Extreme learning machine-based field-oriented feedback linearization speed control of permanent magnetic synchronous motors

Abstract

An extreme learning machine (ELM)-based field-oriented feedback linearization speed control (ELMFOFLC) is proposed to enhance the robustness and tracking performance of a permanent magnetic synchronous motor (PMSM) system. First, the field-oriented control (FOC) is adopted to control the electromagnetic torque and the stator magnetic flux of PMSM independently with a detailed discussion on effects especially brought by the model parameter uncertainties on a FOC-based PMSM model. Then, three field-oriented feedback linearization controllers (FOFLCs) are designed to control the electromagnetic torque loop, the stator magnetic flux loop and the outer speed loop, respectively, and cancel nonlinearities in these three loops. Furthermore, a specific ELM is proposed based on the analysis of the characteristics and the uncertainties of PMSM with FOFLC. The stability is proved using the Lyapunov method. Finally, comprehensive simulations and experiments demonstrate that the proposed control is robust to various uncertainties with a superior speed tracking performance.

Appendix

The proof of Lemma 1 is given as follows:

According to Eq. (2):

$$\dot{e}_{q} = \dot{I}_{q} - \dot{I}_{q}^{*} = \left( { - \omega_{e} I_{d} - \frac{{R_{s} }}{L}I_{q} - \frac{1}{L}\psi_{r} \omega_{e} + \frac{1}{L}U_{q} + \Delta_{d2} } \right) - \dot{I}_{q}^{*}$$

(38)

Substitute Eq. (23) into Eq. (38):

$$\begin{aligned} \dot{e}_{q} = & - \omega _{e} I_{d} - \frac{{R_{s} }}{L}I_{q} - \frac{1}{L}\psi _{r} \omega _{e} + \frac{1}{L}[L\left( - \omega _{e} I_{d} - \frac{{R_{s} }}{L}I_{q} - \frac{{\psi _{r} }}{L}\omega _{e}\right) \\ & + L\left(\dot{I}_{q}^{*} - K_{{p2}} e_{q} - K_{{i2}} \int e_{q}\right) + \Delta _{{d2}} - \dot{I}_{q}^{*} = - K_{{p2}} e_{q} - K_{{i2}} \int e_{q} + \Delta _{{d2}} \\ \end{aligned}$$

(39)

According to Eq. (2) with ${\mathrm{I}}_{\mathrm{d}}^{*}=0$:

$$\dot{e}_{d} = \dot{I}_{d} - \dot{I}_{d}^{*} = - \frac{{R_{s} }}{L}I_{d} + \omega_{e} I_{q} + \frac{1}{L}U_{d} + \Delta_{d1}$$

(40)

Substitute Eq. (23) into Eq. (40):

$$\dot{e}_{d} = - \frac{{R_{s} }}{L}I_{d} + \omega_{e} I_{q} + \frac{1}{L}\left[ {L\left( {\omega_{e} I_{q} - \frac{{R_{s} }}{L}I_{d} - K_{p3} e_{d} - K_{i3} \int e_{d} } \right)} \right] + \Delta_{d1} = - K_{p3} e_{d} - K_{i3} \int e_{d} + \Delta_{d1}$$

(41)

Choose Lyapunov functions with ${\text{K}}_{\text{p2}}>0$, ${\text{K}}_{\text{p3}}>0$, ${\text{K}}_{\text{i2}}>0$, ${\text{K}}_{\text{i3}}>0$:

$$\left\{ {\begin{array}{*{20}c} {V_{1} = \frac{1}{2}e_{d}^{2} + \frac{{K_{i3} }}{2}(\int e_{d} )^{2} \ge 0} \\ {V_{2} = \frac{1}{2}e_{q}^{2} + \frac{{K_{i2} }}{2}\left(\int e_{q}\right )^{2} \ge 0} \\ \end{array} } \right.$$

(42)

Then, the time derivatives of Eq. (42) are:

$$\left\{ {\begin{array}{*{20}c} {\dot{V}_{1} = e_{d} \dot{e}_{d} + e_{d} K_{i3} \int e_{d} } \\ {\dot{V}_{2} = e_{q} \dot{e}_{q} + e_{q} K_{i2} \int e_{q} } \\ \end{array} } \right.$$

(43)

Substitute Eqs. (39), (41) into (43):

$$\left\{ {\begin{array}{*{20}c} {\dot{V}_{1} = e_{d} \left( { - K_{p3} e_{d} + \Delta_{d1} } \right)} \\ {\dot{V}_{2} = e_{q} \left( { - K_{p2} e_{q} + \Delta_{d2} } \right)} \\ \end{array} } \right.$$

(44)

According to A2, we have ${|\Delta }_{\mathrm{d}1}|\le \mathrm{M}$ and ${|\Delta }_{\mathrm{d}2}|\le \mathrm{M}$,

$$\left\{ {\begin{array}{*{20}c} {\dot{V}_{1} \le - K_{p3} e_{d}^{2} + \left| {e_{d} } \right||\Delta_{d1} | \le - K_{p3} e_{d}^{2} + M\left| {e_{d} } \right|} \\ {\dot{V}_{2} \le - K_{p2} e_{q}^{2} + \left| {e_{q} } \right|\left| {\Delta_{d2} } \right| \le - K_{p2} e_{q}^{2} + M\left| {e_{q} } \right|} \\ \end{array} } \right.$$

(45)

For ${\dot{\mathrm{V}}}_{1}<$ 0 and ${\dot{\mathrm{V}}}_{2}<$ 0, it is sufficient that

$$\left\{ {\begin{array}{*{20}c} {\left| {e_{d} } \right| > \frac{M}{{K_{p3} }}} \\ {\left| {e_{q} } \right| > \frac{M}{{K_{p2} }}} \\ \end{array} } \right.$$

(46)

Therefore, the system will first converge and then confine within

$$\left\{ {\begin{array}{*{20}c} {\left| {e_{d} } \right| \le \frac{M}{{K_{p3} }}} \\ {\left| {e_{q} } \right| \le \frac{M}{{K_{p2} }}} \\ \end{array} } \right.$$

(47)

Here completes the proof.