Aircraft Pitch Control Via Parametric Identification and PID Optimization

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Introduction Contemporary aviation technology demands enhanced safety, stability, and controllability [1]. The precise characterization of aircraft short-period motion, defined by the coupled dynamics of the angle of attack, pitch rate, and normal load factor, is fundamental to achieving these objectives [2; 3]. Accurate modeling of aerodynamic phenomena, including the transitional states of complex con-figurations, is essential [3]. Traditional analytical methods have limitations when addressing non-linear dynamics and stochastic disturbances [4]. Consequently, numerical integration techniques critically influence simulation fidelity and com-putational efficiency [4; 5]. Euler’s method offers advantages for real-time applications in onboard flight control computers [6], whereas Runge - Kutta methods enable higher-fidelity aircraft motion modeling [7]. The relevance of this study is the need to improve the accuracy of short-period motion models, particularly under nonlinear dynamics and stochastic disturbances. Traditional analytical modeling methods described in studies on flight dynamics are often limited to simple cases, whereas modern computational approaches, such as Runge - Kutta and Euler methods, allow solving complex problems with controlled error. The reliable estimation of aerodynamic parameters from flight data underpins effective system identification [8; 9]. Parametric identification methods, including the least-squares method (LSM) and neural net-work algorithms, remain key tools for minimizing errors in estimating aerodynamic coefficients. Experimental data processing methods, such as noise filtering and error estimation, also play an important role, which is particularly important for complex aerodynamic systems. The estimated coefficients enable a robust flight control design in which PID controllers are universally em-ployed for aircraft attitude regulation[18] [10-12]. Computational aero-dynamic modeling provides the foundation for flight dynamics simulation [13], although validation against experimental data remains imperative [14]. PID synthesis leverages transfer function fitting [5], optimization techniques [10], and specialized approaches for nonlinear systems [13]. Adaptive control strategies [14; 15] and delay-compensation methods [16] address practical implementation constraints, whereas rotor dynamics simulations [17] and director algorithms for landing [18] further demonstrate the dependency on precise models. PID tuning methodologies [19] complete this essential framework. The scientific novelty of this work consists of the integration of Runge - Kutta methods of the 4th order and adaptive LSM for joint modeling of aircraft dynamics and parameter identification in conditions of nonlinear disturbances, the development of an algorithm for automatic adjustment of the PID controller, taking into account the relationship between aerodynamic coefficients and transient characteristics of the system, and the creation of a universal methodology for parametric studies in MATLAB/Simulink. This study syste-matically evaluates how numerical integration errors propagate into aerodynamic coefficient esti-mation and subsequently affect PID controller performance. This analysis provides critical insights into aviation systems that require balanced com-putational efficiency and control fidelity. 1. Problem Statement In aircraft dynamics, the short-period mode of the aircraft motion refers to pitch motion in which the aircraft rotates around its lateral axis. This motion includes changes in the pitch angle, pitch rate, and airspeed. In this short-period mode, the aircraft experiences rapid pitch angle fluctuations owing to interference, such as turbulence or pilot actions. The stability and controllability characteristics of the aircraft in this mode are crucial for flight safety and efficiency. The aim of this study is to develop an inte-grated approach to model the short-period motion of aircraft, combining numerical methods for solving differential equations (Euler and Runge - Kutta methods), algorithms for parametric identifi-cation, and optimization of control systems based on PID controllers. This study solves the following tasks: comparative analysis of the accuracy of the Euler and Runge - Kutta methods, identification of aerodynamic coefficients using LSM, and Repetition. Revision required synthesis and automatic adjustment of the PID controller in the Simulink environment to improve the stability of the pitch control system. Presents an integrated methodology comprising: Dynamics simulation of short-period motion using Euler and Runge - Kutta methods, Aerodynamic parameter identification, and Automatic PID controller tuning for pitch control. The methodological basis of the study includes a three-stage approach: dynamics modeling using Euler and Runge - Kutta methods to solve a system of nonlinear differential equations, parameter identification using LSM based on experimental data, as well as control optimization through automatic adjustment of the PID controller. 2. Modeling of Aircraft Short Period Motion This study employed numerical modeling to simulate the aircraft’s short-period longitudinal motion, specifically capturing the coupled dynamics of the angle of attack (AoA), pitch rate, and normal load factor. This approach minimizes the costs and risks associated with traditional methods that rely on wind tunnel experiments or flight tests. Elevator deflection served as the input ex-citation, while the outputs such as AoA, pitch rate, and normal load factor were selected as standard parameters defining the short-period mode for longitudinal dynamics analysis. The governing system of nonlinear differential equations was solved using two numerical inte-gration techniques: the explicit Euler method and the 4th-order Runge-Kutta (RK4) method. The choice of the numerical method critically influences the solution stability and local error magnitude, particularly under nonlinear aerody-namic conditions, as established in prior research. 2.1. Euler Method This is the most elementary numerical method for solving systems of ordinary differential equa-tions (ODEs). It was first described by Leonhard Euler in 1768 in his work “Institutiones Calculi Integralis”. The Euler method is an explicit single-step first-order accuracy scheme. It approximates the solution curve with a piecewise linear function termed the Euler polygon. Moreover, it represents the simplest numerical technique for solving first-order ODEs. This method is employed for the approximate solution of the initial value problems (Cauchy problems) and determines the values of a function defined by a differential equation on a specified point grid. The Euler method approximates the solution at each step using local linearization. At each iteration, a new function value is computed based on the current valueand its derivative. The integration step is denoted as h. At every step, the x-value increments by h, whereas the y-value is updated according to the following formula: , (1) where is the current function value; is the next function value; is the derivative at a point ; is the integration step. The process is repeated for each step until the endpoint of the integration interval is reached. 2.2. Runge-Kutta Method The Runge - Kutta method represents one of the numerical techniques used for solving ordinary differential equations (ODEs). The most widely adopted variant is the fourth-order Runge - Kutta method (RK4), which delivers a high accuracy and is extensively utilized in practice. The RK4 method employs four intermediate points to compute the slope (derivative) at each integration step. This approach enables a signifi-cantly more accurate solution approximation com-pared to the Euler method. The integration step is denoted as h. At each iteration, the x-value increments by h, whereas the y-value is updated as follows: (2) where are intermediate coefficients calculated at each step. The intermediate coefficients are calculated as follows: (3) The new value is calculated as a weighted sum of these coefficients. 2.3. Modeling the Dynamics of the Angular Motion of an Aircraft as a Control Object The presented system of ordinary differential equations describes the dynamics of the angular motion of the aircraft in the longitudinal channel, considering it as an object of control. (4) In short-period motion, the change in the airspeed of the aircraft can be negligible, and in this flight mode, and . Then, the mathematical model of the aircraft can be described as follows: (5) Simulations in the presence of a stepwise input signal of the angle of attack and pitch rate were performed using the Euler method and the fourth-order Runge - Kutta method. The pro-cessing time was 10s, in increments of 0.01. The simulation results of the short-period motion of the aircraft in the longitudinal channel are shown in Figures 1, 2, and 3. Figure 1. Angle of attack S o u r c e: by San Lin Aung in MATLAB Figure 2. Pitch rate S o u r c e: by San Lin Aung in MATLAB Figure 3. Normal load factor S o u r c e: by San Lin Aung in MATLAB A key feature of this approach is the utiliza-tion of simulated data instead of experimental data for the subsequent parametric identification. This enables the estimation of aerodynamic coefficients via the Least Squares Method (LSM) under controlled conditions, eliminating the influence of external noise and measurement inaccuracies The simulation results demonstrate that the Runge - Kutta method yields a reduced dis-cretizetion error compared to the Euler method, which is critical for identifying high-frequency flight regimes. The accuracy of the numerical method is crucial for analyzing short-duration dynamic modes. The obtained data provides a foundation for developing digital twins of aero-dynamic systems corresponding to trends in aviation engineering digitalization. Thus, combining Euler and Runge - Kutta methods in simulation frameworks not only reproduces short-period motion, but also evaluates the applicability boundaries of each method for specific problem classes. This contributes to the advancement of aviation system design methodologies. 3. Identification of Aerodynamic Coefficients Based on Numerically Simulated Data The identification of aerodynamic parameters constitutes a critical stage in the design and analysis of aircraft, as the accuracy of determining these parameters directly impacts the effectiveness and reliability of control systems. The obtained coeffi-cient estimates were utilized for developing and tuning automatic control systems that ensure aircraft motion stabilization and control. Conse-quently, the accurate determination of aerodynamic characteristics plays a pivotal role in creating safe and high-performance control systems capable of adapting to various flight regimes and external conditions. 3.1. Estimation of Aerodynamic Coefficients Using the Least Squares Method To estimate the aerodynamic coefficients, the traditional least-squares method was used in this work. The least-squares method is undeniably more effective for linear systems. The mathematical model of an object can be described as: (6) where is the output vector; is the vector of regressors or state vector; is the vector of unknown parameters to be estimated. In this case, is of dimension and is of dimension: Unlike maximum likelihood, when using the least squares method, the parameters to be estimated must occur in expressions for the average values of observations. When the parameters are displayed linearly in these expressions, the least-squares estimation problem can be solved in a closed form, and it is relatively simple to obtain statistical properties for the resulting parameter estimates. The least-squares method is described as: (7) where - the vector of unknown parameters to be estimated; - the object model matrix; - the output vector. 3.2. Estimation of Aerodynamic Coefficients Based on Numerically Simulated Data This study presents the estimation of two force coefficients and four moment coefficients asso-ciated with the angle of attack, pitch rate, and normal load factor. To estimate the two force coefficients, the normal load factor is required to form the output signal vector. The object model matrix incorporates angle of attack and elevator deflection. To estimate the four moment coefficients, a de-rivative of the pitch rate is required to form the output signal vector. The object model matrix in-corporates the angle of attack, pitch rate, derivative of the angle of attack, and the elevator deflection. Thus, the object model matrices, output signal vectors, and unknown parameter vectors take the following form: For the estimation of two force coefficients and For the estimation of four moment coefficients and . To analyze the efficacy and stability of the Least Squares Method (LSM) for estimating aero-dynamic coefficients, a numerical experiment was conducted with varying measurement noise levels. Measurement noise was modeled using random variables with normal distribution, characterized by zero mean and different standard deviation values (σ = 0.01, 0.02, 0.03). The quantitative results of the aerodynamic coefficient estimation, reflecting the error dependence on measurement noise intensity, are presented in Figures 4 and 5. The research results demonstrate that the data obtained using the Runge - Kutta method provide significantly higher accuracy for estimating aero-dynamic force coefficients, whereas Euler method-derived data yield superior accuracy for estimating aerodynamic moment coefficients across various measurement noise levels. These findings under-score the critical importance of selecting appropri ate numerical methods for aerodynamic modeling, as they directly impact parameter estimation reliability and, consequently, control system design. Figure 4. The results of the estimation of aerodynamic coefficients using data modeled by the Euler method S o u r c e: by San Lin Aung in Excel Figure 5. The results of the evaluation of aerodynamic coefficients using the data modeled by the Runge Kutta method S o u r c e: by San Lin Aung in Excel 4. Aircraft Pitch Control System Using PID Controller The aerodynamic coefficients were estimated using the Least Squares Method (LSM), which en-abled the acquisition of the necessary parameters for designing an automatic pitch angle control system. The obtained coefficients are utilized to derive the transfer function relating the pitch rate to the elevator deflection, which represents a critical stage in control system development. This transfer function is derived from equations describing the angle of attack (α) and pitch rate, establishing a mathematical relationship that quantifies how changes in elevator deflection (δ) affect the pitch dynamics (Figure 6). This phase is essential for creating a reliable and precise control system that ensures effective regulation of aircraft angular motion. A diagram of a block diagram AI-generated content may be incorrect. Figure 6. Structure diagram of the aircraft's automatic pitch control system S o u r c e: by San Lin Aung in Simulink Aircraft transfer functions via Laplace trans-formation can be expressed as: In this study, a model that does not consider the elevator lift is used as a mathematical model of the first approximation. (8) The transfer function from the elevator to the pitch rate through the Laplace transform can be written as: (9) The initial PID controller coefficients were set empirically, followed by a linear analysis of the transient response of the system to step elevator deflection (Figure 7). The investigation results revealed the following dynamic performance indicators: § Overshoot σ = 22.9%, indicating pronounced oscillatory components during the transients; § Rise time , reflecting high initial responsiveness; § Settling time , demonstrating in-sufficient damping in the initial controller configuration resulting in prolonged stabilization within the 2% tolerance band. To address these deficiencies, automatic parametric optimization of the PID controller was implemented using gradient descent algorithms in Simulink, targeting settling-time minimization while maintaining acceptable overshoot levels. These findings underscore the necessity of adaptive ap-proaches to tuning controllers in dynamically com-plex aerospace systems. Figure 7. Transient function of the aircraft's automatic pitch control system using a PID controller at preset coefficient values S o u r c e: by San Lin Aung in Simulink The initial PID controller coefficients were adjusted using an automatic tuning algorithm implemented in Simulink (Figure 8). For the simplified mathematical model, excluding the actuator dynamics and physical constraints, the transient characteristics demonstrate high controller efficacy: Figure 8. Transient function of the aircraft’s automatic pitch control system using a PID controller with automatic tuning coefficient values S o u r c e: by San Lin Aung in Simulink § Overshoot , reflecting moderate oscillatory components during transients within acceptable limits for aviation systems; § Rise time , indicating minimal system inertia when responding to control inputs; § Settling time , confirming the rapid attainment of steady state within the 2% tolerance band. These parameters demonstrate the high system responsiveness and stability achieved through gra-dient-based optimization methods. However, the model limitations stemming from neglected actuator dynamics and real operational constraints necessitate further research for validation under near-physical conditions. These results establish a foundation for developing adaptive algorithms that account for nonlinear effects and external disturbance characteristics of aircraft control systems. Conclusion This study confirms the efficacy of integrating Euler and Runge - Kutta methods for solving aerodynamic modeling, parametric identification, and pitch control system synthesis tasks. It was established that the Runge - Kutta method pro-vides enhanced accuracy for estimating aero-dynamic force coefficients, whereas the Euler method demonstrates advantages for estimating moment coefficients under distinct measurement noise conditions. The developed pitch angle PID controller, characterized by a rise time of 0.0709 s, overshoot σ = 11.6%, and settling time of 0.608 s, validates the feasibility of achieving high-quality control despite noise disturbances in systems neglecting actuator dynamics. The proposed methodology integrating nu-merical simulation, parameter identification, and adaptive controller synthesis demonstrates the potential for creating digital twins and their application in preliminary design stages, notwithstanding simplifications in the aerodynamic model. These results underscore the critical role of numerical method selection, which directly impacts estimation credibility and designed system reliability.

About the authors

Lin Aung San

Higher Education Center of the Defence Services Academy

Author for correspondence.
Email: sunlinaung91788@gmail.com
ORCID iD: 0009-0002-2475-2875

Master student

Pyin Oo Lwin, Myanmar

References

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