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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.

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 [

The goal of the research for this dissertation was to develop a rotor position/speed/load torque sensorless control system with performance comparable to the sensor-based control systems for PMSMs over their entire operating range.

The naturally structure of non-linear multivariable state of PMSM models induces the use of robust feedback linearization method [

However, this feedback control technique requires the knowledge of the instantaneous speed which is often difficult to access or not usually measurable in practice. Also parameters variations (more specifically the stator resistance and load torque variation) and noises injected by the inverter in the PMSM can induce a lack of field orientation and a state-space “coupling”, which can involve a performance degradation of the system.

In order to achieve better system dynamic performance, the approach proposed in this paper consists to design extended observers allowing an on-line estimation of speed, position, load torque and stator resistance.

The Extended Kalman Filter (EKF) presented in [

Another approach proposed in [

Thus, this paper proposes a sixth-order Discrete-time Extended Sliding Mode Observer (DESMO) to provide not only Speed/rotor position estimation but also the load torque and stator resistance reconstruction for the PMSM

After a brief review of the PMSM model, the simulation results for a 1.6 kW PMSM drive system are presented to validate the high robustness of the proposed DESMO approach against parameter variations, measurement and system noises.

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 [

with

This relation (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 [

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.

Let us consider the dynamic model of the PMSM given by the system (1). Assume that among the state variable, only the currents

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 [

with

where the parameters (t, e) 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: (a, b 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 [

with

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.

Finally, the proposed scheme (

The nominal parameters of the PMSM are given in the _{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:

P_{mn} = 1.6 kW | U_{n} = 220/380V | f_{n} = 0.0162 N.m.sec.rad^{−1} |
---|---|---|

p = 3 | W_{n} = 1000 rpm | J_{n} = 0.0049 kg∙m^{2} |

R_{sn} = 2.06 W | Phif_{n} = 0.29 Wb | L_{qn} = L_{dn} = 9.15 mH |

・ _{L} = 2 N.m);

・ _{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

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.

In this research, a robust feedback linearization strategy and a DESMO algorithm are used not only to decouple and then control independently the currents of the PMSM in a field-oriented (d, q) coordinate but also to provide the unmeasurable state variable estimation (speed, position, stator resistance and load torque). A series of simulations tests have been achieved on the PMSM. The results obtained have demonstrated a good performance of this robust decoupling controland DESMO algorithm against stator resistance variations, measured noise and load torque.

Thus, in the industrial applications, one will appreciate very well the experimental implement of this robust estimator for the reconstitution of the speed, position and the torque as well as the stator resistance.

PierreTety,AdamaKonaté,OlivierAsseu,EtienneSoro,PamelaYoboué, (2016) An Extended Sliding Mode Observer for Speed, Position and Torque Sensorless Control for PMSM Drive Based Stator Resistance Estimator. Intelligent Control and Automation,07,1-8. doi: 10.4236/ica.2016.71001

T_{em}, T_{l}: Electromagnetic and load torques (N.m).

I_{d}, I_{q}: (d, q)-axis stator currents (A).

p, J, f: p: pole number; J: inertia (kg∙m^{2}); f: Damping coefficient (Nm.s/rad).

L_{d}, L_{q}: (d, q)-axis inductances (H).

R_{s}, T_{e}: Stator resistance (W) and Sampling period (s).

V_{d}, V_{q}: D-axis and q-axis stator voltage (V).

F_{f}, q: Rotor magnet flux linkage (Wb);q: Rotor position at electrical angle (rpm).

w_{r}, W, w_{r}: Rotor electrical radian speed; W: Mechanical rotor speed (rad/s).