Design of a vehicle motion control system based on a PID controller

Foreword

With the continuous development of science, technology, and the economy, automobiles have become an indispensable means of transportation in people's daily lives. The automotive industry plays a significant role in the contemporary world economy, being one of the world's largest and most important industrial sectors and a pillar industry for most countries. The level of automotive industrial development and the average number of cars owned per household have become indicators of a country's industrial development. However, at the same time, the rapid increase in demand for petroleum resources and the serious negative impact on the environment brought about by the development of the automotive industry have increasingly attracted people's attention. Research on vehicle motion performance generally starts with handling, stability, ride comfort, and handling characteristics. However, in recent years, with the increasing complexity of transportation systems, consideration of road conditions and other factors has begun to be emphasized in vehicle motion performance. Therefore, research on vehicle motion control systems has become particularly important.

In the field of modern control engineering, MATLAB is the most popular computer-aided design and teaching software. It is a general-purpose programming language for scientific computing, graphical interactive systems, and control system simulation. It can perform calculations and graphical displays in several areas, including numerical analysis, optimization, statistics, automatic control, signal and image processing. It is highly suitable for computer-aided design in modern control theory. MATLAB also provides a series of control statements whose syntax and usage rules are similar to high-level languages ​​such as FORTRAN and C, but are more concise. It has become the most popular computer-aided design and teaching software in the international control community, possessing unparalleled advantages over other languages ​​in scientific and engineering computing.

With the continuous development of computer technology, electronic technology, and motor magnetic materials, servo control has gradually become the mainstream of motion control in factory automation. With the development of modern AC speed control technology, various new control algorithms have emerged, such as adaptive control, expert systems, and intelligent control. Theoretically, many control strategies can achieve good dynamic and static characteristics of motors; however, due to the complexity of the algorithms themselves and the difficulty in model identification of the system, implementation in practical systems is challenging. For traditional PID controllers, the greatest advantages of positional PID controllers lie in their simple algorithm, easy parameter tuning, strong robustness, high adaptability, and high reliability. These characteristics have led to the widespread application of positional PID controllers in industrial control.

1. Vehicle Motion Control Analysis

If we ignore the rotational inertia of the wheels during the car's movement, and assume that the magnitude of the frictional resistance experienced by the car is proportional to its speed and its direction is opposite to the direction of the car's motion, then the system can be simplified into a simple mass damping system. Figure 1 shows the car operation control system.

Figure 1. Schematic diagram of car motion

2. Establishment of motion control system model

Figure 2. Structure diagram of the motion control system

3. Design of PID controller for motion control system

A traditional PID controller mainly consists of three parts: a proportional element, an integral element, and a derivative element. It is also the most widely used controller in the control field. The mathematical model of a traditional PID controller can be expressed as:

Where: u(t) is the controller output; Kp is the proportional gain; e(t) is the controller input deviation signal; Ti is the controller integral response time; and Td is the controller derivative response time.

Traditional PID controllers mostly use data sampling for control. The signal entering the controller must be sampled and quantized into a digital quantity that the controller can recognize before it can enter the controller's registers and memory. During data processing and calculation, the controller can only use numerical approximation methods. Therefore, in digital PID controllers, to achieve the PID control law, a numerical approximation method must be used. When the data sampling period is short, summation is used instead of integral action, and difference quotient is used instead of derivative action, discretizing the PID control algorithm. This is the digital PID position control algorithm, which can be expressed as follows:

In the formula: u(k) is the output of the PID at sampling period k; e(k) is the system deviation at sampling period k; Ts is the system sampling period.

This is a typical negative feedback structure based on deviation control, where e is the deviation, i.e., the difference between the output and the setpoint; u is the control quantity, which acts on the controlled object and causes a change in the output. Kp is the proportional coefficient, whose control effect is to reduce the rise time and static error of the corresponding curve, but it cannot eliminate the static error. Therefore, simple P correction is a differential control and is generally not used alone. Ki is the integral gain coefficient, whose control effect is to eliminate the static error. I is zero-error control, but it will prolong the transient response time, so it is generally not used alone. Kd is the integral gain system, whose control effect is to enhance the stability of the system, reduce the transient response time, and reduce the overshoot. The relationship between Kp, Ki, Kd and the system time-domain performance indicators is shown in Table 1 below.

Table 1. Relationship between PID control parameters and system time-domain performance indicators

This table illustrates the changes in system performance metrics as PID parameters increase. However, the relationships between these parameters and performance metrics are not absolute; they represent relative relationships within a certain range. This is because the parameters influence each other; a change in one parameter will alter the control effect between the other two.

4. PID calibrator

Starting from the initial state of the system, based on the step response curve, using the principle of series compensation, and the influence of parameter changes on the system response, we conduct a detailed analysis of the static and dynamic performance indicators, and finally design a control system that meets our needs.

4.1 Uncorrected system step response

Based on the preceding analysis, the transfer function of the system without any correction circuitry was obtained by plotting the original step response curve of the system using MATLAB scripting language, as shown in Figure 3.

Figure 3 shows the step response curve of the system without correction.

As shown in Figure 3, the open-loop response curve of the system does not oscillate, indicating overdamped behavior. Such curves generally have slow response times. Indeed, the figure and program show that the system's rise time is approximately 100 seconds, and the steady-state error reaches 98%, far from meeting the requirement of following the setpoint. This is because the constant term in the denominator of the system's transfer function is 50, meaning the DC component gain is 1/50. Therefore, as time approaches infinity and the angular frequency approaches zero, the steady-state value of the system equals 1/50 = 0.02. To significantly reduce the system's steady-state error and simultaneously decrease the rise time, we aim for satisfactory performance across all aspects of the system. This necessitates a comprehensive proportional-integral-derivative (PID) correction, i.e., employing a typical PID controller.

4.2 Design of PID Correction Device

For the engineering control system described in this paper, PID correction generally yields satisfactory control results. The closed-loop transfer function of the system is then:

The selection of Kp, Ki, and Kd is generally based on experience to determine a rough range first, and then gradually corrected using graphs generated by MATLAB. Here we choose Kp=700, Ki=100, and Kd=100. The program code is as follows:

holdon

Kp=700;

Ki=100;

Kd=100;

num=[KdKpKd];

den=[m+Kdb+kpKi];

disp('The closed-loop transfer function after PID correction is:')

printsys(num, den);

t=0:0.01:50;

step(u*num,den,t);

axis([011050]);

title('SyetemStepResponseafterPID

CRRECTION');

xlabel('Time-sec');

ylabel('Response-value');

gridon;

text(25,9.5,'Kp=700Ki=100Kd=100');

Maxpid = max(c);

disp('PID overshoot is:')

Mppid=(Maxpid-10)/10

The closed-loop step response of the system after adding PID correction is shown in Figure 4. From Figure 4 and the program execution results, it is clear that the static and dynamic performance indicators of the system have well met the design requirements. The rise time is less than 5s, and the overshoot is less than 8%, approximately 6.67. Specific values ​​can be calculated by the program.

Figure 4. Closed-loop step response curve of the system after PID correction.

5. Conclusion

This design demonstrates that PID control is quite effective for general control systems, requiring minimal analysis of the controlled object's mechanism. Design can be based solely on the parameter characteristics of Kp, Ki, and Kd, and the step response curve plotted using MATLAB. Within the MATLAB environment, we can select PID parameters based on simulation curves. By considering the system's performance indicators and experience with basic tuning parameters, we can simulate different PID parameters and ultimately determine the optimal settings. This approach is intuitive, computationally efficient, and easy to adjust.

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