Parameters Identification
for motor PMSM
dsPIC embedded
parts comes from joint work with Dr. R.Delpoux from INSA Lyon (Fr)
2 steps
- Stand-still Identification (coarse)
- {R, L}
- Starts Sliding Mode Observer with current control loop
- Runtime Identification (refined in situe)
- {R, L and Ke}
Stand-still Identification
R during rotor “alignement” (800ms)
L during current relaxation phase (300ms)
Assuming: $Ts << \tau$
with $Ts$ Sample time and $\tau = \frac{L}{R}$
relaxation phase
$A \exp(-\frac{nT_s}{\tau})$
with
$ \exp(-\frac{nT_s}{\tau}) = 1 -\frac{nT_s}{\tau} + \frac{(\frac{nT_s}{\tau})^2}{2!} + \ldots$
initialisation sequence
relaxation phase
$ \frac{x_{n+1} - x_n}{x_n} = \exp(-\frac{Ts}{\tau})- 1$
$ \frac{x_{n+1} - x_n}{x_n} \approx -\frac{Ts}{\tau} + \frac{(\frac{Ts}{\tau})^2}{2!} - \frac{(\frac{Ts}{\tau})^3}{3!} + \ldots$
$ \frac{x_{n+1} - x_n}{x_n} \approx -\frac{Ts}{\tau} + O(\frac{Ts}{\tau}^2)$
relaxation phase
$ \frac{x_{n+1} - x_n}{x_{n+1} + x_n} = -\tanh(\frac{Ts}{2\tau}) $
$ \frac{x_{n+1} - x_n}{x_{n+1} + x_n} \approx -\frac{Ts}{2\tau} + \frac{Ts^3}{24\tau^3} + \ldots$
- Increased precision (Mitigate denominator noise)
- Increased accuracy ($O(\frac{Ts}{\tau}^3$)
relaxation phase
Averaging on one or more relaxation phase (for 300ms)
$ \frac{1}{N}\sum_n \frac{x_{n+1} - x_n}{x_{n+1} + x_n} \approx -\frac{Ts}{2\tau} + O(\frac{Ts}{\tau}^3)$)
initialisation sequence
Static ident results
Long Hurst motor
- R = 0.671 Ohm L = 0.229 mH
- R = 0.678 Ohm L = 0.271 mH
- R = 0.663 Ohm L = 0.198 mH
- R = 0.678 Ohm L = 0.271 mH
- R = 0.670 Ohm L = 0.227 mH
- R = 0.671 Ohm L = 0.237 mH
- R = 0.647 Ohm L = 0.185 mH
Static ident results
Short Hurst motor
- R = 2.561 Ohm L = 1.741 mH
- R = 2.582 Ohm L = 1.610 mH
- R = 2.699 Ohm L = 1.871 mH
- R = 2.588 Ohm L = 2.022 mH
- R = 2.776 Ohm L = 2.180 mH
- R = 2.672 Ohm L = 1.606 mH
- R = 2.709 Ohm L = 1.629 mH
Next Step
Parameters {R,L} are good enough to start:
- current control loop
- Sliding Mode Observer
-> refine parameteres during runtime
running identification
Motor electric equations when set
$\begin{bmatrix} v_d\\ v_q \end{bmatrix} = \begin{bmatrix} i_d & -\omega i_q & 0 \\ i_q & \omega i_d & \omega \end{bmatrix} \begin{bmatrix} R \\ L \\ Ke \end{bmatrix} $
following part from joint work with Dr. R.Delpoux from INSA Lyon (Fr, Lab ampere)
running identification
$\underbrace{\begin{bmatrix} v_d\\ v_q \end{bmatrix}}_y = \underbrace{\begin{bmatrix} i_d & -\omega i_q & 0 \\ i_q & \omega i_d & \omega \end{bmatrix}}_A \underbrace{\begin{bmatrix} R \\ L \\ Ke \end{bmatrix}}_x $
Minimum Mean Square Error (MMSE): $\min_x\left(y-Ax\right)^2$
best fit for the model might not be the best real parameters values
running identification
$\underbrace{\begin{bmatrix} v_d\\ v_q \end{bmatrix}}_y = \underbrace{\begin{bmatrix} i_d & -\omega i_q & 0 \\ i_q & \omega i_d & \omega \end{bmatrix}}_A \underbrace{\begin{bmatrix} R \\ L \\ Ke \end{bmatrix}}_x $
$\min_x\left(y-Ax\right)^2$
$\frac{d}{dx}\left( y^ty - 2y^tAx + x^tA^tAx \right) = 0$
$ (A^tA)x_{best} = y^tA$
$x_{best} = -(A^tA)^{-1}*y^tA$
implementation
$\underbrace{A^tA}_{\text{3x3 matrix}} = \sum_n \begin{bmatrix} i_d^n & i_q^n \\ -\omega^n i_q^n & \omega i_d^n \\ 0 & \omega^n \end{bmatrix} \begin{bmatrix} i_d^n & -\omega^n i_q^n & 0 \\ i_q^n & \omega i_d^n & \omega^n \end{bmatrix} $
$\underbrace{y^tA}_{\text{3x1 vector}} = \sum_n \begin{bmatrix} v_d^n & v_q^n \end{bmatrix} \begin{bmatrix} i_d^n & -\omega^n i_q^n & 0 \\ i_q^n & \omega i_d^n & \omega^n \end{bmatrix} $
Solve $Ax=b$ linear system with Cholesky factorization
Simulink block Cholesky
Solve $Ax=b$ linear system with Cholesky factorization
remarks
- requires > 3 states (speed, load, $I_d injection$…)
- use data history from the beginning
- better results with higher current (i.e. load)
$(y-Ax)$ error indicate parameters mismatch/changes
runtime ident results
short hurst
- R = 2.655 Ohm L = 2.554 mH Phif = 4.320 V/Krpm
- R = 2.699 Ohm L = 2.604 mH Phif = 4.290 V/Krpm
- R = 2.696 Ohm L = 2.630 mH Phif = 4.293 V/Krpm
- R = 2.772 Ohm L = 2.592 mH Phif = 4.277 V/Krpm
- R = 2.750 Ohm L = 2.609 mH Phif = 4.281 V/Krpm
- R = 2.666 Ohm L = 3.885 mH Phif = 4.278 V/Krpm
- R = 2.789 Ohm L = 2.583 mH Phif = 4.252 V/Krpm
runtime ident results
long hurst
- R = 0.571 Ohm L = 0.305 mH Phif = 4.199 V/Krpm
- R = 0.648 Ohm L = 0.338 mH Phif = 4.192 V/Krpm
- R = 0.604 Ohm L = 0.248 mH Phif = 4.199 V/Krpm
- R = 0.640 Ohm L = 0.271 mH Phif = 4.197 V/Krpm after reset (remove standstill data history)
- R = 0.487 Ohm L = 0.242 mH Phif = 4.245 V/Krpm
- R = 0.472 Ohm L = 0.257 mH Phif = 4.250 V/Krpm
- R = 0.480 Ohm L = 0.280 mH Phif = 4.246 V/Krpm
- R = 0.481 Ohm L = 0.280 mH Phif = 4.245 V/Krpm
go further
Other method tested without angle requirement
requires ${v_\alpha, v_\beta, i_\alpha, i_\beta}$
Works in open-loop during startup.
Require solving system with constraints.
et voilà !