Unscented Kalman Estimator for Estimating the State of Two-phase Permanent Magnet Synchronous Motor

This paper pr esents t he unscente d Kalman filters (UKF) for estimating the states (winding curre nts, rotor speed and r otor angular position) of two-phase Permanent Magnet Synchronous Motor ( PMSM). The UKF is based on first ly specifying a minimal set of c arefully ch osen s ample points. T hese sample poi nts completely ca pture the true mean and c ovariance of the Gaussian Random Variable (GRV), and when pr opagated through the true nonlinear system (motor model), capture the posterior mean a nd covariance accurately to the second order (Taylor series expansion). The results showed that the UK estimator could successively estimate the states of PMSM without need any Jacobian matrix.


Introduction:
There are two reasons that one might want to know the states of a system: q q First, one might need to estimate states in order to control the system.
q q Second, one might need to estimate system states because they are interesting in their own right.Some systems need to schedule their activities, to evaluate their q q performance or to predict failure probabilities.The Kalman filter is a tool that can estimate the variables of a wide range of processes.The standard Kalman filter is an effective tool for estimation, but it is limited to linear systems.Most real-world systems are nonlinear, in which case standard Kalman filters do not directly apply [1,2].
The extended Kalman filter (EKF) is the most widely applied state estimation algorithm for nonlinear systems.However, the EKF can be difficult to tune and often gives unreliable estimations if the system nonlinearities are severe.This is because the EKF relies on linearization to propagate the mean and covariance of the state [3].
The unscented Kalman filter (UKF), proposed by Julier et al. [4], is an extension of the Kalman filter that reduces the linearization errors of the EKF.The use of the UKF can provide significant improvement over the EKF.
The basic difference between the EKF and UKF stems from the manner in which Gaussian random variables (GRV) are represented for propagating through system dynamics.In the EKF, the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system.This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to suboptimal performance and sometimes divergence of the filter.The UKF address this problem by using a deterministic sampling approach.The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points.These sample points completely capture the true mean and covariance of the GRV, and, when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to second order (Taylor series expansion) for any nonlinearity.The EKF, in contrast, only achieves first-order accuracy.No explicit Jacobian or Hessian calculations are necessary for the UKF [4,5].

Motor state estimation
The main objective of the work is to use the UKF for estimating the states of a two-phase permanent magnet synchronous motor.The state estimation may be necessary to regulate them with a control algorithm, or to know the position or velocity of the motor for some other reason.Let 's suppose that it is possible to measure the motor winding currents, and we want to use the UKF to estimate the rotor position and velocity.The system equations are [4,6] Where a i and b i are the currents in the two motor windings, θ and ω are the angular position and velocity of the rotor, R and L are the motor winding's resistance and inductance, λ is the flux constant of the motor, F is the coefficient of viscous friction that acts on the motor shaft and its load, J is the moment of inertia of the motor shaft and its load.

The unscented transformation
The unscented transformation (UT) can be summarized as follows [4,5,6 Approximate the mean and covariance of y as follows: The block diagram illustrating the steps in performing the UT is shown in Figure (1).
To apply the UKF to the motor, one need to define the states of the system.The state vector x can be defined as The unscented transformation is the milestone of the UKF algorithm, which is abbreviated as follows [7, 8, 9]. 1.The system equations are given as Where Q , R are the covariance of the process noise k ω and the measurement noise k v , respectively.

2.
The UKF is initialized as follows.
The a priori error covariance is estimated.However, one should add This step can be omitted if desired.That is, instead of generating new sigma points one can reuse the sigma points that were obtained from the time update.This will save computational effort, but would sacrifice performance [10,11] The measurement update of the state estimate can be performed using the normal Kalman filter equations [12]:

Results
It will be assumed that the measurement noise terms, ak v and bk v , are zero-mean random variables with PDF created with pdfFactory Pro trial version www.pdffactory.comThe initial conditions of the system and the estimator are given as A rectangular integration with a step size of T=1 msec is used to simulate the system (the extended Kalman filter and the unscented Kalman filter) for 2 seconds.
Figure (2) shows the true and estimated winding currents, rotor velocity and rotor position of synchronous machine.One can easily see that the UKF could estimate all the states of the motor.
Figure (3) shows the standard deviation of state estimation errors obtained from the filter with six sigma points (since we chose W(0) = 0).
The P matrix quantifies the uncertainty in the state estimates.In other words, the P matrix should give us an idea of how accurate our estimates are. Figure (4) gives the behavior of the sum of diagonal elements (trace) of matrix P for the UK filter.The figure shows that the confidence of the filter with its estimates is low at start of filtering, and then the filter will become more certain with its estimate as time pass beyond 0.2 seconds.

Conclusions:
q q The simulated results shows that the unscented filter could successively estimate all motor states.q q The UKF does not use Jacobians.
For systems with analytic process and measurement equations, it is easy to compute Jacobians.But some systems are not given in analytical form and it is numerically difficult to compute Jacobians.
q q The confidence of the filter with its estimates has been measured by the trace of the covariance matrix P. The transient behavior of this matrix shows that the filter is not certain of its estimates, while the steady state of matrix P will settles to a low value (0.5).This indicates that the filter will have more accurate estimations than its transient phase.
q q One can figure out the rotor position and velocity without using an encoder.Instead of an encoder to get rotor position, one just needs a couple of sense resistors to measure the winding currents, and a Kalman filter algorithm.PDF created with pdfFactory Pro trial version www.pdffactory.com

a u and b u are the
voltages that are applied across the two motor windings, two winding PDF created with pdfFactory Pro trial version www.pdffactory.comEng.& Tech.Journal ,Vol.28,No.performed by sense resistors.The measurements are distorted by measurement noises a v and b v , which are due to things like sense resistance uncertainty, electrical noise or quantization errors.
The following time update equations are used to propagate the state estimate and covariance from one measurement time to the next.(a) To propagate from time step (k -1) to k, first choose sigma points

Q
to the end of the equation to take the process noise into account: Now that the time update equations are done to implement the measurement update equations: The covariance of the predicted measurement is estimated.However, one should add k R to the end of the equation to take the measurement noise into account:

Figure ( 2 )
Figure (3) Currents, velocity and position estimation error magnitudes resulting from filter

Eng.& Tech. Journal ,Vol.28, No.15,2010 Unscented Kalman Estimator for Estimating the State of Two-phase Permanent Magnet Synchronous Motor 5074
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