Position Sensorless Control of a Permanent Magnet Stepping Motor Based on Sliding Mode Observer

Abstract This paper deals with a position sensorless control of a permanent magnet stepping motor. The proposed controller is designed to reduce the studied actuator oscillations and overshoot which can induce an erratic working. As the mechanical solutions are expensive and cumbersome, the control ones are considered to be the most suitable. Closed loop control of these motors offers significant performance advantages. But, the use of position sensors complicates the drive design and its control system. To avoid the use of position sensors which are costly and cumbersome, we propose in this paper a sliding mode observer allowing the estimation of the motor position from the statoric voltages and currents. It is proven that a sliding mode observer is robust against observed systems parametric uncertainties and well adapted to a large variety of nonlinear systems such as electrical motors. The performances of the sensorless position proportional derivative controller and the robustness of the sliding mode observer are tested by numerical simulations. The obtained results show the efficiency of the proposed control law.


Introduction
A stepping motor is an electromechanical actuator which produces incremental movements called steps. It is known by its positioning accuracy. Moreover, it is an actuator designed to operate in open-loop motion control [1,2].
The stepping motor position evolution presents some oscillations and a long settling time which can disturb some applications requiring high positioning precision. Moreover, the oscillations can induce the motor, at some control frequencies, to an erratic working.
The mechanical solutions consist into connecting a mechanical reductor to the motor which is often costly or increasing friction at the expense of machine output.
The control solutions, more flexible than the mechanical ones, consist of closed-loop control solutions which require position sensor considered to be cumbersome and expensive [3]. Consequently, for the case of the stepping motors it is interesting to apply a position sensorless control.
In this paper, a position sensorless control based on a proportional derivative control is proposed to reduce the position oscillations of a stepping motor. The position of the studied actuator is estimated by a sliding mode observer. This paper is organized as follows. In section 2, the studied stepping motor model is presented. The proposed Sliding Mode Observer is designed in section 3. In section 4, the proposed position sensorless stepping motor Proportional Derivative control is described.

Studied Stepping Motor Model
The studied bipolar permanent magnet stepping motor is described by the following differential nonlinear dynamical equations [1,4]: The application of the DQ transformation allows the transformation of the stationary coordinate system to the rotating one.
For the stationary voltages and currents, such transformation is defined by the following two expressions [4]: The application of the DQ transformation yields to the following expressions: The studied motor is characterized by the following electrical and mechanical parameters: R 18 = Ω ; L 12 2 mH , = ; p 12 = ;

Proposed Sliding Mode Observer
Several methods of observer design have been proposed for different types of nonlinear models. Moreover, there are no systematic methods for designing a nonlinear observer. Sliding mode observers are known to be robust nonlinear observers [8][9][10][11][12]. In order to avoid the use of a position sensor a sliding mode observer allowing the estimation of the motor position and speed is proposed in the following.

Principle
With reference to the stepping motor model (1) and considering the stator currents as the system outputs, the dynamics of the proposed sliding mode observer can be constructed as follows [13]: where ∧ denotes estimated value and 1 K , 2 K and 3 K are the observer gains which are determined by Lyapunov stability analysis.
Let the error difference between real state and the observer estimate defined as follows: For a good estimation errors tend toward zeros. From (5) and (6), the estimation errors dynamics of the sliding mode observer can be expressed as: The sliding surface is defined as follows: Consider the following positive definite Lyapunov function candidate [14,15]: From equation (7) and equation (9) the time derivative of V can be expressed as: If the derivative of the chosen Lyapunov function is definite negative, the estimated state space variables converges asymptotically to the real ones [9,10]. where:  We can also establish the following inequalities: By considering the inequalities (11), (12), (13), (14) and (15) As V  must be negative definite to insure the convergence of the estimated space variables to the real ones, it is sufficient to choose the gains 1 K , 2 K and 3 K so that:

Proposed Observer Validation
The proposed sliding mode observer is tested by numerical simulations. The first test is carried out by application of the nominal voltage to the motor. Thus the rotor moves with a step of 0.13 rad which is the studied motor elementary incremental movement. The obtained results are presented in Fig. 1. Fig.1-a and Fig. 1-b presents respectively the real and estimated rotor positions.
We can deduce that the observer allows the good estimation of the motor position and speed from the speed error estimation and the position error estimation Fig. 1-c and Fig.1

Principle
In order to apply a PD control to eliminate the position oscillations, we propose to simplify the mathematical model of the studied stepping motor by considering a current control. For such case, the electrical equations are expressed as follows: Thus, by neglecting the dry friction, the mechanical equations describing the slow motor dynamic are given by the following expressions: The application of the DQ transformation to the eq. (19) leads to the following linear equation: From the eq. (20) we can define the following transfer function: The PD-controller transfer function is expressed as follows: where: p K and d K are respectively the proportional and integral gains, ref θ is the desired reference position.
The closed loop control system of model (21) can then be written as: The determination of the p K and d K parameters can be done by a pole placement technique.
To avoid the use of a position sensor the PD control will be associated with the proposed sliding mode observer, fig.  4.

Conclusion
This paper presents a position senseless PD control for a permanent magnet stepping motor. The designed control is based on a sliding mode observer allowing the estimation of the motor position, speed and currents.
The proposed control approach has the advantage of not requiring a position sensor which is cumbersome and expensive. The simulation results prove the efficiency and the robustness of the sliding mode observer. Moreover, when the PD control is associated to the sliding mode observer it allows the elimination of the motor oscillations.