Abstract : This paper addresses the optimal control problem of a linear system with quadratic functional indices as performance indicators, and utilizes linear quadratic optimal control theory to achieve smooth control of a magnetic levitation system. Due to the special indices and the linear nature of the system, the dynamic optimization problem with equality constraints can obtain linear state feedback based on the Riccati equations. A corresponding mathematical model is established based on this model. Furthermore, comparisons between MATLAB simulations and PID control simulations demonstrate that this proposed scheme yields more satisfactory results.

Keywords: linear quadratic form, magnetic levitation, optimal control, MATLAB

1. Introduction

In recent years, magnetic levitation technology has developed rapidly and is being applied more and more widely. Due to its contactless nature, magnetic levitation avoids friction and wear between objects, extends the service life of equipment, and improves operating conditions, thus having broad application prospects in transportation, metallurgy, machinery, electrical appliances, materials, and other fields. Currently, international research on magnetic levitation is mainly focused on magnetic levitation trains, where progress is fastest, having moved from the experimental research stage to the test operation stage.

The optimal control problem based on linear quadratic performance indices is a method for designing control systems that was developed in the late 1950s. It combines the obtained optimal feedback control with nonlinear open-loop optimal control, which can reduce the error of open-loop control and achieve more accurate control.

This paper uses the Googol Technology magnetic levitation teaching experimental equipment as a model, and combines the general theory of linear quadratic optimal control to achieve smooth control of the magnetic levitation system. Through comparison with classical PID control, theoretical analysis, and MATLAB simulation, more satisfactory results are obtained.

2. Mathematical Model of Magnetic Levitation System

The magnetic levitation ball control system is a typical platform for studying magnetic levitation technology; it is a typical suction-float levitation system.

Its system structure diagram is shown in Figure 1. It mainly consists of components such as LED light source, electromagnet, photoelectric position sensor, power supply, amplification and compensation device, data acquisition card, and controlled object (steel ball).

Figure 1. Structural diagram of the magnetic levitation experimental system

2.1 System Working Principle

A current flowing through the windings of an electromagnet generates an electromagnetic force F. By controlling the current in the electromagnet windings to balance the electromagnetic force generated by the force against the weight mg of the steel ball, the steel ball can levitate in the air and remain in equilibrium. To achieve a stable equilibrium system, closed-loop control must be implemented to ensure the entire system is stable and has a certain degree of anti-interference capability. This system uses a non-contact measuring device composed of a light source and a photoelectric position sensor to detect changes in the distance x between the steel ball and the electromagnet. The magnitude of the control current in the electromagnet serves as the input quantity for the magnetically levitated controlled object.

2.2 Mathematical Model of the System

The model parameters of the actual system are as follows:

Since the input is directly the control current of the electromagnet, the influence of inductive reactance on the system is not considered; instead, the analysis and modeling are performed from the perspective of energy storage in inductive components. Furthermore, it is assumed that the output current of the power amplifier has a strictly linear relationship with the input voltage and that there is no delay.

The system can be described by the following equations:

After Laplace transform, we get:

From boundary equations Substituting into the equation, we obtain the open-loop transfer function of the system:

Let the input of the system object be the input voltage of the power amplifier, i.e., the control voltage, and the output of the system object be the reflected output voltage (output voltage of the sensor post-processing circuit). Then, the model of the system control object can be written as:

Based on the above, the system state variables are taken respectively The system's state equations are as follows:

Substituting the above parameters yields the following result.

As can be seen from the above, the rank of the system's state controllability matrix is ​​equal to the dimension of the system's state variables, and the rank of the system's output controllability matrix is ​​equal to the dimension of the system's output vector. Therefore, the magnetic levitation experimental system is both controllable and observable, and thus a controller can be designed to stabilize the system.

3. Linear Quadratic Optimal Control Method

3.1 Structure of a Linear Quadratic Optimal Controller

Its structural block diagram is shown in Figure 2.

Figure 2. Structure diagram of the linear quadratic optimal controller

3.2 Linear Quadratic Optimal Control Method

Let the state-space equation of a linear time-invariant system be:

in, It is an n -dimensional state vector. It is an m- dimensional control vector . yes peacekeeping 3D matrix vector.

The purpose of designing a state regulator, specifically linear quadratic optimal control, is to design... This makes the performance index of linear quadratic forms...

Minimum, where _and are weighted matrices for the state variables and input vectors, respectively. For optimal control, It is a positive semi-definite symmetric constant matrix. Let F be a positive semi-definite symmetric constant matrix, and let F be a positive definite symmetric matrix. Construct the Hamiltonian function.

Solve the Riccati matrix differential equation:

Achieving optimal control:

In the formula The state feedback coefficient matrix can be described by the following formula:

4. MATLAB Simulation of the System

The system model obtained from Figure 2 is as follows:

Now assume the optimal control law of the system is Find the feedback gain matrix. This results in the following performance metrics:

Select A program was written in the Matlab environment to simulate the design of a linear quadratic optimal controller. The value of the feedback gain matrix K of the optimal controller was obtained as: K = [10.3998 1.0042]. The simulation results of the impulse response are shown in Figure 3.

Figure 3. Simulation curves of impact response (I)

The optimal controller feedback gain matrix K is _selected as follows: K = [32.0173 1.0127]. The simulation results of the impulse response are shown in Figure 4.

Figure 4. Simulation curves of impact response (II)

Select The feedback gain matrix K of the optimal controller is obtained as: K = [100.3929 1.0394]. The simulation results of the impulse response are shown in Figure 5.

Figure 5. Simulation curves of impact response (Part III)

Comparing the simulation results, it is found that the shock response in Figure 5 requires a shorter settling time, and the amplitude change of the system is small within the settling time, which satisfies the system's smooth control well. Therefore, it can be seen that the optimal control method in this paper has well met the system requirements.

When the system uses a PID controller, the system structure is shown in Figure 6:

Figure 6 PID system structure diagram

After multiple simulations with different PID parameters, the system control performance was found to be optimal when the PID parameters were set correctly. The simulation results are shown in Figure 7.

Figure 7. Simulation curves of impact response (IV)

While the control results achieved by using a PID controller can achieve the control objective, the control performance is not ideal, and the parameter testing time is relatively long. Optimal control, by optimizing the feedback gain, reduces the time required and achieves more satisfactory control performance.

Conclusion

This paper combines the general theory of linear quadratic optimal control to achieve smooth control of a magnetic levitation system. Through theoretical analysis and MATLAB simulation, and by comparing the simulation results with those of classical PID control, it is found that the optimal control method requires a shorter settling time and exhibits smaller amplitude changes within the settling time, thus better satisfying the smooth control requirements of the system. Therefore, the optimal control method presented in this paper effectively meets the system requirements.