Event-triggered consensus control method with communication faults for multi-UAV

2.1. Problem formulation

Consider a group of UAVs $$ i $$, $$ i \in N = \left\{1, …, N\right\} $$, facing communication faults in that partial position sensors are damaged. The dynamic of each UAV $$ i $$ is described by

where $$ x_{i}(t) \in \mathbb{R}^{n} $$, $$ v_{i}(t) \in \mathbb{R}^{n} $$, and $$ u_{i}(t) \in \mathbb{R}^{n} $$ are the position velocity and control input of UAV $$ i $$ at time $$ t $$, respectively, in the inertial frame. $$ m_{i} > 0 $$ is the mass of UAV $$ i $$, which can also be generalized as the inertia index or decision weight.

Remark 1. Referring to hierarchical interaction mechanisms, the decision weight is influenced by individual attribute, which is determined by social relationships and interaction patterns. Higher decision weight means that an individual is less susceptible to the influence of neighbors. Therefore, the conclusion of this paper can be extended to the heterogeneous system. In addition, this paper focuses on the consistency proof of large-scale network topology based on the graph theory. Using traditional drone models will make the proof process obscure and cumbersome. The control input in this paper can be considered as the expected acceleration. Therefore, the dynamic model of UAVs has been simplified during the proof process.

Definition 1. The heterogeneous multi-UAV system (1) is said to reach consensus for any initial conditions, when and only when we have $$ \lim\limits_{t\to+\infty} \parallel{x_{i}-x_{j}}\parallel = 0 $$ and $$ \lim\limits_{t\to+\infty} \parallel{\dot{x}_{i} - \dot{x}_{j}}\parallel = 0 $$ for $$ \forall i $$, $$ j \in N $$.

To achieve urgent task objectives, an event-triggered consensus protocol will be proposed based on the following second-order consensus protocol:

where $$ k $$ and $$ b $$ are stiffness gain and damping gain, respectively. $$ w_{ij} $$ is the coupling coefficient of position information interaction, and $$ v_{ij} $$ is the coupling coefficient of velocity information interaction. If $$ w_{ij} > 0 $$ or $$ v_{ij} > 0 $$, it means that the relevant information of UAV $$ j $$ can be captured by UAV $$ i $$. How to achieve consensus in system (1) based on the above protocol and event triggering mechanism is the problem that needs to be addressed in this paper.

Remark 2. The communication and sensor faults assumed in this paper refer to the inability of individuals to obtain information sent by neighbors through wireless data transmission or other means. Therefore, in order to cope with situations where wireless data transmission cannot be utilized due to strong interference, the method of individuals acquiring information through sensors, such as position and velocity, is widely adopted. We further assume that position sensors of some individuals are damaged, and they are unable to obtain the position information of surrounding individuals (in fact, the processing methods for damaged position sensors and speed sensors are generally similar, and this article only discusses the former), which is reflected in the Laplacian matrix that contains all-zero rows.

2.2. Preliminaries

Lemma 1. Communication topology can be represented as a weighted directed (undirected) graph $$ G = (V_{n}, \varepsilon, A) $$ of order $$ n $$ with a vertex set $$ V_{n} = \left\{1, 2, …, n\right\} $$ and edge set $$ \varepsilon \subset V_{n} \times V_{n} $$ and a non-negative symmetric matrix $$ A =[a_{ij}]_{n \times n} $$. $$ (s_{i}, s_{j}) \in \varepsilon \Leftrightarrow a_{ij} > 0 \Leftrightarrow $$ the information of individual $$ j $$ can be captured by individual $$ i \Leftrightarrow j $$ is the neighbor member of individual $$ i $$. We assume $$ a_{ii} = 0 $$. The neighbor set of individual $$ i $$ is represented by $$ N_{i} = \left\{j|(i, j) \in \varepsilon \right\} $$. The Laplacian matrix of the weighted diagraph is defined as $$ L = [l_{ij}] $$, where $$ l_{ij} = -a_{ij} $$ and $$ l_{ii} = \Sigma_{j \neq i}a_{ij} $$.

Lemma 2^[25]. If graph $$ G $$ contains at least one directed spanning tree, its corresponding Laplacian matrix $$ L $$ satisfies the following properties:

(a) $$ rank(L) = n - 1 $$;

(b) 0 is an eigenvalue of matrix $$ L $$, and $$ [l, l, …, l]^T $$ is its corresponding eigenvector;

(c) $$ \mathrm{Re}(\lambda_l) \ge 0 $$, $$ \forall i \in \left\{1, 2, …, n\right\} $$; and there is only one eigenvalue of 0;

(d) Laplacian matrix $$ L $$ related to the strongly connected graph $$ G $$ is an irreducible matrix.

Laplacian matrix $$ L = [l_{ij}]_{n \times n} $$ and $$ H = [h_{ij}]_{n \times n} $$ are defined as:

The mass matrix of the system is recorded as $$ M = diag(m_{1}, m_{2}, …, m_{n}) > 0 $$. Describe system (1) by using matrices as follows:

Remark 3. Actual physical meaning in the formula denoted by $$ I_{N} $$ is the dimension of state space, which is usually defined as 3.

Due to the limited refresh rate and sampling frequency of sensors and processors, the event triggering mechanism is proposed to reduce the pressure on sensors and save processor resources while ensuring that they can still react quickly in the face of unexpected situations.

State error is an important decision factor in event-triggered consensus control. Define $$ \xi_{i}(t) = x_{i}(t) - x_{1}(t) $$ and $$ \eta_{i}(t) = v_{i}(t) - v_{1}(t) $$, which denote position error and state error, respectively. Apart from this, define $$ \xi(t) = col(\xi_{2}(t), …, \xi_{n}(t)) $$ and $$ \eta(t) = col(\eta_{2}(t), …, \eta_{n}(t)) $$, $$ \varepsilon(t) = col(\xi(t), \eta(t)) $$. Suppose the incremental time column that satisfies the event-triggered condition is $$ \left\{t_{1}, …, t_{k}\right\} $$. System (1) is equivalent to

We define

And we have

Therefore, system (5) can be converted to the form in continuous time gives

Different from the previous consensus control methods [Similar to the form of System (5)] for the UAV system (1), the individuals are supposed to guarantee the interaction of velocity through independent information collection of position and velocity [the form of System (4)] when extreme cases, such as partial damage to position sensors, are considered, which is also the difficulty and focus of this study.

3.1. Linear transformation of the system

First, System (4) can be transformed into the form of system (5) based on the lemma as follows:

Lemma 3^[26]. For Laplacian matrix related to the directed graph, there exists a non-singular matrix

so that

where $$ h \in \mathbb{R}^{n-1} $$ and $$ H \in \mathbb{R}^{(n-1) \times (n-1)} $$. $$ U^{-1} $$ has the form as

where $$ \sum_{i=1}^{n}\gamma_{i}=1 $$.

Therefore, non-singular linear transformations are built as $$ \xi=U_{-1}\otimes I_{N}x $$, and $$ U^{-1} $$ is defined as

Thus, $$ \xi_{1}=\sum_{i=1}^{n}\gamma_{i}x_{i} $$ and $$ \xi_{k}=x_{k}-x_{1} $$, where $$ k=\left\{2, 3, \dots, n\right\} $$. The state vector of the system is transformed into the error vector by linear transformations. System (4) is equivalent to the system as follows:

Lemma 4. For Laplacian matrix $$ L = [l_{ij}]_{n\times n} $$ related to the strongly connected graph $$ G $$, there exists positive vector $$ \gamma = [\gamma_{1}, \gamma_{2}, \dots, \gamma_{n}] $$ that is the eigenvector corresponding to the eigenvalue 0 of the matrix $$ \Lambda L $$, where $$ \Lambda = M^{-1} > 0 $$ and $$ \sum_{i=1}^{n}\gamma_{i} = 1 $$.

Proof: Define $$ A = [a_{ij}]_{n\times n} = [-l_{ij}]_{n\times n} $$ and $$ w = \max_{i\in N}|a_{ii}/m_{i}| $$. Since Laplacian matrix $$ L $$ is related to the strongly connected graph, matrix $$ \Lambda A + wI $$ is semi-positive definite. Based on the Gershgorin circle theorem, the eigenvalues of the matrix $$ \Lambda A $$ and $$ \Lambda A + wI $$ satisfy conditions as follows:

where $$ i\in N $$. Since the eigenvalues of the matrix will not be altered after transposition, for $$ \forall i\in N $$

Since $$ A^{T}\Lambda^{T} + wI \ge 0 $$, there exists a non-negative vector $$ \xi^{T} $$ corresponding to the right eigenvector of the eigenvalues $$ w $$. Thus, one has that

then

Since $$ A = -L $$, there exists a non-negative vector $$ \xi $$ satisfies the condition as follows:

On the other hand, since matrix $$ L $$ is related to the strongly connected graph, and according to Lemma 2, $$ L $$ is an irreducible matrix. Therefore, according to the Perron-Frobenius theorem, vector $$ \xi $$ is a positive vector. Then, the vector $$ \gamma $$ to be found is expressed as

The proof is, thus, completed.

According to Lemma 4, the positive vector $$ \gamma = [\gamma_{1}, \gamma_{2}, \dots, \gamma_{n}] $$ is defined as corresponding to matrix $$ M^{-1}L $$ in system (16), where $$ \sum_{i=1}^{n}\gamma_{i} = 1 $$. Define $$ \Gamma = diag(\gamma_{1}, \gamma_{2}, \gamma_{3}, \dots, \gamma_{n}) $$, and $$ \Gamma $$ is an invertible matrix. Then, one has inferences as follows:

Furthermore, system (13) is equivalent to the system as follows:

Therefore:

Similarly, one has that

Define $$ \xi = col(\xi_{1}, \xi_{e}) $$, where $$ \xi_{e} = col(\xi_{2}, \xi_{3}, \dots, \xi_{n}) $$. System (22) is equivalent to the system as follows:

From system (26), together with (5), one has that

where $$ F_{1} = -k\widetilde{\Gamma}^{-1}\widetilde{H} $$, $$ F_{2} = -b\widetilde{\Gamma}^{-1}\widetilde{L} $$, and $$ \varepsilon(t) = col(\xi_{e}(t), \eta_{e}(t)) $$. Therefore, the event-triggered consensus protocol in this paper can be written as:

According to (6), (7), and (8), Converting system (27) to the form in continuous time gives:

where $$ e(t) $$ is defined as

Thus, the proof of consensus in system (1) is transformed into the proof of stability of system (27).

Remark 4. The stability of system (27) implies that the state errors between the UAVs are zero. According to Definition 1, these two propositions are equivalent.

3.2. Analysis of stability

Now, the main result of this paper can be given as follows.

Theorem 1. Consider system (27) and event-triggered consensus protocol (28), sufficient conditions for the stability of the system are given as follows:

where

Proof: Choose the Lyapunov function as

in which

Define $$ \widetilde{P} = T^{T}PT $$, where

The contract transformation does not change the positivity of the matrix, and

Since condition (31) implies that $$ \Gamma $$ is a positive definite matrix, one has that $$ \widetilde{P} > 0 $$. Therefore, $$ P $$ is a positive definite matrix.

Differentiate $$ V $$ to obtain that

in which

And $$ Q = Q^{T} $$. Performing the contract transformation on matrix $$ Q $$, one can obtain that

where

According to conditions (31) and (32), matrix is a positive definite matrix. One has that

in which

From condition (33), one has that

Define an event-triggered function as follows:

At an event time $$ t \in \left\{t_{1}, \dots, t_{k}\right\} $$, one can obtain:

which denotes that condition (33) can always be satisfied.

Therefore, based on Lyapunov stability principles, for an arbitrary initial state $$ \xi(0) $$ and $$ \eta(0) $$ of system (27), one can achieve that

which are equivalent to that

where $$ i, j \in \left\{1, 2, \dots, n\right\} $$. The stability of system (27) can be achieved, which denotes that the consensus of system (1) can be achieved.

The proof is thus completed.

Remark 5. According to the event-triggered function (50), the event-triggered condition is met when the error $$ \parallel e \parallel $$ exceeds the threshold, which means that the accumulated error in the information received at the previous triggering time has exceeded the stable range, and the control input needs to be updated immediately. Apparently, this event-triggered condition is easier to achieve in emergency situations due to the rapid and drastic state changes of neighbors.

Theorem 2. Consider system (27) and event-triggered consensus protocol (28), the system will not exhibit the Zeno behavior, which means that the time interval between any two events will not be less than

in which

$$ \beta = \phi(\tau, 0) = \frac{\sigma}{2}\frac{\lambda_{\mathrm{min}}(Q)}{\parallel \bar{P}\parallel} $$

Proof: Similar to the proof in^[32], we define

$$ y = \frac{\parallel e\parallel}{\parallel \varepsilon\parallel} $$

And one has that

$$ \dot{y} = (\frac{-e^{T}}{\parallel e \parallel}-\frac{\varepsilon^{T}\parallel e \parallel}{\parallel\varepsilon\parallel^{2}})\frac{\dot{\varepsilon}}{\parallel\varepsilon\parallel}\\ \leq (1+\frac{\parallel e \parallel}{\parallel\varepsilon\parallel})\frac{\parallel\dot{\varepsilon}\parallel}{\parallel\varepsilon\parallel}\\ = (1 + y)(\parallel F\parallel + \parallel J\parallel y) $$

and $$ y $$ satisfies that

$$ y(t) \leq \phi(t, \phi_{0}) $$

in which $$ \phi(t, \phi_{0}) $$ is the solution of

$$ \dot{\phi} = (1 + \phi)(\parallel F\parallel + \parallel J\parallel\phi)\\ \phi(0, \phi_{0}) = \phi_{0} $$

From (33), the solution of the equation above also satisfies that

$$ \phi(\tau, 0) = \frac{\sigma}{2}\frac{\lambda_{\mathrm{min}}(Q)}{\parallel \bar{P}\parallel} $$

so that

$$ \tau = \frac{1}{\parallel F\parallel - \parallel J\parallel}\ln\frac{1 + \phi(\tau, 0)}{1 + \frac{\parallel J\parallel}{\parallel F\parallel}\phi(\tau, 0)} $$

The proof is, thus, completed.

Theorem 3. Consider system (27) and event-triggered consensus protocol (28), for any positive definite matrix $$ Q $$, if there exists a positive definite matrix $$ P $$ satisfying $$ Q = -(F^{T}P + PF) $$, suitable parameter $$ \sigma $$ and trigger functions $$ f(e) $$ can always be designed so that the system achieves consensus under the event-triggered conditions based on the above proof process.

Theorem 4. For the multi-UAV system, appropriate distance should be maintained between individuals. Consensus protocol (2) can be transformed into

$$ u_{i}(t) = -\sum_{j\in N_{i}}\left\{kw_{ij}[(x_{i} - d_{i}) - (x_{j} - d_{j})] + bv_{ij}[(\dot{x}_{i} - \dot{d}_{i}) - (\dot{x}_{j} - \dot{d}_{j})]\right\} \\ = -\sum_{j\in N_{i}}[kw_{ij}(x_{i} - x_{j} - d_{ij}) + bv_{ij}(\dot{x}_{i} - \dot{x}_{j})] $$

in which $$ d_{i} $$ is the constant offset. We define $$ x_{i}^{'} = x_{i} - d_{i} $$ and $$ \dot{x}_{i}^{'} = \dot{x}_{i} $$ so that

$$ u_{i}(t) = -\sum_{j\in N_{i}}\left\{kw_{ij}[(x_{i} - d_{i}) - (x_{j} - d_{j})] + bv_{ij}[(\dot{x}_{i} - \dot{d}_{i}) - (\dot{x}_{j} - \dot{d}_{j})]\right\}\\ = -\sum_{j\in N_{i}}[kw_{ij}(x_{i}^{'} - x_{j}^{'}) + bv_{ij}(\dot{x}_{i}^{'} - \dot{x}_{j}^{'})] $$

Therefore, offset $$ d_{i} $$ will not affect the consensus of the system.

Authors' contributions

Made significant contributions to the research direction and design and conducted theoretical analysis, proof, and explanation: Guo Z, Wei C, Shen Y

Providing administrative, technical, and material support: Yuan W

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported by the Science and Technology Innovation 2030-Key Project of "New Generation Artificial Intelligence" (No. 2018AAA0102403) and the National Natural Science Foundation of China under grants (No. T2121003, No. U20B2071, No. 91948204, and No. U19B2033).

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

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Copyright

© The Author(s) 2023.