In this paper, we address the problem of fixed-time attitude tracking control for a quadcopter system subjected to inertial uncertainties and external disturbances. The quadcopter, as a versatile unmanned aerial vehicle, has garnered significant attention due to its applications in surveillance, payload delivery, and environmental monitoring. However, achieving precise attitude control under dynamic uncertainties and disturbances remains a challenge. Traditional control methods, such as PID and finite-time controllers, often fail to guarantee convergence times independent of initial conditions, limiting their practicality. To overcome this, we propose a novel adaptive fixed-time fuzzy sliding mode control strategy that ensures rapid and robust attitude tracking. Our approach integrates a fixed-time disturbance observer, a non-singular terminal sliding mode surface, and a fuzzy logic system to adaptively adjust control gains, thereby suppressing chattering and enhancing performance. Through rigorous Lyapunov-based analysis and simulations, we demonstrate the effectiveness of our method in achieving fixed-time stability with minimal energy consumption.
The dynamics of a quadcopter are inherently nonlinear and coupled, making attitude control a complex task. The attitude system can be described using Euler angles and angular velocities. Let $\Omega = [\phi, \theta, \psi]^T$ represent the roll, pitch, and yaw angles, and $\omega = [p, q, r]^T$ denote the angular velocities in the body frame. The kinematic equation relates the attitude angles to the angular velocities as follows:
$$ \dot{\Omega} = J(\Omega) \omega $$
where $J(\Omega)$ is the transformation matrix. The dynamic equation, considering inertial uncertainties $\Delta I$ and external disturbances $\mathbf{d}$, is given by:
$$ (I + \Delta I) \dot{\omega} + \omega \times (I + \Delta I) \omega = \tau + \mathbf{d} $$
Here, $I = \text{diag}(I_x, I_y, I_z)$ is the nominal inertia matrix, and $\tau = [\tau_1, \tau_2, \tau_3]^T$ represents the control torques. By defining a lumped disturbance $\mathbf{D}_a$ that encompasses both uncertainties and external effects, the dynamics can be rewritten as:
$$ \dot{\mathbf{x}}_1 = \mathbf{x}_2 $$
$$ \dot{\mathbf{x}}_2 = R_a \tau + \mathbf{D}_a $$
where $\mathbf{x}_1 = \omega$, $\mathbf{x}_2 = \dot{\omega}$, and $R_a$ is a known matrix derived from the inertia properties. This formulation facilitates the design of a robust controller that compensates for unknown perturbations.
To estimate the lumped disturbance $\mathbf{D}_a$ in fixed time, we develop a fixed-time disturbance observer. The observer structure is designed to ensure that the estimation error converges to zero within a time bound independent of initial conditions. Define an auxiliary vector $\mathbf{H}_a = \lambda_a \mathbf{x}_2 + \mathbf{D}_a$, where $\lambda_a > 0$ is a design parameter. The observer dynamics are constructed as:
$$ \dot{\hat{\mathbf{x}}}_e = -\gamma_{a,1} \text{sign}^{\alpha_{a,1}}(\hat{\mathbf{x}}_e) – \gamma_{a,2} \text{sign}^{\alpha_{a,2}}(\hat{\mathbf{x}}_e) + \hat{\mathbf{H}}_a $$
$$ \hat{\mathbf{D}}_a = \hat{\mathbf{H}}_a – \lambda_a \hat{\mathbf{x}}_e $$
where $\hat{\mathbf{x}}_e$ is the estimate of the observer error, and $\gamma_{a,1}$, $\gamma_{a,2}$, $\alpha_{a,1}$, $\alpha_{a,2}$ are positive constants with $\alpha_{a,1} > 1$ and $0 < \alpha_{a,2} < 1$. The sign function is defined component-wise. Using Lyapunov analysis, we prove that the estimation error $\tilde{\mathbf{D}}_a = \mathbf{D}_a – \hat{\mathbf{D}}_a$ converges to zero in fixed time $T_f$, where $T_f$ is upper-bounded by a constant dependent only on the observer parameters.
For attitude tracking, we define the tracking error as $\mathbf{e}_a = \Omega_d – \Omega$, where $\Omega_d$ is the desired attitude trajectory. A non-singular terminal sliding surface is designed to avoid singularity issues commonly encountered in terminal sliding mode control. The sliding variable $\mathbf{s} = [s_1, s_2, s_3]^T$ is given by:
$$ \mathbf{s} = \sigma_a \mathbf{e}_a + \beta_a \eta(\mathbf{e}_a) $$
where $\sigma_a > 0$, $\beta_a > 0$ are gains, and $\eta(\mathbf{e}_a)$ is a switching function defined to prevent singularity when the error approaches zero. The function $\eta(\mathbf{e}_a)$ is specified as:
$$ \eta(e_{a,i}) = \begin{cases}
c_1 e_{a,i}^{k_1} + c_2 e_{a,i}^{k_2} & \text{if } |e_{a,i}| < \epsilon \\
\text{sign}(e_{a,i}) |e_{a,i}|^{k} & \text{if } |e_{a,i}| \geq \epsilon
\end{cases} $$
where $c_1$, $c_2$, $k_1$, $k_2$, and $\epsilon$ are design parameters chosen to ensure smooth transitions and non-singularity. The derivative of the sliding surface leads to the control law design.
The control law is derived using a double power-rate reaching law, which enhances convergence speed and reduces chattering. The reaching law is expressed as:
$$ \dot{\mathbf{s}} = -k_3 \text{sign}(\mathbf{s}) – k_4 \text{sign}^{r_1}(\mathbf{s}) – k_5 \text{sign}^{r_2}(\mathbf{s}) $$
where $k_3, k_4, k_5 > 0$, $r_1 > 1$, and $0 < r_2 < 1$. The control torque $\tau$ is computed as:
$$ \tau = R_a^{-1} \left[ J(\Omega) \left( \dot{\omega}_d – \sigma_a^{-1} \beta_a \dot{\eta}(\mathbf{e}_a) \right) + k_3 \text{sign}(\mathbf{s}) + k_4 \text{sign}^{r_1}(\mathbf{s}) + k_5 \text{sign}^{r_2}(\mathbf{s}) – \hat{\mathbf{D}}_a \right] $$
To further mitigate chattering, we incorporate a fuzzy logic system that adaptively adjusts the gain $k_4$. The fuzzy system uses the sliding variable $\mathbf{s}$ and its derivative $\dot{\mathbf{s}}$ as inputs, and $k_4$ as the output. The fuzzy rules are designed based on expert knowledge to minimize control efforts while maintaining robustness. The adaptive law for updating the fuzzy parameters is given by:
$$ \dot{\hat{\Gamma}}_i = \varrho_i \mathbf{s} \xi_i(\mathbf{v}) – \varrho_i \hat{\Gamma}_i $$
where $\hat{\Gamma}_i$ is the estimate of the optimal fuzzy parameter vector, $\varrho_i > 0$ is the adaptation rate, and $\xi_i(\mathbf{v})$ is the fuzzy basis function. This adaptive mechanism ensures that the control gains are tuned online, reducing the need for high constant gains and thus attenuating chattering.
The convergence analysis is conducted using Lyapunov stability theory. Consider the Lyapunov function $V = \frac{1}{2} \mathbf{s}^T \mathbf{s} + \frac{1}{2} \sum_{i=1}^3 \frac{1}{\varrho_i} \tilde{\Gamma}_i^T \tilde{\Gamma}_i$, where $\tilde{\Gamma}_i = \Gamma_i^* – \hat{\Gamma}_i$ is the parameter estimation error. The time derivative of $V$ yields:
$$ \dot{V} \leq -\rho V + \zeta $$
where $\rho > 0$ and $\zeta$ is a bounded term. According to fixed-time stability theory, the sliding variable $\mathbf{s}$ converges to a small neighborhood of zero within a fixed time $T_s$, and the attitude tracking error $\mathbf{e}_a$ converges to zero asymptotically thereafter. The fixed-time bound $T_s$ is computed as:
$$ T_s \leq \frac{1}{\rho (1 – \kappa)} \ln \left( \frac{\rho V(0) + \zeta}{\zeta} \right) $$
for some $0 < \kappa < 1$, demonstrating that the convergence time is independent of initial conditions.
To validate the proposed controller, we conduct simulations using a quadcopter model with parameters listed in Table 1. The quadcopter is subjected to sinusoidal disturbances and inertia variations. The desired attitude trajectory is set to $\Omega_d = [2\sin(0.5t), 2\sin(0.5t), 2\cos(0.5t)]^T$, and the initial conditions are $\Omega(0) = [0,0,2]^T$, $\omega(0) = [0,0,0]^T$. The controller parameters are chosen as in Table 2 to ensure fair comparison with existing methods.
| Parameter | Value | Description |
|---|---|---|
| $I_x$ | 0.039 kg·m² | Roll inertia |
| $I_y$ | 0.039 kg·m² | Pitch inertia |
| $I_z$ | 0.078 kg·m² | Yaw inertia |
| $l$ | 0.2 m | Arm length |
| $b_t$ | 1.5e-5 N·s² | Thrust coefficient |
| $b_m$ | 2.5e-7 N·m·s² | Moment coefficient |
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $\sigma_a$ | 4.2 | $\beta_a$ | 4.2 |
| $k_1$ | 2 | $k_2$ | 0.23 |
| $r_1$ | 2 | $r_2$ | 0.2 |
| $k_3$ | 5 | $k_4$ | Adaptive |
| $\lambda_a$ | 4 | $\gamma_{a,1}$ | 1 |
| $\gamma_{a,2}$ | 2 | $\alpha_{a,1}$ | 3 |
| $\alpha_{a,2}$ | 0.4 | $\varrho_i$ | 2 |
The simulation results demonstrate that the proposed fixed-time fuzzy sliding mode controller achieves faster convergence and higher tracking accuracy compared to conventional fixed-time sliding mode control. The attitude tracking errors for roll, pitch, and yaw converge to zero within 2 seconds, whereas the benchmark method requires over 3 seconds. The control torques generated by our method exhibit significantly reduced chattering, as shown in the time responses. The disturbance observer accurately estimates the lumped disturbances within 0.5 seconds, validating its fixed-time convergence. Performance metrics, including integral squared error (ISE), integral absolute error (IAE), and integral squared control effort (ISV), are summarized in Table 3. Our method yields lower ISE and IAE values, indicating superior tracking performance, and lower ISV, reflecting reduced energy consumption.
| Method | Metric | Roll | Pitch | Yaw |
|---|---|---|---|---|
| Proposed | ISE | 0.0182 | 0.0276 | 0.0224 |
| IAE | 0.0843 | 0.0988 | 0.0870 | |
| ISV | 0.2307 | 1.0694 | 1.5178 | |
| Benchmark | ISE | 0.0253 | 0.1910 | 0.0656 |
| IAE | 0.1198 | 0.4056 | 0.2317 | |
| ISV | 1.4758 | 1.0944 | 3.0125 |
In conclusion, we have developed a fixed-time adaptive fuzzy sliding mode control scheme for quadcopter attitude tracking. The integration of a fixed-time disturbance observer, a non-singular sliding surface, and a fuzzy gain scheduler ensures robust performance against uncertainties and disturbances. The theoretical analysis guarantees fixed-time convergence of tracking errors, while simulations confirm the effectiveness and efficiency of the proposed method. Future work will extend this approach to scenarios involving actuator faults, output constraints, and payload variations, further enhancing the applicability of quadcopter systems in real-world environments.
The mathematical foundation of our approach relies on fixed-time stability theory, which ensures that the system trajectories reach equilibrium within a time bound independent of initial conditions. Consider a nonlinear system $\dot{x} = f(x)$, and a Lyapunov function $V(x)$ satisfying:
$$ \dot{V} \leq -a V^p – b V^q $$
where $a, b > 0$, $p > 1$, and $0 < q < 1$. Then, the system is fixed-time stable, and the settling time $T$ is bounded by:
$$ T \leq \frac{1}{a(p-1)} + \frac{1}{b(1-q)} $$
This principle underpins our controller design, ensuring predictable and rapid response for the quadcopter attitude dynamics. The use of fuzzy logic adapts the control gains dynamically, optimizing performance without manual tuning. Overall, this work contributes to the advancement of robust control strategies for quadcopter applications, promising improved reliability and efficiency in autonomous operations.