CONSTRUCTION AND SURVEY OF DIGITAL CONTROL CHARACTERISTICS OF AIRCRAFT AUTOMATIC CONTROL SYSTEM

ABSTRACT

The paper presents the development and investigation of digital signal control in an automatic flight control system. The research aims to determine the influence of the sampling period and signal conversion on the control quality, stability, and accuracy of the system. Based on the mathematical model of the flight control system, the study involves designing, simulating, and analyzing various cases with different sampling periods. The results show that the optimal conversion period plays a crucial role in maintaining stability, reducing error, and ensuring a fast system response. The study confirms that selecting an appropriate sampling frequency and digital signal conversion technique is a key factor in improving the performance and effectiveness of automatic control in modern flight systems.

АННОТАЦИЯ

В статье представлены разработка и исследование цифрового управления сигналами в системе автоматического управления самолета. Целью исследования является определение влияния периода дискретизации и преобразования сигнала на качество управления, устойчивость и точность системы. На основе математической модели системы управления полетом исследование включает в себя проектирование, моделирование и анализ различных случаев с различными периодами дискретизации. Результаты показывают, что оптимальный период преобразования играет решающую роль в поддержании устойчивости, снижении ошибок и обеспечении быстрого отклика системы. Исследование подтверждает, что выбор подходящей частоты дискретизации и метода цифрового преобразования сигнала является ключевым фактором повышения производительности и эффективности автоматического управления в современных системах управления самолетом.

Keywords: sampling cycle, automatic control system; aircraft; digital.

Ключевые слова: цикл отбора проб, автоматическая система управления; самолет; цифровой.

Introduction

In the automatic control system of modern aircraft, the conversion process from analog signals to digital signals plays an important role in ensuring control accuracy and stability. The conversion cycle, or sampling cycle, determines the speed of information update and directly affects the quality of system response. Too long a cycle can cause delay and instability, while too short a cycle increases unnecessary computational load. This paper focuses on building a model and investigating the impact of the sampling cycle to determine the optimal value, ensuring stability, accuracy and efficiency for the digital flight control system, contributing to improving the automation and safety of modern aircraft.

Construction of a digital signal conversion process in an aircraft automatic control system

The control system of the aircraft, in which the on-board digital computer is directly fed into the control loop of the angular coordinates or the tractor of the aircraft, belongs to the type of discrete systems. Direct digital control includes the following operations:

- obtaining the measured value of the control parameter from the corresponding sensor and converting the analog signal into a digital signal;

- calculating the error signal (deviation of the current value of the measured parameter from its required value);

- converting the non-conforming signal according to the corresponding control algorithm into a control signal;

- converting the control signal from digital to analog form;

- processing the control signal by the aircraft control equipment.

Thus, in a certain part of the aircraft's automatic control system, the operation of technical signals – is discrete in time and quantized in level.

From the point of view of ensuring the best flow of dynamic processes, the sampling period of the signal in time T should be as small as possible. In fact, the sampling period T is determined by the fact that during this time the on-board digital computer must calculate n algorithms that perform the conversion of the non-conforming signal into a control signal, then:

                                                                 (1)

here  - is the duration of the ith processing algorithm, that is, the value of the sampling interval must be greater than the total computation time of all algorithms.

When converting an analog signal to a digital signal, both sampling and quantization occur. The degree of quantization depends on the bit depth of the analog-to-digital converter and the microprocessor. With modern converters of 12–16 bit depth, quantization errors are so small that they can be neglected in dynamic analysis at the first approximation. However, if the sampling period T is equal to or greater than the characteristic time constants of the control system, the discrete behavior of the digital controller will be significantly affected, degrading the dynamics and stability of the system.

In the paper, we will simulate the digital signal in the aircraft automatic control system in static autopilot mode.

The structure of all the aircraft angular coordinate control channels is similar. In the simplest case, the control is performed by static autopilot mode, Fig. 1.

Figure 1. Static channel structure controlling aircraft angle coordinates

  - one of the angular coordinates of the aircraft ( -  yaw,  - roll and  - pitch); Wла(p) - transfer function of the aircraft control channel along the given coordinate;, k – gain factor for the angle, rate of change and acceleration respectively; δ – deflection angle of the corresponding steering gear.

First, we examine the automatic control system of the pitch channel in the form of a continuous signal:

Consider the transfer function of the pitch channel as follows [1]:

                                                    (2)

In there:

w²=n₃₂+n₂₂n₃₃; 2d₀w₀=n₀+n₂₂+n₃₃                                                  (3)

The coefficients n_ik present in the equations are known functions of time. For short time intervals not exceeding the aerodynamic time constant t_a by more than one order of magnitude, they can be considered constants.

For a light aircraft and a flight altitude of 11 km, the aerodynamic constant , We obtain the following values ​​of the coefficients [1]:

Từ công thức (3) ta có:

w²=n^’₃₂+n’₂₂n’_{33 =} 3.039; 2d₀w₀=n’₀+n’₂₂+n’₃₃ = 1.382     

Amplification factor for angle:

                                                        (4)

With the calculated values ​​of the coefficients, the transfer function (2) has the following form:

                                        (5)

For a continuous system (5), the transient characteristics obtained by investigating on VisSim software [2, 3, 4] are shown in Fig. 2.

Figure 2. Transient characteristics in the aircraft pitch control channel

The results of the survey for continuous signals are shown in Table 1:

Table 1.

Results of the survey for continuous signals

Next, we proceed to investigate the characteristics of the system in the form of digital signals. In the study of discrete systems, the Z-transform method is often used to solve the difference equations thanks to the ability to inherit experience from modeling analog systems. When constructing the discrete transfer function, it is necessary to consider the characteristics of the output signal from the digital-analog converter, which is usually generated by a zero-order extrapolator, that is, the signal is kept constant in each sampling period. In this case, the discrete transfer function has the form:

                                          (6)

In there:  - Z-transform.

Using the MathCAD package [5], the discrete transfer function of the pitch channel can be obtained by first performing the inverse Laplace transform [6] from the transfer function W(p)/p as follows:

  (7)

Note: when using the Z-transform function of the MathCAD package, the expression for the discrete transfer function is obtained for a discrete period T = 1s. In a real system, the discrete period will be different from 1 second and the resulting expression cannot be used for T ≠ 1s. To obtain a more general expression for the discrete transfer function, suitable for all discrete periods, the Z-transform table must be used or, before performing the Z-transform in MathCAD, the time in the original expression must be multiplied by the discrete period T.

We obtain a more general expression by multiplying the time t by the discrete period T before the Z-transform in expression (6):

                         (8)

Then performing the Z-transform of expression (8), combined with expression (6) we get the resulting expression of the discrete transfer function W(z,T), valid for any discrete period T:

     (9)

We consider 3 cases with the value of discrete period T respectively as T=1s; T=0.5s; T=0.1s, then from (9), the corresponding discrete transfer function is obtained as follows:

Case T=1s:

                                 (10)

Case T=0.5s:

                               (11)

Case T=0.1s:

                                (12)

The simulation results of the transient characteristics for different values ​​of the discrete period T along the channel pitch are shown in Figure 4 and Table 2.

Figure 4. Transient characteristics in the discrete control channel for aircraft pitch

Table 2.

Simulation results of the transient characteristics for different values

The simulation results show that the quality of the digital control system depends on the discrete period T. Its increase often contributes to the increase of the transient function overshoot and may eventually lead to system instability, that is, with the same structure and system parameters, the digital signal control system is often less stable than the continuous control system. Therefore, the system modeled at T = 1 s can be considered unsatisfactory.

At T = 0.1 s, the indices of the discrete system are close to those of the analog system and in this case they indicate a quasi-continuous system.

Frequency characteristic construction of digital aircraft automatic control system

The analysis of the characteristics of digital automatic control systems can also be performed in the frequency domain according to its frequency characteristics.

From a discrete transfer function, it is possible to obtain the frequency transfer function of a discrete system, and then use it to construct the frequency characteristics of the frequency response and phase response. These are constructed depending on the so-called pseudo-frequency, denoted by λ.

The discrete frequency transfer function W(λ) is constructed using a special bilinear transformation, where in the discrete transfer function z is replaced by the expression:

                                                       (13)

And:

                                                     (14)

And its logarithmic frequency characteristic is calculated by the formula:

.                                                        (15)

In the cases T=1s, T=0.5s, T=0.1s, with frequency transfer function  , has the corresponding frequency characteristics , , .

Similarly, the initial continuous system frequency transfer function has the form:

                                  (16)

Its frequency characteristics :

.                        (17)

Figure 5 shows the frequency response of a continuous system and the frequency response of a discrete system for three different discrete periods.

Figure 5. Amplitude-frequency characteristics of the continuous and discrete automatic control systems for aircraft pitch

It can be seen from the plots that the frequency response of the discrete system A2(λ) at T=0.1 s practically coincides with the frequency response of the continuous system Q(ω) over the entire frequency range. While with increasing discrete period the system properties, starting from the middle frequency (in this case, at λ > 2-3 rad/s), deteriorate, the gain increases, leading to an increase in overshoot and the duration of the transient.

Conclusion

The simulation results show that the quality of the digital control system depends on the discrete period T. In the aircraft automatic control system, when applying the digital control system, the stability of the control system is only guaranteed when the discrete period T is small. When the discrete period is small, the digital automatic control system basically guarantees the same as the continuous automatic control system.

References:

1. V.I. Petunin, "Các hệ thống điều khiển tự động máy bay", - Ufa, 2006. - 45 tr. (В.И. Петунин, “Системы автоматического управления летательными аппаратами”. – Уфа, 2006. – 45 с.).
2. Automatic control theory textbook, Hanoi University of Technology, link: https://www.studocu.com/vn/document/truong-dai-hoc-cong-nghiep-ha-noi/ly-thuyet-o-to/giao-trinh-ltdktd-v1-ly-thuyet-dieu-khien-tu-dong/18427341
3. Modeling and analysis of automatic control systems in the VisSim program (Моделирование и анализ систем автоматического управления в программе VisSim),link: http://istasgsv.ru/docs/gsv/disciplini/modelsys/rab_vissim.pdf
4. Download VisSim 6.0 + Addons, link: https://en.lbsite.org/vissim-6-0-addons/
5. Engineer Huynh Vuong Thu Minh, "MATHCAD TEXTBOOK", link: https://voer.edu.vn/c/giao-trinh-mathcad/dc3f6c61
6. Laplace transform, link: https://math.semestr.ru/tau/laplas.php