A Derivative Estimation Toolbox based on Algebraic Methods –Theory and Practice

Josef Zehetner, Johann Reger and Martin Horn

Abstract— In this paper, the implementation and usage ofa so-called Derivative Estimation Toolbox is demonstrated. Bymeans of this toolbox, time derivatives of sampled, noisy timesignals may be determined in realtime, all based on a recentlypresented algebraic derivative estimation method. The maincontribution is a possible implementation on a prototypingcontrol unit. The performance of the Derivative EstimationToolbox is experimentally validated on a brake-testbench. Inparticular, the friction force and the drive torque are estimatedin realtime.

I. I NTRODUCTION

The fast and robust estimation of derivatives of measured(or sampled) time signals is a very important issue in a vari-ety of applications in control engineering, failure diagnostics,signal processing and transmission. Due to the fact that, inpractice, signals are affected by measurement noise, filteringis inevitable. A common approach is based on linear time-invariant filters. More sophisticated approaches [1] are forexample the Savitzky-Golay differentiation scheme [2], aver-aged finite difference methods [3] and wavelet differentiationschemes [4]. In [5]–[7] some of these approaches are usedfor state estimation of nonlinear systems.

A novel method on derivative estimation that is based onLaurent Schwarz’ calculus of distributions and operationalcalculus, respectively, was introduced by Fliess and Sira-Ramırez [8], recently. Preliminary results of the so-calledalgebraic derivative estimationwere presented at three in-ternational summer schools: theWorkshop on Identification,State Reconstruction, and Generalized PI-Control, July 4–8,2005 in Munich, the summer schoolFast Estimation Methodsin Automatic Control and Signal Processing, July 18–20,2005 in Paris, and the summer schoolFast Estimation andIdentification Methods in Control and Signal, September 11–15, 2006, in Grenoble, and published in [9], [10]. Basedon this new method, in [11] a realtime implementationwas presented. This so-calledDerivative Estimation Toolbox(DET) offers the possibility to use the presented derivativeestimation technique in control and signal-processing appli-cations in realtime for the first time.

The main contribution of this work is to show how theDET may beneficially be used for the estimation of the

Josef Zehetner is with the Institute of Automation and Control, GrazUniversity of Technology, 8010 Graz, Austria and with MagnaSteyr, 8041Graz, Austriajosef.zehetner@tugraz.at

Johann Reger is with the EECS Control Laboratory at the University ofMichigan, Ann Arbor, MI 48109-2122, USAreger@ieee.org

Martin Horn is with the Institute of Automation and Control, Graz Uni-versity of Technology, 8010 Graz, Austriamartin.horn@tugraz.at

This work was partially funded by the FIT-IT Project VehDynCtrl 809186granted by bmvit.

friction force and the drive torque on a laboratory brake-testbench. This will be done in realtime using a prototypingcontrol unit.

The paper is organized as follows: in Section II the fun-damentals of the derivative estimation method are recalled.A brief overview of a possible realtime implementation,the DET, is given in Section III. In Section IV we showexperimental results of a friction force and drive torqueestimation on a brake-testbench, based on theDET. Finally,Section V concludes the presented work and outlines futureactivities.

II. A LGEBRAIC DERIVATIVE ESTIMATION BASED ON A

RECEDING HORIZON STRATEGY

The basics of the receding horizon approach to algebraicderivative estimation shall be recalled from [9], [8] and [12].An alternative approach was taken in [10], [13] where time-varying linear filters were used.

Consider a real-valued, analytic function of time,y(t),which for time instantst > 0 shall be approximated by itsTaylor-series expansion

y(t) ≈ yN (t) =

N∑

i=0

y(i)(0)

i!ti (1)

of orderN .For simplicity of notation, we set

ai := y(i)(0), i = 0, 1, . . . , N . (2)

Referring to elementary operational calculus we transformyN (t) � YN (s) and obtain

YN (s) =

N∑

i=0

ai

si+1. (3)

It is the key idea of the algebraic derivative estimationmethod to equivalently modify the above equation by a left-multiplication by an appropriate operator that helps isolatethe j-th coefficientaj = y(j)(0), j = 0, 1, 2, . . . , N .

To this end, Equation (3) is premultiplied by the operatorsN+1, thus

sN+1 YN (s) =N

∑

i=0

ai sN−i , (4)

which may be differentiatedN − j times with respect to theoperators. In doing so, we have eliminated all coefficientsaj+1, . . . , aN , hence

dN−j

dsN−j

(

sN+1YN (s))

=

j∑

i=0

ai

(N − i)!

(j − i)!sj−i . (5)

The remaining coefficientsa0, . . . , aj−1 may be canceled, aswell. To this end, we premultiply by the operator1/s

1

s

dN−j

dsN−j

(

sN+1YN (s))

=(N−j)!

saj+

j−1∑

i=0

ai

(N − i)!

(j − i)!sj−i−1

(6)and differentiate the latter equationj times with respect tothe operators in order to cancel the sum on the right handside. This yields

dj

dsj

(

1

s

dN−j

dsN−j

(

sN+1YN (s))

)

=(−1)j j! (N − j)!

sj+1aj . (7)

The expression on the left hand side of (7) containssN asoperator monomial of maximal degree, which is equivalent toanN -fold derivation with respect to time. Thus, we premul-tiply the entire equation by1/sN+ν+1 with the consequencethat any time signal is at least integrated once1. Therefore,we obtain

1

sN+ν+1

dj

dsj

(

1

s

dN−j

dsN−j

(

sN+1YN (s))

)

=(−1)jj! (N−j)!

sN+j+ν+2aj

(8)which may be transferred back to the time domain employingLeibniz’ formula for the differentiation of products. In lightof this, we process the following steps:

1sN+ν+1

(

dj

dsj

(

1s

dN−j

dsN−j (sN+1 YN(s))))

=

= 1sN+ν+1

(

dj

dsj

(

N−jP

κ1=0

(

N−jκ1

) (N+1)!(N−κ1+1)!s

N−κ1dN−j−κ1YN (s)

dsN−j−κ1

))

= 1sN+ν+1

(

N−jP

κ1=0

(

N−jκ1

) (N+1)!(N−κ1+1)!

dj

dsj

(

sN−κ1dN−j−κ1YN (s)

dsN−j−κ1

)

)

= 1sN+ν+1

N−jP

κ1=0

jP

κ2=0

(

N−jκ1

)(

jκ2

) (N+1)! (N−κ1)!(N−κ1+1)! (N−κ1−κ2)!

×

sN−κ1−κ2dN−κ1−κ2YN (s)

dsN−κ1−κ2=

N−jP

κ1=0

jP

κ2=0

(

N−jκ1

)(

jκ2

)

×

(N+1)!(N−κ1−κ2)! (N−κ1+1)

1sν+κ1+κ2+1

dN−κ1−κ2YN (s)dsN−κ1−κ2

.

Due to (8) we conclude that

1sN+j+ν+2 aj = (−1)j

j! (N−j)!

N−jP

κ1=0

jP

κ2=0

(

N−jκ1

)(

jκ2

)

×

(N+1)!(N−κ1−κ2)! (N−κ1+1)

1sν+κ1+κ2+1

dN−κ1−κ2YN (s)dsN−κ1−κ2

. (9)

There is no difficulty now to carry out the backward trans-form to the time domain: The left hand side represents apolynomial in the time domain, the right hand side expres-sion of s matches a respective convolutional integral; hence

aj =

∫ t

0

ΠjNν(t, τ) yN (τ) dτ (10)

with

ΠjNν(t, τ) = (N+j+ν+1)! (N+1)! (−1)j

tN+j+ν+1

N−jP

κ1=0

jP

κ2=0

(t−τ)ν+κ1+κ2 (−τ)N−κ1−κ2

κ1! κ2! (N−j−κ1)! (j−κ2)! (N−κ1−κ2)! (ν+κ1+κ2)! (N−κ1+1) .(11)

1Instead of multiple integration with respect to time arbitrary filters ofrespective order may be used.

In this formula, instead ofyN (t) we may resort naturally tothe measured (noisy) signaly(t). Furthermore, an arbitrarysmall time windowT may be used for the determinationof aj . In course of time, the Taylor-series expansion (1)becomes inaccurate since it holds only in the vicinity oft = 0. On the contrary, usually one is interested in thederivative estimates at a time instantt ≫ 0. Therefore, wehave to adapt the method for arbitrary time instants. For thispurpose, we flip the values ofy, i. e. integrate backwards intime, so as to expand at the time instant of interest, that ist. Finally, we obtain

y(j)(t) = (−1)j

∫ T

0

ΠjNν(T, τ) y(t − τ) dτ. (12)

The interval of integration with lengthT can be interpretedas the window width of a receding horizon strategy. Thewindow width T should be chosen small so as to calculatethe derivative estimate within an acceptable short delay, ithas to be chosen large enough in order to sustain the lowpass filtering property for suppressing measurement noiseson y(t) (adjustable withν) which improves the quality ofthe estimate.

In Section IV, the following polynomials are used forestimating the derivatives in a realtime application:

N = 1, j = 0, ν = 0:

y(0)(t) =

∫ T

0

−6τ + 4T

T 2y(t − τ) dτ (13)

N = 1, j = 1, ν = 0:

y(1)(t) =

∫ T

0

−12τ + 6T

T 3y(t − τ) dτ (14)

N = 2, j = 1, ν = 0:

y(1)(t) =

∫ T

0

180τ2 − 192τT + 36T 2

T 4y(t − τ) dτ (15)

The value of the integral within (12) and (13), (14),(15), respectively, may be approximated by a trapezoidalnumerical integration. For discrete time values ofyk =

y(kTs) andy(j)k = y(j)(kTs) we obtain the approximation

y(j)k ≈ (−1)j Ts

2

M∑

i=1

(Πi−1 yk−i+1 + Πi yk−i), (16)

wheretk−i = (k − i)Ts and ti = i Ts, yk−i = y(tk−i) andΠi = ΠjNν(T, ti). Ts is the sample time andM denotes thenumber of summation steps, it holdsM = T/Ts.

III. DERIVATIVE ESTIMATION TOOLBOX

The ideas presented in Section II can be used forofflineanalysis, i.e. after a measurement has been finished. Forthis purpose, a vast literature on different filter algorithmsis available. More challenging is theonline calculation ofderivatives, in particular, in a realtime setting during mea-surement for control tasks. Based on (16) a toolbox forSimulink from Mathworks was developed. The main idea isto solve themoving integralfor every simulation step usinga so-calledS-function[14]. The S-function is written in C.Therefore it can be used for both simulation and realtimeapplications. A detailed overview of the implementation andfirst results are presented in [11].

A. Data structure

The S-function has to be parameterized using the follow-ing parameters:

j: order of derivative for the input signalT : interval of integration in [sec]N : order of the Taylor-series expansionν: number of additional integralsTs: sample time in [sec]

The M = T/Ts last sample values of the timewindow(t−T, t] are used to calculate the actual estimated derivativey(j)k at time t. This window ismovingat every time step by

adding the actual sample valueyk and removing the oldestvalueyk−M . For this purpose the data can be stored in a socalled ring-buffer in a very efficient manner. A ring-bufferis a linear array of sizeM. A circular memory-structureis created by usingmodulo-operations2 for every read- andwrite-access.

B. Code structure

The flow diagram of the S-function which has to beevaluated for every time step is shown in Figure 1. Thecritical section concerning the computing time of the S-function is the loop calculating the integral. In Figure 2the pseudo-code representation is shown. The ring-buffer ofthe sizeRBSZ is stored in the arrayRB. The sample valueyk at time t is written to the positionRBPOS in RB. Theuser-parametersPARAM are fetched and checked at everytime step (line2), i.e. they can be adjusted online withoutresetting the application. Lines5 to 11 form the loop tocalculate the integral. In6 and 7 the sample valuesyk−i

andyk−i+1 are read fromRB. In line 8 and9 the functionderivate evaluates the appropriate equation forPARAMusingi andRB1, RB2, respectively. In10 the trapezoidalnumerical integration is carried out and summed up.

C. Remarks

Some interesting features of the presented code structureconcerning realtime applications should be mentioned. Mul-tiple time derivatives (of different order) from one signal

2Themodulo-operation gives the remainder of the division of two integernumbers, e.g.p = 34 modulo 20 = 14.

Initialization of the S-function

Read new sample

Save new sample to the ring- buffer, discard oldest

Select polynomial depending on parameter

Evalute polynomial using actual

values

Output estimated value

Summation (Integration)

M-times

For each simulation step

Fig. 1. Flow diagram of the S-Function

1: get SAMPLE2: get and check PARAM3: increment RBPOS4: save SAMPLE to RB5: for i = 1 to RBSZ6: RB1 = RB((RBPOS-i+1) modulo RBSZ)7: RB2 = RB((RBPOS-i) modulo RBSZ)8: TRAP1 = derivate(RB1,i-1,PARAM)9: TRAP2 = derivate(RB2,i,PARAM)10: OUTPUT ++= trap(TRAP1,TRAP2)11: endfor12: write OUTPUT

Fig. 2. Pseudocode

may be calculated using the same S-function. The overheadgenerated by thefor-loop is produced only once for allresults. In the same manner, the time derivatives of multipleinput signals may be calculated once, e.g. the first derivativeof four wheel-speed-sensor signals.

On realtime systems only a limited number of calculationsper time step are possible, i.e. for higher sample rates theintegration intervalT has to be chosen rather small. Inthis case the interval can be widened in two different waysto improve the suppression of noise. On the one hand,only everyL-th (L ≥ 2) sample value may be taken intoaccount, i.e. for a constant number of summation stepsM,the integration intervalT is multiplied by L. On the otherhand, the number of summation stepsM can be partitionedinto P time steps, i.e. the number of summations per timestep is reduced toM = M/P.

IV. APPLICATIONS

The DET is used for estimating time derivatives on abrake-testbench. A detailed description of the testbench is

given in [15]. For the sake of completeness a short explana-tion of the testbench is given here.

A. Brake-testbench

The testbench is located at theInstitute of Automation andControl at Graz University of Technology, Austria. It is usedto implement and verify control strategies for antiblock andanti-spin systems for passenger cars. The testbench consistsof two main parts, a wheel with a rubber tire and a bearing-supported solid steel roll. A schematic is given in Figure3. The wheel is connected to an electric motor and to

Fig. 3. Schematic of the brake-testbench

a conventional automotive brake system. By acceleratingor braking the wheel the roll is accelerated and braked,respectively, due to the friction forces between the rubbertire and the steel roll.

Index “1” marks variables associated to the wheel, index“2” denotes variables with respect to the roll. The equationsof motion for both parts of the brake-testbench are given as

J1dω1

dt= Td − Tb − Tf,1 + Fr1 (17)

for the wheel and

J2dω2

dt= −Tf,2 − Fr2 (18)

for the roll. The moments of inertia with respect to theaxes of rotation are given asJ1 and J2, ω1 and ω2 arethe angular velocities. The drive torque generated by themotor is denoted byTd and the brake torque applied isTb.Tf,1 andTf,2 represent respective friction torques.F is thelongitudinal friction force andr1, r2 are the radii of thewheel and the roll, respectively. The differential equations(17) and (18) are coupled via the friction forceF whichis a nonlinear function of the angular velocitiesω1 andω2,hence, renders the entire dynamic system nonlinear.

The angular velocitiesω1 and ω2 are measured withinductive wheel-speed sensors. The parametersJ1, J2, Tf,1,Tf,2, r1 and r2 are supposed to be known. The estimationof these parameters can be found in [15].

The brake-testbench is equipped with aMicroAutoBoxDS 1401/1501(MABX) from dSpace. This is a prototypingcontrol unit specially designed for automotive applications.The MABX is based an a PowerPC 750 FX processor andprovides numerous I/O-ports, e.g. two CAN controllers, four

0 2 4 6 8 100

10

20

30

40

50

60

70

80

90angular velocity ω [1/sec]

time [sec]

ω [1

/sec

]

ω1

ω2

Fig. 4. Acceleration of wheel without anti-spin control

input channels for frequency measurement or 16 analogueand up to 32 digital inputs. A target compiler for Simulinkis available. Therefore, applications for the MABX may bebuilt directly from Matlab/Simulink. A detailed descriptioncan be found in [16].

A target wheel speed can be set to control the testbench.The angular velocities of the wheel and the roll are measured.The measured signals and the results from theDET aretransferred to a PC via a highspeed serial interface inrealtime.

For the following experiments a simple test cycle wasgenerated. Therefore a target angular velocity ofω1 =80 [1/ sec] is set. The anti-spin control is disabled, i.e. thewheel-slip can reach values close to1. The measured angularvelocitiesω1 andω2 for this maneuver are depicted in Figure4.

The longitudinal friction forceF is a function of thewheel-slip

λ :=ω1r1 − ω2r2

max(ω1r1, ω2r2),

where the wheel-slipλ is calculated by estimating the zero-order-derivative forω1 and ω2 using theDET as a zero-order-derivative filter. To this end, polynomial (13) is usedwith parametersN = 1, j = 0, ν = 0. For the estimation ofω1 an integration interval ofT = 0.2 sec is used, concerningω2 the window size is reduced toT = 0.1 sec since thereis less noise on the measurement signal forω2. In Figure 5the algebraically estimated wheel-slipλ is compared to thewheel-slip calculated from the unfiltered signals.

B. Friction force estimator

Based on (18) a simple estimator for the friction forceFcan be derived as

F =1

r2

(

−Tf,2 − J2dω2

dt

)

. (19)

Values forr2, Tf,2 and J2 are well known in our case.ω2

is measured by an inductive wheel-speed sensor mounted tothe roll axle. Due to measurement and quantization noise

0 2 4 6 8 10

0

0.5

1

Wheel slip λ [−]

time [sec]

λ [−

]

0 2 4 6 8 10

0

0.5

1

time [sec]

λ [−

]

λ unfiltered

λ estimated

Fig. 5. Unfiltered wheel-slipλ vs. algebraically estimated wheel-slipλ

0 2 4 6 8 10

−300

−200

−100

0

Friction force F [N]

time [sec]

F [N

]

F filtered

0 2 4 6 8 10

−300

−200

−100

0

time [sec]

F [N

]

F estimated

tc

Fig. 6. Comparison of filtered and algebraically estimatedF over time

the direct expressiondω2/dt yields no acceptable result. Asa consequence, theDET shall be used to calculate the firstderivative with respect to time ofω2 in realtime. Equation(14) is used for this estimation. The parameters arej = 1,T = 0.2 sec, N = 1, ν = 0 andTs = 0.002 sec.

For a comparison of the results obtained with theDET afiltered “dirty” first derivative ofω2 with respect to time iscalculated by the transfer function

P (s) =s

0.05s + 1. (20)

Figure 6 shows the filtered friction force compared to thealgebraically estimated friction force.

In Figure 7, the algebraically estimated and the filteredfriction forces are plotted with respect to the estimated andthe unfiltered wheel-slips. It has to be mentioned that onlyvalues from the time-intervaltc (see Figure 6) are usedfor this plots. Note that the friction forces are plotted withnegative sign.

C. Drive torque estimator

Using (17) and the results from above an estimator forthe drive torqueTd (the brake torqueTb) follows straightforward as

Td − Tb = J1dω1

dt+ Tf,1 − Fr1. (21)

0 0.2 0.4 0.6 0.80

50

100

150

200

250Friction Force F [N] over λ [−]

λ [−]

−F

[N]

0 0.2 0.4 0.6 0.80

50

100

150

200

250

λ [−]

−F

[N]

F filtered

F estimated

Fig. 7. Comparison of filtered and algebraically estimatedF over λ

0 2 4 6 8 10

0

50

100

Drive torque Td [Nm]

time [sec]

Td [N

m]

0 2 4 6 8 10

0

50

100

time [sec]

Td [N

m]

Td filtered

Td estimated

Fig. 8. Comparison of filtered and algebraically estimated drive torqueTd

In the case of acceleration, the brake torqueTb is assumedto be zero. Again the determination ofdω1/dt is the crucialpart. To this end, again, the first derivative ofω1 with respectto time is algebraically estimated referring to Equation (15)with parametersj = 1, T = 0.4 sec, N = 2, ν = 0 andTs = 0.002 sec. Again a filtered dirty derivativedω1/dt iscalculated with transfer function (20). In Figure 8 the resultsof the corresponding laboratory experiment are shown.

V. CONCLUSIONS AND FUTURE WORKS

A. Conclusions

In this work, the mathematical framework recalled inSection II is tested in a realworldand realtime environment.It was shown that theDerivative Estimation Toolboxoffersthe possibility to design simple and considerably robustestimators for realtime applications. The results indicate thepotential of the method in an impressive way. The imple-mentation on a prototyping electronic control unit closes thegap between theoretic brilliance and applied sciences.

B. Future Works

In light of the high quality of the results theDerivativeEstimation Toolboxwill be tested in more complex appli-cations, e.g. in prototype vehicles. The mathematical theorystill offers potential for optimization concerning exactness

and calculation effort. This will lead to even better resultsand a faster online calculation.

ACKNOWLEDGEMENTS

The authors would like to thank Cedric Join and HeberttSira-Ramırez for their invaluable discussions during thesummer schools in Munich, Paris and Grenoble.

REFERENCES

[1] S. Diop, J. Grizzle, and F. Chaplais, “On numerical differentiationalgorithms for nonlinear estimation,” inProceedings of the 39th IEEEConference on Decision and Control, vol. 2, 2000, pp. 1133–1138.

[2] W. Gander and U. von Matt, “Smoothing filters,” inSolving Problemsin Scientific Computating Using Maple and Matlab, 3rd ed., W. Ganderand J. Hrebicek, Eds. Berlin, Germany: Springer, 1997, pp. 121–139.

[3] R. Anderssen, F. de Hoog, and M. Hegland, “A stable finite differenceansatz for higher order differentiation of non-exact data,” Bull. Austral.Math. Soc., vol. 58, pp. 223–232, 1998.

[4] C. Burrus, R. Gopinath, and H. Guo,Introduction to Wavelets andWavelet Transforms. A Primer. New Jersey: Prentice-Hall, 1998.

[5] S. Diop, V. Fromion, and J. Grizzle, “A global exponentialobserverbased on numerical differentiation,” inProceedings of the 40th IEEEConference on Decision and Control, vol. 4, 2001, pp. 3344–3349.

[6] S. Diop, J. Grizzle, P. Moraal, and A. Stefanopoulou, “Interpolationand numerical differentiation for observer design,” inAmerican Con-trol Conference, vol. 2, 1994, pp. 1329–1333.

[7] F. Plestan and J. Grizzle, “Synthesis of nonlinear observers viastructural analysis and numerical differentiation,” inProceedings ofthe 5th European Control Conference, Sept. 1999.

[8] M. Fliess and H. J. Sira-Ramırez, “Control via state-estimation ofsome nonlinear systems,” inIFAC Symp. on Nonlinear Control Systems(NOLCOS), Stuttgart, Germany, 2004.

[9] M. Fliess, C. Join, M. Mboup, and H. Sira-Ramırez, “Analyse etrepresentation de signaux transitoires: applicationa la compression, audebruitage, eta la detection de ruptures,” inGRETSI 2005, Louvain-la-Neuve, Belgium, 2005.

[10] J. Reger, H. J. Sira-Ramırez, and M. Fliess, “On non-asymptoticobservation of nonlinear systems,” inIEEE Int. Conf. on Decisionand Control (CDC), Seville, Spain, 2005.

[11] J. Zehetner, J. Reger, and M. Horn, “Echtzeitimplementierung einerAbleitungsschatzung mit Methoden der differentiellen Algebra,”at -Automatisierungstechnik, submitted for publication.

[12] J. Jouffroy and J. Reger, “An algebraic perspective to single-transponder underwater navigation,” inIEEE Conf. on Control Ap-plications (CCA), Munich, Germany, 2006.

[13] J. Reger, M. Mai, and H. J. Sira-Ramırez, “Robust algebraic stateestimation of chaotic systems,” inIEEE Conf. on Control Applications(CCA), Munich, Germany, 2006.

[14] Simulink Help: Writing S-Functions, The MathWorks Inc., 1994–2006.[15] M. Horn and J. Zehetner, “A brake-testbench for research and educa-

tion,” in Proc. IEEE Conference on Control Applications (CCA) 2007,Singapore, submitted for publication.

[16] MicroAutoBox Features, Release 5.1, dSpace GmbH, Paderborn, Ger-many, May 2006.