Nonlinear Control of a Permanent Magnet Synchronous Motor Based on State Space Neural Network Model Identification and State Estimation by Using a Robust Unscented Kalman Filter

Abstract

This work proposes a nonlinear modeling of a permanent magnet synchronous motor (PMSM) based on state space neural networks. The state space neural network is trained and the state variables (currents in a direct–quadrature frame and the rotational speed) are estimated by considering a robust Unscented Kalman Filter (UKF). Two contributions are presented in this work: the fist one is a nonlinear modeling structure for a PMSM based on a state space neural network that allows real-time parameter identification, and the second one is PMSM neural network training and state estimation based on a robust UKF. The robustness of the UKF is obtained by using a singular value decomposition of the covariance matrix. A comparison analysis is performed over a real PMSM motor by considering the proposed approach and a linear approximation of the nonlinear model where the states and parameters are computed by using an Extended Kalman Filter. The identified model is validated in closed loop by considering a nonlinear control strategy based on state feedback linearization.

1. Introduction

The control of permanent magnet synchronous motors (PMSMs) requires a detailed modeling of the system. A PMSM is a nonlinear system and therefore the best performance can be achieved by applying nonlinear controllers. Even when field-oriented control (FOC) techniques based on PID are widely used, state feedback controllers are more suitable to modify the inner structure of the system in order to obtain the decoupled behavior of the system. However, the state feedback controllers require the estimation of the states of the system. For example, in [1] is proposed a nonlinear optimal controller based on exact feedback linearization, but considering that the state variables are directly measured. However, in several cases, the state variables are not directly measured, and therefore the state variables need to be estimated.

The UKF has been widely used for estimation of the states of nonlinear systems, where an improvement is obtained for the covariance estimation in comparison to linearized approaches, like the Extended Kalman Filter (EKF) [2]. In [3], a dual UKF for state and parameter estimation is proposed for synchronous generators with unknown parameters, by considering a constrained iterated approach.

In [4], model predictive control is presented based on an identified model where the parameters are estimated for a PMSM. In [5], several state estimators of a PMSM are proposed, based on a UKF, and are compared by considering load torque and flux linkage identification, where the superiority of the augmented model based on the UKF is validated in simulations, and also in [6] a comparison of the EKF and UKF is presented for position sensorless state estimation in a PMSM where the load torque and flux linkage are identified.

In [7], sensorless speed control of a PMSM based on a UKF is proposed where the state variables are estimated based on measurements. In [8], a UKF-based sensor fault diagnosis of PMSM drives in electric vehicles is proposed, where three UKFs are used to detect and isolate current sensor and position sensor faults of the PMSM drive system.

In [9], several UKF variations are evaluated for PMSM state estimation in order to perform sensorless control. In [10], sensorless direct torque control of a PMSM based on a UKF is proposed for torque control. In [11], a robust adaptive UKF for nonlinear estimation with uncertain noise covariance is proposed. In addition, in [12] a nonlinear model predictive control based on an EKF and UKF is proposed for a brushless direct current (BLDC) motor with state estimation.

In this work, a nonlinear modeling of a PMSM is proposed based on state space neural networks, where the neural network is trained and the state variables (currents in a direct–quadrature frame and the rotational speed) are estimated by considering a robust Unscented Kalman Filter (UKF). A comparison analysis is performed by considering a linear approximation of the nonlinear model where the states and parameters are computed by using an Extended Kalman Filter. The model is evaluated in simulation and validated by using an exact feedback linearization control structure. Figure 1 presents a block diagram that summarizes the proposed approach.

The main contributions of this work can be summarized as follows: the fist one is a nonlinear modeling structure for a PMSM that allows real-time parameter identification, and the second one is the PMSM nonlinear model’s neural network training process based on a robust UKF, which allows real-time implementation. This paper is organized as follows: in Section 2, a theoretical framework where a dual UKF for state and parameter estimation is proposed. In Section 3, the experimental setup, results, and conclusion are evaluated by comparing the proposed approach by using an EKF. Finally, in Section 4 the conclusions and future works are presented.

2. Theoretical Framework

2.1. Discrete State Space Nonlinear Modeling of a PMSM

In [1] is presented a discrete-time nonlinear model of a PMSM described in a direct−quadrature framework, which can be represented as follows:

$$\begin{array}{c}\hfill x^{+}=f_{PMSM}(x,u,w,{\tau }_L)+\eta \end{array}$$

(1)

with $f_{PMSM}(x,u,w,{\tau }_L)$ being the state space neural network, defined as

$$\begin{array}{c}\hfill f_{PMSM}(x,u,w,{\tau }_L)=C_{w}w\end{array}$$

(2)

and where $C_w$ is defined as follows

$$\begin{array}{c}\hfill C_w=\left[\begin{array}{cccccccccc}{x}_1& x_2{x}_3& u_d& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& x_2& x_1{x}_3& x_3& u_q& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& x_2& x_3& {\tau }_L\end{array}\right]\end{array}$$

(3)

the state vector x is defined as

$$\begin{array}{c}\hfill x={\left[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}\right]}^{\top }\end{array}$$

(4)

and where w is the vector of PMSM model parameters, defined as

$$\begin{array}{c}\hfill w={\left[\begin{array}{cccccccccc}{w}_1& w_2& w_3& w_4& w_5& w_6& w_7& w_8& w_9& w_{10}\end{array}\right]}^{\top }\end{array}$$

(5)

with $x_i$ being the $i$-th state at sample k ($x_i\left[k\right]$) and $x_i^{+}$ the $i$-th state at sample $k+1$ ($x_i[k+1]$), and where $x_1=i_d$ and $x_2=i_q$ are the direct and quadrature currents, and $x_3=\omega$ is the rotational speed. The input vector u is defined as

$$\begin{array}{c}\hfill u=\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right]\end{array}$$

(6)

where $u_d$ and $u_q$ are the direct and quadrature voltages, respectively. Finally, the external disturbance ${\tau }_L$ is the torque load. In addition, $\eta$ is the additive Gaussian noise with zero mean and covariance matrix $Q_{\eta }={\sigma }_{\eta }^{2}I$ with variance ${\sigma }_{\eta }^2$.

It is worth mentioning that (2) describes a bilinear model where $C_w$ are the inputs and cross-correlated variables of the neural network and w are the weights of the neural network, assuming that the activation functions are linear with no bias. This simplified structure corresponds to an adaptive linear network (ADALINE), but can be generalized to any other neural network structure. By assuming that the output is the rotational speed $\omega$, the measurement equation of the nonlinear system can be defined by

$$\begin{array}{c}\hfill \omega =Cx+\epsilon \end{array}$$

(7)

where $\epsilon$ is the additive Gaussian noise with zero mean and covariance matrix $Q_{\epsilon }={\sigma }_{\epsilon }^{2}I$ with variance ${\sigma }_{\epsilon }^2$, and where C is defined as

$$\begin{array}{c}\hfill C=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]\end{array}$$

(8)

It is worth mentioning that the rotational speed $\omega$ is the state space variable $x_3$, which is assumed to be the only measured state variable of the PMSM. Moreover, it is assumed that the external disturbance ${\tau }_L$ is also measured and therefore the state estimation and the model identification must be computed based only on the measurements of $\omega$ and ${\tau }_L$.

2.2. Robust Neural Network Training and State Estimation by a Dual UKF

The structure of the state space neural network proposed in (1) and (2) allows us to adequately model the dynamics of the PMSM. However, the main assumption is that the PMSM model is unknown and must be identified, which involves the neural network training process to find w. In order to design a robust state space-based nonlinear control of the PMSM system by considering the additive noise $\eta$ and $\epsilon$, an additional estimation of the states of the PMSM is required. Therefore, two processes must be considered: state estimation and model identification. In this work, a dual UKF is designed for state estimation and model identification.

In order to perform the state space neural network training of the nonlinear PMSM model, a dual estimation based on a UKF [13] is proposed. The proposed approach includes a robust computation of the sigma points based on a singular value decomposition (SVD) for the state estimation stage [14].

The KF for parameter estimation of the nonlinear PMSM model is performed by considering that the evolution of the parameters w is defined as follows:

$$\begin{array}{c}\hfill w^{+}=w+ϵ\end{array}$$

(9)

with $ϵ$ being the additive Gaussian noise with zero mean and covariance matrix $Q_{ϵ}$, and where $P_w$ is the covariance matrix of the parameters w.

It is worth noting that the covariance matrix of the states  $P_x$ is decomposed by using the SVD as follows:

$$\begin{array}{cc}\hfill U\Lambda V^{\top }& =P_x\hfill \end{array}$$

(10)

with $\Lambda$ being the singular values, and where the columns of U and V are called the left singular vectors and right singular vectors of $P_x$, respectively. Therefore, the sigma points X are computed by considering x and the expressions $\gamma U\sqrt{\Lambda }$ and $x-\gamma U\sqrt{\Lambda }$.

The aforementioned methodology is computed by a using a sequential approach, where the time update equations for parameters and states are performed as follows:

$$\begin{array}{cc}\hfill w^{-}& =w\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill P_w^{-}& =P_w+Q_{ϵ}\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill U\Lambda V^{\top }& =P_x\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill X& =\left[\begin{array}{ccc}x& x+\gamma U\sqrt{\Lambda }& x-\gamma U\sqrt{\Lambda }\end{array}\right]\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill X^{*}& =f_{PMSM}(X,u,w^{-},{\tau }_L)\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill x^{-}& ={\displaystyle \sum _{i=0}^{2L}}{W}_i^{\left(m\right)}{X}_i^{*}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill P_x^{-}& ={\displaystyle \sum _{i=0}^{2L}}{W}_i^{\left(c\right)}(X_i^{*}-x^{-}){(X_i^{*}-x^{-})}^{\top }+Q_{\eta }\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill Y^{*}& =C{X}^{*}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\omega }^{-}& ={\displaystyle \sum _{i=0}^{2L}}{W}_i^{\left(m\right)}{Y}_i^{*}\hfill \end{array}$$

(19)

with $P_x^{-}$ being the prior covariance matrix of the states, and ${\omega }^{-}$ the prior rotational output, and where the measurement update equations are defined by

$$\begin{array}{cc}\hfill P_{yy}=& ={\displaystyle \sum _{i=0}^{2L}}{W}_i^{\left(c\right)}(Y_i^{*}-{\omega }^{-}){(Y_i^{*}-{\omega }^{-})}^{\top }+Q_{\epsilon }\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill P_{xy}=& ={\displaystyle \sum _{i=0}^{2L}}{W}_i^{\left(c\right)}(X_i^{*}-x^{-}){(Y_i^{*}-{\omega }^{-})}^{\top }\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill K_x& =P_{xy}{P}_{yy}^{-1}\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill x& =x^{-}+K_x(\omega -{\omega }^{-})\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill P_x& =P_x^{-}-K_x{P}_{yy}{K}_x^{\top }\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill K_w& =P_w^{-}{C}_w^{\top }{(C_w{P}_w^{-}{C}_w^{\top }+Q_{\eta })}^{-1}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill w& =w^{-}+K_w(x-C_w{w}^{-})\hfill \end{array}$$

(26)

with $P_{yy}$ being the covariance matrix of the output $\omega$, $P_{xy}$ the cross-correlated covariance matrix between the output and the states, and $K_x$ and $K_w$ the Kalman Filter gains for th states and parameters, respectively. In addition, $C_w$, as described in (3), is computed at each iteration at $x^{-}$ as follows:

$$\begin{array}{c}\hfill C_w=\left[\begin{array}{cccccccccc}{x}_1^{-}& x_2^{-}{x}_3^{-}& u_d& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& x_2^{-}& x_1^{-}{x}_3^{-}& x_3^{-}& u_q& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& x_2^{-}& x_3^{-}& {\tau }_L\end{array}\right]\end{array}$$

(27)

where $W^{\left(m\right)}$ and $W^{\left(c\right)}$ are defined as

$$\begin{array}{cc}\hfill W_0^{\left(m\right)}& ={\displaystyle \frac{\lambda }{L+\lambda }}\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill W_0^{\left(c\right)}& ={\displaystyle \frac{\lambda }{L+\lambda }}+1-{\alpha }^2+\beta \hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill W_i^{\left(m\right)}& =W_i^{\left(c\right)}={\displaystyle \frac{1}{2(L+\lambda )}},i=1,\dots ,2L\hfill \end{array}$$

(30)

with L being the number of state variables of x, $\lambda ={\alpha }^2(L+\kappa )-L$, the spread of the sigma points X is defined by $\alpha$, $\kappa =3-L$, $\beta =2$ for Gaussian distributions, and $\gamma =L+\lambda$.

It is worth mentioning that compared to traditional Cholesky decomposition [13], the SVD presented in (13) provides a more robust computation of the sigma points, which is a desirable property for systems with sequential estimation of states and parameters [15].

2.3. Exact Feedback Linearization in Discrete Time

In discrete time, the exact feedback linearization control scheme for the PMSM is described in [1]; the control law for $u_d\left[k\right]$ and $u_q\left[k\right]$ can be computed in function of the PMSM model parameters and states variables as follows:

$$\begin{array}{cc}\hfill u_d& ={\displaystyle \frac{1}{w_3}}\left(-k_{d1}{x}_1-w_1{x}_1-w_2{x}_2{x}_3\right)\hfill \end{array}$$

(31)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill u_q& ={\displaystyle \frac{1}{w_7{w}_8}}\left(k_{di}{e}_i-k_{d2}{x}_3-k_{d3}(w_8{x}_2+w_9{x}_3)\right.\hfill \\ \hfill & \left.-(w_4{w}_8{x}_2+w_5{w}_8{x}_1{x}_3+w_6{w}_8{x}_3+w_8{w}_9{x}_2+w_9^2{x}_3)\right)\hfill \end{array}\end{array}$$

(32)

with $e_i$ being the integral tracking error, computed as

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

(33)

with $T_s$ being the sampling time, ${\omega }_{ref}$ the desired output, and $\omega =x_3$ the measured output.

It is worth mentioning that the vector of w parameters defined in (5) and the vector of x states defined in (4) are estimated by the robust dual UKF described in (11) to (26).

3. Results

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(Tecknic, Inc., Victor, NY, USA) is considered.

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

Parameter	Value
Rated torque	$0.274\mathrm{N}\mathrm{m}$
Continuous current	$7.1\mathrm{A}$
Number of pole pairs	$p=4$
Stator resistance per phase	$r_s=0.3643\Omega$
Inductance, phase to phase	$0.40\mathrm{mH}$
Electrical Time Constant	$0.56\mathrm{ms}$
Back EMF	$4.64\mathrm{Vpeak}/\mathrm{krpm}$
Encoder resolution	$4000\mathrm{counts}/\mathrm{rev}$

shows the PMSM Teknic M-2310P-LN-04K parameters from the manufacturer.

By considering a sampling time $T_s=0.00005$ s, the following discrete model for the PMSM is obtained:

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

(34)

It is worth noting that (34) is the simulated PMSM model, where the state variable $x_3$ (rotational speed $\omega$) is the only state space variable that is measured and it is assumed that the torque load is also measured.

The following controller gains are considered [1]:

$$\begin{array}{cc}\hfill k_{d1}& =-2.4588\times {10}^{-15}\hfill \\ \hfill k_{d2}& =0.005\hfill \\ \hfill k_{d3}& =0.005\hfill \\ \hfill k_{di}& =99.75\hfill \end{array}$$

The outputs of the closed-loop responses, considering a UKF with a $\eta$ variance of 5 and $\epsilon$ variance of 100, are shown in Figure 2. The term “Measured” in the figure caption corresponds to the variables that are measured in the simulated PMSM model. On the other hand, the term “Estimated” corresponds to the variables that are computed by the Kalman Filter.

The states $x_1$, $x_2$, and $x_3$ of the closed-loop responses, considering the UKF of Figure 2, are shown in Figure 3.

The inputs $u_d$ and $u_q$ of the closed-loop responses, considering the UKF of Figure 2, are shown in Figure 4.

And finally, the evolution of the parameters of the PMSM model, which are estimated from the closed-loop responses by considering the UKF of Figure 2, are shown in Figure 5.

It is noticeable that the currents and voltages of the PMSM shown in Figure 4 and Figure 3 are over the limits of the PMSM parameters shown in Table 1. However, in order to reduce the magnitude of those variables, the controller gains must be modified, which is out of the scope of this work.

A comparison is performed by considering a dual EKF, as proposed in [16]. The corresponding equations for the EKF adapted for the PMSM identification and state estimation are described as follows:

$$\begin{array}{cc}\hfill w^{-}& =w\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill P_w^{-}& =P_w+Q_{ϵ}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill x^{-}& =f_{PMSM}(x,u,w^{-},{\tau }_L)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill P_x^{-}& =A{P}_x{A}^{\top }+Q_{\eta }\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill {\omega }^{-}& =C{x}^{-}\hfill \end{array}$$

(39)

with $A={\left.{\displaystyle \frac{\partial f_{PMSM}(x,u,w^{-},{\tau }_L)}{\partial x}}\right|}_{x,w^{-}}$ being computed as follows:

$$\begin{array}{c}\hfill A=\left[\begin{array}{ccc}{w}_1^{-}& w_2^{-}{x}_3& w_2^{-}{x}_2\\ w_5^{-}{x}_3& w_4^{-}& w_5^{-}{x}_1+w_6^{-}\\ 0& w_8^{-}& w_9^{-}\end{array}\right]\end{array}$$

(40)

and the measurement update equations being defined by

$$\begin{array}{cc}\hfill K_x& =P_x^{-}{C}^{\top }(C{P}_x^{-}{C}^{\top }+Q_{\epsilon })\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill x& =x^{-}+K_x(\omega -{\omega }^{-})\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill P_x& =(I-K_{x}C)P_x^{-}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill K_w& =P_w^{-}{C}_w^{\top }{(C_w{P}_w^{-}{C}_w^{\top }+Q_{\eta })}^{-1}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill w& =w^{-}+K_w(x-C_w{w}^{-})\hfill \end{array}$$

(45)

In Figure 6 are shown closed-loop responses considering an EKF with a $\eta$ variance of 5 and $\epsilon$ variance of 100.

The states $x_1$, $x_2$, and $x_3$ of the closed-loop responses, considering the EKF of Figure 6, are shown in Figure 7.

The inputs $u_d$ and $u_q$ of the closed-loop responses, considering the EKF of Figure 6, are shown in Figure 8.

And finally, the evolution of the parameters of the PMSM model, which are estimated from the closed-loop responses by considering the EKF of Figure 6, are shown in Figure 9.

An additional evaluation is performed by considering several levels of noise (considering a variance $\eta$ of 10, 25, 50, and 100 and a variance $\epsilon$ of 100, 250, 500, and 1000) for the dual UKF estimation with closed-loop response.

In Figure 10 are shown the corresponding outputs for the closed-loop response for each one of the noise variances. It can be seen that the outputs of the closed-loop system effectively track the set-points ${\omega }_{ref}$.

In

Table 2. RMSE noise comparison.

Noise Variance	RMSE
${\sigma }_{\eta }^2=10$, ${\sigma }_{\epsilon }^2=100$	$53.72$
${\sigma }_{\eta }^2=25$, ${\sigma }_{\epsilon }^2=250$	$57.18$
${\sigma }_{\eta }^2=50$, ${\sigma }_{\epsilon }^2=500$	$64.73$
${\sigma }_{\eta }^2=100$, ${\sigma }_{\epsilon }^2=1000$	$68.30$

is shown a comparison of the root mean squared error (RMSE) between the speed reference set-point and the estimated rotational speed of the results presented in Figure 10 for each level of $\eta$ and $\epsilon$.

From Table 2, it can be seen that the RMSE has a small increase with the noise levels but reference tracking is still achieved even under the several noise conditions. This result is consistent with the results presented in Figure 10.

In Figure 11 is shown the estimation of the corresponding parameters for the closed-loop response for each of the noise variances. It can be seen that the parameters of the closed-loop system effectively tend to a steady-state value even under noise conditions.

An additional test with higher speed is performed in order to evaluate the proposed approach by considering ${\sigma }_{\eta }^2=10$ and ${\sigma }_{\epsilon }^2=100$. A speed change from 800 to 1000 RPM is considered with the same torque variation shown in Figure 2. This additional test is shown in Figure 12, where the UKF is used by assuming initial states and parameters equals to zero.

It can be seen that the proposed approach is still valid for higher speed tracking requirements.

4. Conclusions

In this work is presented a novel nonlinear modeling of a PMSM where the model parameters are identified and the state variables (currents in a direct–quadrature frame and the rotational speed) are estimated by considering a robust Unscented Kalman Filter (UKF) in real time. The proposed nonlinear model structure allows us to identify the model’s parameters in real time. In addition, the robust UKF structure is designed specifically for the PMSM nonlinear model in order to allow its application in real time. It is worth noting that since the UKF allows a state estimation of the model, the validation of the proposed approach is performed in closed loop based on a nonlinear control method, exact feedback linearization. It can be seen in the results that the proposed approach effectively estimates the nonlinear model as well as the state estimation, which makes the proposed approach suitable for real-time implementation. It is worth noting that due to the noise, the control signals are over the limits of the PMSM operational range, and therefore the controller gains must be recalculated. This task can be considered in a future work. However, by considering the performance of the controller, it can be seen that the PMSM model parameters and the state vectors ($x_1$, $x_2$, and $x_3$) are estimated by considering only the rotational speed $\omega$ and the torque load ${\tau }_L$ measurements, and the PMSM system is effectively controlled in terms of speed tracking and torque variation. The proposed approach is validated by comparing it to the EKF and it is shown that the control signals and states estimated by the EKF show higher levels of noise in comparison with the UKF state estimate. In addition, it can be seen that the proposed approach is valid over several levels of noise.