State Estimation of Permanent Magnet Synchronous Motor Using Improved Square Root UKF

Bo Xu 1, Fangqiang Mu 1, Guoding Shi 1, Wei Ji 2,* and Huangqiu Zhu 1 1 The School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China; xubo@ujs.edu.cn (B.X.); mfqjsdx@163.com (F.M.); sgd18260622040@163.com (G.S.); zhuhuangqiu@ujs.edu.cn (H.Z.) 2 Key Laboratory of Facility Agriculture Measurement and Control Technology and Equipment of Machinery Industry, Jiangsu University, Zhenjiang 212013, China; jwhxb@163.com * Correspondence: jwhxb@163.com; Tel.: +86-138-6244-5908


Introduction
The extended Kalman filter (EKF) has been successfully implemented as a state observer for induction motor (IM) drives in various areas [1][2][3][4][5][6][7][8].However, EKF needs to calculate the nonlinear equation of the Jacobian matrix, which is sub-optimal and can easily lead to divergence.In Ref. [9], Unscented Kalman filter yielded performance equivalent to the Kalman filter for linear systems, yet generalized elegantly to nonlinear systems without the linearization steps required by the EKF.Also in this paper, a symmetric sigma point solution, which used 2n + 1 points to match the mean and covariance of an n-dimensional random variable, was presented.With this set of points, the unscented transform guaranteed the same performance as the truncated second order filter, with the same order of calculations as an EKF but without the need to calculate any approximations or derivatives.In real-time applications, it is critical that both the computational costs and storage requirements are minimized.In [10], the minimal skew simplex points were introduced, which used the minimum number of sigma points, and reduced the computation time.In [11], UKF was used to estimate the synchronous generator state.Compared with EKF, this method can effectively improve the precision of state estimation.However, covariance asymmetry and non-positive definite defects all exist in both EKF and UKF.A square root unscented Kalman filter (SRUKF) algorithm solves the problem of filtering divergence caused by non-positive of error covariance matrix in general EKF and UKF, and the stability of the algorithm is improved.In [12], SRUKF was used to estimate the speed and rotor

Square-Root UKF
Consider the following a general discrete-time nonlinear dynamic system: where f (,) and h(,) denote the nonlinear function with one-order continuous partial derivative.x k represents the unobserved state of the system.u k is a known exogenous input.y k is the observed measurement signal.w k and v k are uncorrelated from each other with zero-mean and covariance, respectively.Q k and R k are the noise covariance of the process and measurement, respectively.

The Minimal Skew Simplex Points
The minimal skew sigma points are a simplex set, which is chosen to match the mean, covariance and minimize their skew.These points have the important property that they can be constructed recursively.The point selection algorithm for the simplex unscented transform is as follows: (1) Choose the initial weight value: (2) Choose weight sequence: (3) Initialize the vector sequences of sigma points as: Energies 2016, 9, 489 3 of 14 (4) Expand vector sequences for j = 2, . . .,n, according to:

Square-Root UKF for State-Estimation
The square-root UKF makes use of three powerful linear algebra techniques, QR decomposition, Cholesky factor updating and efficient least squares.The algorithmic description of SRUKF is as follows [12]: (1) Initialize with x0 " Erx 0 s (6) where x0 denotes an initial guess for the initial state x 0 , and P 0 is the initial estimation variance.Chol (¨) represents the matrix square-root using lower triangle Cholesky decomposition.(2) Sigma point calculation and time update χk{k´1 " f pχ i,k´1 , u k´1 q (10) xk " where qr{¨} and cholupdate{¨} represents QR decomposition and Cholesky factor updating.

Proof of Lemma 1.
Epγ k`j γ k T q " Erpy k`j ´yḱ where So Lemma 1 is satisfied.In the proof, the noise and signal is statistical independence, and the three times and above error are ignored.
In order to weaken the effect of old data, and achieve fast tracking to the abrupt status, time-varying fading factor λ k is introduced as follows Energies 2016, 9, 489 5 of 14 where λ k " From Lemma 1, if P x k y k ´Kk{k C k " 0, i.e., I ´pλ k P y k q ´1C k " 0, then the following equation can be deduced.
Solving the trace of Equation ( 27), In order to avoid excessive regulation of λ k , and achieve more smooth of the state estimation, softening factor η is introduced, then Equation (28) becomes where , ρ is forgetting factor, 0 < ρ ď 0.95.Generally, ρ = 0.95.When the model is more accurate, λ k = 1, the improved SRUKF degrades into SRUKF.

Mathematical Model of PMSM
The mathematical model of PMSM in α-β coordinate system can be represented as follows where u α , u β and i α , i β are the stator voltage and current respectively.Ψ f is magnet flux linkage.L d and L q are the stator inductance in d-q coordinate system, respectively.F is the friction coefficient of rotor and load.R is the stator resistance.T e and T m are the electromagnetic torque and mechanical torque, respectively.P 1 is the number of pole pairs in torque winding.J is the total moment of inertia of the mechanical system.θ is the angle of rotor position.ω is the rotor speed.Assuming the sampling period T s , the Equations ( 30) and (31) are discretized.We set the state variable From the discretization Formula (32), .

System Implementation
The basic configuration of speed sensorless vector control system for PMSM based on the improved SRUKF is shown in Figure 1.The space vector pulse width modulation (SVPWM) inverter is used.Parameters of PMSM used in simulation experiment are shown in Table 1.

System Implementation
The basic configuration of speed sensorless vector control system for PMSM based on the improved SRUKF is shown in Figure 1.The space vector pulse width modulation (SVPWM) inverter is used.Parameters of PMSM used in simulation experiment are shown in Table 1.

Simulation Results
The performance of the improved SRUKF observer is validated subject to a speed command.Experimental results are shown in Figures 2-16.In which, "Real" indicates the measurement speed, "SRUKF", and "improved SRUKF" indicate the speed or space position estimation and error with the SRUKF and improved SRUKF, respectively.∆θ=θ´θ, θ is the measurement space position of the rotor, θ is the estimation space position, ∆θ is the space position estimation error.In Figure 2, the rotor speed ω reference is 50 rad/s, during this test, the motor is loaded at 3 N¨m.The estimated and measured speeds match well, which indicates that the observer provides a good estimate at low speeds.The corresponding space position estimation and error are shown in Figures 3 and 4. The error of improved SRUKF is smaller than SRUKF.

Error Analysis
Set the root mean square error (RMSE) as the system estimation quality evaluation criteria

Error Analysis
Set the root mean square error (RMSE) as the system estimation quality evaluation criteria

Error Analysis
Set the root mean square error (RMSE) as the system estimation quality evaluation criteria error covariance square root matrix.During steady-state operation at a constant speed, time-varying fading factor equals 1, and the improved SRUKF filter degrades to ordinary SRUKF.700 rad/s load disturbance 0.0458 0.0156 500 rad/s to 100 rad/s to 500 rad/s 0.1541 0.0187 parameters disturbance 0.1084 0.0451 Figure 17 shows the values of the time-varying fading factor at speed accelerated from 500 rad/s to 100 rad/s, then to 500 rad/s.Figure 18 shows the values of the time-varying fading factor at parameters disturbance.Time-varying fading factor is adaptive, which can adjust the filter gain and error covariance square root matrix.During steady-state operation at a constant speed, time-varying fading factor equals 1, and the improved SRUKF filter degrades to ordinary SRUKF.700 rad/s load disturbance 0.0458 0.0156 500 rad/s to 100 rad/s to 500 rad/s 0.1541 0.0187 parameters disturbance 0.1084 0.0451 Figure 17 shows the values of the time-varying fading factor at speed accelerated from 500 rad/s to 100 rad/s, then to 500 rad/s.Figure 18 shows the values of the time-varying fading factor at parameters disturbance.Time-varying fading factor is adaptive, which can adjust the filter gain and error covariance square root matrix.During steady-state operation at a constant speed, time-varying fading factor equals 1, and the improved SRUKF filter degrades to ordinary SRUKF.

Conclusions
This paper has proposed and investigated an improved SRUKF filter for state estimation in sensorless PMSM drives.The main contribution is the combination of the SRUKF and a strong tracking filter.The SRUKF reduces the computational errors by propagating the SR of matrices instead of the matrices themselves.In order to realize the residuals orthogonality and force the SRUKF filter to track the real state rapidly, the time-varying fading factor and softening factor are introduced to self-adjust gain matrices and the state forecast covariance square root matrix.
A speed sensorless vector control system for PMSM based on the improved SRUKF is implemented.The simulation results illustrate that the proposed method has higher nonlinear approximation accuracy, stronger numerical stability, and computational efficiency.Strong robustness is achieved under the conditions of low speed, high speed, step response, and load disturbance.It can achieve a precise estimate of the speed, space position, and ensure the rotor suspends stably.Going forward, learning how to apply this algorithm on an actual system based on DSP will be the next project.

Figure 1 .
Figure 1.Configuration of the vector control system based on improved SRUKF.Figure 1. Configuration of the vector control system based on improved SRUKF.

Figure 8 .
Figure 8. Speed estimation at speed step response.

Figure 8 .
Figure 8. Speed estimation at speed step response.

Figure 8 .
Figure 8. Speed estimation at speed step response.

Figure 12 .
Figure 12.Space position estimation at load disturbance.

Figure 15 .
Figure 15.Space position estimation error at parameters disturbance.

Figure 16 .
Figure 16.Space position estimation error at parameters disturbance.

Figure 15 .
Figure 15.Space position estimation error at parameters disturbance.

Figure 16 .
Figure 16.Space position estimation error at parameters disturbance.

Figure 15 .
Figure 15.Space position estimation error at parameters disturbance.

Figure 16 .
Figure 16.Space position estimation error at parameters disturbance.

Figure 16 .
Figure 16.Space position estimation error at parameters disturbance.

Figure 17 .
Figure 17.Fading factors of speed step response.

Figure 17 .
Figure 17.Fading factors of speed step response.
Define εpkq " x k ´x k , xk is the estimated state with the improved SRUKF algorithm, if Or|εpkq| 2 s ăă Or|εpkq|s, then the following equation exist.