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Design and Optimization of a Piecewise Flight Controller for VTOL UAVs

Shanfei Su

https://orcid.org/0000-0002-3913-2436

Southern University of Science and Technology, 518055 Shenzhen, People’s Republic of China

*Ph.D. Student, Department of Mechanics and Aerospace Engineering; .

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Peng Yu

https://orcid.org/0000-0003-2073-7512

Southern University of Science and Technology, 518055 Shenzhen, People’s Republic of China

†Associate Professor, Guangdong Provincial Key Laboratory of Turbulence Research and Applications, Department of Mechanics and Aerospace Engineering; .

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Hao Wang

https://orcid.org/0000-0002-9683-0919

Southern University of Science and Technology, 518055 Shenzhen, People’s Republic of China

‡Research Associate Professor, Department of Mechanics and Aerospace Engineering; (Corresponding Author).

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and

Xiaowen Shan

https://orcid.org/0000-0002-6350-9248

BNU-HKBU United International College, 519087 Zhuhai, People’s Republic of China

§Chair Professor, Institute for Advanced Study; .

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Published Online:2 Dec 2024https://doi.org/10.2514/1.C037806

Abstract

The dynamic systems of unmanned aerial vehicles (UAVs) are susceptible to parametric uncertainties, unmodeled dynamics, and external vibrational disturbances during motion control. Although proportional–integral–derivative (PID) controllers are widely used in UAVs, the manual tuning process is time-consuming, and the resultant control performance is vulnerable to uncertainties in varying flight conditions, leading to suboptimal flight control throughout the entire hovering flight envelope. To address this issue, the present study designs and optimizes a novel piecewise flight controller based on a hybrid data-driven approach combining gradient-based methods and pattern search algorithms. The optimization process is validated through comprehensive flight testing, comparing the piecewise controller with the baseline controller across a wide range of flight conditions. The study examines the objective functions and design parameters, demonstrating the robustness of the optimized piecewise controller in the presence of disturbances. The results reveal significant variations in the system model under different flight conditions, and the piecewise optimized controller demonstrates a response that is more than twice as fast as the benchmark, with acceptable overshoots and steady-state errors under the predefined trajectories. The salient features of the proposed piecewise flight controller and design optimization process are verified through flight testing, which is expected to provide a framework for future UAV flight control designs.

Nomenclature
$A, B, C, D$	model state-space representation matrices
$a$	acceleration vector, $m / s^{2}$
$b$	span, m
$e$	error
FF	feedforward gain
$h$	height, m
KP, KI, KD	rate controller proportional–integral–derivative gains
$O - X Y Z$	body axis
PP	angle controller P gain
$q$	vector part of quaternion
$r$	position vector, m
$T$	thrust force, N
$t_{p}$	peak time, s
$t_{r}$	rise time, s
$t_{s}$	settling time, s
$v$	velocity vector, $m / s$
$x$	independent variable
$y$	response amplitude
$y_{p}$	response amplitude at $t_{p}$
$y_{r}$	response amplitude at $t_{r}$
$δ$	aerodynamic control surface angular deflection, deg or rad
$θ$	pitch angle, deg or rad
$ϕ$	roll angle, deg or rad
$ψ$	yaw angle, deg or rad
$Ω$	angular rate vector, $deg/ s$ or $rad / s$

I. Introduction

The widespread accessibility of cost-effective and powerful embedded systems, along with compact sensor technology, has significantly impacted the evolution of unmanned aerial vehicle (UAV) designs and their diverse applications, leading to a notable increase in the proliferation of UAVs in recent years [1–3]. Achieving trajectory tracking control for UAVs poses a formidable challenge due to the system’s highly nonlinear rigid-body dynamics and significant cross-coupling [4]. Existing literature and practical applications demonstrate the widespread and successful utilization of proportional–integral–derivative (PID) controllers, although with some inherent limitations [5–7]. Nonetheless, tuning the PID controller parameters can be challenging, as it must be tailored to facilitate effective dynamic behavior of the vehicle near its equilibrium point [8].

Various researchers have effectively employed model-based adaptive methodologies for the fine-tuning of PID controller parameters either in the design phase or as an offline experimental procedure [9]. Leveraging the dynamic model of the quadrotor in an X-mode configuration obtained through the Newton–Euler formulation, Noordin et al. [6] conducted simulations of an adaptive PID control (APIDC) system, incorporating a sliding mode approach. The results of these simulations successfully demonstrated enhanced the system’s ability to counteract challenges posed by parameter uncertainties and external disturbances [6]. By substituting the fixed-gain controllers within the autopilot, Spencer et al. developed an adaptive autotuner featuring controllers that dynamically adjusted through retrospective cost optimization [10], applying the particle swarm optimization (PSO) method proposed by Kennedy and Eberhart [11]. The effectiveness of the autotuner was validated through the execution of a test trajectory, with its performance subsequently compared to that of a meticulously tuned autopilot. Mac et al. conducted research involving an AR-drone quadrotor to fine-tune the parameters of a PID controller using multi-objective particle swarm optimization (MOPSO) [12]. This approach entailed modifying the velocity formula employed in conventional PSO systems to enhance both search and execution time efficiency. To encompass the intrinsic tradeoffs among diverse objectives and mitigate the risk of converging into local minima, the integration of Pareto optimality principles with the MOPSO algorithm is employed. This amalgamation facilitates the swift identification of optimal sets of PID parameters [13].

Advanced control theories, such as linear quadratic regulator (LQR) control [14], model predictive control (MPC) [15], backstepping control [16], and sliding mode control [17], among others, have been investigated in the context of maneuver control tasks for UAVs. Based on the system’s physical model, controller tuning methods include pole placement and loop shaping. In the pole placement method, control loops are designed by introducing additional control elements and adjusting the pole locations based on their distribution within the system [18]. This method allows for precise control over the system’s dynamic behavior by strategically placing poles to meet the desired control objectives. The loop shaping method relies on frequency-domain analysis of the model, often using tools like the Nyquist criterion [19]. It involves introducing supplementary control elements to shape the system’s response curve, ensuring alignment with the predefined control requirements. This method is particularly effective in achieving the desired response characteristics. However, it is noteworthy that these methodologies predominantly rely on explicit system modeling, which renders them susceptible to constraints in terms of their adaptability and generalizability [13].

The fully model-based adaptive nonlinear control for UAVs has undergone extensive theoretical analysis, establishing it as a mature discipline with established theoretical guarantees [2,20]. Nevertheless, its dependence on model-based formulations restricts the scope of results to specific known model types, presenting significant challenges in extending these approaches to more general models [21]. Data-driven controllers operate independently of any pre-existing mathematical models of the system [22]. Controllers rooted in system identification, for example, depend solely on real-time measurements obtained directly from the system. This characteristic enables them to manage uncertainty adeptly and adapt to changing conditions. The process of system identification primarily furnishes a comprehensive dissection of the constituent elements responsible for the observed responses, thereby imparting a holistic comprehension of the dynamics inherent to the aerial platform. Additionally, system identification engenders a precise and comprehensive repository of data for utilization in flight simulators and offline digital simulations. These simulation tools find extensive application in pilot training programs and serve to mitigate risk during experimental testing, a realm characterized by significant cost implications [23].

Several system identification methodologies have demonstrated effectiveness in the context of small, cost-effective UAVs [24]. A noteworthy contribution in this domain is the Comprehensive Identification from Frequency Responses (CIFER) software package developed by Tischler, designed to facilitate system identification of flight vehicle models using frequency-response methods. It is particularly suited for handling noisy flight-test data and capturing the dynamics of unstable bare airframes [25]. While CIFER’s primary application is in helicopter systems, existing literature provides evidence of its adaptability to fixed-wing UAVs [26–28]. Typically, a scaled-down prototype of a helium balloon quadcopter was assembled by Graham [28]. The dynamic models governing longitudinal and lateral-directional behavior during hovering were identified within the frequency domain. This process was facilitated by employing simulated aerostatic lift forces and the above software package. These identified models are intended to establish the foundational framework for the subsequent design of full-scale control systems and the initiation of flight tests.

A flight vehicle inherently embodies dynamic characteristics, and a time-domain state-space framework closely approximates the underlying physical reality more accurately than frequency-domain transformation methods [23]. Liu et al. identified a model-based nonlinear system of a quad tilt-rotor UAV in the time domain to propose an accurate model, while unstructured nonlinearities are ignored [29]. Simmons et al. [30] undertook the system identification of a nonlinear flight dynamic model for a small, low-cost, fixed-wing unmanned aircraft equipped with a limited instrumentation system. Their parameter estimation approach utilized the time-domain formulation of the output-error method, applying nonlinear flight dynamic equations for a rigid aircraft with a nonlinear aerodynamic model [30]. Additionally, their team advanced system identification research through the general system identification process, incorporating the application of uncorrelated input signals, the development of an aeropropulsive model, and a spin aerodynamic model for small fixed-wing aircraft [31–33]. Verling et al. introduced a VTOL tail-sitter UAV with a high-fidelity model that utilizes parameter estimation from wind tunnel data and applied a nonlinear model-based controller for stable and efficient flight [34]. Rohr et al. presented a comprehensive approach to modeling and system identification for a VTOL tiltwing UAV, deriving a 3-degree-of-freedom rigid-body dynamic model and identifying key parameters from flight data. The model serves as the foundation for a nonlinear model predictive control (NMPC) strategy that optimizes throttle, tilt rate, and pitch angle to efficiently track velocity trajectories [35]. These nonlinear controllers rely on simplified physics models, aerodynamic derivatives, and system identification based on flight test data. Parameter estimation and system identification methods, based on conventional nonlinear rigid-body dynamic equations, are well-established for vehicles in the cruise state. However, when VTOL UAVs operate under hovering conditions, conventional nonlinear controllers face significant challenges due to the lack of accuracy in modeling complex nonlinear aerodynamics, wind gusts, flow separation, and unsteady propeller slip flow.

To achieve more accurate stability control over a wide range of the nonlinear flight envelope, the present work proposes a novel piecewise flight controller based on data-driven system identification. The piecewise concept allows precise control at each subregion, and the data-driven system identification overcomes the challenges due to the lack of accuracy in physics modeling. State-space models using the subspace identification method are applied as a foundation for the identification. For the controller parameter optimization, the proposed approach integrates a hybrid optimization technique combining gradient-based methods and pattern search algorithms. This hybrid approach effectively balances exploitation and exploration during optimization, which is particularly beneficial in dealing with the complex and nonlinear nature of VTOL UAV dynamics.

The subsequent sections of the paper are structured as follows: Section II introduces the testing vehicle and the piecewise flight controller designed to enhance the flight stability across a wide range of angles of attack. Section III proposes the piecewise processing procedure, including piecewise data acquisition, system identification, objective functions, and parameter optimization methods. Data validation, optimization results, and detailed analysis are discussed in Sec. IV. Finally, the authors conclude the paper in Sec. V.

II. Testing Vehicle

A. UAV Model

The Egretta is a vertical takeoff and landing (VTOL), fixed-wing UAV designed for the purposes of long-range surveillance and urgent cargo delivery by the Southern University of Science and Technology (SUSTech), located in China [36]. The fundamental physical dimensions of this vehicle are listed in Table 1. As shown in Fig. 1, the tail-sitter configuration of the UAV includes a unique installation of propellers situated behind the primary wing. The two elevons receive mixed control signals from yaw rotation around the $X$ axis and pitch rotation around the $Y$ axis, allowing them to deflect independently to stabilize and control motion along the $X$ and $Y$ axes. Simultaneously, the rotational speeds of the two propellers are independently controlled to stabilize rotation around the $Z$ axis and regulate vertical height. This innovative design feature not only prevents slip flow from passing over the primary wing, thereby increasing the overall takeoff weight, but also enhances control effectiveness by directing the propellers' slip flow onto the elevons, significantly improving controllability.

The onboard data acquisition system includes various sensors, such as an accelerometer, GPS, gyroscope, and barometer, all of which are systematically summarized in Table 2. A visual representation of the flight data transmission system is shown in Fig. 2.

Fig. 2
Data transmission system of the testing vehicle.

The proposed flight control is developed based on the PX4¶ cascaded architecture. The illustration of applied control architecture is shown in Fig. 3. The angular rate control is an inner loop controller, coupled with an outer loop angle controller, forming the proportional-proportional-integral-derivative (P-PID) architecture in the body frame. The tail-sitter VTOL UAV lacks inherent stability while hovering; however, dynamic flight stability can be achieved by applying the control architecture for active control. The position loop is only used when holding position or when the requested velocity along an axis is null. Among the various controllers that make up a UAV autopilot system, it is crucial to emphasize the pivotal role of attitude tracking control [37].

Fig. 3
The illustration of applied control architecture.

The current PX4 flight controller employs a baseline P-PID controller, which uses a single set of parameters to manage the entire hovering flight condition. The manual iteration of parameter tuning does not account for uncertainties such as wind perturbation, structural vibration, and propeller slip-flow perturbation at large pitch angles. This makes the system vulnerable to uncertainties in varying flight conditions, leading to suboptimal flight control across the entire hovering flight envelope.

B. Proposed Piecewise Controller

To achieve more accurate stability control over a wide range of the nonlinear flight envelope, Fig. 4. shows the framework of the proposed a piecewise flight controller. Initial baseline flight tests, for different flight conditions, are set up to obtain corresponding high-resolution control signals and attitude data. During the offline processing phase, system identification for different flight conditions is prioritized. The system takes into account the vehicle’s dynamic model, nonlinear perturbations, structural vibration, etc. The model obtained from the identification process is then applied to controller optimization. Following dynamic performance index limitations and controller constraints, control parameters for different flight conditions are determined, leading to the generation of a comprehensive control database. In the online processing loop, the sensor-estimated attitude data are transmitted to the classifier, which determines the real-time flight conditions based on the data. The controller supervisor responds accordingly and applies the control parameters to the controller to achieve control optimization under different flight conditions.

To ensure satisfactory flight quality, the optimization of the piecewise controller gains for both the angle and rate controllers is necessary to achieve the desired control performance. Figure 5 displays the dynamic response model with the optimization constraint, which is derived based on the identified system. The reference attitude signal generates control signals through the attitude control loop and the attitude rate control loop, and the identified system model outputs the attitude signal. During this process, the optimization module inversely determines the feasible parameter domain based on the dynamic response performance. The parameters of all controllers are tuned to search for optimal objectives. Detailed optimization problem formulation will be emphasized in Sec. III.

Fig. 5
Dynamic response with optimization constraint.

III. Methodology

A. Piecewise Flight Test Procedure

The dynamics of UAVs vary across different flight conditions due to disturbances such as wind gusts, turbulence, and mechanical vibrations from the propellers and motors. These disturbances can cause fluctuations in the UAV’s orientation and position, affecting its stability and control. Effective control strategies must account for these disturbances to maintain consistent performance. To facilitate piecewise system identification and optimization, full flight phase data of the tail-sitter VTOL UAV will be collected under the accurate control of baseline outdoor flight tests. These tests are conducted under acceptable wind conditions to minimize their impact on the UAV’s control performance. The environmental conditions on the testing date were a sunny day, a flight altitude of approximately 90 m, an average wind speed of 2 m/s (wind force level 2), and a prevailing wind direction of 44 deg east of north.

The baseline flight tests cover a spectrum of flight conditions, ranging from 0 to 70 deg in the pitch direction, with data collected for system identification under each of these conditions (see Fig. 6). The detailed flight test procedure is as follows:

Step 1: The setup for a single flight condition is depicted in Fig. 7. A preset 0.3 s step response with a 10 deg amplitude is applied. To achieve the desired response, the attitude command is programmed into the board using the PX4 flight stack as the foundation.
Step 2: Before issuing the step perturbation command, the UAV is aligned with the wind direction and allowed to drift along the wind direction until reaching a steady state, resulting in zero effective wind speed from the UAV’s body-fixed reference frame.
Step 3: The operator can initiate specific system identification maneuvers by toggling a switch, enabling the overwriting of control inputs with the predetermined functions set in step 1.
Step 4: Steps 2 and 3 are repeated, with the step perturbation command uniformly tested three times for each flight condition.

To initiate system identification maneuvers during flight testing, the autopilot is engaged to supplant manual control inputs with predefined functions. This approach allows the operator to maintain control authority over all remaining flight control channels, except for the specific channel activated for the given maneuver. For instance, when performing a pitch adjustment, the operator retains control over throttle, roll, and yaw inputs. Notably, the entire procedural methodology incorporates pitch angles ranging from 0 to 70 deg, making this approach instrumental in enhancing the safety of the flight test. High-rate actuator control data (up to 250 Hz) and IMU data (up to 1 kHz) are logged for system identification and comparison.

Fig. 7
Predefined step maneuvers of condition $θ = 0$ deg.

B. System Identification Processing

The diagram for the decoupled cascaded P-PID controllers for roll, pitch, and yaw is developed separately [38]. The pitch and roll controllers share the same structure, where the angular error is converted to an angular rate command through the angle controller. Subsequently, the angular rate command is transformed into an actuator control signal by the angular rate controller. The models used for system identification, depicted in Fig. 8, are based on high-resolution flight test data, taking into account external disturbances and structural vibrations. The actuator control signal serves as input data, while the estimated angle serves as output data. In response to external disturbances, the weather vane function is activated, orienting the VTOL UAV toward the relative wind to mitigate the wind’s effects on the vehicle. Although this does not directly enhance the UAV’s inherent stability, it helps maintain a more manageable control environment. This functionality reduces the likelihood of crosswinds impacting the wings, which can lead to significant lateral forces and moments. These forces can cause the UAV to roll or yaw uncontrollably, potentially resulting in divergence in a coupled pitch and yaw direction. By aligning with the crosswinds through commanding a yaw rate based on roll angle variation, the weather vane function minimizes these destabilizing effects, thereby enhancing the overall control environment. Consequently, the stability and control of the UAV’s pitch direction become paramount. The piecewise system identification and optimization in the present work focus on the pitch controller.

The identification process concentrates on the angle and angular rate components. Based on physical understanding and model plausibility, a second-order state-space model is employed. Utilizing subspace algorithms to identify combined deterministic-stochastic systems, the subspace identification method Numerical Algorithms for Subspace State Space System Identification (N4SID) [39] is applied for system identification with the above time-domain data.

C. Flight Control Parameter Optimization

1. Objective Formulation

Starting from the system input, the control signal output is processed through the controller, enters the identified system model, and, in conjunction with the feedback channel, constitutes a control loop. The problem at hand revolves around the holistic optimization of system dynamic responses. This optimization entails the consideration of dynamic performance metrics, including system rise time, settling time, and overshoot, among others. The goal is to derive the optimal control parameters within the feasible control region. The objective of the present work is to minimize the error $e$ between the reference and the measured value by optimizing the controller. The objective function in the controller model is generated by computing errors between the time-based reference signal and the simulated output of the plant over time.

For tracking problems, the reference value is defined as a sequence of time-amplitude pairs:

y_{ref} (t_{ref}), t_{ref} \in {t_{ref 0}, t_{ref 1}, \dots, t_{ref N}}

(1)

Simultaneously, the simulated output is computed as another sequence, with specific values of $t_{sim}$ potentially matching those of $t_{ref}$ :

y_{sim} (t_{sim}), t_{sim} \in {t_{sim 0}, t_{sim 1}, \dots, t_{sim N}}

(2)

Then, a new time base $t_{new}$ is formed by amalgamating the elements of $t_{sim}$ and $t_{ref}$ :

t_{new} = {t : t_{sim} \cap t_{ref}}

(3)

Employing this revised time series $t_{new}$ , the model computes $y_{sim}$ and $y_{ref}$ at $t_{new}$ and then derives the scaled error $e (t)$ :

e (t_{new}) = \frac{y_{sim} (t_{new}) - y_{ref} (t_{new})}{\max_{t_{new}} | y_{ref} |}

(4)

Finally, the objective function is determined by computing the weighted integral square error:

f = \int_{t_{1}}^{t_{2}} w (t) e {(t)}^{2} d t

(5)

where

w (t)

is a weight factor with a default value of 1, and

t_{1}

and

t_{2}

are the starting and ending points of the time series

t_{new}

, respectively.

2. Dynamic Response Constraints

A typical dynamic response with required characteristics is shown in Fig. 9. The response characteristic constraints define the feasible region of the objective solution.

By combining the objective function and constraints, the optimization problem can be formulated:

Min

f (K) = \int_{t_{1}}^{t_{2}} w (t_{new}) {(\frac{y_{sim} (t_{new}) - y_{ref} (t_{new})}{\max_{t_{new}} | y_{ref} |})}^{2} d t

(6)

subject to

{\begin{matrix} c_{1} (K) = t_{r} \leq 0.5 \\ c_{2} (K) = t_{s} \leq 1.5 \\ c_{3} (K) = overshoot \leq 0.1 \\ c_{4} (K) = | settling amplitude | \leq 0.05 \\ c {eq}_{1} (K) = initial value = 0 \\ c {eq}_{2} (K) = final value = 1 \\ K \in [lb, ub] \end{matrix}

where

K

is the set of design parameters of the controller with appropriate bounds:

K = (FF, PP, KP, KI, KD)

and

lb

and

ub

are the lower bound and upper bound of

K

, respectively. The constraints can be adjusted according to the required dynamic response metrics.

In Fig. 10, the signal constraint block (SCB) is depicted, encompassing the desired signal and associated constraints. The region highlighted in white within the bold black boundary delineates the feasible parameter space for the respective problem instances. When selecting the optimal parameters, the feasibility of dynamic performance metrics and their practical impact on flight outcomes should be considered comprehensively.

Based on empirical dynamic stability requirements and the baseline values of the PX4 controller, Tables 3 and 4 show the optimization constraints and bounds of controller gains, respectively. The baseline PID controller utilizes a single set of parameters to manage the entire hovering flight condition. This set of parameters is initially tuned at a zero pitch angle in an indoor environment and then verified and fine-tuned at larger pitch angles based on outdoor flight tests. The baseline controller is designed to handle the entire hovering flight envelope, ensuring that the UAV remains stable and controllable. Based on manual tuning, the baseline controller used in this work is so far the best in terms of controllability, stability, overshoot, and wind resistance, covering the entire hovering flight envelope.

D. Optimization-Based Tuning

The default approach for minimizing the objective function is the gradient descent method. Despite being a local method, it can be deemed suitable if it yields reasonable results. However, if the results are not satisfactory, the global optimization method, specifically the pattern search algorithm, will be employed instead. This optimization strategy aims to balance computational cost with optimization accuracy.

Gradient descent is an iterative optimization algorithm used to find a local minimum, as depicted in Fig. 11a. The core concept involves taking successive steps in the direction opposite to the gradient (or its approximation) of the function at the current point. This strategy is effective because it aligns with the steepest descent direction [40]. Unlike traditional optimization techniques that leverage information about the gradient or higher derivatives for seeking an optimal point, a pattern search algorithm, as depicted in Fig. 11b, explores a set of points around the current location. The goal is to identify a point where the objective function value is lower than that at the current location [41].

The present analysis reveals that the computational expense involved in obtaining the optimal solution through a global optimization approach (pattern search) is notably high, especially when trying to achieve accuracy levels similar to those obtained with the gradient-based method. Therefore, it is suggested that the optimal approach may involve first navigating to the feasible region using a global optimization technique and then determining the minimum using a gradient-based optimization method, under the condition that the feasible region exclusively contains a single minimum value.

A tolerance value of $1 e - 6$ will be applied for all constraints, functions, and parameter values. Furthermore, the parameters to be optimized are controller gains without units.

IV. Results and Discussion

A. System Identification

1. System Identification Result

Time-domain state-space models serve as the fundamental framework for conducting system identification using actual flight data. Specifically, a second-order state-space model is employed for the identification of pitch models. Consistently high fitting accuracy, exceeding 90%, is achieved across a range of flight conditions. The response of the identified system model for a 15 deg condition is compared with raw data, as shown in Fig. 12. The real test data and extended validation data exhibit fitness levels of 94.11 and 90.93%, respectively, underscoring the reliability of the identified model. Notably, the identified model accurately predicts the system output within both the original time range and in an extended time range.

Fig. 12
Identified system validation of flight condition $θ = 15$ deg.

2. Comparison of Piecewise Identified Models

Figure 13 illustrates the responses of system models identified under various flight conditions when exposed to the same input signal. Although the response trends across different conditions are similar, variations in response time and amplitude are clearly observed. This observation indicates that a single set of flight control parameters is inadequate for achieving optimal control across the full 0–70 deg pitch angle range. The data analysis confirms the presence of variations in the system models identified under different flight conditions. The variations observed in the identified models exhibit strong nonlinear effects under the investigated flight conditions, which conventional PID controllers fail to resolve or capture. This emphasizes the necessity of developing corresponding control parameters to maintain UAV stability across varying flight conditions. High-resolution system identification is essential for accurately capturing the dynamic variations across different flight modes, thereby enhancing the precision of the piecewise controller.

Fig. 13
Comparison of identified model of various flight conditions.

B. Validation of the Control Framework

Alongside ensuring the fitting accuracy of the identification model, it is essential to verify the feasibility of the control framework employed for optimization. In this process, normalized control signals derived from actual flight data are fed into the control system. These signals pass through the identification phase and the complete control loop, generating simulation output control signals, as shown in Fig. 14. The feasibility of the entire control framework is validated by rigorously comparing the control signals produced by the system with those observed in actual flight test scenarios. For instance, Fig. 15 shows the comparison between the control signals generated by the model and the actual control signals when real pitch angle data for a 0 deg pitch angle are input. The alignment between these two signals confirms the control framework’s feasibility. Discrepancies may arise due to less than 100% fitting accuracy in system identification and potential disturbances in measurement feedback.

Fig. 14
The control framework applied for comparison.

Fig. 15
Comparison of simulation and real control signal.

C. Piecewise Parameter Optimization

Control parameters satisfying optimization constraints and dynamic performance criteria have been successfully derived. Figures 16 and 17 present a comparative analysis of simulation results for various flight conditions, demonstrating significant improvements in achieving the objective.

Fig. 16
Optimized response comparison, $θ : 0 \overset{}{\to} 10$ deg.

A detailed examination of the control parameters and dynamic response characteristics for a specific condition is presented. As depicted in Fig. 16, the rise time is reduced from 0.33 to 0.15 s, while the settling time decreases from 0.75 to 0.38 s, nearly halving the overall response time. Furthermore, the optimized overshoot stands at 3.45%, well within the defined limits of dynamic response criteria. Table 5 presents the control parameters and dynamic response metrics for the 0 deg condition.

D. Flight Test Verification

After a systematic optimization process, the results are validated through practical flight testing. The control parameters derived from piecewise optimization are applied to their corresponding flight conditions. A comparison of the data for the 0 deg condition is depicted in Fig. 18. Starting with initial baseline data and baseline control parameters, a system model applied for the simulation control framework is derived through system identification. Subsequently, control parameters that meet dynamic response criteria are optimized through detailed analysis. Finally, data validation is achieved through actual flight tests. The system response accelerates significantly, with the rise time reducing from 0.33 to 0.15 s and the settling time decreasing from 0.75 to 0.39 s, effectively halving their original values, consistent with the simulation results. The key factor lies in the consistency between simulation and flight test results, ensuring that the specified system performance is achieved.

Fig. 18
Flight test validation, $θ : 0 \overset{}{\to} 10$ deg.

V. Conclusions

This paper presents the development of a piecewise flight controller and its corresponding parameter optimization using a novel hybrid data-driven approach, followed by a comparative analysis with the baseline PX4 controller across various flight test missions. The proposed piecewise flight controller demonstrates enhanced control accuracy for VTOL UAVs across a wide range of nonlinear flight envelopes. The study investigates objective functions and design parameters, employing a hybrid approach that integrates gradient-based methods with pattern search algorithms for piecewise optimization. The results reveal that the optimized controller achieves the desired rise time, settling time, and overshoot for the specified trajectories.

The system identification is accurate, achieving a similarity rate exceeding 90%. Empirical evidence from data analysis confirms the existence of variations among the system models identified under different flight conditions. These variations serve as essential prerequisites for the subsequent optimization of the piecewise flight controller.
The three critical components, flight control, system identification, and optimization, are integrated into a closed-loop parameter optimization framework and validated through flight testing. This process also establishes a feasible control framework for future optimization of control performance.
The influence of parameter uncertainties on simulation outcomes and subsequent predictions is observable to a certain extent. However, the effectiveness of the piecewise flight controller parameter optimization methodology has been validated through flight testing, demonstrating its capability to achieve precise control across a broad spectrum of flight conditions. This methodology holds significant potential for application across various engineering technologies.

Y. XuAssociate Editor

¶ PX4, “Controller Diagrams,” https://docs.px4.io/main/en/flight_stack/controller_diagrams.html [retrieved 31 May 2020].

Acknowledgments

The authors disclose receipt of the following financial support for the research of this paper: This study was supported by the Department of Science and Technology of Guangdong Province (2019B121203001, 2020B1212030001, 2023B1212060001), the National Science and Technology Major Project of China (J2019-II-0006-0026) and the Shenzhen Science and Technology Innovation Bureau (SGDX20230116091648011, KCXFZ20211020174803005). The authors thank Xizhi Qiu and LiuYong Xie, their indispensable pilots, for their full support and dedication in the ground handling and flight test work.

Appendix: Introduction Video

There is a short but complete video that shows the flight status of the tail-sitter vertical takeoff and landing UAV named Egretta, including takeoff, transition, cruise flight, wind resistance, landing, and other working conditions, which is available at https://youtu.be/mnnj9KuEMPU. Another video demonstrates flight tests aimed at data acquisition, which is available at https://youtu.be/bs4aa6vNCqQ.

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Tables

**Table 1 Basic specifications of Egretta**
Parameter	Value
$b$	2.0 m
$h$	1.2 m
Payload weight	1.0 kg
Payload volume	$0.12 * 0.1 * 0.15, m^{3}$
Maximum takeoff weight (MTOW)	5.5 kg

**Table 2 Model and accuracy of the applied onboard sensors**
SN	Description	Model	Accuracy
1	Barometer	MS5611	$\pm 1.5 mbar$
2	GPS	NEO V3	$\pm 0.5 - 2 m$
3	Accelerometer and gyroscope	ICM-20689	$\pm 2 %$ , $\pm 2 %$
		ICM-20602	$\pm 1 %$ , $\pm 1 %$
		BMI055	0.98 mg, $\pm 0.004 \deg / s$
4	Magnetometer	IST8310	$\pm 0.3 uT$

**Table 3 Optimization constraints**
Monitoring index	Pitch	Roll
$t_{r}$ , s	0.3	0.3
Rise, %	85	85
Undershoot, %	3	3
Overshoot, %	10	10
$t_{s}$ , s	1.5	1.5
Settling, %	5	5

**Table 4 Bounds of controller gains**
Parameter	Baseline	Min	Max
FF	0	0	1
PP	3.5	0.1	7
KP	0.6	0.01	1.2
KI	0.05	0	0.6
KD	0.001	0	0.02

**Table 5 Comparison of dynamic response metrics, $θ : 0 \overset{}{\to} 10$ deg**
Pitch	Baseline	Optimized
Feedforward FF	0	0.9622
Pitch PP	3.5	4.38
Pitch rate [KP, KI, KD]	[0.6, 0.05, 0.001]	[1.09, 0.02, 0.005]
$t_{r}$ , s	0.33	0.15
$t_{s}$ , s	0.75	0.38
Overshoot	——	3.45%
Steady-state error, deg	0.9	0.7

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Received16 November 2023
Accepted16 September 2024
Published online2 December 2024

Design and Optimization of a Piecewise Flight Controller for VTOL UAVs

Abstract

I. Introduction

II. Testing Vehicle

A. UAV Model

B. Proposed Piecewise Controller

III. Methodology

A. Piecewise Flight Test Procedure

B. System Identification Processing

C. Flight Control Parameter Optimization

1. Objective Formulation

2. Dynamic Response Constraints

D. Optimization-Based Tuning

IV. Results and Discussion

A. System Identification

1. System Identification Result

2. Comparison of Piecewise Identified Models

B. Validation of the Control Framework

C. Piecewise Parameter Optimization

D. Flight Test Verification

V. Conclusions

Acknowledgments

Appendix: Introduction Video

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