Quadrotor drones, characterized by their simple mechanical structure and agile maneuverability, have found extensive applications in fields such as intelligent inspection, aerial photography, and precision agriculture. However, the control of a quadrotor drone presents significant challenges due to its inherent properties of underactuation, strong nonlinearity, and coupled dynamics. These characteristics make it difficult to maintain stable and high-precision flight, especially in the presence of external disturbances and system uncertainties. The trajectory tracking problem, wherein the drone is required to follow a predefined path in three-dimensional space, is a fundamental and critical task. Achieving asymptotic tracking, where the tracking error converges to zero over time, is a desirable but non-trivial control objective. This paper delves into a novel control strategy designed to address this very challenge.
Traditional control methods for quadrotor drones often face limitations. While the backstepping technique offers a systematic design framework for nonlinear systems, it suffers from the “explosion of complexity” issue due to repeated differentiations of virtual control laws. Dynamic surface control (DSC) mitigates this by using first-order filters, but it typically ignores the filtering errors introduced, which can degrade performance. Furthermore, many adaptive and robust control schemes only guarantee uniformly ultimately bounded (UUB) stability, meaning the tracking error remains within a small bound but does not necessarily converge to zero. The use of static control gains in many controllers also limits their adaptability; a fixed gain may not provide optimal disturbance rejection across different flight regimes. This work proposes a comprehensive solution that integrates several advanced techniques to overcome these limitations for a quadrotor drone.
The core contribution lies in the synthesis of a gain-varying adaptive asymptotic tracking control scheme. The method guarantees that all signals in the closed-loop system are bounded and, more importantly, that the trajectory tracking errors converge to zero asymptotically. This is achieved by employing a command-filtered backstepping approach to circumvent complexity explosion, incorporating a novel error compensation mechanism to eliminate filter errors, utilizing fuzzy logic systems (FLS) to approximate unknown nonlinear functions and disturbances, and most distinctively, implementing a time-varying function gain in place of traditional static gains. This variable gain enhances the system’s resilience and adaptability to varying operational conditions and disturbances. The stability of the entire closed-loop system for the quadrotor drone is rigorously proven using Lyapunov theory.
1. Dynamic Model of the Quadrotor Drone
The quadrotor drone is modeled as a rigid body with six degrees of freedom (position and orientation) controlled by the thrusts generated by four independent rotors. The dynamic equations, derived from the Newton-Euler formalism, are typically separated into positional (altitude) and rotational (attitude) subsystems. Under standard assumptions (small attitude angles, symmetrical structure), the simplified model is given by:
$$
\begin{aligned}
\ddot{z} &= \frac{\cos\phi \cos\theta}{m} u_1 – g + f_z(\mathbf{x}) + d_z(t) \\
\ddot{\phi} &= \frac{l}{J_x} u_2 + f_\phi(\mathbf{x}) + d_\phi(t) \\
\ddot{\theta} &= \frac{l}{J_y} u_3 + f_\theta(\mathbf{x}) + d_\theta(t) \\
\ddot{\psi} &= \frac{l}{J_z} u_4 + f_\psi(\mathbf{x}) + d_\psi(t)
\end{aligned}
$$
where:
- $z$, $\phi$, $\theta$, $\psi$ represent the altitude, roll, pitch, and yaw angles, respectively.
- $m$ is the mass of the quadrotor drone.
- $g$ is the gravitational acceleration.
- $l$ is the distance from the center of mass to each motor.
- $J_x, J_y, J_z$ are the moments of inertia.
- $u_1$ is the total thrust control input, and $u_2, u_3, u_4$ are the rolling, pitching, and yawing moment control inputs.
- $f_z, f_\phi, f_\theta, f_\psi$ are unknown smooth nonlinear functions encompassing aerodynamic damping effects (like drag coefficients $G_{(\cdot)}$) and Coriolis/centripetal terms. For example, $f_\phi = \dot{\theta}\dot{\psi}(J_y – J_z)/J_x – (G_\phi l / J_x) \dot{\phi}$.
- $d_z, d_\phi, d_\theta, d_\psi$ are unknown but bounded external disturbances.
To facilitate the controller design using the backstepping procedure, we define the state variables for each subsystem $i$ ($i=1$ for altitude, $i=2,3,4$ for roll, pitch, yaw):
$$
\begin{aligned}
x_{i,1} &= \begin{cases} z & i=1 \\ \phi & i=2 \\ \theta & i=3 \\ \psi & i=4 \end{cases} \\
x_{i,2} &= \dot{x}_{i,1}
\end{aligned}
$$
The system can then be expressed in a strict-feedback form suitable for backstepping:
$$
\begin{aligned}
\dot{x}_{i,1} &= x_{i,2} \\
\dot{x}_{i,2} &= g_i u_i + f_i(\mathbf{x}) + d_i(t), \quad i=1,2,3,4
\end{aligned}
$$
where $g_1 = (\cos\phi \cos\theta)/m$, $g_2 = l/J_x$, $g_3 = l/J_y$, $g_4 = l/J_z$ are known control coefficients, and $f_i(\mathbf{x})$ consolidates the unknown internal dynamics.
The control objective is to force the output $x_{i,1}$ to asymptotically track a smooth, bounded desired trajectory $x_{i,d}(t)$, i.e., $\lim_{t\to\infty} (x_{i,1}(t) – x_{i,d}(t)) = 0$, in the presence of unknown $f_i(\mathbf{x})$ and disturbances $d_i(t)$.
2. Control Scheme Design for the Quadrotor Drone
The proposed control scheme is built upon the command-filtered backstepping framework but is significantly enhanced with gain-varying and asymptotic tracking capabilities. The core ideas include: (1) Using a command filter to produce filtered virtual controls and their derivatives, avoiding analytic differentiation. (2) Designing an error compensation system to counteract the effects of filtering errors. (3) Employing a Fuzzy Logic System (FLS) as a universal approximator for the unknown nonlinear functions $f_i(\mathbf{x})$. (4) Incorporating a time-varying, state-dependent function gain $G(\cdot)$ to replace static linear feedback gains.
2.1 Key Preliminaries and the Gain-Varying Function
Several lemmas and definitions form the basis of the stability proof and controller design.
Lemma 1: For any variable $\rho \in \mathbb{R}$ and any time-varying parameter $\delta(t) > 0$ satisfying $\int_0^\infty \delta(\tau)d\tau < \infty$, the following inequality holds:
$$ |\rho| \le \frac{\rho^2}{\sqrt{\rho^2 + \delta^2(t)}} + \delta(t) $$
Lemma 2 (Barbalat’s Lemma): If a function $x(t)$ is uniformly continuous and $\int_0^\infty x^2(\tau)d\tau < \infty$, then $\lim_{t\to\infty} x(t) = 0$.
Lemma 3 (Universal Approximation of FLS): For any continuous unknown function $f(\mathbf{x})$ defined on a compact set $\Omega$, there exists an FLS such that:
$$ \sup_{\mathbf{x} \in \Omega} | f(\mathbf{x}) – \mathbf{W}^T \mathbf{S}(\mathbf{x}) | \le \epsilon $$
where $\mathbf{W}$ is the ideal weight vector and $\mathbf{S}(\mathbf{x})$ is the fuzzy basis function vector.
The central innovative element is the gain-varying function $G(\sigma)$. Instead of a constant gain $k$, we use a function $G(\sigma)$ that is monotonic and satisfies $G(\sigma) \ge 0$ for $\sigma \ge 0$. This function operates on the absolute value of its argument, effectively creating a nonlinear, variable gain. Several candidate functions can be chosen, as summarized in the table below. This gain-varying mechanism allows the controller to apply more aggressive correction for larger errors and finer adjustment for smaller errors, inherently improving transient performance and disturbance rejection for the quadrotor drone.
| Type | Function Form $G(\sigma)$, $\sigma=|\rho|$ | Characteristics |
|---|---|---|
| Power | $p_1 \sigma^{p_2}$ | Simple, provides polynomial scaling. |
| Hyperbolic Tangent | $p_1 [\tanh(p_2 \sigma)]$ | Smooth, bounded gain. |
| Exponential | $p_1 (\lambda^{p_2 \sigma} – 1)$ | Provides rapid gain increase. |
| Logarithmic | $p_1 \log^{p_2}(\lambda \sigma + 1)$ | Provides strong initial gain that saturates slowly. |
| Fractional Power | $p_1 \left( (1+\lambda)^{-\sigma} – 0.5 \right)^{p_2}$ | Complex, adaptable shape. |
| Rational | $p_1 \left( \frac{1}{\Pi + \sigma} – \frac{1}{\Pi} \right)^{p_2}$ | Gain decreases from a maximum as error increases. |
where $p_1>0, p_2>0, \lambda>0, \Pi>0$ are design parameters. The choice depends on the desired performance for the specific quadrotor drone application.
2.2 Command Filter and Error Compensation
To avoid the differentiation of virtual controls, a command filter is employed. For each step $j$ in subsystem $i$, with virtual control input $\alpha_{i,j}$, the filter is:
$$
\begin{aligned}
\dot{\xi}_{i,j,1} &= \omega_n \xi_{i,j,2} \\
\dot{\xi}_{i,j,2} &= -2\zeta\omega_n \xi_{i,j,2} – \omega_n (\xi_{i,j,1} – \alpha_{i,j})
\end{aligned}
$$
The outputs are $x_{i,c} = \xi_{i,1,1}$ and $\dot{x}_{i,c} = \omega_n \xi_{i,1,2}$, which approximate $\alpha_{i,1}$ and $\dot{\alpha}_{i,1}$, respectively. $\omega_n$ and $\zeta$ are positive filter parameters.
The filtering error $(x_{i,c} – \alpha_{i,1})$ can degrade performance. To eliminate its influence, an error compensation system is constructed:
$$
\begin{aligned}
\dot{\xi}_{i,1} &= -G(|\xi_{i,1}|)\xi_{i,1} + (x_{i,c} – \alpha_{i,1}) + \xi_{i,2} – \frac{l_{i,1}\xi_{i,1}}{\sqrt{\xi_{i,1}^2 + \delta^2(t)}} \\
\dot{\xi}_{i,2} &= -G(|\xi_{i,2}|)\xi_{i,2} – \xi_{i,1} – \frac{l_{i,2}\xi_{i,2}}{\sqrt{\xi_{i,2}^2 + \delta^2(t)}}
\end{aligned}
$$
where $\xi_{i,1}, \xi_{i,2}$ are compensation signals, $l_{i,1}, l_{i,2}>0$ are constants, and $\delta(t)$ is a time-varying parameter satisfying Lemma 1.
2.3 Controller Design via Adaptive Command-Filtered Backstepping
The design proceeds for each subsystem of the quadrotor drone. Define the tracking error and the compensated tracking error:
$$
\begin{aligned}
z_{i,1} &= x_{i,1} – x_{i,d} \\
z_{i,2} &= x_{i,2} – x_{i,c} \\
v_{i,1} &= z_{i,1} – \xi_{i,1} \\
v_{i,2} &= z_{i,2} – \xi_{i,2}
\end{aligned}
$$
The goal is to stabilize $v_{i,1}$ and $v_{i,2}$.
Step 1: Consider the Lyapunov function $V_{i,1} = \frac{1}{2} v_{i,1}^2$. Its derivative is:
$$ \dot{V}_{i,1} = v_{i,1} \dot{v}_{i,1} = v_{i,1}(z_{i,2} + (x_{i,c} – \alpha_{i,1}) + \alpha_{i,1} – \dot{x}_{i,d} – \dot{\xi}_{i,1}) $$
Choose the virtual control law $\alpha_{i,1}$ as:
$$ \alpha_{i,1} = -G(|v_{i,1}|)v_{i,1} + \dot{x}_{i,d} – \frac{1}{2}G(|\xi_{i,1}|)v_{i,1} – \frac{l_{i,1}\xi_{i,1}}{\sqrt{\xi_{i,1}^2 + \delta^2(t)}} $$
Substituting $\alpha_{i,1}$ and the dynamics of $\dot{\xi}_{i,1}$, and using Young’s inequality, we obtain:
$$ \dot{V}_{i,1} \le v_{i,1}v_{i,2} – G(|v_{i,1}|)v_{i,1}^2 + \frac{1}{2} G(|\xi_{i,1}|)\xi_{i,1}^2 $$
Step 2: Now consider the Lyapunov function $V_{i,2} = V_{i,1} + \frac{1}{2} v_{i,2}^2 + \frac{1}{2\Gamma_i} \tilde{\Theta}_i^2 + \frac{1}{2\Upsilon_i} \tilde{\epsilon}_i^2$, where $\tilde{\Theta}_i = \Theta_i – \hat{\Theta}_i$ and $\tilde{\epsilon}_i = \epsilon_i – \hat{\epsilon}_i$. Here, $\Theta_i$ is the upper bound of the FLS weight norm ($||\mathbf{W}_i||^2 \le \Theta_i$), and $\epsilon_i$ is the upper bound of the approximation error plus disturbance ($|e_i + d_i| \le \epsilon_i$). Their estimates are $\hat{\Theta}_i$ and $\hat{\epsilon}_i$.
The derivative is:
$$ \dot{V}_{i,2} = \dot{V}_{i,1} + v_{i,2}\left( g_i u_i + f_i(\mathbf{x}) + d_i – \dot{x}_{i,c} – \dot{\xi}_{i,2} \right) – \frac{1}{\Gamma_i}\tilde{\Theta}_i \dot{\hat{\Theta}}_i – \frac{1}{\Upsilon_i}\tilde{\epsilon}_i \dot{\hat{\epsilon}}_i $$
The unknown function $f_i(\mathbf{x})$ is approximated by the FLS: $f_i(\mathbf{x}) = \mathbf{W}_i^T \mathbf{S}_i(\mathbf{X}) + e_i(\mathbf{X})$, where $|e_i| \le \bar{e}_i$. Using Lemma 1 and 3, we have:
$$ v_{i,2} f_i(\mathbf{x}) \le |v_{i,2}| \Theta_i^{1/2} ||\mathbf{S}_i|| + |v_{i,2}| \bar{e}_i \le \frac{\Theta_i v_{i,2}^2 ||\mathbf{S}_i||^2}{\sqrt{v_{i,2}^2 ||\mathbf{S}_i||^2 + \delta^2}} + \Theta_i \delta + \frac{\epsilon_i v_{i,2}^2}{\sqrt{v_{i,2}^2 + \delta^2}} + \epsilon_i \delta $$
where $\epsilon_i \ge \bar{e}_i + |d_i|$.
Choose the actual control input $u_i$ and the adaptation laws:
$$
\begin{aligned}
u_i &= \frac{1}{g_i} \Bigg[ -G(|v_{i,2}|)v_{i,2} – v_{i,1} – \frac{1}{2}G(|\xi_{i,2}|)v_{i,2} – \frac{l_{i,2}\xi_{i,2}}{\sqrt{\xi_{i,2}^2 + \delta^2}} \\
&\quad – \frac{\hat{\Theta}_i v_{i,2} ||\mathbf{S}_i||^2}{\sqrt{v_{i,2}^2 ||\mathbf{S}_i||^2 + \delta^2}} – \frac{\hat{\epsilon}_i v_{i,2}}{\sqrt{v_{i,2}^2 + \delta^2}} + \dot{x}_{i,c} \Bigg] \\
\dot{\hat{\Theta}}_i &= \Gamma_i \left( \frac{v_{i,2}^2 ||\mathbf{S}_i||^2}{\sqrt{v_{i,2}^2 ||\mathbf{S}_i||^2 + \delta^2}} – \sigma_{\Theta_i} \hat{\Theta}_i \right) \\
\dot{\hat{\epsilon}}_i &= \Upsilon_i \left( \frac{v_{i,2}^2}{\sqrt{v_{i,2}^2 + \delta^2}} – \sigma_{\epsilon_i} \hat{\epsilon}_i \right)
\end{aligned}
$$
where $\sigma_{\Theta_i}, \sigma_{\epsilon_i} > 0$ are small leakage modification parameters to ensure boundedness of estimates.
Substituting all terms into $\dot{V}_{i,2}$ and after extensive manipulation using inequalities, we arrive at the final bound for each subsystem of the quadrotor drone:
$$ \dot{V}_{i,2} \le -\sum_{j=1}^{2} G(|v_{i,j}|) v_{i,j}^2 + \frac{1}{2} \sum_{j=1}^{2} G(|\xi_{i,j}|) \xi_{i,j}^2 + \mu_i \delta(t) $$
where $\mu_i$ is a positive constant aggregating the bounds related to $\Theta_i$ and $\epsilon_i$.
3. Stability Analysis for the Quadrotor Drone System
The stability of the overall closed-loop quadrotor drone system is established by considering the composite Lyapunov function $V = \sum_{i=1}^{4} V_{i,2} + V_\xi$, where $V_\xi = \frac{1}{2} \sum_{i=1}^{4} \sum_{j=1}^{2} \xi_{i,j}^2$ accounts for the compensation errors.
Theorem (Asymptotic Tracking): Consider the quadrotor drone system governed by the dynamics in Section 1, subject to unknown nonlinearities and bounded disturbances. Under the proposed control law $u_i$, the virtual control $\alpha_{i,1}$, the command filter, the error compensation system, and the adaptation laws, the following results hold for all $i=1,2,3,4$:
- All signals in the closed-loop system, including states $x_{i,j}$, control inputs $u_i$, and parameter estimates $\hat{\Theta}_i$, $\hat{\epsilon}_i$, are uniformly ultimately bounded.
- The tracking errors $z_{i,1}(t) = x_{i,1}(t) – x_{i,d}(t)$ converge to zero asymptotically, i.e., $\lim_{t \to \infty} z_{i,1}(t) = 0$.
Proof Sketch:
- Boundedness of $\xi_{i,j}$: Analyze $V_\xi$. Using its derivative and the properties of the gain-varying function $G(\cdot)$ and Lemma 1, it can be shown that $\dot{V}_\xi \le -\sum_{i,j} G(|\xi_{i,j}|)\xi_{i,j}^2 + \bar{l} \delta(t)$, where $\bar{l}$ is a positive constant. Integrating this and using the property of $\delta(t)$ from Lemma 1, we prove $\int_0^\infty G(|\xi_{i,j}(\tau)|) \xi_{i,j}^2(\tau) d\tau < \infty$. Since $G(|\xi_{i,j}|)\xi_{i,j}^2$ is uniformly continuous (as its derivative is bounded), by Lemma 2 (Barbalat’s), we conclude $\lim_{t\to\infty} G(|\xi_{i,j}(t)|)\xi_{i,j}(t) = 0$. Given that $G(\sigma)$ is monotonic and $G(\sigma)=0$ only when $\sigma=0$, it follows that $\lim_{t\to\infty} \xi_{i,j}(t) = 0$.
- Boundedness of $v_{i,j}$ and Asymptotic Convergence of $z_{i,j}$: From the inequality for $\dot{V}_{i,2}$ and summing over all $i$, we get a bound for the derivative of the total Lyapunov function $V$:
$$ \dot{V} \le -\sum_{i=1}^{4}\sum_{j=1}^{2} G(|v_{i,j}|) v_{i,j}^2 + \frac{1}{2}\sum_{i=1}^{4}\sum_{j=1}^{2} G(|\xi_{i,j}|) \xi_{i,j}^2 + \mu \delta(t) $$
Integrating this inequality from $0$ to $t$, and using the facts that $V$ is bounded from below and that the terms involving $\xi_{i,j}^2$ and $\delta(t)$ have finite integrals, we obtain $\int_0^\infty G(|v_{i,j}(\tau)|) v_{i,j}^2(\tau) d\tau < \infty$. Again, by the uniform continuity of $G(|v_{i,j}|)v_{i,j}^2$ and Lemma 2, we have $\lim_{t\to\infty} G(|v_{i,j}(t)|)v_{i,j}(t) = 0$, which implies $\lim_{t\to\infty} v_{i,j}(t) = 0$. - Since $z_{i,j} = v_{i,j} + \xi_{i,j}$ and both $v_{i,j}$ and $\xi_{i,j}$ converge to zero, we finally have $\lim_{t\to\infty} z_{i,j}(t) = 0$. Specifically for $j=1$, this proves the asymptotic tracking of the desired trajectory by the quadrotor drone: $\lim_{t\to\infty} (x_{i,1}(t) – x_{i,d}(t)) = 0$.
4. Simulation Validation
The effectiveness of the proposed gain-varying adaptive asymptotic tracking control scheme is verified through numerical simulations in the Matlab/Simulink environment. The physical parameters of the simulated quadrotor drone are listed below:
| Parameter | Symbol | Value |
|---|---|---|
| Mass | $m$ | 1.79 kg |
| Moment of Inertia (Roll/Pitch) | $J_x, J_y$ | 0.03 kg·m² |
| Moment of Inertia (Yaw) | $J_z$ | 0.04 kg·m² |
| Arm Length | $l$ | 0.2 m |
| Gravity | $g$ | 9.81 m/s² |
| Air Drag Coefficients | $G_z, G_\phi, G_\theta, G_\psi$ | 0.1, 0.12, 0.15, 0.18 N·s/m |
| Command Filter ($\omega_n, \zeta$) | 50 rad/s, 0.9 | |
| Gain-Varying Function (Type I) | $G(\sigma)$ | $2 \sigma^{0.5}$ |
| Adaptation Gains | $\Gamma_i, \Upsilon_i$ | 1, 0.5 |
| Leakage Coefficients | $\sigma_{\Theta_i}, \sigma_{\epsilon_i}$ | 0.01 |
The desired trajectories for the quadrotor drone are chosen as:
$$
\begin{aligned}
z_d(t) &= 0.7 \, \text{m} \quad \text{(constant altitude)} \\
\phi_d(t) &= 0.5 \sin\left(\frac{2\pi}{5}t\right) \, \text{rad} \\
\theta_d(t) &= 0.5 \sin\left(\frac{2\pi}{5}t + \frac{\pi}{3}\right) \, \text{rad} \\
\psi_d(t) &= 0.5 \sin\left(\frac{2\pi}{5}t + \frac{2\pi}{3}\right) \, \text{rad}
\end{aligned}
$$
External disturbances are added as $d_i(t) = 0.1 \sin(0.5\pi t) + 0.05 \cos(2t)$ for all channels.
The simulation results demonstrate the superior performance of the proposed controller. The altitude and attitude angles of the quadrotor drone accurately track their respective desired trajectories after a brief transient period. The key observation is that the tracking errors $z_{i,1}(t)$ do not merely remain within a bounded region but visibly decay and converge close to zero, validating the asymptotic tracking property. The control inputs $u_1$ to $u_4$ remain smooth and within realistic bounds throughout the maneuver. The adaptation parameters $\hat{\Theta}_i$ and $\hat{\epsilon}_i$ converge to steady-state values, indicating successful learning of the system’s unknown dynamics and disturbance bounds. The incorporation of the gain-varying function $G(\cdot)$ is evident in the controller’s responsive behavior, applying stronger corrective action when errors are large and finer adjustments as they diminish, leading to improved transient performance and robustness for the quadrotor drone against the applied disturbances.
5. Conclusion
This paper has presented a comprehensive gain-varying adaptive asymptotic tracking control solution for the trajectory tracking problem of a quadrotor drone subject to model uncertainties and external disturbances. The proposed scheme successfully integrates several advanced control techniques to achieve performance goals that are often difficult to realize simultaneously. By employing the command-filtered backstepping framework, the inherent “explosion of complexity” issue is resolved. The dedicated error compensation mechanism ensures that filtering errors do not degrade the final tracking precision. The use of fuzzy logic systems provides a robust means to adaptively estimate and cancel unknown nonlinear dynamics. The hallmark of the design is the introduction of a time-varying, state-dependent function gain $G(\cdot)$, which replaces conventional static gains, granting the controller enhanced flexibility and disturbance rejection capability.
The most significant theoretical contribution is the rigorous proof, via Lyapunov analysis, that all closed-loop signals are bounded and that the trajectory tracking errors converge to zero asymptotically. This is a stronger result than the typical uniformly ultimately bounded (UUB) stability guaranteed by many nonlinear adaptive controllers. Numerical simulations confirm the theoretical findings, showing precise and asymptotically convergent tracking of desired altitude and attitude trajectories for the quadrotor drone under simulated disturbances. The control strategy is general and can be tailored to different quadrotor drone platforms or other similar nonlinear systems by selecting appropriate gain-varying functions and tuning parameters. Future work may focus on extending this method to formation control of multiple quadrotor drones or incorporating state constraints and actuator saturation limits into the design.