Implementation and Evaluation of Online System Identification of Electromechanical Systems Using Adaptive Filters Parag Kshirsagar, Member IEEE, Dong Jiang, Member IEEE, Zhendong Zhang, Member IEEE United Technologies Research Center 411 Silver Ln, East Hartford, CT-06108
[email protected]
Abstract− Online system identification method provides useful insight into the electromechanical plant behavior during commissioning phase as well as health monitoring. In addition to controller tuning, the methods can be applied for identifying un-modelled mechanical resonances, sensor vibrations, bearing friction, etc. Accordingly, this paper focuses on the implementation and evaluation of real time frequency response of electromechanical systems using adaptive filters. Some of the challenges during implementation include selecting the structure of adaptive filters and understanding influence of convergence gain on accuracy of frequency response. In this work, online system identification is demonstrated on two setups (a) magnetic suspension system and (b) sensorless permanent magnet motor drive. The experimental result validate that the method elaborated is simple, easy to implement and accurate when compared with commercial off the shelf spectrum analyzer. Index Terms—System Identification, Electromagnetic System, Perturbation.
Adaptive
Filter,
I. INTRODUCTION For electromechanical systems, model based analysis for controller design is a widely adopted method [1]. However, variation in the plant model potentially caused by mechanical vibration of position sensors, installation resonances, bearing friction, gear backlash, and electric motor parameter variation may lead to undesired controller response and in worst case, loss of operation. Consequently, online system identification methods can provide better insight into the plant during installation phase and maintenance. The system identification methods are of two types (i) parametric and (ii) nonparametric. While parametric methods require prior knowledge of the plant model, nonparametric methods do not as they are based on input-output frequency response of the system under study. Frequency response analysis is advantageous for system identification methods due to features such as noise rejection, reduction in data size, and better model validation [2]. It is obtained by using the measured input and output signals of the plant, perturbed by a known excitation. The linear model of the dynamic system can then be evaluated real time over the frequency range of interest.
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Fig 1. Block diagram of online frequency response analysis using adaptive filtering method.
With the given background, this paper will evaluate nonparametric online system identification of electromechanical systems using adaptive filters. Block diagram of such a scheme is shown in Fig 1. In this method, input and output signals of either the plant or the controller can be reconstructed using adaptive weights of the filters thereby providing insight into magnitude and phase of the system under study. In the illustration shown, the feedback signal x’ and the controller error e due to periodic excitation signal (sine sweep) are used to generate adaptive weights which in turn are used to estimate the closed loop frequency response of the system. For frequency response analysis, apt selection of excitation signal is critical. As a result, white noise and its derivation such as filtered Gaussian white noise, random binary noise, and pseudo random binary sequence (PRBS) are commonly used [3]. Sinusoidal signals such as multi-sines, and swept sinusoids are other forms of periodic excitation signals [4]. In most of the reported power electronic applications, frequency response is analyzed by fast Fourier transform (FFT) or discrete Fourier transform (DFT) [4], [5], [6]. The Fourier transform method uses a linear combination of dilated trigonometric signals to represent the original data. In [4], a pseudo random binary signal (PRBS) is used as an excitation signal for system identification and cross-correlation is applied to process the DFT. While FFT provides frequency information, other methods such as wavelet transformation breaks up a signal into frequency components as a function of time [7]. Wavelet analysis has been applied in power systems
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for disturbance detection that affects power quality [8] and detection of broken-bar in induction machines [9]. For online system identification, the excitation and FFT based frequency analysis function are usually embedded in the DSP/FPGA software. The known drawback of such an approach is that the system must remain unchanged during the frequency response measurement. Consequently, the method has limitations for time varying systems [10]. Other limitations of such an approach are - buffer and array size allocation for performing calculations, noise rejection capability, data post processing requirement, and accuracy with wide side lobes [11]. The above limitations can be addressed through adaptive filters which have been widely applied for system identification problems [12]. The adaptive filters use least mean square (LMS) algorithm which is computationally less intensive and therefore favorable for real-time applications. Use of adaptive filters for frequency response has been discussed for communication applications in [13] and for parametric plant identification and control of DC/DC power converters in [14]. An online recursive least square (RLS) based parametric identification method for electromechanical system has been elaborated in [15]. As discussed earlier, parametric methods require prior knowledge of the plant and may require continuous parameter estimation for desired convergence. Therefore, a simple non-parametric transfer function method for online system identification of electromechanical systems without resort to FFT, wavelet transform or parametric identification is worthy of research. It is essential to address selection of adaptive filter structure and selection of adaptation or convergence gain on accuracy of frequency response. Accordingly, this paper is organized as follows: Principle of system identification using adaptive filters is elaborated in section II. Influence of convergence gain of adaptive filters is discussed in same section. In section III, online system identification method is demonstrated experimentally on two setups (a) magnetic suspension system and (b) permanent magnet motor drive operating in sensorless mode. Finally advantages and limitations of the method are discussed in section IV. II. FREQUENCY RESPONSE USING ADAPTIVE FILTERS Frequency response of electromechanical plant can be identified using parametric and nonparametric modeling approaches. In both cases the system is assumed to be linear and continuous in time. The following sections will discuss the same.
(a)
(b) Fig.2 Block diagram of online system identification methods using adaptive filter (a) parametric (b) non-parametric.
periodic excitation signal. The adaptive parametric transfer function of the plant is then given as [2] ∑ ∑
(1)
where k is the polynomial order of the transfer function. The outputs of plant and adaptive model are compared and the and error ek is used to continuously update the weights of the adaptive model in (1) until they converge to the desired value. In this method, an initial estimation of the polynomial order is necessary. B. Nonparametric model identification using adaptive filter Fig 2(b) shows block diagram of nonparametric system identification method using adaptive filter. As seen in the figure, both input and output variables xk and gk of the plant are reconstructed by the adaptive filter as hxk and hdk respectively. The input signal can be represented in terms of adaptive weights w1x and w2x as xk = w1x cos(ωot) + w2x sin(ωot) while the output signal can be represented as dk = w1d cos(ωot) + w2d sin(ωot). With this formulation, the adaptive weights can be used to estimate the magnitude and phase response as
A. Parametric model identification Fig 2(a) shows block diagram of parametric online system identification method. As illustrated, the input to the unknown plant and the adaptive model is periodically excited through a sine sweep signal xk resulting in output signals gk and yk respectively. In the illustration, ωo is the frequency of the
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(2) tan
tan
where is the magnitude response and is the phase response of the transfer function. This approach does not require prior estimation of the order of the mode. For good frequency response, it is essential to understand the operation of adaptive filter and is discussed in following section.
Illustration of tracking of input signal dk by the reconstructed or estimated signal yk is shown in Fig 4. Once the steady state has been reached, the magnitude and phase of the input signal is calculated at the excitation frequency using (2). For the filter shown in Fig 3, the closed loop transfer function of the narrow band pass filter in continuous and discrete form is given as
C. Structure of adaptive filter and tracking response
2 2 2 2 1
(4) 2
cos cos
1
2
respectively [12] ,[16]. Where, ωn is the center frequency, 2μ is its bandwidth, and Ts is the sampling time. Then the corresponding transfer function of the adaptive notch filter in continuous and discrete form is given as 1- H(s) and 1-H(z) respectively.
Fig.3 Adaptive notch filter for active noise cancellation [6]
Adaptive filter is essentially a notch filter with adjustable bandwidth around the center frequency and can adaptively track the input signal. Its structure is shown in Fig. 3 [6]. As shown in the figure, the reconstructed or estimated signal yk is compared with the input signal dk. Any error ek in estimation of the input signal is fed into an accumulator using Least Mean Square (LMS) algorithm. For convergence, the error ek is driven to zero through continuous adaptation of the weights determines the rate of w1k and w2k. The adaptation gain convergence and also stability of the filter. The reconstructed signal yk based on input signal is expressed as cos
sin Fig. 5. Frequency response of adaptive filter
cos
2
sin
2
(3)
where, ω0 is the known periodic excitation frequency. The weights are essentially Fourier coefficients at the excitation frequency of the reference signal and hence feasible for reconstructing input signal [6].
= 628 Frequency response of H(s) and H(z) for = 0.5, rad/sec and Ts = 100 sec is shown in Fig 5. From the closed loop transfer function, it can be observed that adaptive filters have very good noise rejection capability at the excitation frequency. Also, the discrete and continuous time models overlaps in magnitude and a small deviation in phase due to 10 kHz sampling rate.
Amplitude
1V/div
estimated signal yk
input signal dk
50ms
Fig 6. Step response of adaptive filter for various values of μ
Fig 4. Experimental results: Adaptive filter tracking of input signal
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The adaptation gain μ, must be chosen to yield faster convergence, smaller overshoot and noise rejection. Selection of this gain is important for accurate and stable tracking performance especially in case of sine sweep input signal. Influence of adaptation gain based on step response of closed loop transfer function of adaptive filter is shown in Fig 6. As seen in Fig 6, for the case with adaptation gain μ=6, the settling time (convergence rate) is 0.5 sec and the overshoot is 30 percent, while for μ=1, the settling time is around 1 sec and the overshoot is only 10 percent. The result implies that for higher values of adaptation gain, the settling time is much lower and vice versa. It is critical to consider the tradeoff in selecting the gain μ during estimation of magnitude and phase response of the input and output signals. This is because the settling time will determine the choice of the sweep rate and the frequency. For periodic excitation signal, the sweep frequency is calculated as 10∆
A single axis magnetic suspension system illustrated in Fig 7 comprises of a control object (magnetic rotor) which is suspended in an air gap xo using two electromagnets, power amplifiers and a closed loop position controller. The control currents ic1 and ic2 in the electromagnets are controlled to balance the top and bottom forces in the coils thereby achieving rotor levitation.
1
∆
(5)
where, fn is the sweep frequency at the current point, fstart is the start frequency of the sweep, fstop is the sweep stop frequency, N is total number of points during the sweep and n is the current swept point. The new sweep frequency must be updated only after the adaptive filter converges. Failure to do so may lead to erroneous frequency response. Influence of the adaption gain on plant noise will be discussed in section III.B. With the above background, experimental results of online system identification using adaptive filters for frequency response is discussed in the following sections. III. NONPARAMETRIC SYSTEM IDENTIFICATION ON SELECTED APPLICATIONS
Fig. 8. Control block diagram of single axis magnetic suspension system.
The system is inherently unstable and position control is essential. The position closed loop transfer function is determined by the controller and the plant model. In order to tune the controller, information of the closed loop transfer function identification is important. Controller block diagram of the system in Fig 7 is shown in Fig 8. The closed loop system comprises of cascaded models of PID controller, low pass filter, amplifier, electromagnet coils, mechanical system and the position feedback sensor. The variables in digital controller are used for frequency response analysis using adaptive filtering method. Before evaluating the results, simplified analytical transfer functions of the control blocks are given as follows: 1. PID controller model
A. Single axis magnetic suspension system
(6)
where, , and are the proportional, derivative and integral gains of the controller. 2. Filter model 1
(7)
1
where, is the low pass filter time constant. The low pass filter is essential to reduce the noise due to derivative gain of the PID controller. 3. The amplifier and electromagnets are modeled together as a first order delay element due to limited current control bandwidth. It is expressed as 1 1
Fig.7. Illustration of single axis magnetic suspension system.
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(8)
where, 1/ bandwidth.
and
is the closed loop current control
4. The linear mechanical model of the plant is given as (9)
where, m is the mass of rotor, is the force to current factor is the force to of the electromagnetic coils while displacement factor also known as negative stiffness. The sensor feedback transfer function R(s) is assumed to be unity due to its high bandwidth compared to the electromechanical system. Based on the analytical plant model, the position controller gains for initial levitation are derived as [17]. 2
,
1.414 (9)
where, is the desired cross over frequency and is chosen as 30Hz. The proportional, derivative and integral gains are then re-tuned experimentally to achieve stable performance. The parameters of the plant and controller are listed in Table I. Table I. Parameters of single axis suspension system
Name Proportional gain Differential gain Integral gain Filter time constant Current control bandwidth Force to current factor Negative stiffness Mass of rotor DC link voltage
Symbol
m Vdc
Value 28 A/mm 0.246 A/sec-mm 5.32 A-sec/mm 1/(2π200) sec 2π(600) rad/sec 316 N/A 2.37e6 N/m 22 lb 160V
The objective in this evaluation is to compare the position closed loop transfer function x(s)/x(s)* analytically and experimentally using spectrum analyzer as well as adaptive filter based approach. In Fig. 8, the sinusoidal excitation signal is swept from 1 Hz to 200 Hz to evaluate the dynamic electromechanical performance of the system. For synthesizing frequency response using spectrum analyzer, the sine sweep is sampled by the analog to digital input of the controller and then added to the reference position. The feedback position is tied to the input channel of the spectrum analyzer and thus the closed loop transfer function is identified as a ratio of feedback to reference signal. The spectrum analyzer uses FFT method for the frequency response. For digital implementation of frequency response using adaptive filter, the sinusoidal periodic excitation signal is generated using equation (5). The sampling frequency is 10 kHz while the adaptation gain μ is set to 10. The input-output transfer function is estimated online during the sine sweep using equation (2).
Fig. 9. Frequency response comparison of position closed loop transfer function using analytical model, spectrum analyzer and adaptive filter based approach.
Fig. 9 shows frequency response comparison of position closed loop transfer function x(s)/x(s)* using analytical model, spectrum analyzer and adaptive filter based approach. Following observations are made from Fig 9: 1. Within the frequency range of interest, both magnitude and phase response of the analytical model (black trace) shows good match with experimental models. The analytical model provides better insight into the behavior of the system and hence used as a benchmark. 2. The spectrum analyzer (blue trace) shows various mechanical resonances between 40Hz and 60Hz. These resonances are not captured by the analytical model. 3. The online adaptive filter based frequency response mostly overlays on the spectrum analyzer result. It is able to track the mechanical modes between 40Hz and 60Hz very well. Overall, the adaptive filter method works fairly well in comparison to spectrum analyzer given that the plant is considered as an unknown. Application of system identification approach on field oriented sensorless control of permanent magnet synchronous motor (PMSM) drive is discussed in the following section. B. PMSM sensorless field oriented control Implementation of sensorless filed oriented control (FOC) for PMSM for fan type application is elaborated in [18]. In the same work, various controllers and their performances was evaluated using spectrum analyzer. Block diagram of the sensorless control scheme is shown in Fig.10. The method is based on Luenberger observer for back-EMF estimation and then tracking controller for position/speed estimation. The estimated position is used for co-ordinate transformation to close the loop on current and speed regulators. The performance if close to classical field oriented vector control. In this section, the effectiveness of online adaptive filter based frequency response will be demonstrated by estimating
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iqrˆ
ωref
ωˆ r Ddq
Ddq
iq _ ref
FW
max
vq _ decoupling
id _ ref
vd _ decoupling
idrˆ
Ddq
∫ ωˆ r
θˆr
rˆ vdq
rˆ idq
eˆdrˆ
θˆr eˆqrˆ
Fig.10 Diagram of sensorless FOC for PMSM [18]
the transfer function of speed regulator ( /ωr*) and tracking controller ( ̂ / ) as an example. These controllers are chosen since they represent the electromechanical time constants of the system. Since the analytical models have been discussed in [18], only models based on experimental results will be discussed henceforth. a.
Experimental setup
The experimental set-up of sensorless control of PMSM is shown in Fig.11. It consists of a 10-kW motor drive on the load side and a 7.5-kW motor drive on the test side. A PSM 1600 spectrum analyzer is used for evaluating the transfer functions at various points in the control block diagram. As discussed in section III.A, the online adaptive filter method is embedded in the DSP. For all the test cases, the periodic signal injection amplitude is limited to 5 percent of the maximum reference amplitude while the frequency is swept from 1Hz to 20Hz. In case of spectrum analyzer, the number of cycles for every sweep frequency is 8 while in case of online adaptive filter method, it is 20.
The total sweep time required in both cases is less than 120 seconds. b.
Speed regulator
The speed loop of PMSM drive is shown in Fig.12. The speed closed loop comprises of a PI regulator, torque constant KT , rotor inertia Jm and number of poles P. As shown in the figure, an online adaptive filtering algorithm is added to estimate the closed loop frequency response of the speed regulator. The periodic excitation signal is added to the reference speed and the corresponding variables for closed loop transfer function are monitored.
Fig.12 Speed loop of PMSM drive
Fig.11 Experimental set-up of sensorless control of PMSM
Fig.13 shows the experimental results of closed loop frequency response ( /ωr*) and comparison between the spectrum analyzer and online adaptive filter based approach. For both cases, the speed closed loop bandwidth is 1.2 Hz. The results show that the frequency response using online adaptive filter is comparable with that of the spectrum analyzer. The peaking of the magnitude and phase response of the spectrum analyzer near 5.2Hz is due to absence of averaging function. For spectrum analyzers, averaging is commonly used in case of noisy input signals to achieve a filtered frequency response.
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c.
0
Fig.15 shows the diagram of the phase locked loop for the rotor position estimation also known as tracking controller. It comprises of a PI regulator that forces the estimated d-axis back EMF voltage to zero by aligning the estimated rotor position to the actual one. The back EMF is estimated by a state observer in rotor reference frame which in turn is dependent on the estimated rotor position. The phase lock loop ensures that the observer model is tracking the PMSM drive model. The periodic excitation signal is added to the reference voltage (zero in this case) and the corresponding variables for closed loop transfer function are monitored.
1
0
Tracking controller
10
Frequency / Hz
Fig.13 Frequency response comparison of speed closed loop transfer function between spectrum analyzer and online adaptive filter based approach.
The aforementioned results are reported based on a single sine sweep and no averaging. While the aforementioned closed loop transfer function results show a good match, it is essential to evaluate the influence of the adaptation gain µ. Accordingly, Fig.14 shows the speed closed loop response with µ=1 and µ=100, when compared with spectrum analyzer. It is seen that with µ=100, the magnitude response of online adaptive filter overlays on the spectrum analyzer response implying less phase delay. Since µ also determines the pass band frequency of the adaptive filter, higher values will result in more noise into the system. This can be seen from the phase response near 4Hz. For µ=1, frequency response of online adaptive filter is smooth however with phase delay. Therefore, as discussed in section II.c, there exists a tradeoff between noise rejection capability and convergence time (phase delay) based on choice of the adaptation gain.
Fig. 15 PLL-based rotor speed and position estimation scheme using a PI controller
0
1
0
10
Frequency / Hz
10
1
Frequency / Hz
Fig. 16 Frequency response comparison of tracking controller transfer function between spectrum analyzer and online adaptive filter based approach
Fig.14 Frequency response comparison of speed closed loop transfer function between spectrum analyzer and online adaptive filter based approach for various adaptation gains.
For the tracking controller, Fig 16 shows the frequency response comparison between spectrum analyzer and online adaptive filter based approach. It is seen that the magnitude and phase response identified by adaptive filter matches the result from spectrum analyzer very well. The variation in phase near 1.8Hz is due to frequency wrapping. Below the closed loop bandwidth of the controller, the phase curve stays
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around 180 degrees because the gain of the back EMF observer for d-axis is -1. This gain is compensated in block Glin of the controller.
[2]
Frequency response analysis is commonly used for nonparametric system identification. Based on the theory described in section II, the experimental results validate the methodology and effectiveness on use of online adaptive filters for frequency response analysis. Distinct advantages of the online implementation method are; no buffer and arrays required for performing calculations and better noise rejection capability based on choice of adaptation gain. The known limitation of adaptive filters is its convergence time and accuracy tradeoff based on the choice of adaptation gain. This limitation requires the frequency sweep to be slow. Consequently, the overall response time using adaptive filter method can be slower than PRBS DFT based approach. Although the requirement of system identification response time is dependent on the particular application, opportunity for improvement exists.
[3] [4]
[5]
[6]
[7]
[8]
IV. CONCLUSIONS Online nonparametric system identification of electromechanical system using adaptive filters can potentially eliminate use of external spectrum analyzer because of its simplicity, noise rejection capability and simultaneous multichannel output capability. It is also computationally less intensive and therefore favorable for real-time applications. Following are the key conclusions and contributions of this work: 1.
2.
3.
4.
Online frequency response analysis using adaptive filters is evaluated for electromechanical systems. It is easy to implement frequency response of input output signals for generating transfer functions on state of art micro-controllers. In this work, method for online frequency response analysis using adaptive filters is described systematically. The influence of adaptation gain on accuracy of frequency response is highlighted through closed loop transfer functions. The online system identification using adaptive filters was demonstrated experimentally on a single axis magnetic suspension system. Experimental results are compared with analytical models to validate the approach used. The results show a good match between the frequency response of adaptive filter based approach and the spectrum analyzer. Influence of adaptation gain on convergence time and accuracy of frequency response has been demonstrated experimentally on PMSM drive system operating in sensorless mode. The results validate the effectiveness of the approach used.
[9]
[10]
[11]
[12] [13]
[14]
[15] [16]
[17]
[18]
REFERENCES [1]
Astrom, K.J. and Hagglund, T., “Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins” Automatica, Vol. 20, No.5, pp. 645-651, 1984.
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R. Pintelon, P. Guillaume, Y. Rolain, J. Schoukens, H. Van hamme, “Parametric Identification of Transfer Functions in the Frequency Domain-A Survey,” IEEE Trans. Automatic Control, vol. 39, no. 11, pp. 2245 – 2260, Nov. 1994. L.Ljung, System Identification: Theory for the User, 2nd ed.: Upper Saddle River, NJ: Prentice Hall, 1999. Botao Miao; Zane, R.; Maksimovic, D., "System identification of power converters with digital control through crosscorrelation methods," Power Electronics, IEEE Transactions on, vol.20, no.5, pp.1093,1099, Sept. 2005 N. Kong, A. Davoudi, M. Hagen, E. Oettinger, M. Xu, D. S. Ha, and F. C. Lee, “Automated System Identification of Digitally-Controlled Multi-phase DC-DC Converters,” in Proc. 2009 IEEE APEC, pp. 259-263, April. 2009. D. Martin, E. Santi, and A. Barkley, “Wide Bandwidth System Identification of AC System Impedances by Applying Perturbations to an Existing Converter,” in Proc. 2011 IEEE ECCE, pp. 2549-2556, Sept. 2011. K. F. Alvin, A. N. Robertson, G. W. Reich, K. C. Park, “Structural System Identification: from Reality to Models.” Computers and Structures 81 (2003) 149~1176. M. Karimi, H. Mokhtari, and M. R. Iravani, “Wavelet based on-line disturbance detection for power quality applications,” Power Delivery IEEE Transactions on., vol.15, no.4, pp.12121220, Oct. 2000 J. Antonino, M. Riera, J. Roger-Folch, and M. P. Molina, “Validating of a new method for the diagnosis of rotor bar failures via wavelet transform in industrial induction machine,” IEEE Trans. Ind. Appl., vol. 42, no.4, pp. 990-996, Jul./Aug. 2006. Rife, D.D. and Vanderkooy, J. (1989) Transfer Function measurement with Maximum-Length Sequences”, J. Audio Eng. Soc., Vol. 37, No. 6, pp.419-443. Jian Li ; Stoica, Petre, “An adaptive filtering approach to spectral estimation and SAR imaging”, IEEE Transactions on Signal Processing, Vol: 44 , Issue: 6 pp 1469-1484, 1996 Bernard Widrow; Samuel D. Stearns, “Adaptive Signal Processing”, Prentice-Hall Press N.A.S. Alwan and M.A.R. Sahib, “Frequency response identification using exact adaptive frequency sampling filters ”, Emirates Journal for Engineering Research, Vol. 11, No.2, 2006 M. Algreer, M. Armstrong, and D. Giaouris, “Active On-Line System Identification of Switch Mode DC-DC Power Converter Based on Efficient Recursive DCD-IIR Adaptive Filter”, IEEE Transactions on Power Electronics, vol.27, pp.4425-4435, Nov. 2012. Ilyas Eker, “Experimental on-line identification of an electromechanical system”, ISA Transactions 43 (2004) 13–22. V. Blasko, L. Arnedo, P. Kshirsagar and S. Dwari, “Control and elimination of sinusoidal harmonics in power electronics equipment: A system approach, IEEE Energy Conversion Congress Exposition (ECCE) Annu. Meeting, 2011 pp- 2827– 2837 Akira Chiba, Tadashi Fukao, Osamu Ichikawa and Masahide Oshima, “Magnetic Bearings and Bearingless Drives”, Elsevier Press, 2005 P. Kshirsagar, R.P Burgos, J. Jang, A.Lidozzi, , F. Wang, D. Boroyevich, S. K. Sul, “Implementation and Sensorless Vector-Control Design and Tuning Strategy for SMPM Machines in Fan-Type Applications”, IEEE Transactions on Industry Applications Special Issue pp. 2402-2413, Nov/Dec 2012.
