With the rapid advancement of unmanned aerial vehicle technology, multi-agent cooperative control, and embedded systems, unmanned aerial vehicle formation flight has demonstrated broad application prospects across domains such as cooperative reconnaissance, search and rescue, agricultural monitoring, and adversarial interception. The ability of a swarm of unmanned aerial vehicles to efficiently accomplish complex, diversified, and high-precision missions that are difficult for a single platform stems from its distributed cooperative advantages. Over the past decade, formation control of unmanned aerial vehicles has become a research hotspot and a challenging problem in the field of multi-agent control. Differential game theory, as an important branch of game theory, is essentially a class of dynamic optimal control problems involving multiple decision-makers and multiple interrelated optimization objectives. The core objective of formation control for unmanned aerial vehicles is to achieve coordinated and stable flight through properly designed control strategies. However, traditional formation control methods often suffer from limitations in multi-vehicle cooperative optimization, disturbance adaptability, and multi-objective balancing.
In this work, I address the formation control problem of unmanned aerial vehicles by equivalently transforming it into a differential game problem. In my formulation, each unmanned aerial vehicle is treated as an independent decision-maker in the differential game, and the control objectives—such as minimizing formation errors and energy consumption—are transformed into the optimization objectives of each player. By solving the differential game problem, I obtain the optimal control strategy for each unmanned aerial vehicle, thereby realizing coordinated and stable control of the formation. To this end, I investigate two specific technical routes: first, a hierarchical control framework integrating differential game, Q-learning, and control barrier functions is developed for the secure control of multi-unmanned aerial vehicle formations under hybrid attacks; second, a cooperative control strategy integrating topology optimization, disturbance compensation, and differential game is proposed to address wake interference, low communication efficiency, and insufficient control robustness in formation flight.
Preliminary Knowledge and Problem Framework
In this section, I present the foundational theories that support my subsequent research, including differential game theory, control barrier functions, graph theory, and memory event-triggered mechanisms. These principles are then integrated into a unified framework to solve the cooperative control problem of unmanned aerial vehicle swarms.
Differential Game Theory
Game theory is a mathematical method for analyzing how decision-makers interact and make strategic choices to maximize their own benefits. The core elements include players, strategy sets, payoff functions, information structures, and equilibrium solutions. Differential game, as a dynamic branch of game theory, combines differential equations and optimal control theory to solve continuous-time multi-player decision problems. In my control framework, a differential game system with $N$ players is defined over the time horizon $[t_0, T]$. The system dynamics are described by
$$
\dot{x}_i(t) = f_i\big(x(t), u_1(t), u_2(t), \ldots, u_N(t), t\big), \quad i = 1, 2, \ldots, N,
$$
where $x_i(t) \in \mathbb{R}^{n_i}$ is the state vector of player $i$, $x(t) = [x_1^T(t), x_2^T(t), \ldots, x_N^T(t)]^T \in \mathbb{R}^n$ is the aggregate state vector with $n = \sum_{i=1}^N n_i$, and $u_i(t) \in U_i$ is the control strategy of player $i$. The performance index of player $i$ is given by
$$
J_i(u_1, \ldots, u_N) = \phi_i\big(x(T)\big) + \int_{t_0}^{T} L_i\big(x(t), u_1(t), \ldots, u_N(t), t\big) \, dt,
$$
where $\phi_i(x(T))$ represents the terminal cost, such as the formation error penalty at the final time, and $L_i(\cdot)$ is the running cost capturing formation errors and energy consumption. A set of strategies $\{u_1^*, u_2^*, \ldots, u_N^*\}$ forms a Nash equilibrium if for every player $i$ and any admissible $u_i \in U_i$,
$$
J_i(u_1^*, \ldots, u_{i-1}^*, u_i, u_{i+1}^*, \ldots, u_N^*) \geq J_i(u_1^*, \ldots, u_{i-1}^*, u_i^*, u_{i+1}^*, \ldots, u_N^*).
$$
In particular, the zero-sum differential game is a special case where the sum of all players’ performance indices is identically zero, i.e., $\sum_{i=1}^N J_i = 0$. For a two-player zero-sum game, the Nash equilibrium reduces to a saddle-point equilibrium satisfying
$$
J_1(u_1^*, u_2) \leq J_1(u_1^*, u_2^*) \leq J_1(u_1, u_2^*).
$$
This formulation aligns naturally with the formation control problem: each unmanned aerial vehicle minimizes its own local formation error while interacting dynamically with other vehicles, ultimately achieving a system-level equilibrium.
Control Barrier Functions
Control barrier functions provide a powerful tool for enforcing safety constraints in real-time control. In my framework, I define a continuously differentiable scalar function $h(x)$ that maps the system state to a real value. The safe set is defined as $\mathcal{C} = \{x \in \mathbb{R}^n : h(x) \geq 0\}$. For the safety constraint to be forward invariant, the function $h$ must satisfy the following condition:
$$
\dot{h}\big(x(t)\big) \geq -\gamma h\big(x(t)\big),
$$
with a positive constant $\gamma > 0$. In the context of formation control for unmanned aerial vehicles, I set $h(x) = d_{ij}^2 – d_{\text{safe}}^2$, where $d_{ij} = \|p_i – p_j\|$ is the Euclidean distance between aircraft $i$ and $j$, and $d_{\text{safe}}$ denotes the minimum safe distance. The CBF condition ensures that all pairwise distances remain above the safety threshold throughout the mission.
Graph Theory and Communication Topology
The communication topology of a swarm of unmanned aerial vehicles is characterized by an undirected graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, where $\mathcal{V} = \{1, 2, \ldots, N\}$ is the node set and $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the edge set. The adjacency matrix $A = [a_{ij}] \in \mathbb{R}^{N \times N}$ is defined by $a_{ij} = 1$ if $(i,j) \in \mathcal{E}$ and $a_{ij} = 0$ otherwise. The Laplacian matrix $L = [l_{ij}]$ is defined as
$$
l_{ij} = \begin{cases} \sum_{q \neq i, q=1}^N a_{iq} & \text{if } j = i, \\ -a_{ij} & \text{if } j \neq i. \end{cases}
$$
For a connected graph, $L$ has a single zero eigenvalue and all other eigenvalues are positive. Furthermore, there exists a unitary matrix $U$ such that $U^T H U = \Lambda = \operatorname{diag}\{\lambda_1, \lambda_2, \ldots, \lambda_N\}$, with $0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_N$, where $H$ is a positive semidefinite matrix derived from $L$ in my specific formulations.
Memory Event-Triggered Mechanism
To reduce communication burdens in formation control, I employ a memory event-triggered mechanism that combines historical transmission data with current sampling data for trigger decision-making. Let $\{t_0^i, t_1^i, \ldots, t_k^i, \ldots\}$ denote the sequence of trigger instants for the $i$-th unmanned aerial vehicle. The next trigger instant is determined by
$$
t_{k+1}^i = \inf\big\{ mh > t_k^i : e_i^T(t) \Phi e_i(t) > \theta_i(t) z_i^T(t) \Phi z_i(t) \big\},
$$
where $e_i(t)$ includes both the state error and the control input error, $\Phi$ is a positive definite weighting matrix, $\theta_i(t)$ is a dynamic threshold, and $\tau_i(t)$ is an internal dynamic variable satisfying
$$
\dot{\tau}_i(t) = -\varepsilon_1 \tau_i(t) + \varepsilon_1 e_i^T(t) \Phi_1 e_i(t),
$$
with $\varepsilon_1 > 0$ and $\Phi_1 > 0$. The threshold $\theta_i(t)$ evolves according to
$$
\theta_i(t) = \delta + \rho \exp\left(-\sigma \sum_{s=1}^{S} \|e_{is}(t)\| \right),
$$
where $0 < \delta + \rho < 1$, and $S$ represents the number of recent transmissions incorporated into the trigger logic. This mechanism endows the formation with an adaptive communication pattern: when the system is far from equilibrium, the threshold is small and communication is frequent; when the system approaches stability, the threshold grows and transmission tasks are signally reduced.
Secure Cooperative Control of UAV Formations under Hybrid Attacks
In this chapter, I focus on the secure formation control problem of a swarm of unmanned aerial vehicles under adversarial environments involving both denial-of-service (DoS) attacks and false data injection (FDI) attacks. I propose a hierarchical control framework integrating differential game theory, Q-learning, and control barrier functions.
Modeling of the Cyber-Physical System and Attack Signals
I consider $N$ unmanned aerial vehicles, where each vehicle $i$ has the following point-mass dynamic model:
$$
\begin{aligned} \dot{x}_i &= V_i \cos \gamma_i \cos \chi_i, \\ \dot{y}_i &= V_i \cos \gamma_i \sin \chi_i, \\ \dot{h}_i &= V_i \sin \gamma_i, \\ \dot{V}_i &= \frac{T_i – D_i}{m_i} – g \sin \gamma_i, \\ \dot{\gamma}_i &= \frac{L_i \cos \phi_i – m_i g \cos \gamma_i}{m_i V_i}, \\ \dot{\chi}_i &= \frac{L_i \sin \phi_i}{m_i V_i \cos \gamma_i}. \end{aligned}
$$
After feedback linearization, the model reduces to the double-integrator form
$$
\ddot{x}_i = u_{x_i}, \quad \ddot{y}_i = u_{y_i}, \quad \ddot{z}_i = u_{z_i}.
$$
Defining $z_i = [p_i^T, v_i^T]^T$ with $p_i$ the position vector and $v_i$ the velocity vector, I obtain the state-space representation
$$
\dot{z}_i = A z_i + B u_i,
$$
with $A = \begin{bmatrix} 0 & I_3 \\ 0 & 0 \end{bmatrix} \otimes I_3$ and $B = \begin{bmatrix} 0 \\ I_3 \end{bmatrix} \otimes I_3$.
Both DoS attacks and FDI attacks may occur on the input channels. I model the attacked control input as
$$
\tilde{u}_i(k) = a_i(k) u_i(k) + b_i(k) \psi_i(k),
$$
where $a_i(k)$ and $b_i(k)$ are Bernoulli-distributed random variables. Specifically, $a_i(k) = 0$ indicates that a DoS attack blocks the control input at time step $k$, while $b_i(k) = 1$ indicates that an FDI signal $\psi_i(k)$ is maliciously injected into the system. The attacked cyber-physical system is then described by
$$
\begin{aligned} x(k+1) &= C x(k) + D \tilde{u}(k), \\ y(k) &= G x(k), \end{aligned}
$$
where $\tilde{u}(k)$ reflects the aggregated effect of the hybrid attacks.
Zero-Sum Differential Game for Attack Mitigation
I formulate a zero-sum differential game to model the confrontation between the controller $u(k)$ (defender) and the FDI attack $\psi(k)$ (attacker). The attacker is designated as the leader who attempts to maximize the cost function, while the controller acts as the follower who minimizes the same cost. The value function for the $i$-th vehicle is defined as
$$
J_i(x(k)) = \mathbb{E} \sum_{t=k}^{\infty} \eta^{t-k} \Big[ x^T(t) Q x(t) + a(t) u^T(t) R_1 u(t) – b(t) \psi^T(t) R_2 \psi(t) \Big],
$$
with symmetric weighting matrices $Q \geq 0$, $R_1 > 0$, $R_2 > 0$, and a discount factor $0 \leq \eta < 1$. The corresponding Bellman equation is
$$
J_i(x(k)) = x^T(k) Q x(k) + a(k) u^T(k) R_1 u(k) – b(k) \psi^T(k) R_2 \psi(k) + \eta J_i(x(k+1)).
$$
Assuming a quadratic value function $J(x(k)) = x^T(k) P x(k)$, the Hamiltonian becomes
$$
\begin{aligned} H\big(x(k), u(k), \psi(k)\big) &= x^T(k) Q x(k) + a(k) u^T(k) R_1 u(k) – b(k) \psi^T(k) R_2 \psi(k) \\ &\quad + x^T(k) P x(k) – \eta x^T(k+1) P x(k+1). \end{aligned}
$$
By applying the stationary conditions
$$
\frac{\partial H}{\partial u} = 0, \qquad \frac{\partial H}{\partial \psi} = 0,
$$
I obtain the worst-case FDI attack and the optimal control input as follows.
$$
\psi^*(k) = \big(\theta – \alpha\beta\eta D^T P D \rho^{-1} D^T P D\big)^{-1} \big(\beta\eta D^T P C x(k) – \alpha\beta\eta D^T P D \rho^{-1} D^T P C x(k)\big),
$$
$$
u^*(k) = -\rho^{-1} \big(\eta D^T P C x(k) + \beta \eta D^T P D \psi^*(k)\big),
$$
where $\rho = R_1 + \eta D^T P D$ and $\theta = R_2 – \eta D^T P D$. The matrix $P$ satisfies the generalized algebraic Riccati equation (GARE)
$$
P = Q + \eta C^T P C – \eta^2 \begin{bmatrix} \alpha C^T P D & \beta C^T P D \end{bmatrix} \begin{bmatrix} \rho & -\alpha\beta\eta D^T P D \\ -\alpha\beta D^T P D & \beta\theta \end{bmatrix}^{-1} \begin{bmatrix} \alpha D^T P C \\ \beta D^T P C \end{bmatrix}.
$$
The following theorem formalizes the optimality condition.
Theorem 1. In the zero-sum game framework, if $P$ satisfies the GARE and the inequality $R_2 – \eta D^T P D – \alpha\beta\eta D^T P D \rho^{-1} \rho^{-1} \rho R_1 \rho^{-1} D^T P D > 0$ holds, then the strategies in the above equations constitute a saddle-point equilibrium of the zero-sum differential game.
To circumvent the explicit dependence on the system dynamics, I employ a Q-learning algorithm to iteratively approximate the optimal control policy. The Q-function is defined as
$$
\mathcal{Q}(z(k)) = \phi(k) + \eta x^T(k+1) P x(k+1),
$$
where $\phi(k)$ includes the instantaneous cost terms. In the iterative learning process, the Q-matrix is updated according to Algorithm 1. I summarize the main algorithmic steps in the following table.
| Step | Operation |
|---|---|
| 1 | Initialize $P^{(0)}=0$, $K^{(0)}=0$, $L^{(0)}=0$, $M^{(0)}=0$. |
| 2 | Policy evaluation: solve $z_k^T M^{(t)} z_k = \phi(k) + \eta z_{k+1}^T M^{(t)} z_{k+1}$ using least squares. |
| 3 | Policy improvement: update $u^{(t+1)}$ and $\psi^{(t+1)}$ using the Q-function decomposition. |
| 4 | Iteration: repeat steps 2–3 until convergence $\|M^{(t+1)} – M^{(t)}\| < \epsilon$. |
Control Barrier Function-Based Safety Constraint
To guarantee collision-free operation, I augment the optimal controller with a safety layer based on control barrier functions. For any pair of unmanned aerial vehicles $i$ and $j$, I define the barrier function as
$$
h_{ij}(z) = \|p_i – p_j\|^2 – d_{\text{safe}}^2.
$$
The safety constraint imposed on the control input is
$$
\dot{h}_{ij} \geq -\lambda h_{ij}, \quad \lambda > 0.
$$
Combining the CBF constraint with the quadratic programming (QP) formulation, the actual control input for each unmanned aerial vehicle is obtained by solving the following optimization problem:
$$
\begin{aligned} \min_{u_i} \quad & \frac{1}{2} \|u_i – u_i^*\|^2 \\ \text{s.t.} \quad & \dot{h}_{ij}(z) + \lambda h_{ij}(z) \geq 0, \quad \forall j \neq i. \end{aligned}
$$
This QP formulation ensures that the unmanned aerial vehicle tracks the nominal optimal control from the game layer while strictly respecting the safety distance constraints.
Formation Strategy and Nash Equilibrium
I now design the formation control law. Let $d_{ij}$ denote the desired displacement vector from aircraft $j$ to aircraft $i$. The performance index for aircraft $i$ is
$$
J_i = \sum_{j \in \mathcal{N}_i} l_{ij} \Big( \|p_i(t_f) – p_j(t_f) – d_{ij}\|^2 + \|v_i(t_f) – v_j(t_f)\|^2 \Big) + r_i \int_0^{t_f} \|u_i^2\| \, dt,
$$
where $l_{ij}$ are the elements of the Laplacian matrix, $r_i$ is a positive scalar, and $t_f$ is the terminal time. The performance index indicates that each unmanned aerial vehicle seeks to minimize the terminal formation error and the terminal velocity error while minimizing its control effort, yielding a multi-player differential game.
The open-loop Nash equilibrium is derived through the Pontryagin minimum principle. I introduce the transformed state variables
$$
s_{p_i}(t) = (t_f – t) v_i(t) + p_i(t), \qquad s_{v_i}(t) = v_i(t),
$$
which lead to the transformed dynamics
$$
\dot{s}_{p_i}(t) = B_p u_i(t), \qquad \dot{s}_{v_i}(t) = B_v u_i(t),
$$
where $B_p = (t_f – t) I_3$ and $B_v = I_3$. After some algebraic manipulations, I obtain the following open-loop Nash solution:
$$
u_i^*(t) = \frac{1}{r_i} F_i \Big( M^{-1} \begin{bmatrix} s_p(0) \\ s_v(0) \end{bmatrix} + \mathcal{W} d \Big) + \frac{1}{r_i} B_p^T B_v^T d_i.
$$
In this expression,
$$M = \begin{bmatrix} w_{pp} & w_{pv} \\ w_{vp} & w_{vv} \end{bmatrix} + \begin{bmatrix} I_3 \\ -I_3 \end{bmatrix} R^{-1} \begin{bmatrix} I_3 & -I_3 \end{bmatrix} \otimes L,
$$
with $R = \operatorname{diag}\{r_1, \ldots, r_N\}$, and $w_{pp}, w_{pv}, w_{vp}, w_{vv}$ defined as
$$
w_{pp} = \int_0^{t_f} B_p^T B_p \, dt, \quad w_{pv} = \int_0^{t_f} B_p^T B_v \, dt, \quad w_{vp} = \int_0^{t_f} B_v^T B_p \, dt, \quad w_{vv} = \int_0^{t_f} B_v^T B_v \, dt.
$$
The matrix $M$ is shown to be nonsingular under the connectivity assumption of the communication graph. However, the direct implementation of the open-loop Nash strategy requires global knowledge of the initial states, which is impractical in distributed scenarios. I therefore design a distributed estimator for each unmanned aerial vehicle to estimate its own terminal position and velocity, exchanging these estimates with neighboring vehicles according to the information graph.
The distributed estimation law for the $i$-th unmanned aerial vehicle is given by
$$
\begin{aligned} \begin{bmatrix} \dot{\zeta}_{p_i} \\ \dot{\zeta}_{v_i} \end{bmatrix} &= k_i \Big( \begin{bmatrix} w_{pp} \\ w_{vp} \end{bmatrix} \begin{bmatrix} s_p(0) \\ s_v(0) \end{bmatrix} + \begin{bmatrix} s_{p_i} \\ s_{v_i} \end{bmatrix} \Big) \\ &\quad – k_i W^{-1} \sum_{j \in \mathcal{N}_i} l_{ij} \Big( \begin{bmatrix} \zeta_{p_i} \\ \zeta_{v_i} \end{bmatrix} – \begin{bmatrix} \zeta_{p_j} \\ \zeta_{v_j} \end{bmatrix} \Big), \end{aligned}
$$
where $k_i$ is a positive scalar and $W$ is derived from the Lyapunov matrix. The estimation law is fully distributed: unmanned aerial vehicle $i$ only needs its own private information, the broadcast estimates from its neighbors, and its own state measurements.
Finally, the distributed Nash equilibrium control law is
$$
u_i^*(t) = \frac{1}{r_i} \sum_{j \in \mathcal{N}_i} l_{ij} B_p^T (\zeta_{p_i} – \zeta_{p_j} – d_{ij}) – \frac{1}{r_i} \sum_{j \in \mathcal{N}_i} l_{ij} B_v^T (\zeta_{v_i} – \zeta_{v_j}),
$$
which relies only on locally available information. The simulation results for a five-unmanned aerial vehicle example are summarized in Table 2.
| Parameter | Value | Post-CBF Verification |
|---|---|---|
| Number of AAVs | 5 | — |
| Mass of each AAV | 20 kg | — |
| Safety distance $d_{\text{safe}}$ | 30 m | Minimum distance achieved |
| Attack probability (DoS/FDI) | 0.2 / 0.2 | No collision observed |
| Formation time | 30 s | Formation achieved at 30 s |
| GARE convergence | Fast convergence | Stable $P, K, L$ values |
The three-dimensional trajectory plot and the top-view trajectory plot both confirm that the five unmanned aerial vehicles successfully form and maintain the prescribed leader–follower formation under the proposed differential game-based controller. The distance profile between consecutive vehicles, as presented in the lower panel, indicates that the CBF-based safety layer maintains all pairwise distances above the safety threshold even in the presence of mixed attacks. The DoS and FDI attack inputs applied to the leader show the random nature of the adversarial events, modeled with probability 0.2. The GARE convergence plot demonstrates that the Riccati matrix $P$, the controller gain $K$, and the attacker gain $L$ all converge rapidly, confirming the numerical reliability of the game-theoretic solution.
Cooperative Formation Control with Memory Event-Triggered Sliding Mode Control and Differential Game
In this chapter, I address the wake interference, communication efficiency, and robustness challenges in multi-unmanned aerial vehicle formation flight. I propose an integrated control strategy that combines topology optimization, disturbance compensation, and differential game theory.
Wake-Interference-Affected Dynamics
For the follower aircraft, the wake generated by the leader significantly disturbs the nominal aerodynamic state. I model the wake-affected dynamics of the $i$-th follower aircraft in the form
$$
\begin{aligned} \dot{x}_i &= V_i \cos \gamma_i \cos \chi_i + \varsigma_{x_i}, \\ \dot{y}_i &= V_i \cos \gamma_i \sin \chi_i + \varsigma_{y_i}, \\ \dot{h}_i &= V_i \sin \gamma_i + \varsigma_{h_i}, \\ \dot{V}_i &= \frac{V_{i,l} – V_i}{\tau_V} + \varsigma_{V_i}, \\ \dot{\gamma}_i &= \frac{\gamma_{i,l} – \gamma_i}{\tau_\gamma} + \varsigma_{\gamma_i}, \\ \dot{\chi}_i &= \frac{\chi_{i,l} – \chi_i}{\tau_\chi} + \varsigma_{\chi_i}, \end{aligned}
$$
where $\varsigma_{x_i}, \varsigma_{y_i}, \varsigma_{h_i}, \varsigma_{V_i}, \varsigma_{\gamma_i}, \varsigma_{\chi_i}$ represent the six components of bounded wake disturbance integrated into each state channel. These disturbance terms capture the coupled aerodynamic interactions among the unmanned aerial vehicles.
Topology Control Layer Design
To optimize the communication topology and reduce the transmission power consumption, I design a distributed topology control layer using spherical space partitioning and minimum spanning trees. The topology control algorithm operates in parallel processes. The first process periodically updates the positions of neighboring vehicles by broadcasting beacon messages. The second process selects the links to retain based on the spatial partition vectors S, as summarized in Algorithm 2.
| Step | Procedure |
|---|---|
| 1 | Initialize neighbor position vector set $P \leftarrow \emptyset$. |
| 2 | In the neighbor-update function: broadcast own position and receive beacon messages; update $P$. |
| 3 | In the link-selection function: compute $d = p_0 – p$, select $s^* = \arg\max_{s \in S} s \cdot d$. |
| 4 | Group nodes into partitions $\Pi_i$ and construct MST in each partition. |
| 5 | Select the nearest neighbor link in each partition; assign the corresponding transmission range. |
By controlling the available link count and the transmission range per link, the topology control layer effectively reduces redundant communication and energy expenditure while preserving network connectivity. The average number of communication links in a five-unmanned aerial vehicle formation is reduced from 10 to approximately 4.5, confirming the effectiveness of the switching topology scheme.
Sliding Mode Disturbance Observer Design
To compensate for the wake disturbances, I design six independent sliding-mode disturbance observers (SMDOs), one for each disturbance channel. For example, the SMDO for the x-channel disturbance $\varsigma_{x_i}$ is designed as
$$
\begin{aligned} \dot{\hat{x}}_i &= V_i \cos \gamma_i \cos \chi_i + \hat{v}_i, \\ \dot{\hat{v}}_i &= -\lambda_{1_i} \operatorname{sig}^{1/2}(\hat{x}_i – x_i) + \varsigma_{x_i} – \lambda_{2_i} \operatorname{sign}(\hat{\varsigma}_{x_i}), \\ \dot{\hat{\varsigma}}_{x_i} &= -\lambda_{2_i} \operatorname{sign}(\hat{\varsigma}_{x_i} – v_i), \end{aligned}
$$
where $\lambda_{1_i} > 0$ and $\lambda_{2_i} > 0$ are observer gains, and $\operatorname{sig}^{\alpha}(\cdot) = |\cdot|^{\alpha} \operatorname{sign}(\cdot)$. The remaining five observers are constructed following the same structure. The estimates $\hat{\varsigma}_{x_i}, \hat{\varsigma}_{y_i}, \hat{\varsigma}_{h_i}, \hat{\varsigma}_{V_i}, \hat{\varsigma}_{\gamma_i}, \hat{\varsigma}_{\chi_i}$ are then used in the controller design to compensate for the disturbance in real time.
Sliding Mode Controller Design with METM
Let $z_i(t) = [p_i^T(t), v_i^T(t)]^T$ be the full state of the $i$-th unmanned aerial vehicle. The tracking error is $e_i(t) = z_i(t) – z_i^d(t)$, and the aggregate error dynamics are
$$
\dot{e}(t) = \mathcal{A} e(t) + \mathcal{B} u(t),
$$
with $\mathcal{A} = I_N \otimes A$ and $\mathcal{B} = I_N \otimes B$.
I design the sliding-mode surface as
$$
\begin{aligned} \varsigma_i(t) &= \mathcal{C} z_i(t) + \int_{mh}^{t} \sum_{j \in \mathcal{N}_i} \mathcal{C} \mathcal{B} K a_{ij} \big( z_i(\rho) – z_j(\rho) \big) d\rho \\ &\quad + \int_{mh}^{t} \sum_{j \in \mathcal{N}_i} \mathcal{C} \mathcal{B} a_{ij} \big( u_i(mh) – u_j(mh) \big) d\rho \\ &\quad – \int_{mh}^{t} \sum_{j \in \mathcal{N}_i} \mathcal{C} \mathcal{B} K a_{ij} \big( z_i(mh) – z_j(mh) \big) d\rho, \end{aligned}
$$
where $\mathcal{C}$ is chosen such that $\mathcal{C}\mathcal{B}$ is nonsingular, and $K$ is the controller gain to be determined. Under the memory event-triggered mechanism, the sliding mode controller is given by
$$
u_i(t) = -(\mathcal{C}\mathcal{B})^{-1} \mathcal{C} \mathcal{A} z_i(t) – K z_i(t – \eta(t)) – K e_i(t) + (\mathcal{C}\mathcal{B})^{-1} \mathcal{C} e_i^u(t) – \theta (\mathcal{C}\mathcal{B})^{-1} \varsigma_i(t) – \kappa (\mathcal{C}\mathcal{B})^{-1} \operatorname{sign}(\varsigma_i(t)),
$$
where $\theta > 0$ and $\kappa > 0$ are constants. The linear term $-\theta(\mathcal{C}\mathcal{B})^{-1}\varsigma_i(t)$ accelerates the convergence to the sliding surface, while the sign-function term $-\kappa(\mathcal{C}\mathcal{B})^{-1}\operatorname{sign}(\varsigma_i(t))$ enhances robustness against the residual disturbances.
The reachability condition is established as
$$
\varsigma_i^T(t) \dot{\varsigma}_i(t) \leq -\theta \varsigma_i^T(t) \varsigma_i(t) – \kappa \|\varsigma_i(t)\| < 0,
$$
showing that the sliding surface is reached in finite time. On the sliding surface, the equivalent control is
$$
u_{\text{eq}}(t) = -(\mathcal{C}\mathcal{B})^{-1} \mathcal{C}\mathcal{A} z(t) – K z(t – \eta(t)) – K e(t) + (\mathcal{C}\mathcal{B})^{-1} \mathcal{C} e^u(t),
$$
which yields the reduced-order sliding mode dynamics
$$
\dot{z}(t) = \bar{A} z(t) – \bar{K} z(t – \eta(t)) – \bar{K} e(t) + \bar{B} e^u(t).
$$
To guarantee the exponential stability of the closed-loop sliding mode dynamics, I present the following theorem. Let $\lambda_1$ and $\lambda_N$ be the smallest non-zero and largest eigenvalues of the Laplacian-related matrix $H$, respectively.
Theorem 2. Under the sliding mode observer layer, the METM conditions, and the sliding mode controller, the closed-loop system is exponentially stable if there exist positive definite matrices $\mathcal{P}>0$, $\mathcal{Q}>0$, $\mathcal{R}>0$, $\Phi > 0$, $\Phi_1 > 0$, $\mathcal{X}$, $\mathcal{Y}_1$, $\mathcal{Y}_2$ such that the following LMI holds for $i = 1, 2$ and $j = \lambda_1, \lambda_N$:
$$
\begin{bmatrix} \Theta_{11} & * & * & * & * & * & * \\ \Theta_{21} & \Theta_{22} & * & * & * & * & * \\ \Theta_{31} & \Theta_{32} & \Theta_{33} & * & * & * & * \\ \Theta_{41} & \Theta_{42} & 0 & \Theta_{44} & * & * & * \\ 0 & 0 & 0 & \Theta_{54} & \Theta_{55} & * & * \\ \Theta_{61} & 0 & 0 & \Theta_{64} & 0 & \Theta_{66} & * \\ 0 & 0 & 0 & 0 & 0 & 0 & -\mathcal{R} \end{bmatrix} < 0.
$$
The controller gain and the event-triggered weighting matrix are
$$
K = \mathcal{Y}_1 \mathcal{X}^{-1}, \qquad \Phi = \mathcal{X}^{-1} \Phi \mathcal{X}^{-1}.
$$
The proof of Theorem 2 is based on the Lyapunov–Krasovskii functional method, the sector-bound condition of the nonlinear triggering mechanism, and the Jensen’s inequality. Due to space limitations, the detailed proof is omitted here.
Distributed Nash Equilibrium for Formation Control
Following the differential game formulation presented in the previous chapter, I now adapt the Nash equilibrium strategy to the topology-optimized and event-triggered framework. The performance index for each unmanned aerial vehicle is
$$
J_i = \sum_{j \in \mathcal{N}_i} l_{ij} \Big( \|s_{p_i}(t_f) – s_{p_j}(t_f) – d_{ij}\|^2 + \|s_{v_i}(t_f) – s_{v_j}(t_f)\|^2 \Big) + r_i \int_0^{t_f} \|u_i\|^2 \, dt.
$$
By solving the Hamiltonian system defined by the transformed dynamics, I obtain the open-loop Nash equilibrium. To make the strategy implementable in a distributed manner, I employ the following estimation law:
$$
\begin{aligned} \begin{bmatrix} \dot{\zeta}_{p_i} \\ \dot{\zeta}_{v_i} \end{bmatrix} &= k_i \Big( \begin{bmatrix} s_{p_i}(0) \\ s_{v_i}(0) \end{bmatrix} – \begin{bmatrix} \zeta_{p_i} \\ \zeta_{v_i} \end{bmatrix} \Big) \\ &\quad – k_i W^{-1} \sum_{j \in \mathcal{N}_i} l_{ij} \Big( \begin{bmatrix} \zeta_{p_i} \\ \zeta_{v_i} \end{bmatrix} – \begin{bmatrix} \zeta_{p_j} \\ \zeta_{v_j} \end{bmatrix} \Big), \end{aligned}
$$
which enables each unmanned aerial vehicle to asymptotically estimate the terminal position and terminal velocity. The distributed Nash formation control law is then
$$
u_i^*(t) = -\frac{1}{r_i} \sum_{j \in \mathcal{N}_i} l_{ij} B_p^T (\zeta_{p_i} – \zeta_{p_j} – d_{ij}) – \frac{1}{r_i} \sum_{j \in \mathcal{N}_i} l_{ij} B_v^T (\zeta_{v_i} – \zeta_{v_j}).
$$
The complete architecture of the proposed formation control framework is summarized in the following table.
| Layer | Component | Function |
|---|---|---|
| Topology layer | Switching topology with spherical partition + MST | Optimize link distribution and reduce transmission power |
| Disturbance layer | Six independent SMDOs | Estimate wake disturbances in each channel |
| Communication layer | Memory event-triggered mechanism | Reduce data transmission burden |
| Control layer | Sliding mode controller | Guarantee exponential stability and robustness |
| Optimization layer | Differential game and Nash equilibrium | Achieve distributed cooperative formation control |
Simulation Results and Discussion
To validate the proposed strategies, I conducted comprehensive MATLAB simulations for both scenarios. The simulation setup for the second scenario is summarized in Table 5.
| Parameter | Symbol | Value |
|---|---|---|
| Total simulation time | $T$ | 40 s |
| Time step | $dt$ | 0.01 s |
| Topology update interval | — | 0.2 s |
| Number of spatial partitions | $n$ | 8 |
| Maximum transmission range | — | 50 m |
| Minimum transmission power | — | 0.1 W |
| SMDO gains | $\lambda_1, \lambda_2$ | 5.0, 3.0 |
| Safety distance | $d_{\text{safe}}$ | 30 m |
The simulation results can be summarized as follows. First, the communication topology evolution plot (depicted as the upper left subplot in the figure) shows that the dynamic switching topology stabilizes after the unmanned aerial vehicles achieve the desired formation, effectively reducing the average number of links while maintaining global connectivity. Second, the cumulative topology-change count plot confirms that the topology converges to a steady-state configuration. Third, the X-direction SMDO observation plot for unmanned aerial vehicle 2 demonstrates the observer’s ability to accurately capture the wake disturbance, enabling targeted compensation in the control input. The event-triggered transmission count statistics, presented in the lower panel, show that the number of communication events is reduced by 40%–54% compared to the periodic sampling approach, confirming the communication efficiency of the METM. Finally, the three-dimensional flight trajectory plot shows that the swarm successfully forms the prescribed formation within the required time, with smooth and continuous trajectories free of jerky motions. The formation error plot shows rapid convergence during the initial phase and steady-state accuracy in the formation maintenance phase. The Nash equilibrium convergence plot verifies the stability and convergence of the differential game-based algorithm, with the steady-state gap approaching zero.
These results collectively confirm that the proposed topology optimization, disturbance compensation, and differential game-based sliding mode control strategy effectively addresses the wake interference, communication efficiency, and robustness challenges in multi-unmanned aerial vehicle formation flight. The strategy enables rapid formation acquisition, stable trajectory tracking, and energy-efficient communication, providing a viable technical pathway for cooperative formation control of unmanned aerial vehicle swarms in complex environments.
Conclusions and Future Outlook
In this paper, I have systematically investigated the formation control problem of unmanned aerial vehicles through the lens of differential game theory. The principal contributions are twofold.
First, I proposed a hierarchical control framework integrating differential game theory, Q-learning, and control barrier functions to address the secure formation control problem under hybrid attacks. The Q-learning algorithm trains the anti-attack policy offline to handle both DoS attacks and FDI attacks. The differential game-based controller, augmented with CBF constraints, guarantees prescribed formation acquisition and collision avoidance. The quadratic programming formulation optimizes control effort while strictly enforcing the safety distance. Simulation results for a five-unmanned aerial vehicle system confirmed the effectiveness of the proposed framework in maintaining safe and stable formation flight in adversarial environments.
Second, I developed a cooperative control strategy that integrates topology optimization, disturbance compensation, and differential game theory for formation flight. The graph-theoretic switching topology with spherical space partitioning and minimum spanning trees reduces the number of communication links and lowers the transmission power. Six independent sliding-mode disturbance observers accurately estimate the wake disturbances. The memory event-triggered sliding mode controller reduces the data transmission burden while guaranteeing exponential stability of the closed-loop system. The distributed terminal state estimation and Nash equilibrium control strategy enable each unmanned aerial vehicle to achieve cooperative formation control using only local information.
Several topics merit further investigation. Future research can incorporate multi-source composite disturbances such as sensor noise and aerodynamic parameter perturbations to enhance practical applicability. The extension to heterogeneous formations, including fixed-wing and rotary-wing unmanned aerial vehicles, combined with ground and marine vehicles, presents a challenging but promising direction for cooperative control. Furthermore, the advanced reinforcement learning approaches, including multi-agent actor-critic algorithms and proximal policy optimization, can be explored to improve the adaptability and scalability of the anti-attack strategy. The combination of deep learning and differential game theory may open new avenues for solving large-scale formation control problems in increasingly complex operational scenarios.