In recent years, the application of drone formations has expanded significantly, driven by advancements in networked systems. Coordinating multiple drones in a leader-follower structure presents a powerful paradigm for complex tasks. However, as the scale of a drone formation increases, the likelihood and potential severity of faults, particularly in actuators, escalate. Real-time and accurate fault diagnosis is therefore critical for ensuring the safety and reliability of the entire networked system. This article presents a method for estimating actuator faults within a drone formation characterized by a directed communication topology, where the upper bound of the fault is unknown.
The core challenge lies in designing a diagnostic scheme that respects the distributed nature of information exchange in a drone formation, is robust to disturbances and nonlinearities, and does not require prior knowledge of the fault’s magnitude. To address this, we develop a distributed adaptive sliding mode observer (SMO). The observer for each drone incorporates a relative output estimation error, which aggregates information from neighboring agents based on the directed graph topology. An adaptive law is embedded to estimate the unknown bound of the actuator fault online. The stability of the global estimation error system and the reachability of the sliding surface are rigorously proven using Lyapunov theory. All observer gain matrices are computed efficiently via Linear Matrix Inequalities (LMIs). Numerical simulations on a nonlinear drone formation model demonstrate the effectiveness and rapid convergence of the proposed fault estimation scheme.
1. Introduction and Problem Formulation
Coordinated control of drone formations is a quintessential example of a cyber-physical system within the broader Internet of Things (IoT) paradigm. Each drone acts as an intelligent agent, exchanging data over a network to achieve a collective objective. While substantial research focuses on formation control and optimization, the problem of fault diagnosis in such distributed, interconnected systems has received comparatively less attention, especially for realistic scenarios involving directed communication and complex nonlinear dynamics.
Existing fault diagnosis methods for single vehicles are often centralized and do not scale well or account for the networked interactions in a drone formation. Some distributed approaches for multi-agent systems focus only on fault detection and isolation (FDI). However, for effective risk management and subsequent fault-tolerant control, precise real-time fault estimation is indispensable. Few studies address fault estimation for multi-agent systems, and they typically assume simpler linear dynamics or undirected communication graphs. This work advances the state-of-the-art by considering a nonlinear drone formation operating under a directed communication topology, which is more general and practically relevant.
The primary contributions of this work are:
- We address the fault estimation problem for a nonlinear drone formation system under a directed graph topology, a more challenging and realistic setting than undirected or fully connected networks.
- A novel distributed sliding mode observer is designed for each drone. It utilizes the relative output estimation error from neighboring drones to achieve cooperative diagnosis.
- An adaptive mechanism is integrated to handle actuator faults with unknown upper bounds, enhancing the observer’s robustness and practicality.
- The observer design is formulated as an LMI problem, ensuring systematic computation of gains. The design also allows for optimizing the observer’s tolerance to system nonlinearities.
2. Graph Theory and Drone Formation Model
2.1 Network Communication Topology
The communication within the drone formation is modeled by a directed graph $\mathcal{G}=(\mathcal{V}, \mathcal{E}, \mathcal{A})$, where $\mathcal{V}=\{V_1, V_2, …, V_N\}$ is the set of $N$ follower drones, $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of edges, and $\mathcal{A}=[a_{ij}]_{N \times N}$ is the adjacency matrix. An edge $(V_j, V_i) \in \mathcal{E}$ indicates that Drone $i$ can receive information from Drone $j$. The set of neighbors of node $i$ is $\mathcal{N}_i = \{j \mid (V_j, V_i) \in \mathcal{E}\}$. The element $a_{ij}$ of $\mathcal{A}$ is 1 if $(V_j, V_i) \in \mathcal{E}$, and 0 otherwise. The Laplacian matrix $\mathbf{L}$ is defined as $\mathbf{L} = \mathbf{D} – \mathcal{A}$, where $\mathbf{D} = \text{diag}(d_i)$ is the in-degree matrix with $d_i = \sum_{j \in \mathcal{N}_i} a_{ij}$.
The leader-follower structure is encoded using a pinning matrix $\mathbf{G} = \text{diag}(g_1, g_2, …, g_N)$. If follower $i$ has direct access to the leader’s state information, then $g_i = 1$; otherwise, $g_i = 0$. We assume at least one follower is pinned to the leader, ensuring the matrix $(\mathbf{L}+\mathbf{G})$ is nonsingular, which is crucial for the stability of the observer error dynamics.
2.2 Dynamics of the Drone Formation
Consider a formation of $N$ follower drones and one leader. The $i$-th follower drone, subject to actuator faults and disturbances, is modeled by the following state-space representation:
$$
\begin{aligned}
\dot{\mathbf{x}}_i(t) &= \mathbf{A} \mathbf{x}_i(t) + \mathbf{B} \mathbf{u}_i(t) + \mathbf{g}(\mathbf{x}_i(t)) + \mathbf{E} \mathbf{f}_i(t) + \mathbf{D} \boldsymbol{\phi}_i(t), \\
\mathbf{y}_i(t) &= \mathbf{C} \mathbf{x}_i(t),
\end{aligned}
$$
where $i = 1, 2, …, N$. The vectors $\mathbf{x}_i(t) \in \mathbb{R}^n$, $\mathbf{u}_i(t) \in \mathbb{R}^m$, and $\mathbf{y}_i(t) \in \mathbb{R}^p$ represent the state, control input, and output of the $i$-th drone, respectively. The nonlinear term $\mathbf{g}(\mathbf{x}_i(t))$ satisfies the Lipschitz condition $||\mathbf{g}(\mathbf{x}_1) – \mathbf{g}(\mathbf{x}_2)|| \le \gamma ||\mathbf{x}_1 – \mathbf{x}_2||$. The actuator fault $\mathbf{f}_i(t) \in \mathbb{R}^r$ is bounded but its upper bound $\alpha$ (where $||\mathbf{f}_i(t)|| \le \alpha$) is unknown. The external disturbance $\boldsymbol{\phi}_i(t) \in \mathbb{R}^q$ is bounded with a known bound $||\boldsymbol{\phi}_i(t)|| \le \beta$. The matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{E}$, and $\mathbf{D}$ are constant, real, and of appropriate dimensions. The pair $(\mathbf{A}, \mathbf{C})$ is observable, and $\mathbf{C}$ and $\mathbf{E}$ are of full rank. The leader acts as a command generator with a known, ideal state.
| Symbol | Dimension | Description |
|---|---|---|
| $\mathbf{x}_i(t)$ | $\mathbb{R}^n$ | State vector of the $i$-th drone |
| $\mathbf{u}_i(t)$ | $\mathbb{R}^m$ | Control input vector |
| $\mathbf{y}_i(t)$ | $\mathbb{R}^p$ | Output vector |
| $\mathbf{g}(\mathbf{x}_i(t))$ | $\mathbb{R}^n$ | Lipschitz nonlinear function |
| $\mathbf{f}_i(t)$ | $\mathbb{R}^r$ | Actuator fault vector (unknown bound) |
| $\boldsymbol{\phi}_i(t)$ | $\mathbb{R}^q$ | External disturbance (known bound) |
| $\mathcal{A}, \mathbf{L}$ | $\mathbb{R}^{N \times N}$ | Adjacency and Laplacian matrices |
| $\mathbf{G}$ | $\mathbb{R}^{N \times N}$ | Pinning matrix for leader-follower structure |
3. Distributed Adaptive Sliding Mode Observer Design
3.1 Observer Structure
For each drone in the formation, we propose the following distributed adaptive sliding mode observer:
$$
\begin{aligned}
\dot{\hat{\mathbf{x}}}_i(t) &= \mathbf{A} \hat{\mathbf{x}}_i(t) + \mathbf{B} \mathbf{u}_i(t) + \mathbf{g}(\hat{\mathbf{x}}_i(t)) – \mathbf{v}_i(t) – \mathbf{K} \boldsymbol{\xi}_i(t), \\
\hat{\mathbf{y}}_i(t) &= \mathbf{C} \hat{\mathbf{x}}_i(t),
\end{aligned}
$$
where $\hat{\mathbf{x}}_i(t)$ and $\hat{\mathbf{y}}_i(t)$ are the estimated state and output. $\mathbf{K} \in \mathbb{R}^{n \times p}$ is the observer gain matrix to be designed. The term $\boldsymbol{\xi}_i(t)$ is the relative output estimation error, crucial for information fusion across the drone formation:
$$
\boldsymbol{\xi}_i(t) = \sum_{j \in \mathcal{N}_i} a_{ij} \left[ (\hat{\mathbf{y}}_i(t) – \mathbf{y}_i(t)) – (\hat{\mathbf{y}}_j(t) – \mathbf{y}_j(t)) \right] + g_i (\hat{\mathbf{y}}_i(t) – \mathbf{y}_i(t)).
$$
The sliding mode control input $\mathbf{v}_i(t)$ is designed as:
$$
\mathbf{v}_i(t) =
\begin{cases}
\frac{\rho_0 + \hat{\rho}_i(t)}{||\mathbf{F} \mathbf{e}_{yi}(t)|| + \omega} \mathbf{P}^{-1} \mathbf{C}^T \mathbf{F}^T \mathbf{F} \mathbf{e}_{yi}(t), & \text{if } \hat{\mathbf{y}}_i(t) \neq \mathbf{y}_i(t), \\
0, & \text{if } \hat{\mathbf{y}}_i(t) = \mathbf{y}_i(t),
\end{cases}
$$
where $\mathbf{e}_{yi}(t) = \hat{\mathbf{y}}_i(t) – \mathbf{y}_i(t)$ is the local output estimation error, $\mathbf{P}$ and $\mathbf{F}$ are design matrices, $\rho_0 > 0$ is a constant, and $\omega$ is a small positive scalar to mitigate chattering. The adaptive parameter $\hat{\rho}_i(t)$, estimating the combined effect of the fault and its bound, is updated by:
$$
\dot{\hat{\rho}}_i(t) = \eta ||\mathbf{F} \mathbf{e}_{yi}(t)||,
$$
with $\eta > 0$ as the adaptive gain.
3.2 Stability Analysis and Observer Synthesis
Defining the state estimation error as $\mathbf{e}_{xi}(t) = \hat{\mathbf{x}}_i(t) – \mathbf{x}_i(t)$ and aggregating errors for the entire drone formation, we obtain the global error dynamics:
$$
\dot{\mathbf{e}}_x(t) = \left[ \mathbf{I}_N \otimes \mathbf{A} – (\mathbf{L}+\mathbf{G}) \otimes (\mathbf{K} \mathbf{C}) \right] \mathbf{e}_x(t) + \mathbf{e}_g(t) + \mathbf{v}(t) – (\mathbf{I}_N \otimes \mathbf{E}) \mathbf{f}(t) – (\mathbf{I}_N \otimes \mathbf{D}) \boldsymbol{\phi}(t),
$$
where $\otimes$ denotes the Kronecker product, and $\mathbf{e}_x(t)$, $\mathbf{e}_g(t)$, $\mathbf{v}(t)$, $\mathbf{f}(t)$, $\boldsymbol{\phi}(t)$ are the concatenated global vectors.
Theorem 1 (Stability): For the drone formation system (1) and the observer (2)-(5), if there exist symmetric positive definite matrices $\mathbf{P} \in \mathbb{R}^{n \times n}$, matrices $\mathbf{Y} \in \mathbb{R}^{n \times p}$, $\mathbf{F} \in \mathbb{R}^{r \times p}$, and a scalar $\theta > 0$ such that the following Linear Matrix Inequalities (LMIs) are feasible:
$$
\begin{bmatrix}
\mathbf{\Psi} & \mathbf{I}_N \otimes \mathbf{P} & \mathbf{0} \\
* & -\gamma^{-2} \mathbf{I}_{nN} & \mathbf{0} \\
* & * & -\mathbf{I}_{nN}
\end{bmatrix} < 0, \quad \text{and} \quad \begin{bmatrix}
-\theta \mathbf{I} & \mathbf{E}^T \mathbf{P} – \mathbf{C}^T \mathbf{F}^T \\
* & -\theta \mathbf{I}
\end{bmatrix} < 0,
$$
where $\mathbf{\Psi} = \mathbf{I}_N \otimes (\mathbf{P}\mathbf{A} + \mathbf{A}^T\mathbf{P}) – (\mathbf{L}+\mathbf{G}) \otimes (\mathbf{Y}\mathbf{C} + \mathbf{C}^T\mathbf{Y}^T)$, then the global state estimation error $\mathbf{e}_x(t)$ is uniformly ultimately bounded. The observer gain is given by $\mathbf{K} = \mathbf{P}^{-1} \mathbf{Y}$, and $\gamma$ represents the maximum allowable Lipschitz constant.
Proof Sketch: Consider the Lyapunov function candidate $V_1(t) = \mathbf{e}_x^T(t) (\mathbf{I}_N \otimes \mathbf{P}) \mathbf{e}_x(t) + \eta^{-1} \tilde{\boldsymbol{\rho}}^T(t) \tilde{\boldsymbol{\rho}}(t)$, where $\tilde{\boldsymbol{\rho}}(t)$ is the adaptive parameter error. Using the Lipschitz condition, the SMO structure, and the adaptive law, the derivative $\dot{V}_1(t)$ can be shown to be negative outside a compact region around the origin, proving ultimate boundedness. The LMIs are derived by applying the Schur complement lemma to enforce this negativity condition.
Theorem 2 (Sliding Mode Reachability): If the gain $\rho_0$ in the SMO input (4) is selected such that:
$$
\rho_0 \ge \frac{||\mathbf{A}_{21}|| \vartheta + \gamma \vartheta + \beta ||\mathbf{I}_N \otimes (\mathbf{C}\mathbf{D})|| + \sigma}{||\mathbf{I}_N \otimes (\mathbf{C}\mathbf{P}^{-1}\mathbf{C}^T\mathbf{F}^T)||},
$$
where $\vartheta$ is the radius of the ultimate bound from Theorem 1 and $\sigma > 0$, then the output estimation error $\mathbf{e}_y(t)$ will reach the sliding manifold $\mathcal{S} = \{ \mathbf{e}_y(t) | \mathbf{e}_y(t) = \mathbf{0} \}$ in finite time.
Proof Sketch: A transformation matrix is applied to the error dynamics to separate the output error subspace. A second Lyapunov function $V_2(t) = \frac{1}{2} \mathbf{e}_y^T(t) (\mathbf{I}_N \otimes (\mathbf{C}\mathbf{P}^{-1}\mathbf{C}^T)^{-1}) \mathbf{e}_y(t) + …$ is analyzed. Using the condition on $\rho_0$, it can be shown that $\dot{V}_2(t) \le -\sigma ||\mathbf{e}_y(t)||$, guaranteeing finite-time convergence to the sliding surface.
3.3 Actuator Fault Estimation
Once the sliding mode is established and maintained ($\mathbf{e}_y(t) = \dot{\mathbf{e}}_y(t) = \mathbf{0}$), the equivalent control signal $\mathbf{v}_{eq}(t)$ can be extracted by solving the algebraic equation derived from the error dynamics in the sliding phase. This equivalent control contains information about the fault. The actuator fault for the entire drone formation can then be estimated as:
$$
\hat{\mathbf{f}}(t) \approx – \left[ (\mathbf{I}_N \otimes (\mathbf{C}\mathbf{E}))^T (\mathbf{I}_N \otimes (\mathbf{C}\mathbf{E})) \right]^{-1} (\mathbf{I}_N \otimes (\mathbf{C}\mathbf{E}))^T (\mathbf{I}_N \otimes \mathbf{C}) \mathbf{v}_{eq}(t).
$$
This provides a direct, online estimate of the fault signals affecting each drone in the formation.
| Step | Action | Tool/Result |
|---|---|---|
| 1 | Model the drone formation communication as a directed graph $\mathcal{G}$ with pinning matrix $\mathbf{G}$. | Graph $\mathcal{G}$, matrices $\mathbf{L}$, $\mathbf{G}$. |
| 2 | Formulate the dynamics of each drone including nonlinearity $\mathbf{g}(\cdot)$, fault $\mathbf{f}_i(t)$, and disturbance $\boldsymbol{\phi}_i(t)$. | System matrices $\mathbf{A, B, C, E, D}$, Lipschitz constant $\gamma$. |
| 3 | Solve the LMIs from Theorem 1 for matrices $\mathbf{P, Y, F}$ and scalar $\theta$. | LMI Solver (e.g., MATLAB’s feasp). Gains $\mathbf{K}=\mathbf{P}^{-1}\mathbf{Y}$. |
| 4 | Select adaptive gain $\eta$ and compute $\rho_0$ using the condition from Theorem 2. | Parameters $\eta$, $\rho_0$. |
| 5 | Implement the distributed observer (2) for each drone, using Eqs. (3)-(5). | Online estimation of states $\hat{\mathbf{x}}_i(t)$ and fault $\hat{\mathbf{f}}_i(t)$. |
4. Simulation Results and Analysis
To validate the proposed methodology, a simulation of a drone formation consisting of 1 leader and 5 followers with the directed communication topology shown in the figure is conducted. The lateral dynamics of each drone are considered, with the system matrix $\mathbf{A}_{lat}$ and input matrix $\mathbf{B}_{lat}$ as given below. A nonlinear term $g_1(\mathbf{x}_i(t)) = -6.3541 \times \sin(x_{i4}(t))$ is added to the first state channel.
The LMI conditions are solved using MATLAB, yielding the gain matrix $\mathbf{K}$. The simulation parameters are: step size = 0.001 s, disturbance $\boldsymbol{\phi}_i(t) = 0.05 \cos(7t)$, adaptive gain $\eta = 1$, and $\omega = 0.01$. Different types of actuator faults are injected into the followers at various times:
- Drone 1: A step fault of magnitude 0.3 at t=2s.
- Drone 2: A sinusoidal fault $0.3\sin(t)$ at t=2s.
- Drone 3: A sinusoidal fault $0.5\sin(t)$ and a constant bias of 0.6 at t=2s.
- Drone 4: A complex fault $0.5\sin(0.3\cos(t))$ at t=3s.
- Drone 5: A persistent sinusoidal fault $0.3\sin(t)$ from the start.
| Matrix | Value |
|---|---|
| $\mathbf{A}_{lat}$ | $\begin{bmatrix} -0.732 & -0.0143 & 0.996 & 0.0706 \\ 893.000 & -9.059 & 0 & -2.044 \\ 101.637 & 0.0186 & -1.283 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}$ |
| $\mathbf{B}_{lat}$ | $\begin{bmatrix} 0 & 0.244 \\ 328.653 & 308.498 \\ 47.528 & -102.891 \\ 0 & 0 \end{bmatrix}$ |
| $\mathbf{C}_{lat}$ | $\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$ |
| $\mathbf{E}$ | $\mathbf{B}_{lat}$ |
The simulation results demonstrate the effectiveness of the distributed adaptive SMO. The fault estimates $\hat{\mathbf{f}}_i(t)$ for all drones converge rapidly to the true fault profiles $\mathbf{f}_i(t)$ within approximately 0.5 seconds of the fault occurrence. The estimation error is negligible for constant faults. For time-varying faults, a very small transient error is observed immediately after fault inception, but it is quickly eliminated. Notably, even when multiple faults are injected simultaneously into the drone formation (e.g., in Drones 1, 2, and 3 at t=2s), the observer accurately estimates all faults with minimal cross-coupling effects. This robustness is a key advantage of the distributed design, where information sharing among neighboring drones enhances the diagnostic capability of the entire formation.
5. Conclusion
This article has presented a comprehensive solution for the problem of actuator fault estimation in a nonlinear drone formation operating under a directed communication topology. The proposed distributed adaptive sliding mode observer successfully addresses the challenges of unknown fault bounds, external disturbances, and system nonlinearities. By leveraging the relative output estimation error from the network, each drone’s observer benefits from cooperative information exchange, leading to accurate and robust fault diagnosis for the entire formation. The theoretical framework, grounded in Lyapunov stability and LMI-based synthesis, guarantees the convergence of the estimation errors. Simulation studies on a realistic drone model confirm the practical viability and rapid performance of the method.
Future work will focus on extending this approach to handle sensor faults, which are equally critical for drone formation safety. Furthermore, investigating fault diagnosis and fault-tolerant control for heterogeneous drone formations, where agents have different dynamics and capabilities, presents a significant and challenging direction for research. Integrating the developed fault estimation scheme with a distributed fault-tolerant controller to achieve resilient drone formation control is another promising avenue.