An Extended Sliding Mode Observer for Speed , Position and Torque Sensorless Control for PMSM Drive Based Stator Resistance Estimator

This paper presents a robust sixth-order Discrete-time Extended Sliding Mode Observer (DESMO) for sensorless control of PMSM in order to estimate the currents, speed, rotor position, load torque and stator resistance. The satisfying simulation results on Simulink/Matlab environment for a 1.6 kW PMSM demonstrate the good performance and stability of the proposed ESMO algorithm against parameter variation, modeling uncertainty, measurement and system noises.


Introduction
Drive applications with PMSM are receiving more and more interest because of their better performance in dynamic and steady state responses, from their greater power density, larger torque/ampere, best efficiency, lower cost and easier maintenance [1] [2].To achieve high-performance field oriented control, accurate rotor position information, which is usually measured by rotary encoders or resolvers, is indispensable.However, the use of these sensors increases the cost, size, weight, and wiring complexity and reduces the mechanical robustness and the reliability of the overall PMSM drive systems.

Model of PMSM
By assuming that the saturation of the magnetic parts and the hysteresis phenomenon are neglected; by considering the case of a smooth-air-gap PMSM (where the inductances are equal: L d = L q ) and according to the field oriented principle where the direct axis current (I d ) is always forced to be zero which simplifies the dynamics and achieve maximum electromagnetic torque per ampere, the PMSM model in the rotor reference (d, q) frame are as follows [2] [5]: (1) shows that the PMSM dynamic model can be represented as a non-linear function of speed and stator resistance which varies with temperature.A variation of this parameter can induce, for the PMSM, a lack of field orientation, performance and stability.Thus, to preserve the reliability and robustness stability under the stator resistance variation, a robust input-output linearization via feedback control, proposed by [3] [4], is used to provide a good regulation and convergence of the currents for the PMSM drive.However, the resolution of the feedback control for the PMSM requires an on-line estimation of the speed value that is not measurable.
Thus, in order to take into account the load torque and stator resistance variations, this work uses a full sixth-order Discrete-time ESMO method to provide an on-line estimation of currents, speed, rotor position, load torque and stator resistance in a PMSM.

Discrete-Time ESMO Model
Let us consider the dynamic model of the PMSM given by the system (1).Assume that among the state variable, only the currents ( ) ( ) , , ˆ, z z are the estimates of the currents and denote ( ) ˆ, x x the estimates of the speed (Ω) and position (θ).Thus, In order to solve at the same time the problem of the load torque and stator resistance estimations in a PMSM, a six-dimensional extended state vector defined by [ ] has been introduced.Thus the proposed ESMO structure is a copy of the model ( 1), extended to the load torque and stator resistance equation, and by adding corrector gains with switching terms [8]: with where the parameters (τ, ε) present the slow variation of (T r , R s ); K is the observer gain matrices and the switching "J s " that depends on the estimated currents, is given by: ( ) ( ) Setting the estimation error dynamics is given by: The condition for convergence is verified by chosen the following observer gain matrices K 1 , K 2 , K 3 , K 4 , K 5 and K 6 : From the expression of K, it can be seen that there are three adjusting gains: (α, β and n) > 0, which play a critical role in the potential stability of the scheme with respect to stator resistance, speed and load torque estimation.These three adjusting gains must be chosen so that the estimator satisfies robustness properties, global or local stability, good accuracy and considerable rapidity.
In order to implement the ESMO algorithm in a DSP for real-time applications, the proposed extended sliding mode observer must be discretized using Euler approximation (1 st order) proposed in [11].The Discrete-time Extended Sliding Mode Observer (DESMO) should be written as: ˆê where k means the k th sampling time, i.e. t = k•T e with T e the adequate sampling period chosen without failing the stability and the accuracy of the discrete-time model.

Simulation Results and Discussion
Finally, the proposed scheme (Figure 1), a combination nonlinear feedback control and DESMO approach, is carried out for a 1.6 kW PMSM by the simulation on SIMULINK /MATLAB in order to evaluate its robustness and effectiveness in the presence of measurement noise and parameter variations.
The nominal parameters of the PMSM are given in the Table 1.The sampling period is T e = 1 ms.Two kinds of tests have been performed (with nominal and non-nominal parameters) in order to compare the behaviour of the DESMO algorithm with respect to parameter variation and the presence of about 20% noise on the simulated currents:     orientation (the current I q converges very well to zero) which is due to a favourable stator resistance estimation.Also we can see an absence or a rejection of noises on the speed, position and load torque values in the both figure cases.Furthermore A variation in load torque cannot influence on the speed/position response that remains acceptable.
All those good waveforms show that the agreement between the observation dynamic performance and the simulated ones is demonstrated.

•Figure 2
P mn = 1.6 kW U n = 220/380V f n = 0.0162 N.m.sec.rad−1 p = 3 Ω n = 1000 rpm J n = 0.0049 kg•m 2 R sn = 2.06 Ω Phif n = 0.29 Wb L qn = L dn = 9.15 mH shows the simulation results with nominal parameters for a load torque (T L = 2 N.m); • Figure 3 illustrates the results where the stator resistance varies (R s = 2.R sn ) with a load torque T L =3 N.m and a step variation in current I d (4 to 3 A).For each test, the comparative simulation and estimated results are presented.Better estimation performance yielded by the proposed DESMO is obvious from the observation results.Thus it can be seen that the estimation waves are quite similar to the simulation ones.The observed speed, position and load torque indicate the good

Figure 2 .
Figure 2. Nominal case (R s = R sn ): Comparison between estimated and simulated values for T L = 2 N.m in the presence of measurement noise.

Figure 3 .
Figure 3. Non Nominal case (R r = 1.5 × R rn ): Comparison between estimated and simulated values for T L = 3 N.m in the presence of measurement noises.

Table 1 .
Nominal parameters of the PMSM.