(1) C = T θ. 1 FIR = finite impulse response

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1 Comparison of Speed Acquisition Methods based on Sinusoidal Encoder Signals Alexander Baehr, Peter Mutschler Member IEEE Darmstadt University of Technology, Department of Power Electronics and Control of Drives Landgraf-Georg-Str. 4, Darmstadt, Germany Abstract The PI-speed-control using sinusoidal encoder signals for feedback is the standard configuration in highly precise and dynamic servo control. In design, there is necessarily a tradeoff between the design goals of fast rejection of load changes (high controller gain) and uniform turning (moderate controller gain), with especially mechanical resonances limiting the controller gain. This tradeoff can be solved well only if a precise and low-noise speed signal is available. This paper investigates different speed acquisition schemes concerning their performance. In addition to the industry-standard filtered derivative, observers based on the rigid-body, two-mass, and three-mass models are considered. The methods are implemented in a 2.2kW permanent magnet servo motor plant, and a comparison of experimental results is given. Index Terms PI speed control, resonant load, sinusoidal encoder, observer, two inertia system, three inertia system, two mass system, three mass system I. INTRODUCTION This paper investigates different speed acquisition schemes concerning their closed loop performance. It presents results that were measured on a setup which consists mainly of two coupled 2.2kW servo motors, power electronics and a control PC. Either PI-speed control or P-position / PI-speed-control were implemented in the PC software for one servo motor. Performance is measured on the one hand as the r.m.s. deviation of the speed signal during speed control with constant reference speed of 1 rev/s and no load. For an ideal controller, the speed would be exactly constant, so its r.m.s. deviation is a measure for the non-ideality of the control loop. It is calculated by off-line time discrete derivation of an 8 seconds record of the encoder position, so the data regarded are not smoothed by the respective filter or observer. To make comparison easier, all control loops were designed such that the r.m.s. deviation was approximately 0.1 rad/s. For the setup regarded, this is the subjective limit beyond which significant acoustic noise occurs. Tuning of the speed deviation was done by varying the delay time constant of the speed acquisition filter or observer, which changes the speed quality by itself and additionally affects the controller design; see the next sections for details. On the other hand, performance is measured regarding the load step rejection with P-position- / PI-speed-control. Several defined stepwise load torque changes of amount T were executed by the load machine, and the maximum deviation θ from the reference position was used to compute the dynamic stiffness of the control loop C as The standard speed acquisition method for industrial drives, deriving and low-pass filtering the position signal was investigated first. The investigated setup has its lowest mechanical resonant frequency at 970 Hz, so the filter s task is to passively damp this frequency in order to allow high controller gains. In addition, a notch filter can be used that damps especially the resonant frequency in the control loop spectrum. Standard notch filters are derived from a low pass filter through the lowpass-bandstop transformation. As an alternative, [1] proposes a simple FIR 1 band stop filter. A second method for speed acquisition is an observer based on the simple rigid-body model of the mechanical system (see fig. 2). Such observers are tested in [3] and [4] in comparison to the filtered derivative. Both theses state that observers lead to a better uniform run. In both cases, experimental setups of linear motors without load machines are regarded, so that data for the disturbance rejection are missing and mechanical resonances should be negligible. Besides, [3] uses only an incremental position encoder, causing significant quantization noise. The good performance of the rigid-body observer for incremental encoder signal processing is confirmed in other papers [11]. A more advanced method is an observer modeling the mechanical resonance as a two- or three-inertia system. Nearly all papers on this subject are limited to two-inertia models, pointing out that higher order systems can be approximated this way [5]. The frequencies to be actively damped are usually significantly lower than the lowest resonance of the setup investigated here, which is 970 Hz. As a consequence, the encoder is assumed to be stiffly mounted, while its oscillating behavior is considered in this paper. Many papers deal with the active damping through changes in the controller structure, such as acceleration or shaft torque feedback [9], [10], [5]; such changes are not subject of this paper. Where observers for multi inertia systems are used, they are often designed by placing all poles to one location in the s-plane [5] or as a Kalman filter [6]. [7] uses the poles of the state control loop, left-shifted by a constant distance. In [8], a setup regarded as a three inertia system with eigenfrequencies of 400 and 860 Hz is investigated. An observer for speed acquisition is designed, and it is shown that good quality of the speed signal allows active damping even through the standard PI controller. Further investigations of a state control revealed that state control leads to no better performance. This was confirmed for the setup regarded here. C = T θ (1) 1 FIR = finite impulse response

2 II. SPEED ACQUISITION USING THE FILTERED DERIVATIVE OF THE POSITION SIGNAL The standard method for industrial drives is using the low pass filtered derivative of the position signal as speed feedback. For the investigated setup, main task for the filter was to passively damp the 970 Hz resonant mode in order to allow high controller gains. This paper is limited to investigation of IIR 2 low pass digital filters that are achieved from continuous-time filters through the bilinear transformation. FIR low pass filters are not investigated because they need a higher order and delay for an equal falling edge rate. Filters of 1st to 3rd order according to the Butterworth and Chebyscheff optimizations were tried (results see fig. 5, no. 1-5). These filters are optimized for a steep falling edge rate. For Butterworth filters, the absolute value bode plot is monotonous, while for Chebyscheff filters it is allowed to fluctuate in the passband by a specified amount. Which amount to choose surely depends on the application; Chebyscheff filters with 3 db fluctuation are investigated here as an example. For each filter, a delay time constant was calculated through approximation of the denominator polynomial by its constant and linear parts. They were made derivation filters by multiplication with s, and then transformed to time discrete IIR filters. To passively damp especially the 970 Hz resonance, a notch filter can be added (fig. 5, 6-7). It is designed independently and then added to the low pass by multiplication of the transfer functions. Standard notch filters are achieved from the lowpass-to-bandstop transformation of first order lowpass filters. For design, the notch frequency (here: 970 Hz) and the width of the stop band have to be specified. Since it was never possible to extend the low pass filter s cutoff frequency beyond 970 Hz, its cutoff frequency was used as the lower border frequency for the notch filter. The time constant of the filter was chosen by varying the cutoff frequency of the low pass filter. The notch filter turned out to contribute only a little delay. As an alternative, a simple FIR bandstop filter (fig. 5 no. 8) is proposed in [1]. It is located at the output of the speed controller, but can be moved to the feedback path with no changes to the load behavior which is investigated here. This filter superposes its input with a copy delayed by a number of time steps k. For the frequency ω S fulfilling ω S k T = π (with sample time T), this means a phase shift of 180, so it is damped out. The best filter design in z-domain to damp out a 970 Hz oscillation at 10 khz sample rate is then F (z) = 1 + z 5 2 with a delay time of 2.5 T = 250µs. chained with a 1st order low pass filter. 2 IIR = infinite impulse response (2) This filter was Fig. 1. Amplification vs. frequency of different derivation filters and one observer (see section III); performance see fig. 5 Fig. 2. *! E G J 6 Structure of the rigid-body observer * J I F A J H A H Ω * θ θ A H III. SPEED ACQUISITION USING OBSERVERS BASED ON THE ONE INERTIA MODEL ( RIGID-BODY OBSERVER ) Figure 2 shows the structure of the rigid-body observer (with plant inertia J, torque constante k T, load torque t L, reference current in the quadrature axis i q, speed Ω, encoder angle θ, and feedback constants k B1,2,3 ). Estimated values are marked as ˆx. To observe the load torque, an integrator is used which gets its input only from the feedback signal. Load and motor torque are integrated two times to get the estimated position angle, which is compared to the sensor angle to compute the observer feedback. For implementation in the control PC, the corresponding time discrete observer has to be used. For a time discrete observer, there is generally the possibility to use the estimated values of one step ahead; this improved the observer s performance a little. For observer design, two methods for pole placing and the Kalman filter were considered. As to pole placing, the desired poles were specified in s-domain, then converted to z-domain to calculate the feedback values. The Kalman filter design algorithm works directly in z-domain. The first observer (fig. 5 no. 9) was designed by placing all three poles to one negative real value. This method is recommended in several papers, such as [3] and [4]. The delay time constant was chosen by changing this value. A second way is to place the observer s eigenvalues to those of a 3rd order Butterworth filter (fig. 5 no. 10). The goal of this placement is to achieve an optimum attenuation for distur-

3 A H E? H? K F E C A H? JH IAHL IAHL Ω? Ω? Ω Ω 0 θ 0! I B Fig. 3. Three mass model for the regarded setup? bances beyond the filter s cutoff frequency. The time constant is altered by changing the cutoff frequency. Finally, the static Kalman filter design method was used (fig. 5 no. 11). This method is based on the idea of a state space system that is disturbed by white noise added to the system equations and sensor outputs. The covariance matrices of system noise Q and sensor noise R are specified, and the feedback coefficients are chosen for a statistically optimal estimation. Usually, only the main diagonal of Q is set to values which stand for the amount of all kinds of disturbances the designer expects for the respective equation. R is a scalar for single-sensor systems. As a first approach, only the load torque was considered disturbed. If the state vector x is defined as x T = (θ Ω t L ) this means that only R and the matrix element q 33 are given a value different from zero. As the resulting filter does not change if the Q and R matrices are scaled equally, there is only one degree of freedom left. This was used to adjust the observer s time constant: The lower R and the larger q 33, the faster the observer. In a second step, it was tried to consider the speed fluctuations due to slot latching as disturbances in the ˆΩ-equation, so q 22 was set to a values computed from recorded data. This led to no better results. IV. OBSERVERS FOR TWO AND THREE INERTIA SYSTEMS This section deals with the modelling of the eigenfrequencies through a two or three inertia system. The model shown in figure 3 is considered a valid model for the experimental setup. For a two-mass approach, J 1 and J 2 are joined to one inertia J 1. As opposed to other papers, considering the encoder fixing a separate oscillating inertia was plausible and lead to good results. The system model is based on the identification of the controlled system. In an experiment, the encoder signal was measured when exciting the plant by a current waveform consisting of several sinusoids. With an 8 second record of both signals, a least squares estimation was done using the AR- MAX model [12]. This revealed three lightly damped poles of the transfer function at 970, 1370 and 2330 Hz as well as two lightly damped zeros at 772 and 1730 Hz. The Fouriertransformed response signal shows an increased noise level between about 1000 and 1300 Hz, which is caused by oscillations of the encoder. Because of their light damping, the poles and zeros were approximated as being undamped, and regarded as those of an E G 6 H Ω 1 J I F A H J H A Fig. 4. Structure of the two inertia system model. The observer s feedback paths were omitted for clarity undamped multi mass system. The models were designed to represent the lowest poles and zeros of the system. First model is a two inertia system (results see fig. 5 no ) with a resonant frequency of 970 Hz. Given the two inertias J 0 and J 1 from geometrical data of the plant, the spring constant can be calculated from the resonant frequency. The second model is a three inertia system (fig. 5 no ) consisting of encoder fixing, controlled servo and load servo. The resulting transfer function in s-domain has two conjugate complex zeros and two times two conjugate complex poles, which were matched to 772, 970, and 1370 Hz, respectively. With these equations, only one degree of freedom remains for the calculation of the system parameters. Here, the one solution was used where the computed inertias are closest to the values calculated from geometrical setup data. The structure of the two inertia model is shown in fig. 4; the observer feedback paths were omitted for clarity. It consists of the two inertias (J 0 : encoder mount, J 1 : machines), the elastic coupling with spring constant c 01, the integrator for modeling the encoder angle and the integrator estimating the load torque. The load torque was modeled as affecting both inertias equally. This is not physically correct, but decouples the load torque observer from system oscillations, leading to better observer performance. The proportional part of the PI speed controller was implied in the system model as if it was part of the system; this path is depicted as a dashed line in fig. 4. The three mass model is developed by adding one more inertia J 2, and one more elastic coupling; the motor torque drives J 1, and the load torque affects all three inertias. Based on this model, the first step is to design an observer similar to [8]: The encoder angle is derived and compared to the modeled encoder speed ˆΩ 0, the difference is used for feedback; this structure will be called the speed-only observer (fig. 5 no. 12,13,16,17). For design, the method of eigenvalue placing was used. Though the eigenvalues of an observer can be placed arbitrarily in general, the usual pole placing strategies mentioned

4 TABLE I SUMMARY OF POLE CONFIGURATIONS USED Version 1: 20 damping Version 2: 45 damping -3000, e ±j e ±j135 additional poles for three inertia observer: 9000 e ±j100 additional pole for complete observer: in section III led to poor results. A number of approaches to design a Kalman filter were tried, but did not work either. Better results were obtained when the poles were placed in a configuration not too far away from the system s natural pole locations. See table I for a summary of the pole configurations used. The two mass speed-only observer is 4th order. The system s natural poles are two poles at s = 0 and one conjugate imaginary pole pair, representing the resonant frequency. The observer feedback s goal is to move the poles to the left, making the system stable. For the resonant poles, the absolute value was changed only little, in order to move it away from the encoder oscillation range at 1000 to 1300 Hz. The poles were turned around the origin to the left by either 20 or 45 degrees. Turning them further, in order to get a faster observer, resulted in too much noise in the estimated state variables. The two poles at the origin could be moved far to the left to s = The corresponding three mass system is of 6th order, with one more imaginary pole pair representing the second resonant frequency. These poles absolute value was enlarged a little, then they were turned to the left by 10 degrees. Further turning was not necessary, since the effect of this resonance is not important for the practical controlled system (see fig. 6). As an alternative observer structure, the estimated encoder angle as shown in fig. 4 can be compared to the measured encoder angle, with the difference used for feedback. This structure is called the complete observer (fig. 5 no. 14,15,18,19), its order is one more than the speed-only observer. It is good to place the additional pole considerably faster than the others, as the observer will show low frequency oscillations otherwise. The third variant tested is the reduced observer [14] (fig.5 no. 20,21). This observer dispenses with observing the measurable states, reducing its order by that number. For state control, the observer s states then have to be used together with the measured values. There is no longer the option of predicting the system states one step ahead. All observers were designed as stated here. In order to tune the control loop for the given speed quality of 0.1 rad/s, only the PI controller gain was changed (see next section). V. EXPERIMENTAL SETUP The speed acquisition methods introduced above were tested on an experimental setup which consists mainly of two coupled 2.2 kw permanent magnet servo machines, two inverters with a coupled DC link, signal processing electronics, and a PC in which the control algorithms are implemented. The torque constant for one motor is k T = 0.533Nm/A in the sense of fig. 2 and 4, the overall inertia of the setup is J = 20.05kgcm 2. The controlled servo has an incremental encoder and an acceleration sensor mounted on its b-side using a short extension sleeve; only the encoder signal is used here. The encoder is a Huebner HOGS 80 type hollow shaft sinusoidal encoder with 2048 periods/rev. Its signals are digitalized by 12-bit ADconverters. The position is then computed using the usual arctangens method, showing an overall precision of 9 valid bits per encoder period. Speed acquisition and control algorithms are implemented in a standard PC. The load servo is used to simulate a mechanical load. Its inverter is switched off for smooth turning experiments, and run in current control mode to get quick load changes for the disturbance rejection experiments. The time constant of its current control loop is approx. 500µs. While position and speed control are realized in the PC software, current control is done by an analog circuit designed according to [13]. This control method works with hysteresis comparators; the PC provides an analog current command in stator-fixed coordinates. The usual cascade of P-position and PI-speed controllers is used. The speed controller was designed according to the well-known symmetrical optimum, based on the load acquisition filter s or observer s delay time constant together with an additional time constant of T S = 234µs for digital control and current control loop. For design of the position controller, the optimum of magnitude was used. There are two possibilities to design a speed controller using feedback from an observer. First, the observer s characteristic polynomial can be approximated by its linear part in order to get an approximate time constant, which is then used for speed controller design. Second, the method from state control can be used. In state control, the controller feedback is usually designed independently from the observer, i. e. assuming that all state variables were measurable without delay. Care must only be taken that the observer poles are far left from the control system s poles. This method is based on the separation theorem, which states that the eigenvalues of the closed controller loop with observer will be those of the control loop plus the observer s. For the proof of the separation theorem [14], no assumptions about the feedback vector need to be made, so it extends to the proportional controller feeding back only one state variable. If the PI speed controller is designed according to the symmetrical optimum assuming a delay-free observation, the gain is too high to produce working results. Thus, an extension of the symmetrical optimum was used where the proportional gain is chosen arbitrarily, and the integral gain is computed such that in the bode plot, the maximum phase occurs where the logarithmic absolute value graph crosses zero. For an open speed loop transfer function PI speed controller controlled system { (}}{ F O (s) = K R ) {}}{ K S st I s (1 + st S ) (3) the rule is to set the integration time constant to T I = 1 (K R K S ) 2 T S (4)

5 ' &! " # $ % & ' % $ # "! I J A H B EJ A H # # A H * K J J A H M H J D B EJ A H # # 0! A H * K J J A H M H J D B EJ A H % # A H O+ I D? A D> A B B B EJ A * # 0! A H + D A > O I? D A B B B EJ A * ' & 0 I J A H B EJ A H & # 0 J? D B EJ A H & # ' A H * K J J A H M H J D B EJ A H '! 0 J? D B EJ A H '! 0 I J A H B EJ A H $ > I J F B EJ A H I A A H EC E@ O > I A H L A H F A I F =? J I " H EC E@ O > I A H L A H F A I F =? E A = " 0 * K J J A H M H J D B EJ A H H EC E@ O I J = J E? = = B EJ A H $ G!! 4 E A H J E= I F A O > I A H L A = F E C E A H J E= I F A O > I A H L A H " = F E C E A H J E=? F A J A > I A H L A = F E C E A H J E=? F A J A > I A H L A H " = F E C! E A H J E= I F A O > I A H L A = F E C! E A H J E= I F A O > I A H L A H " = F E C! E A H J E=? F A J A > I A H L A = F E C! E A H J E=? F A J A > I A H L A H " = F E C! E A H J E= H K? > I A H L A = F E C! E A H J E= H K? > I A H L A H " = F E A HEL = JEL A M = AI HI EL = J EL A M B EJ A H J? D F B EJ = I AI H B EJ A H H EC E@ O > I A H L A H JM E A HJE= > I A H L A H J D H A A E A H J E= > I A H L A H # # # F A I F A F C = E O = E? = I J EB B A I I H Fig. 5. Investigated speed acquisition structures, used open loop gain, and achieved dynamic stiffness no longer regarding the observer s delay. This method was used for the two and three mass observers. Because of their high order and thus high delay time constant, the standard symmetrical optimum would have led to inacceptably low gains. VI. EXPERIMENTAL RESULTS Twenty one different speed acquisition schemes were investigated experimentally for their closed loop performance (see fig. 5). The comparison is based on equal steady state behavior, i. e. in all cases the control was tuned to produce a 0.1 rad/s r.m.s. speed deviation at steady state. Performance is measured as the dynamic stiffness when stepwise load changes are applied to the position controlled system. The open speed loop gain that resulted from the tuning, and the measured dynamic stiffness are shown in fig. 5. It can be seen that all low pass filters (no. 1-5) lead to similar results. Performance shrinks with increasing filter order, thus filters of higher order than 3 were not investigated. Obviously for the used setup, a steeper stop band behavior at the expense of a higher delay time constant does not pay, because the resonance frequencies allow a high control loop bandwidth even when lightly damped. The control loop without any filter was unusable, though. The low pass plus notch filters (no. 6,7) perform significantly better than simple low pass filters. The reason is that the notch filter achieves a much better suppression of the 970 Hz resonant frequency (see fig. 1) while contributing only few to the delay time constant. The FIR band stop filter according to (2) (no. 8) shows a weak stop band behavior, as can be seen in figure 1, in spite of its rather high delay time contribution. This explains the poor performance of this filter in the experiments. The results using rigid-body observers (no. 9-11) are very poor. Generally speaking, an observer compared to a filter implements knowledge about the controlled system, so that the modeled behavior can be observed without delay. The expense, as figure 1 shows, is a much poorer low pass characteristic; please note that the observer is 3rd order. The rigid body observer neither implements knowledge about the 970 Hz resonance, nor suppresses it sufficiently, which is an explanation for the poor performance. The observers for two and three inertia systems (no ) do implement this knowledge in a physically quite correct way. As a result, their performance is significantly better than that of a filter. Best performance is achieved using the speedonly observer for a three mass system. Compared to the reduced observer, it has the advantage of realizing a prediction of the state variables. The complete observer is inferior because has a higher order and tends more to low frequency oscillations, most likely due to model imperfections. These oscillations were the limiting factor for the control loop gain using complete three mass observers; their speed quality was better than demanded. The speed-only two inertia observer no. 12 shows a little inferior performance, while other structure variations lead to much inferior results. So, the approximation of the system as a two mass system is sufficient if the observer is designed carefully, but it is shown here that the results are better and more reliable using the three mass system. It has to be added that for the two mass system, identification and modeling are much easier. Concerning the pole placing, placing to stronger damped locations was not successful. It enlarged the noise in the feedback signal without achieving a better damping effect of the whole control. The Kalman filter method did not work for the two and three mass observers. It was tried, but did not produce useful result. This is because the main disturbances of the setup are the encoder oscillations in the range of Hz. This frequency information was used for pole placing (the poles absolute values were put well away from that range), but can hardly be formulated as white noise covariances. It must be admitted that, if a mechanical oscillating system is regarded as the correct model for the setup, the deviation of the encoder speed is surely not a valid quality measure. In real applications, the speed quality at the load mass would be important. It could not be used here because the load servo s encoder signal is not yet available for the control PC. As a

6 Fig. 6. FFT graphs of machine speed Ω m and reference current i q during uniform turning at 1 rev/s: 1st order filtered derivative used as speed signal (performance see fig.5 no. 1) promising fact, oscillations in a multi mass system usually affect each mass; if they can be measurably damped at one location, they are damped out everywhere. Fig. 6 shows the frequency spectra of machine speed and reference current for a control loop with speed acquisition using derivation of the position signal and a 1st order low pass filter. The resonant frequencies at 970 and 1370 Hz can be seen as well as the 1st, 2nd and, aliased, 3rd harmonic of the encoder frequency. Certain low frequencies have high amplitudes due to slot latching. The encoder frequency harmonics are even feeded through the controller and can be seen in the reference current signal. The magnitudes in the speed signal are nearly equally high for all control loops. Using higher order filters reduces them a little, and suppresses those frequencies in the current signal. Though, the overall performance of the higher order filters is poorer. VII. CONCLUSION In this paper, different speed acquisition schemes based on sinusoidal encoder signals are tested for their ability to solve the tradeoff between a fast rejection of load changes and a low deviation of speed while turning, i. e. to allow high gain control loops with the ability to turn smoothly. For the setup regarded, the limiting problem is a 970 Hz mechanical resonance. In addition to the industry-standard filtered derivative, observers based on the rigid-body, two-mass, and three-mass models are investigated. The investigations revealed that the low pass filtered derivative of the encoder signal is yet a rather good speed signal, allowing a dynamical stiffness around 1000 Nm/rad together with a rather smooth turning. For the setup regarded, higher order filters perform weaker. Performance is improved by adding a notch filter, preventing the controller from exciting the plant s lowest resonant frequency. The simple rigid-body observer achieves much poorer results, i. e. only half the stiffness compared to a control loop with first order filter. This can be explained because the observer neither considers the resonance in its model nor achieves a transfer function with a good low pass behavior. Observers based on two or three inertia mechanically oscillating system models do implement knowledge about the resonance, and produce much better results, allowing a stiffness as high as 2200 Nm/rad. Three inertia observers show a little better performance, at the expense of a much more complicated identification and modeling procedure. Further work will be to add an encoder signal correction which eliminates the encoder frequency harmonics from the spectrum of the feedback signal. ACKNOWLEDGMENT This investigation is part of a research supported by the DFG Deutsche Forschungsgemeinschaft under Project No. MU 1109/6-1 REFERENCES [1] S. Vukosavic, M Stojic: Suppression of Torsional Oscillations in a High-Performance Speed Servo Drive; IEEE Transactions on Industrial Electronics, Vol.45 No.1 pp , February 1998 [2] A. Gees: Accelerometer-enhanced Speed Estimation for Linear-Drive Machine Tool Axes; Diss. Ecole Polytechnique Federale de Lausanne 1996 [3] U. Brahms: Regelung von Lineardirektantrieben fuer Werkzeugmaschinen; Diss. Universitaet Hannover 1998 [4] S. Brueckl: Regelung von Synchron-Linearmotoren fuer hochgenaue Vorschubantriebe bei Werkzeugmaschinen; Diss. TU Muenchen 2000, VDI Fortschritt-Berichte Reihe 8, Nr. 831 [5] Y. Hori, H. Iseki, K. 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