Generalized proportional integral disturbance observer-based fuzzy sliding mode control for active magnetic bearing system

This paper presents a generalized proportional integral disturbance observer-based a combination of a fuzzy logic control and sliding mode control for the suspension active magnetic bearing system. Firstly, a generalized disturbance observer based on proportional and integral is built to estimate the parameters variation and the outside disturbances. In order to guarantee the system-state will be stabilized on pre-defined states, a disturbance feedback is constructed with a low pass filter. Subsequently, the proportional integral derivative surface sliding mode control is designed. Finally, the Fuzzy logic controller is constructed to determine the hitting controller gain. The stability analysis is given following the Lyapunov law. The proposed control structure can guarantee the system transient response quite good, no overshoot value, and settling time is quite narrow. Two testing cases of the proposed controller with and without the generalized disturbance observe is compared.


Introduction
The active magnetic bearing system offer a noncontact working between the stator and rotor [1][2][3][4]. However, the active magnetic bearing system is a highly unstable system. In order to control this system previous published paper as our research [3] proposed the fuzzy sliding mode control for the suspension active magnetic bearing system. The fuzzy is used to regulate the controller current value, this research given good chattering rejection method, with small overshoot, narrow settling time, freechattering value, but the research ignored the outside disturbance and uncertainty values. Chen et al. [5] proposed the neural network to approximate the uncertainty value of the active magnetic bearing system, their proposed method is very good at tracking with non-periodic trapezoidal and periodic trapezoidal signals. Zad et al. [6] presented a design and adaptive sliding-mode control of hybrid magnetic bearings, the results of their paper are good but the settling time still high. Due to the sensitive with the output disturbance of the suspension active magnetic bearing system, this paper uses a general proportional integral to estimate the output disturbance and inside uncertainty values. The convergence of the system state is guaranteed by a low-pass-filter. The concept of disturbance observer was presented and has been developed by many researchers. Kim et al. [7] said that the disturbance is estimated by a general disturbance observer, the original work of their paper is based on the concept of the friction observer. Chen et al. [8] applied the general proportional integral observerbased composition control method for robotic thermal tactile sensor with disturbances, their paper is well achieved with the regulation temperature of the robotic thermal tactile under unknown time varying disturbances. In [9] Wang et al. present the same method to estimate the output disturbance for IOP Publishing doi:10.1088/1757-899X/1113/1/012006 2 the induction motor. The results of their control method are very good at reject the output disturbance and parameter uncertainty. This paper proposes the sliding mode control with a proportional-integralderivative surface is used to construct the controller for the active magnetic bearing system.
The concept of the sliding mode was initially developed in mid-1950s. In [10] Utkin presented the concept and the stability of a sliding mode controller. Sliding mode control is a type of the variable structure controller [11]. The stability of system should be satisfied the Lyapunov condition. The control signals of sliding mode controller are consisted the equivalent value and the switching value. The switching control value is used to force the system-state converge on the pre-defined surface, and the equivalent control signal is used to stabilize the system-state on that surface. In order to reduce the chattering value of the sliding mode control, this paper use the saturation function to replace the sign function of the switching control part. The fuzzy output is used to approximate the hitting control gain of the switching control part. The archived results will be shown in the subsequent part.

Mathematical modeling of the SAMB system
The control structure of the suspension active magnetic bearing system is included two opposite poles, one rotor, and one embedded thrust disk, one computer includes a multi channels analog to digital card with 16 bits revolution. Multi channels digital to analog card with 16 bits revolution, an eddy-current position sensor, and an amplifiers 0.5 A/V. The details of the system are presented as Fig. 1 below.
where F  and F  are the upper and lower forces of upper and lower the magnet coils, respectively. K is the coefficient of the force values. 0 i and x i are the initial current and controlled current values.
Using Taylor expansion of the Eq. (1) leads to With , k p k i are the amplification factors of the rotor position and the magnet coil currents, respectively. Where Follows the Newton II law: where m is the mass of the inside rotor and an embedded thrust disk, F x is the Lorentz force, and is the unexpected output disturbance. The Eq. (3) can be written as Then the Eq. (4) can be re-written as n and C n are known system state matrixs. The values of are unknown parameters, it represents the uncertainties values of the system and parameter variation. the system model is represented as The Eq. (6) can be written as upper boundary of the unknown disturbance of the system. The system parameters are shown in Table  1 below.

Proposed approach
A highly unstable like the suspension active magnetic bearing system could be provided a robust controller. The disturbance observer can be used to reject the outside disturbance and system

Generalized proportional integral disturbance observer
In facts, the disturbance rejection is one of the most importance thing to estimate the controller performance. This paper presents a GPIDO as follows the a m are unknown constant values. The observer system mathematical model is described as The estimate disturbance value is asymptomatically tracking the lump of disturbance and uncertainty when the disturbance gain is resulted the ( For reducing the noise of high frequency term, this study used the low-pass-filter gain to estimate disturbance value. The low-pass-filter is A suitable low-pass-filter time constant need to be chosen.

The stability of proposed observer
In this section the detail analysis of the stability of the disturbance estimation will be revealed. By denoting ˆ, d d d  then the derivative of the estimation disturbance error is are Hurwitz. A filter with approximate bandwidth as a low-pass-filter is used to guarantee the disturbance error value goes to zero. The stability is proved as The disturbance and uncertainty are accurately approximated by The Eq. (17) then could be The transformation of the term

Sliding mode control
The sliding mode control are included an equivalent control value, and a switching control value. Every system-state will slide and converge on a pre-defined state. This paper proposes the sliding surface by   Taking the derivative of the Eq. (21) Combining Eq. (10) and Eq. (20), the current is calculated as The Eq. (22) can be written to obtain the disturbance and uncertainty value as The inequality (34) is satisfied when the hitting control gain should be defined as Then the inequality 28 could be written as The hitting control gain should by approximate for the chattering as small as possible. The paper provides the fuzzy logic controller to implement the work. Fuzzy logic is another practical mathematical addition to classic Boolean logic [12]. The Fuzzy rules were chosen somehow the output of the fuzzy is satisfied the Lyapunouv condition. The hitting control gain is output of the fuzzy system. The sliding mode stability provides (t) (t) (t)

Fuzzy logic control
, with f k is the fuzzy output. In case (t) 0 s  the f k increase will lead to (t) s decrease and vice versa. Otherwise, in case (t) 0 s  the  Table 2 and Fig. 2 below.   The proposed method results are given by the next part. The control parameters are in Table 3.

Proposed approach
The approximation fuzzy system used centroi-weight defuzzification to operate the fuzzy system. The stability of the system now is combination of Eqs. (24) and (36) with a hitting control gain is determined by .
k f With a case of testing the outside disturbance. This section brief describes the effectiveness of the proposed control method by these performances. An illustrative example of distance tracking response is most important impact, their properties is including the settling time, overshot value, and steady state, the disturbance response is important also.

The Fuzzy sliding mode control performance
This section gives some result of the case without generalized proportional integral disturbance observer, a disturbance signal is given from outside to test the performance of the proposed method was implemented. The given output signal determined that the proposed control method is good at tracking the flexible signal. The uncertainty and disturbance are rejected, but not completely rejected the output disturbance signal. The given output are shown in Fig. 4 below.  10 This section will present the performance of the GPI disturbance observer-based fuzzy sliding mode control for the active magnetic bearing system. The outside disturbance is completely rejected by a feedback gain. The results are represented as The given output signals figure out that the performance of the proposed method is quite good at tracking the sinusoidal reference signal, the settling time just approximately 3ms in case of no disturbance observer, and 2ms in case of disturbance observer is equipped, respectively. There are no overshoot values in both case, and the average distance tracking error values are equal to 1.344um. The maximum of the distance tracking error value is approximately equalled 17.366mm and 11.662mm in case 1 and case 2, respectively. The testing disturbance is responded by a lump of disturbance observer, which means the exogenous disturbance is automatically rejected.

Proposed approach
This study used a proportional integral derivative to construct the sliding mode control surface, a fuzzy logic control is used to construct the hitting control gain. There are somehow the Lyapunov is guaranteed for system stability. A general proportional integral disturbance observer is equipped to estimate the unknown outside disturbance and unknown parameters variation. The design methodology is utilized to control the suspension active magnetic bearing system. The archived result are figured out that proposed methodology is good at tracking the dynamic input signal, the proposed controller can reject the output disturbance and the variation of the parameters. The main advantages are, the settling time of proposed controller is very small, no overshot value, and the average of the distance tracking error value is quite small. The energy of control value is strongly small. The proposed fuzzy controller is suitable for building the hitting control gain. In comparison of two cases is aimed to figure out that the proposed controller is good at rejecting the outside disturbance and system parameters variation.