Position Sensorless Control of PMSM Based on a Sliding Mode Observer

Sliding mode observer (SMO) for permanent magnet synchronous motor (PMSM) speed and rotor position is used in this paper to realize sensorless control. According to the PMSM mathematical model and the sliding mode control theory, the SMO model was given. The stability condition of SMO is studied to make sure that the observer is stable in converging to the sliding mode plane. This paper analyzes the structure and performance of the proposed SMO strategy with SIMULINK based simulation. Simulation results are presented to verify the proposed sensorless control algorithm.


Introduction
Permanent magnet synchronous machines (PMSM) have been increasingly applied for drive applications due to their simple structure, high power density, high torque to inertia ratio and high efficiency.To achieve the efficient vector control of a PMSM, knowledge of rotor position is necessary.Usually the rotor position angle is measured by a Resolver, or other absolute encoder.However, the presence of such sensors increases the cost and encumbrance of the overall system as well as reduces its robustness and reliability.Furthermore the sensors are expensive and very sensitive to the environmental constraints such as vibration and temperature.To overcome these problems, instead of using the position sensors, the sensorless control method has been developed to control the motor using the estimated values of the position and velocity of the rotor (1)(2)(3) .
Such as high-frequency injection method, extended Kalman filter, observer estimation method, flux estimation, artificial intelligence estimation method, adaptive reference control method has been extensively studied.High-frequency injection method at high speed, the back-electromotive force is too large, furthermore the speed and position detection accuracy of the rotor is deteriorated , system stability is poor; Extended Kalman filter method has a large calculate volume, the algorithm requires high carries for the chip, the stator voltage is very small Near zero speed, estimation error of the status will be affected by increases of measurement error and uncertainty of motor model; Artificial intelligence estimation method and technology issues is not mature yet, it need specialized hardware support and more difficult, therefore, it is difficult for the practical application; adaptive reference control method exist a problem how to choose the model reference adaptive rational adaptive law, to ensure system stability and robustness of the parameters at the same time improving the convergence rate of this method (4)(5)(6) .Compared with other methods, sliding mode variable structure can response fast, have good robustness and ensure that the system is asymptotically stable, at the same time its algorithm is simple and easy to achieve on project (7)(8) .
To cope with the foregoing problems, this paper presents a sensorless vector control algorithm for PMSM.The rotor position angle is estimated by a sliding mode observer.Simulation results are used to verify its validity.

Mathematical Model Of PMSM
In order to create a mathematical model of PMSM sine wave, make the following assumptions: the effect of magnetic saturation of the rotor core and stator are ignored, the hysteresis loss and eddy current of motor are ignored; Armature reaction magnetic field generated by the permanent magnet excitation field and three-phase windings in the air gap are sinusoidal distribution; Steady-state operation, the phase windings induced electromotive force waveform is a sine wave.
The current state equation of surface mounted PMSM in the synchronous rotating coordinate (d-q coordinates) of: In the equation: i d , i q , u d , u q are currents and voltages of synchronous rotating coordinate system; R, L are the stator resistance and inductance of motor; P n , ψ f are pole pairs number of motor and the rotor flux, ω r are mechanical angular velocity of the motor.
In the stationary-phase permanent magnet synchronous motor coordinates, coordinates and static two-phase model of the two-phase synchronous rotating coordinate system is shown in Figure 1.
The equation ( 1) is converted to the current state of the equation of stationary coordinate (α-β coordinates): Where: In the equation: i α , i β , u α , u β are currents and voltages of the stationary coordinate system; e α , e β are electromotive force of the stationary coordinate system; ω e , θ are rotor electrical angular velocity and electrical angle of the motor.

Design of Control System Based On Sliding Mode Observer
By equation (3) shows the back EMF contains a rotor position signal, the observer can be extracted.The sliding mode observer which is designed in this article puts the stator current under static coordinates system as input of observer, through the observation of the motor back EMF, to extract rotor speed and position information of the measurements.
According to equation (2) given a current state equation of permanent magnet synchronous motor under stationary coordinate system, based on the theory of sliding mode variable structure control, we can construct the current sliding mode observer equation:   Then, we can get current error equations as follows: The key of sliding mode variable structure control design is to control the function u(x) and design of switching surface s(x), here we choose constant switch control function u=u o sgn(s(x)) as control function, u o is taken as -l 1 , to ensure that the condition SS=0 of sliding mode reaching is established, the value of l 1 is related to the stability of the system, here's an analysis of the range of the l 1 .
By the previous conclusions, we have: Similarly, we have: Arrival condition is: That is to satisfy the 0 di i dt    From the comparison of above waveform, the rotational speed value can be estimated by using phase-locked loop, estimated speed waveform is more smooth, estimate speed value fluctuates up and down along the actual speed value, can fast track speed's changes, and the error between rotor estimation speed value and the actual value is less, and system run more stable.
Then makes the motor in the sliding mode observer model with a phase-locked loop start up with no-load at speed of 400r/min, and set the given speed is from 400 r/min up to 800 r/min at 0.  Finally makes the motor in the sliding mode observer with a phase-locked loop model start up with no-load at speed of 400r/min, the given speed is from 400 r/min up to 800 r/min at 0.1s.The simulation waveforms of speed's real and estimated values and the simulation waveforms of torque and the simulation waveforms of the air-gap magnetic field track as shown below.

Conclusions
In this paper, a sliding mode observer-based position sensorless control scheme has been presented for PMSM drives.The stability of the proposed sliding mode observer has been proved by a Lyapunov stability analysis.
The proposed system comprises a sliding mode observer and a field-oriented PI current controller for the speed control loop.The SMO is used to estimate the rotor position and speed of the PMSM due to its strong robustness.Simulation results indicate the feasibility and effectiveness of the proposed control system.

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conditions are met, so that we can guarantee the stability of the error equation.In practice, l 1 can't take too much, otherwise it will increase the chattering noise, causing unnecessary estimation error.Switching surface s(x) only to select current error value, namely   , , visible when the sliding mode motion occurs, as   0 S x  and , the equivalent   0 S x  

Fig. 5 .
Fig.5.The sensorless control system simulation model based on the sliding mode observer with phase-locked loop

Fig. 9 .
Fig.9.The speed based on the SMO and PLL control system

Fig. 10 .
Fig.10.Rotor position based on SMO control system Fig.13.The rotor position partial enlarged based on SMO and PLL control systemFrom the comparison of above waveform, the rotational speed value can be estimated by using phase-locked loop, estimated speed waveform is more smooth, estimate speed value fluctuates up and down along the actual speed value, can fast track speed's changes, and the error between rotor estimation speed value and the actual value is less, and system run more stable.Then makes the motor in the sliding mode observer model with a phase-locked loop start up with no-load at speed of 400r/min, and set the given speed is from 400 r/min up to 800 r/min at 0.1 s.The simulation waveforms of speed's real and estimated values and the simulation waveforms of counter electromotive force estimates values under static coordinates as shown in the figure below.