An adaptive series control for an interior permanent magnet synchronous motor with actuator compensation

Abstract

An adaptive series speed control system for an interior permanent magnet synchronous motor (IPMSM) drive is presented in this paper. This control system consists of a current and a speed control loop, and it is intended to improve the drive’s speed tracking performance as well as to compensate for voltage distortions caused by non-ideal characteristics of the drive’s actuator, which is a voltage source inverter (VSI). To achieve these goals, a simple model that captures these characteristics of the VSI is developed and embedded in the motor’s electrical model. Then, based on the resulting model, an adaptive proportional-integral (PI) control for the current loops is designed, allowing for state regulation and actuator compensation. Additionally, to improve the drive’s speed tracking performance, a proportional-model-reference adaptive controller (MRAC) is designed for the speed loop. Techniques from machine learning are used for designing the MRAC to effectively address nonlinearities and uncertainties in the speed dynamic. Finally, simulation results are presented to illustrate the outstanding performance of the proposed multi-loop controller.

Appendix

1.1 Proof of Lemma 1

Following the work [30], we take a Lyapunov approach to prove this lemma. Consider the Lyapunov function candidate

$$\begin{aligned} V_e(t)=\dfrac{1}{2}e_r^2(t). \end{aligned}$$

The time derivative of this function along the speed tracking error dynamic (37) is given by

$$\begin{aligned} \dot{V}_e(t)&=-a_me_r^2(t)+g_{\lambda }\Big (-k_1e_r^2(t)\\&\quad +\tilde{W}^\mathrm{T}(t)S(z)e_r(t) - \varepsilon (z) e_r(t)\Big ). \end{aligned}$$

Therefore, we can show that we have

$$\begin{aligned} \begin{aligned} \dot{V}_e(t)&\le -\big (a_m+\dfrac{1}{2}\bar{r}k\big )e_r^2(t)\\&\quad +\dfrac{1}{2}g_{\lambda }\Big (ke_r^2(t)-\dfrac{{{\varepsilon ^{2}(z)}}}{{\eta _1 }}-\dfrac{{s^{*2} \tilde{W}^\mathrm{T}(t)\tilde{W}(t)}}{\eta _2}\Big ), \end{aligned} \end{aligned}$$

(43)

where $k=k_1-\eta _1/2-\eta _2/2$ for any arbitrary scalars $\eta _1>0$ and $\eta _2>0$, and $\bar{r}>0$ is a lower bound on $g_\lambda$. It is noted that the inequalities

$$\begin{aligned}&- e_r\varepsilon \le \dfrac{{\eta _1}{e_r^2}}{2} + \dfrac{\varepsilon ^{*2}}{2\eta _1 },\end{aligned}$$

(44)

$$\begin{aligned}&\tilde{W}^TS(z)e_r \le \dfrac{{\eta _2}{e_r^2}}{2} + \dfrac{s^{*2} {\tilde{W}}^\mathrm{T}\tilde{W}}{2 \eta _2} \end{aligned}$$

(45)

have been used to arrive at (46). Now, the inequality (46) implies that for

$$\begin{aligned} \mid e_r\mid >\dfrac{\mid \varepsilon (z)\mid }{\sqrt{k\eta _1}}+\dfrac{s^{*} \sqrt{\tilde{W}^\mathrm{T}\tilde{W}}}{\sqrt{k\eta _2}}, \end{aligned}$$

it can be guaranteed that

$$\begin{aligned} \dot{V}_e(t) \le -\Big (a_m+\dfrac{1}{2}\bar{r}k\Big )e_r^2(t), \end{aligned}$$

(46)

implying the input-to-state stability of the error dynamic (37) with the input-to-state gain functions as in (39).