Adaptive Two-Stage Extended Kalman Filter Theory in Application of Sensorless Control for Permanent Magnet Synchronous Motor

Abstract

Extended Kalman filters (EKF) have been widely used for sensorless field oriented control (FOC) in permanent magnet synchronous motor (PMSM). The first key problem associated with EKF is that the estimator requires all the plant dynamics and noise processes are exactly known. To compensate inaccurate model information and improve tracking ability, adaptive fading extended Kalman filtering algorithms have been proposed for the nonlinear system. The second key problem is that the EKF suffers from computational burden and numerical problems when state dimension is large. The two-stage extended Kalman filter (TSEKF) with respect to this problem has been extensively studied in the past. Combining the advantages of both AFEKF and TSEKF, this paper presents an adaptive two-stage extended Kalman filter (ATEKF) for closed-loop position and speed estimation of a PMSM to achieve sensorless operation. Experimental results demonstrate that the proposed ATEKF algorithm for PMSMs has strong robustness against model uncertainties and very good real-time state tracking ability.

1. Introduction

Owing to their characteristics of high efficiency, high power density, and reliability, AC machines and more recently especially permanent magnet synchronous machines (PMSMs) have obtained dominance [1]. However, a PMSM cannot be easily controlled because of the uncertainties such as parameter variations and load-torque variations. Therefore, the linear-control methods such as PID control cannot guarantee high performance. To solve this problem, many researchers have proposed various design methods, for example, adaptive control [2, 3], robust control [4], sliding mode control [5], nonlinear feedback linearization control [6], and fuzzy control [7]. Recently, several authors [8–11] have proposed disturbance-observer-based PMSM control methods that can effectively suppress parameter variations or load-torque variations.

In most PMSM drives, some types of shaft sensors such as an optical encoder or a resolver are connected to the rotor shaft. However, such sensors cause several disadvantages such as high drive cost, low reliability, low noise immunity, and increase in machine size and maintenance requirements. Therefore, the interest toward sensorless FOC of PMSMs has grown in order to increase the reliability and to reduce the costs [12, 13].

Various methods of indirect (or sensorless) position and speed estimation have been investigated for PMSMs. One of major methods is based on extended Kalman filter (EKF) [14–17]. The EKF is an optimal estimator in the least-square sense for estimating the states of dynamic nonlinear systems, and it is, thus, a viable and computationally efficient candidate for the online determination of rotor position and speed of a PMSM.

In spite of its successful use, extended Kalman filter still has some drawbacks. This extended Kalman filtering technique requires complete specifications of both dynamical model parameters and statistic noise levels of the system [18, 19]. In a number of practical situations, the models contain parameters that may deviate from their nominal values. The statistic noise levels of the model are given before the filtering process and will be maintained unchanged during the whole recursive process. Commonly, this a priori information is determined by test analysis and certain knowledge about the observation type beforehand. However, a priori information of this kind is often unavailable. Inaccuracy in system models or poor estimates of noise statistics may seriously degrade the performance of the filter and sometimes even leads to filtering divergence.

To overcome these drawbacks, several adaptive extended Kalman filtering algorithms have been proposed for the nonlinear system. Most of the work reported in this area has concentrated on innovation based adaptive estimation (IAE), which utilizes new statistical information from the innovation sequence to improve the estimation of the covariance matrices. IAE is originally proposed in [20] and later utilized in combination with fuzzy logic and neurofuzzy logic in [21, 22], for linear and nonlinear systems, respectively. One of the IAEs is called the adaptive fading extended Kalman filter (AFEKF) [23], which employs suboptimal fading factors. A weighting factor, which enhances the influence of innovation information, may be incorporated as a multiplier for improving the tracking capability in high dynamic control of PMSM.

On the other hand, Kalman filter (KF) appears to be very complex because of the high order of the mathematical models. The applicability of the KF to real-time state estimation problems is generally limited by the complex mathematical operations. To solve this problem, Friedland [24] proposed to employ the two-stage Kalman filter (TSKF) to decouple the KF into two parallel reduced-order filters. Many researchers [25, 26] have contributed to this problem. However, the results they obtained are suboptimal. The two-stage Kalman estimator in [27] can be optimal when satisfying an algebraic constraint, but almost all practical systems will not satisfy this algebraic constraint.

To improve the performance of TSKF, Hsieh and Chen propose an optimal two-stage kalman estimator (OTSKE) [28], in which the algebraic constraint [27] is removed and the optimal performance is guaranteed. OTSKE is mathematically equivalent to KF without requiring system constraints. Although the proposed OTSKE is slightly more complex than the TSKF, it prevents the performance degradation inherent in the TSKF. Therefore, the proposed OTSKE is the best balance between the performance and the computational complexity. It is known that many practical processes require nonlinear observers. In some works [29, 30], two-stage extended Kalman filter (TSEKF) has been developed by extending TSKF to nonlinear systems. In this paper, the structure of TSEKF will be employed in sensorless algorithm to reduce the computational complexity.

Combining advantages of TSEKF and AFEKF, we present an adaptive two-stage extended Kalman filter (ATEKF) for estimating rotor position and speed by the following two-step procedure: (1) in order to enhance robustness, astringency, and tracking ability, we introduce a fading factor into the conventional EKF to formulate AFEKF; (2) to solve the problem of computational complexity, an ATEKF is obtained based on proposed AFEKF algorithm. The proposed ATEKF is effective implementation of AFEKF. The complete equations of this filter are presented and compared with straight implementation of the AFEKF equations.

The paper is organized in six sections. In Section 2, according to the discrete model of the PMSM, a conventional EKF algorithm for estimating rotor position and speed is designed. In Section 3, an AFEKF is proposed, which uses the conventional EKF algorithm. In Section 4, ATEKF are developed by the same approach used for the OTSKE, and its stability is analyzed. In Section 5, to verify the performance of the ATEKF, experimental results are discussed. Finally, a conclusion wraps up the paper.

2. Conventional EKF Theory

2.1. The Model of PMSM

As elaborated in [6], the machine equations in the rotor (𝑑𝑞) reference frame are as follows: 𝑋_𝑘+1=𝐴_𝑘𝑋_𝑘+𝐵^Θ_𝑘Θ_𝑘+𝐵^𝑢_𝑘𝑈_𝑘+𝑊^𝑥_𝑘,Θ_𝑘+1=𝐺_𝑘Θ_𝑘+𝑊^Θ_𝑘,𝑌_𝑘=𝐶_𝑘𝑋_𝑘+𝜂_𝑘, (1) with 𝑋_𝑘=𝑖_𝑑𝑖_𝑞^𝑇,Θ_𝑘=𝜔_𝑟𝜃_𝑟^𝑇,𝑈_𝑘=𝑢_𝛼𝑢_𝛽^𝑇,𝐴_𝑘=𝑅1−^𝑠𝐿_𝑑𝑇_𝑠𝜔_𝑟𝐿_𝑞𝑇_𝑠𝐿_𝑑−𝜔_𝑟𝐿_𝑑𝑇_𝑠𝐿_𝑞𝑅1−^𝑠𝐿_𝑞𝑇_𝑠,𝑌_𝑘=𝑖_𝛼𝑖_𝛽^𝑇,𝐵^𝑢_𝑘=cos𝜃_𝑟𝐿_𝑑𝑇_𝑠sin𝜃_𝑟𝐿_𝑑𝑇_𝑠−sin𝜃_𝑟𝐿_𝑞𝑇_𝑠cos𝜃_𝑟𝐿_𝑞𝑇_𝑠,𝐺_𝑘=𝑇10_𝑠1,𝐵^Θ_𝑘=−𝜓00_𝑟𝑇_𝑠𝐿_𝑞0,𝐶_𝑘=cos𝜃_𝑟−sin𝜃_𝑟sin𝜃_𝑟cos𝜃_𝑟, (2) where 𝑢_𝛼, 𝑢_𝛽 are the stator voltages in the (𝛼𝛽) reference frame, 𝑖_𝑑, 𝑖_𝑞 are stator flux currents in the (𝑑𝑞) reference frame, 𝐿_𝑑 and 𝐿_𝑞 are the machine 𝑑𝑞 axes inductances, 𝑅_𝑠 is the stator winding resistance, and 𝜓_𝑓 is the flux produced by the magnets. The angular velocity 𝜔_𝑟 is measured in electrical radians per second. 𝜃_𝑟 is the electrical position. 𝑇_𝑠 is sampling period.

2.2. The EKF Algorithm

When treating 𝑋_𝑘 as the full order state and Θ_𝑘 as the augmented system state, the state vector is chosen to be 𝑋^𝑎_𝑘=𝑋_𝑘Θ_𝑘^𝑇. 𝑢_𝛼, 𝑢_𝛽 and 𝑖_𝛼, 𝑖_𝛽 are chosen as input and output vectors because these quantities can be easily obtained from measurements of stator phase currents and voltage construction using DC link voltage and switching status. Considering the noise and parameter errors, the state space model in the rotor (𝑑𝑞) reference frame is described by 𝑋^𝑎_𝑘+1=𝐴_𝑘𝑋^𝑎_𝑘+𝐵_𝑘𝑈_𝑘+𝑊_𝑘,𝑌_𝑘+1=𝐶_𝑘𝑋^𝑎_𝑘+𝜂_𝑘, (3) with 𝐴_𝑘=𝐴_𝑘𝐵^Θ_𝑘0𝐺_𝑘,𝐵_𝑘=𝐵^𝑢_𝑘0,𝐶_𝑘=𝐶_𝑘0,𝑊_𝑘=𝑊^𝑥_𝑘𝑊^Θ_𝑘, (4) where 𝑊_𝑘 and 𝜂_𝑘 are zero-mean noise and are independent from the system state 𝑋^𝑎_𝑘. The system noise 𝑊_𝑘 takes into account system disturbance and model inaccuracies, while 𝜂_𝑘 represents the measurement noise. The noise covariance matrices are defined as follows: 𝐸𝑊^𝑥_𝑘𝑊^𝑥_𝑘−𝜏^𝑇=𝑄^𝑥_𝑘𝛿_𝜏,𝐸𝑊^Θ_𝑘𝑊^Θ_𝑘−𝜏^𝑇=𝑄^Θ_𝑘𝛿_𝜏,𝐸𝑊^𝑥_𝑘𝑊^Θ_𝑘−𝜏^𝑇=𝑄_𝑘^𝑥Θ𝛿_𝜏,𝐸𝜂_𝑘𝜂_𝑘−𝜏^𝑇=𝑅_𝑘𝛿_𝜏,𝐸𝑊^𝑥_𝑘𝜂_𝑘−𝜏^𝑇𝐸𝑊=0,_Θ𝑘𝜂_𝑘−𝜏^𝑇=0. (5)

Based on discretized machine equations, an EKF is constructed to estimate the rotor position and speed of the PMSM. The overall structure of the EKF is well known by employing a two-step prediction and correction algorithm [15]. The first step (prediction step) performs a prediction of both quantities based on the previous estimates 𝑋^𝑎_{𝑘−1∣𝑘−1} and the input vector 𝑈_𝑘−1 actually applied to the system. The second step is the innovation step, correcting the predicted state estimation and its covariance matrix through a feedback correction gain 𝐾_𝑘 that makes use of the actual measured quantities. Hence, the filter is given by 𝑋^𝑎_{𝑘∣𝑘−1}=𝐴_𝑘−1𝑋^𝑎_{𝑘−1∣𝑘−1}+𝐵_𝑘−1𝑈_𝑘−1,𝑃_{𝑘∣𝑘−1}=𝐹_𝑘−1𝑃_{𝑘−1∣𝑘−1}𝐹^𝑇_𝑘−1+𝑄_𝑘−1,𝐾_𝑘=𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘𝐻_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘+𝑅_𝑘⁻¹,𝑋^𝑎_𝑘∣𝑘=𝑋^𝛼_{𝑘∣𝑘−1}+𝐾_𝑘𝑌_𝑘−𝐶_𝑘𝑋^𝑎_{𝑘∣𝑘−1},𝑃_𝑘∣𝑘=𝑃_{𝑘∣𝑘−1}−𝐾_𝑘𝐻_𝑘𝑃_{𝑘∣𝑘−1}, (6)(7)(8)(9)(10) with 𝑋^𝑎_{𝑘−1∣𝑘−1}=𝑖_𝑑𝑖_𝑞𝜔_𝑟𝜃_𝑟^𝑇,𝐵_𝑘−1=𝐵^𝑢_𝑘−10,𝐴_𝑘−1=𝐴_𝑘−1𝐵^Θ_𝑘−10𝐺_𝑘−1,𝐹_𝑘−1=𝐹_𝑘−1𝐸_𝑘−10𝐺_𝑘−1,𝐻_𝑘=𝐻¹_𝑘𝐻²_𝑘,𝐾_𝑘=𝐾^𝑥_𝑘𝐾^Θ_𝑘,𝐹_𝑘−1=𝜕𝐴𝜕𝑋^𝑘−1𝑋_𝑘−1+𝐵^𝑢_𝑘−1𝑈_𝑘−1+𝐵^Θ_𝑘−1Θ_𝑘−1,𝐸_𝑘−1=𝜕𝐴𝜕Θ^𝑘−1𝑋_𝑘−1+𝐵^𝑢_𝑘−1𝑈_𝑘−1+𝐵^Θ_𝑘−1Θ_𝑘−1,𝐻¹_𝑘=𝜕𝐶𝜕𝑋^𝑘−1𝑋_𝑘−1,𝐶_𝑘=𝐶_𝑘0,𝐻²_𝑘=𝜕𝐶𝜕Θ^𝑘−1𝑋_𝑘−1(⋅)=𝑃,𝑃^𝑥[⋅]𝑃^𝑥Θ[⋅]𝑃^𝑥Θ[⋅]^𝑇𝑃^Θ[⋅],𝑄(⋅)=𝑄^𝑥[⋅]𝑄^𝑥Θ[⋅]𝑄^𝑥Θ[⋅]^𝑇𝑄^Θ[⋅]. (11)

3. AFEKF Algorithm Based on Innovation Covariance Estimation

Innovation of the filter, which is a directly observable parameter, can be used as a reference for the filter performance by observing the covariance of the innovation sequence. From the incoming measurement 𝑌_𝑘 and the optimal prediction 𝑋^𝑎_{𝑘∣𝑘−1} obtained in the previous step, the innovation sequence is defined as 𝜂_𝑘=𝑌_𝑘−𝐶_𝑘𝑋^𝑎_{𝑘∣𝑘−1}. (12) Then, the innovation covariance of the EKF is 𝑉_𝑘=𝐸𝜂_𝑘𝜂^𝑇_𝑘=𝐸𝑌_𝑘−𝐶_𝑘𝑋^𝑎_{𝑘∣𝑘−1}𝑌_𝑘−𝐶_𝑘𝑋^𝑎_{𝑘∣𝑘−1}^𝑇=𝐻_𝑘𝑃_{𝑘+1∣𝑘}𝐻^𝑇_𝑘+𝑅_𝑘, (13) where 𝑉_𝑘 is referred to as the theoretical innovation covariance. Moreover, according to Shademan and Sharifi [31], an innovation covariance can be calculated by 𝑉_𝑘=1𝑀−1^𝑘_{𝑖=𝑘−𝑀+1}𝜂_𝑖𝜂^𝑇_𝑖, (14) where 𝑀 is a window size. This 𝑉_𝑘 is the estimated innovation covariance.

If the system dynamic can be modeled exactly, the innovation of the filter should be a white noise sequence with zero mean. However, the statistical characteristics of the observed innovation sequence will become complex due to the fact that the prior knowledge of the process and measurement noise is not known exactly. This means that the theoretical error covariance 𝑉_𝑘 is inconsistent with the estimated error covariance 𝑉_𝑘 in practical applications. We can use a scale factor to weigh the relation between 𝑉_𝑘 and 𝑉_𝑘. This factor is defined as 𝑉_𝑘=𝛼_𝑘𝑉_𝑘. Then, the scalar variable 𝛼_𝑘 can be estimated by 𝛼_𝑘1=max1,𝑚tr𝑉_𝑘𝑉_𝑘⁻¹. (15) Considering complexity of matrix inversion, the above equation can be replaced by 𝛼_𝑘=max1,tr𝑉_𝑘tr𝑉_𝑘. (16)

In order to compensate the effect of unmodeled dynamic, the approach envisaged by the fading memory is based on applying a scale factor 𝜆_𝑘 to the a priori estimate error covariance to deliberately increase the variance of the predicted state vector, thus resulting in more “weight” given to the actual measurements. Thus, when the innovation covariance is increased by unaccounted errors, the increased predicted error covariance 𝑃_{𝑘∣𝑘−1} can be utilized to compensate the effect of an inexact dynamic equation. 𝑃_{𝑘∣𝑘−1} is obtained by 𝑃_{𝑘∣𝑘−1}=𝜆_𝑘𝑃_{𝑘∣𝑘−1}. (17)

Here, the scale factor 𝜆_𝑘 is called a fading factor and 𝜆_𝑘≥1. Different fading memory approaches employ several algorithms to calculate the fading factor. One simple approach is to assign the fading factor as a constant, but this leads to some drawbacks. In this paper, we propose a fading memory algorithm using a variable fading factor that will be determined based on the innovation sequence associated with the dynamic and observation model accuracy. According to the above analysis, 𝑉_𝑘 can be represented by 𝑉_𝑘=𝐻_𝑘𝑃_{𝑘+1∣𝑘}𝐻^𝑇_𝑘+𝑅_𝑘=𝐻_𝑘𝜆_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘+𝑅_𝑘or𝑉_𝑘=𝛼_𝑘𝑉_𝑘=𝛼_𝑘𝐻_𝑘𝑃_{𝑘+1∣𝑘}𝐻^𝑇_𝑘+𝑅_𝑘. (18) In (18), we can obtain the following equations: 𝐻_𝑘𝜆_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘+𝑅_𝑘=𝛼_𝑘𝐻_𝑘𝑃_{𝑘+1∣𝑘}𝐻^𝑇_𝑘+𝑅_𝑘. (19) Then 𝜆_𝑘 can be approximated by the following equation: 𝜆_𝑘≈tr𝛼_𝑘𝐻_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘+𝛼_𝑘−1𝑅_𝑘tr𝐻_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘. (20)

It can be inferred from (3) that time-varying motor parameters, such as stator resistance and inductance, are not included in (3). This means that the measurement dynamic equation does not have unaccounted errors. So the innovation covariance is mainly affected by predicted error covariance not by the measurement covariance. Therefore, the ratio of innovation covariances 𝜆_𝑘 is mainly determined by 𝛼_𝑘. In this paper, our hypothesis is that 𝛼_𝑘 is equal to 𝜆_𝑘; then the ratio between the error covariance can be replaced by 𝛼_𝑘. Thus, (7) in conventional EKF can be rewritten as follows to build AFEKF by employing the multiplier 𝜆_𝑘: 𝑃_{𝑘∣𝑘−1}=𝜆_𝑘𝐹_𝑘−1𝑃_{𝑘−1∣𝑘−1}𝐹^𝑇_𝑘−1+𝑄_𝑘−1. (21)

As stated in [32], this type of AEKF is called as “the AFEKF with rescaling-𝑃_{𝑘∣𝑘−1}.” Compared to the proposed adaptive algorithms with excessive computational load in [21, 22], the AFEKF with rescaling-𝑃_{𝑘∣𝑘−1} is simple and easy to realize.

4. The Adaptive Two-Stage Extended Kalman Filter

4.1. The ATEKF Algorithm

Following the same coordinate transformation as used in OTSKE [28], the ATEKF is obtained by decoupling AFEKF into two parallel filters: one for full order states and another one for the augmented states. So it is necessary to define a transformation matrix 𝑇(⋅), and 𝑇(⋅) is specified as follows: 𝑇(𝐽)=𝐼𝐽0𝐼. (22)

The main advantage of using the 𝑇 transformation is that the inverse transformation 𝑇⁻¹(𝑀)=𝑇(−𝑀) involves only a change of sign. Two blending matrices 𝑀_𝑘 and 𝑁_𝑘 are defined, respectively, by 𝑀_𝑘=𝑃^𝑥Θ_{𝑘∣𝑘−1}(𝑃^Θ_{𝑘∣𝑘−1})⁻¹ and 𝑁_𝑘=𝑃^𝑥Θ_𝑘∣𝑘(𝑃^Θ_𝑘∣𝑘)⁻¹. The transformation operation can be achieved by two transformation matrices 𝑇(𝑀_𝑘) and 𝑇(𝑁_𝑘) so that the variance-covariance matrices 𝑃(⋅) in new base are block diagonal: 𝑃(⋅)=𝑃^𝑥(⋅)00𝑃^Θ(⋅). (23)

Using the two transformation matrices defined above, overlined expressions that correspond to vectors and matrices in the new base can be obtained: 𝑋^𝑎_{𝑘∣𝑘−1}=𝑇𝑀_𝑘𝑋^𝑎_{𝑘∣𝑘−1},𝑃_{𝑘∣𝑘−1}=𝑇𝑀_𝑘𝑃_{𝑘∣𝑘−1}𝑇𝑀_𝑘^𝑇,𝑋^𝑎_𝑘∣𝑘=𝑇𝑁_𝑘𝑋^𝑎_𝑘∣𝑘,𝐾_𝑘=𝑇𝑁_𝑘𝐾_𝑘,𝑃_𝑘∣𝑘=𝑇𝑁_𝑘𝑃_𝑘∣𝑘𝑇𝑁_𝑘^𝑇, (24) where 𝑋^𝑎_𝑘=𝑋_𝑘Θ_𝑘,𝐾_𝑘=𝐾^𝑥_𝑘𝐾^Θ_𝑘. (25) So the following relations are obtained: 𝑃^𝑥_{𝑘∣𝑘−1}=𝑃^𝑥_{𝑘∣𝑘−1}+𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑀^𝑇_𝑘,𝑃^Θ_{𝑘∣𝑘−1}=𝑃^Θ_{𝑘∣𝑘−1},𝑃^𝑥Θ_{𝑘∣𝑘−1}=𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1},𝑃^𝑥_𝑘∣𝑘=𝑃^𝑥_𝑘∣𝑘+𝑁_𝑘𝑃^Θ_𝑘∣𝑘𝑁^𝑇_𝑘,𝑃^Θ_𝑘∣𝑘=𝑃^Θ_𝑘∣𝑘,𝑃^𝑥Θ_𝑘∣𝑘=𝑁_𝑘𝑃^Θ_𝑘∣𝑘. (26)(27)(28)(29)(30)(31) Considering characteristic of matrix 𝑇(⋅), (24) becomes 𝑋^𝑎_{𝑘∣𝑘−1}=𝑇−𝑀_𝑘𝑋^𝑎_{𝑘∣𝑘−1},𝑃_{𝑘∣𝑘−1}=𝑇−𝑀_𝑘𝑃_{𝑘∣𝑘−1}𝑇−𝑀_𝑘^𝑇,𝑋^𝑎_𝑘∣𝑘=𝑇−𝑁_𝑘𝑋^𝑎_𝑘∣𝑘,𝐾_𝑘=𝑇−𝑁_𝑘𝐾_𝑘,𝑃_𝑘∣𝑘=𝑇−𝑁_𝑘𝑃_𝑘∣𝑘𝑇−𝑁_𝑘^𝑇. (32) To obtain a full order filter and an augmented states filter, the following two-step iterative substitution method is used.

Step 1. Substituting (6)–(10) into (32), we have 𝑋^𝑎_{𝑘∣𝑘−1}=𝑇−𝑀_𝑘𝐴_𝑘−1𝑋^𝑎_{𝑘−1∣𝑘−1}+𝑇−𝑀_𝑘𝐵_𝑘−1𝑈_𝑘−1,𝑃_{𝑘∣𝑘−1}=𝜆_𝑘𝑇−𝑀_𝑘𝐹_𝑘−1𝑃_{𝑘−1∣𝑘−1}𝐹^𝑇_𝑘−1𝑇−𝑀_𝑘^𝑇+𝑇−𝑀_𝑘𝑄_𝑘−1𝑇−𝑀_𝑘^𝑇,𝐾_𝑘=𝑇−𝑁_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘𝐻_𝑘𝑃_{𝑘∣𝑘−1}𝐻^𝑇_𝑘+𝑅_𝑘⁻¹,𝑋^𝑎_𝑘∣𝑘=𝑇−𝑁_𝑘𝑋^𝛼_{𝑘∣𝑘−1}+𝐾_𝑘𝑌_𝑘−𝐶_𝑘𝑋^𝑎_{𝑘∣𝑘−1},𝑃_𝑘∣𝑘=𝑇−𝑁_𝑘𝑃_{𝑘∣𝑘−1}𝑇−𝑁_𝑘^𝑇−𝑇−𝑁_𝑘𝐾_𝑘𝐻_𝑘𝑃_{𝑘∣𝑘−1}𝑇−𝑁_𝑘^𝑇. (33)

Step 2. Substituting (32) into the right-hand side of (33), the following equations are written: 𝑋^𝑎_{𝑘∣𝑘−1}=𝑇−𝑀_𝑘𝐴_𝑘−1𝑇𝑁_𝑘−1𝑋^𝑎_{𝑘−1∣𝑘−1}+𝑇−𝑀_𝑘𝐵_𝑘−1𝑈_𝑘−1,𝑃_{𝑘∣𝑘−1}=𝜆_𝑘𝑇−𝑀_𝑘𝐹_𝑘−1𝑇𝑁_𝑘−1𝑃_{𝑘−1∣𝑘−1}×𝑇𝑁_𝑘−1^𝑇𝐹^𝑇_𝑘−1𝑇−𝑀_𝑘^𝑇+𝑇−𝑀_𝑘𝑄_𝑘−1𝑇−𝑀_𝑘^𝑇,𝐾_𝑘=𝑇𝑀_𝑘−𝑁_𝑘𝑃_{𝑘∣𝑘−1}𝑇𝑀_𝑘^𝑇𝐻^𝑇_𝑘×𝐻_𝑘𝑇𝑀_𝑘𝑃_{𝑘∣𝑘−1}𝑇𝑀_𝑘^𝑇𝐻^𝑇_𝑘+𝑅_𝑘⁻¹,𝑋^𝑎_𝑘∣𝑘=𝑇𝑀_𝑘−𝑁_𝑘𝑋^𝑎_{𝑘∣𝑘−1}+𝐾_𝑘𝑌_𝑘−𝐶_𝑘𝑇𝑀_𝑘𝑋^𝑎_{𝑘∣𝑘−1},𝑃_𝑘∣𝑘=𝑇𝑀_𝑘−𝑁_𝑘𝑃_{𝑘∣𝑘−1}𝑇𝑀_𝑘−𝑁_𝑘^𝑇−𝐾_𝑘𝐻_𝑘𝑇𝑀_𝑘𝑃_{𝑘∣𝑘−1}𝑇𝑀_𝑘−𝑁_𝑘^𝑇. (34)(35)(36)(37)(38) Supposed variance-covariance matrices 𝑃(𝑀) are block diagonal. The following relation is obtained by using (35) and (38): 0=𝜆_𝑘𝑀_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^𝑥Θ_𝑘−1−𝑀_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^Θ_𝑘−1,0=𝑀_𝑘−𝑁_𝑘−𝐾^𝑥_𝑘𝑆_𝑘, (39) where 𝑀_𝑘=𝐹_𝑘−1𝑁_𝑘−1+𝐸_𝑘−1𝐺⁻¹_𝑘−1,𝑆_𝑘=𝐻¹_𝑘𝑀_𝑘+𝐻²_𝑘. (40) The above equations lead to 𝑀_𝑘=𝑀_𝑘+𝜆_𝑘𝑄_𝑘^𝑥Θ−𝑀_𝑘𝑄^Θ_𝑘𝑃^Θ_{𝑘∣𝑘−1}⁻¹,𝑁_𝑘=𝑀_𝑘−𝐾^𝑥_𝑘𝑆_𝑘. (41) Then, based on corresponding relationship in (34)~(38), the ATEKF algorithm can be organized by the next two parts [33, 34]. The first part of ATEKF for state and parameter prediction is as follows: 𝑃^Θ_{𝑘∣𝑘−1}=𝜆_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^Θ_𝑘−1,𝐸_𝑘=𝜕𝜕Θ_Θ𝑘|𝑘𝐴_𝑘𝑋_𝑘+𝑁_𝑘Θ_𝑘+𝐵^Θ_𝑘Θ_𝑘+𝐵^𝑢_𝑘𝑈_𝑘,𝑀_𝑘=𝐹_𝑘−1𝑁_𝑘−1+𝐸_𝑘−1𝐺_𝑘−1⁻¹,𝑀_𝑘=𝑀_𝑘+𝜆_𝑘𝑄^𝑥Θ_𝑘−1−𝑀_𝑘𝑄^Θ_𝑘−1𝑃^Θ_{𝑘|𝑘−1}⁻¹,𝑄^𝑥_𝑘=𝑄^𝑥_𝑘−1−𝑄^𝑥Θ_𝑘−1𝑀^𝑇_𝑘−𝑀_𝑘𝑄^𝑥Θ_𝑘−1−𝑀_𝑘𝑄^Θ_𝑘−1^𝑇,𝑃^𝑥_{𝑘∣𝑘−1}=𝜆_𝑘𝐹_𝑘−1𝑃^𝑥_{𝑘−1∣𝑘−1}𝐹^𝑇_𝑘−1+𝑄^𝑥_𝑘−1,Θ_{𝑘∣𝑘−1}=𝐺_𝑘−1Θ_{𝑘−1∣𝑘−1},𝑢_𝑘−1=𝐴_𝑘−1𝑁_𝑘−1+𝐵^Θ_𝑘−1−𝑀_𝑘𝐺_𝑘−1Θ_{𝑘−1∣𝑘−1},𝑋_{𝑘∣𝑘−1}=𝐴_𝑘−1𝑋_{𝑘−1∣𝑘−1}+𝐵^𝑢_𝑘−1𝑈_𝑘−1+𝑢_𝑘−1,𝑆_𝑘=𝐶_𝑘𝑀_𝑘,𝑆_𝑘=𝐻¹_𝑘𝑀_𝑘+𝐻²_𝑘, (42)(43)(44)(45)(46)(47)(48)(49)(50)(51)(52) with 𝑉_𝑘=𝐻¹_𝑘𝑃^𝑥_{𝑘−1∣𝑘−1}𝐻¹_𝑘^𝑇+𝐻¹_𝑘𝑁_𝑘−1+𝐻²_𝑘𝑃^Θ_{𝑘−1∣𝑘−1}×𝐻¹_𝑘𝑁_𝑘−1+𝐻²_𝑘^𝑇+𝑅_𝑘,𝜂_𝑘=𝑌_𝑘−𝐶_𝑘𝑋_{𝑘∣𝑘−1}−𝑆_𝑘Θ_{𝑘∣𝑘−1},𝑉_𝑘=1𝑀−1^𝑘_{𝑖=𝑘−𝑀+1}𝜂_𝑖𝜂^𝑇_𝑖,𝜆_𝑘=max1,tr𝑉_𝑘tr𝑉_𝑘. (53) The second part for state and parameter correction is as follows: 𝐾^Θ_𝑘=𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇+𝑅_𝑘+𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘⁻¹,𝑃^Θ_𝑘∣𝑘=𝑃^Θ_{𝑘∣𝑘−1}−𝐾^Θ_𝑘𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1},Θ_𝑘∣𝑘=Θ_{𝑘∣𝑘−1}+𝐾^Θ_𝑘𝜂_𝑘,𝐾^𝑥_𝑘=𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇+𝑅_𝑘⁻¹,𝑃^𝑥_𝑘∣𝑘=𝑃^𝑥_{𝑘∣𝑘−1}−𝐾^𝑥_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1},𝑋_𝑘∣𝑘=𝑋_{𝑘∣𝑘−1}+𝐾^𝑥_𝑘𝑌_𝑘−𝐶_𝑘𝑋_{𝑘∣𝑘−1}+𝑆_𝑘−𝑆_𝑘Θ_{𝑘∣𝑘−1},𝑁_𝑘=𝑀_𝑘−𝐾^𝑥_𝑘𝑆_𝑘. (54)(55)(56)(57)(58)(59)(60) Using (26), the original state 𝑋 can be obtained by the sum of the state 𝑋 with the augmented state Θ: 𝑋_{𝑘∣𝑘−1}=𝑋_{𝑘∣𝑘−1}+𝑀_𝑘Θ_{𝑘∣𝑘−1},𝑋_𝑘∣𝑘=𝑋_𝑘∣𝑘+𝑁_𝑘Θ_𝑘∣𝑘. (61) The initial conditions of this ATEKF are established with the initial conditions of the AFEKF (𝑋_0|0,Θ_0|0,𝑃^𝑥_0|0,𝑃^Θ_0|0,𝑃^𝑥Θ_0|0), so that 𝑁₀=𝑃^𝑥Θ_0∣0𝑃^Θ_0∣0⁻¹,𝑋_0∣0=𝑋_0∣0−𝑁₀Θ_0∣0,Θ_0∣0=Θ_0∣0,𝑃^𝑥_0∣0=𝑃^𝑥_0∣0−𝑁₀𝑃^Θ_0∣0𝑁^𝑇₀,𝑃^Θ_0∣0=𝑃^Θ_0∣0. (62)

By using the number of arithmetic operations (multiplications and additions) as the measure of computational complexity, the computational advantage of the two-stage Kalman filter over the conventional Kalman Filter has been demonstrated in [28, 33]. The analysis of computational complexity of ATEKF has rather similar process and does not need to be discussed here.

4.2. The Stability Analysis of the ATEKF

Theorem 1. The discrete-time adaptive fading extended Kalman filter ((6), (8), (9), (10), and (21)) is equivalent to the adaptive two-stage extended Kalman filter ((42)~(62)).

Proof. Before proving the theorem, the following relationships are needed.
Using (42) and (45), 𝑀_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1=1𝜆_𝑘𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}−𝑄^𝑥Θ_𝑘−1. (63) Using (54) and (57), 𝐾^𝑥_𝑘𝑍_𝑘=𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇+𝐾^𝑥_𝑘𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘,𝐾^Θ_𝑘𝑍_𝑘=𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘, (64)(65) where 𝑍_𝑘=𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇+𝐻¹_𝑘𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘+𝐻²_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘+𝑅_𝑘. (66) Using (55), (64), and (65), 𝑃^Θ_𝑘∣𝑘𝑆^𝑇_𝑘𝐾^𝑥_𝑘^𝑇=𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘−𝐾^Θ_𝑘𝑍_𝑘𝐾^𝑥_𝑘^𝑇+𝐾^Θ_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}=𝐾^Θ_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}. (67) Assume that at time 𝑘, the estimated 𝑋 and Θ are, respectively, equal to the 𝑋 and Θ. By recurrence reasoning, we show that the ATEKF is equivalent to the AFEKF because these properties are still true at time 𝑘+1: 𝑋_𝑘∣𝑘=𝑋_𝑘∣𝑘,Θ_𝑘∣𝑘=Θ_𝑘∣𝑘,𝑃^𝑥_𝑘∣𝑘=𝑃¹¹_𝑘∣𝑘,𝑃^𝑥Θ_𝑘∣𝑘=𝑃¹²_𝑘∣𝑘,𝑃^Θ_𝑘∣𝑘=𝑃²²_𝑘∣𝑘, (68) where𝑃^𝑥𝑃^𝑥Θ(𝑃^𝑥Θ)^𝑇𝑃^Θand𝑃¹¹𝑃¹²(𝑃¹²)^𝑇𝑃²²represent the variance-covariance matrices of the system and estimated variables, respectively.
Using (6), (48)~(50), (61), and (68), we obtain 𝑋_{𝑘∣𝑘−1}=𝐴_𝑘−1𝑋_{𝑘−1∣𝑘−1}+𝑁_𝑘−1Θ_{𝑘−1∣𝑘−1}+𝐵^Θ_𝑘−1Θ_{𝑘−1∣𝑘−1}+𝐵^𝑢_𝑘−1𝑈_𝑘−1=𝑋_{𝑘∣𝑘−1}−𝑢_𝑘−1+𝐴_𝑘−1𝑁_𝑘−1Θ_{𝑘−1∣𝑘−1}+𝐵^Θ_𝑘−1Θ_{𝑘−1∣𝑘−1}=𝑋_{𝑘∣𝑘−1}+𝑀_𝑘Θ_{𝑘∣𝑘−1}=𝑋_{𝑘∣𝑘−1}. (69) Using (6), (48), and (68), we obtain Θ_{𝑘∣𝑘−1}=𝐺_𝑘−1Θ_{𝑘−1∣𝑘−1}=𝐺_𝑘−1Θ_{𝑘−1∣𝑘−1}=Θ_{𝑘∣𝑘−1}. (70) Using (21), (26)~(28), (42), (43), and (45)~(47), we obtain 𝑃^𝑥_{𝑘∣𝑘−1}=𝜆_𝑘𝐹_𝑘−1𝑃^𝑥_{𝑘−1∣𝑘−1}𝐹^𝑇_𝑘−1+𝐸_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐸^𝑇_𝑘−1+𝐹_𝑘−1𝑃^𝑥Θ_{𝑘−1∣𝑘−1}𝐸^𝑇_𝑘−1+𝐸_𝑘−1𝑃^𝑥Θ_{𝑘−1∣𝑘−1}^𝑇𝐹^𝑇_𝑘−1+𝑄^𝑥_𝑘−1=𝜆_𝑘𝐹_𝑘−1𝑃^𝑥_{𝑘−1∣𝑘−1}𝐹^𝑇_𝑘−1+𝑄^𝑥_𝑘−1+𝐹_𝑘−1𝑁_𝑘−1+𝐸_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}×𝐹_𝑘−1𝑁_𝑘−1+𝐸_𝑘−1^𝑇=𝑃^𝑥_{𝑘∣𝑘−1}+𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑀^𝑇_𝑘=𝑃^𝑥_{𝑘∣𝑘−1}=𝑃¹¹_{𝑘∣𝑘−1}. (71) Using (21), (27), and (42), we obtain 𝑃^Θ_{𝑘∣𝑘−1}=𝜆_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^Θ_𝑘−1=𝜆_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^Θ_𝑘−1=𝑃^Θ_{𝑘∣𝑘−1}=𝑃²²_{𝑘∣𝑘−1}. (72) Using (21), (28), (30), (31), and (44), 𝑃^𝑥Θ_{𝑘∣𝑘−1}=𝜆_𝑘𝐹_𝑘−1𝑃^𝑥Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝐸_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^𝑥Θ_𝑘−1=𝜆_𝑘𝐹_𝑘−1𝑁_𝑘−1+𝐸_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^𝑥Θ_𝑘−1=𝜆_𝑘𝑀_𝑘𝐺_𝑘−1𝑃^Θ_{𝑘−1∣𝑘−1}𝐺^𝑇_𝑘−1+𝑄^𝑥Θ_𝑘−1=𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}=𝑃¹²_{𝑘∣𝑘−1}. (73) Using (10), (51), and (71)~(73), 𝐾^𝑥_𝑘=𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇+𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘𝑍_𝑘⁻¹=𝑃^𝑥_{𝑘∣𝑘−1}𝐻¹_𝑘^𝑇+𝐾^𝑥_𝑘𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘+𝑁_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘𝑍_𝑘⁻¹. (74) Using (64), (65), and (74), 𝐾^𝑥_𝑘=𝐾^𝑥_𝑘+𝑁_𝑘𝐾^Θ_𝑘. (75) Using (8), (51), (65), (72), and (73), 𝐾^Θ_𝑘=𝑃^𝑥Θ_{𝑘∣𝑘−1}^𝑇𝐻¹_𝑘^𝑇+𝑃^Θ_{𝑘∣𝑘−1}𝐻²_𝑘^𝑇𝑍_𝑘⁻¹=𝑃^Θ_{𝑘∣𝑘−1}𝑆^𝑇_𝑘𝑍_𝑘⁻¹=𝐾^Θ_𝑘. (76) Using (9), (51), (56), (60), (61), (69), (70), and (75), 𝑋_𝑘∣𝑘=𝑋_{𝑘∣𝑘−1}+𝐾^𝑥_𝑘𝑟_𝑘=𝑋_{𝑘∣𝑘−1}+𝐾^𝑥_𝑘𝑌_𝑘−𝐶_𝑘𝑋_{𝑘∣𝑘−1}+𝑀_𝑘−𝐾^𝑥_𝑘𝑆_𝑘Θ_{𝑘∣𝑘−1}+𝑁_𝑘𝐾^Θ_𝑘𝑟_𝑘=𝑋_𝑘∣𝑘+𝑁_𝑘Θ_𝑘∣𝑘=𝑋_𝑘∣𝑘, (77) where 𝑟_𝑘=𝑌_𝑘−𝐶_𝑘𝑋^𝑎_{𝑘∣𝑘−1}=𝑌_𝑘−𝐶_𝑘𝑋_{𝑘∣𝑘−1}−𝑆_𝑘Θ_{𝑘∣𝑘−1}. (78) Using (9), (70), (76), and (78), Θ_𝑘∣𝑘=Θ_{𝑘∣𝑘−1}+𝐾^Θ_𝑘𝑟_𝑘=Θ_{𝑘∣𝑘−1}+𝐾^Θ_𝑘𝑟_𝑘=Θ_𝑘∣𝑘. (79) Using (10), (29), (47), (51), (55), (60), (71), and (75), 𝑃^𝑥_𝑘∣𝑘=𝑃^𝑥_{𝑘∣𝑘−1}−𝐾^𝑥_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}+𝐻²_𝑘𝑃^𝑥Θ_{𝑘∣𝑘−1}^𝑇=𝑃^𝑥_{𝑘∣𝑘−1}+𝑀_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑀^𝑇_𝑘−𝐾^𝑥_𝑘+𝑁_𝑘𝐾^Θ_𝑘×𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}+𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑀^𝑇_𝑘=𝑃^𝑥_{𝑘∣𝑘−1}+𝐾^𝑥_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}+𝑁_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑀^𝑇_𝑘−𝐾^Θ_𝑘𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}𝑀^𝑇_𝑘−𝐾^Θ_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}=𝑃^𝑥_𝑘∣𝑘+𝑁_𝑘𝑃^Θ_𝑘∣𝑘𝑀^𝑇_𝑘−𝐾^Θ_𝑘𝐻¹_𝑘𝑃^𝑥_{𝑘∣𝑘−1}=𝑃^𝑥_𝑘∣𝑘+𝑁_𝑘𝑃^Θ_𝑘∣𝑘𝑁^𝑇_𝑘. (80) Using (10), (31), (51), (55), (58), (60), (71), and (72), 𝑃^𝑥Θ_𝑘∣𝑘=𝑃^𝑥Θ_{𝑘∣𝑘−1}−𝐾^𝑥_𝑘𝐻¹_𝑘𝑃^𝑥Θ_{𝑘∣𝑘−1}+𝐻²_𝑘𝑃^Θ_{𝑘∣𝑘−1}=𝑀_𝑘−𝐾^𝑥_𝑘𝑆_𝑘−𝑁_𝑘𝐾^Θ_𝑘𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}=𝑁_𝑘𝑃^Θ_𝑘∣𝑘. (81) Using (10), (30), (51), (55), (58), (72), (73), and (76), 𝑃^𝑥_𝑘∣𝑘=𝑃^Θ_{𝑘∣𝑘−1}−𝐾^Θ_𝑘𝐻¹_𝑘𝑃^𝑥Θ_{𝑘∣𝑘−1}+𝐻²_𝑘𝑃^Θ_{𝑘∣𝑘−1}^𝑇=𝑃^Θ_{𝑘∣𝑘−1}−𝐾^Θ_𝑘𝑆_𝑘𝑃^Θ_{𝑘∣𝑘−1}=𝑃^Θ_{𝑘∣𝑘−1}. (82) Finally, show that (68) holds at time 𝑘=0. This can be verified by the initial parameters in (62).

5. Experimental Results

The experiment of this paper aims at making a comparison among ATEKF, AFEKF, and EKF to verify validity and feasibility of ATEKF algorithm. The proposed PMSM sensorless control strategy based on ATEKF is shown in Figure 1. The drive system consists of a three-phase, eight-pole, 1.2 kW PMSM, driven from a three-phase insulated gate bipolar transistor (IGBT) inverter. The PMSM is mechanically coupled to a magnetic clutch, which provides rated torque, even at very low speeds. The drive system is implemented by an Expert3 control system of Myway company. The core DSP processor in Expert3 control system is TMS320C6713 with a clock speed of 225 MHz. The actual rotor position and speed are obtained from an incremental encoder with 10000 pulses per revolution. The inverter space-vector PWM control, current regulator, and observer algorithm operate with a 100 μs sampling/integration time step. The currents flowing in the stator windings are measured with two hall effect current sensors. The parameters of the PMSM are shown in Table 1.

Table 1 

PMSM prototype specification.

Figure 1 

System diagram of the proposed ATEKF sensorless scheme.

In the experiments, real-time parameter estimations observed by ATEKF are used to formulate the closed loop, such as rotor speed and position. The estimations obtained by conventional EKF or AFEKF are not included in the sensorless FOC and only evaluated in open loop.

Experiment 1 (equivalence between AFEKF and ATEKF). In this experiment, a comparison is made to verify the equivalence between AFEKF and ATEKF. The machine is accelerated from 0 rpm to 600 rpm at 0 s and the torque load is set to 2 Nm. The results are shown in Figure 2. During the transient, the maximum difference in speed and position measured by the two estimators is 0.0039 rpm and 0.0000037 rad, respectively. When PMSM reaches the given speed, the difference is rather null as we can see in Figure 2. Due to limited computer word, the differences can be seen as finite-word-length errors caused by single-precision arithmetic used in C code. These experimental results demonstrate that the two observers are mathematically equivalent.

(a) Speed estimation

(b) Difference of speed estimation

(c) Position estimation

(d) Difference of position estimation

Figure 2 

Experimental results for speed and position estimation (ATEKF and AFEKF).

Experiment 2 (dynamics in speed step). From Experiment 2 to Experiment 4, a comparison is made between ATEKF and EKF to verify the performance of the former.
The machine is accelerated from 0 rpm to 600 rpm at 0 s and the torque load is set to 2 Nm. Figures 3(a) and 3(c) show that ATEKF gets better dynamic speed response performance than EKF when PMSM is at the stage of start-up towards expected speed command. Figures 3(b) and 3(d) show the speed and position error in the ATEKF and the EKF for the sensorless control system. As expected, the steady errors of the ATEKF are smaller than those of the EKF.

(a) Speed estimation

(b) Speed error

(c) Position estimation

(d) Position error

Figure 3 

Experimental results with step change of speed reference.

Experiment 3 (dynamics in load-torque steps). Figure 4 shows the dynamic response of the sensorless drive with a load disturbance. The load torque varies from 2 Nm to 4 Nm, at a constant speed of 600 rpm. In response to disturbance acting in the electrical system, the actual velocity deviates significantly from the reference. It should be remarked that when the load changes, the speed response with ATEKF-based observer is faster than the one with EKF-based observer.

(a) Speed estimation

(b) Speed error

(c) Position estimation

(d) Position error

Figure 4 

Experimental results with step change of load torque.

Experiment 4 (dynamics in parameter mismatch). In order to further verify the robustness against model-plant parameter mismatches of the ATEKF, the change in stator resistance is considered. When the motor is running, stator resistance increases due to temperature. To simulate this condition, the stator resistances in ATEKF and EKF module are reduced to 80%. This is equivalent to the stator resistances which increased to 125% in PMSM. The machine is still accelerated from 0 rpm to 600 rpm at 0 s and the torque load is set to 2 Nm.
It is obvious that the motor speed and position observed by ATEKF can track their real value faster than EKF during the transient. A comparison is made between results of Experiments 2 and 4. Compared to Figure 3, Figure 5 shows that the increases in dynamic and steady errors of the ATEKF are obviously smaller than those of the EKF. This means that the performance, which results from the conventional EKF, is influenced heavily by increase in uncertainty, while in the proposed ATEKF the influence of the uncertainty is very small. Good robustness of ATEKF is verified.
The experimental results show that the ATEKF has strong robustness against model-plant parameter mismatches and good real-time state tracking ability.

(a) Speed estimation

(b) Speed error

(c) Position estimation

(d) Position error

Figure 5 

Experimental results with parameter variation.

6. Conclusion

To obtain satisfactory tracking results of EKF, the designers are required to have good knowledge about both the dynamic process and measurement models. Moreover, the real-time digital implementation of extended Kalman filters algorithm for sensorless FOC requires a very fast signal processor to perform complex mathematical calculation. The above demands are widely regarded as main limitations to apply EKF to PMSMs in the industrial field. So this paper proposes a novel extended Kalman filter called adaptive two-stage extended Kalman filter, which has the characteristics of both AFEKF and TSEKF. To design the ATEKF, the AFEKF is firstly designed. Then AFEKF is decoupled into two parallel filters to formulate ATEKF. Experimental results verified the robustness and strong tracking ability of ATEKF. On the other hand, the ATEKF algorithm provides a new idea to solve computational complexity and allows us to use cheaper DSP in practical application.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant no. 51377058.