Adaptive Recursive Deconvolution and

Adaptive Noise Cancellation

Pandolfi, L.∗

Politecnico di TorinoDipartimento di MatematicaCorso Duca degli Abruzzi, 24

10129 Torino — ItalyTel. +39-11-5647516

E–Mail LUCIPAN@POLITO.IT

Abstract

In this paper we apply a recursive deconvolution method to ActiveNoise Cancellation (ANC) in a linear system: the observation of theoutput of a linear system of relative degree one, red at discrete timeinstants, is fed to a deconvolution algorithm which identify the distur-bance (with the delay of one step). This information is used in orderto reduce the effect of the disturbance itself. Deconvolution being anill posed problem, a regularization parameter is to be introduced. Thechoice of the value of the parameter is a delicate issue. We show that,when studying ANC, the discrepancy principle (applied recursively) isa feasible method for the choice of the parameter.

Key words: Active noise cancellation, linear systems, recursive decon-volution.

1 Introduction

Deconvolution problems are ubiquitous in the applications of mathematicsand have many different aspects. Here we distinguish between “off line” and“on line” i.e. “recursive” deconvolution algorithms. Off line deconvolutionalgorithms accumulate all the available pieces of information which are thenelaborated at the end of the process. Recursive deconvolution instead eval-uates an estimates u of the input function u in real time (or with a delay

∗This paper fits the research program of GNAMPA.

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which in principle can be made as small as wanted). Only algorithms of thissecond class can be used for control or regulation, see [2].

The difficulty of the deconvolution problem stems from the fact thatit is “ill posed”, i.e. the solution is not a continuous function of the data.Hence, “regularization” methods have to be used, which provide approximatesolutions continuously depending on the available data. These regularizationmethods do depend on the introduction of a new “regularization parameter”which we shall denote α. Moreover, in practice on line computations areperformed at discrete time instants so that a time step τ has to be introducedas a second parameter of the algorithm which then depends on the twoparameters α and τ . The parameter τ can be seen as a second regularizationparameter, see the formulas (8) and (10). A third parameter which entersthe algorithm is the tolerance h of the errors of the measures.

Recursive deconvolution algorithms have been proposed for example in [7,8, 11] and these algorithms can be used in systems theory.

An interesting approach to “on line” deconvolution is described in [7]. Amodification of this algorithm, which is a recursive application of Tikonovregularization, is studied in details in [3] and it is described below. Thisalgorithm depends on the three parameters mentioned above, the regular-ization parameter α, the error tolerance h and the time step τ and constructsa candidate approximants u of the unknown input u which converges to uprovided that α, h and τ converges to zero while respecting certain “com-patibility conditions” (which depends on the type of convergence which isstudied) and which require in particular that α fades away slower then hand τ .

In general, the parameters α and τ should be “small” in order to guaran-tee fast convergence but in this case both the noise and the round off errorsare amplified so that if α and τ are too small with respect to h then wildoscillations appear which may prevent every reasonable use of the algorithm.

In the case of “off line” reconstruction algorithms, several methods havebeen proposed for an acceptable choice of the regularization parameters,see [6]. The most commonly used is the “discrepancy principle” introducedby Morozov, which essentially requires that the signal produced by u shouldapproximate the observed signal (produced by u) so that the error be of theorder of (a multiple of) h. So, the parameters α and τ of the algorithm aredeemed acceptable when the output produced by u is “close” to the measurestaken on the real output. This is precisely the criteria to be used for noisecancellation, the problem we are going to study.

We note that quality of the reconstruction of the output is not the soleparameter to be considered in practical problems. For example, in practicefast oscillations should be avoided. This goal could be achieved by penalizingalso the derivatives. However, in this paper we are not going to consider

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additional goals like this.

1.1 Deconvolution and Active Noise Control

A recent approach proposed to perform noise reduction has been describedas follows: the plant is subject to a known input f0, corrupted by the noiseω, and to a control u. The observed output y is filtered so to reconstruct anapproximation ω of the disturbance (this is the “deconvolution” problem)which is then fed back to the system (with a “small” delay) so to approx-imately cancel the effect of the noise. This approach has been successfullyused, often on a purely intuitive basis, in ANC (Active Noise Cancellation),see [4, 13] with interesting applications to mobile communications.

Most of the algorithms used in ANC resembles those studied in [2, 3]where and the application of the method has been completely justified. Inthis context, the regularization parameter α has the role of a high gain: ifit is too small then disturbances and round off errors in computations areamplified.

The choice of the regularization parameters is most delicate since it musttake into account conflicting goals. Often it is made on the basis of off-lineexperience, on the basis of the “a priori” information on the disturbances tobe canceled. In this case α is chosen before the process starts and it is keptconstant in time. Instead, an on-line version of the discrepancy principlefor the adaptive choice of α has been proposed in [9, 10]. In this paper weare going to show that the idea of an adaptive choice of the regularizationparameter α can be used to solve ANC for systems of relative degree one(the reason of this restriction will be discussed in section 2.4.1.) The timestep τ is still to be “a priori” chosen.

We note that when the regularization parameter α is chosen in an adap-tive way then it will not be constant in time: the “regularization parameter”is now a function.

For future reference we explicitly note:

Remark 1 The reason for an on-line determination of the regularizationparameter α, based on the discrepancy principle, is now explained. If suffi-cient information on the class of possible errors and disturbances is available,off-line experience may show that very small values of α and τ can be usedso to achieve a very accurate reconstruction of the disturbance. However,unexpected errors or disturbances can destabilize the system. The goal ofthe adaptive determination of α is a more robust and stable algorithm, atthe expenses of the fidelity of the reconstruction.

The plan of the paper is as follows: we shall study linear finite dimen-sional systems, possibly time varying. In section 2 (and subsections) we

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study the deconvolution problem in details. The obtained results are thenapplied to the ANC in Section 3. Simulations of a realistic case, from [2, 14],are given in Section 4.

The structure of Section 2 is as follows: the algorithm is described first inthe (nonrealistic) case of the full state observation and the technical proofsare in subsection 2.2. Subsection 2.4 shows that the algorithm can be ex-tended to systems of relative degree one and discuss the reasons of thislimitation.

2 The algorithm

As we said, the algorithm we use has been precisely studied (with α fixedonce and for all the times) in [3].

We shall assume that the unknown input signal is bounded but knowledgeof the value of the bound of the input is not required. It is only used in theproofs that this bound is finite.

The system which elaborates the signal is the input-output control sys-tem

x = Ax + Bu , 0 ≤ t ≤ T (1)

where x and u are respectively n and m–vectors while A and B denotesmatrices of suitable dimensions, which are continuous function of time t(the matrices are often constant).

The system is initialized at zero, x(0) = 0 so that

x(t) =∫ t

0X(t, s)B(s)u(s) ds (2)

where X(t, s) is the evolution matrix generated by A. If A is constant,X(t, s) = eA(t−s) = X(t− s) and the input-output relation (2) is of convolu-tion type.

We assume that the observation is red at discrete times and contaminatedby errors of known tolerance h. Hence available data are

ξk = Cx(τk) + θk , |θk| < h .

It is not restrictive to assume h < 1.It will be C = I in subsections 2.1 and 2.2.The time instants τk are equispaced for simplicity, τk = kτ .We put

Ik = [τk, τk+1) , τk = kτ .

Our goal is the construction of a signal v such that:

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• the vector v(t) depends on the available data, the fixed step τ of theobservations and a penalization parameter α which may change atevery step;

• the vector v(t) at time t is constructed only using the data ξk availableat previous time instants;

• we want that when h → 0, τ → 0 then automatically α → 0 and insuch a way that

||v − v∗||L2(0,T ) → 0 , where v∗ satisfies Bv∗ = Bu.

We are going to describe an algorithm for the online determination of α.

2.1 Full state observation

In this subsection we assume C = I.In practical applications, in particular applications to control theory, the

full state x is rarely measured. In general, only an output y(t) = Cx(t) ismeasured. The study of the case C = I is instrumental for understandingthe general case.

Clearly, two different inputs u1(t) and u2(t) might produce the sameevolution x(t) of the state. This happens when their difference belongs tokerB(t) a.e. t ∈ (0, T ). So we can replace Eq. (2) with the new input-outputrelation

x = Ax + b , x(0) = 0 , ξk = x(τk) + θk (3)

and we can approximate the function b(t) = B(t)u(t) with a function v(t).After that, the input u(t) = B†v(t) can be singled out, even if in generalv(t) /∈ imB(t).

Remark 2 When n−m is large, as it usually is, the practical computationof B†v will require the use of a penalization method. This is a standardprocedure in the computation of the pseudoinverse of a matrix (see [5]) andit is not considered here.

As in [3, 7], we associate a “model” to our system

w = Aw + v , 0 ≤ t ≤ T . (4)

Now we initialize the reconstruction algorithm as follows: we fix thesampling time τ and we fix two exponents γ and ε, 0 < ε < γ < 1. Moreover,we fix a coefficient µ ≥ 1, whose role is discussed in Remark 3.

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Now we describe the operation to be performed at every time instantτk = kτ in order to construct the function v(t). The notation v(k) is usedfor the restriction of v to the interval Ik,

v(k)(t) = v(t) , t ∈ Ik = [τk, τk+1) .

The operations are recursively described as follows:

i) at time τ0 = 0 we initialize Eq. (4) with w(0) = 0.

We impose the condition

v(0) = arg min{∣∣∣∣

∣∣∣∣∫ τ

0X(τ, s)v(s) ds− ξ1

∣∣∣∣∣∣∣∣2

+ α

∫ τ

0||v(s)||2 ds

}. (5)

This function depends on the still unspecified value α.

We compute

w(t) =∫ t

0X(t, s)v(0)(s) ds , t ∈ I0 . (6)

Also this function depends on the still unspecified value α.

We compare w(τ) and ξ1 and we apply the algorithm described belowso to identify the value α0 of α, to be used on I0. The function v(0)

which corresponds to this value of α is denoted v(0) while w(t) is thecorresponding function, given by (6), when v is replaced by v(0).

ii) Once we have reached time τk = kτ , the functions v(t) and w(t) havebeen computed on [0, τk) and we proceed as follows in order to extendv(t) to the next time interval Ik. First we compute

v(k) = arg min{∣∣∣∣

∣∣∣∣X(τk+1, τk)w(τk) +∫ τk+1

τk

X(τk+1, s)v(s) ds− ξk+1

∣∣∣∣∣∣∣∣2

+α

∫ τk+1

τk

||v(s)||2 ds

}. (7)

The value to be used on Ik for the penalization parameter α has notyet been specified.

We compute now w(τk+1) using this function v(k), which we comparewith ξk+1 and we apply the algorithm (described below) so to identifythe value αk of α, to be used on Ik. The function v(k) which correspondsto this value of α is denoted v(k) and we extend w to Ik:

w(t) = X(t, τk)w(τk) +∫ t

τk

X(t, s)v(k)(s) ds , t ∈ Ik .

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We note that ξk+1 is available at the time τk+1 so that only at this timeit will be possible to define v(k): the function v(t) which approximate b(t) isreconstructed with one step delay.

Now the algorithm for the choice of the value of α at each step, inspiredby the discrepancy principle, see [5, 12], is as follows:

Algorithm: at the time step τk+1 we compare w(τk+1) and ξk+1. If itturns out that for every α > 0 we have

||w(τk+1)− ξk+1|| ≤ µhγ

then the value of α on Ik is

α = µhγ−ε .

Otherwise, we shall see the existence of a unique value of α such that

||w(τk+1)− ξk+1|| = µhγ .

This is the value αk of α we shall use on Ik.

Remark 3 First of all, in order to clarify the algorithm, we anticipate(see [5]) that

α −→ ||α[αI + Rk]−1[Γkwk − ξk+1]||is increasing with limits equal to zero for α → 0+ and ||[Γkwk − ξk+1]|| forα → +∞. Hence we have

||wk+1 − ξk+1|| ≤ µhγ for all α iff ||[Γkwk − ξk+1]|| ≤ µhγ .

In this case αk = µhγ−ε. Otherwise there exists only one value of α for which

||wk+1 − ξk+1|| = µhγ .

The role of µ is now clear: it represent the additional error that it isadmissible in the reconstruction process.

A last observation is that the algorithm we presented penalizes the L2

norm of the input v to the model system. We could add a penalization of theincremental quotient [v(τk+1) − v(τk)]/h in order to reduce the oscillationsin the reconstructed signal, hence also in the regulated output. As we saidalready we are not going to consider this problem here.

We shall prove the following consistency result:

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Theorem 4 Let u be bounded and T < +∞. We relate τ to h so that τ → 0when h → 0, in such a way that

h

τ→ 0 ,

τ

hγ→ 0 .

Under these conditions, we have that v converges to b in L2(0, T ).

As a consequence, u = B†v converges to the input of minimal L2 normwhich produce the evolution x(t) of the state.

This is a consistency result, whose conditions are satisfied if we choose,for example, τ = hστ0, τ0 fixed and γ < σ < 1 but, of course, in practiceneither h → 0 nor τ → 0 and we must content ourselves with suitably smallvalues of them.

Remark 5 The case T = +∞ is of utmost importance for the applications.Theorem 4 can be reformulated also when T = +∞ under additional (stabil-ity) conditions. See section 2.3 for this case. In order to be able to treat thecase T = +∞, we keep track of all the constants in the proof of Theorem 4,even if this leads to a bit involved formulas.

2.2 The proof of the consistency theorem

It is convenient to introduce the following notations. We denote wk, xk thevectors w(τk), x(τk) and we introduce the following operators: the operatorΛk acts from L2(Ik) to IRn,

Λku =∫ τk+1

τk

X(τk+1, s)u(s) ds

so that

(Λk∗x)(s) = X∗(τk+1, s)x , ΛkΛk

∗ =∫ τk+1

τk

X(τk+1, s)X∗(τk+1, s) ds = Rk .

The operator Rk does not depend on k if A is constant.The operator Γk: IRn → IRn is given by

Γkw = X(τk+1, τk)w .

In this section k ≤ N = E[T/τ ] + 1 where E[·] denotes the integer part.Hence there exist constants m, M which do not depend on k and such that

{||Γkw|| ≤ (1 + Mτ)||w||τ(1−mτ)I ≤ Rk ≤ τ(1 + Mτ)I

(8)

where I denotes the identity matrix.

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The upper bound of the unknown input u has no role in the algorithmwe described in the previous section and it needs not be known for theimplementation of the algorithm. However, it will be used in the proofs thatthis upper bound is finite. More precisely we introduce the numbers:

BT = supt∈[0,T ]

||b(t)|| , Bk = supt∈Ik

||b(t)|| . (9)

The following formulas are easily derived:

v(k) = −[αI + Λk∗Λk]−1Λk

∗[Γkwk − ξk+1] ,= −Λk

∗[αI + Rk]−1(Γkwk − ξk+1) ,wk+1 = α[αI + Rk]−1Γkwk + [αI + Rk]−1Rkξk+1 ,wk+1 − ξk+1 = α[αI + Rk]−1[Γkwk − ξk+1] .

(10)

We note that in these formulas the use of ˜ is consistent with the onlinedetermination of α: at the time τk we already know the values of α, v and wfor t ≤ τk; hence, ˜ does not appear on the right hand side of the formulas.Instead, α is to be determined on Ik so that the ˜ notation is used on theleft.

Remark 6 It is seen from (8) that ||(αI + Rk)−1|| is of the order of 1/(α +mτ) so that the last addendum of wk+1 is of the order of h/(α + mτ). Inorder to avoid excessive amplification of the noise (and round off errors) bothα and τ should not be too small.

A preliminary computation is as follows. Here b(k) is the restriction of bto Ik so that

||Λkb(k)|| ≤ τ(1 + Mτ)Bk .

||(αI + Rk)−1{Γkwk − ξk+1}|| ≤||(αI + Rk)−1Γk(wk − ξk)||+ ||(αI + Rk)−1{Γkθk − θk+1}||+||(αI + Rk)−1Λkb(k)|| ≤(1 + Mτ)||wk − ξk||+ [(2 + Mτ)h + τ(1 + Mτ)Bk]

α + τ(1−mτ). (11)

We first prove that our algorithm gives a lower bound for α:

Lemma 7 Let αk be the value of α produced by the algorithm at the instantτk. We have either α = µhγ−ε or

αk ≥ µhγ

Bk + M(hγ + h/τ)(1−mτ) . (12)

The number M depends neither on τ nor on k. See Remark 8 for the de-pendence of M on T .

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Proof. In the first interval,

||w1 − ξ1|| = ||α[αI + R1]−1ξ1|| < h < µhγ

so that α0 = µhγ−ε.Let the assertion be true at k − 1 and let us see what’s going on at the

next step. We noted that

α −→ ||α[αI + Rk]−1[Γkwk − ξk+1]|| = ||wk+1 − ξk+1||is increasing with limits equal to zero for α → 0+ and ||[Γkwk − ξk+1]|| forα → +∞. Hence either

||wk+1 − ξk+1|| ≤ µhγ ∀α ,

so that αk = µhγ−ε, or there exists only one value of α for which

||wk+1 − ξk+1|| = µhγ .

In both the cases, we have ||wk − ξk|| ≤ µhγ and, from (11) the chosenvalue of α gives

||wk+1 − ξk+1|| = µhγ = ||α(αI + Rk)−1[Γkwk − ξk+1]||≤ α

α + τ(1−mτ){(1 + Mτ)||wk − ξk||+ (2 + Mτ)h + τ(1 + Mτ)Bk} .

We noted that ||wk − ξk|| ≤ µhγ . Hence we have

α ≥ µhγ(1−mτ)Mµhγ + (2 + Mτ)(h/τ) + (1 + Mτ)Bk

as wanted.

Note that if τ is too small (with respect to h) then this lower bound forα is small too.

Remark 8 The number M in (12) is explicitly given by

M = max{2, Mµ + MBkτ

hγ+ Mh1−γ} .

Our assumption is that τ/hγ → 0, h → 0 so that when T < +∞ we canchoose (for h and τ/hγ small enough)

M = min{2,M(1 + µ + Bk)} .

If instead T = +∞ it might be impossible to chose a finite value for M . Wefind that M < +∞ if T = +∞ when u is bounded and X(t, s) is boundedin the angle 0 ≤ s ≤ t. This is a stability condition.

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We observe that the estimate from below on α implies that

1α + τ(1−mτ)

≤ 1µhγ−ε + τ(1−mτ)

(13)

when α = µhγ−ε. Otherwise we have

1α + τ(1−mτ)

≤ Bk + M(hγ + h/τ)µhγ(1−mτ)

. (14)

These estimates are now used to study the convergence of v in L2(0, T ).We prove first boundedness of the set of the functions v constructed by

the algorithm. We use

v(k)(s) = X∗(τk+1, s)(αI + Rk)−1[Γkwk − ξk+1] ,

the decomposition (11) and the estimate (12) so to obtain

||v(k)(t)|| ≤1 + Mτ

α + τ(1−mτ)

{||Γk(wk − ξk)||+ ||Γkθk − θk+1||+ ||Λkb(k)||

}

≤ 1 + Mτ

α + τ(1−mτ){(1 + Mτ)µhγ + (2 + Mτ)h + τ(1 + Mτ)Bk} .

When (14) holds we have

||v(k)(t)|| ≤Bk

1−mτ+ o(1) (15)

where o(1) denotes

MτBk

1−mτ+

M(hγ + (h/τ))(1 + Mτ)1−mτ

+Bk + M(hγ + h/τ)

µhγ(1−mτ)[(2 + Mτ)h + τ(1 + Mτ)Bk] .

Otherwise, when inequality (13) holds, we have

||vk(t)|| ≤ o(1) (16)

and

o(1) =1 + Mτ

µ + (τ/hγ)hε(1−mτ)·

·{(1 + Mτ)hε + (2 + Mτ)h1−γ+ε + (τ/hγ)hε(1 + Mτ)Bk

}.

So, boundedness of the function b implies boundedness of the set of thefunctions v constructed by the algorithm, both uniformly and in L2(0, T ).

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We note that v depends on τ and the errors θ, |θ| < h. We choosearbitrary sequences {hn}, {τn} which converges to zero while satisfying theconditions in Theorem 4 and any sequence of errors θ, of tolerance h. Foreach value of n the algorithm gives a function v on [0, T ], which we denotevn. Let {wn} be the corresponding sequence produced by (4).

We already noted that {vn} is bounded. We are going to prove that {vn}converges to b in L2(0, T ).

Boundedness of {vn} implies equicontinuity of the bounded sequence{wn}. From this and the condition

||wn(τk)− ξk|| ≤ µhγ

it follows thatlimwn = x

uniformly on [0, T ].Consequently the set V has a unique weak limit point, which is b.In order to prove that b is also the limit in the norm topology we use a

standard property of the Hilbert spaces: the conditions

vn ⇀ b , ||vn|| → ||b||imply that {vn} converges to b in norm. So, it remains to be proved

||vn||L2(0,T ) → ||b||L2(0,T ).

We note that, by lower weak semicontinuity of the norm,

||b||L2(0,T ) ≤ lim inf ||vn||L2(0,T )

while, using the estimates (15) and (16),

||vn||2L2(0,T ) ≤ τN−1∑

k=0

||vn(k)||2L2(Ik) ≤ τ

N−1∑

k=0

[ Bk

1−mτ+ o(1)

]2

≤{

o(1) + τN−1∑

k=0

B2k

}.

Hence we have

||b||2L2(0,T ) ≤ lim inf ||vn||2L2(0,T ) ≤ lim sup ||vn||2L2(0,T )

≤ lim sup

{o(1) + τ

N−1∑

k=0

B2k

}= ||b||2L2(0,T )

so that we also have

lim v = b in L2(0, T ).

This proves the consistency of the algorithm. We repeat that a suitablechoice for τ which satisfies the assumptions of the theorem is for exampleτ = hστ0, τ0 “a priori” given. The determination of the coefficient τ0 is amatter of practice.

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2.3 Infinite horizon problems

In every practical application T is finite, but often not “a priori” known. Inorder to include also this case in the theory, it is usual to study the caseT = +∞, a case we are going to consider now.

We observe that the condition that u be square integrable on [0,+∞)is not realistic, in particular when u is a disturbance, which does not fadeswith time. A realistic assumption is that u is bounded and every bounded(measurable) u belongs to the Hilbert space K whose elements are the (equiv-alence classes of) functions such that

||u||2 = limT→+∞

1T

∫ T

0||u(s)||2 ds .

This number is the norm of K, which turns out to be a Hilbert space.It is easily seen that theorem 4 holds also in the Hilbert space K provided

that u is bounded on [0, +∞), (finite) values of the constants M and M canbe found on [0, +∞) and furthermore the estimate from below in (8) holdson [0, +∞).

We noted in Remark 8 that the constant M can be found also on [0,+∞)provided that X(t, s) is bounded on 0 ≤ s ≤ t. This is a Liapunov stabilitycondition which always holds in the applications when the system is linear(because when the null solution is not stable then the system will enter anonlinear regime).

The estimate from below in (8) is more delicate. Once the step τ and Thave been fixed, then the best choice for (1 −mτ) is the supremum of thenumbers m such that

mI ≤ 1τ

∫ τk+1

τk

X(τk+1, s)X∗(τk+1, s) ds =1τRk (17)

(in fact we must also take m < 1). Inequality (17) must hold for each k suchthat kτ < T . Hence for each natural number k if T = +∞.

The condition to be checked is that we can find m > 0 which satisfiesthe previous inequality

It is important to note that if T = +∞, the number m defined as aboveis strictly positive in the important case that A is a constant matrix since,as we noted, in this case Rk does not depend on k. If instead the systemis time dependent and T = +∞ then m can be zero, as it is seen in thefollowing example: the system is scalar and A(t) = −t. In this case

X(t, s) = e(t2−s2)/2 , Rk =∫ τk+1

τk

e−(τ2k+1−s2) ds .

A standard computation shows that[∫ τk+1

τk

es2ds

]2

≤ π

4

[e2kτeτ − 1

]

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so that

Rk ≤√

π

2e−k2τ/2ekτ/2e−τ/2 → 0

when k → +∞, with fixed τ . Hence, in this case the conditions used in theproof of Theorem 4 are not satisfied.

2.4 Systems of relative degree one

We consider now systems of relative degree one. This means that the systemis time-invariant, i.e. the matrices A, B and C are constant, and that

kerCB = 0 ,

see [1]. This case is of great importance in robustness theory. We are goingto show that this case can be reduced to the case of full state transformationand, in principle, treated as previously described. Of course, in concreteexamples, different procedures might be more convenient, see the commentsin Remark 10 and the example in Section 4. As in [3], we consider the matrix

J = BB∗C∗C .

It is seen in [3] that this matrix has nonnegative eigenvalues and it is diag-onalizable. So, in a suitable coordinate system we have

J =

[J 00 0

], A =

[A11 A12

A21 A22

]

B =

[B+

B−

], C =

[C+ C−

].

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Here J is diagonal with positive diagonal elements. The condition kerCB =0 implies that B− = 0, see [3, Th. 23]. In fact, we have more:

Lemma 9 We have kerC+ = 0, kerB+ = 0 and C∗+C− = 0.

Proof. As B− = 0 kerCB = 0 we see that kerC+B+ = 0 so that we havealso kerB+ = 0. We compute J and we find that it has the form

J =

[B+B∗

+C∗+C+ B+B∗

+C∗+C−

0 0

]

Hence,J = B+B∗

+C∗+C+ .

We know that J is invertible so that B+ is surjective so that kerB+B∗+ = 0.

From 0 = B+B∗+C∗

+C− we get C∗+C− = 0.

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This lemma has an important consequence: we can multiply the outputy by C∗

+ and we get a new output, which does not “see” the component y−.Moreover, the property kerC+ = 0 shows that C∗

+C+ is invertible: a suitabletransformation of coordinates reduces the system to the following form:

{x+ = A11x+ + A12x− + B+ux− = A21x+ + A22x−

y = x+ .

Noisy observation is nowξk = x+(τk) + θ

where the error θ is of the order of h. Hence, x+ is known (wit tolerance h)and the initial condition is zero. Consequently, the output y is given by

y(τk) =[∫ τk

0eA11(τk−s)b(s) ds + ζk

]

where b(s) = B+u(s) as in the previous sections and

ζk =∫ τk

0eA11(τk−s)[A12x−(s)] ds

can be computed (with an error of the order of h). This can be subtractedfrom the noisy observation so to obtain a new observation

ξk = ξk − ζk

which is the full state (noisy) observation for a system of the form

x+ = A11x+ + B+u .

The observation error is still of the order of h and we are so reduced to thecase we solved in the previous sections.

Remark 10 We observe that even if the matrix A is stable, the submatrixA11 might be unstable so that the methods just described might be practi-cally not feasible. This problem can be handled with the device describedin [2, Sect. B].

2.4.1 Why relative degree one

It is well known that any system of the form (1) is equivalent to its Morsequasicanonical form. This observation has been used in [3], where it is provedthat the identification problem in the general case C 6= I can be reduced tothe solution of a (finite) chain of identification problems as those describedin Sect. 2. The case treated in [3] is the case that the value of α is fixed onceand for all.

15

The arguments in [3] (and applied to the disturbance reduction in [2])can be applied to every linear time invariant system. The restriction thatthe system be of relative degree one was not used. In order to understandthis point, let us consider the simple case

x′1 = x2 , x′2 = u , y = x1 .

The goal is the reconstruction of u which is achieved in two steps: we use yand our algorithm in order to obtain an estimate y1 = x2 of the “input” x2

which is then seen as a new “observation” in order to estimate u. This canbe done on-line (with the delay of two steps). Thanks to Morse canonicalform, this example contains all the features of the general case.

Of course, this two-steps algorithm requires a very precise reconstructionof x2 in order to achieve a reasonable reconstruction of u and, as seen in [2,Formula (42)], this require small values of α (i.e. of the error say h1 in the“observation” of x2) which cannot be achieved with the adaptive method,due to the estimate in Lemma 7. In fact, we repeat that the goal of thisrecursive choice of α is not sharp reconstruction of the input signal, butrobustness in the case of poor information on the class of possible inputs,see Remark 1.

3 Active noise cancellation

We assume now that the input u to the systems is

(f0 + ω)− u

where f0 is a known signal and ω is the disturbance. the control u has tobe constructed so to reduce the effect of ω. We proceed as described in [2].On the first interval we put u = u(0) = 0 and we feed Bf0(t) + u(0) + v(t) tothe model system. The signal v has to be determined. It is determined fromcondition (5). The term which is penalized is the norm of v and not thatof Bf0 + v since, due to the linearity, the contribution of Bf0 in the systemand in the model cancel out.

The algorithm provides a function v(0) and ω(t) = B†v(0)(t). In the nextinterval I1 = [τ1, τ2) we feed

u(1)(t) = B†ω(t− τ)

to the system andBf0(t)−Bu(1)(t) + v

to its model. The function v is then penalized as in (7) in order to constructω(t) on I2. We then choose

u(2)(t) = B†ω(t− τ) , t ∈ I3 .

16

We proceed in a similar way in each one of the next steps. Alternatively, asin [2] we can consider u(1)(t) as an “approximant” of the noise. We feed itto the model and not to the system. In this way the system and the modelare subject to similar noises which is then “estimated” in the next step. Wethen subtract this estimate of the noise, i.e. B†ω(t − 2τ), both from thesystem and its model. This has the role of the control which suppress thenoise with a two-steps delay both from the system and from its model. Thesame proof as in [2] shows that asymptotically, for τ → 0, α → 0+ whilerespecting suitable relations the output y converges to the nominal outputof the system, i.e. the output driven by f0 without noise. The condition onα is automatically achieved when h → 0+ if the adaptive algorithm is used.

4 Example and simulations

Simulations of realistic problems have been presented in [2]. In that pa-per however the choice of the regularization parameter α was off-line. Weconsider now one of the systems already studied in [2], but now we usethe adaptive algorithm as studied in this paper for the determination ofα. The interest of this system to flight control is described in [14]. Herewe confine ourselves to present the equations. the system is scalar, i.e.dimu = dim y = 1 while the state has dimension 3. It has the form

x = Ax + B{ζf0 − u} , y = Cx

where the nominal value of ζ is 1. The value of ζ may change due to thefailure of some component and the effect of u should be to restore the nominalvalue of ζf0. This problem fits into our framework once we represent

ζf0 − u = f0 + (ζ − 1)f0 − u

where now (ζ − 1)f0 is seen as a disturbance to be evaluated and canceled.The matrices of the system are

A =

−0.5162 26.96 178.9−0.6896 −1.225 −30.380 0 −13

B =

−175, 6014

, C =

[1 12.43 0

].

This system is of relative degree one. The matrix A is exponentially stablebut the transformation described in Section 2.4.1 is not convenient sincethe submatrix corresponding to the matrix J (i.e. the matrix that after

17

the transformation will appear in position (1, 1), now of dimension 1) is notstable. So, we proceed as in [2] where it is shown that a suitable coordinatetransformation can be used to represent the system as

x = −0.0385x− {0.7101x2 − 0.0902x3}+ ζf0 − u[x2

x3

]′=

[7.3467 −2.4221124 −29.8436

] [x2

x3

]+

[1.732520.7556

]x .

In an appropriate scale, the output is y = x.The matrix of the second subsystem is stable, with eigenvalues of the

order of −17.8 and −4.6 so that the values of x1 and x2 get soon verysmall (in spite of the fact that the eigenvectors of the matrix are almostcolinear). This suggests that we proceed as if the system were describedonly by its first component, the effect of the second and third componentsbeing approximately computed from the output. So, we proceed as follows:we use as a model the system

w = −0.0385x− {0.7101w2 − 0.0902w3}+ ζf0 − u[w2

w3

]′=

[7.3467 −2.4221124 −29.8436

] [w2

w3

]+

[1.732520.7556

]ξ

where ξ is the function of the observations, ξ(t) = ξk for t ∈ [τk, τk+1).The output y is read on a time interval of 50 time units, 0 ≤ t ≤ 50 = T .

Preliminarily we fix the sample rate. We decide to sample twice each timeunit, τ = .5. At each time τk = kτ we read the output corrupted by noise,

ξk = y(τk)(1 + hω)

where ω is a random error uniformly distributed in [−0.5, 0.5]. Hence hrepresents here a relative error.

Instead then choosing γ and ε these parameters are determined from thetolerance h on the measures and the tolerance µh admitted in the recon-struction, as

γ = log(µh)/ log h , ε = 0.9γ

(the choices are such that h and µh are less then 1 so that γ ∈ (0, 1)).The nominal value of ζ is ζ = 1 and we consider the case that, due to

the failure of an actuator,

ζ(t) =

1 if t < T/3−10 if T/3 ≤ t < 2T/3−20 if t > 2T/3 .

The function f0(t) is f0(t) = 1/(t + 1).

18

Figure 1: Left: desired output (tick) and uncompensated output. Right: thejump function.

0 5 10 15 20 25 30 35 40 45 50−15

−10

−5

0

5

10

0 5 10 15 20 25 30 35 40 45 50−25

−20

−15

−10

−5

0

5

time

Figure 1 plots the desired and the uncompensated evolution of the outputon the left and the jump function ζ(t) on the right.

The figures we present next show the effect of the compensator. Weconsider first µ = 3 and h = 0.01 (i.e. 1% of relative error on each individualmeasure). In Fig. 2 we represent the compensated evolution and the desiredevolution of the output on the left, the estimate of ζ(t)f0(t) on the right.In order to present a case which seems not acceptable, we represent the

Figure 2: Error tolerance h = 0.01, µ = 3.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

time 0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5

2

same plots in Fig. 3, but with h = 0.1 (i.e. 10% error) and µ = 10. Theunacceptable feature of this simulation is not so much high error (µ = 10,this was our choice) but the fact that cancellation has been obtained at theexpenses of fast oscillations. In fact, in this paper we did not make any

19

Figure 3: Error tolerance h = 0.1, µ = 10.

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

time 0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5

attempt to attenuate the derivative of the reconstruction of the signal. Suchfast oscillations are due to poor reconstruction of the jump function ζ(t). Thenext figure (Fig. 4) shows in fact the functions ζ(t) and its reconstructionson the left for the case examined in Fig. 2 and on the right for the caseexamined in Fig. 3.

Figure 4: The jump function and its reconstruction.

0 5 10 15 20 25 30 35 40 45 50−30

−25

−20

−15

−10

−5

0

5

0 5 10 15 20 25 30 35 40 45 50−35

−30

−25

−20

−15

−10

−5

0

5

Fidelity of reconstruction is not a key point of ANC but it is clear thattoo poor a reconstruction is not compatible with a satisfactory performance.

Finally, Fig. 5 shows the function α(t) on the left for the case examinedin Fig. 2 and on the right for the case examined in Fig. 3.

20

Figure 5: The values of α and µhγ .

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

time0 5 10 15 20 25 30 35 40 45 50

1

1.5

2

2.5

3

3.5

4

4.5

time

References

[1] V. Dragan, A. Halanay, Stabilization of linear systems, Birkhauser,Boston, 1999.

[2] F. Fagnani, V. Maksimov, L. Pandolfi, A recursive deconvolution ap-proach to disturbance reduction, IEEE Trans. Automat. Control, 49907-921, 2004.

[3] F. Fagnani, L. Pandolfi, A singular perturbation approach to a recur-sive deconvolution problem, SIAM J. Control Optim. 40 (5), 1384-1405(2002).

[4] W.S. Gan, S.M. Kuo, An integrated audio and active noise headset,IEEE Trans. Consumer Electronics, 48 242-247, 2002.

[5] C.W. Groetsch, The theory of Tikhonov regularization for Fredholmequations of the first kind, Pitman, Boston, 1984.

[6] M. Hanke, P.C. Hensen, Regularization methods for large-scale prob-lems, Surv. Math. Ind., 3 253–315, 1993

[7] Yu. S. Osipov, A.V. Kryazhimskii Inverse problems for ordinary dif-ferential equations: dynamical solutions, Gordon and Breach, London,1995.

[8] P.K. Lamm, Future-sequential regularization methods for ill-posedVolterra equations, J. Mathematical Analysis Appl. 195 465–494, 1995.

[9] P.K. Lamm, Full convergence of sequential local regularization methodsfor inverse problems, Inverse Problems, 21 785-803, 2005.

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[10] P.K. Lamm, Variable smoothing regularization methods for inverseproblems, in Theory and Practice of Control and Systems, A. Tornambe,G. Conte, A.M. Perdon Ed.s, World Scientific, Singapore, 1999.

[11] V.I. Maksimov, Dynamical inverse problems of distributed systems VSP,Utrecht, the Netherlands, 2002.

[12] V.A. Morozov, Methods for solving incorrectly posed problems, Springer,New York, 1984

[13] Y. Shiang-Hwua, H. Jwu-Sheng, Controller design for active noise can-cellation headphones using experimental row data, IEEE/ASME Trans.Mechatron., 6 483-490, 2001.

[14] J.-S. Yee, J. L. Wang, B. Jiang, A fault tolerant flight controller de-sign for actuator faults, in Proc. 3th Int. Conference in Nonlinear Prob-lems in Aviation and Aerospace, S. Sivasundaram Ed., Cambridge U.K.,2001, pp. 723-730.

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