Real-Time Identification and Nonlinear Control of a Permanent-Magnet Synchronous Motor Based on a Physics-Informed Neural Network and Exact Feedback Linearization

Abstract

This work proposes a novel method for the real-time identification and nonlinear control of a permanent-magnet synchronous motor (PMSM) based on a Physics-Informed Neural Network (PINN) and the exact feedback linearization approach. The proposed approach is presented in a direct-quadrature framework, where the quadrature current and the rotational speed are selected as outputs and the direct and quadrature voltages are selected as inputs. A nonlinear difference equation is selected to describe the physical dynamics of the PMSM, and a PINN is designed based on the aforementioned structure. A simplified training scheme is designed for the PINN based on a least-squares structure to facilitate online training in real time. A nonlinear controller based on exact feedback linearization is designed by considering the nonlinear model of the system identified based on the PINN. Therefore, the proposed approach involves identification and control in real time, where the PINN is trained online. In order to track the reference for the rotational speed, a nonlinear controller with integral action based on exact feedback linearization is designed based on a linear quadratic regulator. As a result, the proposed approach can be used to identify the system to be controlled in real time, and it is able to track any small change in the real model; in addition, it is robust to both external and internal disturbances, such as variations in torque load and resistance. The proposed approach is evaluated through simulation and using a real PMSM, and the results of reference tracking are evaluated under disturbances. The identification performance is evaluated by using a Taylor diagram under closed-loop and open-loop structures, where ARX and NARX structures are used for comparison. It is thereby verified that this novel proposed control approach involving a PINN-based model can adequately track the dynamics of a PMSM system, where the performance of the proposed nonlinear control is maintained even when using the identified model based on the PINN.

1. Introduction

Permanent-magnet synchronous motors (PMSMs) are preferred over other types of motors due to their high efficiency, simple structure, and high power density [1,2,3]. In addition, PMSMs are generally used in electric vehicles due to their lower rotor inertia, more straightforward construction, and simpler control [4,5]. The permanent-magnet brushless DC motor (PMBLDC) is another important type of motor. Its main features include high power density, high efficiency, and low electromagnetic interference in addition to requiring less maintenance. It is used in aerospace, medical applications, servo appliances, robotics, and especially electric vehicle (EV) applications due to its driving performance [6,7,8].

PMSMs have inherent nonlinear behavior; therefore, more suitable controllers are those that take into account nonlinear dynamics, as proposed in [9], through the use of a model predictive control structure. To obtain an adequate nonlinear model, the use of artificial intelligence models, such as neural networks, has been considered. For example, in [10], a model predictive current control based on the Adaline Neural Network for a PMSM is proposed for improving the parameter robustness by estimating the PMSM inductance; however, only one parameter is estimated. Another approach involves approximating the nonlinear model with a linear model estimated from data, as proposed in [11]; however, the nonlinearities of the system are no longer considered when using this approach.

Classical nonlinear control techniques, such as state feedback linearization, can also be used for performance stabilization and tracking [12]. However, these techniques require a detailed nonlinear model where the model dynamics are parameterized. In this case, a Physics-Informed Neural Network (PINN) can be used to obtain a model where the physical laws of the dynamical model are imposed on the learning stage of the neural network [13]. In addition, the authors of [14] propose a novel PINN structure designed for control applications that is called Physics-Informed Neural Nets for Control (PINC). In [15], PINNs are proposed as a suitable solution for several real-world optimization tasks, and, in [16], a strategy is presented for linearizing a system based on discrete-time nonlinear feedback linearization via physics-informed machine learning. Furthermore, in [17], an additional strategy to design a nonlinear observer based on a PINN is also proposed. Alternatively, field-oriented control techniques are based on three PID controllers, which are tuned by trial-and-error techniques or by considering transient responses and stability analysis [18].

This work introduces an innovative exact feedback linearization technique tailored for the multivariable nonlinear control of a permanent-magnet synchronous motor (PMSM), leveraging Physics-Informed Neural Networks (PINNs) for real-time identification. The novel approach is framed within the direct-quadrature coordinate system, where the quadrature current and rotational speed are designated as output variables, while the direct and quadrature voltages serve as inputs. To effectively track the reference for rotational speed, we designed a sophisticated nonlinear controller with integral action grounded in exact feedback linearization. The main contribution of this work is the design of a PINN for the identification of a nonlinear PMSM model in real time and its coupling with a multivariable nonlinear controller with integral action and based on exact feedback linearization. The proposed approach is evaluated through simulation by considering a real PMSM, and the results of reference tracking are evaluated for two control approaches based on the exact feedback linearization nonlinear technique by considering a PMSM model with known parameters and an identified PMSM model based on the PINN structure. In addition, the performance of the PINN-based PMSM model is verified under closed-loop and open-loop structures by using a Taylor diagram, wherein the standard deviation, centered root-mean-square difference, and correlation are presented simultaneously, with a dataset of 100 trials being generated for the closed-loop and open-loop structures. The proposed approach is also validated for a real PMSM model. This paper is organized as follows: In Section 2, we present the nonlinear model of the PMSM in a direct-quadrature frame, the structure of the PINN, and the training stage and the mathematical formulation of the multivariable nonlinear controller based on the exact feedback linearization approach with integral action. In Section 3, we present the simulated and experimental results and the comparison of reference tracking under external and internal disturbances for the proposed approach by using a PMSM model with known parameters and a PINN-based model trained online. Finally, in Section 4, we present our conclusions and plans for future work.

2. Theoretical Framework

2.1. State Space Nonlinear Modeling of a PMSM in Continuous Time

Consider a nonlinear model of a PMSM described by the following set of equations in a direct-quadrature framework [19]:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{x}}_1& =c_1{x}_1+c_2{x}_2{x}_3+c_3{u}_d\hfill \\ \hfill {\dot{x}}_2& =c_4{x}_2+c_5{x}_1{x}_3+c_6{x}_3+c_7{u}_q\hfill \\ \hfill {\dot{x}}_3& =c_8{x}_2+c_9{x}_1{x}_2+c_{10}{x}_3+c_{11}{\tau }_L\hfill \end{array}\end{array}$$

(1)

where

$$\begin{array}{c}\hfill x_1\left(t\right)=i_d\left(t\right)\end{array}$$

(2)

$$\begin{array}{c}\hfill x_2\left(t\right)=i_q\left(t\right)\end{array}$$

(3)

$$\begin{array}{c}\hfill x_3\left(t\right)=\omega \left(t\right)\end{array}$$

(4)

and $u_d\left(t\right)$ and $u_q\left(t\right)$ are the inputs, and external disturbance ${\tau }_L\left(t\right)$ is the torque load. It is assumed that the state variables $x_1\left(t\right)=i_d\left(t\right)$ and $x_3\left(t\right)=\omega \left(t\right)$, as outputs of the PMSM system, are as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_1& =h_1\left(x\right)=x_1\hfill \\ \hfill y_2& =h_2\left(x\right)=x_3\hfill \end{array}\end{array}$$

(5)

It is worth noting that, because the aforementioned PMSM is a multivariable nonlinear system, a multivariable nonlinear controller is required. In this work, the nonlinear state feedback controller is designed based on the exact feedback linearization methodology proposed in [12].

2.2. Discrete State Space Nonlinear Modeling of a PMSM

Presented in [19] is a nonlinear model of a PMSM described in direct-quadrature framework, which can be represented in discrete time as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_1^{+}& =d_1{x}_1+d_2{x}_2{x}_3+d_3{u}_d\hfill \\ \hfill x_2^{+}& =d_4{x}_2+d_5{x}_1{x}_3+d_6{x}_3+d_7{u}_q\hfill \\ \hfill x_3^{+}& =d_8{x}_2+d_9{x}_1{x}_2+d_{10}{x}_3+d_{11}{\tau }_L\hfill \end{array}\end{array}$$

(6)

where $x_i$ is the $i-$th state at sample k ($x_i\left[k\right]$) and $x_i^{+}$ the $i-$th state at sample $k+1$ ($x_i[k+1]$). $x_1=i_d$, $x_2=i_q$, and $x_3=\omega$, with inputs of $u_d$ and $u_q$, and the external disturbance ${\tau }_L$ is the torque load, with outputs $i_d=x_1$ and $\omega =x_3$, where $d_j$ parameters are defined as

$$\begin{array}{cc}\hfill d_1& =1+T_s{c}_1,d_2=T_s{c}_2,d_3=T_s{c}_3\hfill \\ \hfill d_4& =1+T_s{c}_4,d_5=T_s{c}_5,d_6=T_s{c}_6\hfill \\ \hfill d_7& =T_s{c}_7,d_8=T_s{c}_8,d_9=T_s{c}_9\hfill \\ \hfill d_{10}& =1+T_s{c}_{10},d_{11}=T_s{c}_{11}\hfill \end{array}$$

if considering a forward difference approximation method with sampling time $T_s$.

The model can be simplified by considering that the parameters $c_9=0$ and $d_9=0$, and it thus follows that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_1^{+}& =d_1{x}_1+d_2{x}_2{x}_3+d_3{u}_d\hfill \\ \hfill x_2^{+}& =d_4{x}_2+d_5{x}_1{x}_3+d_6{x}_3+d_7{u}_q\hfill \\ \hfill x_3^{+}& =d_8{x}_2+d_{10}{x}_3+d_{11}{\tau }_L\hfill \end{array}\end{array}$$

(7)

In order to obtain adequate parameters for application in a real system, an estimation scheme based on measurements is proposed. Therefore, the parameters $d_i$ are estimated by using a PINN-based identification scheme.

2.3. PINN-Based Identification

In order to perform the identification of the nonlinear model based on a PINN, the structure of the neural network is obtained by considering the mathematical relations of the state variables presented in the difference equation shown in (7). It is worth noting that (7) describes the dynamics of the PMSM in discrete time, so the PINN will also be able to describe the PMSM dynamics. In addition, as a result of the selected PINN-based structure, the weight parameters of the neural network have physical meaning and can be used to design the exact feedback nonlinear controller. In this case, a tailored-design recurrent neural network structure is obtained with one hidden layer containing only two neurons, one input layer containing six neurons (three of which are recurrent from the output layer), and one output layer containing three neurons. Since (7) contains no independent coefficients, the structure of the neural network is selected without bias. In this case, linear activation functions for the neurons are considered. Depicted in Figure 1 is the resulting tailored-design PINN-based structure with linear activation functions and no bias.

It is worth noting from Figure 1 that the number of neurons and their connections in the hidden layer are constrained to the dynamical model of the PMSM described in (7). The initial conditions for the PINN-based structure are defined as $x_1\left[0\right]$, $x_2\left[0\right]$, and $x_3\left[0\right]$.

In order to perform online training of the PINN, the model described in Figure 1 is rewritten as follows:

$$\begin{array}{c}\hfill \underset{x^{+}}{\underbrace{\left[\begin{array}{c}{x}_1^{+}\\ x_2^{+}\\ x_3^{+}\end{array}\right]}}=\underset{{\Theta }_{nl}^T}{\underbrace{\left[\begin{array}{cccccccc}{d}_1& 0& 0& 0& d_2& d_3& 0& 0\\ 0& d_4& d_6& d_5& 0& 0& d_7& 0\\ 0& d_8& d_{10}& 0& 0& 0& 0& d_{11}\end{array}\right]}}\underset{{\Phi }_{nl}}{\underbrace{\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_1{x}_3\\ x_2{x}_3\\ u_d\\ u_q\\ {\tau }_L\end{array}\right]}}\end{array}$$

(8)

where ${\Theta }_{nl}$ are the PINN weights, and the PINN outputs can be computed in a simplified form as follows:

$$\begin{array}{c}\hfill x^{+}={\Theta }_{nl}^T{\Phi }_{nl}\end{array}$$

(9)

It is worth mentioning that, since ${\tau }_L$ is not measured, it is considered an external disturbance. Therefore, during PINN training, the ${\tau }_L$ parameters are computed by considering the estimation error $e_{\tau }$, computed as

$$\begin{array}{c}\hfill e_{\tau }=\left[\begin{array}{ccc}0& 1& 0\end{array}\right]\left(x-{\Theta }_{nl}^T{\Phi }_{nl}\right)\end{array}$$

(10)

Therefore, (8) can be rewritten as follows:

$$\begin{array}{c}\hfill \left[\begin{array}{c}{x}_1^{+}\\ x_2^{+}\\ x_3^{+}\end{array}\right]=\left[\begin{array}{cccccccc}{d}_1& 0& 0& 0& d_2& d_3& 0& 0\\ 0& d_4& d_6& d_5& 0& 0& d_7& 0\\ 0& d_8& d_{10}& 0& 0& 0& 0& d_{11}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_1{x}_3\\ x_2{x}_3\\ u_d\\ u_q\\ e_{\tau }\end{array}\right]\end{array}$$

(11)

A simplified training scheme based on recursive least-squares is proposed, where the matrices ${\Theta }_{nl}$ and ${\Phi }_{nl}$ from (11) are considered, resulting in the following learning equation [20]:

$$\begin{array}{cc}\hfill {\Theta }_{nl}^{+}& ={\Theta }_{nl}+{\displaystyle \frac{{\Omega }_{nl}{\Phi }_{nl}}{1+{\Phi }_{nl}^T{\Omega }_{nl}{\Phi }_{nl}}}\left(x^T-{\Phi }_{nl}^T{\Theta }_{nl}\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill {\Omega }_{nl}^{+}& ={\Omega }_{nl}-{\displaystyle \frac{{\Omega }_{nl}{\Phi }_{nl}{\Phi }_{nl}^T{\Omega }_{nl}}{1+{\Phi }_{nl}^T{\Omega }_{nl}{\Phi }_{nl}}}\hfill \end{array}$$

(13)

2.4. Exact Feedback Linearization in Discrete Time

The exact feedback linearization procedure in discrete time is similar to the one proposed in continuous time [12]. Assuming that the first output is $y_1=i_d$, it follows that

$$\begin{array}{cc}\hfill y_1^{+}& =x_1^{+}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill y_1^{+}& =d_1{x}_1+d_2{x}_2{x}_3+d_3{u}_d\hfill \end{array}$$

(15)

Since the input $u_d\left[k\right]$ appears, the control signal $v_1$ is defined as

$$\begin{array}{cc}\hfill y_1^{+}& =v_1\hfill \end{array}$$

(16)

Assuming that the second output is $y_2=\omega$, it follows that

$$\begin{array}{cc}\hfill y_2^{+}& =x_3^{+}=y_{2p}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill y_2^{+}& =d_8{x}_2+d_{10}{x}_3\hfill \end{array}$$

(18)

Since there are no outputs in the resulting equation, the procedure continues as follows:

$$\begin{array}{cc}\hfill y_{2p}^{+}=& d_8{x}_2^{+}+d_{10}{x}_3^{+}\hfill \end{array}$$

(19)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_{2p}^{+}=& d_8(d_4{x}_2+d_5{x}_1{x}_3+d_6{x}_3+d_7{u}_q)+d_{10}(d_8{x}_2+d_{10}{x}_3)\hfill \end{array}\end{array}$$

(20)

Since the input $u_q\left[k\right]$ appears, the control signal $v_2$ is defined as

$$\begin{array}{cc}\hfill y_{2p}^{+}& =v_2\hfill \end{array}$$

(21)

The resulting discrete state space system is

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_1^{+}\\ y_2^{+}\\ y_{2p}^{+}\end{array}\right]& =\underset{F}{\underbrace{\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]}}\left[\begin{array}{c}{y}_1\\ y_2\\ y_{2p}\end{array}\right]+\underset{G}{\underbrace{\left[\begin{array}{cc}1& 0\\ 0& 0\\ 0& 1\end{array}\right]}}\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]& =\underset{C}{\underbrace{\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]}}\left[\begin{array}{c}{y}_1\\ y_2\\ y_{2p}\end{array}\right]\hfill \end{array}$$

(23)

It can be seen that the system of (22) is decoupled and can therefore be written as two subsystems. For the output $y_1$, it follows that

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_1\end{array}\right]& =\underset{f_1}{\underbrace{\left[\begin{array}{c}0\end{array}\right]}}\left[\begin{array}{c}{y}_1\end{array}\right]+\underset{g_1}{\underbrace{\left[\begin{array}{c}1\end{array}\right]}}\left[\begin{array}{c}{v}_1\end{array}\right]\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_1\end{array}\right]& =\underset{c_1}{\underbrace{\left[\begin{array}{c}1\end{array}\right]}}\left[\begin{array}{c}{y}_1\end{array}\right]\hfill \end{array}$$

(25)

and, for the output $y_2$, it follows that

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_2^{+}\\ y_{2p}^{+}\end{array}\right]& =\underset{F_2}{\underbrace{\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]}}\left[\begin{array}{c}{y}_2\\ y_{2p}\end{array}\right]+\underset{G_2}{\underbrace{\left[\begin{array}{c}0\\ 1\end{array}\right]}}\left[\begin{array}{c}{v}_2\end{array}\right]\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_2\end{array}\right]& =\underset{C_2}{\underbrace{\left[\begin{array}{cc}1& 0\end{array}\right]}}\left[\begin{array}{c}{y}_2\\ y_{2p}\end{array}\right]\hfill \end{array}$$

(27)

Similar to the continuous case, the control law for $v_1\left[k\right]$ is defined by

$$\begin{array}{cc}\hfill v_1& =-k_{d1}{y}_1\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill v_1& =-k_{d1}{x}_1\hfill \end{array}$$

(29)

Following design, the control law for $u_d\left[k\right]$ can be computed as follows:

$$\begin{array}{c}\hfill u_d={\displaystyle \frac{1}{d_3}}\left(-k_{d1}{x}_1-d_1{x}_1-d_2{x}_2{x}_3\right)\end{array}$$

(30)

Similarly, the control law for $v_2\left[k\right]$, considering integral action, can be defined as

$$\begin{array}{cc}\hfill v_2& =-k_{d2}{y}_2-k_{d3}{y}_{2p}+k_{di}{e}_i\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill v_2& =-k_{d2}{x}_3-k_{d3}\left(d_8{x}_2+d_{10}{x}_3\right)+k_{di}{e}_i\hfill \end{array}$$

(32)

where $e_i$ is the integral error, which is defined in discrete time as

$$\begin{array}{c}\hfill e_i^{+}=e_i+T_s({\omega }_{ref}-x_3)\end{array}$$

(33)

with $T_s$ being the sampling time.

By considering (33), an augmented discrete state space system can be obtained as follows:

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{y}_2^{+}\\ y_{2p}^{+}\\ e_i^{+}\end{array}\right]& =\underset{F_a}{\underbrace{\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ -T_s& 0& 1\end{array}\right]}}\left[\begin{array}{c}{y}_2\\ y_{2p}\\ e_i\end{array}\right]+\underset{G_a}{\underbrace{\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]}}\left[\begin{array}{c}{v}_2\end{array}\right]+\underset{G_r}{\underbrace{\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]}}\left[\begin{array}{c}{\omega }_{ref}\end{array}\right]\hfill \end{array}$$

(34)

with $v_2$ defined as

$$\begin{array}{c}\hfill v_2=\underset{K_a}{\underbrace{\left[\begin{array}{ccc}{k}_{d2}& k_{d3}& -k_{di}\end{array}\right]}}\left[\begin{array}{c}{y}_2\\ y_{2p}\\ e_i\end{array}\right]\end{array}$$

(35)

Following design, the control law for $u_q\left[k\right]$ can be computed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_q& ={\displaystyle \frac{1}{d_7{d}_8}}\left(k_{di}{e}_i-k_{d2}{x}_3-k_{d3}(d_8{x}_2+d_{10}{x}_3)\right.\hfill \\ \hfill & \left.-(d_4{d}_8{x}_2+d_5{d}_8{x}_1{x}_3+d_6{d}_8{x}_3+d_8{d}_{10}{x}_2+d_{10}^2{x}_3)\right)\hfill \end{array}\end{array}$$

(36)

A discrete linear quadratic regulator (dLQR) controller can be designed based on the state feedback structure. For control law $v_1$, it follows that a quadratic cost function $J_1$ can be defined as follows:

$$\begin{array}{c}\hfill J_1=\sum _N(q_1{x}_1^2+r_1{v}_1^2)\end{array}$$

(37)

Similarly, for a control law $v_2$, it follows that a quadratic cost function $J_2$ can be defined as follows:

$$\begin{array}{c}\hfill J_2=\sum _N(x_a^T{Q}_a{x}_a+r_2{v}_2^2)\end{array}$$

(38)

with the augmented states $x_a$ defined as

$$\begin{array}{c}\hfill x_a=\left[\begin{array}{c}{y}_2\\ y_{2p}\\ e_i\end{array}\right]\end{array}$$

(39)

The main advantage of this design is that the controller gains $k_{d1}$, $k_{d2}$, $k_{d3}$, and $k_{di}$ are time-invariant and can be computed offline since the matrices $f_1$ and $F_a$ are also time-invariant and thus not dependent on the PMSM model. By contrast, the control laws for $u_d$ and $u_q$ are computed through PINN-based identification of the PMSM parameters.

A diagram of the optimal controller based on nonlinear exact feedback linearization that includes PINN-based identification is shown in Figure 2.

3. Results

3.1. Experimental Setup

In order to evaluate the performance of the proposed control approach, a low-voltage three-phase PMSM Teknic M-2310P-LN-04K system (developed by Tecknic, Inc. at Victor, NY, USA) is considered. The proposed approach involving a PINN-based structure for the identification stage is evaluated through simulation by considering an online training scheme in open-loop (only the PMSM system) and closed-loop (PMSM system and nonlinear optimal controller based on exact feedback linearization) structures. In addition, the proposed approach involving a PINN-based model of the PMSM with online training is compared to the simulated model of the PMSM with known parameters by using, in both cases, nonlinear optimal control based on exact feedback linearization in discrete time. The proposed approach is also validated for a real PMSM Teknic M-2310P-LN-04K system by considering a BOOSTXL-DRV8301 motor driver and a LaunchPad XL F28379D with a C2000 Delfino dual-core processor (where the motor driver and the Launchpad are developed by Texas Instruments Incorporated at Dallas, TX, USA). The PMSM Teknic M-2310P-LN-04K has two feedback sensors (hall and encoder), and the motor parameters are listed in

Table 1. PMSM Teknic M-2310P-LN-04K parameters.

Parameters	Unit	Value
Rated voltage	V	40
Rated speed	RPM	6000
Rated torque	N·m	0.274
Rated power	W	170
Continuous current	A	7.1
Pair poles ($n_p$)	-	4
Stator resistance per phase (R)	$\Omega$	0.3643
Inductance, phase to phase	mH	0.40
Electrical time constant	mS	0.56
Back EMF	Vpeak/kRPM	4.64
Encoder resolution	counts/rev	4000

.

In addition to the parameters presented in Table 1, the following values are also considered: stator d-axis inductance $L_d=0.0002$H, stator q-axis inductance $L_q=0.0002$H, motor inertia $J=7.0616\times {10}^{-6}$ Kg·m², friction constant $B=2.6369\times {10}^{-6}$ N·m·s, $F=2.6369\times {10}^{-6}$ N·m·s, and permanent flux linkage constant ${\lambda }_{pm}=0.0064$Wb.

The parameters $c_j$ of the PMSM are defined as follows:

$$\begin{array}{cc}\hfill c_1& =-{\displaystyle \frac{R}{L_d}},c_2={\displaystyle \frac{L_q{n}_p}{L_d}},c_3={\displaystyle \frac{1}{L_d}}\hfill \\ \hfill c_4& =-{\displaystyle \frac{R}{L_q}},c_5=-{\displaystyle \frac{L_d{n}_p}{L_q}},c_6=-{\displaystyle \frac{{\lambda }_{pm}{n}_p}{L_q}}\hfill \\ \hfill c_7& ={\displaystyle \frac{1}{L_q}},c_8={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p{\lambda }_{pm}}{J}},c_9={\displaystyle \frac{3}{2}}{\displaystyle \frac{n_p}{J}}(L_d-L_q)\hfill \\ \hfill c_{10}& =-{\displaystyle \frac{F}{J}},c_{11}=-{\displaystyle \frac{1}{J}}\hfill \end{array}$$

Assuming $L_d=L_q$ and the parameter $c_9=0$ results in the following set of $c_j$ parameters:

$$\begin{array}{cc}\hfill c_1& =-1.8\times {10}^3,c_2=4,c_3=5000\hfill \\ \hfill c_4& =-1.8\times {10}^3,c_5=-4,c_6=-127.9083\hfill \\ \hfill c_7& =5000,c_8=5.434\times {10}^3,c_9=0\hfill \\ \hfill c_{10}& =-0.3734,c_{11}=-1.4165\times {10}^5\hfill \end{array}$$

By considering $T_s=0.00005$ seconds, the $d_j$ parameters can be computed as

$$\begin{array}{cc}\hfill d_1& =0.91,d_2=2\times {10}^{-4},d_3=0.25\hfill \\ \hfill d_4& =0.91,d_5=-2\times {10}^{-4},d_6=-0.0064\hfill \\ \hfill d_7& =0.25,d_8=2\times {10}^{-4},d_9=0\hfill \\ \hfill d_{10}& =0.999981,d_{11}=-7.0805\hfill \end{array}$$

resulting in the following discrete model:

$$\begin{array}{cc}\hfill x_1^{+}& =0.91x_1+2\times {10}^{-4}{x}_2{x}_3+0.25u_d\hfill \\ \hfill x_2^{+}& =0.91x_2-2\times {10}^{-4}{x}_1{x}_3-0.0064x_3+0.25u_q\hfill \\ \hfill x_3^{+}& =0.999981x_3+2\times {10}^{-4}{x}_2-7.0805{\tau }_L\hfill \end{array}$$

The block diagram that represents the PMSM system in an open-loop structure in discrete time is shown in Figure 3.

The block diagram that represents the PMSM system in a closed-loop structure through including the nonlinear optimal controller based on exact feedback linearization is shown in Figure 4.

The diagram that represents the exact feedback nonlinear control of the PMSM system in the identification stage is shown in Figure 5.

It is worth noting that the discrete controller gains are obtained by using a discrete LQR controller. To this end, the weight constraints are selected for the optimal LQR controller as follows:

$$\begin{array}{cc}\hfill q_1& =100\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill r_1& =1\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill Q_a& =\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 10000\end{array}\right]\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill r_2& =1\hfill \end{array}$$

(43)

This results in the following controller gains:

$$\begin{array}{cc}\hfill k_{d1}& =-2.4588\times {10}^{-15}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill k_{d2}& =0.005\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill k_{d3}& =0.005\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill k_{di}& =99.75\hfill \end{array}$$

(47)

3.2. Training Dataset

In order to generate a training dataset for the online training scheme in an open-loop structure, the variables $i_d$, $i_q$, and $\omega$ of the PMSM system shown in Figure 3 are measured by considering the signals $u_d$, $u_q$, and ${\tau }_L$ as inputs, with the step structure shown in Figure 6.

The outputs corresponding to the inputs in Figure 6 are shown in Figure 7.

A total number of 100 trials are generated by randomly modifying the time step of each input in Figure 6, beginning with a range of between $0.1$ and $0.9$ seconds. The corresponding outputs, like the ones shown in Figure 7, are considered as the ground truth for evaluating the performance of PINN-based identification.

For the online training scheme in a closed-loop structure, the inputs are now the references for the outputs $i_d$ (which is zero) and $\omega$ (which is ${\omega }_{ref}$) and the non-controlled torque load input ${\tau }_L$. Since the main objective of the closed-loop structure is reference tracking and robustness under disturbances (in this case, the torque load ${\tau }_L$), the dataset for the evaluation of the proposed approach is generated by considering the inputs ${\omega }_{ref}$ and ${\tau }_L$, as shown in Figure 8.

In order to evaluate the performance of the proposed approach in the online training scheme, 100 trials are generated by considering the same inputs of Figure 8 for each trial but selecting random initial values in a range between 0 and 1 for the PINN weights.

The closed-loop outputs from using the exact feedback nonlinear control technique and optimal control and by considering the inputs ${\omega }_{ref}$ and ${\tau }_L$ from Figure 8 are shown in Figure 9, with the corresponding control inputs shown in Figure 10.

In order to evaluate the proposed approach, the results shown in Figure 9 are selected as the ground truth since they are obtained by using the PMSM system parameters.

Examples of the closed-loop outputs in the identification stage from using the exact feedback nonlinear control technique and optimal control and by considering the inputs ${\omega }_{ref}$ and ${\tau }_L$ from Figure 8 are shown in Figure 11, with the corresponding control inputs shown in Figure 12.

It is worth mentioning that, from the comparison of Figure 11 and Figure 12 to Figure 9 and Figure 10, it is shown that the tracking performance is maintained when including the online identification stage. However, a small variation can be observed during the first 10 milliseconds that corresponds to the first samples of the system identified by the PINN.

3.3. Training Process

The process for online training of the PINN-based model is performed by using the recursive least-squares method proposed in (12). The PINN-based model weights are defined by ${\Theta }_{nl}$ and computed for each sample. It is worth noting that, in the closed-loop approach, the corresponding control signals are also computed for each sample by considering the ${\Theta }_{nl}$ weights and the nonlinear exact feedback linearization controller for $u_d$ and $u_q$.

An example of the weights’ evolution during online training, related to Figure 11, is shown in Figure 13.

It is worth noting that the parameters of the PINN-based model shown in Figure 13 are the parameters $d_1$ to $d_{11}$ taken directly from (7).

A detailed view of the training weights’ evolution over the first 12 milliseconds is shown in Figure 14.

The estimated outputs and the measured outputs are presented in Figure 15 and correspond to the parameter evolution in Figure 13.

In order to evaluate the performance of the PINN-based modeling by using the closed-loop structure, a Taylor diagram is obtained, as shown in Figure 16. This is derived from 100 trials considering the same inputs in Figure 8 but selecting random initial values in a range between 0 and 1 for the PINN weights.

It can be seen that the correlation coefficient between the PINN-based model using the closed-loop structure and the real model is close to the unity, which means that the PINN-based model adequately describes the PMSM system dynamics.

The PINN-based modeling performance is also computed by using the open-loop structure. This is performed by considering 100 randomly generated trials with the structure shown in Figure 6 and Figure 7 as the ground truth. Two additional systems are considered for evaluating the PINN-based modeling performance: a linear autoregressive exogenous model (ARX) and the nonlinear ARX model (NARX) with linear activation functions. The structure of the ARX model is depicted in Figure 17.

The structure of the NARX model is depicted in Figure 18.

It is worth noting that the same learning algorithm of (12) is used to obtain the parameters for the ARX and NARX models. By considering the ARX and NARX models, a Taylor diagram including the PINN, ARX, and NARX models is obtained, as shown in Figure 19.

It can be seen that, as well as in the closed-loop structure, the correlation coefficient between the PINN-based model and the real model is also close to the unity (over 0.99). It is also worth mentioning that the NARX model has almost the same correlation coefficient as the PINN-based model, while the ARX model has a correlation coefficient around 0.97. However, in this case, the PINN-based modeling is preferred since the weight parameters of the neural network have physical meaning and can therefore be directly used in designing the exact feedback nonlinear controller.

An additional experiment is performed to validate the capability of the PINN-based method in tracking parameter variations. In this case, the time-varying behavior of the $d_1$ and $d_4$ parameters is considered, which are directly related to stator resistance. As a result, the parameters $d_1$ and $d_4$ are increased by $10\%$ of their value at time $t=0.6$ seconds, as shown in Figure 20. The ${\omega }_{ref}$ and ${\tau }_L$ values from Figure 8 are also used.

The corresponding closed-loop outputs are shown in Figure 21. It can be seen that there is a small variation at time $t=0.6$ seconds in the $\omega$ output, which corresponds to the parametric disturbance of Figure 20. However, the PINN model tracks temporal variations in the PMSM model, and the proposed nonlinear controller thereby also adapts to the newly identified parameters.

Presented in Figure 22 is the learning behavior for one of the weights of the PINN-based model related to the $d_1$ parameter, which is used to verify that the PINN-based model tracks temporal variations in the PMSM model.

3.4. Experimental Results

Shown in Figure 23 is the experimental workbench with a real PMSM Teknic M-2310P-LN-04K system using a BOOSTXL-DRV8301 motor driver and a Texas Instruments LaunchPad XL F28379D with a C2000 Delfino dual-core processor.

Speed tracking by applying the proposed approach is shown in Figure 24. It can be seen that the PMSM motor can track the speed, with similar results to the simulation.

Figure 25 shows the corresponding measured phase currents $i_A$ and $i_B$ corresponding to a time of 41.51 seconds for Figure 24.

4. Conclusions

This paper presents work on the novel real-time identification and nonlinear control of a permanent-magnet synchronous motor (PMSM) based on a Physics-Informed Neural Network (PINN) and the exact feedback linearization approach. The nonlinear difference equation that describes the physical dynamics of the PMSM was adequately identified using the proposed PINN structure. It is worth mentioning that the simplified training scheme for the PINN based on a least-squares structure also adequately estimated the system parameters in real time. In addition, the proposed approach can identify the system to be controlled in real time, and it is thus able to track any small changes in the real model, such as torque load disturbances. The proposed approach was compared to a fixed controller with known parameters, with the performances found to be quite similar if the first 20 milliseconds of training are ignored. By considering the Taylor diagrams for the closed-loop and open-loop identification structures, it can be seen that the correlation coefficient between the PINN-based model using the closed-loop structure and the real model is close to the unity, which means that the PINN-based model adequately describes the PMSM system dynamics. In addition, by considering the ARX and NARX structures, it was verified that the proposed PINN-based model and the NARX-based model both accurately track the real model, with a correlation coefficient close to one; however, the PINN-based model has a significantly lower number of parameters. The proposed approach was compared to a PMSM model with known parameters, where it was verified that the PMSM can be successfully identified and controlled in real time using the PINN-based model and exact feedback linearization based on an optimal control design in discrete time. In addition, the proposed approach was validated for a real PMSM Teknic M-2310P-LN-04K system, where the model was shown to adequately track the reference. The PINN model can thus clearly be trained in real time and is therefore ready to track any variations in the PMSM parameters.