Adaptive fuzzy sliding mode controller design for PMLSM position control

ABSTRACT


INTRODUCTION
Linear motions of higher speed and accuracy are needed by modern control systems as equipment of manufacturing, transporting and robots. With the help of rotary motors with the mechanism of transmission, the linear motion is implemented traditionally in such a way that there is a reduction in gears and leading screws. Like mechanical transmission, there are a lot of industrial products that fail to reduce the position of linear motion along with dynamic response. On the other hand, they introduce backlash, the loads with large friction and inertia [1]. Most of the times, high speed and modern and most newly constructed semi conductors need higher accuracy motions and other equipments for manufacturing, X-Y driving gadgets, artificial hearts and robots. [2]- [4].
The permanent, magnetic synchronous and linear motions have plenty of advantages and this is the reason of their extensive use in making industrial scale robots, equipments for manufacturing, instruments for various machines, photolithographic machines etc. High controllability and higher thrust density are some of the main features of PMLSM along with responsive dynamics, lesser losses, superior dynamic working, and precision in its high positioning with the simplicity of their mechanical structure. On the other hand, the significant effects of model uncertainties and external disturbances are caused by the suppression of mechanical transmission parts. [1], [5], [6]. Hence, we can say that for PMLSM devices a significant role is played by the design of a controller and load disturbance compensator. For the improvement of PMLSM many investigative workers have created advanced control methods and they all are expected to enhance the  [1], [3], [7], [8]. In the same way, various kinds of observational algorithms are added as well to maintain the equilibrium of the external disturbance load and its evaluation. For the PMSLM control of position, some methods are highly successful such as back stepping, exact linearization and adaptive controlling methods. [3], [5].
For the higher performance of control methods and permanent magnetic linear and synchronous motor the field-oriented technique is extensively used and this technique can provide the best and similar performance, which one can get via DC machines excitedly separate. [5], [9]. For this objective to be achieved, we need to keep the value of rotor force equal to normal value in such a way that the maximum power efficiency is ensured by the optimal magnetic circuit and its contemporarily exploitation. On the other hand, the dependency of PMLSM in IFOC scheme is the bigger disadvantage. The PMLSM is dependent on motor parameter variations due to heat, level of saturation and effects on skin. [5]. Specifically speaking, the IFOC control scheme is influenced to a great extent by the primary resistance as a main parameter and due to ohm's heat, it is changing widely. [3], [10].
Based on order reduction the sliding mode control due to the characteristics of its strong robustness and rejection disturbances are obtained. These make the implementation of power converters easier and some of the electrical machines depend on its prospective control methods. [11], [12]. Since the communication is implied between the controllers, SMC is a type of variable structure scheme. Two steps are there in all the designed controllers for variable structures and these phases are sliding and reaching phases. [13], [14]. With the help of feedback control law at initial stage, the system is moved in the direction of switching surface. To maintain the sliding motion a term switching is used and in this by the choice of sliding surface, the system's dynamics are determined as well. Parametric uncertainties and load disturbances are some of the factors on which the sliding mode is independent of in terms of its motion. For position control of PMLSM the SMC is employed as well [8]. In the end, SMC has a discontinuous kind of switching feature nature and the control system is chattering is caused due to it. [11], [12], [15], [16]. To reduce or mitigate the chattering phenomenon effects, numerous approaches have been proposed [17]- [20].
Adaptive fuzzy controller with sliding mode is there in this paper. It has a power to combine the fuzzy logic and adaptive sliding mode for the position control of PMLSM. At the position, tracking objective under uncertainty parameter the suggested scheme is applied to arrive. The load thrust disturbance is observed as well in the magnetic linear permanent synchronous motor and this is done to reduce the phenomenon of chattering and problems of tracking trajectory. As the stability is ensured, the machine performance and its robustness are improved. The other parts of this investigative paper include the following.
The mathematical model is reviewed in the section 2 and various IFOC principles (indirect fieldoriented control) are studied for the permanent magnetic linearly synchronous motor. The development of suggestive mode control design for adaptive fuzzy sliding mode is demonstrated in Section 3 for the position control of PMLSM. The outcomes and discussions are explained in section 4 and some diagrams and results are mentioned as follows.

MATHEMATIC MODEL AND VECTOR CONTROL OF PMLSM
For the permanent magnetic linearly synchronous motor we present the mathematical model with the help of rotating frame in synchronous pattern d-q in the form of differential equation mentioned below: (1) to (4) [1], [3], [5].
The d-axis and q axis currents are mentioned as and . In the same way the d and q axis inductance is represented by and , d-axis and q-axis stator voltages are represented by and . The stator resistance is Rs. Polar Pitch is represented as . Maximum flux linkage due to permanent magnet in The position control in the ideal fiend-oriented case can be obtained simply by the control of axis of voltage q if d-axis current [1], [5], [9], [10]. We can describe the analogous force equation to the DC machine as follows: So, the force can be controlled through the current as in a DC machine. This strategy allows the synthesis of the control structure from the decoupled model of the PMLSM. The decoupling control method chooses the reference voltages must be.
The block diagram has shown in the Figure 1 presents the indirect field-oriented control (IFOC) [3], [6]- [8] of a permanent magnet linear synchronous motor. Figure 1. Block diagram of IFOC for a permanent magnet linear synchronous motor

THE PROPOSED FOR ADAPTIVE FUZZY-SLIDING MODE REGULATOR 3.1. Sliding mode control of PMLSM
With the sliding control mode there are some efficient and non-linear approaches which act as a strong regulator due to their structural variations. With the invariance properties of uncertainties, the system dynamics are introduced especially in the sliding mode when the dynamics of this system are controlled. [13], [14]. In the start, a sliding surface is selected for SMC design and in state variable space, the models that are desired for closed loop performances are selected. System state trajectories are forced to move ahead and the control is specially designed in this way so that the stay is possible on its surface. In the period, the trajectory of system state before even reaching to its surface of sliding is said to be the reaching phase. When the sliding phase is obtained by the trajectory of the system, the system can stay on its surface and slides to the origin of it. In the sliding model, we can see some insensitivity of the regulator device but this is not seen in the case of reaching phase. There are some uncertainties in this reaching phase and hence this phase is impacted directly by these uncertainties [7], [8], [11], [12], [15], [16], [21]. Employing a sign function often reasons chattering in practice. To solve this issue is by introduce a boundary layer around the switching surface [11], [15], [21]: eq n u u u + = (8) Where: Indicate the discontinuous control law part.
( )  sat is a saturation function and the constant  presents the thickness of the boundary layer are showed in (10).

Position and current controllers design by sliding mode control
Firstly, we define sliding surface for position control.
Taking into account the equations of the PMLSM, the time derivative of the position sliding surface is.
We take: , the equivalent control action can be defined as (14).
And the discontinuous regulator law written as (15), We choose conditions ( 0 ) to prove the system stability, it is adequate to select the gain q k strictly positive and sufficiently large.

Design of sliding mode current controllers
In this design we used two sliding mode controllers to regulate the d-axis and q-axis secondary currents respectively. The design consists of two steps. One, we define sliding surfaces   (1) and (2), the reference voltages control laws can be writing as (19), (20) and (21).
We take: In the sliding mode and at steady-state conditions, we have , , and the equivalent control actions presents as (22) and (23).
The Figure 2 presents the block diagram of PMLSM control using sliding mode control.

Adaptive fuzzy-based sliding mode control
The disadvantage of sliding mode controllers is that the discontinuous control signal produces chattering dynamics. In order to eliminate the chattering phenomenon, different schemes have been proposed in the literature [13], [14], [17]- [19], [22]- [24]. The most known solutions are those which propose to replace the signum function by a saturation function or a hyperbolic tangent [18]. However, we must find a compromise between the tracking error and the thickness of the boundary layer [6] , [20], [25], [26]. In order to overcome the disadvantages of the SMC controller and to ensure good responses in closed loop control with high possible attenuation of the chattering phenomenon, we propose in this section a combination approach between the Fuzzy logic and SMC. The new adaptive SMC controller based on fuzzy logic has the same control law as the classical SMC, but the parameters k and  of the switching control given in (9) are adjusted by a fuzzy logic inference system. The adaptive fuzzy sliding mode controller works in such way that when s and s  are big, k and  take a high values, when s and s  are small, k and  take a low values.
The scheme of the proposed adaptive sliding mode using fuzzy logic for PMLSM position control is showns in Figure. 3. The fuzzy logic controller works as an adaptive system of the switching control part of the SMC. In the proposed fuzzy-SMC scheme, the sliding surface (s) and its time derivative s  form the input space of the fuzzy logic adapter whereas the adaptive gain  represents the output of the fuzzy adapter. Therefore, the discontinuous control law of (9). The load disturbance is rejected by the fuzzy sliding controller mode with error that is negligibly called steady state error. The clear observation is seen in the suggested Adaptive Fuzzy-SMC controller offers favorable tracking results with respect to the application of external force, response time and overshoot, we can also see that the chattering phenomen on is very reduced in the control effort as a function of the adjustment of the limit band values and the gain of the discontinuous command while keeping the stability of the system and the setting precision.
The time variation of the switching control law parameters (  Figure 9, it can be seen that perfect control performances are given by the suggested schemes and this is how the robustness of a controller can be judged and the chattering phenomenon is highly reduced while ensuring stability, robustness and improved machine performance. Figure 10 shows a comparison between the classical Siding and Adaptive Fuzzy Sliding mode regulator. It shows clearly that the Fuzzy Sliding mode control provide a good control performance compared to the classical SMC and the phenomenon of chattering is very reduced by one way remarkably. The load disturnances are rejected by the FSMC adaptive controllers and no overshoot is seen rapidly with the error state. In Figure 11, the classical sliding mode controller is shown:    Figure 11. Comparison between classical SMC and adaptive FSMC mode position control of PMLSM

CONCLUSION
There were some disadvantages of SMC controllers, and it has been proven in this paper that they hybrid fuzzy controller and SMC controller can be applied on SMC controllers to overcome all these disadvantages. In the initial state, an adaptive SMC is obtained on the basis of fuzzy logical tuning. The designing and implementing the adaptive FSMC is attested successfully in this current study for the PMSLM position control. The perfect performances are seen in the schemes suggested as there are some key benefits of using it such as there is a minimal time to rise, no overshoot, better disorder rebuffing, decoupling in excellent form and due to all these factors, the suggested scheme works efficiently for the disturbances in external forces that are timely variable. Higher performance and controller robustness is obtained by the outcomes of different simulation. A dynamic response is obtained by the position control and there is no zero-state steady error and no overshoot as well.
In the end, there is a significant decline in the chattering performance. This also ensures the improved and enhanced performance of machines, stability and robustness with the help of FSMC suggested adaptive technique in contrast to the traditional and classical control of the sliding mode.