The coordinated flight of multiple Unmanned Aerial Vehicles (UAVs), or drone formation, represents a significant advancement in aerial systems. By sharing information and working cooperatively, a formation can dramatically enhance overall mission efficiency, robustness, and capability. This technology has found extensive applications in both military and civilian domains, including surveillance, payload delivery, search and rescue, and aerial light shows. The core challenge in drone formation control lies in maintaining a specific geometric configuration among the vehicles while following a desired trajectory, especially in the presence of real-world imperfections and threats.
During flight, a drone formation is inevitably subjected to various adverse factors. Externally, gusts and wind shear act as persistent disturbances. Internally, parameter uncertainties, unmodeled dynamics, and aging components like propellers introduce additional complexities. Furthermore, the failure of actuators, such as motors losing efficiency, poses a direct threat to flight safety. Beyond these reliability concerns, the dynamic process of adjusting the formation itself can be hazardous. Significant overshoot or large transient errors during maneuvers in a tight drone formation can lead to inter-vehicle collisions. Therefore, a high-performance formation controller must simultaneously address three critical aspects: the rapid attenuation of multi-source disturbances (including faults), fault tolerance, and the guaranteed satisfaction of transient performance constraints on formation errors to ensure safety.
Traditional methods like Proportional-Integral-Derivative (PID) or H∞ robust control offer basic formation keeping but often lack precision and speed in disturbance rejection. Sliding mode control improves robustness but may introduce chattering. While adaptive control can handle uncertainties, its response is generally slow. These methods primarily rely on error feedback to passively suppress disturbances, which may be inadequate under strong, time-varying interference. A more proactive approach is Active Disturbance Rejection Control (ADRC), which employs an observer to estimate the “total disturbance” – an aggregate of external disturbances, internal uncertainties, and nonlinearities – and then actively compensates for it in the control law. This framework leads to faster and more precise disturbance attenuation. However, a key limitation of standard disturbance observers, like the Extended State Observer (ESO), is their difficulty in accurately estimating and compensating for high-order time-varying disturbances, which commonly arise when faults and complex external disturbances are coupled within the system dynamics.
Concurrently, Prescribed Performance Control (PPC) has emerged as a powerful technique to enforce transient and steady-state performance on tracking errors. By confining the error evolution within a pre-defined, exponentially decaying envelope, PPC can prevent excessive overshoot and guarantee minimum convergence rate, thereby directly addressing the collision avoidance problem in tight drone formations. The conventional PPC method, however, has a stringent requirement: it needs prior knowledge of the initial error bounds to set the performance function’s initial value. If the initial formation error falls outside this preset envelope, the control scheme can fail. In practical scenarios, the exact initial state of a drone formation may not be precisely known, limiting the applicability of traditional PPC.
To overcome these intertwined challenges, this paper presents a novel composite control strategy for a leader-follower drone formation system subjected to external wind disturbances and actuator faults. The main contributions are twofold:
- We introduce a novel prescribed performance function based on the hyperbolic cotangent function. This function inherently possesses an extremely large value at the initial time, which can be tuned via a simple parameter. This property ensures that the initial formation tracking error, regardless of its actual (finite) value, always starts within the performance envelope, effectively removing the need for prior knowledge of the initial error bounds.
- We employ a Generalized Proportional Integral Observer (GPIO) for disturbance estimation and compensation. Unlike a standard ESO, the GPIO is capable of estimating not just the total disturbance but also its higher-order derivatives. This capability allows it to accurately track and compensate for high-order time-varying disturbances, significantly improving the formation accuracy and robustness against complex coupled disturbances and faults.
By integrating the GPIO-based active disturbance compensation with the modified PPC scheme, we develop a composite dynamic inversion controller. This controller guarantees that the formation tracking errors converge asymptotically to zero while their entire transient evolution remains within the user-prescribed performance bounds, ensuring safe and high-precision formation flight even under significant uncertainties and faults.
System Modeling and Problem Formulation for Drone Formation
We consider a leader-follower drone formation architecture. The leader’s kinematics in the ground frame \(\{O_g X_g Y_g\}\) are given by:
$$\begin{cases}
\dot{x}_L = v_L \cos \phi_L \\
\dot{y}_L = v_L \sin \phi_L
\end{cases}$$
where \(x_L, y_L\) are the leader’s positions, \(v_L\) is its speed, and \(\phi_L\) is its heading angle.
To describe relative positions clearly, we define a leader-body frame \(\{O_L X_L Y_L\}\), with its origin at the leader’s center of mass, the \(X_L\)-axis aligned with the leader’s velocity direction, and the \(Y_L\)-axis perpendicular to the left. The longitudinal distance (\(L_x\)) and lateral distance (\(L_y\)) of a follower relative to the leader in this frame define the formation geometry. For a triangular formation with two followers (F1, F2) and side length \(a\), the desired relative positions in the leader-body frame are:
$$
L_{x}^{F1} = \frac{\sqrt{3}}{2}a, \quad L_{y}^{F1} = -\frac{a}{2}; \quad L_{x}^{F2} = \frac{\sqrt{3}}{2}a, \quad L_{y}^{F2} = \frac{a}{2}.
$$
These are transformed to the ground frame as:
$$
\begin{bmatrix} L_x^i \\ L_y^i \end{bmatrix} = \begin{bmatrix} \cos \phi_L & -\sin \phi_L \\ \sin \phi_L & \cos \phi_L \end{bmatrix} \begin{bmatrix} L_x^{Fi} \\ L_y^{Fi} \end{bmatrix}, \quad i=1,2.
$$
Thus, the desired trajectory for follower \(i\) in the ground frame is:
$$\begin{cases}
x_{gr}^{Fi} = x_L – L_x^i = x_L – L_x^{Fi}\cos\phi_L + L_y^{Fi}\sin\phi_L \\
y_{gr}^{Fi} = y_L – L_y^i = y_L – L_x^{Fi}\sin\phi_L – L_y^{Fi}\cos\phi_L
\end{cases}$$
The dynamics of follower \(i\), considering actuator partial failure factors \(k_a^f, k_p^f \in [0, 0.8)\) and wind disturbances \(d_{\text{wind1}}^{Fi}, d_{\text{wind2}}^{Fi}\), are:
$$\begin{cases}
\dot{x}^{Fi} = v^{Fi} \cos \phi^{Fi} \\
\dot{y}^{Fi} = v^{Fi} \sin \phi^{Fi} \\
\dot{v}^{Fi} = (1 – k_a^f) a^{Fi} + d_{\text{wind1}}^{Fi} \\
\dot{\phi}^{Fi} = (1 – k_p^f) p^{Fi} + d_{\text{wind2}}^{Fi}
\end{cases}$$
where \(a^{Fi}\) and \(p^{Fi}\) are the acceleration and heading rate control inputs.
The formation tracking errors for follower \(i\) are defined as \(e_x^{Fi} = x_{gr}^{Fi} – x^{Fi}\) and \(e_y^{Fi} = y_{gr}^{Fi} – y^{Fi}\). After differentiating twice and combining with the leader’s dynamics, we obtain the error dynamics:
$$\begin{cases}
\ddot{e}_x^{Fi} = u_x^{Fi} + D_x^{Fi} \\
\ddot{e}_y^{Fi} = u_y^{Fi} + D_y^{Fi}
\end{cases}$$
where the virtual control inputs \(u_x^{Fi}, u_y^{Fi}\) are related to the actual inputs by:
$$
\begin{bmatrix} u_x^{Fi} \\ u_y^{Fi} \end{bmatrix} = \begin{bmatrix} -\cos\phi^{Fi} & \sin\phi^{Fi} \\ -\sin\phi^{Fi} & -\cos\phi^{Fi} \end{bmatrix} \begin{bmatrix} a^{Fi} \\ v^{Fi}p^{Fi} \end{bmatrix}.
$$
The term \(D^{Fi} = [D_x^{Fi}, D_y^{Fi}]^T\) represents the total disturbance, which aggregates the effects of the leader’s unknown acceleration/heading rate, wind disturbances, and actuator faults:
$$
D_x^{Fi} = f_1(v_L, \phi_L, L^{Fi}, \dot{\phi}_L, \ddot{\phi}_L) – d_{L1}^{Fi}\cos\phi^{Fi} + d_{L2}^{Fi} v^{Fi} \sin\phi^{Fi}
$$
$$
D_y^{Fi} = f_2(v_L, \phi_L, L^{Fi}, \dot{\phi}_L, \ddot{\phi}_L) – d_{L1}^{Fi}\sin\phi^{Fi} – d_{L2}^{Fi} v^{Fi} \cos\phi^{Fi}
$$
with \(d_{L1}^{Fi} = d_{\text{wind1}}^{Fi} – k_a^f a^{Fi}\) and \(d_{L2}^{Fi} = d_{\text{wind2}}^{Fi} – k_p^f p^{Fi}\). Clearly, \(D^{Fi}\) is a complex, high-order time-varying signal. The core control problem is to design \(u^{Fi}\) to drive \(e^{Fi}\) to zero while respecting transient constraints, despite \(D^{Fi}\).
Prescribed Performance Control Scheme for Drone Formation
For safety in a tight drone formation, we must ensure the formation tracking error remains within safe bounds during transients. The prescribed performance requirement is formulated as:
$$
-\rho_j(t) < e_j(t) < \rho_j(t), \quad \forall t \ge 0, \quad j \in \{x, y\}
$$
where \(\rho_j(t)\) is a smooth, positive, and decreasing performance function.
Innovative Prescribed Performance Function: To remove the dependency on initial error knowledge, we propose a new performance function based on the hyperbolic cotangent:
$$
\rho_j(t) = \coth(k_j t + l_j) – 1 + \rho_{j\infty}
$$
where \(\rho_{j\infty} > 0\) is the ultimate bound, \(k_j > 0\) governs the convergence rate, and \(l_j > 0\) is a small positive constant. The key property is: \(\lim_{l_j \to 0^+} \rho_j(0) = +\infty\). By choosing a sufficiently small \(l_j\), we can ensure \(\rho_j(0)\) is large enough to encompass any practical finite initial error \(e_j(0)\), thus satisfying \(-\rho_j(0) < e_j(0) < \rho_j(0)\) automatically.
Error Transformation: To transform the constrained problem into an unconstrained one, we define a normalized error \(\tilde{e}_j(t) = e_j(t) / \rho_j(t)\). The constraint \(|\tilde{e}_j(t)| < 1\) is equivalent to the original performance requirement. We then introduce a transformed error variable \(\varepsilon_j(t)\) via a smooth, strictly increasing function:
$$
\varepsilon_j(t) = \frac{1}{2} \ln\left(\frac{1 + \tilde{e}_j(t)}{1 – \tilde{e}_j(t)}\right) \quad \Leftrightarrow \quad \tilde{e}_j(t) = \tanh(\varepsilon_j(t)).
$$
If we can design a controller to ensure \(\varepsilon_j(t)\) is bounded and converges to zero, then \(\tilde{e}_j(t)\) will remain within \((-1, 1)\) and \(e_j(t)\) will satisfy the prescribed performance and converge to zero. The dynamics of the transformed error are derived as:
$$
\ddot{\varepsilon}_j = n_j (u_j + D_j) + f_j(e_j, \dot{e}_j, \rho_j, \dot{\rho}_j, \ddot{\rho}_j)
$$
where \(n_j = 1/[\rho_j (1+\tilde{e}_j)(1-\tilde{e}_j)] > 0\) and \(f_j(\cdot)\) is a known nonlinear function. The control objective is now to design \(u_j\) to stabilize the \(\varepsilon_j\)-dynamics.
Composite Controller Design and Stability Analysis
Our control strategy has two main components: a GPIO to estimate the total disturbance \(D_j\) and the error derivative \(\dot{e}_j\), and a dynamic inversion controller that uses these estimates and enforces the prescribed performance via the transformed error \(\varepsilon_j\).
Generalized Proportional Integral Observer (GPIO) Design
Assuming the total disturbance \(D_j\) is second-order differentiable with bounded second derivative (\(\ddot{D}_j\)), we design the following fourth-order GPIO for each channel (\(j=x,y\)) of each follower:
$$\begin{cases}
\dot{z}_{j1} = z_{j2} – \beta_{j1}(z_{j1} – e_j) \\
\dot{z}_{j2} = z_{j3} – \beta_{j2}(z_{j1} – e_j) + u_j \\
\dot{z}_{j3} = z_{j4} – \beta_{j3}(z_{j1} – e_j) \\
\dot{z}_{j4} = -\beta_{j4}(z_{j1} – e_j)
\end{cases}$$
The estimates are provided by:
$$
\hat{\dot{e}}_j = z_{j2}, \quad \hat{D}_j = z_{j3}.
$$
The observer gains \(\beta_{j1}, \beta_{j2}, \beta_{j3}, \beta_{j4}\) are chosen such that the characteristic polynomial \(s^4 + \beta_{j1}s^3 + \beta_{j2}s^2 + \beta_{j3}s + \beta_{j4}\) is Hurwitz. Defining the estimation errors as \(e_{j1}=z_{j1}-e_j\), \(e_{j2}=z_{j2}-\dot{e}_j\), \(e_{j3}=z_{j3}-D_j\), \(e_{j4}=z_{j4}-\dot{D}_j\), their dynamics are:
$$
\begin{bmatrix} \dot{e}_{j1} \\ \dot{e}_{j2} \\ \dot{e}_{j3} \\ \dot{e}_{j4} \end{bmatrix} =
\begin{bmatrix}
-\beta_{j1} & 1 & 0 & 0 \\
-\beta_{j2} & 0 & 1 & 0 \\
-\beta_{j3} & 0 & 0 & 1 \\
-\beta_{j4} & 0 & 0 & 0
\end{bmatrix}
\begin{bmatrix} e_{j1} \\ e_{j2} \\ e_{j3} \\ e_{j4} \end{bmatrix} +
\begin{bmatrix} 0 \\ 0 \\ 0 \\ -\ddot{D}_j \end{bmatrix}.
$$
Given the Hurwitz property of the system matrix and the boundedness of \(\ddot{D}_j\), the estimation errors \(e_{j2}, e_{j3}\) are globally uniformly ultimately bounded. If \(\ddot{D}_j\) converges to zero, the estimates converge asymptotically.
Composite Prescribed Performance Dynamic Inversion Controller
Using the estimates from the GPIO, we design the following composite control law for each channel:
$$
u_j = -\frac{k_p}{n_j} \varepsilon_j – \frac{k_d}{n_j} \dot{\varepsilon}_j – \frac{\dot{n}_j}{n_j}(\hat{\dot{e}}_j – \tilde{e}_j \dot{\rho}_j) + \dot{\tilde{e}}_j \dot{\rho}_j + \tilde{e}_j \ddot{\rho}_j – \hat{D}_j.
$$
Here, \(k_p > 0\) and \(k_d > 0\) are controller gains. This control law has a clear structure: the first two terms form a PD controller on the transformed error \(\varepsilon_j\) to drive it to zero; the next three terms compensate for the known nonlinearities \(f_j(\cdot)\) derived from the performance function transformation; and the final term \(-\hat{D}_j\) actively cancels the estimated total disturbance.
Substituting this controller into the transformed error dynamics \(\ddot{\varepsilon}_j = n_j(u_j + D_j) + f_j\) yields the closed-loop dynamics:
$$
\ddot{\varepsilon}_j = -k_d \dot{\varepsilon}_j – k_p \varepsilon_j + \dot{n}_j e_{j2} + n_j e_{j3}.
$$
Combining this with the observer error dynamics, we get the augmented closed-loop system:
$$
\frac{d}{dt}
\begin{bmatrix} e_{j1} \\ e_{j2} \\ e_{j3} \\ e_{j4} \\ \varepsilon_j \\ \dot{\varepsilon}_j \end{bmatrix} =
\begin{bmatrix}
-\beta_{j1} & 1 & 0 & 0 & 0 & 0 \\
-\beta_{j2} & 0 & 1 & 0 & 0 & 0 \\
-\beta_{j3} & 0 & 0 & 1 & 0 & 0 \\
-\beta_{j4} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & \dot{n}_j & n_j & 0 & -k_p & -k_d
\end{bmatrix}
\begin{bmatrix} e_{j1} \\ e_{j2} \\ e_{j3} \\ e_{j4} \\ \varepsilon_j \\ \dot{\varepsilon}_j \end{bmatrix} +
\begin{bmatrix} 0 \\ 0 \\ 0 \\ -\ddot{D}_j \\ 0 \\ 0 \end{bmatrix}.
$$
The system matrix is block triangular. The observer block is Hurwitz by design, and the lower-right PD control block \( \begin{bmatrix} 0 & 1 \\ -k_p & -k_d \end{bmatrix} \) is also Hurwitz. Furthermore, \(n_j\) and \(\dot{n}_j\) are bounded due to the properties of \(\rho_j(t)\) and \(\tilde{e}_j(t)\). Therefore, the overall augmented system is input-to-state stable with respect to the bounded input \(\ddot{D}_j\). Consequently, the transformed error \(\varepsilon_j(t)\) and the observer errors \(e_{j2}(t), e_{j3}(t)\) are globally uniformly ultimately bounded.
Prescribed Performance Guarantee: Since the controller ensures \(\varepsilon_j(t)\) is bounded, and by the properties of the transformation \(\tilde{e}_j = \tanh(\varepsilon_j)\), it follows that \(|\tilde{e}_j(t)| < 1\) for all \(t \ge 0\). This directly implies the original prescribed performance condition is satisfied:
$$
-\rho_j(t) < e_j(t) < \rho_j(t).
$$
Furthermore, if the disturbance derivatives vanish in steady-state (\(\ddot{D}_j \to 0\)), then \(\varepsilon_j(t) \to 0\), which implies \(e_j(t) \to 0\) asymptotically. The actual control commands for acceleration \(a^{Fi}\) and heading rate \(p^{Fi}\) are recovered by inverting the virtual control relationship:
$$
\begin{bmatrix} a^{Fi} \\ v^{Fi} p^{Fi} \end{bmatrix} = \begin{bmatrix} -\cos\phi^{Fi} & -\sin\phi^{Fi} \\ \sin\phi^{Fi} & -\cos\phi^{Fi} \end{bmatrix} \begin{bmatrix} u_x^{Fi} \\ u_y^{Fi} \end{bmatrix}.
$$
Simulation Analysis and Results
To validate the proposed method (labeled PPC+NDIC+GPIO), we conduct numerical simulations for a triangular leader-follower drone formation with two followers. We compare it against three other controllers: Basic Nonlinear Dynamic Inversion Control (BNDIC), Composite NDIC with an ESO (NDIC+ESO), and Composite NDIC with a GPIO but without PPC (NDIC+GPIO).
Simulation Setup: The leader executes a maneuver comprising acceleration, constant velocity, a turn, and deceleration. Followers start with initial position errors. A shear wind disturbance acts on the followers, and actuator partial failure occurs at \(t = 15\) s. The desired formation is an equilateral triangle with side \(a=4\) m. The proposed controller parameters are set as follows:
| Component | Parameter | Value (Follower 1 & 2) |
|---|---|---|
| Controller | \(k_p\) | 2.25 |
| \(k_d\) | 3.0 | |
| GPIO Gains (\(\beta\)) | \(\beta_1\) | 28 |
| \(\beta_2\) | 294 | |
| \(\beta_3\) | 1372 | |
| \(\beta_4\) | 2401 | |
| PPC Function (F1, x) | \(l_x\) | 0.05 |
| \(k_x\) | 0.4 | |
| \(\rho_{x\infty}\) | 0.3 |
Results and Discussion: The simulation results conclusively demonstrate the advantages of the proposed integrated approach for drone formation control.
1. Formation Tracking Accuracy: The formation tracking error plots for both followers show that all three active disturbance rejection methods (NDIC+ESO, NDIC+GPIO, PPC+NDIC+GPIO) achieve significantly smaller steady-state errors compared to the baseline BNDIC. During the turning maneuver (a high-disturbance phase), the error magnitude for the proposed method is the smallest among all, showcasing its superior disturbance rejection capability. The NDIC+GPIO method performs better than NDIC+ESO, highlighting the GPIO’s advantage in handling higher-order disturbance dynamics arising from the coupled faults and wind.
2. Guaranteed Transient Performance: This is the most distinctive feature. The plots clearly show that only the proposed PPC+NDIC+GPIO method guarantees the formation errors (both \(e_x\) and \(e_y\)) remain strictly within the prescribed performance bounds (the dashed envelopes) at all times, including the initial transient. The innovative hyperbolic cotangent function creates a very large initial bound, successfully encapsulating the unknown initial errors. Other methods, including NDIC+GPIO, violate these bounds during the initial convergence and the turn, indicating a risk of collision in a physically tight formation.
3. Disturbance Estimation Performance: The disturbance estimation error plots (\(D_j – \hat{D}_j\)) provide insight into the observer’s performance. The GPIO-based methods (both with and without PPC) show faster convergence and significantly smaller estimation errors compared to the ESO-based method, especially after the actuator fault occurs at \(t=15\) s. This accurate and rapid estimation is the key enabler for the precise cancellation in the control law.
4. Control Effort: The control input histories for all methods are of comparable magnitude, indicating that the performance improvement of the proposed method is not achieved through excessive control action but through more intelligent disturbance estimation and compensation coupled with transient shaping.
| Method | Disturbance/Fault Rejection | Transient Performance Guarantee | Dependency on Initial Error Info | Suitability for Tight Formation |
|---|---|---|---|---|
| BNDIC | Poor | No | N/A | Low |
| NDIC+ESO | Good | No | N/A | Medium |
| NDIC+GPIO | Very Good | No | N/A | Medium |
| PPC+NDIC+GPIO (Proposed) | Excellent | Yes | No | High |
Conclusion
This paper has addressed the critical challenge of safe and precise drone formation control in the presence of high-order time-varying disturbances, including external wind and actuator faults. The proposed solution is a composite control strategy that synergistically combines an advanced Prescribed Performance Control (PPC) scheme with a Generalized Proportional Integral Observer (GPIO)-based active disturbance rejection framework.
The introduced hyperbolic cotangent-based performance function successfully eliminates the practical limitation of traditional PPC by ensuring the initial formation error is always within the performance envelope, without requiring any prior knowledge of its bounds. The GPIO provides high-fidelity estimates of the complex, aggregated disturbances, enabling their direct and effective cancellation in the control law. The resulting composite dynamic inversion controller guarantees that the formation tracking errors not only converge asymptotically but do so within rigorously pre-defined transient performance bounds, effectively mitigating the risk of inter-vehicle collision during aggressive maneuvers.
Simulation studies on a leader-follower triangular drone formation confirm the superiority of the proposed method over basic dynamic inversion and observer-based methods without performance constraints. It demonstrates excellent disturbance rejection, precise fault accommodation, and guaranteed safe transient error evolution. Future work will focus on extending this approach to address practical communication constraints within the formation, such as time delays and packet losses, to further enhance the robustness and applicability of autonomous drone formation systems in real-world scenarios.