We propose a gain-varying adaptive asymptotic tracking control scheme to address trajectory tracking challenges in quadrotor Unmanned Aerial Vehicle (UAV) systems subject to external disturbances and parametric uncertainties. This approach ensures asymptotic convergence of tracking errors while enhancing disturbance rejection capabilities, leveraging the flexibility of function gains over static gains. The integration of command-filtered backstepping resolves computational complexity issues inherent in traditional methods, with error compensation mechanisms eliminating filtering errors. Stability is rigorously proven via Lyapunov analysis, and simulations validate the method’s efficacy.
1. Dynamic Model of Quadrotor UAV
The quadrotor dynamics, characterized by underactuation and nonlinearity, are expressed as:
$$ \begin{aligned}
\ddot{Z} &= \frac{\tau_H}{m} (\cos\phi \cos\theta) – g – \frac{G_H}{m} \dot{Z} + d_H \\
\ddot{\phi} &= \frac{l}{J_x} \tau_\phi + \dot{\theta}\dot{\psi} \frac{J_y – J_z}{J_x} – \frac{G_\phi l}{J_x} \dot{\phi} + d_\phi \\
\ddot{\theta} &= \frac{l}{J_y} \tau_\theta + \dot{\phi}\dot{\psi} \frac{J_z – J_x}{J_y} – \frac{G_\theta l}{J_y} \dot{\theta} + d_\theta \\
\ddot{\psi} &= \frac{l}{J_z} \tau_\psi + \dot{\phi}\dot{\theta} \frac{J_x – J_y}{J_z} – \frac{G_\psi l}{J_z} \dot{\psi} + d_\psi
\end{aligned} $$
where \(Z\) denotes altitude; \(\phi, \theta, \psi\) represent roll, pitch, and yaw angles; \(l\) is motor-to-center distance; \(g\) is gravity; \(\tau_{(\cdot)}\) are control inputs; \(J_x, J_y, J_z\) are moments of inertia; \(G_{(\cdot)}\) are drag coefficients; and \(d_{(\cdot)}\) are disturbances. Defining state variables \(x_{1,1}=Z, x_{1,2}=\dot{Z}, x_{2,1}=\phi, x_{2,2}=\dot{\phi}, x_{3,1}=\theta, x_{3,2}=\dot{\theta}, x_{4,1}=\psi, x_{4,2}=\dot{\psi}\) simplifies the model to:
$$ \begin{cases}
\dot{x}_{1,1} = x_{1,2} \\
\dot{x}_{1,2} = g_1 u_1 – g + f_1(\mathbf{x}) + d_1 \\
\dot{x}_{2,1} = x_{2,2} \\
\dot{x}_{2,2} = g_2 u_2 + f_2(\mathbf{x}) + d_2 \\
\dot{x}_{3,1} = x_{3,2} \\
\dot{x}_{3,2} = g_3 u_3 + f_3(\mathbf{x}) + d_3 \\
\dot{x}_{4,1} = x_{4,2} \\
\dot{x}_{4,2} = g_4 u_4 + f_4(\mathbf{x}) + d_4
\end{cases} $$
with \(g_1 = \frac{\cos\phi \cos\theta}{m}, g_2 = \frac{l}{J_x}, g_3 = \frac{l}{J_y}, g_4 = \frac{l}{J_z}\), and unknown terms \(f_i(\mathbf{x})\). Key assumptions include bounded reference trajectories \(x_{i,d}\) and disturbances \(d_i \leq d_{i,\text{max}}\).
2. Control Design Methodology
2.1 Command Filter and Error Compensation
A command filter avoids computational explosion in backstepping:
$$ \begin{aligned}
\dot{\xi}_{i,1} &= \omega_n \xi_{i,2} \\
\dot{\xi}_{i,2} &= -2\eta\omega_n \xi_{i,2} – \omega_n (\xi_{i,1} – \alpha_i)
\end{aligned} $$
where \(\omega_n > 0\) and \(\eta > 0\). Filter outputs are \(x_{i,c} = \xi_{i,1}\), \(\dot{x}_{i,c} = \omega_n \xi_{i,2}\). Tracking errors \(z_{i,1} = x_{i,1} – x_{i,d}\), \(z_{i,2} = x_{i,2} – x_{i,c}\) and compensated errors \(v_{i,j} = z_{i,j} – \xi_{i,j}\) are constructed. To mitigate filtering errors \((x_{i,c} – \alpha_i)\), an error compensation system is designed:
$$ \begin{aligned}
\dot{\xi}_{i,1} &= -G(\text{sign}(\xi_{i,1})\xi_{i,1}) \xi_{i,1} + (x_{i,c} – \alpha_i) + \xi_{i,2} – \frac{l_{i,1} \xi_{i,1}}{\sqrt{\xi_{i,1}^2 + \delta^2}} \\
\dot{\xi}_{i,2} &= -G(\text{sign}(\xi_{i,2})\xi_{i,2}) \xi_{i,2} – \xi_{i,1} – \frac{l_{i,2} \xi_{i,2}}{\sqrt{\xi_{i,2}^2 + \delta^2}}
\end{aligned} $$
where \(l_{i,j} > 0\), and \(\delta(t)\) satisfies \(\lim_{t \to \infty} \int_0^t \delta^2(\tau)d\tau < \infty\).
2.2 Gain-varying Adaptive Control
Virtual control signals \(\alpha_i\) and actual inputs \(u_i\) employ variable gains \(G(\cdot)\) to replace static gains. For altitude control:
$$ \begin{aligned}
\alpha_1 &= -G(\text{sign}(v_{1,1})v_{1,1})v_{1,1} – \frac{1}{2} G(\text{sign}(\xi_{1,1})\xi_{1,1})v_{1,1} \\
&\quad – \frac{l_{1,1} \xi_{1,1}}{\sqrt{\xi_{1,1}^2 + \delta^2}} + \dot{x}_{1,d} \\
u_1 &= \frac{1}{g_1} \left( -G(\text{sign}(v_{1,2})v_{1,2})v_{1,2} – \frac{1}{2} G(\text{sign}(\xi_{1,2})\xi_{1,2})v_{1,2} – z_{1,1} \right. \\
&\quad \left. – \frac{l_{1,2} \xi_{1,2}}{\sqrt{\xi_{1,2}^2 + \delta^2}} – \frac{\dot{\Theta}_1 v_{1,2} \|S(\mathbf{X})\|^2}{\sqrt{v_{1,2}^2 \|S(\mathbf{X})\|^4 + \delta^2}} – \frac{\dot{\varepsilon}_1 v_{1,2}}{\sqrt{v_{1,2}^2 + \delta^2}} + g + \dot{x}_{1,c} \right)
\end{aligned} $$
Attitude controllers follow similarly. Fuzzy Logic Systems (FLS) approximate unknown nonlinearities \(f_i(\mathbf{x}) = \mathbf{W}^T \mathbf{S}(\mathbf{x}) + e_i\) with \(|e_i| \leq q\). Adaptive laws update \(\Theta_i = \max \|\mathbf{W}\|^2\) and \(\varepsilon_i = e_i + d_i\):
$$ \begin{aligned}
\dot{\Theta}_i &= c_i \frac{v_{i,2}^2 \|S(\mathbf{X})\|^2}{\sqrt{v_{i,2}^2 \|S(\mathbf{X})\|^4 + \delta^2}} – c_i \Theta_i \delta \\
\dot{\varepsilon}_i &= r_i \frac{v_{i,2}^2}{\sqrt{v_{i,2}^2 + \delta^2}} – r_i \varepsilon_i \delta
\end{aligned} $$
2.3 Function Gain Selection
Monotonic function gains \(G(\text{sign}(\rho)\rho)\) enhance disturbance rejection. Options include:
| Type | Function Gain \(G(\text{sign}(\rho)\rho)\) |
|---|---|
| I | \(p_1 (\text{sign}(\rho)\rho)^{p_2}\) |
| II | \(p_1 \tanh^{p_2}(\text{sign}(\rho)\rho)\) |
| III | \(p_1 (\lambda^{\text{sign}(\rho)\rho} – 1)^{p_2}\) |
| IV | \(p_1 \log^{p_2} \lambda (\text{sign}(\rho)\rho + 1)\) |
| V | \(p_1 \left( \frac{1}{1 + \lambda^{-\text{sign}(\rho)\rho}} – 0.5 \right)^{p_2}\) |
| VI | \(p_1 \left( \frac{1}{\Pi + \text{sign}(\rho)\rho} – \frac{1}{\Pi} \right)^{p_2}\) |
Parameters \(p_1 > 0\), \(p_2 > 0\), \(\Pi > 0\), \(\lambda > 0\) are application-dependent.
3. Stability Analysis
Consider the Lyapunov function candidate:
$$ V = \frac{1}{2} \sum_{i=1}^{4} \sum_{j=1}^{2} v_{i,j}^2 + \frac{1}{2} \sum_{i=1}^{4} \left( \frac{1}{c_i} \tilde{\Theta}_i^2 + \frac{1}{r_i} \tilde{\varepsilon}_i^2 \right) + \frac{1}{2} \sum_{i=1}^{4} \sum_{j=1}^{2} \xi_{i,j}^2 $$
where \(\tilde{\Theta}_i = \Theta_i – \hat{\Theta}_i\), \(\tilde{\varepsilon}_i = \varepsilon_i – \hat{\varepsilon}_i\). Differentiating \(V\) and substituting control laws yields:
$$ \dot{V} \leq -\sum_{i=1}^{4} \sum_{j=1}^{2} G(\text{sign}(v_{i,j})v_{i,j}) v_{i,j}^2 + \frac{1}{2} \sum_{i=1}^{4} \sum_{j=1}^{2} G(\text{sign}(\xi_{i,j})\xi_{i,j}) \xi_{i,j}^2 + \delta \sum_{i=1}^{4} \sigma_i $$
with \(\sigma_i = \Theta_i(\Theta_i + 1) + \varepsilon_i(\varepsilon_i + 1)\). Integrating over \([t_0, t)\):
$$ V(t) \leq V(t_0) – \sum_{i=1}^{4} \sum_{j=1}^{2} \int_{t_0}^{t} G(\text{sign}(v_{i,j})v_{i,j}) v_{i,j}^2 d\tau + \frac{1}{2} \sum_{i=1}^{4} \sum_{j=1}^{2} \int_{t_0}^{t} G(\text{sign}(\xi_{i,j})\xi_{i,j}) \xi_{i,j}^2 d\tau + \delta \sum_{i=1}^{4} \int_{t_0}^{t} \sigma_i d\tau $$
Boundedness of \(\xi_{i,j}\) and \(\lim_{t \to \infty} \int_{t_0}^{t} \delta(\tau) d\tau < \infty\) imply:
$$ \lim_{t \to \infty} G(\text{sign}(\xi_{i,j})\xi_{i,j}) \xi_{i,j}(t) = 0 \quad \Rightarrow \quad \lim_{t \to \infty} \xi_{i,j}(t) = 0 $$
Similarly, \(\lim_{t \to \infty} v_{i,j}(t) = 0\). Since \(z_{i,j} = v_{i,j} + \xi_{i,j}\), asymptotic convergence \(\lim_{t \to \infty} z_{i,j}(t) = 0\) is achieved.
4. Simulation Results
Simulations validate the controller using parameters:
| Parameter | Value |
|---|---|
| Mass (\(m\)) | 1.79 kg |
| Moments of Inertia (\(J_x, J_y, J_z\)) | 0.03, 0.03, 0.04 kg·m² |
| Motor Arm Length (\(l\)) | 0.2 m |
| Gravity (\(g\)) | 9.8 m/s² |
| Desired Trajectories | \(Z_d = 0.7\text{m}, \phi_d = 0.5\sin\left(\frac{2\pi}{5}t\right), \theta_d = 0.5\sin\left(\frac{2\pi}{5}t\right), \psi_d = 0.5\sin\left(\frac{2\pi}{5}t\right)\) |
| Control Gains (\(l_{i,j}\)) | \(l_{1,1}=0.1, l_{1,2}=0.12, l_{2,2}=0.15, l_{3,2}=0.2, l_{4,1}=0.22, l_{4,2}=0.9\) |
Trajectory tracking results demonstrate asymptotic convergence:
$$ \begin{aligned}
&\lim_{t \to \infty} |Z – Z_d| = 0, \quad \lim_{t \to \infty} |\phi – \phi_d| = 0 \\
&\lim_{t \to \infty} |\theta – \theta_d| = 0, \quad \lim_{t \to \infty} |\psi – \psi_d| = 0
\end{aligned} $$
Tracking errors \(z_i\) converge to zero within 5 seconds, confirming the controller’s effectiveness under disturbances and uncertainties.
5. Conclusion
This work presents a gain-varying adaptive asymptotic tracking control framework for quadrotor Unmanned Aerial Vehicle systems. By integrating command-filtered backstepping with error compensation, computational complexity is reduced while filtering errors are mitigated. Variable function gains enhance robustness against disturbances compared to static gains. Lyapunov-based stability guarantees ensure asymptotic convergence of tracking errors, validated through simulations. Future work will address experimental implementation and multi-UAV coordination, further advancing drone technology in complex environments.