97.14	97.14	97.14	97.14
FRR			96.67	96.67	96.67
RC			100	28	20

Conclusion

To enhance UAV reliability and autonomous fault reconfiguration, this paper presents an optimal design strategy based on structurally redundant sensor configuration and the Improved Crow Search Algorithm (ICSA). First, through structural analysis, the correlation between Minimal Structurally Overdetermined (MSO) sets and fault reconfigurability is established, with reconfigurability evaluated using a fault reconfiguration matrix. Next, sensor configuration and MSO search algorithms are developed based on the system’s structural model: Algorithm 1 determines the minimal sensor set for maximum fault detectability in structurally exactly determined subsystems. Algorithm 2 identifies the minimal sensor set for maximum fault isolability in structurally overdetermined subsystems. Algorithm 3 integrates Algorithms 1 and 2 to achieve maximum diagnosability. Algorithm 4 extracts the system’s MSOs. Algorithm 5 derives reconfigurable MSOs by leveraging the preceding algorithms. Finally, to optimize reconfigurability while minimizing design cost, an ICSA-based reconfigurability algorithm is proposed to select the best-performing MSOs. Experimental results demonstrate that ICSA reduces the reconfiguration cost of $M_{uav}^{'}$ by 80%, an 8% improvement over the pre-optimized solution. This approach ensures efficient, cost-effective, and reliable UAV fault reconfiguration.

The key contributions of this study are fourfold:

Reconfigurability Assessment Method: A qualitative evaluation framework based on structural analysis is proposed, along with a reconfigurability assessment index.

Structural Redundancy Enhancement: A sensor configuration method is designed to increase structural redundancy, thereby improving system reconfigurability.

Reconfigurable MSO Search Algorithm: An algorithm is developed to identify reconfigurable MSO sets efficiently.

Cost-Aware Optimization Strategy: An ICSA-based design strategy is introduced to optimize reconfigurability while considering system requirements and design costs.

These contributions collectively enhance UAV fault tolerance through systematic structural and computational optimization.

Building upon the qualitative evaluation approach to system reconfigurability enhancement presented in this study, future research directions should expand into quantitative reconfigurability assessment methodologies, development of fault-tolerant control algorithms utilizing reconfigurable MSOs, investigation of diagnosability and reconfigurability in structurally underdetermined subsystems, and comprehensive analysis of interference and noise impacts on reconfigurability performance, while additionally exploring reconfigurable design applications for diverse UAV types and evaluating alternative intelligent optimization algorithms for improved reconfigurable system design.

Footnotes

ORCID iD

XuPing Gu

Author contribution

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.

Funding

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

Data availability statement

All data supporting this study’s findings are included in this manuscript and its supplementary information files.

References

Telli

Kraa

Himeur

, et al. A comprehensive review of recent research trends on unmanned aerial vehicles (UAVs). Systems 2023; 11: 400.

Gargalakos

The role of unmanned aerial vehicles in military communications: application scenarios, current trends, and beyond. J Defense Model Simul Appl Methodol Technol 2024; 21: 313–321.

Shakhatreh

Sawalmeh

Al-Fuqaha

, et al. Unmanned aerial vehicles (UAVs): a survey on civil applications and key research challenges. IEEE Access 2019; 7: 48572–48634.

Aslan

Durdu

Sabanci

, et al. A comprehensive survey of the recent studies with UAV for precision agriculture in open fields and greenhouses. Appl Sci 2022; 12: 1047.

Yang

Ding

Wang

The programming model of Air-Ground cooperative patrol between Multi-UAV and Police car. IEEE Access 2021; 9: 134503–134517.

Napoleone

Pozzetti

Macchi

Core characteristics of reconfigurability and their influencing elements. IFAC-PapersOnLine 2018; 51: 116–121.

Yang

Hua

Hongzhuan

, et al. Control reconfigurability of nonlinear system based on control redundancy. In: IEEE 10th International conference on industrial informatics, 2012, pp.815–820. IEEE. DOI: 10.1109/INDIN.2012.6301366.

Steffen

Control reconfiguration of dynamical systems. Springer-Verlag, 2005. Vol. 320.

Yang

Jiang

Staroswiecki

Fault recoverability analysis of switched systems. Int J Syst Sci 2012; 43: 535–542.

10.

Bhattacharyya

Ghosh

Basu

Design of an active compliant liquid column damper by LQR and wavelet linear quadratic regulator control strategies. Struct Control Health Monit 2018; 25: e2265.

11.

Arıcı

Kara

Robust adaptive fault tolerant control for a process with actuator faults. J Process Control 2020; 92: 169–184.

12.

Xian

Hao

Nonlinear robust fault-tolerant control of the tilt trirotor UAV under rear servo’s stuck fault: theory and experiments. IEEE Trans Ind Inform 2019; 15: 2158–2166.

13.

Van

SS.

Adaptive fuzzy integral sliding-mode control for robust fault-tolerant control of robot manipulators with disturbance observer. IEEE Trans Fuzzy Syst 2021; 29: 1284–1296.

14.

Amin

Sajid Iqbal

Hamza Shahbaz

Development of intelligent fault-tolerant control systems with machine learning, deep learning, and transfer learning algorithms: a review. Expert Syst Appl 2024; 238: 121956.

15.

Shi

A review of research on diagnosability of control systems based on structural analysis. Appl Sci 2023; 13: 12241.

16.

Schmid

Gebauer

Endisch

Structural analysis in reconfigurable battery systems for active fault diagnosis. IEEE Trans Power Electron 2021; 36: 8672–8684.

17.

Frisk

Krysander

Jung

A toolbox for analysis and design of model based diagnosis systems for large scale models. IFAC-PapersOnLine 2017; 50: 3287–3293.

18.

Reissig

Hartung

Svaricek

Strong structural controllability and observability of linear time-varying systems. IEEE Trans Automat Contr 2014; 59: 3087–3092.

19.

Tang

Liu

Pan

A review on representative swarm intelligence algorithms for solving optimization problems: applications and trends. IEEE/CAA J Automat Sin 2021; 8: 1627–1643.

20.

Zhong

, et al. Swarm intelligence-based optimization algorithms: an overview and future research issues. Int J Autom Control 2020; 14: 656–693.

21.

Cuong-Le

Minh

H-L

Khatir

, et al. A novel version of Cuckoo search algorithm for solving optimization problems. Expert Syst Appl 2021; 186: 115669.

22.

Valian

Tavakoli

Mohanna

, et al. Improved cuckoo search for reliability optimization problems. Comput Ind Eng 2013; 64: 459–468.

23.

NMohd

Khan

Rehman

MZ.

A new back-propagation neural network optimized with cuckoo search algorithm. In: Murgante

Misra

Carlini

, et al. (eds) Computational science and its applications – ICCSA 2013. Springer, 2013, pp.413–426.

24.

Song

P-C

Pan

J-S

Chu

S-C.

A parallel compact cuckoo search algorithm for three-dimensional path planning. Appl Soft Comput 2020; 94: 106443.

25.

Dulmage

Mendelsohn

NS.

Coverings of bipartite graphs. Can J Math 1958; 10: 517–534.

26.

Krysander

Aslund

Nyberg

An efficient algorithm for finding minimal overconstrained subsystems for model-based diagnosis. IEEE Trans Syste Man Cybernet Part A Syst Humans 2008; 38: 197–206.

27.

Shi

Diagnosability-optimized design of unmanned aerial vehicles based on structural analysis and maximum mean covariance differences. Measurement 2024; 238: 115334.

28.

Gehin

A-L

Assas

Staroswiecki

Structural analysis of system reconfigurability. IFAC Proc Volumes 2000; 33: 297–302.

29.

Wang

, et al. Reconfiguration incremental control allocation for tailless aerial vehicle with control surface faults. J Phys Conf Ser 2023; 2477: 012094.

30.

Mareli

Twala

An adaptive cuckoo search algorithm for optimization. Appl Comput Inform 2018; 14: 107–115.

31.

Zhang

Suresh

Jiang

, et al. Reconfigurable control allocation against aircraft control effector failures. In: 2007 IEEE International conference on control applications, 2007, pp.1197–1202. IEEE. DOI: 10.1109/CCA. 2007.4389398.

32.

Chen

Cheng

An efficient method for determining fault isolability properties based on an augmented system model. Eur J Control 2021; 58: 90–100.