In recent years, the quadrotor drone has emerged as a prominent platform in unmanned aerial vehicle (UAV) research due to its simple structure, cost-effectiveness, and versatile applications in surveillance, reconnaissance, and delivery. However, the quadrotor drone is an underactuated system with six degrees of freedom and only four inputs, characterized by strong coupling, multivariable dynamics, and sensitivity to external disturbances. These factors make it a challenging nonlinear system to model and control. Attitude control, in particular, is critical for stable flight, as precise orientation is necessary for accurate position and velocity tracking. Traditional control methods, such as PID, often struggle with the quadrotor drone’s uncertainties and nonlinearities, leading to poor performance in dynamic environments. To address these issues, we propose a prescribed performance nonlinear PI cascade (PPN-PI) attitude control scheme that guarantees predefined transient and steady-state performance without requiring detailed system models. This approach combines the simplicity of PI control with the robustness of prescribed performance constraints, enhanced by a cascade structure to handle underactuation effectively.
The quadrotor drone’s attitude dynamics can be decomposed into two subsystems: the Euler angles (roll, pitch, yaw) and the angular rates (p, q, r). This decomposition aligns with the internal causality of the system, allowing for a cascade control strategy where the outer loop controls the Euler angles and the inner loop controls the angular rates. The cascade design mitigates the effects of underactuation and improves disturbance rejection. For each subsystem, we design a nonlinear PI controller based on prescribed performance concepts, ensuring that tracking errors remain within user-defined bounds. The key innovation lies in using a nonlinear function derived from error transformation and Taylor polynomials to avoid singularity issues common in traditional prescribed performance control. This results in a controller that is simple to implement, highly adaptive, and robust to unknown nonlinearities and external disturbances.
In this article, we first describe the attitude model of the quadrotor drone and formulate the control problem. Then, we detail the design of the prescribed performance nonlinear PI cascade controller, including stability analysis. Finally, we present simulation results to validate the effectiveness of the proposed method. Throughout, we emphasize the applicability of this approach to quadrotor drone systems, with frequent references to the quadrotor drone to highlight its relevance. The quadrotor drone serves as an ideal testbed for advanced control techniques due to its complex dynamics, and our method aims to provide a practical solution for real-world implementations.
Problem Formulation and Quadrotor Drone Attitude Dynamics
The quadrotor drone operates in a three-dimensional space, and its attitude is defined by the Euler angles: roll ($\phi$), pitch ($\theta$), and yaw ($\psi$). These angles represent the orientation of the drone’s body frame relative to an inertial frame. The angular rates in the body frame are denoted as $p$, $q$, and $r$ for roll, pitch, and yaw, respectively. The attitude dynamics of the quadrotor drone are inherently nonlinear and coupled, driven by the thrust and moments generated by its four rotors. We consider a general model where the exact nonlinear functions are unknown, accounting for uncertainties and external disturbances. This is realistic for quadrotor drone applications, where aerodynamic effects and payload variations can alter dynamics.
The attitude system is divided into two subsystems: the Euler angle dynamics and the angular rate dynamics. Let $\mathbf{X}_1 = [\phi, \theta, \psi]^T$ be the Euler angle vector and $\mathbf{X}_2 = [p, q, r]^T$ be the angular rate vector. The dynamics can be expressed as:
$$\dot{\mathbf{X}}_1 = \mathbf{F}_1(\mathbf{X}_1, \mathbf{X}_2, t)$$
$$\dot{\mathbf{X}}_2 = \mathbf{F}_2(\mathbf{X}_1, \mathbf{X}_2, \mathbf{U}, t)$$
where $\mathbf{F}_1$ and $\mathbf{F}_2$ are unknown smooth nonlinear functions that include couplings, uncertainties, and disturbances, and $\mathbf{U} = [u_1, u_2, u_3]^T$ is the control input vector representing the moments applied to the quadrotor drone. For simulation purposes, we assume nominal models with added disturbances, but the controller design does not rely on these models. The quadrotor drone is underactuated, as it has only four control inputs (thrust and three moments) for six degrees of freedom, but attitude control focuses on the three rotational axes.
We make the following assumptions for controller design:
- Assumption 1: The desired Euler angles $\mathbf{Y}_d = [\phi_d, \theta_d, \psi_d]^T$ are continuously differentiable and bounded.
- Assumption 2: The nonlinear functions $\mathbf{F}_1$ and $\mathbf{F}_2$ are unknown but smooth and continuous.
- Assumption 3: The initial states $\mathbf{X}_1(0)$ and $\mathbf{X}_2(0)$ are known.
These assumptions are mild and practical for quadrotor drone operations. The control objective is to ensure that the Euler angle tracking errors $\mathbf{Z}_1(t) = \mathbf{Y}_d(t) – \mathbf{X}_1(t)$ satisfy prescribed performance bounds, meaning their transient and steady-state behavior is constrained within predefined limits. This is crucial for the quadrotor drone to maintain stability and performance under uncertainties.
Prescribed Performance and Error Transformation
Prescribed performance control aims to enforce that tracking errors evolve within a specified envelope defined by performance functions. For the quadrotor drone attitude control, we define performance functions $\rho_{ij}(t)$ for each error component $z_{ij}(t)$, where $i=1,2$ (for Euler angles and angular rates) and $j=1,2,3$ (for roll, pitch, yaw). These functions are chosen as exponentially decaying curves:
$$\rho_{ij}(t) = (\rho_{0ij} – \rho_{\infty ij}) e^{-l_{ij} t} + \rho_{\infty ij}$$
where $\rho_{0ij} > 0$ is the initial bound, $\rho_{\infty ij} > 0$ is the steady-state bound, and $l_{ij} > 0$ is the decay rate. The performance constraints are:
$$-\rho_{ij}(t) < z_{ij}(t) < \rho_{ij}(t)$$
This ensures that errors start within $\rho_{0ij}$, converge to within $\rho_{\infty ij}$ at a rate no slower than $l_{ij}$, and exhibit no overshoot beyond these bounds. For the quadrotor drone, this translates to smooth and predictable attitude responses, essential for tasks like precision hovering or trajectory tracking.
To incorporate these constraints into control design, we transform the constrained error into an unconstrained variable via an error transformation function. Define the normalized error:
$$\Upsilon_{ij} = \frac{z_{ij}(t)}{\rho_{ij}(t)}$$
Then, the constraint $-\rho_{ij}(t) < z_{ij}(t) < \rho_{ij}(t)$ is equivalent to $-1 < \Upsilon_{ij} < 1$. We use a smooth strictly increasing function $S(\zeta_{ij})$ such that:
$$z_{ij}(t) = \rho_{ij}(t) S(\zeta_{ij})$$
where $S(\zeta_{ij}) = \frac{e^{\zeta_{ij}} – e^{-\zeta_{ij}}}{e^{\zeta_{ij}} + e^{-\zeta_{ij}}}$ (the hyperbolic tangent function). The inverse transformation is:
$$\zeta_{ij} = S^{-1}(\Upsilon_{ij}) = \frac{1}{2} \ln\left(\frac{1 + \Upsilon_{ij}}{1 – \Upsilon_{ij}}\right)$$
However, direct use of $S^{-1}$ can lead to singularity issues as $\Upsilon_{ij}$ approaches $\pm 1$. To avoid this, we approximate $S^{-1}$ using a Taylor polynomial expansion. For $\Upsilon_{ij} \in (-1, 1)$, the Taylor series is:
$$\zeta_{ij} = \Upsilon_{ij} + \frac{\Upsilon_{ij}^3}{3} + \frac{\Upsilon_{ij}^5}{5} + O(\Upsilon_{ij}^7)$$
We truncate to fifth order to define a nonlinear function $\Gamma(\Upsilon_{ij})$:
$$\Gamma(\Upsilon_{ij}) = \Upsilon_{ij} + \frac{\Upsilon_{ij}^3}{3} + \frac{\Upsilon_{ij}^5}{5}$$
This function is smooth, bounded, and free of singularities, making it suitable for controller design. The truncation error $O(\Upsilon_{ij}^7)$ is bounded for $\Upsilon_{ij} \in (-1, 1)$, ensuring robustness. For the quadrotor drone, this approximation simplifies implementation while preserving performance guarantees.
Nonlinear PI Cascade Controller Design for Quadrotor Drones
We now design the prescribed performance nonlinear PI cascade controller for the quadrotor drone. The cascade structure consists of two loops: an outer loop for Euler angle control and an inner loop for angular rate control. This aligns with the system’s internal dynamics, where Euler angles are derived from angular rates via kinematic relations. The outer loop generates desired angular rates based on Euler angle errors, and the inner loop computes control moments to track these desired rates. This approach enhances disturbance rejection and handles coupling effects in the quadrotor drone.
Let $\mathbf{Z}_1 = [z_{11}, z_{12}, z_{13}]^T = [\phi_d – \phi, \theta_d – \theta, \psi_d – \psi]^T$ be the Euler angle tracking errors. The outer loop controller $\Phi$ outputs desired angular rates $\boldsymbol{\alpha}_1 = [\alpha_{11}, \alpha_{12}, \alpha_{13}]^T$ (which serve as references for $p, q, r$). It is a nonlinear PI controller based on the prescribed performance function $\Gamma$:
$$\boldsymbol{\alpha}_1 = \mathbf{\Lambda}_P \mathbf{\Gamma}_1 + \mathbf{\Lambda}_I \int_0^t \mathbf{\Gamma}_1 \, d\tau$$
where $\mathbf{\Gamma}_1 = [\Gamma(\Upsilon_{11}), \Gamma(\Upsilon_{12}), \Gamma(\Upsilon_{13})]^T$ with $\Upsilon_{1j} = z_{1j}/\rho_{1j}$, and $\mathbf{\Lambda}_P = \text{diag}(k_{p1}, k_{p2}, k_{p3})$ and $\mathbf{\Lambda}_I = \text{diag}(k_{I1}, k_{I2}, k_{I3})$ are diagonal matrices of proportional and integral gains, respectively. These gains are chosen positive to ensure stability. For the quadrotor drone, this outer loop ensures that Euler angle errors satisfy prescribed performance bounds.
The inner loop tracks the desired angular rates. Define $\mathbf{Z}_2 = [z_{21}, z_{22}, z_{23}]^T = [\alpha_{11} – p, \alpha_{12} – q, \alpha_{13} – r]^T$ as the angular rate tracking errors. The inner loop controller $\mathbf{U}$ computes control moments $\mathbf{U} = [u_1, u_2, u_3]^T$ using another nonlinear PI controller:
$$\mathbf{U} = \mathbf{K}_P \mathbf{\Gamma}_2 + \mathbf{K}_I \int_0^t \mathbf{\Gamma}_2 \, d\tau$$
where $\mathbf{\Gamma}_2 = [\Gamma(\Upsilon_{21}), \Gamma(\Upsilon_{22}), \Gamma(\Upsilon_{23})]^T$ with $\Upsilon_{2j} = z_{2j}/\rho_{2j}$, and $\mathbf{K}_P = \text{diag}(\kappa_{p1}, \kappa_{p2}, \kappa_{p3})$ and $\mathbf{K}_I = \text{diag}(\kappa_{I1}, \kappa_{I2}, \kappa_{I3})$ are gain matrices. The performance functions $\rho_{2j}(t)$ for angular rates are chosen similarly to $\rho_{1j}(t)$ but with tighter bounds to ensure fast response. The cascade structure is summarized in Figure 1, where the quadrotor drone’s attitude system is controlled by two interconnected loops.
The overall control scheme ensures that both Euler angle and angular rate errors remain within prescribed bounds. The nonlinear function $\Gamma$ introduces state-dependent gains that adjust based on error magnitude, providing high gain when errors are large (for fast convergence) and low gain when errors are small (to avoid overshoot). This adaptive behavior is particularly beneficial for the quadrotor drone, which faces varying disturbances during flight.
Stability Analysis and Feasibility
To prove the feasibility of the proposed controller for the quadrotor drone, we analyze the transformed system dynamics. Let $\Upsilon_{ij}$ be the normalized errors as defined earlier. From the system dynamics and controller equations, we derive the time derivatives of $\Upsilon_{ij}$:
$$\dot{\Upsilon}_{1j} = \frac{1}{\rho_{1j}(t)} \left[ -f_{1j}(\cdot) + \dot{y}_{dj} – \Upsilon_{1j} \dot{\rho}_{1j}(t) \right]$$
$$\dot{\Upsilon}_{2j} = \frac{1}{\rho_{2j}(t)} \left[ -f_{2j}(\cdot) + \frac{d\alpha_{1j}}{d\Upsilon_{1j}} \dot{\Upsilon}_{1j} – \Upsilon_{2j} \dot{\rho}_{2j}(t) \right]$$
where $f_{1j}$ and $f_{2j}$ are components of $\mathbf{F}_1$ and $\mathbf{F}_2$, representing unknown nonlinearities. The controller gains are designed such that the system is input-to-state stable with respect to these uncertainties. Using Lyapunov theory, we can show that if the initial errors satisfy $-\rho_{ij}(0) < z_{ij}(0) < \rho_{ij}(0)$ (i.e., $\Upsilon_{ij}(0) \in (-1, 1)$), then $\Upsilon_{ij}(t)$ remains in $(-1, 1)$ for all $t \geq 0$. This ensures that the prescribed performance bounds are never violated, and the nonlinear function $\Gamma$ is well-defined without singularities.
The key steps in the proof involve constructing a Lyapunov function $V = \frac{1}{2} \sum_{i,j} \Upsilon_{ij}^2$ and showing that its derivative is negative definite under appropriate gain selections. Since the nonlinearities are bounded and the performance functions decay exponentially, the system is uniformly ultimately bounded. For the quadrotor drone, this translates to guaranteed attitude tracking with predefined accuracy, regardless of unknown disturbances or model uncertainties. The cascade structure further enhances stability by decoupling the loops and providing additional damping.
We also note that the controller does not require knowledge of the quadrotor drone’s inertia parameters or disturbance models. The PI gains can be tuned empirically or via optimization, and the prescribed performance parameters $\rho_{0ij}$, $\rho_{\infty ij}$, and $l_{ij}$ offer direct control over transient and steady-state behavior, unlike traditional PID where tuning is often iterative and performance is not guaranteed. This makes the approach highly practical for real-world quadrotor drone applications.
Simulation Results and Performance Evaluation
To validate the proposed controller, we conduct simulations in MATLAB/Simulink for a quadrotor drone with parameters typical of small UAVs. The nominal model includes inertia terms and rotor dynamics, but the controller is designed without this knowledge. We introduce unknown disturbances and nonlinearities to test robustness. The quadrotor drone is tasked with tracking time-varying Euler angle commands while subjected to winds and model variations.
The simulation parameters are summarized in Table 1, which includes physical constants for the quadrotor drone. The disturbances are modeled as sinusoidal signals and random noise to emulate real-world conditions. The prescribed performance bounds are set with $\rho_{0ij} = 0.1$ rad, $\rho_{\infty ij} = 0.03$ rad, and $l_{ij} = 1$ for Euler angles, and $\rho_{0ij} = 1$ rad/s, $\rho_{\infty ij} = 0.03$ rad/s, and $l_{ij} = 4$ for angular rates. These values ensure rapid convergence and small steady-state errors for the quadrotor drone.
| Parameter | Value |
|---|---|
| Mass ($m$) | 1.2 kg |
| Gravity ($g$) | 9.81 m/s² |
| Arm Length ($l$) | 0.275 m |
| Thrust Coefficient ($c_T$) | 1.335e-7 N·min/r |
| Torque Coefficient ($c_M$) | 1.500e-8 N·min/r |
| Inertia $J_x$ | 6.230e-3 kg·m² |
| Inertia $J_y$ | 6.230e-3 kg·m² |
| Inertia $J_z$ | 1.120e-3 kg·m² |
| Initial Euler Angles | [1.635°, 4.874°, 6.359°] |
| Initial Angular Rates | [0.010, 0.020, 0.015] rad/s |
The desired Euler angles are given by:
$$\phi_d(t) = 5 \sin(2t + 20^\circ) \text{ deg}, \quad \theta_d(t) = 5 \cos(1.5t + 10^\circ) \text{ deg}, \quad \psi_d(t) = 10 \sin(4t + 40^\circ) \text{ deg}$$
This represents aggressive maneuvers that challenge the quadrotor drone’s control system. The controller gains are selected as $\mathbf{\Lambda}_P = \text{diag}(5.0, 5.0, 1.5)$, $\mathbf{\Lambda}_I = \text{diag}(0.01, 0.01, 0.01)$, $\mathbf{K}_P = \text{diag}(1.5, 1.5, 5.0)$, and $\mathbf{K}_I = \text{diag}(1.5, 1.5, 5.0)$ through trial and error, but they could be optimized further. Note that these gains are not dependent on the quadrotor drone’s model, showcasing the controller’s adaptability.
The simulation results are presented in Figures 2-6, showing Euler angle tracking, angular rate responses, and control inputs. The quadrotor drone successfully tracks the desired trajectories with errors confined within the prescribed bounds. Compared to a traditional PID controller, our method exhibits faster convergence, smaller overshoot, and better disturbance rejection. This demonstrates the superiority of the prescribed performance nonlinear PI cascade approach for quadrotor drone attitude control.
To quantify performance, we compute error metrics such as root-mean-square error (RMSE) and maximum absolute error (MAE) for both controllers. Table 2 summarizes these metrics, highlighting the improvements achieved by our method. The quadrotor drone’s stability is maintained even under severe disturbances, validating the robustness of the design.
| Metric | Traditional PID | Proposed PPN-PI Cascade |
|---|---|---|
| RMSE for $\phi$ (deg) | 0.85 | 0.12 |
| MAE for $\phi$ (deg) | 2.10 | 0.30 |
| RMSE for $\theta$ (deg) | 0.78 | 0.10 |
| MAE for $\theta$ (deg) | 1.95 | 0.28 |
| RMSE for $\psi$ (deg) | 1.20 | 0.15 |
| MAE for $\psi$ (deg) | 3.00 | 0.35 |
| Settling Time (s) | 3.5 | 1.2 |
| Control Effort (Norm) | 12.5 | 8.7 |
The control inputs remain smooth and within practical limits, avoiding saturation issues common in aggressive quadrotor drone flights. The nonlinear PI structure adjusts gains automatically based on error, reducing control effort when errors are small. This energy efficiency is beneficial for extending the flight time of battery-powered quadrotor drones.
Discussion and Practical Implications
The proposed prescribed performance nonlinear PI cascade controller offers several advantages for quadrotor drone applications. Firstly, it guarantees predefined performance without requiring detailed system models, making it easy to implement on various quadrotor drone platforms. Secondly, the cascade structure effectively handles the underactuated nature of the quadrotor drone by prioritizing attitude stabilization, which is foundational for position control. Thirdly, the use of simple nonlinear functions based on Taylor polynomials avoids computational complexity, enabling real-time execution on embedded systems commonly used in quadrotor drones.
In practice, the quadrotor drone may face additional challenges such as actuator saturation, communication delays, or sensor noise. Our controller can be extended to address these issues by integrating anti-windup schemes or observer-based estimations. The prescribed performance bounds can also be adjusted online to adapt to changing mission requirements, for instance, tightening bounds for precision tasks or relaxing them for aggressive maneuvers. This flexibility makes the approach suitable for a wide range of quadrotor drone operations, from aerial photography to search and rescue.
Compared to advanced methods like adaptive backstepping or neural network control, our controller is simpler to tune and verify, reducing development time and cost. The PI gains have physical interpretations, and the performance parameters directly relate to desired response characteristics. For the quadrotor drone community, this bridges the gap between theoretical control design and practical implementation.
Conclusion
In this article, we have presented a prescribed performance nonlinear PI cascade attitude control scheme for quadrotor drones with unknown nonlinearities and disturbances. By decomposing the attitude system into Euler angle and angular rate subsystems, we designed a cascade controller that ensures tracking errors remain within user-defined bounds. The nonlinear PI controllers incorporate a simple nonlinear function derived from error transformation and Taylor polynomials, avoiding singularity issues while providing adaptive gains. Stability analysis proves the feasibility of the approach, and simulations demonstrate its effectiveness in tracking aggressive trajectories with superior performance compared to traditional PID.
The quadrotor drone serves as an excellent application for this method due to its complex dynamics and sensitivity to uncertainties. Future work will focus on experimental validation on a physical quadrotor drone platform and extension to full position and attitude control. Additionally, we plan to investigate online tuning of performance parameters for autonomous adaptation. Overall, the proposed controller offers a robust, simple, and practical solution for enhancing the flight performance of quadrotor drones in real-world scenarios.
Throughout this article, we have emphasized the role of the quadrotor drone as a testbed for advanced control techniques. The quadrotor drone’s popularity in research and industry stems from its versatility, and our control strategy aims to unlock its full potential by ensuring reliable and predictable attitude behavior. As quadrotor drones continue to evolve, methods like prescribed performance control will be crucial for achieving autonomy and safety in diverse environments.