Synthesis of control laws for magnetic levitation systems based on serial invariant manifolds

ABSTRACT


INTRODUCTION
A magnetic levitation system (MLS) is an electromagnetic device that lifts and suspends ferromagnetic objects using electromagnetic principles. MLS technology eliminates mechanical contact between moving and stationary parts thereby reducing friction. MLS brings many advantages such as low noise, the ability to work in a high vacuum environment, high-precision positioning platform. MLS typically operates on three types of forces: lift, thrust, and drag [1]. Therefore, MLS technology is used in many drivetrains: high-speed trains, magnetic bearings, semiconductor technology, precision positioning, magnetic suspension and non-contact haptic interactions, non-pollution, and multi-directional degrees of freedom (DOF) [2]- [4]. Due to the high nonlinearity and existence of many model parameters of MLS, MLS is a standard object to study and test control methods and control laws.
Currently, there is a lot of research on MLS. In order to improve the quality of control. Liu et al. [5] and Pradhan and Singh [6] used the classical control algorithms proportional-integral-derivative (PID) and proportional-derivative (PD) to stabilize MLS. However, the PID controller is based on the error, so the control quality for high nonlinear systems is not good. In [7], a serial multilayer neural network is used to model the system in which learning and control are performed concurrently. In addition, the adaptive controllers studied in [8] have good results. Pradhan and Singh [6] presented the optimization of the PD controller using a bat-swarm algorithm for the MLS system. However, tuning the control parameters of these algorithms is not simple to apply in real-time systems. Besides, the uncertain components from the mathematical model of the system as well as the deviation of the measuring tools affect the quality of the controller. A fuzzy logic controller for MLS is presented in [9]. In [10], quasi-time optimal controllers based on dynamic adaptation are proposed and tested. However, these controllers suffer from the limitation of being very sensitive to the initial conditions, as well as to noise effects and changes in the parameters of the system. In [11]- [14], the sliding control method and its variations are used to stabilize MLS. The obvious disadvantage of a sliding mode controller (SMC) is that the control signals are discontinuous and have a similar shape to a bipolar square wave with high bias switching frequency. These studies also try to overcome these disadvantages by changing the switching signal form and adapting to the parameters of the sliding surface.
In this paper, a model and controller are designed based on synergetic control theory (SCT) with sequential asymptotic stable serial invariant manifolds presented by Kolecnikov et al. [15]- [21]. The quality of the proposed control law is illustrated by simulation results. The responses of the system are compared with the sliding mode controller to prove the effectiveness of the proposed control law.

MATHEMATICAL MODELS OF THE MLS
The model of MLS is shown in Figure 1, where the gravity P has a downward direction, U is the control input, changed to control the electromagnetic force F to raise or lower the ball a xb distance from the electromagnet. The x distance between the ball and the electromagnet is the output control. The distance between the ball and the electromagnet is determined by the Hall effect sensor. The inductance of the coil is a function of the position of the ball, determined as (2) [10], where L1 is a parameter of the system. Set the state variable 1 = , 2 = , 3 = , the state equation of system (1) is written as (3).
The control objective is to ensure the position of the ball to the desired position when the preset value xsp is changed.

SYNTHESIS OF SYNERGETIC CONTROL LAW FOR MLS USING SERIAL INVARIANT MANIFOLD 3.1. Diagram of control structure for MLS
To control the balance of the ball at a given position, the diagram of the control structure of MLS is shown in Figure 2. The controller gives a voltage u acting on the inductor in MLS from signals about the position, velocity, and current in the electromagnet.

Synthesis of synergetic control law based on serial invariant manifold
The purpose of the control problem is to keep the magnetized object stable at a specified position by varying the voltage supplied to the magnetic coil. From the point of view of SCT, this means that it is necessary to synthesize the control signal ( 1 , 2 , 3 ), a function that depends on the phase coordinates. The control signal will move the magnetized object from the initial position (within the controllable value range) to the final state according to the set value, when it satisfies the required quality criteria [15]- [21].
Since the control problem of MLS is to stabilize the magnetized object in the desired position, we introduce the first invariant technology corresponding to the control goal: In the first step, based on reality and mathematical model, when the control signal u changes, it will affect the change of the current 3 , so the first manifold is selected in the form of (5).

SYNERGETIC CONTROL LAW
In the manifold (7) containing the function 1 ( 1 , 2 ), this function determines the desired characteristics of the change in current 3 at the intersection with the invariant manifold 1 = 0. The function 1 ( 1 , 2 ) is determined in the process of synthesizing the control law, derived from the invariant condition (4).
According to the method of analytical design of aggregated regulators (ADAR) [16]- [18], the macro variable 1 must satisfy the root of the basic functional equation: where 1 > 0 ensures the asymptotic stability of the system motion. Substitute (5) into (6): Substitute ̇3 in the state (3): From (8) we find the control law u of the form: When the system enters the manifold, the performance point of the system touches the intersection of the 1 = 0 manifold, then the system will have a dynamic decomposition of the system (3) and the dynamics of the closed loop system are described by (10).
The function 1 ( 1 , 2 ) in the decomposition system (7) can be thought of as an internal control signal.
In the second step of the synthesis, to search for the control and to determine the function 1 ( 1 , 2 ), an additional invariant manifold is introduced, which will ensure the stability of the closed-loop system and the response of invariant technology (4). We choose a second manifold: The system dynamics on this manifold can be rewritten as: From the dynamics equation (9) the relative stability condition of the system at 1 = is < 0. To satisfy the condition 2 = 0, macro variable 2 must satisfy the root of the equation: where T2>0 is the condition for asymptotic stability of the motion of the system with the invariant manifold. Substitute (11) into (13) to find the internal control signal 1 ( 1 , 2 ).
Furthermore, the equations of the decomposition system (10) are substituted into (14), giving the expression as in (15).
From (15), we find the internal control signal 1 ( 1 , 2 ): Gravity pulls the magnetic object down, so to keep the object in the desired position it is necessary to supply the electromagnet with a voltage of a fixed direction. In this paper, a positive control signal value is selected and creates upward force. The desired control law is found as a common root of the (9) and (16) and has the following form: (18):

Design of sliding mode controller for MLS Represent the (3) in the form
where = . Then, it is necessary to determine the control voltage so that → ,̇,̈→ 0 when → ∞ to → as the initial control target.
Set the output of the system: From (18) and (19), the relationship between the output and input of the system in the initial coordinates is established as (20).
Select the sliding surface of the controller as (21).
With the coefficients a1, a0 is chosen so that the characteristic equation 2 + 1 + 0 = 0 is a Hurwitz polynomial. Then, the sliding mode control law is defined as follows: where K is a positive constant. To avoid chattering, use the sat function instead of the sign function in the control law (22).
The simulation results of the sliding mode controller and the synergetic controller with the ladder signal are shown in Figures 3 to 6. The efficiency of the position response of the magnetized object is shown in Figure 3. It can be seen that the response of SMC gives a poorer response. The position response of STC is no oscillation occurs and no overshoot when changing the set value. The SMC gives good results, but the overshoots when the set value changes are 6%, 3.5%, and the oscillations occur in the first stage. The velocity response of STC is better when the maximum amplitude is smaller and no oscillation occurs as shown in Figure 4. The variation of the control input u and the current intensity is shown in Figure 5 and Figure 6, where control input u is always within the limits and the time response is faster when using STC.  Figure 7, the position response of STC law is better than SMC at the amplitude of oscillation, the time to reach and the position error at the peaks of the set value. Figure 8 shows the results of the velocity of the magnetized object. It is clear that the amplitude of oscillation when using STC is much smaller when using SMC and the stabilization time is also smaller. The current intensity and control voltage of both control laws are within the original technical limitation. Therefore, the simulation results show that the system can have better control quality by using the STC and partial sequential invariant manifolds with fast response frequency (electrical part) to stabilize the slower response frequency part (mechanical part).

CONCLUSION
On the basis of the nonlinear model of the magnetic levitation system, using the method of synergetic control theory with serial invariant manifolds, the control law has been synthesized to ensure the motion of the object to the desired position for ahead or follow a desired trajectory. An important advantage of synergetic nonlinear control laws is the ability to obtain control laws in analytical form, by solving macro variables and functional equations to ensure their asymptotic stability. Analysis of transient characteristics of the nonlinear controller, synthesized by SCT and SMC, allows the authors to conclude that SCT is performed more efficiently, with no unwanted output overshoot, better response time, fewer oscillations, and smaller tracking error. Further studies will consider the dynamics of the manifolds to choose the law of the optimal manifold for different objects.