IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006 937

A Simple Improved Velocity Estimation for Low-Speed RegionsBased on Position Measurements OnlyY. X. Su, C. H. Zheng, P. C. Mueller, and B. Y. Duan

Abstract—High-quality low-speed motion control calls forprecise position and velocity signals. However, velocity estimationbased on simple numerical differentiation from only the positionmeasurement may be very erroneous, especially in the low-speedregions. A simple efficient high-quality instantaneous velocity esti-mation algorithm is developed in this paper, by using the positionmeasurements only. The proposed estimator is constructed basedon the fact that numerical integration can provide more stable andaccurate results than numerical differentiation in the presence ofnoise. The main attraction of the new algorithm is that it is veryeffective as far as in low-speed ranges, high robustness againstnoise, and easy implementation with simple computation. Bothextensive simulations and experimental tests have been performedto verify the effectiveness and efficiency of the proposed approach.

Index Terms—Integration, measurement noise, motion control,tracking filters, velocity estimation.

I. INTRODUCTION

BOTH extensive theoretical and experimental resultsdemonstrate that the performance of control systems can

often be enhanced by including some type of velocity feedbackcontrol techniques [1], [2]. However, it may not always bepossible to measure velocities, or it may not even be desirableto do it. For example, the signals delivered by tachometersmight be contaminated with noise. Also, costs and possibilityof failure are increased when more sensors are used. In practice,the optical encoder is still the most popular position sensorused in industrial fields because of its simple detection circuit,high resolution, high accuracy, and relative ease of adaptationin digital control systems. Therefore, it is necessary to numeri-cally reconstruct a high-quality velocity signal from the noisyposition measurement.

Many researchers contributed their efforts on the accuratevelocity estimation based on the measured position only, andvarious methods have been proposed. Brown et al. [3] providedan overview, classification, and comparison of the existingtechniques for velocity approximation, including Taylor seriesexpansions, backward difference expansions and least-squarecurve fits. It is well known that these methods are inaccurateat low- and high-velocity ranges. To obtain a high-quality ve-locity signal, various filtering techniques have been developed[4]–[10]. This method, however, suffers from the drawback of

Manuscript received January 26, 2005. Manuscript received in final formMarch 7, 2006. Recommended by Associate Editor S. Weibel.

Y. X. Su and B. Y. Duan are with the School of Electro-Mechanical Engi-neering, Xidian University, Xi’an 710071, China (e-mail: yxsu@mail.xidian.edu.cn).

C. H. Zheng is with the School of Electronic Engineering, Xidian University,Xi’an 710071, China.

P. C. Mueller is with the School of Safety Control Engineering, University ofWuppertal, D-42097 Wuppertal, Germany.

Digital Object Identifier 10.1109/TCST.2006.876917

additional delay, which is particularly harmful when the filteredvelocity is to be used for time-critical feedback or synchroniza-tion purposes. Therefore, Valiviita and Vainio [8] presenteda so-called delayless recursive finite impulse response (FIR)filter, Janabi-Sharifi et al. [9] developed a class of adaptive FIRvelocity estimation techniques, and Zhang et al. [10] designeda dynamic nonlinear filter for derivative estimation, targetingoptimal attenuation of white noise and delay. As an alternativeto the use of filter techniques, some researchers employed aKalman filter to estimate the velocity accurately [4], [11], [12].Since Kalman filter is a model-based approach that requiresthe target velocity trajectory be sent to the filter, it cannot beapplied to the case where arbitrary velocity is measured. Withthe development of control theory, many approaches using theobserver theory have been extensively studied [13]–[21]. Othervelocity estimation methods based on the position measurementonly can be found in [22]–[24].

In fact, velocity belongs to the chain of kinematic quantities:position and velocity. They are interrelated by successive differ-entiations and integrations. Therefore, all the kinematic quan-tities could be derived from a single quantity. However, onlyintegration is widely used to process these kinematic quanti-ties in practice, since integration typically provides advanta-geous noise attenuation. Differentiation, on the other hand, isnoise amplifying by its nature. Motivated by this fact, a non-linear tracking differentiator (TD) [25]–[28] is developed forhigh-quality derivative estimation. Though the developed TDcan select a better derivative signal in steady state over the con-ventional backward difference, it has no favorable transitioncharacteristics, and the convergence is slow.

In this brief, an enhanced differentiator (ED) is proposed foran improved velocity estimation in low-speed regions. The mainadvantages of the proposed method are twofold. First, unlikeother available Kalman filter or observer-based approaches, theproposed velocity estimator is model free and easy to imple-ment with simple calculation. Second, accurate instantaneousvelocity estimation can be selected even in very low-speed re-gions. The effectiveness of the proposed approach is verified byboth numerical simulations and real-time experimental results.

II. ENHANCED TRACKING DIFFERENTIATOR

The proposed ED is constructed based on the fact that numer-ical integration can provide more stable and accurate results thannumerical differentiation in the presence of noise. In particular,given a reference signal , the system can provide two sig-nals and , such that and ,respectively.

First of all, the following lemmas are given.Lemma 1: Suppose is a continuous function defined in

that satisfies . If

1063-6536/$20.00 © 2006 IEEE

938 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

, then for an arbitrarily given , the following expressionholds:

(1)

Proof: See [30].Lemma 2: If the following system is asymptotically stable at

the origin

(2)

then for any arbitrarily constants , and , thesolution to the system

(3)

makes the following expression hold:

(4)

Proof: See [30].Lemma 3: If the following system is asymptotically stable at

the origin:

(5)

then for any arbitrarily bounded integrable function , andgiven constants and , the solution to thesystem

(6)

satisfies the following equation:

(7)

Proof: See [25] and [30].Lemma 4 [28], [30]: The following system is asymptotically

stable at the origin:

(8)

if the parameters are chosen to , andare chosen to positive odd numbers.

Proof: Let us define a Lyapunov-like function candidate as

(9)

Since and are chosen as positive odd numbers, isan even number. As a result, . Therefore, it isstraightforward that .

Differentiating (9) with respect to time and using (8), we have

(10)

Notice that is an even number, so . Asa result, we can conclude that

(11)

Hence, is a nonnegative Lyapunov-like functionwhose time derivative is negative in .Since means , using (8), we have

. By the fundamental Lyapunov Theorem, we have, as .

This completes the proof.According to Lemmas 1–4, an enhanced differentiator (ED)

can be formulated as

(12)where the state variables and represents the es-timated states, that is,

, and are positive odd numbers.

III. SIMULATIONS

The following simulations were performed to validate thebetter performance of the proposed enhanced differentiator(ED) in comparison with a nonlinear tracking differentiator(TD) used in [27] and an extended state estimator (ESO) pre-sented in [29] and [30]. For this velocity estimation application,the discrete ED with Euler method can be described as

(13)

where and represent the estimated position and velocity, re-spectively, is the position estimation error,

is the reference position, is the sampling period, and de-notes the th sampling instant.

The discrete TD with Euler method can be expressed as [27]

(14)

where denotes a velocity factor to determine the transition,and a filtering factor to cancel out the noise. Large mayprovide a fast convergence and better tracking and large ishelpful to cancel out the noise. Usually a small sampling time

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006 939

requires large to guarantee the fast convergence. The non-linear function is defined as

(15)

where denotes a standard sign function, and and canbe determined as follows:

(16)

with

(17)

The discrete extended states estimator (ESO) with Eulermethod can be expressed as [29], [30]

(18)

where represents the estimated real action of the unknown dis-turbance, which is very helpful to realize the disturbance com-pensation without a model [29], [30], and , andare design parameters. The nonlinear function is de-fined as

(19)

It is presumed that the reference position profile is corruptedby an additive white noise component with the maximum ampli-tude of 0.01 rad. The sampling period was determined asms. The parameters were determined as follows:

, and for the ED,and T for the TD, and

, and for the ESO, respec-tively. All the initial values are set as zero.

The simulations of the proposed ED, TD, and ESO at zero ve-locity case were first performed. The velocity estimation errorsobtained by using the ED, TD, and ESO are shown in Fig. 1,respectively. It can be seen that the two differentiators (ED andTD) have almost the same transition, but the proposed ED ob-tains a more improved velocity estimation precision in compar-ison with the TD. The proposed ED obtains a comparative re-sult over the ESO with a rapid transition. Second, suppose themotion control system performs a unit cosinoidal position pro-file, that is, (rad). The velocity estimation errors areshown in Fig. 2. From the comparisons, it can be seen that theproposed ED has much better estimation performance than theTD, and a comparative estimation with a rapid transition overthe ESO. Third, to verify the improved velocity estimation in

Fig. 1. Velocity estimation errors of ED, TD, and ESO from a unit step positionprofile.

Fig. 2. Velocity estimation errors of ED, TD, and ESO from a unit cosinoidalposition profile.

low-speed ranges, comparisons from a very low cosinoidal po-sition profile (rad) with the same noise wereconducted again. Fig. 3 illustrates the velocity estimation errors.It can be seen that the better velocity estimation is also obtainedby the proposed ED. Furthermore, to validate the high robust-ness of the proposed ED, TD, and ESO against measurementnoise, simulations from a step unit position and a very low cosi-noidal position profiles with a large noise amplitude of 0.02 rad,were also performed. With this large noise, the velocity estima-tion errors obtained by using ED, TD, and ESO are shown inFigs. 4 and 5, respectively. It can be seen that the proposed EDalso retains the advantage of the TD and ESO that have higherrobustness against noise.

940 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Fig. 3. Velocity estimation errors of ED, TD, and ESO from a very low cosi-noidal position profile.

Fig. 4. Velocity estimation errors of ED, TD, and ESO from a unit step positionprofile with a large noise.

Therefore, from these simulations, we can conclude that theproposed ED has a higher precision velocity estimation in com-parison with the TD, based only on the position measurementsas well as in low-speed ranges. Compared with the ESO, the pro-posed ED obtains a comparative estimation result with a rapidtransition and simple calculation. Moreover, the proposed EDstill retains the advantage of having higher robustness againstnoise.

IV. EXPERIMENTAL RESULTS

A. Velocity Estimation

Experiments were further carried out to verify the effec-tiveness of the proposed enhanced differentiator in practice.The experimental setup consists of a six-channel D/A output

Fig. 5. Velocity estimation errors of ED, TD, and ESO from a very low cosi-noidal position profile with a large noise.

Fig. 6. Experimental tracking errors of a step position profile.

card PCL-726, a three-axis quadrature encoder card PCL-833,and a permanent magnet synchronous motor (PMSM) withthe matched servo drive made by Panasonic Inc. The feedbackposition signal is measured by an incremental encoder with theresolution of 4000 pulse/rev. The high-precision position ofthis PMSM system is achieved by using the nonlinear controlmethod presented in [27], and implemented in C on an Advan-tech industrial computer.

The experimental position tracking error for a step positionprofile is shown in Fig. 6. With this position error, the velocityestimation errors are shown in Fig. 7, respectively. Obviously,better velocity estimation is obtained by using the proposed EDover that of the TD. Compared with the ESO, the proposed EDobtains a comparative estimation result with a rapid transition.After that, comparative velocity estimations from a low sinu-soidal position profile of rad were performed again.Fig. 8 illustrates the position tracking error. The steady-statetracking error is less than rad. With this position error,

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006 941

Fig. 7. Experimental velocity estimation errors of ED, TD, and ESO from aunit step position profile.

Fig. 8. Experimental tracking error of a very low sinusoidal position profile.

the obtained velocity estimation errors are shown in Fig. 9, re-spectively. It can be seen that an improved velocity estimationis also obtained by using the proposed ED.

As a result, we can conclude that the developed ED can obtaina much better velocity estimation in comparison with the TDand ESO as well as in low-speed ranges, based on the positionmeasurements only.

B. Motion Performance Improvement

The estimated high-quality velocity can be used in the feed-back loop of the motion controller for an improved performance.Using the high-quality velocity obtained by the developed dif-ferentiator, a simple PD position controller is formulated andcan be expressed as

(20)

Fig. 9. Experimental velocity estimation errors of ED, TD, and ESO from avery low sinusoidal position profile.

Fig. 10. Experimental position tracking errors by using ED and TD for a unitstep position profile.

where is the control signal to the servo drive of the PMSMsystem, and are the desired position and velocity, respec-tively, and and are proportional and derivative gains,respectively. For this application, they are determined as

and , respectively.To illustrate the improved performance of using the high-

quality velocity obtained by the proposed ED and TD, positiontracking of a step position profile was first performed. The posi-tion tracking errors are shown in Fig. 10. It can be seen that thesystem with the proposed ED has a faster transition and a bettertracking performance over that of the TD. In addition, the posi-tion tracking of a sinusoidal position profile radwas also conducted. Fig. 11 illustrates the position tracking er-rors of this periodical sinusoidal position profile. From the com-parison, it can be seen that the ED control obtains a much better

942 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 14, NO. 5, SEPTEMBER 2006

Fig. 11. Experimental tracking errors by using ED and TD for a low sinusoidalposition profile.

position tracking in comparison with the TD control. Also, notethat there are more high-frequency components with TD, whichjust demonstrates the high-quality velocity selected by the pro-posed ED.

V. CONCLUSION

A high-quality instantaneous velocity estimation, as well asin low-speed regions, is developed by using position measure-ments only. The proposed method is constructed based on thefact that numerical integration can provide more stable and ac-curate results than numerical differentiation in the presence ofnoise. The best attraction of the developed velocity estimator isthat it is very simple and easy to implement in practice. Fur-thermore, it has high robustness against noise. Both simula-tions and experiments demonstrate the effectiveness of the pro-posed approach. Moreover, the improved performance by usingthe high-quality velocity signal is also verified by real-time ex-perimental results in the control of a PMSM. The proposedmethod can select an accurate instantaneous velocity as well asin low-speed ranges, which has built a solid base for the highdynamic performance of the system.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor,Dr. S. P. Weibel, and the anonymous referees for their pertinentcomments on this paper.

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