Quadrotor unmanned aerial vehicles (UAVs), commonly known as quadrotor drones, have become ubiquitous in modern applications ranging from aerial photography and surveying to security and infrastructure inspection due to their simple mechanical structure, vertical take-off and landing (VTOL) capability, and autonomous hovering. However, the flight control system design for a quadrotor drone presents significant challenges. Its dynamics are inherently nonlinear, strongly coupled, and underactuated. Furthermore, their typically small size and low mass make them highly susceptible to complex internal model uncertainties and external disturbances, which can severely degrade control performance and even lead to system instability. Therefore, the development of high-performance, robust flight control strategies for quadrotor drones remains a critical research focus.
Traditional linear control methods, such as PID, often fall short in managing the nonlinear and coupled dynamics effectively. Among nonlinear control techniques, backstepping control offers a systematic, recursive design procedure. A major drawback of conventional backstepping is the “explosion of complexity,” where the repeated differentiation of virtual control laws leads to overly complex controller expressions. While dynamic surface control (DSC) mitigates this by using first-order filters, it often neglects filter error compensation, sacrificing performance. Command-filtered backstepping addresses this by explicitly compensating for filter errors.
More critically, control laws designed using classical Lyapunov theory typically guarantee only asymptotic stability, meaning convergence time can be unbounded and is heavily dependent on initial conditions. For time-sensitive applications, finite-time control offers faster convergence, but its settling time upper bound still depends on the initial state. Fixed-time stability theory provides a superior alternative, guaranteeing that the system states converge to an equilibrium within a time upper bound that is independent of the initial conditions, set solely by the controller parameters. This property is highly desirable for quadrotor drone operations where predictable, fast response is crucial regardless of initial deviations.
This article presents a comprehensive framework for the attitude tracking control of a quadrotor drone subject to both model uncertainties and external disturbances. The core contributions are threefold. First, a novel fixed-time command filter based on a smooth hyperbolic tangent function is proposed. This filter estimates the derivatives of virtual control laws within a fixed time, effectively circumventing the “explosion of complexity” inherent in backstepping while eliminating the singularity and chattering issues prevalent in traditional fixed-time filters that employ fractional-order terms. Second, a composite adaptive Radial Basis Function Neural Network (RBFNN) is employed to approximate unknown system dynamics. Unlike standard gradient-based updates, the adaptive law for the neural network weights is designed using both tracking error and prediction error from a parallel estimation model, significantly improving online approximation accuracy and convergence. Third, an adaptive disturbance observer (DO) is integrated to estimate and compensate for external disturbances in real-time. By synthesizing these components within a backstepping framework, a fixed-time control law is derived. Rigorous Lyapunov analysis proves that all closed-loop signals are uniformly ultimately bounded and that the attitude tracking error converges to a small neighborhood of the origin within a fixed time, independent of initial conditions. Extensive numerical simulations validate the effectiveness and superiority of the proposed control scheme over conventional methods.
Theoretical Preliminaries and Quadrotor Model
Mathematical Model of Quadrotor Dynamics
The kinematic and dynamic model of a quadrotor drone subject to disturbances is derived using the Newton-Euler formalism. Consider the earth-fixed inertial frame $\mathcal{F}_e(O_eX_eY_eZ_e)$ and the body-fixed frame $\mathcal{F}_b(O_bX_bY_bZ_b)$. The attitude is described by the Euler angle vector $\boldsymbol{\Theta} = [\phi, \theta, \psi]^T$, representing roll, pitch, and yaw, respectively. The angular velocity vector in the body frame is $\boldsymbol{\sigma} = [p, q, r]^T$. The nonlinear model is:
$$
\begin{aligned}
\dot{\phi} &= p + q \sin\phi \tan\theta + r \cos\phi \tan\theta \\
\dot{\theta} &= q \cos\phi – r \sin\phi \\
\dot{\psi} &= q \frac{\sin\phi}{\cos\theta} + r \frac{\cos\phi}{\cos\theta} \\
J_{xx}\dot{p} &= (J_{yy} – J_{zz})qr + \tau_x + d_x \\
J_{yy}\dot{q} &= (J_{zz} – J_{xx})pr + \tau_y + d_y \\
J_{zz}\dot{r} &= (J_{xx} – J_{yy})pq + \tau_z + d_z
\end{aligned}
$$
where $\mathbf{J} = \text{diag}(J_{xx}, J_{yy}, J_{zz})$ is the inertia matrix, $\boldsymbol{\Gamma} = [\tau_x, \tau_y, \tau_z]^T$ is the control torque vector, and $\mathbf{d} = [d_x, d_y, d_z]^T$ represents external disturbances. Defining the state vectors $\mathbf{x}_1 = \boldsymbol{\Theta}$, $\mathbf{x}_2 = \boldsymbol{\sigma}$, and the control input $\mathbf{u} = \boldsymbol{\Gamma}$, the model can be written in a compact state-space form suitable for controller design:
$$
\begin{aligned}
\dot{\mathbf{x}}_1 &= \mathbf{B}(\mathbf{x}_1)\mathbf{x}_2 \\
\dot{\mathbf{x}}_2 &= \mathbf{J}^{-1}\left( -\mathbf{x}_2 \times \mathbf{J}\mathbf{x}_2 + \mathbf{u} + \mathbf{F}(\mathbf{x}) + \mathbf{d} \right) \\
\mathbf{y} &= \mathbf{x}_1
\end{aligned}
$$
Here, $\mathbf{B}(\mathbf{x}_1)$ is the kinematic transformation matrix, and $\mathbf{F}(\mathbf{x})$ represents lumped model uncertainties.
Assumptions and Supporting Lemmas
The control design and stability analysis are based on the following standard assumptions and mathematical lemmas.
Assumption 1: The external disturbance $\mathbf{d}$ and its rate are bounded, i.e., $\|\mathbf{d}\| \leq \bar{d}$ and $\|\dot{\mathbf{d}}\| \leq \bar{d}_1$, where $\bar{d}$ and $\bar{d}_1$ are unknown positive constants.
Assumption 2: The roll ($\phi$) and pitch ($\theta$) angles operate within the range $(-\pi/2, \pi/2)$ to avoid singularities in the rotation matrix $\mathbf{B}(\mathbf{x}_1)$.
Assumption 3: All system states are measurable. The desired attitude trajectory $\mathbf{y}_r(t)$ and its first and second derivatives $\dot{\mathbf{y}}_r(t)$, $\ddot{\mathbf{y}}_r(t)$ are smooth, known, and bounded.
Lemma 1 (RBF Neural Network): For any continuous unknown function $f(\mathbf{Z}): \mathbb{R}^n \rightarrow \mathbb{R}$ over a compact set $\Omega_Z$, there exists an RBFNN such that:
$$ f(\mathbf{Z}) = \mathbf{W}^{*T}\mathbf{S}(\mathbf{Z}) + \varepsilon^{*}(\mathbf{Z}) $$
where $\mathbf{W}^*$ is the ideal bounded weight vector, $\mathbf{S}(\mathbf{Z})$ is the basis function vector, and $\varepsilon^{*}(\mathbf{Z})$ is the minimal approximation error satisfying $|\varepsilon^{*}| \leq \bar{\varepsilon}$.
Lemma 2 (Property of Hyperbolic Tangent): For any $\kappa > 0$ and $\eta \in \mathbb{R}$, the following inequality holds:
$$ 0 \leq |\eta| – \eta \tanh\left(\frac{\eta}{\kappa}\right) \leq \iota \kappa $$
where $\iota = 0.2785$.
Lemma 3 (Young’s Inequality): For any $a, b \geq 0$, $p>1$, $q>1$ with $(1/p)+(1/q)=1$, we have:
$$ ab \leq \frac{a^p}{p} + \frac{b^q}{q} $$
Lemma 4 (Fixed-Time Stability): Consider the system $\dot{x} = f(x; t)$, $x(0)=x_0$. If there exists a positive definite Lyapunov function $V(x)$ satisfying
$$ \dot{V}(x) \leq -\alpha V^p(x) – \beta V^q(x) + \varrho $$
where $\alpha, \beta, \varrho > 0$, $0 < p < 1$, and $q > 1$, then the system is practically fixed-time stable. The trajectory converges to the set $\{ x | V(x) \leq \min\{ (\varrho/((1-\zeta)\alpha))^{1/p}, (\varrho/((1-\zeta)\beta))^{1/q} \} \}$ within a settling time $T_s$ bounded by:
$$ T_s \leq \frac{1}{\alpha \zeta (1-p)} + \frac{1}{\beta \zeta (q-1)} $$
for any $0 < \zeta < 1$. This bound is independent of $x_0$.
Controller Design Methodology
The proposed control architecture is a synthesis of a fixed-time command filter, a composite adaptive RBFNN, an adaptive disturbance observer, and a backstepping control law. The overall structure is depicted in the following conceptual block diagram, illustrating the flow from desired trajectory to control torque.
1. Fixed-Time Command Filter with Hyperbolic Tangent Function
A key innovation is the design of a non-singular, chattering-free fixed-time command filter. Traditional fixed-time filters often use fractional-power terms (e.g., $\text{sig}^{1/2}(\cdot)$), which can cause singularity when the argument is zero and induce chattering. To overcome this, we propose a filter based on the smooth hyperbolic tangent function:
$$
\dot{\hat{\boldsymbol{\zeta}}}(t) = -\varrho \left( \tanh\left(\frac{\hat{\boldsymbol{\zeta}}(t) – \boldsymbol{\zeta}(t)}{\kappa}\right) + \alpha \tanh^{3}\left(\frac{\hat{\boldsymbol{\zeta}}(t) – \boldsymbol{\zeta}(t)}{\kappa}\right) \right)
$$
where $\boldsymbol{\zeta}(t)$ is the input signal (e.g., a virtual control law), $\hat{\boldsymbol{\zeta}}(t)$ is the filter output, and $\varrho, \alpha, \kappa > 0$ are design parameters. This filter guarantees that the estimation error $\tilde{\boldsymbol{\zeta}} = \hat{\boldsymbol{\zeta}} – \boldsymbol{\zeta}$ converges to a small neighborhood of zero within a fixed time $T_f$, i.e., $\|\tilde{\boldsymbol{\zeta}}\| \leq \Delta$ for all $t \geq T_f$. Moreover, the filter output $\hat{\boldsymbol{\zeta}}$ and its derivative $\dot{\hat{\boldsymbol{\zeta}}}$ remain bounded. This filter is applied in the backstepping procedure to obtain the derivative of virtual controls without causing complexity explosion.
2. Composite Adaptive RBF Neural Network Design
To handle the lumped model uncertainty $\mathbf{F}(\mathbf{x})$, a composite adaptive RBFNN is employed. The key idea is to update the neural network weights using information from both the tracking error and a parallel estimation model prediction error, leading to faster and more accurate parameter convergence.
The network approximates the uncertainty as:
$$ \hat{\mathbf{F}}(\mathbf{x} | \hat{\mathbf{W}}) = \hat{\mathbf{W}}^T \mathbf{S}(\mathbf{x}) $$
where $\hat{\mathbf{W}}$ is the estimate of the ideal weights $\mathbf{W}^*$. A serial-parallel estimation model is constructed:
$$ \dot{\hat{\mathbf{x}}}_2 = \mathbf{J}^{-1}\left( -\mathbf{x}_2 \times \mathbf{J}\mathbf{x}_2 + \mathbf{u} + \hat{\mathbf{W}}^T \mathbf{S}(\mathbf{x}) \right) – \mathbf{K}_1 \tilde{\mathbf{x}}_2 – \mathbf{K}_2 \text{sig}^{1/2}(\tilde{\mathbf{x}}_2) $$
with $\tilde{\mathbf{x}}_2 = \hat{\mathbf{x}}_2 – \mathbf{x}_2$ and $\mathbf{K}_1, \mathbf{K}_2 > 0$. The prediction error $\tilde{\mathbf{x}}_2$ provides rich information about the approximation quality. The composite adaptive law is then designed as:
$$ \dot{\hat{\mathbf{W}}} = \boldsymbol{\Gamma}_w \left( \mathbf{S}(\mathbf{x}) (\mathbf{e}_2^T \mathbf{J}^{-1} + \gamma \tilde{\mathbf{x}}_2^T) – \sigma_w \hat{\mathbf{W}} \right) $$
where $\boldsymbol{\Gamma}_w > 0$ is a gain matrix, $\mathbf{e}_2$ is a tracking error to be defined later, $\gamma > 0$ is a weighting factor, and $\sigma_w > 0$ is a $\sigma$-modification term for robustness. This law ensures that the weight estimates $\hat{\mathbf{W}}$ converge rapidly, improving the overall system’s adaptability.
3. Adaptive Disturbance Observer Design
An adaptive disturbance observer (DO) is designed to estimate the time-varying external disturbance $\mathbf{d}$ in fixed time. Let $\hat{\mathbf{d}}$ be the estimate of $\mathbf{d}$. Define an auxiliary variable $\mathbf{z} = \hat{\mathbf{d}} – \mathbf{L} \mathbf{x}_2$, where $\mathbf{L} > 0$. The observer dynamics are proposed as:
$$ \begin{aligned}
\dot{\mathbf{z}} &= -\mathbf{L}\mathbf{J}^{-1}\left( -\mathbf{x}_2 \times \mathbf{J}\mathbf{x}_2 + \mathbf{u} + \hat{\mathbf{F}} + \hat{\mathbf{d}} \right) – \boldsymbol{\Lambda}_1 \text{sig}^{1/2}(\tilde{\mathbf{d}}) – \boldsymbol{\Lambda}_2 \tilde{\mathbf{d}} \\
\hat{\mathbf{d}} &= \mathbf{z} + \mathbf{L} \mathbf{x}_2
\end{aligned} $$
where $\tilde{\mathbf{d}} = \hat{\mathbf{d}} – \mathbf{d}$ is the disturbance estimation error, and $\boldsymbol{\Lambda}_1, \boldsymbol{\Lambda}_2 > 0$. It can be shown that the error dynamics $\dot{\tilde{\mathbf{d}}}$ are fixed-time stable, guaranteeing $\|\tilde{\mathbf{d}}\| \rightarrow 0$ within a time $T_d$ independent of initial estimation error.
4. Fixed-Time Backstepping Control Law Synthesis
The controller is designed using the backstepping technique in two steps, incorporating the aforementioned components.
Step 1 (Attitude Kinematics Control): Define the tracking error for the attitude angles: $\mathbf{e}_1 = \mathbf{x}_1 – \mathbf{y}_r$. A virtual control law $\boldsymbol{\alpha}_1$ for the angular velocity $\mathbf{x}_2$ is designed to stabilize $\mathbf{e}_1$:
$$ \boldsymbol{\alpha}_1 = \mathbf{B}^{-1}(\mathbf{x}_1) \left( -\mathbf{K}_{1p} \mathbf{e}_1 – \mathbf{K}_{1f} \text{sig}^{a_1}(\mathbf{e}_1) + \dot{\mathbf{y}}_r \right) $$
where $\mathbf{K}_{1p}, \mathbf{K}_{1f} > 0$, and $0 < a_1 < 1$. To avoid differentiating $\boldsymbol{\alpha}_1$, it is passed through the fixed-time command filter (2) to obtain $\boldsymbol{\alpha}_{1f}$ and its derivative $\dot{\boldsymbol{\alpha}}_{1f}$.
Step 2 (Attitude Dynamics Control): Define the angular velocity tracking error: $\mathbf{e}_2 = \mathbf{x}_2 – \boldsymbol{\alpha}_{1f}$. Using the model, the dynamics of $\mathbf{e}_2$ are:
$$ \mathbf{J} \dot{\mathbf{e}}_2 = -\mathbf{x}_2 \times \mathbf{J}\mathbf{x}_2 + \mathbf{u} + \mathbf{F}(\mathbf{x}) + \mathbf{d} – \mathbf{J} \dot{\boldsymbol{\alpha}}_{1f} $$
Substituting the RBFNN approximation $\hat{\mathbf{F}}(\mathbf{x} | \hat{\mathbf{W}})$ and the disturbance estimate $\hat{\mathbf{d}}$, the final control law is designed as:
$$ \begin{aligned}
\mathbf{u} &= \mathbf{x}_2 \times \mathbf{J}\mathbf{x}_2 – \hat{\mathbf{W}}^T \mathbf{S}(\mathbf{x}) – \hat{\mathbf{d}} + \mathbf{J} \dot{\boldsymbol{\alpha}}_{1f} \\
&\quad – \mathbf{K}_{2p} \mathbf{e}_2 – \mathbf{K}_{2f} \text{sig}^{a_2}(\mathbf{e}_2) – \mathbf{J} \mathbf{B}^T(\mathbf{x}_1) \mathbf{e}_1
\end{aligned} $$
where $\mathbf{K}_{2p}, \mathbf{K}_{2f} > 0$, and $a_2 > 1$. The terms $-\mathbf{K}_{if} \text{sig}^{a_i}(\mathbf{e}_i)$ are responsible for enforcing fixed-time convergence.
The complete set of design parameters and their typical roles are summarized in the table below.
| Component | Key Parameters | Primary Function / Tuning Effect |
|---|---|---|
| Fixed-Time Filter | $\varrho, \alpha, \kappa$ | Govern convergence speed and smoothness of derivative estimation. Larger $\varrho, \alpha$ give faster convergence but may increase initial control effort. |
| RBFNN Adaptive Law | $\boldsymbol{\Gamma}_w, \gamma, \sigma_w$ | Control the rate and accuracy of neural weight learning. $\gamma$ balances tracking and prediction error influence. |
| Disturbance Observer | $\mathbf{L}, \boldsymbol{\Lambda}_1, \boldsymbol{\Lambda}_2$ | Determine the speed and accuracy of disturbance estimation. Larger gains yield faster estimation but may increase sensitivity to noise. |
| Backstepping Control | $\mathbf{K}_{1p}, \mathbf{K}_{1f}, a_1, \mathbf{K}_{2p}, \mathbf{K}_{2f}, a_2$ | $\mathbf{K}_{ip}$ terms provide asymptotic damping. $\mathbf{K}_{if}$ and $a_i$ govern fixed-time convergence characteristics. $a_1<1$, $a_2>1$ are essential for fixed-time stability. |
Stability Analysis and Performance Guarantees
The closed-loop stability of the entire system—comprising the quadrotor drone dynamics, the fixed-time filter, the composite adaptive RBFNN, the disturbance observer, and the backstepping controller—is rigorously proven using Lyapunov theory.
Theorem (Fixed-Time Practical Stability): Consider the quadrotor drone system (1) under Assumptions 1-3. With the fixed-time command filter (2), the composite adaptive RBFNN with update law (4), the disturbance observer (5), and the control law (7), all signals in the closed-loop system are uniformly ultimately bounded. Furthermore, the attitude tracking errors $\mathbf{e}_1$ and $\mathbf{e}_2$ converge to the compact sets $\Omega_1$ and $\Omega_2$, respectively, within a fixed time $T^*$, where $T^*$ is independent of the initial conditions. The sets $\Omega_1$ and $\Omega_2$ can be made arbitrarily small by appropriately increasing the control gains $\mathbf{K}_{if}$ and reducing the neural network approximation error bound $\bar{\varepsilon}$.
Proof Sketch: The proof constructs a composite Lyapunov function candidate $V$ encompassing the tracking errors ($\mathbf{e}_1, \mathbf{e}_2$), the neural network weight estimation errors ($\tilde{\mathbf{W}} = \hat{\mathbf{W}} – \mathbf{W}^*$), and the disturbance estimation error ($\tilde{\mathbf{d}}$):
$$ V = \frac{1}{2} \mathbf{e}_1^T \mathbf{e}_1 + \frac{1}{2} \mathbf{e}_2^T \mathbf{J} \mathbf{e}_2 + \frac{1}{2} \text{tr}(\tilde{\mathbf{W}}^T \boldsymbol{\Gamma}_w^{-1} \tilde{\mathbf{W}}) + \frac{1}{2} \tilde{\mathbf{d}}^T \tilde{\mathbf{d}} $$
The time derivative $\dot{V}$ is evaluated along the trajectories of the closed-loop system. By substituting the control law (7), the adaptive laws, and applying the properties of the command filter and disturbance observer, one can derive an inequality of the form:
$$ \dot{V} \leq -\lambda_1 V^{\frac{1+a_1}{2}} – \lambda_2 V^{\frac{1+a_2}{2}} + \varpi $$
where $\lambda_1, \lambda_2, \varpi > 0$ are constants aggregating the various controller gains. Since $0 < \frac{1+a_1}{2} < 1$ and $\frac{1+a_2}{2} > 1$, by Lemma 4 (Fixed-Time Stability), the system is practically fixed-time stable. The residual bound $\varpi$ arises from the bounded neural network approximation error $\bar{\varepsilon}$ and the ultimate bound of the filter error. This completes the proof.
Numerical Simulations and Results
To validate the proposed control framework, extensive numerical simulations were conducted using a high-fidelity model of a standard quadrotor drone. The performance was compared against a conventional finite-time backstepping controller and a PID controller under identical conditions of model uncertainty and external disturbance.
Simulation Setup: The quadrotor drone parameters are: mass $m = 1.4$ kg, inertia matrix $\mathbf{J} = \text{diag}(0.0211, 0.0219, 0.0366)$ kg·m². The desired attitude trajectory is a challenging sinusoidal signal: $\mathbf{y}_r(t) = [3\sin(0.5t), 2\sin(0.5t+\pi/4), \sin(0.5t)]^T$ rad. The initial attitude is set to $[0.2, 0.15, 0.1]^T$ rad. The lumped model uncertainty is $\mathbf{F}(\mathbf{x}) = 0.2 \cdot [\sin(x_2), \cos(x_4), \sin(x_6)]^T$ N·m, and the external disturbance is $\mathbf{d}(t) = [0.3\cos(t), 0.2\sin(0.8t), 0.25\cos(1.2t)]^T$ N·m.
Control Parameters: The parameters for the proposed controller were selected as follows:
- Filter: $\varrho=12, \alpha=5, \kappa=0.1$.
- RBFNN: 27 neurons per output, $\boldsymbol{\Gamma}_w=50\mathbf{I}, \gamma=2, \sigma_w=0.01$.
- Disturbance Observer: $\mathbf{L}=8\mathbf{I}, \boldsymbol{\Lambda}_1=5\mathbf{I}, \boldsymbol{\Lambda}_2=5\mathbf{I}$.
- Backstepping Gains: $\mathbf{K}_{1p}=3\mathbf{I}, \mathbf{K}_{1f}=2\mathbf{I}, a_1=0.8, \mathbf{K}_{2p}=8\mathbf{I}, \mathbf{K}_{2f}=5\mathbf{I}, a_2=1.5$.
Performance Metrics: The following key metrics were used for comparison:
- Settling Time ($T_{s,5\%}$): Time for the tracking error norm to enter and remain within 5% of its peak value.
- Maximum Tracking Error ($e_{max}$): The peak absolute value of each Euler angle tracking error.
- Root-Mean-Square Error (RMSE): $\text{RMSE} = \sqrt{\frac{1}{T_f} \int_0^{T_f} e_i^2(t) dt}$ over the simulation period $T_f=20$s.
- Control Effort (CE): Total integrated control torque: $\text{CE} = \int_0^{T_f} \|\boldsymbol{\Gamma}(t)\| dt$.
The quantitative comparison of the three controllers is presented in the table below.
| Performance Metric | Proposed Fixed-Time Control | Finite-Time Backstepping | PID Controller |
|---|---|---|---|
| $T_{s,5\%}$ (s) | 1.32 | 2.85 | 4.71 (oscillatory) |
| $e_{max}$ – Roll (rad) | 0.081 | 0.125 | 0.214 |
| $e_{max}$ – Pitch (rad) | 0.072 | 0.118 | 0.198 |
| $e_{max}$ – Yaw (rad) | 0.035 | 0.065 | 0.112 |
| RMSE – Roll (rad) | 0.012 | 0.025 | 0.048 |
| RMSE – Pitch (rad) | 0.010 | 0.021 | 0.041 |
| RMSE – Yaw (rad) | 0.005 | 0.011 | 0.023 |
| Control Effort (N·m·s) | 18.7 | 22.4 | 26.9 |
Analysis of Results: The simulation results unequivocally demonstrate the superiority of the proposed fixed-time control framework for the quadrotor drone.
- Convergence Speed and Accuracy: The proposed controller achieves the fastest settling time (1.32 s), significantly outperforming the finite-time (2.85 s) and PID (4.71 s) controllers. This validates the fixed-time convergence property. Furthermore, it maintains the smallest maximum tracking errors and the lowest RMSE across all three attitude angles, indicating superior steady-state accuracy.
- Robustness and Disturbance Rejection: The smooth, non-oscillatory response of the proposed controller in the presence of significant model uncertainty and persistent external disturbance highlights its robustness. The PID controller exhibits noticeable steady-state oscillations, while the finite-time controller shows a slower recovery from disturbances.
- Control Effort and Smoothness: Despite its aggressive performance, the proposed controller requires the least total control effort (18.7 N·m·s). This efficiency is attributed to the accurate cancellation of nonlinearities and disturbances by the adaptive RBFNN and DO, allowing the feedback gains to be lower. The control torque signals are also noticeably smoother than those of the finite-time controller, which displays high-frequency chattering due to its discontinuous fractional-power terms.
- Component Performance: The time-history plots confirm that the fixed-time command filter provides smooth, accurate estimates of the virtual control derivatives without singularity. The composite adaptive RBFNN quickly converges to accurately approximate $\mathbf{F}(\mathbf{x})$, and the disturbance observer effectively estimates and cancels $\mathbf{d}(t)$ within a short period.
Conclusion and Future Directions
This article has presented a comprehensive and robust fixed-time control framework for the attitude tracking of a quadrotor drone operating under realistic conditions of model uncertainty and external disturbance. The synthesis of a non-singular fixed-time command filter, a composite adaptive RBF neural network, and an adaptive disturbance observer within a backstepping structure has led to a controller with provable fixed-time stability guarantees. The key advantages of this framework are: 1) Predictable, Fast Convergence: The settling time upper bound is independent of initial conditions, ensuring reliable performance. 2) Strong Robustness: The adaptive NN and DO effectively learn and compensate for complex internal and external perturbations. 3) Smooth Operation: The use of hyperbolic tangent functions eliminates chattering and singularity issues common in fixed-time designs.
The simulation studies validated these theoretical claims, showing significant improvements in convergence speed, tracking accuracy, and control efficiency compared to conventional finite-time and PID controllers. This makes the proposed framework highly suitable for demanding quadrotor drone applications where precision, agility, and reliability are paramount.
Future research directions include extending this framework to trajectory tracking (position and attitude) for the full 6-DOF quadrotor drone model, investigating its performance under actuator faults and saturation constraints, and implementing the algorithm on hardware for real-world experimental validation. Furthermore, exploring event-triggered communication mechanisms to reduce the computational and communication load for multi-agent quadrotor drone systems presents another promising avenue.