velocity observer for pmsm without rotational...

Abstract—In this paper, a velocity estimator is proposed for the purpose of improving the transient velocity control performance of a permanent magnet synchronous motor (PMSM) operating without a rotor position or velocity transducer. An observer is designed to estimate rotor velocity using the electro-mechanical parameters of the system, measured stator currents, and the estimated rotor angle, where the angle estimate is obtained from a separate observer. Special attention is given to very low speed operation where the rotor angle estimate may be inaccurate due to ill-conditioning of the rotor angle observer under those conditions. Experiments are used to examine the velocity control performance of the resulting sensorless PMSM drive and the robustness of the system to external disturbances and model uncertainty.

Index Terms—brushless machines, digital control, motor control, motor drives, state estimation, velocity estimation, torque control, variable speed drives.

I. INTRODUCTION

UE to reliability concerns or physical constraints associated with the use of rotor position and velocity

sensors, there is considerable interest in the operation of the permanent magnet synchronous motor (PMSM) without rotor or velocity transducers. Many approaches have been proposed for estimating both rotor angle and velocity; but little attention has been given to velocity control of rotor speed/position sensorless PMSM systems for applications where precise velocity tracking performance is required in transient.

High-performance PMSM drives require fast and accurate velocity feedback for optimal velocity control of the machine. The accuracy and speed of response of the velocity feedback directly influences dynamic and steady-state velocity control performance. In a position/speed sensorless PMSM drive, however, the lack of a speed feedback sensor presents a considerable challenge to obtain satisfactory speed tracking in transient, such as during startup, a speed reversal, or tracking a time-varying speed command.

Speed estimation techniques for the PMSM generally fall into one of two categories. One method is to differentiate the estimated rotor position to obtain velocity [1]. Another is to use the ratio of induced stator emf and permanent magnet (PM) flux linkage [2]. The former approach is quite accurate in steady-state, but yields poor response during transient and

T. D. Batzel is with the Department of Electrical Engineering, Penn State Altoona, Altoona, PA 16601 USA (e-mail: [email protected]).

K. Y. Lee is with the Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802 (e-mail: [email protected]).

near zero speed. The latter technique gives a fast response, but is prone to errors arising from measurement noise and machine parameter uncertainty. Several researchers have proposed improvements to velocity estimation where both of the speed estimation methods are used concurrently. The work in [3] uses the derivative of the estimated position over a period of time to eliminate steady-state errors produced by parameter uncertainty in the PM flux linkage and stator resistance. The work in [4] suggests a technique that uses the derivative of rotor angle for steady-state angular velocity determination, but relies primarily on the ratio of emf to PM flux linkage during transient. These works have demonstrated overall improvement in the velocity estimate, but the bandwidth of the velocity estimate after low-pass filtering is still not sufficient for many sensorless PMSM drive applications.

The objective of the work in this paper is to improve the velocity tracking performance of the rotor speed/position sensorless PMSM drive proposed in [5] to address the need for fast and accurate velocity tracking in transient. The proposed improvements to the velocity control performance of the PMSM drive are to be carried out through the application of a velocity observer that uses measured stator currents, the model of the electromechanical system, and the rotor position estimate derived from the observer described in [6]. This rotor position observer was selected due to its very robust operation and accurate position estimation. However, attention must be given to low-speed system performance where the rotor angle is not observable [6]. Experimental results are used to demonstrate the improved velocity tracking performance of the proposed system compared to the technique used in [5].

II. ANGULAR VELOCITY ESTIMATION

There are two major categories of velocity estimators used in sensorless PMSM drives. The position differentiation method obtains angular velocity by differentiating the estimated rotor angle. The back-emf method relies on measured terminal voltage and current and the PMSM model parameters to determine velocity. In this section, both techniques are presented and an adaptive velocity estimator that is a combination of these two techniques is discussed.

A. Differentiation Method

The differentiation method for estimating angular velocity

uses the time derivative of the estimated rotor angle θ̂

dtde θω ˆˆ = . (1)

Velocity Observer for PMSM without Rotational Transducer

Todd D. Batzel, Member, IEEE, and Kwang Y. Lee, Fellow, IEEE

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Clearly, this method requires a stable rotor position observer. In addition, the rotor angle observer should have poles with a natural frequency that is high relative to the angular velocity

eω [7]. Subject to these constraints, the estimated angular

velocity obtained by (1) approaches the actual value when averaged over a sufficient period of time. The differentiation method yields accurate steady-state speed estimates in the medium to high speed operating range; however, is susceptible to noise that results from the differentiation process and the incorrect speed estimates that may result at very low-speed where the operation of the rotor angle observer is problematic. Thus, the differentiation method is of limited utility during low-speed operation, startup, and during speed reversals.

B. Back-emf Method

The back-emf estimator for magnitude of the angular velocity is [6]:

( ) ( )

m

e

RivRiv

λω ββαα

23

ˆ22 −+−

= , (2)

where vαβ and iαβ represent the terminal voltage in the

stationary two-phase reference frame, and R and mλ are the

phase resistance and PM flux linkage constant, respectively. From (2), it can be seen that only the magnitude of the angular velocity can be directly determined. To resolve the direction of rotation at sampling interval k we use:

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

=

−−=

−

)()(

)()(tan)(

ˆ)1()(sgnˆ

1

kRikv

kvkRik

kk ee

ββ

ααφ

ωφφω

. (3)

The back-emf method produces relatively accurate estimates of the magnitude of angular velocity – even at zero and very low-speed. However, determining the direction of rotation at low-speed is problematic, and estimation accuracy is dependent on the modeling certainty of PMSM parameters. Furthermore, any sensing error associated with the stator voltage and current will result in velocity estimation error with an ac component at the fundamental frequency and its second harmonic [5]. To remove the unwanted ac component, low-pass filtering of the velocity estimate is required, which tends to reduce velocity tracking accuracy in transient.

C. Adaptive Velocity Estimation

The differentiation and back-emf method are sometimes combined to form the adaptive velocity estimation method [3]. The back-emf method yields good performance at very low speeds, but is subject to errors arising from PMSM parameter uncertainty and resolution of the direction of rotation. The differentiation method, however, is essentially independent of the parameter estimation errors so long as the observer poles have a natural frequency much higher than the angular velocity and the velocity is sampled over a sufficient period of time. Thus, the strengths of each method may be

combined to form the velocity estimation correction scheme shown in Fig. 1.

To use (1) to correct for velocity estimation errors obtained by (2), a modifier ∆ is included to represent the uncertainty in the permanent magnet flux linkage:

( ) ( )

( )∆+

−+−=

m

e

RivRiv

λω ββαα

23

ˆ22

. (4)

The adaptation process compares the output of (1) with (2) to generate the velocity estimation error. As long as the magnitude of the speed estimate of (4) is above the low-speed threshold where the rotor angle observer is assumed to be stable, a PI controller operates on the estimation error, adjusting the output ∆ to correct for the parameter uncertainty. Time constant of the controller is chosen to be slow to remove noise generated by the differentiation of the rotor angle.

The adaptive velocity estimation technique has demonstrated the ability to estimate velocity accurately in steady-state despite PMSM modeling uncertainty and signal measurement errors. However, the required low-pass filtering of the estimated velocity reduces the accuracy during transient.

III. MODEL-BASED ANGULAR VELOCITY OBSERVER

Here, a model-based angular velocity observer is developed. The technique was originally developed to improve velocity estimation in PMSM servo drives using rotary encoder feedback devices [8]. In the context of this work, however, it shall be used to provide highly accurate rotor position feedback in a position/velocity sensorless PMSM drive.

A. Electro-Mechanical System Model

The electro-mechanical model for the PMSM is given by

mm

d

dt

θ ω= (5)

m m l md T T B

dt J

ω ω− −= (6)

where mθ , mω , mT , lT , B and J represent the mechanical

rotor angle, mechanical angular velocity, motor torque, load torque, viscous friction, and inertia, respectively. To obtain a third state equation, the load torque is assumed to be a slowly varying quantity such that it is constant over the sampling interval:

Fig. 1. Adaptive velocity estimation block.


0ldT

dt= . (7)

The developed motor torque for a non-salient PMSM is written as [9]:

3 ˆ ˆsin cos2m m m mT P i iα βλ θ θ⎡ ⎤= − +

⎣ ⎦, (8)

where P is the number of pole pairs, and m̂θ is the estimated

mechanical rotor angle.

B. Velocity Observer Construction

With the state variables, input vector, and output defined as

[ ]m m lTθ ω ′=x (9)

mT=u (10)

mθ=y , (11)

respectively, the state space definition of the electro-mechanical system can be written

==

x Ax + Bu

y Cx

&

, (12)

where

[ ]0 1 0 0

1 10 ; = ; = 1 0 0

00 0 0

BJJ J

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥= − − ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

A B C . (13)

Using standard observer design techniques for a linear system [10], the form of the proposed angular velocity observer is:

( )e = A - GC e& (14)

ˆe = x - x , (15) where G and e are the observer gain matrix and state estimation errors, respectively. The observer is used to reconstruct the state variables yielding the estimated angular velocity mω as shown in the block diagram of Fig. 2. The

estimated rotor angle mθ is used only to compare with the

value obtained from the separate rotor angle observer.

The observability of the system is determined by the rank of the observability matrix [11]:

′⎡ ⎤⎣ ⎦2C CA CA . (16)

The observability matrix is full rank for finite rotor inertia so that the angular velocity is observable for all practical operating conditions.

C. Observer Pole Placement

The observer eigenvalue locations determine the speed of convergence and the stability of the observer. Let the desired eigenvalue ( λ ) locations be placed according to:

1 1kλ = − (17)

2 2 3k jkλ = − + (18)

*3 2 3 2k jkλ λ= − − = . (19)

The constants 1k and 2k should be positive real numbers for

stability of the observer. The closed loop eigenvalue locations of (17)-(19) are enforced by the gain matrix G.

Following the procedure outlined in [11], the observer gain matrix to place the eigenvalues at the location specified in (17)-(19) is determined:

( )( )

1 2

2 22 3 1 2 1 2

2 21 2 3

2

2 2

Bk k J

B Bk k k k k kJ J

Jk k k

⎡ ⎤+ −⎢ ⎥⎢ ⎥= + + + − −⎢ ⎥⎢ ⎥

− +⎢ ⎥⎣ ⎦

G . (20)

The observer gains given in (20) are included in the overall velocity observer block diagram of Fig. 2.

IV. EXPERIMENTAL RESULTS

To evaluate the performance of the velocity observer in the sensorless drive, simulations were performed where the position/velocity sensorless controller shown in Fig. 3 operates the PMSM. For various operating conditions, comparisons between the adaptive velocity estimator (section II) and model-based velocity observer (section III)

Fig. 2. Velocity observer block diagram.


performance were made. In both cases, the rotor position observer described in [6] was used. In addition, voltage and current sensing error was introduced into the simulation so that low-pass filtering of the adaptive velocity estimate was necessary as described in [5]. For all experiments, the cutoff frequency of the low-pass velocity estimation filter was set to ¼ of the fundamental electrical frequency. For the model-based observer, the rotor inertia and friction used by the model differed from their actual values by a factor of two to intentionally introduce model uncertainty to the simulation.

A. Steady-state Performance

The experiment shown in Fig. 4 demonstrates steady-state velocity track performance. From the figure, the model-based

Fig. 3. Block diagram of sensorless PMSM drive system.

a) adaptive velocity estimator with filter.

b) model-based velocity observer

FIG. 4 STEADY-STATE VELOCITY TRACKING PERFORMANCE COMPARISON.

a) adaptive velocity estimator with filter


Fig. 5. Startup performance comparison.

velocity observer shows increased accuracy as compared to the performance using the adaptive velocity estimator. The


actual and commanded velocities differ by only .01 rpm., compared to about 0.6 rpm difference for the old velocity estimator. It should be noted that lowering the cut-off frequency of the low-pass filter could reduce the adaptive velocity estimation error at the expense of accuracy in transient.

B. Transient Performance

Startup from zero speed and speed reversals of a sensorless system are challenging since rotor angle is not observable at very low speed. In this section, performance under transient conditions is examined.


b) model-based velocity observer Fig. 6. Speed reversal performance comparison.



Fig. 7. Dynamic velocity command tracking comparison.


Fig. 5 shows a startup from zero speed with an initial rotor angle estimation error of approximately 0.12 radians (typical error after a forced alignment). From the figure, the adaptive velocity estimator yields poor tracking during the startup transient compared with the performance using the model-based observer, which tracks the command nearly perfectly.

A speed reversal is shown in Fig. 6. Similar to startup performance, the model-based observer approach produces superior tracking during the transient through zero speed. The adaptive velocity estimator difficulties around zero speed are due to several phenomena, including a lag introduced by filtering, and difficulty in determining the direction of rotation at near-zero speed.

Dynamic performance is further evaluated in Fig. 7, where the tracking of a periodic, time-varying sinusoidal speed command is examined. Due to the required low-pass filter, the adaptive velocity estimator is not able to track the command. Tracking of the velocity command using the model-based observer is reasonable.

V. CONCLUSIONS

This paper has presented the application of a model-based velocity observer intended to increase the ability of a position/velocity sensorless PMSM drive to track velocity commands in both steady-state and transient operating conditions. Simulations have been used to demonstrate the effectiveness of the proposed velocity observer in terms of estimation accuracy and robustness. In transient, the velocity observer performs very well, and the system was shown to perform reliably for both startup and speed reversal transients. Comparisons with another velocity estimation technique (adaptive velocity estimation) indicate that the model-based observer will yield superior performance in transient. In conclusion, the model-based velocity observer offers improvement over the adaptive velocity estimation algorithm in cases where low-pass filtering is required to remove unwanted ac components from the velocity estimate. The ac components arise due to harmonics in the PMSM back-emf waveform, and stator voltage and current sensing errors.

VI. REFERENCES [1] K.W. Lim, K.S. Low, and M.F. Rahman, “A position observer for

permanent magnet synchronous motor drive,” 20th International Conference on Industrial Electronics, Control and Instrumentation (IECON), vol. 2, pp. 1004-1008, 1994.

[2] L. Sicot, S. Siala, K. Debusschere, and C. Bergmann, “Brushless DC motor control without mechanical sensors,” 27th Annual IEEE Power Electronics Specialists Conference, 1996, vol. 1, pp. 375-380, June 1996.

[3] J. Kim and S. Sul, “New approach for high performance PMSM drives without rotational position sensors,” IEEE Transactions on Power Electronics, vol. 12, pp. 904-911, Sept. 1997.

[4] J.X. Shen , Z.Q. Zhu, and D. Howe, “Improved speed estimation in sensorless PM brushless AC drives,” IEEE Trans. on Ind. App., vol. 38, no. 4, pp. 1072-1080. Aug. 2002

[5] T.D. Batzel, “Improved angular velocity estimation for high performance sensorless PMSM,” 8th Annual IEEE Workshop on Power Electronics in Transportation, pp. 89-96, Oct. 2004.

[6] T. D. Batzel and K. Y. Lee, “Slotless permanent magnet synchronous motor operation without a high resolution rotor angle sensor,” IEEE Trans. Energy Conv., vol. 15, no. 4, pp. 366-371, 2000.

[7] T. D. Batzel, Electric propulsion using the permanent magnet synchronous motor without rotor position transducers, Ph.D. dissertation, Dept. of Elect. Eng., Pennsylvania State University, University Park, 2000.

[8] X. Dianguo, W. Hong, and S. Jingzhuo, “PMSM servo system with speed and torque observer,” IEEE 34th Annual Power Electronics Specialists Conference, vol. 1, pp. 241-245, June 2003.

[9] T.D. Batzel and K.Y. Lee, “Commutation torque ripple minimization for permanent magnet synchronous machines with Hall effect position feedback,” IEEE Transactions on Energy Conversion, v. 13, no. 3, pp. 257-262, 1998.

[10] W.L. Brogan, Modern Control Theory, Englewood Cliffs, NJ: Prentice-Hall, 1991.

[11] C. Phillips and R. Harbor, Feedback Control Systems, 4th ed., Englewood Cliffs, NJ: Prentice-Hall, 1999.

VII. BIOGRAPHIES

Todd D. Batzel (M‘00) received the B.S. and PhD. degrees in electrical engineering from the Pennsylvania State University, University Park, in 1984 and 2000, respectively, and the M.S. degree in electrical engineering from the University of Pittsburgh, Pittsburgh, PA, in 1989. Currently, he is Assistant Professor of electrical engineering at Penn State Altoona. His research interests include electric machines, electric motor

controls, power electronics, artificial intelligence applications to control, and embedded control systems. Dr. Batzel is a Member of the IEEE.

Kwang Y. Lee (F’01) received the B.S. degree in electrical engineering from Seoul National University, Korea, in 1964, the M.S. degree in electrical engineering from North Dakota State, Fargo, in 1968, and the Ph.D. degree in systems science from Michigan State University, East Lansing, in 1971. Currently, he is a professor of electrical engineering and is Director of Power Systems Control Laboratory at the Pennsylvania State University,

University Park. He has also been with Michigan State, Oregon State, and University of Houston. His interests include power systems control, operation and planning, and intelligent system applications to power systems. Dr. Lee is and an Associate Editor of IEEE Transactions on Neural Networks and an Editor of IEEE Transactions on Energy Conversion. Dr. Lee is a Fellow of the IEEE.


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