The coordinated control of multi-agent systems has garnered immense attention in recent years, finding critical applications in areas ranging from environmental monitoring to complex logistical operations and strategic defense scenarios. Within this domain, the quadrotor drone has emerged as a particularly versatile platform due to its agility, vertical take-off and landing capability, and relatively low maintenance cost. However, the operational efficacy of a single quadrotor drone is inherently limited. To execute complex missions such as large-scale reconnaissance, collaborative payload transport, or coordinated area denial, the deployment of a quadrotor drone swarm becomes essential. The core technology enabling such collaborative operations is formation control, which allows a group of agents to achieve and maintain a desired spatial configuration autonomously.
Traditional formation control strategies often focus solely on the asymptotic convergence of agents to their target positions. Yet, in dynamic and unpredictable real-world environments, merely achieving convergence is insufficient. The quality of the convergence process—encompassing transient performance, control effort, and robustness to communication changes—is equally vital. For instance, a swarm may need to reconfigure rapidly in response to a threat, demanding fast convergence. Alternatively, for extended surveillance missions, minimizing the cumulative energy expenditure during formation maneuvers is crucial to prolong operational endurance. This necessitates moving beyond basic formation control towards optimized formation control with guaranteed performance bounds.
This article addresses the challenge of prescribed performance optimization formation control for a swarm of quadrotor drones operating under switching communication topologies. The primary objective is to design a distributed control protocol that not only drives the swarm to achieve a desired time-varying formation but also guarantees that a predefined cooperative performance index, quantifying the cumulative formation coordination errors, remains strictly below a user-specified threshold. This approach directly embeds performance specifications into the control design, ensuring predictable and reliable swarm behavior.
1. Problem Formulation and System Dynamics
Consider a swarm comprising N identical quadrotor drones. The communication network among them is represented by an undirected graph \(\mathcal{G}^{(\kappa(t))}=(\mathcal{V}, \mathcal{E}^{(\kappa(t))})\), where \(\mathcal{V}=\{1,2,\ldots,N\}\) is the node set, and \(\mathcal{E}^{(\kappa(t))} \subseteq \mathcal{V} \times \mathcal{V}\) is the edge set at time \(t\), dictated by a switching signal \(\kappa(t)\). The set of all possible connected, undirected topologies is finite. The Laplacian matrix associated with the active topology \(\mathcal{G}^{(\kappa(t))}\) is denoted as \(L^{(\kappa(t))}=[l_{ij}] \in \mathbb{R}^{N \times N}\).
For formation control analysis, the translational dynamics of a quadrotor drone in a planar or 3D space can often be effectively modeled by a double-integrator system. Let the state of the \(i\)-th quadrotor drone be \(x_i(t) = [p_i^T(t), v_i^T(t)]^T \in \mathbb{R}^6\), where \(p_i(t) \in \mathbb{R}^3\) and \(v_i(t) \in \mathbb{R}^3\) represent its position and velocity vectors, respectively. The dynamics are:
$$ \dot{x}_i(t) = A x_i(t) + B u_i(t), \quad i=1,2,\ldots,N $$
with system matrices:
$$ A = \begin{bmatrix} 0 & I_3 \\ 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ I_3 \end{bmatrix} $$
where \(u_i(t) \in \mathbb{R}^3\) is the control input (acceleration command). The pair \((A, B)\) is controllable.
The desired time-varying formation for the swarm is specified by a vector \(h(t)=[h_1^T(t), \ldots, h_N^T(t)]^T\), where \(h_i(t) \in \mathbb{R}^6\) is piecewise continuously differentiable. The control objective is to design a distributed protocol such that the states of the quadrotor drones converge to the formation pattern \(h(t)\) with a prescribed performance guarantee on the coordination errors.
2. Distributed Control Protocol and Performance Index
To achieve the formation objective under switching topologies, we propose the following distributed control protocol for the \(i\)-th quadrotor drone:
$$ u_i(t) = K_c \xi_i(t) + K_u \sum_{j \in \mathcal{N}_i^{(\kappa(t))}} (\xi_i(t) – \xi_j(t)) + s_i(t) $$
where \(\xi_i(t) = x_i(t) – h_i(t)\) is the local formation error, and \(\mathcal{N}_i^{(\kappa(t))}\) is the set of neighbors of agent \(i\) under the current topology. The terms are:
- \(K_c \xi_i(t)\): A self-feedback term used to independently regulate the motion mode of the formation center.
- \(K_u \sum (\xi_i – \xi_j)\): A consensus term that drives the relative states of neighboring quadrotor drones to agreement, enforcing the formation geometry.
- \(s_i(t)\): A formation compensation signal, designed to expand the class of achievable formation vectors \(h_i(t)\) by canceling the influence of uncontrollable modes inherent in the formation specification.
The key innovation lies in coupling this protocol with a strict performance constraint. We define a global cooperative performance index \(J\) that measures the integrated squared formation coordination errors across the entire swarm over time:
$$ J = \frac{1}{N} \sum_{i=1}^{N} \sum_{j=1}^{N} \int_0^{\infty} (\xi_i(t) – \xi_j(t))^T Q (\xi_i(t) – \xi_j(t)) \, dt $$
where \(Q = Q^T > 0\) is a positive definite weighting matrix chosen by the designer. A smaller \(J\) indicates better overall formation coordination (lower total error). The prescribed performance control problem is then formally stated: For any given performance preset \(J^* > 0\) and any bounded initial errors \(\xi_i(0)\), design the gain matrices \(K_c, K_u\) and signals \(s_i(t)\) such that:
- Formation Achievement: \(\lim_{t \to \infty} (\xi_i(t) – \xi_c(t)) = 0\) for all \(i\), where \(\xi_c(t)\) is the formation center trajectory.
- Performance Guarantee: The actual performance index satisfies \(J \le J^*\).
This ensures the swarm not only forms the desired shape but does so with a guaranteed upper bound on the “cost” of coordination, effectively solving an optimization problem online.
3. Theoretical Analysis and Design Methodology
The analysis begins by employing a state transformation to decouple the system dynamics into two subsystems: one describing the motion of the formation center, and the other describing the relative formation errors. Let \(U^{(\kappa(t))} = [\mathbf{1}/\sqrt{N}, \bar{U}^{(\kappa(t))}]\) be an orthonormal matrix where the columns of \(\bar{U}^{(\kappa(t))}\) span the subspace orthogonal to the consensus direction. Define the transformed state \(\delta(t) = ((U^{(\kappa(t))})^T \otimes I_6) \xi(t) = [\delta_1^T(t), \zeta^T(t)]^T\).
The closed-loop dynamics then separate into:
$$ \dot{\delta}_1(t) = (A + B K_c) \delta_1(t) + d_1(t) $$
$$ \dot{\zeta}(t) = (I_{N-1} \otimes (A + B K_c) – \Lambda^{(\kappa(t))} \otimes B K_u) \zeta(t) + d_\zeta(t) $$
where \(\Lambda^{(\kappa(t))}\) is a diagonal matrix of the positive eigenvalues of \(L^{(\kappa(t))}\), and \(d_1(t), d_\zeta(t)\) are terms involving \(h_i(t)\) and \(s_i(t)\). The vector \(\zeta(t)\) encapsulates all relative formation errors. Achieving formation is equivalent to the asymptotic stability of the \(\zeta\)-subsystem: \(\lim_{t \to \infty} \zeta(t) = 0\).
3.1 Formation Feasibility and Center Motion
The formation compensation signal \(s_i(t)\) is designed specifically to cancel the inhomogeneous terms \(d_\zeta(t)\) in the error dynamics, which originate from the time derivatives of the formation vectors \(h_i(t)\). The feasibility condition for the formation \(h(t)\) is that there exists a signal \(s_i(t)\) such that \(d_\zeta(t)=0\). This condition can be expressed as finding a matrix \(B_2\) (the left annihilator of \(B\), i.e., \(B_2 B = 0\)) such that:
$$ B_2 (A h_i(t) – \dot{h}_i(t)) = 0, \quad \forall i $$
If this holds, we can set \(s_i(t) = B_1 (\dot{h}_i(t) – A h_i(t))\) for a suitable \(B_1\) (with \([B_1^T, B_2^T]^T\) being invertible). This eliminates the influence of \(h(t)\) on the error dynamics, greatly expanding the class of achievable formations. Under this condition, the formation center trajectory \(x_c(t)\) is explicitly given by:
$$ x_c(t) = \frac{1}{N} \sum_{i=1}^{N} \left[ e^{(A+BK_c)t} x_i(0) – h_i(t) – \int_0^t e^{(A+BK_c)(t-\tau)} B(K_c h_i(\tau)-s_i(\tau)) \, d\tau \right] $$
Critically, the matrix \(K_c\) appears in the exponent, providing a direct mechanism to independently prescribe the dynamic mode of the formation center (e.g., making it stable, oscillatory, or tracking a reference) without affecting the relative formation stability, as long as \(A+BK_c\) is Hurwitz.
3.2 Prescribed Performance via Linear Matrix Inequalities
The core challenge is to design the consensus gain \(K_u\) to stabilize the error subsystem \(\zeta(t)\) while ensuring \(J \le J^*\). Using Lyapunov stability theory, we construct a candidate Lyapunov function \(V(t) = \zeta^T(t) (I_{N-1} \otimes P) \zeta(t)\) with \(P>0\). The performance index \(J\) can be rewritten in terms of \(\zeta(t)\):
$$ J = 2 \int_0^{\infty} \zeta^T(t) (I_{N-1} \otimes Q) \zeta(t) \, dt $$
The goal is to ensure \( \dot{V}(t) + 2 \zeta^T(t) (I_{N-1} \otimes Q) \zeta(t) < 0 \), which would imply \(J \le V(0)\). To obtain a design condition that incorporates the preset \(J^*\), we relate the initial condition \(V(0)\) to \(J^*\). Let \(\alpha_\xi = \xi^T(0) ((I_N – \frac{1}{N}\mathbf{1}\mathbf{1}^T) \otimes I_6) \xi(0)\), which is a known scalar related to the initial formation errors. The condition \(J \le J^*\) is guaranteed if we can ensure \(V(0) \le J^*\).
This leads to the following sufficient condition formulated as a set of Linear Matrix Inequalities (LMIs). For a given performance preset \(J^*>0\), if there exists a symmetric positive definite matrix \(R > 0\) such that the following LMIs hold:
$$ \begin{bmatrix} (A+BK_c)R + R(A+BK_c)^T – 2BB^T & R \\ R & -\frac{1}{2}Q^{-1} \end{bmatrix} < 0 $$
$$ I_6 – (J^*)^{-1} \alpha_\xi \, R < 0 $$
then the prescribed performance formation control is achievable. The control gain \(K_u\) is then given by:
$$ K_u = \lambda_{\text{min}}^{-1} B^T R^{-1} $$
where \(\lambda_{\text{min}} = \min_{i \in \text{topology set}} \{\lambda_2(L^{(i)})\}\) is the minimum non-zero eigenvalue (algebraic connectivity) across all possible connected communication topologies. This robust choice ensures stability under arbitrary switching among these topologies, provided a minimum dwell time is respected.
The first LMI ensures the exponential stability of the formation error dynamics with a sufficient decay rate, implicitly bounding the integral of \(\zeta^T Q \zeta\). The second LMI explicitly enforces the bound \(V(0) \le J^*\) by linking the Lyapunov matrix \(R\) (\(P=R^{-1}\)), the initial energy \(\alpha_\xi\), and the performance preset \(J^*\). This elegant formulation directly embeds the performance specification into the control gain design process.
| Parameter | Symbol | Role in Control Design |
|---|---|---|
| Self-feedback Gain | \(K_c\) | Configures the dynamics (e.g., eigenvalues) of the formation center motion. Chosen independently based on mission needs. |
| Consensus Gain | \(K_u\) | Drives relative states to agreement, ensuring formation geometry. Calculated as \(K_u = \lambda_{\text{min}}^{-1} B^T R^{-1}\). |
| Lyapunov Matrix | \(R\) | The positive definite solution to the LMIs. Its inverse \(P=R^{-1}\) defines the Lyapunov function for error stability. |
| Performance Weight | \(Q\) | Designer-chosen matrix to weight different components (e.g., position vs. velocity) in the performance index \(J\). |
| Performance Preset | \(J^*\) | The upper bound for the cumulative cooperative error \(J\). A smaller \(J^*\) demands faster/ tighter formation convergence. |
| Compensation Signal | \(s_i(t)\) | Expands feasible formation set; calculated as \(s_i(t)=B_1(\dot{h}_i(t)-Ah_i(t))\) if the feasibility condition holds. |
4. Simulation Validation
To validate the proposed algorithm, a simulation was conducted with a swarm of \(N=5\) quadrotor drones. The desired formation \(h_i(t)\) was defined as a rotating pentagon in 3D space. The communication topology switched every 0.5 seconds among four different connected undirected graphs. The performance weight was set to \(Q = I_6\), and the performance preset was chosen as \(J^* = 1.8 \times 10^4\). The self-feedback gain \(K_c\) was designed to place the eigenvalues of \(A+BK_c\) to achieve desired damping. Solving the LMIs yielded the matrix \(R\) and subsequently the consensus gain \(K_u\).
The swarm’s evolution from random initial states to the desired formation is depicted in the following table, which captures the snapshots of the formation at key time instances, illustrating the convergence process and the final rotating pattern.
| Time (s) | Formation Status Description | Key Observation |
|---|---|---|
| t = 0 s | Initial dispersed positions. | Agents start from arbitrary, unordered locations in 3D space. |
| t = 2 s | Intermediate convergence phase. | The swarm begins to coalesce towards the pentagonal structure under the distributed protocol. |
| t = 10 s | Formation achieved. | The pentagon formation is clearly established and maintained. |
| t = 12 s | Steady-state rotation. | The entire formation rotates as a rigid body around the predefined center, tracking \(h_i(t)\). |
The simulation results confirmed the theoretical predictions. The formation errors \(\xi_i(t) – \xi_c(t)\) for all quadrotor drones converged asymptotically to zero. Most importantly, the calculated performance index converged to a final value of approximately \(J \approx 1.2 \times 10^4\), which is strictly below the prescribed threshold \(J^* = 1.8 \times 10^4\). This successfully demonstrates that the proposed control protocol achieves the dual objectives of accurate formation tracking and guaranteed cooperative performance under switching communication links.
5. Conclusion and Future Directions
This article has presented a comprehensive framework for the prescribed performance optimization formation control of quadrotor drone swarms. The proposed distributed control protocol, integrating self-feedback, consensus, and compensation terms, effectively decouples the formation center dynamics from the relative error dynamics. This allows for independent tuning of the swarm’s macroscopic motion. The core contribution is the formulation of the performance guarantee problem as a set of solvable Linear Matrix Inequalities, which directly incorporate the user-defined performance upper bound \(J^*\) into the gain design process. This ensures that the cumulative coordination error throughout the formation maneuver never exceeds the specified limit.
The method is robust to changes in the communication network topology, provided the network remains connected, making it suitable for real-world applications where links may be intermittent. Future research can extend this work in several promising directions. Firstly, investigating heterogeneous swarms comprising quadrotor drones with different dynamics or capabilities presents a significant challenge. Secondly, integrating adaptive or learning-based techniques (e.g., reinforcement learning) to online tune the gain matrices or the performance weight \(Q\) could enhance adaptability in completely unknown environments. Finally, combining this optimization-based approach with obstacle avoidance and dynamic trajectory planning under stringent real-time constraints would constitute a major step towards deploying fully autonomous, resilient, and performance-guaranteed quadrotor drone swarms in complex scenarios.