The quadrotor drone has become a cornerstone of modern aerial robotics, prized for its mechanical simplicity, vertical take-off and landing (VTOL) capability, and high agility. Its applications span critical fields such as aerial photography, precision agriculture, search and rescue, and payload delivery. The core of its flight autonomy lies in the precise control of its attitude—the orientation defined by roll (φ), pitch (θ), and yaw (ψ) angles. While extensive research exists for controlling quadrotor drones with constant parameters, real-world missions like spraying, firefighting, or transporting slung loads introduce a significant complication: the system’s inertial parameters become time-varying. Furthermore, actuator failures due to motor wear or propeller damage are inevitable in prolonged operations. This work addresses the complex yet practical problem of achieving high-precision, reliable attitude control for a quadrotor drone subject to simultaneous time-varying inertia, actuator faults, and external disturbances.
The primary objective is to design a fault-tolerant control law that guarantees not only asymptotic tracking of desired attitude trajectories but also enforces pre-defined transient and steady-state performance constraints on the tracking error. This means the error’s convergence rate, overshoot, and final precision can be prescribed by the designer beforehand. To this end, we integrate the Prescribed Performance Control (PPC) technique with a novel application of the “freezing variable” method to handle the time-varying dynamics, ensuring all closed-loop signals remain bounded. The proposed control scheme is validated through comprehensive numerical simulations, demonstrating its superiority over conventional robust methods.
1. Problem Formulation and Dynamic Modeling
We consider a quadrotor drone whose mass distribution changes during flight, leading to time-varying moments of inertia. This is typical when the drone is carrying a moving payload, dispensing a liquid, or has articulated arms. The attitude dynamics are derived from the angular momentum theorem in the body-fixed frame B, leading to the following nonlinear model:
$$
\begin{aligned}
\dot{\phi} &= p + \sin\phi \tan\theta \, q + \cos\phi \tan\theta \, r \\
\dot{\theta} &= \cos\phi \, q – \sin\phi \, r \\
\dot{\psi} &= \frac{\sin\phi}{\cos\theta} q + \frac{\cos\phi}{\cos\theta} r
\end{aligned}
$$
For small attitude angles (φ, θ ≈ 0), the kinematics simplify, and the rotational dynamics considering time-varying inertia I(t), aerodynamic damping, actuator faults, and disturbances are:
$$
\begin{aligned}
\ddot{\phi} &= -\frac{k_{f1} + \dot{I}_x(t)}{I_x(t)} \dot{\phi} – \frac{I_z(t) – I_y(t)}{I_x(t)} \dot{\theta}\dot{\psi} + \frac{\kappa_1 u^c_1}{I_x(t)} + d_1′(t) \\
\ddot{\theta} &= -\frac{k_{f2} + \dot{I}_y(t)}{I_y(t)} \dot{\theta} – \frac{I_x(t) – I_z(t)}{I_y(t)} \dot{\phi}\dot{\psi} + \frac{\kappa_2 u^c_2}{I_y(t)} + d_2′(t) \\
\ddot{\psi} &= -\frac{k_{f3} + \dot{I}_z(t)}{I_z(t)} \dot{\psi} – \frac{I_y(t) – I_x(t)}{I_z(t)} \dot{\phi}\dot{\theta} + \frac{\kappa_3 u^c_3}{I_z(t)} + d_3′(t)
\end{aligned}
$$
Here, I(t) = diag([I_x(t), I_y(t), I_z(t)]) is the time-varying inertia matrix. The control inputs u^c_1, u^c_2, u^c_3 are the commanded roll, pitch, and yaw moments. The actuator fault is modeled by the effectiveness factors κ_i(t) ∈ (0, 1] and additive fault terms Δu_i(t). The lumped disturbances d’_i(t) encompass external disturbances (e.g., wind gusts) and the effects of the additive actuator faults.
To facilitate controller design, we transform the model into a state-space form. Define the state vectors for each channel (i = 1,2,3 for φ, θ, ψ respectively): x_{1,i} = angle, x_{2,i} = angular rate. The coupled nonlinear terms are denoted as η_1 = \dot{\theta}\dot{\psi}, η_2 = \dot{\phi}\dot{\psi}, η_3 = \dot{\phi}\dot{\theta}. The dynamics can then be written in a compact, control-oriented form:
$$
\boxed{
\begin{aligned}
\dot{x}_{1,i} &= x_{2,i} \\
\dot{x}_{2,i} &= a_{1,i}(t)x_{2,i} + a_{2,i}(t)\eta_i + b_i(t)u^c_i + d_i(t)
\end{aligned}}
$$
The time-varying parameters a_{1,i}(t), a_{2,i}(t), b_i(t) and the lumped disturbance d_i(t) are defined as follows:
| Parameter | Physical Meaning | Expression |
|---|---|---|
| a_{1,i}(t) | Effective damping coefficient | $- (k_{fi} + \dot{I}_i(t)) / I_i(t)$ |
| a_{2,i}(t) | Coupling coefficient | $- (I_j(t) – I_k(t)) / I_i(t)$ |
| b_i(t) | Control effectiveness | $\kappa_i(t) / I_i(t) > 0$ |
| d_i(t) | Lumped uncertainty | $\Delta u_i/I_i(t) + \text{ext. disturbance}$ |
Assumptions:
1. The desired trajectories x_{1,i,d}(t) and their first two derivatives are smooth, known, and bounded.
2. The time-varying parameters a_{·,i}(t), b_i(t) belong to an unknown compact set Θ.
3. The lumped disturbance d_i(t) and its rate are bounded, i.e., |\dot{d}_i(t)| ≤ D_i, with unknown bound D_i.
The control challenge is to design inputs u^c_i for the aforementioned quadrotor drone model such that:
1. Asymptotic Tracking: $\lim_{t\to\infty} (x_{1,i}(t) – x_{1,i,d}(t)) = 0$.
2. Prescribed Performance: The tracking error e_i(t) = x_{1,i}(t) – x_{1,i,d}(t) strictly evolves within a predefined performance envelope:
$$-\underline{m}_i \mu_i(t) < e_i(t) < \bar{m}_i \mu_i(t), \quad \forall t \geq 0$$
where $\mu_i(t) = (\mu_{i0} – \mu_{i\infty})e^{-h_i t} + \mu_{i\infty}$ is a performance function.
3. Closed-loop Stability: All signals in the system remain uniformly ultimately bounded (UUB).
2. Control Design Methodology
2.1 Prescribed Performance and Error Transformation
The core idea of PPC is to transform the constrained tracking error problem into an unconstrained stabilization problem. The performance function $\mu_i(t)$ and constants $\underline{m}_i, \bar{m}_i$ define the allowable error boundary, governing maximum overshoot ($\bar{m}_i \mu_i(0)$), minimum convergence rate ($h_i$), and maximum steady-state error ($\bar{m}_i \mu_{i\infty}$).
We define the normalized error: $s_i(t) = e_i(t) / \mu_i(t)$. The performance constraint is equivalent to keeping $s_i(t)$ within $(-\underline{m}_i, \bar{m}_i)$. To achieve this, we employ a smooth, strictly increasing transformation function:
$$
\varepsilon_i = T(s_i) = \frac{1}{2} \ln\left( \frac{\bar{m}_i (s_i + \underline{m}_i)}{\underline{m}_i (\bar{m}_i – s_i)} \right)
$$
Its derivative is $\dot{\varepsilon}_i = r_i (\dot{e}_i – e_i \dot{\mu}_i / \mu_i)$, where $r_i = \partial T / \partial e_i = \frac{1}{2\mu_i}\left(\frac{1}{s_i+\underline{m}_i} – \frac{1}{s_i-\bar{m}_i}\right) > 0$. Crucially, if $\varepsilon_i$ is bounded, then $s_i(t)$ remains within $(-\underline{m}_i, \bar{m}_i)$, satisfying the performance constraint. Furthermore, asymptotic tracking ($e_i \to 0$) is achieved if $\varepsilon_i \to 0$. Thus, our goal reduces to stabilizing the transformed error $\varepsilon_i$ to zero.
2.2 Controller Structure and the Freezing Variable Method
We define a combined error variable for each channel to facilitate a second-order design:
$$ S_i = \dot{\varepsilon}_i + k_i \varepsilon_i $$
where $k_i > 0$ is a design constant. Differentiating $S_i$ and substituting the dynamics yields:
$$ \dot{S}_i = \dot{r}_i(\dot{e}_i – \frac{e_i \dot{\mu}_i}{\mu_i}) + r_i\left( \ddot{e}_i – \frac{2\dot{e}_i\dot{\mu}_i + e_i\ddot{\mu}_i}{\mu_i} + \frac{e_i \dot{\mu}_i^2}{\mu_i^2} \right) + k_i r_i (\dot{e}_i – \frac{e_i \dot{\mu}_i}{\mu_i}) $$
Substituting $\ddot{e}_i = \ddot{x}_{1,i} – \ddot{x}_{1,i,d}$ from the quadrotor drone dynamics leads to an expression containing the time-varying parameters $a_{1,i}(t), a_{2,i}(t), b_i(t)$:
$$ \dot{S}_i = \ldots + r_i \left( a_{1,i}(t)x_{2,i} + a_{2,i}(t)\eta_i + b_i(t)u^c_i + d_i(t) – \ddot{x}_{1,i,d} + \ldots \right) $$
Directly designing a controller with these purely time-varying terms is complex. We employ the freezing variable method. The key idea is to treat the time-varying parameters at each instant as constant but unknown values, denoted as $\bar{a}_{1,i}, \bar{a}_{2,i}, \bar{b}_i$. The deviations are considered part of a new lumped uncertainty:
$$
\begin{aligned}
a_{1,i}(t) &= \bar{a}_{1,i} + \Delta a_{1,i}(t), \\
a_{2,i}(t) &= \bar{a}_{2,i} + \Delta a_{2,i}(t), \\
b_i(t) &= \bar{b}_{i} + \Delta b_i(t), \quad \text{with } \bar{b}_i > 0.
\end{aligned}
$$
The nominal constants $\bar{a}_{·,i}, \bar{b}_i$ are unknown, and the variations $\Delta a_{·,i}(t), \Delta b_i(t)$ are bounded due to Assumption 2.
2.3 Adaptive Fault-Tolerant Controller Design
We propose the following adaptive control structure for the quadrotor drone:
$$ u^c_i = \hat{\rho}_i \, \bar{u}^c_i $$
where $\hat{\rho}_i$ is the estimate of $1/\bar{b}_i$ and $\bar{u}^c_i$ is an auxiliary control signal. The auxiliary control is designed using a smooth robust inverse approach:
$$ \bar{u}^c_i = -\frac{S_i r_i \alpha_i^2}{S_i^2 r_i^2 \alpha_i^2 + \sigma_{u,i}^2(t)} $$
$$ \alpha_i = c_i r_i^{-1} S_i + x_{2,i}\hat{a}_{1,i} + \eta_i \hat{a}_{2,i} + \Phi_i \hat{\vartheta}_i – \ddot{x}_{1,i,d} + f_{\text{ppc}}(e_i, \dot{e}_i, \mu_i) $$
Here, $c_i > 0$ is a control gain. The term $f_{\text{ppc}}(\cdot)$ consolidates all terms stemming from the PPC transformation. The regressor $\Phi_i$ and the estimate $\hat{\vartheta}_i$ are designed to handle the lumped uncertainties (including $\Delta a_{·,i}, d_i$, etc.):
$$ \Phi_i = \frac{S_i r_i x_{2,i}}{S_i^2 r_i^2 x_{2,i}^2 + \sigma_{a1,i}^2} + \frac{S_i r_i \eta_i}{S_i^2 r_i^2 \eta_i^2 + \sigma_{a2,i}^2} + \frac{S_i r_i}{S_i^2 r_i^2 + \sigma_{d,i}^2} $$
$$ \hat{\vartheta}_i \text{ estimates } \vartheta_i \geq \max\{\delta, D_i\} $$
where $\delta$ is related to the size of set Θ, and $\sigma_{*,i}(t)$ are positive, integrable time-varying functions satisfying $\int_0^t \sigma_{*,i}(\tau)d\tau \leq \bar{\sigma}_* < \infty$, which play a key role in achieving asymptotic tracking.
The adaptive laws are designed using gradient-based update rules:
$$
\begin{aligned}
\dot{\hat{a}}_{1,i} &= \gamma_{a1,i} S_i r_i x_{2,i} \\
\dot{\hat{a}}_{2,i} &= \gamma_{a2,i} S_i r_i \eta_i \\
\dot{\hat{\rho}}_i &= -\gamma_{\rho,i} S_i r_i \bar{u}^c_i \\
\dot{\hat{\vartheta}}_i &= \gamma_{\vartheta,i} S_i r_i \Phi_i
\end{aligned}
$$
with $\gamma_{·,i} > 0$ being adaptation gains. Note that $\dot{\hat{\rho}}_i \geq 0$, ensuring $\hat{\rho}_i(t) \geq 0$ if initialized positively.
3. Stability and Performance Analysis
Consider the Lyapunov function candidate for the overall closed-loop quadrotor drone system:
$$ V = \sum_{i=1}^3 \left( \frac{1}{2}S_i^2 + \frac{1}{2\gamma_{a1,i}} \tilde{a}_{1,i}^2 + \frac{1}{2\gamma_{a2,i}} \tilde{a}_{2,i}^2 + \frac{\bar{b}_i}{2\gamma_{\rho,i}} \tilde{\rho}_i^2 + \frac{1}{2\gamma_{\vartheta,i}} \tilde{\vartheta}_i^2 \right) $$
where $\tilde{a}_{1,i} = \bar{a}_{1,i} – \hat{a}_{1,i}$, $\tilde{\rho}_i = 1/\bar{b}_i – \hat{\rho}_i$, etc., are parameter estimation errors.
Taking the derivative and substituting the control and adaptation laws, we leverage the inequality $0 \leq |z| – \frac{z^2}{\sqrt{z^2+\sigma^2}} \leq \sigma$ to bound the effects of uncertainties. After detailed manipulation, the derivative can be bounded as:
$$ \dot{V} \leq \sum_{i=1}^3 \left( -c_i S_i^2 + \sigma_{u,i}(t) + \vartheta_i [\sigma_{a1,i}(t) + \sigma_{a2,i}(t) + \sigma_{d,i}(t)] \right) $$
Integrating both sides from 0 to t yields:
$$ V(t) \leq V(0) – \sum_{i=1}^3 \int_0^t c_i S_i^2(\tau) d\tau + \sum_{i=1}^3 \left( \bar{\sigma}_{u,i} + \vartheta_i (\bar{\sigma}_{a1,i}+\bar{\sigma}_{a2,i}+\bar{\sigma}_{d,i}) \right) $$
Since the $\sigma$-terms are bounded by their finite integrals $\bar{\sigma}_*$, we conclude that $V(t)$ is bounded. This implies all signals—$S_i$, $\varepsilon_i$, $\tilde{a}_{·,i}$, $\tilde{\rho}_i$, $\tilde{\vartheta}_i$, and consequently $\hat{a}_{·,i}$, $\hat{\rho}_i$, $\hat{\vartheta}_i$, $e_i$, $\dot{e}_i$—are Uniformly Ultimately Bounded (UUB).
From the inequality, we also have $S_i \in \mathcal{L}_2$. Given that all closed-loop signals are bounded, Barbalat’s lemma allows us to conclude that:
$$ \lim_{t \to \infty} S_i(t) = 0 $$
Since $S_i = \dot{\varepsilon}_i + k_i \varepsilon_i$, this implies $\lim_{t \to \infty} \varepsilon_i(t) = 0$. By the properties of the transformation function $T(s_i)$, $\varepsilon_i \to 0$ implies $s_i \to 0$, which finally proves the asymptotic tracking result:
$$ \boxed{\lim_{t \to \infty} e_i(t) = \lim_{t \to \infty} \left( x_{1,i}(t) – x_{1,i,d}(t) \right) = 0} $$
Furthermore, since $\varepsilon_i$ was proven bounded, the prescribed performance constraint $-\underline{m}_i \mu_i(t) < e_i(t) < \bar{m}_i \mu_i(t)$ is guaranteed for all time $t \geq 0$.
4. Simulation Studies and Comparative Analysis
To validate the proposed control scheme for the quadrotor drone, we conduct numerical simulations with realistic parameters and challenging conditions. The time-varying inertias are set as $I_x(t)=0.5+0.4\sin(2t)$, $I_y(t)=0.5+0.4\sin(2t)$, $I_z(t)=0.7+0.5\sin(2t)$ kg·m². Actuator partial loss of effectiveness (LOE) faults are introduced: $\kappa_1=0.6$, $\kappa_2=0.7$, $\kappa_3=0.8$, along with additive fault signals $\Delta u_i$. External disturbances are modeled as bounded signals with constant offsets. The desired trajectories are aggressive: $\phi_d = 0.5\cos(0.2t)$, $\theta_d = 0.5\sin(0.2t)$, and $\psi_d = 0.5$ rad. The performance bounds are set with $\mu_{i0}=2, \mu_{i\infty}=0.05, h_i=1, \underline{m}_i=0.5, \bar{m}_i=0.5$.
| Parameter Type | Symbol | Value |
|---|---|---|
| Drone Model | $k_{f1}, k_{f2}, k_{f3}$ | 0.3 N·m·s/rad |
| Initial Attitude | $(0.3, -0.3, 0)$ rad | |
| Disturbance $d_i(t)$ | $0.1e^{-0.3t} + 0.1$ (varies per channel) | |
| PPC Parameters | $(\mu_0, \mu_\infty, h)$ | $(2.0, 0.05, 1.0)$ |
| $(\underline{m}, \bar{m})$ | $(0.5, 0.5)$ | |
| Initial Error $e_i(0)$ | ~$(-0.2, -0.8, -0.5)$ rad | |
| Controller Gains | $k_i$ | [20, 15, 15] |
| $c_i$ | [20, 20, 15] | |
| $\gamma_{a1,i}, \gamma_{a2,i}$ | 0.1 | |
| $\gamma_{\rho,i}, \gamma_{\vartheta,i}$ | [0.1, 0.05] |
The proposed controller is compared against a robust adaptive PPC-SMC (Sliding Mode Control) method from the literature. The key comparative results are summarized below:
| Performance Metric | Proposed PPC+FVM | PPC-SMC [6] | Remarks |
|---|---|---|---|
| Steady-State Error | Asymptotically converges to zero | Converges to a small bounded region | Proposed method achieves perfect asymptotic tracking. |
| Transient Overshoot | Strictly within prescribed bounds (see Fig. 2a) | Within bounds, but larger initial peak | Both respect constraints, but proposed method has smoother onset. |
| Convergence Speed (0-2s) | Faster settling observed | Slower initial convergence | The combined error variable $S_i$ accelerates response. |
| Control Effort ($u_i$) | Smooth, moderate magnitude | Higher initial peaks, more chattering | Smooth inverse design in $\bar{u}^c_i$ reduces aggressiveness. |
| Parameter Adaptation | Stable, bounded estimates ($\hat{a}_{1,i}$, $\hat{\rho}_i$, etc.) | Estimates may drift or be more aggressive | Gradient updates with $\sigma$-modification effect ensure boundedness. |
The simulation plots clearly demonstrate the effectiveness of the proposed method for the quadrotor drone. The tracking errors (Fig. 2a) start within the performance funnel and converge asymptotically to zero, never violating the prescribed bounds. The actual attitude angles (φ, θ, ψ) closely follow their demanding reference trajectories (Fig. 2b-d). A zoomed-in view of the first two seconds confirms the superior transient performance of our method—smaller overshoot and quicker convergence—compared to the PPC-SMC baseline. The control inputs (Fig. 2e) are smooth and practically feasible, without excessive chattering. The adaptation parameters (Fig. 2f-h) remain bounded throughout the operation, confirming the theoretical stability analysis.
5. Discussion and Concluding Remarks
This work has presented a novel solution to the integrated problem of fault-tolerant control with guaranteed transient performance for a quadrotor drone afflicted by time-varying inertial parameters. The main theoretical contribution is the seamless integration of two powerful techniques: Prescribed Performance Control (PPC) for managing transient and steady-state error bounds, and the Freezing Variable Method (FVM) for systematically handling the time-varying nature of the system dynamics without requiring known bounds on the rates of change of parameters.
The designed adaptive controller explicitly accounts for partial loss of effectiveness (LOE) actuator faults, treating them as part of the time-varying control effectiveness parameter b_i(t). The stability proof, grounded in Lyapunov theory, conclusively shows that all signals in the closed-loop system are uniformly ultimately bounded, the tracking error respects the user-defined performance envelope at all times, and—importantly—asymptotic tracking is achieved. This last property distinguishes the approach from many standard robust or PPC methods that only guarantee ultimate boundedness.
The simulation studies validate the theoretical claims and highlight the practical advantages: smoother control effort, faster and better-damped transient response, and exact convergence compared to a sliding-mode-based PPC approach. The method is particularly well-suited for quadrotor drone applications involving payload manipulation, aerial manipulation, or any mission where the inertia properties are not constant and high-performance, reliable control is paramount despite potential actuator degradation.
Future work will focus on extending this framework to the full position and attitude control loop of the quadrotor drone, investigating its robustness under time-varying delays in sensor feedback, and experimental validation on a physical platform with a dynamically changing payload.