### Abstract

Introduction As the high field strength neodymium-iron-boron (NdFeB) magnets become commercially available and affordable, the sinusoidal back electromagnetic force (emf) permanent magnet synchronous motors (PMSMs) are receiving increasing attention due to their high speed, high power density and high efficiency. These characteristics are very favourable for high performance applications, e.g., robotics, aerospace, and electric ship propulsion systems Model and behavior of a synchronous motor To aid advanced controller design for PMSM, it is very important to obtain an appropriate model of the motor. A good model should not only be an accurate representation of system dynamics but should also facilitate the application of existing control techniques. Among a variety of models presented in the literature since the introduction of PMSM, the two-axis dq-model obtained using Park's transformation is the most widely used in variable speed PMSM drive control applications The dynamic model of the synchronous motor in d-q-coordinates can be represented as follows: and In and u q (t) are the dq-components of the stator currents and voltages in synchronously rotating rotor reference frame, ω el (t) is the rotor electrical angular speed, the parameters R s , L d , L q , Φ and p are the stator resistance, d-axis and q-axis inductance, the amplitude of the permanent magnet flux linkage, and p the number of couples of permanent magnets, respectively. At the end, M m indicates the motor torque. Considering an isotropic motor with and where pω mech (t)=ω el (t) and M w is an unknown mechanical load. Structure of the decoupling dynamic estimator The present estimator uses the measurements of input voltages, currents and angular velocity of the motor to estimate the "d-q" winding inductance, the rotor resistance and amplitude of the linkage flux. The structure of the estimator is described in Decoupling structure and minimum error variance algorithm To achieve a decoupled structure of the system described in Eq. (5), a matrix F is to be calculated such that, where u(t)=Fx(t) is a state feedback with and V = im([0, 1] T ) of Eq. (8), according to where the parameters F 11 , F 12 , F 21 ,andF 22 are to be calculated in order to guarantee condition (10) and a suitable dynamics for sake of estimation. Condition (10) is guaranteed if Because of the possible inexact decoupling, it follows that: where n(Δ(L dq −L dq )) is the disturbance due to the inexact cancelation. Proposition 1. Considering the disturbance n(Δ(L dq −L dq )) of Eq. (12) as a white noise, then the current minimum variance error σ e i d (t) = σ i d (t) −î d (t) is obtained by minimising the estimation error of the parameters L dq and R s . Proof 1. If Eqs. (12) and (13) are discretised using Implicit Euler with a sampling frequency equal to t s , then it follows that It is possible to assume an ARMAX model for the system represented by (15) and thus as mentioned above, it follows that: where the coefficients a, b, c 1 ,c 2 , are to be estimated, and n(k) is assumed as white noises. The next sample is: The prediction at time "k" is: Considering that: and assuming that the noise is not correlated to the signal e i d (k), it follows: where σ n is defined as the variance of the white noises. The goal is to findî d (k) such that: It is possible to write (17) as Considering the effect of the noise on the system as follows: and using the Z-transform, then: and Using the Z-transform for Eq. Comparing The dynamic estimator of Φ If the electrical part of the system "q" and "d" axes is considered, then, assuming that ω el (t) = 0, i q (t) = 0, and i d (t) = 0, the following equation can be considered: Consider the following dynamic system: where K is a function to be calculated. Eq. (33) represents the estimators of Φ.I ft h ee r r o r functions are defined as the differences between the true and the observed values, then: and If the following assumption is given: Because of Eq. (32), (37) can be written as follows: and considering (34), then K can be chosen to make Eq. (39) exponentially stable. To guarantee exponential stability, K must be To guarantee dΦ(t) dt << dΦ(t) dt ,thenK >> 0. The observer defined in (33) suffers from the presence of the derivative of the measured current. In fact, if measurement noise is present in the measured current, then undesirable spikes are generated by the differentiation. The proposed algorithm must cancel the contribution from the measured current derivative. This is possible by correcting the observed velocity with a function of the measured current, using a supplementary variable defined as where N (i q (t)) is the function to be designed. and let The purpose of Substituting Inserting Eq. (45) into Eq. (33), the following expression is obtained 1 : Letting N (i q (t)) = k app i q (t), where a parameter has been indicated with k app , then from (42) , and Eq. (43) becomes: Finally, substituting 1 Expression (33) works under the assumption (36): fast observer dynamics. Recent Advances in Robust Control -Theory and Applications in Robotics and Electromechanics www.intechopen.com A Robust Decoupling Estimator to Identify Electrical Parameters for Three-Phase Permanent Magnet Synchronous Motors 9 Using the implicit Euler method, the following velocity observer structure is obtained: where t s is the sampling period. Remark 3. Assumption (36) states that the dynamics of the approximating observer should be faster than the dynamics of the physical system. This assumption is typical for the design of observers. Remark 4. Simulation results Simulations have been performed using a special stand with a 58-kW traction PMSM. The stand consists of a PMSM, a tram wheel and a continuous rail. The PMSM is a prototype for low floor trams. The PMSM parameters are: nominal power 58 kW, nominal torque 852 Nm, nominal speed 650 rpm, nominal phase current 122 A and number of poles 44. The model parameters are: Surface mounted NdBFe magnets are used in PMSM. The advantage of these magnets is their inductance, which is as great as 1.2 T, but theirs disadvantage is corrosion. The PMSM was designed to meet B curve requirements. The stand was loaded by an asynchronous motor. The engine has a nominal power 55 kW, a nominal voltage 380 V and nominal speed 589 rpm. Conclusions and future work This paper considers a dynamic estimator for fully automated parameters identification for three-phase synchronous motors. The technique uses a decoupling procedure optimised by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown to identify the amplitude of the linkage flux using the estimated inductance and resistance. It is generally applicable and could also be used for the estimation of mechanical load and other types of electrical motors, as well as for dynamic systems with similar nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Future work includes the estimation of a mechanical load and the general test of the present algorithm using a real motor. 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