Online Identification of a Mechanical System in Frequency Domain ...

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Online Identification of a Mechanical System in Frequency Domain Using Sliding DFT Niko Nevaranta, Student Member, IEEE, Stijn Derammelaere, Member, IEEE, Jukka Parkkinen, Bram Vervisch, Tuomo Lindh, Kurt Stockman, Markku Niemelä, Olli Pyrhönen and Juha Pyrhönen, Member, IEEE  Abstract—A proper real-time system identification method is of great importance in order to acquire an analytical model that sufficiently represents the characteristics of the monitored system. While the use of different time-domain online identification techniques has been widely recognized as a powerful approach to system diagnostics, the frequency-domain identification techniques have primarily been considered for offline commissioning purposes. This paper addresses issues in the online frequency-domain identification of a mechanical system with varying dynamics; particular attention is paid to detect the changes in the system dynamics. A closedloop online identification method is presented that is based on a Sliding Discrete Fourier Transform (SDFT) at a selected set of frequencies. The method is experimentally validated by a closed-loop controlled servomechanism with a limited stroke and time-varying parameters. Index Terms—Nonparametric estimation, identification, Servomechanism, Sliding DFT

Online

I. INTRODUCTION

T

HE monitoring of a mechanical system in electric drives has become an increasingly important feature in different high-performance motion control applications such as robotics, machine tools, material handling, and packaging, to name but a few. In demanding motion control applications, the control performance plays an important role. At the same time, requirements for high reliability are continuously increasing, which significantly motivates to improve methods, tools, and techniques for the diagnostics and condition monitoring of a mechanical system. As the deterioration of mechanical parts Manuscript received September 30, 2015; revised March 1, 2016; accepted April 19, 2016. N. Nevaranta, J. Parkkinen, T. Lindh, M. Niemelä, O. Pyrhönen and J. Pyrhönen are with the Department of Electrical Engineering, Lappeenranta University of Technology (LUT), 53850, Lappeenranta, Finland, (e-mail: [email protected]; [email protected]; [email protected], [email protected]; [email protected]; [email protected]). S. Derammelaere, B. Vervisch and K. Stockman are with the Department of Electrical Energy Systems and Automation, Ghent University Campus Kortrijk, BE-8510, Kortrijk, Belgium, ([email protected]; [email protected]; [email protected]).

over time or other unexpected changes in the system dynamics may lead to the degradation of the control performance or cause unexpected interruptions, it is important to detect the system changes as proactive maintenance before they lead to performance degradation. For these reasons, different realtime system identification techniques for the monitoring of mechanical parts are viable, for example, to enhance the reliability of electrical drives. In general, a mechanical system can be identified either in offline or online mode by time- or frequency-domain observations. In the literature, the identification routines can be classified into two main categories: nonparametric estimation methods [1] and parametric estimation methods [2] –[5]. Especially, for commissioning purposes, the Fourier analysis is a well-known method for the offline nonparametric frequency domain identification of linear systems [6]. Correspondingly, different time domain parametric methods based on a least squares criterion are commonly applied to the identification of mechanical systems in the open-loop [7] or closed-loop control [7], [8]. However, these methods have high memory storage requirements for data acquisition or require extensive calculations that are not usually desirable features for real-time applications. Thus, for online identification purposes, different recursive time-domain parametric estimation methods such as recursive least squares (RLS) have received the most attention because of their easy real-time implementation and computation efficiency [9]–[10]. Similarly, the extended Kalman filter is a widely recognized tool for estimating the parameters of a mechanical system online [11]–[13]. Despite the theoretical development of offline frequencydomain identification methods [1], [14], [15], there are only a few studies available on the issues related to the use of frequency-domain techniques for online monitoring purposes. The real-time frequency domain identification techniques could introduce attractive features for example to the online monitoring of a mechanical system at a selected set of frequencies. In contrast to parametric methods, an adequate online identified nonparametric model can be used not only to analyze how changes affect system performance [16], but also for the design of an adaptive control law, a subject that has been discussed in [17]–[21]. Moreover, in [22], a real-time equation error method based on a finite Fourier transform in the frequency domain is used for linear model identification.


Another method has been introduced in [23], where a FourierLaguerre series is proposed for the open-loop identification of a linear system. However, these methods are parametric identification methods, which, in practice require initial selection of the model complexity. Furthermore, in [24], a Kalman filter has been configured to perform like a Fourier transform. However, this method uses only one measurement sample at a time and gives only a rough estimate of the frequency behavior. Finally, the difficulties related to the realtime frequency domain identification have been discussed in [25] and a modular virtual-instrument based real-time identification procedure have been proposed. The proposed procedure, however, can be regarded as a general idea to use standardized tools for on-line identification rather than an identification method. The Sliding Discrete Fourier Transform (SDFT) is well known as a simple but effective technique for real-time spectral analysis, and it has been used in some industrial applications [26]–[30]. Despite the success in different applications, to the authors’ knowledge, the SDFT has not yet been applied for real-time non-parametric identification purposes. A similar type of online identification approach has been proposed in [17]–[18] that uses a sliding moving window to calculate the online DFT, but the estimator is used to fit the parametric model which increases computational burden. Furthermore, the method has been proposed for the identification for control purposes, and is thus an open-loop identification method. Naturally, as the identified model is used for controller design, then achieving the control performance goals are more important rather than the quality of the obtained identification result. However, it appears no papers have discussed the monitoring and identification of changes of mechanical system with SDFT at a selected set of frequencies by considering persistent multi-sine excitation signal. It is worth pointing out that recent study by [30] applied the SDFT for vibration mode estimation of parameterchanging flexible manipulator using sinusoidal excitation signal, but focuses more on the tracking properties of the SDFT rather than system identification. The objective of this paper is to show that the SDFT can be effectively used for frequency domain based on-line monitoring and parameter estimation of a mechanical system with varying dynamics at a selected set of frequencies. The established standard frequency-domain indirect identification principle for conventional closed-loop offline identification is extended to an online identification scheme. The performance of the proposed online frequency domain identification method is verified by an experimental closed-loop-controlled servomechanism with a limited stroke and time-varying parameters. This paper is organized as follows. Section II discusses frequency-domain identification and the problem statement. Furthermore, the application of Sliding DFT to the real-time spectrum analysis is discussed. After that, in Section III the mechanical system under study is described in brief and the proposed identification method is studied by simulations. In Section IV, the experimental on-line identification results are shown and analyzed, and Section V concludes the paper.

w (t ) r(t )

-

C(z) Control

+ u(t )

y(t )

G(z)

+

System n(t )

Fig. 1. Closed loop system identification. The excitation signal w(t) is added to the controller C(z) output. Noise n(t) affects the system output and feedback.

II. PROBLEM STATEMENT Frequency-domain identification is a well-established and common approach for different systems. The primary disadvantage of the frequency-domain analysis includes the required calculation of a discrete Fourier transform on the measured data, which is essentially a batch process. Compared with the time-domain identification, the advantage of using non-parametric model of the system in the frequency-domain reduces data reduction and the estimation is performed only at the frequencies where excitation is provided. However, extensive calculations are required that are not usually desirable features for real-time applications. For online identification purposes, the monitoring of the mechanical system at a selected set of frequencies is a desirable feature. When the behavior of these frequencies has to be tracked in real-time, it is worth considering a DFT algorithm that provides benefits in the terms of computational efficiency and real-time performance. A. Frequency Domain Identification Typically, mechanical systems are identified either in openloop or closed-loop. In the case of open-loop identification, the direct identification is straightforward as a parametric or nonparametric model can be built from the input u(t) and output y(t) observations. By considering the relation of the spectral density functions Suy(jω) and Suu(jω), the frequency response function estimate becomes S uy ( j ) Gˆ ( j )  S uu ( j )

(1)

The spectral analysis can be achieved in different ways [1], but in general, (1) gives a correct estimation of the real system G(jω) based on the assumption that the measurement noise n(t) and the input u(t) are uncorrelated meaning that Sun(jω) = 0. When closed-loop identification is considered, as depicted in Fig. 1, there is a correlation between the system input u(t) and the noise n(t) as a result of the feedback loop and the controller. This means that the approach described in (1) will give close to correct results if a low bandwidth controller [1], [4], which filters the noise, is used. However, when noise is affecting the system input u(t), the method will give poor frequency response estimations. As the identification of real-world systems must often be performed in closed-loop because of stability, performance, or safety constraints, an indirect closed-loop identification method is


Resonators

e j2πk1 / N

X k1 (n)

Memory/Buffer Fig. 2. An array of discrete frequencies obtained by a full FFT during the measurement time Tmes and with the sample time Ts.

x(n)

e j2πk2 / N

N

  

Resonator

Memory/Buffer Add the most recent sample

+ -

x(n)

z-N

ej2πk/N

+ +

Xk(n)

ej2πki / N

X ki (n)

Fig. 4. Sliding Discrete Fourier Transform tracking different frequencies. The index i represent the number of the frequencies to be considered.

z-1

Xk(n-1)

x(n-N)

Xk2 (n)

remove the oldest sample

(a) Resonator

Memory/Buffer Add the most recent sample

Xk(n)

+ -

x(n)

+ +

ej2πk/N z

-N

x(n-N)

remove the oldest sample

z-1 Initially: 1

z

-1

u

(b)

Fig. 3. a) Sliding Discrete Fourier Transform (SDFT) implementation. b) Modulated SDFT implementation. The notation u̅ is used to represent complex conjugate.

more preferable. In the indirect method, the knowledge of the excitation signal w(t) is used, and thus, the open-loop frequency response can be estimated as the relation of the spectral density functions Swy(jω) and Swu(jω) S wy ( j ) Gˆ ( j )  S wu ( j )

(2)

The spectral analysis methods in (1) and (2) are commonly applied to offline commissioning of closed loop controlled mechanical system [1], [6]. B. Sliding Discrete Fourier Transform The standard method for spectrum analysis is the discrete Fourier transformation, which requires that all the measurement samples over a full measurement window have to be summed and multiplied by an exponential function. At a certain discrete time instant n, the k-th order harmonic within a period of N samples can be calculated as N 1

X k ( n) 

 x(n  ( N 1)  l )e

 jk ( 2 / N )l

(3)

l 0

An array of discrete frequencies, which can be obtained by the FFT, taking into account the Shannon Nyquist criterion, are depicted in Fig. 2. When the behavior of all these frequencies has to be tracked in real-time, this means that the sum in (3) has to be reconsidered N/2 times at each new time instant n, n+1, n+2,…. This implies an undesirable computational burden when real-time calculation is considered. Therefore, in this paper, the Sliding Discrete

Fourier Transform (SDFT) based spectrum analysis is considered as illustrated in Fig. 3 a). The SDFT relies on the fact that at each new time instant n, only one measurement sample is added to the sum, and only the oldest measurement sample is removed from the window. This can be considered as a kind of buffer (see left part of Fig. 3). On the other hand, the right part, where the previous SDFT component Xk(n-1) is taken into account and the multiplication by the exponential function takes place, can be seen as the resonator for the k-th harmonic component. This reveals another advantage of the SDFT as in practice only one buffer with the window length N is needed to track different frequency orders k, as depicted in Fig. 4. For one harmonic order k, the SDFT can be expressed by the following equation [33] that can be derived from (3) X k (n)  ( X k (n  1)  x(n)  x(n  N ))e jk ( 2 / N )

(4)

The major advantage of the Sliding DFT (4) over the traditional DFT (3) is the computational efficiency acquired by the sliding window that is used to calculate a new DFT bin from the previous DFT result. Thus, regardless of the window length N, the SDFT calculations to obtain Xk(n) are constant after the coefficient Xk(n-1) has been obtained. Based on (4) and Fig. 3, the SDFT can easily be written as a transfer function in the z domain H SDFT ( z ) 

1  z e N

1 e

j k 2 N

jk 2 N

z 1

(5)

As can be seen in (5) the pole of the transfer function is located exactly on the unit circle at ejk2π/N , as a result of which the transformation is only marginally stable. Furthermore, because of the limited numerical precision of this coefficient, the pole can be just in- or outside the unit circle causing numerical instability. However, since the introduction of the SDFT in [31], adaptations and optimizations [32], [33] have been proposed to obtain guaranteed stable SDFT results. In this paper, the modulated SDFT (mSDFT) proposed in [33] is used instead of the standard SDFT to avoid possible rounding errors and instabilities. It is worth mentioning that, the online identification method proposed in [17] can be regarded as a standard form of the SDFT, but in practice, the method is not stable. Using the mSDFT structure with the recursive computation of the modulating sequence and the phase


correction proposed in [33] and as depicted in Fig. 3 b), stability can be guaranteed, but the computational burden is slightly increased compared to standard SDFT. It is worth pointing out that in this paper the mSDFT method [33] is used to calculate DFT on a sample-by-sample basis, but it is referred to as the SDFT in later sections of this paper. The reason for this terminology choice is that the online identification method is based on the utilization of sliding window and the theory proposed can be further extended to different SDFT approaches [31] as well. C. Excitation Signal and Frequencies of Interest In general, when considering the identification of a mechanical system the excitation signal should be rich and persistently exciting the system, and thus, should preferably have a white noise characteristics e.g. a signal having a flat modulus spectrum. These type of characteristics can be obtained for example with PRBS and Schroeder multisine. In general, these type of excitation signals are preferable for most of the mechanical loads and especially the PRBS has been widely used in the identification of different mechanical systems [1],[6],[7]. The SDFT approaches are ideally suited to track the system behavior at certain frequencies. When considering system identifi cation in the frequency domain, the possible leakage problems can be considered as a major drawback. In the case of real-time spectral analysis, [31] suggests windowing techniques applied to the SDFT coefficients Xk-1, Xk, Xk+1 to counteract the leakage problem. However, for system identification experiments as depicted in Fig. 1, the user can exactly determine the excitation signal. Therefore, a multisine signal is ideally suited for the excitation signal

w(t ) 



sin 2π NT k

s

  k  . 

(6)

In this paper Schroeder multisine is considered. The phase of the multisine signals should be chosen carefully in order to avoid a poor crest factor, meaning too high a signal peak compared with the RMS content. When the excitation signal becomes too large, this may disturb the closed-loop controller or otherwise excite non-linearity in the system. Therefore, the crest factor optimization should be considered. Recent research on this topic is available in [34]. However, the Schroeder formula described in [35] is very straightforward and gives, although not optimal [14], acceptable crest factor values

k  1 

πk . k max

(7)

The fact that phases are not random over k makes it less suited for nonlinear detection [14]. However, it is worth pointing out that the identification problem is treated in this paper as a linear time-invariant (LTI). Moreover, to avoid leakage in the case of the SDFT, the frequency components in the excitation signal and the array of k factors for the different resonators should match each other. Thus, based on the a

l1(x)

l2(x) mL

φ1

K1(x) r

φ2

K2(x)

x

r

K3

T

l3 a) Keff(x) Js r2

T r

bs

mL

rφ1 b) x 0 0 Fig. 5. System models. a) Model of the linear tooth belt axis. b) Simplified two-mass model with belt damping. The roller on the left is driven and the roller on the right is free-running.

priori determined frequencies to be monitored, the design of the excitation signal directly determines the number of resonators needed and the size of the SDFT window. III. ONLINE IDENTIFICATION WITH SDFT The servomechanism considered in this paper is a tooth-belt drive that includes a movable mass. A typical belt drive guide consists of an actuator (electric motor), two belt pulleys, a moving cart, and a toothed belt, which connects the cart and the belt pulleys together. The guide is directly driven so no speed-reducing gear is applied in the system. The tooth belt axis can be presented as a spring-mass system as illustrated in Fig. 5a), and thus, following differential equations describing the movement of the pulleys, motor, belt and the cart mass without belt damping are

J p  J m 1   f1  T  rK1 xr1  x  K3 r2  r1  (8) 2   f2  rK 2 x x  r 2   K 3 r 2  r1 , J p

mL x  f f  K1 x r1  x  K 2 xx  r2 

(9) (10)

where Jp and Jm are the inertias of the pulley and the motor, φ1 and φ2 are the angular positions of the driven and non-driven pulleys, x is the position of the cart, r is the radius of the pulleys, mL is the total mass of the cart and the load on the axis, T is the torque developed by the motor, K1(x) and K2(x) are the spring constants between pulleys and the cart as a function of the cart position, K3 is the spring constant between pulleys and τf1, τf2 and ff are the disturbance or friction forces directed to the pulleys and the cart, respectively. Moreover, in Fig. 5 a) the l1(x) and l2(x) are the lengths of the tooth belt between pulleys and the cart as a function of cart position, l3 is the length of the belt between the pulleys. By neglecting the free end dynamics [36], [37] the system model can be simplified to a two-mass system

Js  T   f1   bs  (r  x )  K eff  (r  x), r r r mL x  f f  bs  (r  x)  K eff  (r  x),

(11) (12)


Spring constants [N/m]

6

3

x 10

Keff(x)

2

K1(x)

K2(x)

d dt

K3

Identification zone

-

1 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Position [m]

Fig. 6. Spring constants as a function of position. The position axis represents the maximum stroke 1.6 m of the system.

where Js represents the total inertia of driven-end and Keff is the position dependent equivalent spring constant of the axis K eff ( x)  K1 ( x) 

K 2 ( x) K 3 . K 2 ( x)  K 3

(13)

The explicit non-linear behavior of the equivalent spring constant (13) is illustrated in Fig. 6 with the spring constants K1(x), K2(x) and K3 as a function of position. It can be noticed that the driven-end the spring constant K1(x) has the largest influence on the equivalent spring constant, especially close to the driven-end. On the other hand, when the cart is positioned far away of the driven end the effect of increasing spring constant K2(x) is limited. In this paper, the main interest is the monitoring and on-line identification of a closed-loop controlled linear tooth belt system at the selected set of frequencies around the dominant resonance frequency. The identification problem is treated as linear time-invariant (LTI) system identification in different operating points in a well-defined operating range (see Fig. 6). From (11) and (12) the linearized transfer function from the torque reference T to the motor mechanical angular velocity Ω, can be derived. This leads to the following transfer function expression, without friction terms, that has a rigid part and a flexible part mL s 2  bs s  K eff  1 .  2 T J s mL s 2 J s  mL r 2 J  m r L  s  bs s  s K eff Rigid part J s mL J s mL  

(14)

Flexiblepart

The transfer function (14) corresponds to the best linear approximation of the mechanical coupling in a certain TABLE I PARAMETERS OF THE REFERENCE MODEL Symbol

Nominal value1)

Parameter

Variation2)

Inertia 0.0028 – Spring constant 5.0∙105 4.0∙105 – 1.1∙106 Load and cart mass 6.7 – Belt damping constant 130 115 – 190 Resonance frequency 61 54 – 91 Dist. between pulleys 2.14 – Roller radius 0.019 – * Tension of the belt is set to 450 N, K ** Approximated with bs = eff and using Qk = 10 (see [7],[9])

Js (kgm2) Keff (N/m)* mL (kg) bs (Ns/m) ** fres (Hz) l3 (m) r (m)

1)

2πfres Qk

Nominal value in cart position 0.8 m 2) Variation between cart position 0.35 m and 1.4 m (see Fig. 6)

w(t)

Pffv Cpos

+ yvel(t)

Cvel

+

u(t)

ypos(t)

Ctorq

G(z)

d dt

Fig. 7. Servomechanism controlled by a position cascade controller. The excitation signal is superposed to the output of the velocity control. Input and output signals used in the identification are denoted as u and yvel, respectively.

operating point, and thus, the essential system dynamics in different cart positions. This means, that two-mass-system approximation of the dominant resonance frequency can be used to represent the complete mechanical load of the drive in a simplified way. This is a common approximation for many different mechanical systems. Moreover, the absence of static friction such as Coulomb friction effects in the model is justified as the online identification experiments are performed during operation and zero velocity is avoided as much as possible. In this paper, the high-order model (8)–(10) is used in the simulations and correspondingly the transfer function (14) is used as a reference model for the proposed parameter estimation routine. The most important parameters of the system that are obtained from the material properties and the geometrical values of the experimental tooth-belt drive under study are given in Table I. It is pointed out that these parameters, especially the belt properties, are only known with some degree of confidence. D. Indirect Identification As the tooth-belt drive provides linear motion over a limited stroke, it directly limits the identification experiments as a limited operating range must be taken into consideration. Thus, during the identification tests, the system has to be controlled by a position controller. In most motion control applications the control law of preference is the cascaded Pposition and PID-velocity loops with feedforward terms. Thus, the controller considered in this paper has a cascaded structure, where the inner loop is a velocity controller (PI) and the outer loop position controller is a P controller with a velocity feed forward as depicted in Fig. 7. In this paper, the velocity controller (PI) parameters used in the identification are chosen so that the closed loop is slow with moderate disturbance rejection, but the presence of the integrator in the controller is important in order to maintain the desired operating point. The P part is included in the outer loop of cascade control for adequate position control during the tests. In general, during the identification experiment a soft controller which essentially tries to stabilize the desired operating point has to be used [1], [6]–[10] in order to have a valid estimation result. Moreover, the choice of this specific controller structure is based on the electrical drives used in the experimental validation (see section IV) which use commercial motion control software. However, it should be noticed, that the indirect identification method considered is also valid for other controller structures as well.

10

SDFT f1 = 20 Hz Reference model

5

0 0.2

0.4

0.6

0.8

1

1.2

10

Magnitude [dB]

Magnitude [dB]



0

-10 0.2

1.4

0.4

0.6

0.8

0


-10

-20 0.2

0.4

0.6

0.8

1

1.2

1.4

5

0.8

Reference model 1.2 1.4

1

Magnitude [dB]

Magnitude [dB]

SDFT f3 = 60 Hz -20 0.6

Reference model

-5 -10 0.2

0.4

0.6

0.8

0.8

Reference model 1.2 1.4

1

Amplitude [Nm]

Magnitude [dB]

SDFT f4 = 80 Hz -20 0.6

1.2

1.4


-5

-10 0.2

0.4

0.6

0.8

1

1.2

1.4

Position [m]

0

0.4

1

0

Position [m] 20

0.2

1.4

Position [m]

0

0.4

1.2

SDFT f6 = 120 Hz

0

Position [m] 20

0.2

1

Position [m] Magnitude [dB]

Magnitude [dB]

Position [m]

1 0.5 0 0

f1  20

f2  40

f3  60

Position [m]

f4  80

f5  100

f6  120

f7  140

Frequency [Hz]

Fig. 8. Simulated online monitoring of selected set of frequencies with SDFT compared to the linearized reference model (10) in different positions. The frequency contents of the multisine signal with a frequency range of 20 Hz to 140 Hz with a frequency resolution of 20 Hz is considered.

As depicted in Fig. 7 the excitation signal w(t) is superimposed to the velocity controller output, that forms the reference to the torque control. In modern AC servo drives typical torque-control bandwidth is from several hundred hertz up to a few kilohertz, while dominant resonance frequencies of mechanical systems are significantly lower. In this paper, the sample frequency is set significantly below the bandwidth of the torque control, and hence, the effect of the torque control cannot be seen in the identification signal and thus can be omitted. As the system is closed loop controlled, the online identification scheme is based on an indirect identification method that exploits knowledge of the controllers and the excitation signal. First, by considering the basic idea of indirect identification where the closed-loop transfer function Gcl(jω) is estimated between the measured output y(t) and the excitation signal w(t), the open-loop frequency response estimate can be solved using knowledge of the inner-loop of the cascade controller

Gˆ ( j ) 

Gˆ cl ( j )

. 1 - Gˆ cl ( j )Cvel ( j )

(15)

By considering a case where the full feed forward term Pffv = 1 is part of the cascade controller structure depicted in Fig. 7, the influence of the outer loop can be considered (15) by changing the denominator with  C pos ( j )  j 1 - Gˆ cl ( j )C vel ( j )1  e 2   

 .  

(16)

The main advantage of the indirect identification method (15) is that the open loop model G(jω) can be correctly

estimated even without estimating any noise model. By applying the knowledge of the controller structure (16), the online identification is performed by calculating the SDFT for the excitation signal and the measured signal. It is worth pointing out, that in this paper, the indirect estimator (16) is used for the SDFT based online identification and (2) is used for the offline identification. First, the proposed online identification method is studied by simulations. The identification tests are performed so that a trapezoidal velocity profile is used with constant velocity of 0.38 m/s is used and persistent excitation is added to the torque reference. A multisine signal is considered that contains frequencies in the frequency range of 20 to 140 Hz with the frequency resolution of 20 Hz, and the amplitudes are chosen as 1 Nm as depicted in Fig. 8. Thus, the frequency response is only estimated at the excited frequencies using the SDFT. In the simulations, the size of the window is chosen as N = 100. Based on Fig. 3 the order k for the resonators can be denoted as k  N Ts f interest .

(17)

With a sample time Ts equal to 1 ms and in order to track the frequencies of interest in the multisine excitation signal, resonators with k=[2, 4, 6, 8, 10, 12, 14] are needed. The position controller runs at a 500 μs time level with Kpc = 1 1/s, and the velocity controller is at a 250 μs time level with the gain Kvc = 0.2308 1/s and integration time Ti = 0.2 s. The velocity feed forward gain Pffv is set to 1. Furthermore, the linearized transfer function (14) is used as a reference model in order to validate the online identified results at specific frequencies. In each experiment, white noise, zero mean, with a standard deviation 0.1 is added to the output signal y(t). The

 act

10 0 0

1

2

3

4

5

1

0

0.9

-10

0.8

-20

Ref. system

Coherence 

Offline Eq. (2)

-30 1 10

6

Time [s] a)

2 ( ) wy

0.75 0.25 0

2

4

Time [s] b)

6

0.6

2

10

Frequency [Hz] Reference system Offline Eq. (2) Online est.

3

Phase [deg]

Excitation [Nm]

1.25

0.7

Online est.

50

1.6

Position [m]

10

Coherence

 ref

20

Magnitude [dB]

Velocity [rad/s]


1.5 0 -1.5

0 -50 -100

-3 20

40

60

80

100

N samples c)

Fig. 9. Motion profiles and signals used in the identification experiment: a) Trapezoidal velocity reference and actual velocity during identification. b) Change in the position from initial position 0.25m to final position 1.6m. c) Excitation signal during N samples.

motion profiles and signals used in the identification experiment are illustrated in Fig. 9. Fig. 8 shows the online-monitored magnitude estimates at the selected set of frequencies. It can be seen that transients occur at the beginning of the online magnitude estimates, as the SDFT has not reached a steady state after the acceleration profile. Moreover, the SDFT approach needs a complete set of N measurements samples to provide valid results. In practice, this means the first N of SDFT results are not correct. After the transients, the online-identified magnitude estimates give a good correspondence with the linearized reference model behavior in different cart positions, and thus, the positiondependent dynamics are clearly visible. Even though small discrepancies between the online and reference magnitude behavior can be detected, these results are important because the system changes at the selected frequencies can be tracked in real-time. To further validate the online results, the identification method is studied in the chosen cart position of 0.8 m. Thus, the controller and the excitation signal are kept the same, but the identification is performed at a standstill. It is worth pointing out that standstill online identification tests are not practical with actual tooth belt drives, but in this paper, the results are discussed because the linearized system dynamics resembles the two-mass-systems reported in [1], [7]. Thus, the proposed method could be used more straightforwardly in the identification of an electro-mechanical actuator with free movement. In the simulations, the online method is compared with the offline frequency-domain identification method in (2) by exciting the system with a chirp excitation signal that changes continuously over a frequency band [5, 250] Hz during a 5 s experiment with an amplitude of 2 Nm. The online and offline identification results are compared with the reference system in Fig. 10. To demonstrate the accuracy of the online magnitude estimates, their confidence interval is evaluated using 3σ uncertainty bounds after the 5 s identification experiment. Furthermore, the quality of the obtained offline frequency response is evaluated with the coherence γwy2(ω)

1

2

10

10

Frequency [Hz]

Fig. 10. Offline- and online-identified frequency responses compared with the linearized reference model in the cart position 0.8 m. The quality of the offline identified frequency response function is measured with the coherence γwu2(ω), and the online result is shown by error bars that represent the estimated 3σ error bounds.

2  wy ( )



S wy ( )

2

S ww ( ) S yy ( )

.

(18)

It can be observed in Fig. 10 that the offline method (2) yields an acceptable result as good matching with the reference model is obtained and the coherence indicates satisfactory accuracy. More importantly, the characteristic of the two-mass-system approximation (10) are evident in the online-identified amplitude and phase response, and the obtained results are in a good agreement with the reference model and the error in the frequency points is small. E. Frequency-domain Parametric Estimation A Least Squares (LS) based identification criterion is commonly applied for the estimation of a parametric model from time [7], [9] or frequency domain [18] observations. By considering general discrete transfer function for LTI in zdomain G(z)

G( z ) 

q 1 q2  ... b1 z  b0 B( z ) bq 1 z  bq  2 z  q q 1 A( z ) z  aq 1 z  ... a1 z  a0

(19)

where q denotes the order of system with respect to parameters a0,…,am-1 and b0,…,bm-1 of the system. Let z  e jnTs , following representation for the frequency response is obtained

G(e jnTs )  X n  jYn ,

n  1,...,i

(20)

where the notation n = 1…i represents the i frequencies to be considered in the estimation (see Fig. 4). By substituting z m  e jmnTs to (19) the numerator and denominator can be expressed as

B(e jnTs ) 

q 1

b e m

m 0

jmnTs

(21)


A(e jnTs )  e jqnTs 

q 1

a

me

jmnTs

(22)

m 0

Similarly

A(e

jnTs

by

)  G(e

considering

jnTs

) B(e

jnTs

(20)–(22)

in

form

) and Euler’s identity

z m  e jmnTs  cos( mnTs )  j  sin(mnTs ) ,

(23)

the real and imaginary parts of the system can be separated into two equations [38] and the parameter vector θ can be solved iteratively in a Least-Squares sense



θ  ΦT Φ





1

where Φ  φ nR φ nI

ΦT Ψ ,

T2i(2q1) is

(24) the matrix of the real and

imaginary equation pairs denoted as

φ nR  cos(0 nTs ) cos( nTs ) cos( q nTs )  X n cos(0 nTs )  Yn sin(0 nTs )  X n cos( q nTs )  Yn sin(q nTs )T

(25)

φ nI  sin(0 nTs ) sin( nTs )sin(q nTs )  X n sin(0 nTs )  Yn cos(0 nTs )  X n sin(q nTs )  Yn cos( q nTs )T

(26) Moreover, the Ψ represents the real components of the equation pairs in form

and

 X cos qnTs   Yn sinqnTs  Ψ n   X n cos qnTs   Yn sinqnTs  2i1

imaginary

T

(27)

In this paper, the order of the corresponding discrete transfer function of (14) is q = 3, thus the parameter vector can be denoted as



θ  bˆ0 , bˆ2 , bˆ3 aˆ0 , aˆ1 , aˆ 2



T

(28)

The proposed SDFT based non-parametric identification method is validated by estimating parameters from the indirectly identified open-loop response at the selected set of frequencies by using the LS algorithm. Similar type of frequency-domain LS parameter estimation problem has been considered in [18] in order to adapt control parameters based on the measured frequency response of the system. IV. EXPERIMENTAL RESULTS The proposed identification method is validated by an experimental mechanical test setup shown in Fig. 11, that consist of a flexible tooth belt guide directly connected to a 4.5 kW permanent magnet synchronous motor (PMSM) with nominal speed of 3000 rpm and nominal torque of 14.5 Nm. The PMSM is controlled by a frequency converter (ACSM1)

Fig. 11. Experimental test setup consisting of a tooth belt guide, a driving motor, frequency converted and a PLC.

manufactured by ABB, and Beckhoff’s programmable logic controller (PLC) is used to implement the excitation signals and references. It is worth pointing out that the experimental results in this paper have been obtained by using electrical drives with commercial motion control software, thus the cascaded control structure (see Fig. 7) is under consideration. As the actual parameters of the experimental test setup, especially the belt properties, are only known with some degree of confidence, the tooth-belt drive under study is first offline identified in different cart positions. The main purpose of these tests is to support the results of the on-line identification algorithm proposed and to have a more realistic model at the selected frequencies for comparison. The system is identified in six different positions at a standstill. During these tests, th e tooth-belt drive is closed loop controlled, and a chirp signal is used as an excitation signal (see Fig. 6). The chirp signal is the same as in the simulations, but the amplitude is set higher to 3 Nm in order to reduce the effect of static friction during the identification experiment. The open loop frequency response model in different positions is estimated using (2). Correspondingly, the online identification is performed similarly as in the simulations using the same multisine excitation signal and window size of N = 100. In the experimental results the magnitudes are also estimated with window lengths N = 50 and N = 200 in order to show its influence on the magnitude estimation. The experimental results of the online identification using the proposed SDFT method at the selected set of frequencies are presented in Fig. 12 with the corresponding offlineidentified frequency points. It can be noticed in Fig. 12 that the magnitude estimates become somewhat noisy if the window size is decreased. Correspondingly, if the window size is increased the estimates are smoother, but lag between the real system and estimation is introduced. Choosing a value of the window length N for the SDFT is thus a case-specific compromise between computational burden and estimation accuracy. Although there are slight differences between the offline identified frequency response points and online-estimated magnitude estimates in Fig. 12, the position-dependent system dynamics can be clearly detected as the online identified estimates change when the cart moves towards the driven end similarly as in the case of the simulation in Fig. 8. Besides the fundamental difference between the identification methods and the excitation signals, some of the discrepancies can be explained by the nonlinearities affecting the system. It is

f = 20 Hz

4

Magnitude [dB]

Magnitude [dB]


1

2 0 -2

SDFT N = 50

0.3

0.4

0.5

0.6

SDFT N = 100

0.7

0.8

SDFT N = 200

0.9

1

1.1

Offline ident.

1.2

SDFT N = 100

f = 100 Hz 5

0.3

0.4

0.5

0.6

0.7

Magnitude [dB]

Magnitude [dB]

2

-10 -15 -20 SDFT N = 50

0.5

0.6

SDFT N = 100

0.7

0.8

SDFT N = 200

0.9

1

1.1

Offline ident.

1.2

-6

SDFT N = 50

0.4

0.5

SDFT N = 100

0.6

0.7

Magnitude [dB]

Magnitude [dB]

0.8

SDFT N = 200

0.9

1

1.1

Offline ident.

1.2

0.5

0.6

SDFT N = 100

0.7

0.8

SDFT N = 200

0.9

1

1.1

Offline ident.

1.2

f = 140 Hz 7

-5 -6 -7

1.3

SDFT N = 50

0.3

0.4

0.5

SDFT N = 100

0.6

0.7

Position [m]

0.8

SDFT N = 200

0.9

1

1.1

Offline ident.

1.2

1.3

Position [m]

10

Magnitude [dB]

1.3

Position [m]

0

SDFT N = 50

1.3

-4

-4

3

0.4

1.2

f = 120 Hz

0.3

f = 60 Hz

0.3

1.1

6

1.3

-10 -20

1

-2

Position [m] 10

0.9

0

f = 40 Hz

-5

0.4

0.8

Position [m]

0

0.3

Offline ident.

-2

Position [m]

-25

SDFT N = 200

0

-4

1.3

SDFT N = 50

2

f = 80 Hz 4

5 0 -5

SDFT N = 50

0.3

0.4

0.5

0.6

SDFT N = 100

0.7

0.8

SDFT N = 200

0.9

1

1.1

Offline ident.

1.2

1.3

Position [m]

Fig. 12. Online tracking of identified magnitude estimates at a selected set of frequencies as a function of position when the window length N of SDFT is varied. The corresponding offline-identified frequency response values for the selected frequencies are indicated in different cart positions by crosses.

worth pointing out that the direct comparison of these results is problematic because in the offline identification case the tooth-belt drive is identified at a standstill where friction influences the identification result. Correspondingly, in the case of online identification, the influence of friction is reduced by avoiding the zero speed region using a constant velocity profile. Nevertheless, the results clearly show that the frequency response can be identified online at the selected set of frequencies, and the proposed method is suitable for tracking changes in the system. Even though there are differences between the offline- and online-identified results, the results are in a satisfactory agreement, and similar system changes can be monitored by both methods. A. Parameter Estimation From the parameter estimation point of view, the persistent excitation signal should be rich enough to be able to identify the necessary parameters of the system under study. In this paper for parameter estimation purposes the multi-sine excitation signal is changed so that it excite frequencies around the dominant resonant frequency. Thus, a multi-sine signal with a frequency range of 20Hz to 120 Hz with a frequency resolution of 5 Hz is considered. Moreover, the window size of N = 200 is considered and the proposed frequency-domain LS algorithm is used to estimate parameters of discrete model that correspond (14). In Fig. 13 the estimated system parameters are compared to the reference model parameters. In order to emphasize the uncertainty related to the parameters of the reference model, uncertainty of 15% for the inertia terms, 5% for the spring constant and 20 % for the belt damping has been considered and represented with the shaded area in Fig. 13. Although

there are slight differences between the reference and estimated parameters, the position dependent system dynamics can be clearly noticed as the parameters change as the cart moves during the identification experiment. Moreover, the largest deviation from the reference model parameter behaviour can be noticed from the parameter a0 when the assumed parameter uncertainty is considered. In the case of B(z) parameters of the reference model, the uncertainty related to the total inertia of the system emphasizes the uncertainty region around the nominal behavior. Nevertheless, according to the results, the on-line LS estimation captures the changing parameters of the linearized two-mass-system, and positiondependent system dynamics can be clearly observed. However, it can be noticed that there are notable differences between the reference model and estimated result, and thus, statistical validation process is needed in order to validate the on-line estimated model. In Fig. 14 the normalized crosscorrelations between input signal u(k) and residual ε(k) = y(k) − ŷ(k), and between estimated model output and residual are shown. A confidence level of 97% is expressed as 2.17 / L and practical confidence level [7], [9] of 0.15 is also shown. For the statistical cross-validation, independent validation data is used from another identification experiment and L = 3000. It can be noticed in Fig. 14 that the estimated model mostly remains near by the theoretical limit indicating stochastically verified parameter estimates [7]. However, it is worth remarking that in the on-line estimation routine a certain frequency band is considered and a separate noise model is not estimated, and thus, for these reasons discrepancies can be expected. Nevertheless, the cross-correlation results indicates


-2.91

-2.97

0.2

Estimation Ref. model

b2

a2

-2.85

1

1.5

2

2.5

3

0.17

0.14

3.5

1

1.5

2

Time [s]

b1

a1

-0.27

2.88 Estimation Ref. model 1

1.5

2

2.5

3

-0.33

-0.39

3.5

1

1.5

2

Time [s]

2.5

3

Estimation Ref. model 3.5

Time [s]

-0.9

0.2


b0

a0

3

Time [s]

2.97

2.8

2.5

Estimation Ref. model 3.5

-0.95


0.17 0.14

-1

1

1.5

2

2.5

3

3.5

1

Time [s]

1.5

2

2.5

3

3.5

Time [s]

Fig. 13. Estimated system parameters as a function of time compared to the reference model parameters. Parameter uncertainty is considered in the reference model and represented with shaded area.

acceptable parameter estimates and further validate the result in Fig. 13. V. CONCLUSION This paper presented a method to estimate the frequency response of a servomechanism in real time using the SDFTbased identification routine. The discussion covered the working principle and analysis of the SDFT for indirect identification, in conjunction with a known multi-sine input design. The results from the simulations and experimental tests showed that the approach can be used to achieve satisfactory estimates of frequency responses in real-time using a short record of data in the sliding window. Moreover, it was experimentally validated that the method is suitable for tracking system changes and parameter estimation. The identification problem was treated as LTI and Least-Squares algorithm was applied to parameter estimation. Although linear identification is considered, the obtained parameter values show an acceptable agreement with the reference model and position-dependent system dynamics can be clearly observed. Thus, the proposed SDFT-based identification method is valid for online estimation of the tooth drive parameters. Compared with the conventional online identification solutions such as RLS-based time-domain methods, the implementation of the SDFT requires more computation, and at least one full-signal period is needed to determine the 0.2

0.2

Practical limit |R|< 0.15 Theoretical limit |R|< 2.17/sqrt(L)

REFERENCES [1]

[2]

[3]

[4]

[5]

[6]

[7]

0.15

|R , y|

0.15

|R ,u |

Practical limit |R|< 0.15 Theoretical limit |R|< 2.17/sqrt(L)

fundamental Fourier components, which are obvious weaknesses of the proposed method. On the other hand, the SDFT does not require any initialization or other performancerelated parametrization, and thus the user-specified set of frequencies to be monitored together with the excitation signal design defines the number of resonators needed and the window size. Choosing a value of the window length N for the SDFT is thus a case-specific compromise between computational burden and estimation accuracy. Nevertheless, the proposed method provides a good alternative to the conventional online identification solutions.

0.1

0.1

[8] 0.05

0.05

0 -40

-20

0 Lag  samples a)

20

40

0 -40

-20

0 Lag  samples b)

20

40

Fig. 14. a) Cross-correlation between input and residual. b) Crosscorrelation between estimated output of model and residual. The 97 % confidence limit is indicated as red dashed dot line and the practical confidence limit as dashed black line.

[9]

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[10] R. Garrido and A. Concha, “An algebraic recursive method for parameter identification of a servo model,” IEEE Trans. Mech., vol. 18, no. 5, pp. 1572–1580, Dec. 2013 [11] F. Schutte, S. Beineke, A. Rolfsmeier, and H. Grotstollen, “Online identification of mechanical parameters using extended Kalman filters,” in Conf. Rec. IEEE-IAS Annual Meeting, vol. 1, Oct. 1997, pp. 501–508. [12] K. Szabat and T. Orlowska-Kowalska, “Application of the Kalman filters to the high-performance drive system with elastic coupling,” IEEE Trans. Ind. Electron., vol. 59, no. 11, pp. 4226–4235, Nov. 2012 [13] M. Perdomo, M. Pacas, T. Eutebach, and J. Immel, “Sensitivity analysis of the identification of variable inertia with an extended kalman filter,” in IEEE IECON Conf., Nov. 2013, pp. 3102–3107. [14] R. Pintelon, and J. Schoukens. System Identification. A Frequency Domain Approach. Wiley-IEEE-press, Piscataway, 2001. [15] R. Pintelon and J. Schoukens, “FRF measurement of nonlinear systems operating in closed loop,” IEEE Trans. Instrum. and Meas., vol. 62, no. 5, pp. 1334–1345, May 2013 [16] A. Barkley, and E. Santi, “Improved online identification of a DC-DC converter and its control loop gain using cross-correlation methods,” IEEE Trans. on Power. Elect., vol. 24, no. 8, pp. 2021–2031, Aug. 2009 [17] R. LaMaire, L. Valavani, M. Athans and S. Gunter, “ A frequencydomain estimator for use in adaptive control systems”, in Proc. American Control Conf., Minneapolis, USA, 1987, pp. 238–244 [18] T. Hashimoto and Y. Ishida. “An adaptive I-PD controller based on frequency domain system identification,” ISA Transactions, vol. 39, no. 1, pp. 71-78, 2000. [19] Y. Kurita, T. Hashimoto and Y. Ishida, “An application of time delay estimation by ANNs to frequency domain I-PD controller,” in Proc. Int. Joint Conf. on Neural Networks, Washington, USA, 1999, pp. 2164– 2167 [20] G.G. Yen, “Frequency-domain vibration control using adaptive neural network,” in Proc. Int. Joint Conf. on Neural Networks, Houston, USA, 1997, pp. 806–810 [21] J. Balchen, and B. Lie, “An adaptive controller based upon continuous estimation of closed loop frequency response,” Model., Identif. and Contr., vol. 8, DOI 10.4173/mic.1987.4.3, no. 4, pp. 223–240, 1987 [22] E. A. Morelli, “Real-time parameter estimation in frequency domain,” Jour. of Guidance, Contr. and Dynamics., vol. 23, no. 5, pp. 812–818, Oct. 2000 [23] P.D. Olivier, “Online system identification using Laguerre series,” IEEE Proc.—Control Theory Appl., vol. 141, no. 4, pp. 249–254, 1994. [24] N. Nevaranta, J. Parkkinen, T. Lindh, M. Niemelä, O. Pyrhönen and J. Pyrhönen, “Online identification of a mechanical system in the frequency domain with short-time DFT,” Model., Identif. and Contr., vol. 36, DOI 10.4173/mic.2015.3.3, no. 3, pp. 157–165, 2015. [25] J. G. Németh, B. Vargha, and I. Kollár, “Online frequency domain system identification based on a virtual instrument,” IEEE Trans. Instrum. and Meas., vol. 49, no. 6, pp. 1260–1263, Dec. 2000 [26] S. Derammelaere, C. Debruyne, F. De Belie, K. Stockman and L. Vandevelde, “Load angle estimation for two-phase hybrid stepping motors,” IET Electr. Power Appl., vol. 8, no. 7, pp. 257–266, 2014 [27] L. Baghli, I. Al-Rouh and A. Rezzoug, “Signal analysis and identification for induction motor sensorless control,” Control engineering practice, vol. 14, no. 11, pp. 1313–1324, 2006. [28] C. M. Orallo, I. Carugati, S. Maestri, P. G. Carrica, and D. Benedetti, “Harmonics measurement with a modulated sliding discrete Fourier transform algorithm,” IEEE Trans. Instrum. Meas., vol. 63, no. 4, pp. 781–793, Apr. 2014. [29] A. Abed, F. Weinachter, H. Razik, and A. Rezzoug, “Real-time implementation of the sliding DFT applied to on-line’s broken bars diagnostic,” in Proc. IEEE IEMDC, pp. 345–348, Jun. 2001 [30] C. Shitole and P. Sumathi, “Sliding DFT-Based Vibration Mode Estimator for Single-Link Flexible Manipulator,” IEEE/ASME Trans. Mechatronics, vol. 20, no. 6, pp. 3249–3256, Dec. 2015. [31] E. Jacobsen and R. Lyons, “The sliding DFT,” IEEE Signal Process. Mag., vol. 20, no. 2, pp. 74–80, 2003. [32] E. Jacobsen and R. Lyons, “An Update to the Sliding DFT,” IEEE Signal Process. Mag., vol. 21, no. 1, pp. 110–111, 2004. [33] K. Duda, “Accurate, Guaranteed Stable, Sliding Discrete Fourier Transform [DSP Tips & Tricks],” IEEE Signal Process. Mag., vol. 27, no. 6, pp. 124–127, 2010. [34] J. Ojarand, M. Min, and P. Annus, “Crest factor optimization of the multisine waveform for bioimpedance spectroscopy,” Physiol. Meas., vol. 35, no. 6, pp. 1019–1033, 2014.

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Niko Nevaranta (S’14) received the B.Sc. and M.Sc. degrees in electrical engineering from the Lappeenranta University of Technology (LUT), Lappeenranta, Finland in 2010 and 2011, respectively, where he is currently working as a Researcher towards the D.Sc. degree. His research interest includes motion control, system identification, parameter estimation, system monitoring and diagnostics.

Stijn Derammelaere (M’10) was born in Kortrijk Belgium in 1984. He received the Master’s degree in automation from the Technical University College of West-Flanders Belgium in 2006. He received the PhD degree in 2013 at Ghent University where he has continued his research concerning control engineering focused on the motion control of fractional horsepower machines.

Jukka Parkkinen received the B.Sc. and M.Sc. degrees in electrical engineering from the Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 2010 and 2011, respectively. He is currently pursuing the D.Sc. degree in electrical engineering at the same university. His research interest includes motion control, multi-axis systems, centralized and decentralized control systems.

Bram Vervisch was born in Ieper Belgium in 1986. He received his master’s degree in electromechanical engineering in 2008 at the Technical University of West-Flanders. Now he is working as an assistant and PhD student at Ghent University, Belgium, in the department of Industrial Systems and Product Design. His research interests are structural vibration and system identification.


Tuomo Lindh was born in Lappeenranta, Finland, in 1964. He obtained B.Sc. degree from Mikkeli Institute of Technology in 1989. After couple of years work in industry he obtained his M.Sc. in Technology in 1997 and D.Sc. in Technology in 2003 from the Lappeenranta University of Technology (LUT). He currently works as an Associate Professor at LUT. His research interests include generator and motor drives and system engineering especially in the areas of distributed power generation, electric vehicles and in mechatronics.

Kurt Stockman started his career at Hogeschool West-Vlaanderen in 1995. Since October 2013 he is a Professor at Ghent University. During his PhD, he studied the behaviour of adjustable speed drives during voltage dips. After his PhD he concentrated more on the energy efficiency of motor driven systems and mechatronics systems. His research is characterized by a strong experimental basis and intense collaboration with industry. Prof. Kurt Stockman is author of over 100 papers related to electromechanical drive systems. He is also very active in dissemination of his experience related to energy efficiency both on national and international scene. Markku Niemelä received the B.Sc. degree in electrical engineering from the Helsinki Institute of Technology, Helsinki, Finland, in 1990. He received the M.Sc. and D.Sc. (Technology) degrees from the Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 1995 and 1999, respectively. He is currently a Senior Researcher with the Carelian Drives and Motor Centre in LUT. His current interests include motion control, control of line converters and energy efficiency of electric drives.

Olli Pyrhönen received the M.Sc. and D.Sc. degrees in electrical engineering from the Lappeenranta University of Technology (LUT), Lappeenranta, Finland, in 1990 and 1998, respectively. Since 2000, he has been a Professor in applied control engineering at LUT. In 2010, he received further teaching and research responsibility in the wind power technology at LUT. He has gained industrial experience as a R&D Engineer at ABB Helsinki in 1990-1993 and as a CTO of The Switch in 2007-2010. His current research areas include modeling and control of active magnetic bearings, bearing-less machines, renewable power electronics and electrical drive systems.

Juha Pyrhönen (M’06) was born in 1957 in Kuusankoski, Finland, received the Doctor of Science (D.Sc.) degree in electrical engineering from Lappeenranta University of Technology (LUT), Finland in 1991. He became Professor of Electrical Machines and Drives in 1997 at LUT. He is engaged in research and development of electric motors and power-electronic-controlled drives. Prof. Pyrhönen has wide experience in the research and development of special drives for distributed power production, traction drives and high-speed applications. Permanent magnet materials and applying them in machines have an important role in his research. Currently he is also researching new carbon-based materials for electrical machines.