The proliferation of quadrotor drones across diverse sectors, from aerial cinematography and precision agriculture to infrastructure inspection and search-and-rescue operations, has placed a premium on reliable and high-performance flight control. Among the various control challenges, altitude control stands as a fundamental pillar. Precise height regulation is critical not only for stable hovering and smooth landing but also for navigating complex, vertically constrained environments such as indoor warehouses, forest canopies, or between building floors. In these scenarios, the control system must achieve rapid response to commanded height changes while strictly minimizing overshoot to prevent collisions. Furthermore, it must maintain robust performance in the face of ubiquitous practical challenges: uncertain or varying system parameters (e.g., mass changes due to payload pickup/release) and unmodeled external disturbances (e.g., wind gusts, ground effect). Traditional linear control methods like PID often struggle to satisfy all these conflicting requirements simultaneously, exhibiting trade-offs between agility and overshoot, and lacking inherent robustness guarantees. This necessitates the development of more sophisticated, nonlinear control strategies tailored for the quadrotor drone’s dynamics.
This article presents a comprehensive control methodology that decomposes the high-performance altitude control problem for a quadrotor drone into two synergistic stages: optimal trajectory generation and adaptive robust tracking. First, to address the need for fast, low-overshoot maneuvers in constrained spaces, we formulate the trajectory planning problem as a control problem for a linear system with input saturation. We employ a Composite Nonlinear Feedback (CNF) control law to generate an optimal altitude reference trajectory. Second, to ensure precise tracking of this trajectory despite parametric uncertainties and disturbances, we design a novel adaptive nonsmooth tracking controller. This controller leverages fractional-order feedback and online parameter adaptation to guarantee stability and convergence. The integrated approach ensures that the quadrotor drone can execute aggressive yet safe altitude transitions with high accuracy.
Problem Formulation and Dynamic Model
The foundation of any model-based control design is an accurate dynamic representation. We begin with the standard simplified translational dynamics of a quadrotor drone, assuming a rigid body structure and symmetric design. The equations of motion in the inertial frame are:
$$
\begin{aligned}
\ddot{x} &= (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi)\frac{b}{m} U_1 \\
\ddot{y} &= (\cos\phi \sin\theta \sin\psi – \sin\phi \cos\psi)\frac{b}{m} U_1 \\
\ddot{z} &= (\cos\phi \cos\theta)\frac{b}{m} U_1 – g
\end{aligned}
$$
where \( (x, y, z) \) denote the position coordinates, \( (\phi, \theta, \psi) \) are the roll, pitch, and yaw angles, \( m \) is the mass, \( b \) is the thrust coefficient, \( g \) is gravitational acceleration, and \( U_1 \) is the total thrust input. For the dedicated altitude channel, we focus on the \( z \)-axis dynamics. By defining the vertical control input as \( U’_1 = (\cos\phi \cos\theta) U_1 \) and assuming the attitude controller maintains \( \cos\phi \cos\theta \approx 1 \) (a standard assumption in hierarchical control), the nominal altitude dynamics simplify to a second-order system:
$$
m \ddot{z} = b U’_1 – m g
$$
However, for a practical quadrotor drone, we must account for parametric uncertainties \( \Delta m, \Delta b \) and an aggregate bounded external disturbance \( d_z(t) \) (e.g., wind, model inaccuracies). The true dynamics become:
$$
(m_n + \Delta m) \ddot{z} = (b_n + \Delta b) U’_1 – (m_n + \Delta m)g + d_z(t)
$$
Here, \( m_n \) and \( b_n \) are the nominal, known parameters. To streamline the controller design, we rewrite this uncertain model into a linearly parameterized form. Let us define the following composite parameters:
$$
p_z = \frac{m_n + \Delta m}{b_n + \Delta b}, \quad T_z = \frac{(m_n + \Delta m)g – d_z(t)}{b_n + \Delta b}
$$
The altitude dynamics can then be expressed as:
$$
p_z \ddot{z} + T_z = U’_1
$$
We separate the terms into nominal and uncertain parts: \( p_z = p_{zn} + p_{zu} \) and \( T_z = T_{zn} + T_{zu} \), where \( p_{zn} = m_n / b_n \) and \( T_{zn} = m_n g / b_n \) are known. The uncertain parameter vector is \( \mathbf{p}_{zu} = [p_{zu}, \ T_{zu}]^T \). Defining the regressor vector \( \mathbf{\Phi} = [\ddot{z}, \ 1]^T \), the dynamic model is reformulated as:
$$
p_{zn} \ddot{z} + T_{zn} + \mathbf{\Phi}^T \mathbf{p}_{zu} = U’_1
$$
This representation is crucial for the subsequent adaptive control design, as the uncertainty appears linearly with respect to the known regressor \( \mathbf{\Phi} \).
Methodology: A Dual-Stage Control Architecture
Our proposed solution uses a two-stage architecture. The first stage plans an optimal flight path, and the second stage ensures the quadrotor drone faithfully follows it.
Stage 1: Trajectory Generation via Composite Nonlinear Feedback (CNF)
The objective is to generate a reference trajectory \( z_r(t) \) and its derivative \( \dot{z}_r(t) \) that transition the quadrotor drone from an initial height \( z_0 \) to a desired height \( z_{des} \) rapidly and with minimal overshoot, respecting the physical saturation limits of the actuators. The thrust magnitude for a quadrotor drone is bounded, implying a bounded vertical acceleration \( a_{z,min} \le \ddot{z} \le a_{z,max} \). We conservatively model this as \( |\ddot{z}| \le \alpha g \), where \( \alpha \) is a known positive constant.
We can therefore frame the trajectory generation problem as controlling a double-integrator system with input saturation:
$$
\begin{aligned}
\dot{z} &= v_z \\
\dot{v}_z &= \text{sat}(u)
\end{aligned}
$$
where \( \text{sat}(u) = \text{sign}(u) \min(|u|, \alpha g) \). The goal is to design a control law \( u \) that drives \( z \rightarrow z_{des} \). The CNF control law combines a linear feedback part for fast response and a nonlinear feedback part to dampen overshoot as the output approaches the target. Defining the state vector \( \mathbf{x} = [z, v_z]^T \), the CNF controller is:
$$
u = u_L + u_N = F\mathbf{x} + G r + \rho(r, y) B^T P (A+BF)(\mathbf{x} – \mathbf{x}_e)
$$
where:
- \( u_L = F\mathbf{x} + Gr \) is the linear controller. \( F \) is chosen (e.g., via pole placement) to yield a fast, lightly-damped closed-loop system. \( r = z_{des} \), and \( G = -\left[ C (A+BF)^{-1} B \right]^{-1} \) ensures zero steady-state error for the linear system.
- \( u_N \) is the nonlinear part. \( \mathbf{x}_e = (A+BF)^{-1} B G r \) is the target equilibrium. \( P > 0 \) solves the Lyapunov equation \( (A+BF)^T P + P(A+BF) = -W \) for some \( W>0 \).
- \( \rho(r, y) \) is a non-positive nonlinear function of the tracking error. A typical choice is: $$ \rho(r, y) = -\eta e^{-\gamma |r – y|} $$ where \( \eta, \gamma > 0 \) are tuning parameters. As the output \( y = z \) approaches the reference \( r \), \( \rho \) becomes more negative, increasing the damping ratio and curtailing overshoot.
By discretizing the system and simulating the closed-loop CNF controller under the saturation limit, we obtain the optimal reference trajectories \( z_r(t) \) and \( \dot{z}_r(t) \). The tuning parameters \( F, \eta, \gamma \) offer a direct trade-off between rise time and overshoot, as summarized below:
| Parameter | Effect on Trajectory | Typical Adjustment |
|---|---|---|
| Poles from \( F \) | Faster poles increase initial speed but risk more overshoot. | Place for critical damping as a baseline. |
| \( \eta \) | Larger magnitude increases nonlinear damping effect. | Increase to reduce overshoot. |
| \( \gamma \) | Governs how quickly nonlinear damping engages. | Increase for earlier damping intervention. |
Stage 2: Adaptive Nonsmooth Tracking Control
With a feasible reference trajectory \( z_r(t) \) generated, the next task is to design a control law \( U’_1 \) for the actual quadrotor drone (with uncertainties) to track \( z_r(t) \) precisely. We employ an adaptive nonsmooth control technique, which provides finite-time convergence properties and robustness.
Controller Design: Define tracking errors \( e_1 = z – z_r \) and \( e_2 = \dot{z} – \dot{z}_r \). We use the backstepping framework. Let \( \zeta_1 = e_1 \). Introduce a virtual control law \( \beta \) for the derivative of \( \zeta_1 \):
$$
\beta = \dot{z}_r – k_1 \text{sig}(\zeta_1)^\alpha
$$
where \( k_1 > 0 \), \( 0 < \alpha < 1 \), and \( \text{sig}(x)^\alpha = |x|^\alpha \text{sign}(x) \). This nonsmooth function guarantees faster, finite-time-like convergence compared to linear feedback when the error is small. Define the second error variable \( \zeta_2 = \dot{z} – \beta = e_2 + k_1 \text{sig}(\zeta_1)^\alpha \).
Now, consider the Lyapunov function candidate incorporating parameter estimation errors:
$$
V = \frac{1}{2} \zeta_1^2 + \frac{1}{2} p_{zn} \zeta_2^2 + \frac{1}{2} \tilde{\mathbf{p}}_{zu}^T \Gamma \tilde{\mathbf{p}}_{zu}
$$
where \( \tilde{\mathbf{p}}_{zu} = \hat{\mathbf{p}}_{zu} – \mathbf{p}_{zu} \) is the parameter estimation error and \( \Gamma > 0 \) is a diagonal adaptation gain matrix. Differentiating \( V \) and substituting the dynamics \( p_{zn} \ddot{z} = U’_1 – T_{zn} – \mathbf{\Phi}^T \mathbf{p}_{zu} \) yields:
$$
\begin{aligned}
\dot{V} &= \zeta_1 \dot{\zeta}_1 + p_{zn} \zeta_2 \dot{\zeta}_2 + \tilde{\mathbf{p}}_{zu}^T \Gamma \dot{\hat{\mathbf{p}}}_{zu} \\
&= \zeta_1 (\zeta_2 – k_1 \text{sig}(\zeta_1)^\alpha) + \zeta_2 \left( U’_1 – T_{zn} – \mathbf{\Phi}^T \mathbf{p}_{zu} – p_{zn} \dot{\beta} \right) + \tilde{\mathbf{p}}_{zu}^T \Gamma \dot{\hat{\mathbf{p}}}_{zu}
\end{aligned}
$$
We choose the actual control law \( U’_1 \) as:
$$
U’_1 = T_{zn} + p_{zn} \dot{\beta} – \zeta_1 + \mathbf{\Phi}^T \hat{\mathbf{p}}_{zu} – k_2 \text{sig}(\zeta_2)^\alpha
$$
where \( k_2 > 0 \). Substituting this control law into the expression for \( \dot{V} \) simplifies it to:
$$
\dot{V} = -k_1 \zeta_1 \text{sig}(\zeta_1)^\alpha – k_2 \zeta_2 \text{sig}(\zeta_2)^\alpha + \tilde{\mathbf{p}}_{zu}^T \left( \Gamma \dot{\hat{\mathbf{p}}}_{zu} – \zeta_2 \mathbf{\Phi} \right)
$$
Adaptation Law Design: To cancel the last term and ensure \( \dot{V} \le 0 \), we select the adaptation law:
$$
\dot{\hat{\mathbf{p}}}_{zu} = \Gamma^{-1} \zeta_2 \mathbf{\Phi}
$$
This is a standard gradient-type update law, driven by the tracking error \( \zeta_2 \) and the regressor \( \mathbf{\Phi} \). With this choice, the derivative of the Lyapunov function becomes:
$$
\dot{V} = -k_1 |\zeta_1|^{\alpha+1} – k_2 |\zeta_2|^{\alpha+1} \le 0
$$
Stability Analysis: Since \( V \) is positive definite and radially unbounded, and \( \dot{V} \) is negative semi-definite, \( \zeta_1, \zeta_2, \tilde{\mathbf{p}}_{zu} \in \mathcal{L}_\infty \). Using Barbalat’s lemma or its extensions for nonsmooth systems, one can further show that \( \zeta_1 \rightarrow 0 \) and \( \zeta_2 \rightarrow 0 \) as \( t \rightarrow \infty \). Convergence of \( \zeta_1 \) and \( \zeta_2 \) to zero implies the original tracking errors \( e_1 \) and \( e_2 \) also converge to zero. The parameter estimates \( \hat{\mathbf{p}}_{zu} \) will remain bounded but may not converge to their true values unless the regressor \( \mathbf{\Phi} \) is persistently exciting, which is often not required for tracking purposes.
Performance Evaluation and Discussion
The proposed integrated framework was evaluated through detailed numerical simulations. The quadrotor drone parameters were set as: \( m_n = 2.0 \, \text{kg} \), \( b_n = 2.98 \times 10^{-6} \, \text{N·s}^2/\text{rad}^2 \), with substantial uncertainties: \( \Delta m = 0.5 \, \text{kg} \) (25% uncertainty) and \( \Delta b = -0.5 \times 10^{-6} \, \text{N·s}^2/\text{rad}^2 \). A sinusoidal disturbance \( d_z(t) = 4.0 \sin(4\pi t) \, \text{N} \) was applied. The task was to climb from 0 m to 5 m.
CNF Trajectory Planning: With saturation limit \( \alpha g = 0.4g \) and CNF parameters tuned for low overshoot, the generated reference trajectory achieved the climb with a rise time of approximately 1.8 seconds and an overshoot of less than 0.5%, demonstrating its suitability for constrained environments.
Tracking Controller Comparison: The adaptive nonsmooth (ANS) controller (\( \alpha=0.9 \), adaptation on) was compared against three alternatives: a standard backstepping (BS) controller (\( \alpha=1.0 \), adaptation off), a nonsmooth (NS) controller without adaptation (\( \alpha=0.9 \), adaptation off), and an adaptive backstepping (ABS) controller (\( \alpha=1.0 \), adaptation on). The key performance metrics are summarized below:
| Controller | Max Tracking Error (m) | Settling Time (s) | Steady-State Error Band (m) | Robustness to Uncertainty |
|---|---|---|---|---|
| BS (Baseline) | 0.42 | > 4.0 | ±0.08 | Poor |
| NS | 0.35 | 3.2 | ±0.06 | Low |
| ABS | 0.28 | 2.5 | ±0.047 | Good |
| ANS (Proposed) | 0.23 | 2.1 | ±0.027 | Excellent |
The results clearly show the advantages of the proposed ANS controller. The combination of nonsmooth feedback and online parameter adaptation yielded the smallest maximum error, fastest settling, and tightest steady-state error band. The adaptation mechanism effectively compensated for the constant parameter errors, while the nonsmooth term (\( \alpha < 1 \)) provided superior convergence properties once the error became small.
Effect of the Fractional Power \( \alpha \): The parameter \( \alpha \) in the \( \text{sig}()^\alpha \) function is critical. A comparative study of the ANS controller with different \( \alpha \) values was conducted, with results condensed in the following table. The metrics are normalized against the \( \alpha=1.0 \) (linear) case to highlight trends.
| Value of \( \alpha \) | Normalized Max Error (%) | Normalized Steady-State Error Band (%) | Convergence Characteristic |
|---|---|---|---|
| 1.0 (Linear) | 100% | 100% | Asymptotic (exponential) |
| 0.9 | 91% | 81% | Finite-time like, smooth |
| 0.8 | 82% | 57% | Faster finite-time convergence |
| 0.7 | 72% | 35% | Very fast, but may induce chattering if gains are high |
The table demonstrates a clear performance improvement as \( \alpha \) decreases from 1. Smaller \( \alpha \) leads to significantly lower steady-state error bands and reduced maximum overshoot/error. This is because the term \( |\zeta|^\alpha \) for \( \alpha<1 \) provides a steeper gain when the error \( \zeta \) is small, accelerating the final convergence phase. However, excessively small values of \( \alpha \) (e.g., below 0.5) can make the controller sensitive to measurement noise and potentially lead to chattering in practical implementations. A value in the range of \( 0.7 \le \alpha \le 0.9 \) offers an excellent trade-off between performance and robustness for a quadrotor drone altitude controller.
Conclusion
This article has presented a robust and high-performance solution for the altitude control of quadrotor drones operating in demanding conditions. By decomposing the problem into trajectory generation and robust tracking, the approach addresses the critical need for both agility and precision. The CNF-based planner efficiently generates dynamically feasible, low-overshoot reference trajectories ideal for confined spaces. The adaptive nonsmooth tracking controller guarantees robust stability and superior tracking accuracy despite significant parametric uncertainties and external disturbances, leveraging fractional-order feedback for fast convergence and online adaptation for uncertainty compensation. Simulation studies validate that the integrated strategy outperforms conventional and partially enhanced controllers across key metrics like maximum error, settling time, and steady-state precision. The framework is systematic, with clear tuning guidelines for parameters like the fractional power \( \alpha \), making it both theoretically sound and practically applicable for enhancing the operational capability and safety of quadrotor drones in complex real-world environments. Future work will focus on experimental validation and extension of the nonsmooth adaptive paradigm to the full attitude and position control of the quadrotor drone.