Abstract
The highperformance application of highpower permanent magnet synchronous motor (PMSM) is increasing. This paper focuses on the parameter estimation of PMSM. A novel estimation algorithm for PMSM’s dualrate sampleddata system has been developed. A polynomial transformation technique is employed to derive a mathematical model for PMSM’s dualrate sampleddata system. The proposed modiﬁed stochastic gradient algorithm gets more excellent convergence performance for smaller index $\epsilon $. Simulation and experimental results demonstrate the effectiveness and performance improvement of the proposed algorithm.
1. Introduction
Permanent magnet synchronous motor (PMSM) are now widely employed in industrial servo drives, electric/hybrid electric vehicles, wind power generators, etc., due to their high power/torque density, high efficiency, and excellent control performance. And then, high performance control of PMSM drives requires accurate knowledge of the machine parameters. Often, the online estimation of parameters is required, and the control algorithm can use updated parameter information and modify the controller behavior accordingly. It is important to obtain accurate machine parameters for fault detection rotor/stator temperature monitoring, as well as for achieving high control performance. Many methods have been proposed to obtain the parameters with different online estimation strategies from the measured terminal signals, such as recursive least squares (RLS) [13], extended Kalman filter (EKF) [4, 5], neural network (NN) [68] and the model reference adaptive system (MARS) [9, 10].
To the aforementioned methods, the input and output channel of discretetime systems have the same updating and sampling period in general. A high updating and sampling frequency therefore enables the acquisition of better accuracy to estimation strategies. On the other hand, sampling period has limitations relating to hardware performances. Furthermore, sometimes, it is unrealistic to sample all variables in a complex system with a single frequency. Multirate systems arise when several sampling and updating rates coexist in a system, due to some practical limitations [11, 12]. The identification of multirate systems has received much attention in past decade, because many practical applications in industry can be found [13]. It is the first step of parameter estimation to develop an appropriate model structure that is consistent with all different input/output sampling rates. Generally, there are two main methods to transform the multirate model: lifting technique [16] and polynomial transformation technique [11, 1719]. With the two model transformation techniques, existing methods for multirate system parameters estimation include subspace algorithms [17, 19], stochastic gradient (SG) algorithms [19, 20], and least squares (LS) methods [11]. Ding and Chen [21] presented a stochastic gradient algorithms based on the auxiliary models for dualrate systems to estimate simultaneously the system parameters and the unknown inter sampling output. Ding et al. [20] presented a modified SG (MSG) algorithm with better convergence performance than SG for parameter estimation using the dualrate inputoutput data. Li et al. [17] used the least squares algorithms to estimate parameters of the lifted statespace models. J. Ding and F. Ding proposed an LS algorithm to effectively identify systems parameters with noises [11].
In this paper, dualrate system is defined as a special class of multirate system where the input is update data fast rate and the output is sample data slow rate; the two rates are related by an integer multiple. To PMSM electrical subsystem, “input” is deﬁned as ${u}_{d}^{*}$, ${u}_{q}^{*}$, $dq$axis reference voltages (in volts), measured for PI regulators, that is embedded in high performance microprocessor, such as FPGA or DSP. “Output” is then deﬁned as ${i}_{d}$, ${i}_{q}$, actual $dq$axis currents (in amperes), acquired with Hall current sensors and the microprocessor can get the output information by the A/D converter. Due to limitations of hardware, the sampling speed of the output (${i}_{d}$, ${i}_{q}$) is restricted to be slower than that of input (${u}_{d}^{*}$, ${u}_{q}^{*}$). And then, the sampling period for PMSM is selected so as to be equal to the longest of those two. On the other hand, sampling periods are set individually in multirate sampling control [22]. As a result, better performance can be acquired despite hardware limitations. Many studies have been performed on the system in which output information cannot be acquired fast enough, such as computer hard disk drives or visual servo systems [2325].
The subject of this paper is the design of high performance estimation algorithm for PMSM’s dualrate sampleddata system. We consider the design process to involve 1) setting up dualrate model of PMSM, 2) developing a modiﬁed stochastic gradient algorithm, 3) analyzing the convergence of the proposed algorithm.
The rest of this paper is organized as follows. Section 2 discusses the PMSM model under dualrate sampling. Based on this model, Section 3 proposes a novel parameter estimation algorithm for dualrate systems. Section 4 and 5 provide the simulation and experimental validation, respectively. Section 6 summarizes the more important results of this paper.
2. Dualrate model of PMSM
PMSM speed control is usually achieved using Park’s transformation. This method reduces the threephase “$uvw$” machine equations to a 2D model. Fig. 1 shows the relationships among the two reference frames used in the speed control of the PMSM, i.e., the $x\text{}y$ stator reference frame, and the $d\text{}q$ reference frame corresponding to the real rotor (ﬂux). Due to its simplicity, the linear model of the PMSM on the stator reference frame Eq. (1) is used in [26]:
where:
and $\overrightarrow{u}$, $\overrightarrow{i}$, ${\overrightarrow{\psi}}_{m}$ are the stator voltage, stator current and permanent magnet ﬂux space vectors, respectively. $R$ and ${L}_{s}$ denotes the stator resistance and inductance. $\omega $ and $\theta $ are the rotor speed and position in electrical degrees.
By transforming Eq. (1) into the $d\text{}q$ coordinates, the linear model on the rotor reference frame is obtained as follows:
Subscripts $d$ and $q$ denote the components on the $d\text{}q$ axes of the rotor reference frame. The model Eq. (2) can be rewritten in discrete time. The PMSM electrical statespace model is:
where:
and ${x}_{e}={[{i}_{d},{i}_{q}]}^{\mathrm{T}}$, ${u}_{e}={[{u}_{d},{u}_{q}]}^{\mathrm{T}}$, the superscript $\mathrm{T}$ denotes the matrix transpose. ${T}_{s}$ is the sampling period.
Fig. 1Two reference frames used in speed control of PMSM
2.1. Polynomial transformation technique
In this paper, we focus on identification problems of PMSM’s dualrate sampleddata systems. Fig. 2 shows the dualrate sampling case, where ${H}_{T}$ is the zeroorder hold with period $T\text{,}$ converting the discretetime signal $u\left(k\right)$ into a continuoustime signal $u\left(t\right)$, ${S}_{qT}$ is a sampler with period $qT$ which samples continuoustime signal $y\left(t\right)$ to yield a discretetime signal $y\left(kq\right)$, ${P}_{c}$ is a continuoustime process ($q\ge \text{2}$ is an integer).
Fig. 2The dualrate sampleddata system
Assuming the continuoustime process ${P}_{c}$ is the linear time invariant system, the discretetime model of ${P}_{c}$ can be described as:
where $u\left(k\right)$ and $y\left(k\right)$ are the system input and output, ${a}_{i}$ and ${b}_{i}$ are the unknown parameters, and $n$ is the known system order.
Let ${z}^{1}$ be the unit forward shift operator, and $A\left(z\right)$ and $B\left(z\right)$ be polynomials in ${z}^{1}$:
Then Eq. (4) can be written into a compact form:
This model assumes that all input and output data $\left\{u\right(k),y(k\left)\right\}$ are available. However, in the dualrate sampleddata system, we can get all input data $\left\{u\right(k):k=\text{0, 1, 2...}\}$ and scarce output data $\left\{y\left(kq\right):k=\text{0,}\text{}\text{1,}\text{}\text{2...}\right\}\text{;}$ the intersample output or missing outputs $y\left(kq+j\right)\text{,}$$j=\text{1,}\text{}\text{2,}\text{}\text{...,}\text{}(q1)$ are unavailable. We can use polynomial transformation technique [20] to derive the dualrate model from Eq. (5).
Let the roots of $A\left(z\right)$ be ${z}_{i}$ ($i=\text{1,}\text{}\text{2,...}\text{,}\text{}n$), then:
Deﬁning a polynomial with respect to ${z}_{i}$:
Here, we have used the formula:
Multiplying both sides of Eq. (5) by ${\varphi}_{q}\left(z\right)$ yield:
or
with:
From Eq. (7), one can see that the model in Eq. (6) makes full use of all inputoutput data $\left\{u\right(k),y(kq):k=\text{0,}\text{}\text{1,}\text{}\text{2...}\}$.
2.2. The model of PMSM with dualrate sampling data
Here we take $p=\text{1}$, $q=\text{2}$, and assume $T$, $2T$ are input and output sampling period respectively. Exchange polynomial matrix can be expressed as:
PMSM discretetime model Eq. (3) is multiplied by the polynomial matrix Eq. (8), the corresponding dualrate mode of PMSM can then be expressed as:
$=\left[\begin{array}{cc}\frac{T}{{L}_{s}}& 0\\ 0& \frac{T}{{L}_{s}}\end{array}\right]{u}_{e}\left(k1\right)+\left[\begin{array}{cc}\left(1\frac{R}{{L}_{s}}T\right)\frac{T}{{L}_{s}}& \frac{{T}^{2}\omega \left(k1\right)}{{L}_{s}}\\ \frac{{T}^{2}\omega \left(k1\right)}{{L}_{s}}& \left(1\frac{R}{{L}_{s}}T\right)\frac{T}{{L}_{s}}\end{array}\right]{u}_{e}\left(k2\right)$
$+\left[\begin{array}{c}0\\ \frac{{\psi}_{m}}{{L}_{s}}T\omega \left(k1\right)\end{array}\right]+\left[\begin{array}{c}\frac{{\psi}_{m}}{{L}_{s}}{T}^{2}\omega \left(k1\right)\\ \frac{{\psi}_{m}}{{L}_{s}}T\left(1\frac{R}{{L}_{s}}T\right)\end{array}\right]\omega \left(k2\right).$
For the simplification of the problem, we only take into account the $d$axis current that can then be expressed as:
$+{\beta}_{2}\omega \left(k1\right){u}_{q}\left(k2\right){\beta}_{3}\omega \left(k1\right)\omega \left(k2\right){\omega}^{2}\left(k1\right){T}^{2}{i}_{d}\left(k2\right),$
where:
${\beta}_{0}=\frac{T}{{L}_{s}},{\beta}_{1}=\left(1\frac{R}{{L}_{s}}T\right)\frac{T}{{L}_{s}},{\beta}_{2}=\frac{{T}^{2}}{{L}_{s}},{\beta}_{3}=\frac{{\psi}_{m}{T}^{2}}{{L}_{s}}.$
3. Proposed algorithm
We define parameter vector $\mathbf{\vartheta}$ and inputoutput data vector $\mathbf{\varphi}\left(k\right)$ as:
$\mathbf{\varphi}\left(k\right)\u2254\left[{i}_{d}\right(k2),\omega (k1\left){i}_{q}\right(k2),{u}_{d}(k1),{u}_{d}(k2),$
${\omega (k1){u}_{q}(k2),\omega (k1)\omega (k2)]}^{\mathrm{T}},$
and then, with $qt$ substituding $k$, the model Eq. (10) can be transformed into regression form as:
where the superscript $\mathrm{T}$ denotes the matrix transpose. In one output sampling period, we think the motor speed is constant, and then $\mathbf{\varphi}\left(qt\right)$ contains only the available measurement outputs and inputs.
Let $\widehat{\mathbf{\vartheta}}\left(qt\right)$ be the estimate of $\mathbf{\vartheta}$ at time $qt$. It is well known that the SG algorithm can estimate the parameter vector $\mathbf{\vartheta}$ in Eq. (11) (the DRSG algorithm for short) [20, 27]. However SG algorithm has a slower convergence rate compared with the recursive least squares (RLS) algorithm, but the SG algorithm require slower computation load, because RLS need compute the covariance matrix [11]. In order to improve the convergence rate and tracking performance of the DRSG algorithm, we introduce a convergence index $\epsilon $ and present a modiﬁed stochastic gradient algorithm (the DRMSG algorithm for short) for PMSM as follows:
$+\frac{\mathbf{\varphi}\left(qt\right)\left\{y\left(qt\right)\left[{\mathbf{\varphi}}^{\text{T}}\left(qt\right)\widehat{\mathbf{\vartheta}}\left(qtq\right){\omega}_{r}^{2}\left(qt1\right){T}^{2}{i}_{d}\left(qtq\right)\right]\right\}}{{r}^{\epsilon}\left(qt\right)},\mathrm{}\mathrm{}\mathrm{}0<\epsilon <1,$
${\omega (qt1){u}_{q}(qtq),\omega (qt1)\omega (qtq)]}^{\mathrm{T}},$
where $1/{r}^{\epsilon}\left(qt\right)$ is the stepsize and the norm of matrix $\mathbf{X}$ is defined by ${\Vert \mathbf{X}\Vert}^{2}=\text{tr}\left[\mathbf{X}{\mathbf{X}}^{T}\right].$ The initial value is chosen to be a small vector, e.g., $\widehat{\mathbf{\vartheta}}\left(0\right)=1{0}^{6}{1}_{n}$ with ${1}_{n}$ being an $n$dimensional column vector whose elements are all 1.
4. Simulation results
In this section, we present simulation results to demonstrate the effectiveness of the proposed algorithm for PMSM system. The simulation environment is the Matlab/Simulink. The Simulink model of the PMSM parameters identification system with vector control, which includes parameters identification module, two current PI regulators and one speed PI regulator, has been constructed. The block diagram of the overall PMSM parameters estimation system is given in Fig. 3. The parameters of PMSM are given in Table 1. The motor speed response is given in Fig. 4. The simulation experiment includes two parts, i.e. the MSG identification for singlerate and dualrate PMSM system.
Fig. 3The overall diagram of the PMSM parameters estimation system
Fig. 4Speed response under ω*=100 (rad/s)
Table 1Machine parameters
Parameters  Value 
Stator resistance (Ohm)  2.875 
$d$, $q$axis inductance (mH)  8.5 
Rotor inertia (kg/m^{2})  0.0008 
Permanent magnet flux (Wb)  0.175 
Number of pole pairs  1 
4.1. Parameters identification for singlerate system
In the subsection, simulation results have been obtained to verify the effectiveness of MSG algorithm for PMSM singlerate sampleddata system. PMSM system inputs $\text{(}{u}_{d,q}\text{)}$ and outputs (${i}_{d,q}$) will be updated/sampled with period $T=\text{1\xd7}{\text{10}}^{\text{6}}\text{s}$ in Fig. 3.
From the PMSM discretetime model Eq. (3), we assume parameter ${\mathbf{\vartheta}}_{d}$ and input ${\mathbf{\varphi}}_{d}$ vectors as:
To quantify the identification accuracy, we deﬁne the estimation error as $\delta :=\Vert {\widehat{\mathbf{\vartheta}}}_{d}{\mathbf{\vartheta}}_{d}\Vert /\Vert {\mathbf{\vartheta}}_{d}\Vert $, measured in the Euclidean norm. The estimation results of applying the proposed MSG method with $\epsilon =\text{1}$, $\epsilon =\text{0.9}$, $\epsilon =\text{0.79}$ and $\epsilon =\text{0.78}$ in Fig. 4.
From Fig. 5(a), it is clearly observed that for smaller $\epsilon $, the error $\delta $ is becoming smaller (in general) as time increases. Otherwise, when the $\epsilon $ value is smaller than 0.79, the convergence of MSG is becoming worse from Fig. 5(b). Furthermore, there is an optimal $\epsilon $ value to MSG algorithm for PMSM system.
Fig. 5MSG parameter estimation error δ: a) ε=1, 0.9, 0.79, b) ε=0.78
a)
b)
Fig. 6MSG parameter estimation error δ when ω>0: a) ε=1, 0.9, 0.74, b) ε=0.73
a)
b)
According simulation results in Fig. 45, MSG algorithm for PMSM converges slow and the parameters estimation ${\widehat{\mathbf{\vartheta}}}_{d}$ is inaccurate in finite time. In PMSM startup period, especially when motor speed $\omega <\text{0}$, it would result in poor parameter estimates that PMSM is reversible under the load. In order to eliminate the influence of the load, MSG algorithm is not used to estimate parameters until motor speed $\omega >\text{0}$ in the motor startup period. The estimation errors $\delta $ with $\epsilon =\text{1}$, $\epsilon =\text{0.9}$, $\epsilon =\text{0.74}$ and $\epsilon =\text{0.73}$ are illustrated in Fig. 6. Comparing to Fig. 5, it can be observed that the proposed algorithm estimates the parameters with better accuracy, and also with an optimal $\epsilon $ value.
4.2. Parameters identification for dualrate system
In the test, the PMSM system inputs (${u}_{d,q}$) will be updated with period $T=\text{1\xd7}{\text{10}}^{\text{6}}\text{}\text{s}$ and the outputs (${i}_{d,q}$) sampled with period $qT=\text{2\xd7}{\text{10}}^{\text{6}}\text{}\text{s}$.
Here, we can calculate the parameter vector $\mathbf{\vartheta}$ of PMSM dualrate model Eq. (11) as:
$={[\mathrm{0.999,1.999}\times {10}^{6},1.177\times {10}^{4},1.176\times {10}^{4},1.177\times {10}^{10},2.059\times {10}^{11}]}^{\mathrm{T}}.$
From the true value of the parameter of PMSM dualrate model, it is wellknown that the span of the components of the parameter vector $\mathbf{\vartheta}$ is much large, and that will bring premature convergence on DRMSG algorithm. To avoid the premature convergence, we can get the key components from the parameter vector $\mathbf{\vartheta}$. The new parameter vector ${\mathbf{\vartheta}}_{s}$ and the corresponding inputoutput vector ${\mathbf{\varphi}}_{s}\left(qt\right)$ can be expressed respectively as:
Fig. 7DRMSG parameter estimation error δ: a) ε=1, 0.8, 0.75, b) ε=0.74
a)
b)
Fig. 8DRMSG parameter estimation error δ when ω>0: a) ε=1, 0.8, 0.73, b) ε=0.72
a)
b)
Fig. 9Comparison between the proposed DRMSG and MSG
The PMSM parameter estimation errors of DRMSG algorithm are shown in Fig. 7 (whole procedure) and Fig. 8 ($\omega >\text{0}$). The aforementioned results in the singlerate simulation are also obtained. Comparison with MSG in PMSM’s singlerate datasampled system is shown in Fig. 9. The two estimations are with optimal $\epsilon $ values respectively. From the comparison, it is observed that DRMSG is with the higher convergence performance, and what’s more, DRMSG algorithm estimates the motor parameters on the missing outputs.
5. Experimental results
The experimental results have been obtained with a 0.375kW PMSM with a control board base on TMS320F2812 DSP. The conﬁguration of PMSM drive plant is shown in Fig. 10. The experimental equipment includes magnetic powder brake for applying load torque to the motor, an optical encoder for position feedback. The data are sampled with the period $T=$1 ms, and then been loaded into ‘MATLAB’ to analyze the results. The velocity of reference input is 300 r/min, and the load can be adjusted by the magnetic powder brake DC current. To clearly analyze the convergence performance of DRMSG, the estimation errors $\delta $ are illustrated in Fig. 11 with the brake current 20 mA. It is clearly proved that for smaller $\epsilon $, the convergence performance is becoming better. The same results can be obtained as the simulation.
Fig. 10Experiment plant of PMSM driver
Fig. 11DRMSG parameter estimation error δ
Fig. 12Comparison between DRMSG and DRRLS [11]
Fig. 13Comparison between brake current 20 mA and 160 mA
The dualrate recursive least squares algorithm (the DRRLS for short) [11] is applied to the PMSM’s dualrate sampleddata system. Comparing with DRMSG, the estimation results are shown in Fig. 12. From the comparison, it is observed that the convergence accuracy of DRMSG algorithm is very close to that of the DRRLS algorithm [11], and what is more, it has lower computation load than the DRRLS algorithm. Furthermore, the magnetic powder brake DC current is adjusted to 160 mA in order to increase the load. Comparing with the aforementioned experiment, the DRMSG ($\epsilon =$ 0.1) estimation errors $\delta $ are shown in Fig. 13. From the experiment results, it is proved that the load disturbs the parameter estimation of DRMSG algorithm.
6. Conclusions
In this paper, the dualrate model of PMSM is deduced with polynomial transformation technique. A novel modiﬁed stochastic gradient algorithm is presented for the parameter estimation of the PMSM’s dualrate sampleddata system. Simulation and experimental results demonstrate the effectiveness of the proposed DRMSG algorithm. The estimated parameters are closer to the actual ones for smaller convergence index $\epsilon $. Both simulation and experimental results show that the applied load has a direct impact on the parameter estimation.
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About this article
This work was supported by the National Natural Science Foundation of China (Grant No. 51177137).