instantaneous speed and disturbance torque observer using non linearity cancellation of shaft...

8/14/2019 Instantaneous Speed and Disturbance Torque Observer Using Non Linearity Cancellation of Shaft Encoder

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Instantaneous Speed and Disturbance Torque Observer using NonlinearityCancellation of Shaft Encoder

J . Corres, P. GilPublic University of Navarre, Dept. of Electric and Electronics EngineeringCampus Arrosadia sln. 3 1006 Pamplona. Spain

Email: [email protected]

AbstrPct- In this p aper the theoretical analysis, simulationsand practical results of a new speed and disturbance torqueobserver are proposed. The analysis of the nonlinearityintroduced by the encoder quantization allows improving thebehaviour of the fdter in all the speed range. A Kalman fdtertype observer has been developed. The m ain novelty is that thestate estimate observational update takes place at thequantized position signal discontinuities. Unlike the usuallyapplied Kalman fdter, the measurement noise covariance isvariable. In this observer, it is a function of the shaft speedbecause the selection of the sampling instant results in areduction of the output noise covariance. The observer adaptsits gain depending on the state variables. The optimal gaincalculation results in an observer bandwidth increase.Another adv antage of this observer is that the self-sustainedoscillations are suppressed, obtaining asym ptotic stability.

I. INTRODUCTIONThe encoder is the position transducer most widely used

in industrial applications. The speed regulation on a widerange is fundamental in the drives control for industrialapplications as tool machines, robots or electrical vehicles.The ordinary control uses the information of the averagespeed [1] for feedback. However, when a low-resolutionencoder is used, a delay that causes instability in thefeedback loop compels to decrease the controller gain.Speed and disturbance torque observers have become aneffective technique of calculating the instantaneous speedusing Luenberger-type observers [lo] [2] [3] or KalmanFilters [8] [9].

When the elapsed time between encoder pulses is verylarge, the filter applied in its standard form can causeoscillations in steady state. A projection observer that is runat the instants in which the transitions of the quantizedoutput occurs was developed in [5]; it is in those instantswhen the measurement is fkee of quantization noise. It ispossible to reduce the quantization effect by reducing thefilter bandwidth at low speed, however it is difficult to tuneit correctly for a wide speed range and assure the observerstability. In [6] Sakai attempted to improve the behaviour atlow speed adding static fiiction compensation to theobserver. Yang [7] used an interpolation method of theposition using the average speed with the drawback that itintroduces an additional delay producing oscillations in thetransient.

In this paper, a new observer whose main property is theencoder nonlinearity cancellation is proposed. The aim is toachieve asymptotic stability at very low speed and toincrease the global dynamic performance in all the speedrange. The observer output equations are updated as close

to the transitions as possible in order to cancel thequantization, but as it cannot be completely suppressed dueto hardware limitations, the residual measurement noise isincorporated to the optimal observer gain calculation.

This filter can be viewed as an asynchronous multirateobserver executed periodically, without the need to useinterrupts that makes the implementation difficult. Thediscrete time Kalman Filter equations are updated in virtualsampling instants characterized by their best noisemeasurement properties. This is a simple method to takeadvantage of all the information provided by the positiontransducer. Its implementation only requires an additionalcounter and two-plant discretization every period. Theobserver bandwidth increase can be up to ten times higherin our case.

Robust performance with inertia variations has beenextensively studied [l 11 [121. In this study, we suppose thatthe inertia remains constant or that its variations can beidentified fast enough.

11. THEMECHANICALODELThe model of the mechanical part in Fig. 1 is the

classical third order state space model. The disturbancetorque includes all type of external torques, includingviscous fiiction. If friction torque model exists, it can beincluded. The motor torque of the field oriented controlledinduction motor is

I Tdis

Tm

Fig. 1. Mechanical system and observer

0-7803-7262-x/Ml$10.002002 EEE. 540

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State equation of the continuous system:f =Ax + Bu +wy = cx+vc=(o 1 0)

1 )) fJ" J"-0 - (2)

0 0 0 0

State equation of the discretized system:x,,, = Akxk+ B,u +wky = C x + v ,A =eATS

(3)

111.STATEBSERVERSA . The Projection O bserver

The "projection observer" proposed in [ 5 ] can be appliedto M MO LTI systems with quantized outputs. Theobjective of this observer is to extract the estimated stateviewing the quantized measurement as a limitedinformation problem. In spite of the limited informationoutput, the estimated error tends to zero as t +-

In Fig. 2 the state space is the direct sum of I m(CT )andN(C). Given a certain positive matrix M the vectorsbelonging to the subspace Im(M'CT) are orthogonal tothose of N(C). The orthogonal projection matrix on thevector subspaceN(C) is

P = I - M--'CT CM-'CT I c) (4)The conditions that are imposed for the new observationupdate are two:C1: The projections on N(C) of the estimated state

before and after the update are the same i.e.C2: The state output is equal to the real output

Pinm P ii.e. C i - = Cx

I m C )Fig. 2. Discrete update of the projection observer

State estimate extrapolationi = A i + B u V t g t , (5)State estimate observational updateinwi - K ( y - y , ) V t e k (6)where K = M-'CT (CM-'CT -IDynamic stability is assured maintaining the

characteristic polynomial roots within unit circle.IAI - ( I -K C )AI = 0B. The discrete Kalm an Filter

Although the discrete Kalman Filter (KF) [4 ] is generallyapplied at a constant rate, the projection can be done at anytime by having the precaution of extrapolating the state andthe error covariance matrix correctly. Doing this, the KFbecomes a projection observer similar to the proposed in[5],with the additional advantage that the gain filter istuned automatically. The general KF algorithm isreproduced below.

State estimate extrapolationik Aik-' +BuError covariance extrapolatione = Ak-,pk-,Ak-,T GQk-,GTState estimate observational updateZk t k K k Z k - a k )

(7)

(9)Error covariance updatepk = ( I-K ~ C ) ~I -K ~ c ) ~K , R , K , ~ (10)Kalman Gain MatrixK k = pkC/ (CpkC' + Rk 'The solution is uniformly asymptotically stable, thoughnonlinearities as the quantization introduced by the encoder

or unmodelled aspects can produce instability.

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IV.THEPROPOSEDBSERVERAt low speed, the behaviour related to the position in

stationary state is shown in Fig. 3. The speed anddisturbance torque estimates become oscillatory due to theencoder quantization. The output of the observer growsquickly after a new pulse arrives, and decays immediatelyafterwards to maintain the fixed quantized position, whilethe actual position follows a different path.

It can be observed the similarity between equations (6)and (11). The only difference is the presence ofmeasurement noise. The fundamental idea that underliesthis study is that if the delay is sufficiently reduced, theobserver will approach to the projection observer ( R & I )eliminating in practice the nonlinearity. This isaccomplished projecting (updating the measurement) nearthe encoder pulses. If there is no output feedback for manysampling periods, the estimated position can be greater thanone encoder line. Then the state estimate is updated withthe constant position, adjusting the Kalman gain to thestandard filter one. Each sampling period T, the following isrun:

STEP1: TI - Extrapolation{DiscreteMatrix Calculation using TI (3)State Estimate Extrapolation (7)Error covariance Extrapolation (8) }

eest?moderoutput

0 . 1 4 r . , , . , 10.2 0.22 0.24 0.26 0.28 0.30 0.32t ime($

Fig. 3. Quantized position measurement Q[O] and observed position, Omusing the standardKalman ilter

Fig. 4. Time magnitudes employed in the proposed observer

STEP2: Measurement UpdateIF (Newpulse OR Position error>%lax),THEN {Error covariance update (1 1).Kalman Gain Matrix (10)State estimate observational update (9) 1

STEP3: Tz - Extrapolation{Discrete Matrix Calculation using T2(3)State Estimate Extrapolation (7)Error covariance Extrapolation (8)}

A . High-Frequency CounterIn the new observer, the time that elapses from the last

encoder pulse until the present sampling instant isemployed to improve the estimate of the states. In order tomeasure this time an additional counter is necessary. Thiscounter is reset with encoder signal edges and is latchedtogether with the position up/down counter at a constantperiod T,. The timer width W must comply with2w2 ,T, ; e.g. in simulations a high frequency clockf m=1.25 M H z is used under a Ts=200 ps sampling period.Therefore, the timer should have at least eight bits. Callingn to the output of the counter, I ; = ( N- )TH F t is the timesince the previous sampling until the arrival of the lastpulse before the current sampling, and T, = nT , is the timesince the last pulse.B. Covarian ce matrix of the measurement noise

In the Standard Kalman Filter employed in [8] and [9], Rkis a constant usually employed as a design parameter. Themaximum measurement error of the position is thecorresponding to an encoder line e m m = 2 d P P R ,where PPRis the number of encoder pulses per revolution. Uponincorporating the counter information, the maximum erroris reduced to e ( @ T H F ) = d H F , reserving the limitcorresponding to the angle of a line e - . Concerning thedistribution of the quantization noise, it has a uniformdistribution with covariance R=ez/12. In Fig. 5 thedependency of R with the speed, which is recalculated eachtime a new pulse amves, is represented.

6IO

Standard Kahnsn Filter f

.I I I0 1 4IO 10

IO IS& Speed (rpm)10

Fig. 5. Measurement Covariance(R) s. shaft speed for severalhighkq ue nc y clock valuesV;IF)

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101 . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .IO 1 '01 Frcquecy (radts) lo' 1

Fig.6. Bode graph of estimated disturbance toque (T d SJTdieol).?{-7

Fig. 7. Bode graph of shaft speed (&Jwm,)

0.150.1P$ 0.05P

8Bf -0.05

-0.1

c - 0 0

5%I I l l0 0.1 0.2 0. 3 0.4 0 0.1 0.2 0 .3 0.4

time (s ) time (s)Proposed Filter Standard Filtera) Medium speed range

Fig. 6 and Fig. 7 show how the optimal gain calculationresults in an observer bandwidth increase, which can be upto ten times higher, depending on the rotational speed.

v. SJMULATIONRESULTSSimulations using Matlab for different ranges of shaft

speed, contrast the performance of the proposed observerwith the standard one. Speed and disturbance torqueestimates are compared with real variables.

TABLE ISMULATION PARAMETERS

Total Inertia (Jn): 0.012Kgm'Encoder Resolution(PPR): 4096High frequency clock (THF): 83 3 nsDisturbance Torque Peak (Tdis) 0.1 N.mSampling Time (Ts): 200p

THFhave a strong influence on total simulation time. Teperiod employed in this simulation is enough to show theobserver performance and it does not require an excessivelong simulation time (120 seconds). A lower valueimproves the performance due to lower measurement noise.

Fig. 8 (a) shows observer performance below 100 rpm.The proposed filter responds faster to the externaldisturbance. In Fig. 8 (b) asymptotic stability can beobserved at very low speed (around lrpm) for both speedand disturbance torque estimates.

10. 8

0.60.4

0.2

00 0.1 0.2 0. 3 0.4 0 0.1 0.2 0. 3 0.4-0.1 0.2 0. 3 0.4 1...10 0.1 0.2 0 .3 0.4

time (s) time (s)Proposed Filter Standard Filter

, b) Low speed rangeFig.8. Simulation results. Real and estimated states.

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(b)Fig. 9. Pictures of experimentalsetup (a) 1.5 kW Test-rig (b) DSP basedcard

VI. PRACTICALMPLEMENTATIONNDEXPERIMENTALRESULTS.

The observer and vector control algorithm has beenimplemented allowing a 200 ps sampling time. Fig. 9shows two pictures of the experimental setup. The controlof the squirrel cage induction motor was implemented usinga 32-bit digital signal processor (DSP) TMS320C30running at 40-MHz.A vector coprocessor ( A D 2 S100) and aPWh4 generator (TH8001) help the DSP with vectorcontrol. The inverter was constructed using an IGBTintelligent power module (PMRSH120). PWM switchingfiequency was 5 kHz. A 1.5 kW ac motor with 4096 pulsesper revolution (PPR) encoder was used to obtainexperimental results. Table I1 shows the mechanical andvectorial model parameters. Inertia does not vary; an off-line Extended Kalman Filter (EKF) (with Jn as an extrastate) was used for its identification. In order to apply theload torque, another 1.5 kW ac motor was used. It wascontrolled using a commercial vector controller, set in thespeed control mode.Timers and extra logic have been implemented in anALTERA programmable logic device (PLD). It has a 16-bitup/down counter (storing the position) and a 12-bit counter(storing the time from the last encoder pulse to the presentsampling instant). A state machine converts the quadrature

encoder pulses to "up/down" and "clock" inputs to the firstcounter and the reset signal for the second. Synchronizationof the two counters is essential in order to obtain correctmeasurements. This configuration allows us to measure theaverage speed with the M/T method [13 without the need ofextra logic.

TABLE XIPRACTICALhfPLEMENTATION PARAh4ETERS

Totul Inertia Jn 0.012 Kgm' Rs 5.9 aPPR 409615 12 Rr 2.7aTs 200 ps Lm 0.443HTHF 50 ns Ls 0.462 HNo. of poles 4 Lr 0.462 HRated Power 1500 W Rated Current 3.4 ARated Voltage 380 V

A good average speed is needed for comparison withestimated speed. A frequency divider reduces the effectiveencoder resolution for observers down to 512 PPR, whilethe average speed uses the full resolution. In Fig. 10 theobserver gain varies with the shaft speed in the proposedfilter, while the standard one have a lower and constantgain. Fig. 11 shows the proposed and standard observerperformance in open loop. Disturbance torque of k2N.mamplitude is applied periodically during 50 ms aprox by thevector controlled induction motor acting as a load. In Fig.11 (b) the estimated torque is compared with a transducertorque measurement; the difference between them is due tothe unmeasurable friction torque (about 0.35 N.m). Fig. 12shows an enlargement of speed waveform. As expected theproposed observer exhibits a better performance for thetransient (a) and suppressing the undesirable oscillationscaused by the encoder quantization at very low speed.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

. . . . . . . l.50

s o-50 t

0.1 0.2 0. 3 0.4 0.5 0 .6 0.7 0.8 0.9 1time(s)

Fig. 10.KalmanGains ofproposed and standard filters

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instantaneous speed and disturbance torque observer using non linearity cancellation of shaft...

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