an algebraic denoising scheme

An algebraic denoising scheme

J. Cortes-Romero, C. Garca-Rodrguez, A. Luviano-Juarez, R. Portillo-Velez and H. Sira-Ramrez

Abstract In this paper, the noise filtering problem (de-noising problem) is approached via a suitable modification ofthe traditional Luenberger observer approach, also known asthe high gain observer approach (HGO). The HGO observeris, fundamentally, a Luenberger observer with large stableeigenvalues of the output estimation error dynamics. HGO andsome of its modifications have been particularly useful in thelinear based control of perturbed linear and nonlinear systems.HGO are, however, inappropriate to deal with noisy injectionsignals and noisy plants. To overcome this fact, the structureof the Luenberger state estimator reconstructing the timederivatives of the given signal is enhanced against the effects ofnoise by means of an algebraic filtering scheme. The algebraicfiltering consists in a suitable modification of the recentlyintroduced algebraic parameter identification methodology. Theproposed approach is capable of attenuating the noise effectssignificantly. The proposed strategy is illustrated via numericalsimulations.

I. INTRODUCTION

High gain observers constitute an effective signal deriva-tive estimation technique in a variety of application fieldssuch as robotics [1], motion control [2], position/force con-trol, motor load estimation, induction motor control [3], [4],disturbance and fault estimation [5] among many others. Thisclass of observers offers certain degree of design freedomwhich can be utilized to achieve properties such as robustnessagainst external perturbations and other uncertainties. Inaddition, when the observer is designed, the number ofobserved time derivatives can be chosen in connection withthe observer dimension.

Two problems, arising in connection with these HGOobservers, are: the initial peaking phenomenon and theundesired noise amplification. In relation to the peakingphenomenon, there several approaches have been proposedto solve it. See, for instance, [6], where the high-gain differ-entiators are used in combination with a traditional filteringscheme for estimating the higher order time derivatives of asignal.

On the other hand, high-gain observers have been im-plemented to estimate output time derivatives and unknownplant inputs, such as faults and plant perturbations. In appli-cations where environmental disturbances and measurement

J. Cortes-Romero is with Universidad Nacional de Colombia. Facultadde Ingeniera, Departamento de Ingeniera Electrica y Electronica. Carrera30 No. 45-03 Bogota Colombia. [email protected]

J. Cortes-Romero, C. Garca-Rodrguez, A. Luviano-Juarez, R. Portillo-Velez and H. Sira-Ramrez are with Centro de Investigacion y deEstudios Avanzados del Instituto Politecnico Nacional. Departamentode Ingeniera Electrica, Seccion de Mecatronica. Apartado postal14740, Mexico DF C.P. 07360. [email protected],[email protected], [email protected],[email protected] and [email protected]

noises play an important role, the tuning of the observergain becomes an important and difficult issue. The unde-sired amplification tends to grow up in the calculation ofeach consecutive higher order time derivative. A possiblesolution to this problem is using the well known ExtendedKalman Filter, under the assumption that the noise is aGaussian process with known statistics (see [7], [8]). In [9]an alternative is proposed for reducing the noise effects.A class of proportional integral observers are consideredwhich consist of the use of an integral gain in the observerinjection structure to attenuate the effects of the noise inthe reconstruction error dynamics. This filtering scheme mayachieve an effective reduction of the noise effects. Thus,the problem relies on applying adequate filtering techniquesto the signal and its time derivatives. If classical filteringschemes are selected, then, undesired signal time delays oftenarise. On the other hand, numerical algorithms are slow andthey are also known to be computationally complex.

Algebraic techniques have been proposed to solve theproblem of parameter estimation, for linear systems withconstant parameters, and, also, for linear systems with timevarying parameters (see [10], [11]). There also exists analgebraic approach to carry out the state estimation task ina class of algebraically observable nonlinear systems [12].These approaches are based on the, so called, algebraicderivative approach. In this technique, the problem of stateestimation is closely related to the problem of numerical dif-ferentiation which is an active area of research in numerousscience and engineering areas such as control theory andsignal processing. Some numerical differentiation algorithmsas linear differentiation, backward finite differences and theSavitzky-Golay differentiation have been analyzed in [13].Taylor expansion approaches have been implemented toobtain numerical differentiation from a noisy signal, see[14], [15]. However, this method becomes ill-conditioned forhigher order truncation of the Taylor series expansion.

On the other hand, even if the time derivative estimations,obtained from a high-gain observer, are very noisy; theiraverage signals are good in the sense that they tend tofit the actual underlying signal derivative. This fact can beexploited by means of algebraic manipulations similar tothose applied in the linear parameter identification problem.The aim of our approach, is to provide a filtering scheme fora noisy measured or estimated derivative signal. In particular,we take the estimate expressions from a HGO observer[3] and apply an algebraic filtering scheme. This approachis inspired on the algebraic state identification techniquepresented by Fliess and Sira-Ramrez for continuous timesystems (see [10]).

The remainder of the article is organized as follows:In Section II, the problem formulation is presented; then,an example of the filtering methodology is considered inSection III. To illustrate the methodology, in Section IV somenumerical results are given for two different levels of additivenoise and a comparison with other filtering alternatives isdepicted. Finally, the conclusions and suggestions for furtherwork are comprised in Section V.

II. PROBLEM FORMULATIONConsider a smooth signal x(t), affected by an additive,

zero mean high frequency noise, (t)

y(t) = x(t) + (t) (1)Carrying out any signal processing, or implementing a

control law through this noisy measurement is undesirablebecause of the noise effects downstream. There exist lowpass filtering alternatives, but they need a certain knowledgeabout the noise characteristics while producing a significantdelay of the filtered signal.

Problem Formulation. Given a noisy measurement, y(t),of the smooth signal x(t) as in (1), with, (t), being anadditive, zero mean, high frequency measurement noise;devise an online filtering process for the measured signal,y(t), such that it recovers the signal x(t), with no delay.

A. An observer-based filtering algorithm: An algebraic ap-proach

The proposed algorithm is carried out in two steps.The first step consists in devising an asymptotic estimationscheme for the time derivatives of, y(t), via a traditionalhigh gain observer as if y were uncorrupted by noise.Since the additive signal, (t), is a zero mean noise, weknow that the observer based estimated derivatives are alsocrucially affected by the noise, but, their average valuesare asymptotically coincident with the actual derivatives ofx(t). The second step consists of considering the algebraicexpression relating the j th time derivative of y(t) and itsestimated value from the observer equations. This expressionreceives an algebraic treatment to extract the required filteredsignal. One generally proceeds by multiplying the appropri-ate expression by the j-th positive power of the time variable,(t)j , then, the resulting product is integrated by parts jtimes thus obtaining an expression which is free of the initialconditions and of any time derivatives of y(t). From theobtained expression, the signal y(t) can be expressed as thequotient of two time functions. This filtering can be carriedout quite fast and with no involved delays, as it is commonin many of the traditional low pass filtering schemes.

1) Obtaining an average derivative: Consider a Taylorseries expansion of the noise-free signal x(t), which isassumed to be infinitely differentiable in a neighborhood oft0

x(t) = x(t0) +(t t0)

1!x(t0) +

(t t0)2

2!x(t0) + (2)

For an (n1)th order approximation of x(t), it is possibleto consider the following n th order model

dnx(t)

dtn= 0 (3)

Rewriting (3) in state variable form, we obtainx1 = x2x2 = x3

.

.

.

.

.

.

xn = 0

x(t) = x1

(4)

A Luenberger observer can be proposed in order to esti-mate the unknown states of (4) as follows

x1 = x2 + n1x2 = x3 + n2

.

.

.

.

.

.

xn = 0

(5)

where the output estimation error is defined by = y x1.The asymptotic convergence of this estimation error dependson the parameters i, which can be selected according toa pre-specified Hurwitz polynomial governing the resultingoutput estimation error dynamics.

2) Obtaining an on-line smooth signal: As it is known,under noisy output measurement conditions the output timederivative estimates, obtained from a HGO, are significantlydistorted. Nevertheless, their realizations are close, in the av-erage to their actual values, provided a zero mean assumptionhas been established on the nature of the noise. In this paper,a algebraic filtering process is proposed using the estimatedderivatives supplied by an observer of the high gain type.Abusing the notation1, a j th order algebraic filtering isobtained from

(j)tjy(j) =

(j)tj xj+1 for 1 < j < n (6)

where xj+1 is the jth estimated derivative of y(t) suppliedby (5). Integrating by parts (j)0 tjy(j), it is possible to finda filtered version for y(t).

In order to illustrate we consider the particular case whenj = 3.

Integration by parts of the corresponding expression allowsus to check that (3)

(t)3y(3) = 6

(3)y 18

(2)ty + 9

(1)t2y t3y

1We denote the quantity t0

10

j10

(j)dj . . . d1

by the expression: (j)

(t)

and

y =

(3)t3y(3) + 6

(3)y 18

(2)ty + 9

(1)t2y

t3

given that the third time derivative estimate of y is x4, afiltered version of y is given by

yFilt3 =

(3)t3x4 + 6

(3)y 18

(2)ty + 9

(1)t2y

t3

Remark 1: The algebraic filtering expressions are in termsof the time variable t and some of finite power termswhich may produce high numerical values as time elapses.To avoid having numerical computing problems, instead ofmultiplying by tj , the expression can be multiplied by anadmissible bounded time function (i.e, a function whichallows to apply the algebraic manipulations to generate avalid filtering expression free of initial conditions) so thatnumerator and denominator signals remain bounded. Forinstance, multiplying by (1 et)j instead of (t)j .

III. EXAMPLELet y(t) be a signal of the form (1). It is desired to filter the

noisy output signal by means of the proposed combinationof a Luenberger observer and the algebraic filter approach.Proceeding with the methodology, an approximation to thesignal, x(t), can be modelled as

d7x

dt7= 0 (7)

A. Observer designAn associated state space representation of (7) can be

readily conformed as

x1 = x2

x2 = x3

x3 = x4

x4 = x5

x5 = x6

x6 = x7

x7 = 0

y = x1 +

A Luenberger observer for the polynomial model of x(t)is then synthesized as follows:

x1 = x2 + 6(t)

x2 = x3 + 5(t)

x3 = x4 + 4(t)

x4 = x5 + 3(t)

x5 = x6 + 2(t)

x6 = x7 + 1(t)

x7 = 0(t)

(t) = y x1

where the corresponding output estimation error dynamics isclearly given by the linear dynamics,

(7) + 6(6) + 5

(5) + + 1 + 0 = 0

whose corresponding characteristic polynomial is

p(s) = s(7) +6s6 +5s

5 +4s4 +3s

3 +2s2 +1s+0

The gain parameters of the observer are chosen such thatthe characteristic polynomial of the output observation errormatches the known polynomial (s2 + 2ns+ 2n)3(s+ p),with , n, p being positive real values.

B. Filter designIn this subsection, two different algebraic filters are pre-

sented. Each algebraic filter arises from a different timederivative estimate expression found from the observer:Namely, a first order time derivative estimate based filterand a fourth order time derivative estimate based filter willbe examined.

First order time derivative based filter. From the esti-mation of the first order time derivative, x2, we have

yFilt1 =

tx2 +

y

t(8)

Fourth order time derivative based filter. From theestimation of the fourth order time derivative, x5, we have

yFilt4 =n4(t)

d4(t)(9)

with n4(t) = (4)

t5x524 (4)

y+96 (3)

ty72 (2)

t2y+16

t3y, and d4(t) = t4 respectively.

IV. NUMERICAL RESULTSNumerical simulations are performed in order to show

the performance of the proposed approach. A comparisonis also carried out with the performances of two classicfiltering methods: one of them consisting of an eight orderButterworth low pass filter and a second one consisting of aneight order Chebyshev low pass filter, both filtering schemesare provided with a cut-off frequency of 700 [s1]. Thesignal to be taken has the following form

y(t) =10

1 + .5 cos(5t2)+ (t) (10)

with (t) being is a zero mean noise.The observer gain parameters were selected as a function

of the desired characteristic polynomial, with the desiredparameters: n = 200, p = 200 and = 1. Two differentnoise levels were taken for the simulations, one consistingin a 20 [dB] amplitude and another one of 10 [dB]. Figure1 depicts the analyzed signal. Figures 2 and 3 show thecomparison between the different order algebraic filters forthe 20 [dB] additive noise, and the 10 [dB] additive noiserespectively. The fourth order filtering scheme achieves lessharmonic components in relation to the first order filter. Thecomparison regarding the classic filtering approaches is given

in Figures 4 and 5 for the 20 [dB] noise and finally, thesame comparison for the 10 [dB] noise is shown in Figures6 and 7. To highlight the filtering effects the images showa zoom of the filtered response. Notice that classic filteringresponses exhibit a significant delay effect. The delay effectsare absent in the proposed combination of observer plusalgebraic filtering approach.

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

Time [s]

y(t)+(t) y(t)

Fig. 1. Noisy chirp signal.

0 0.2 0.4 0.6 0.8 1

5

10

15

20

25

Time [s]

y(t)+(t) y(t)Filt1

y(t)Filt4

y(t)

Fig. 2. Different order Algebraic Filtering response: SNR=20 dB.

V. CONCLUDING REMARKS

In this paper, a novel algebraic filtering technique for noisysignals has been proposed. The filtering scheme is based ona high-gain observer and algebraic manipulations on someof its expression to obtain estimates of the required signals.The algebraic filtering approach has several advantages overclassical filtering methods. First, there is no delay in the

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

Time [s]

y(t)+(t) y(t)Filt1

y(t)Filt4

y(t)

Fig. 3. Different order Algebraic Filtering response: SNR=10 dB.

0.5 0.6 0.7 0.8 0.9 1 1.15

10

15

20

Time [s]

y(t) y(t)Butter y(t)Cheby y(t)Filt4

Fig. 4. Comparison with classical low pass filters: SNR=20 dB.

0.5 0.6 0.7 0.8 0.9 1 1.15

10

15

20

Time [s]



2.75 2.8 2.85 2.9 2.955

10

15

20

Time [s]



2.75 2.8 2.85 2.9 2.955

10

15

20

Time [s]



filtered signal, which is a major advantage for practical appli-cations. Besides, the algebraic filter does not need any specialfeature, as for classical filter design, like cutoff frequencyor the need for gain tuning. Thus, once the filter has beenbuilt, it can work over a wide range of signal frequencieswithout any redesign. The comparison with some classicalfilters demonstrates the above conclusions. Moreover, thepresented approach can be extended for time derivativesfiltering, which is an still an open problem. In addition, theproposed scheme is easy to implement, then, it is ideal forengineering applications like signal processing and automaticcontrol.

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an algebraic denoising scheme

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