parts comes from joint work with Dr. R.Delpoux from INSA Lyon (Fr)
R during rotor “alignement” (800ms)
L during current relaxation phase (300ms)
Assuming: $Ts << \tau$
with $Ts$ Sample time and $\tau = \frac{L}{R}$
$A \exp(-\frac{nT_s}{\tau})$
with
$ \exp(-\frac{nT_s}{\tau}) = 1 -\frac{nT_s}{\tau} + \frac{(\frac{nT_s}{\tau})^2}{2!} + \ldots$
$ \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)$
$ \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$
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)$)
Parameters {R,L} are good enough to start:
-> refine parameteres during runtime
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)
$\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
$\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$
$x_{best} = -(A^tA)^{-1}*y^tA$
$\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
Solve $Ax=b$ linear system with Cholesky factorization
$(y-Ax)$ error indicate parameters mismatch/changes
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 voilĂ !