Model description:
The equations describing the stepped motor in the attached image are given as follows:
$$\begin{align*} \dot{x}_1 &= -K_1x_1 + K_2x_3\sin{(K_5x_4)} + u_1 \\ \dot{x}_2 &= -K_1x_2 + K_2x_3\sin{(K_5x_4)} + u_2 \\ \dot{x}_3 &= -K_3x_1\sin{(K_5x_4)} + K_3x_2\cos{(K_5x_4)} - K_4x_3 - K_6\sin{(4K_5x_4)} - \tau_L/J \\ \dot{x}_4 &= x_3, \end{align*}$$
where $K_1 = R/L$, $K_2 = K_m/L$, $K_3=K_m/J$, $K_4 = B/J$, $K_5 = N_r$, $K_6 = K_D/J$, $K_5 = N_r$, $K_6 = K_D/J$, $u_1=v_a/L$.
| $i_a, i_b$, and $v_a, v_b$ | currents and voltages in phase $A$ and $B$, respectively. |
| $L$ and $R$ | self-inductance and resistance of each phase winding |
| $K_m$ | motor torque constant |
| $N_r$ | number of rotor teeth |
| $J$ | rotor inertia |
| $B$ | vicious friction constant |
| $\omega$ | rotor speed |
| $\theta$ | motor position |
| $\tau_L$ | Load torque |
The term $K_D\sin{(4N_r\theta)}$ represents the detent torque due to the permanent rotor magnet interacting with the magnetic materia of the stator poles. $K_D$ is typically 5% to 10% of the value of $K_mi_0$, where $i_0$ is the rated current.
The state variables $x_1, x_2, x_3$ and $x_4$ are assigned by $x^{\mathrm T}=[i_a, i_b, i_c, i_d]^{\mathrm T}.$
Type:
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Publication details:
| Title | Position Control of a PM Stepper Motor by Exact Linearization |
| Publication Type | Journal Article |
| Year of Publication | 1991 |
| Authors | Zribi, M., and Chiasson J. |
| Journal | IEEE Transactions on Automatic Control |
| Volume | 36 |
| Start Page | 620 |
| Issue | 5 |
| Pagination | 620-625 |
| Date Published | 05/1991 |
| ISSN | 0018-9286 |
| Accession Number | 3939472 |
| Keywords | feedback, linearisation techniques, machine control, permanent magnet motors, position control, state estimation, stepping motors |
| Abstract | The authors consider the position control of a permanent magnet (PM) stepper motor using the exact linearization method. This nonlinear controller takes into account the full dynamics of the stepper motor. In particular, the phase shift between voltage and current in each phase is automatically taken into account. The feedback linearization controller makes the stepper motor into a fast accurate positioning system. The authors consider the feedback linearization technique for the PM stepper motor and show, when the detent torque is neglected, how it quite naturally leads to the well-known DQ transformation of electric machine theory. The authors indicate how constant load torques may be asymptotically rejected by using a nonlinear observer |
| DOI | 10.1109/9.76368 |