IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 1285

small parameter" > 0, has been established. This proposition isbased on the assumption of the stabilizability of the boundary-layersystem. It was also shown that this connection is only one-directionvalid, i.e., the controllability of the reduced-order and boundary-layersystems always yields the controllability of the original system, butnot vice versa. The criterion of the impulse-freeE0-controllabilityof the reduced-order system is derived in the terms of an auxiliarygain matrixK(t). The invariance of this criterion toK(t) is shown.Due to the duality, similar results can be obtained for the Euclideanspace observability of singularly perturbed linear time-dependentsystems with multiple small delay. In this case, the assumption of thestabilizability of the boundary-layer system has to be replaced by theassumption of the detectability of this system.

REFERENCES

[1] P. V. Kokotovic and A. H. Haddad, “Controllability and time-optimalcontrol of systems with slow and fast modes,”IEEE Trans. Automat.Contr., vol. AC-20, pp. 111–113, 1975.

[2] P. V. Kokotovic, H. K. Khalil, and J. O’Reilly,Singular PerturbationMethods in Control: Analysis and Design. London, U.K.: Academic,1986.

[3] H. K. Khalil, “Feedback control of nonstandard singularly perturbed sys-tems,”IEEE Trans. Automat. Contr., vol. 34, pp. 1052–1060, Oct. 1989.

[4] Y. Y. Wang, P. M. Frank, and N. E. Wu, “Near-optimal control ofnonstandard singularly perturbed systems,”Automatica, vol. 30, pp.277–292, 1994.

[5] H. Krishnan and N. H. McClamroch, “On the connection between non-linear differential-algebraic equations and singularly perturbed controlsystems in nonstandard form,”IEEE Trans. Automat. Contr., vol. 39, pp.1079–1084, May 1994.

[6] V. Kecman and Z. Gajic, “Optimal control and filtering for nonstandardsingularly perturbed linear systems,”J. Guid. Control Dyna., vol. 22, pp.362–365, 1999.

[7] H. Xu and K. Mizukami, “Nonstandard extension ofH -optimal con-trol for singularly perturbed systems,” inAdvances in Dynamic Gamesand Applications. Boston, MA: Birkhauser, 2000, vol. 5, pp. 81–94.

[8] E. Fridman, “A descriptor system approach to nonlinear singularlyperturbed optimal control problem,”Automatica, vol. 37, pp. 543–549,2001.

[9] V. Y. Glizer and E. Fridman, “H control of linear singularly perturbedsystems with small state delay,”J. Math. Anal. Applicat., vol. 250, pp.49–85, 2000.

[10] T. B. Kopeikina, “Controllability of singularly perturbed linear systemswith time-lag,”Diff. Equat., vol. 25, pp. 1055–1064, 1989.

[11] V. Y. Glizer, “Euclidean space controllability of singularly perturbedlinear systems with state delay,”Syst. Control Lett., vol. 43, pp. 181–191,2001.

[12] , “Controllability of singularly perturbed linear time-dependentsystems with small state delay,”Dyna. Control, vol. 11, pp. 261–281,2001.

[13] R. B. Vinter and R. H. Kwong, “The infinite time quadratic controlproblem for linear systems with state and control delays: an evolutionequation approach,”SIAM J. Control Optim., vol. 19, pp. 139–153, 1981.

[14] M. C. Delfour, C. McCalla, and S. K. Mitter, “Stability and the infi-nite-time quadratic cost problem for linear hereditary differential sys-tems,”SIAM J. Control, vol. 13, pp. 48–88, 1975.

[15] V. Y. Glizer, “Asymptotic solution of a singularly perturbed set of func-tional-differential equations of Riccati type encountered in the optimalcontrol theory,”Nonlinear Diff. Equat. Applicat., vol. 5, pp. 491–515,1998.

[16] R. B. Zmood, “The Euclidean space controllability of control systemswith delay,”SIAM J. Control, vol. 12, pp. 609–623, 1974.

[17] A. Halanay, Differential Equations: Stability, Oscillations, TimeLags. New York: Academic, 1966.

[18] I. M. Cherevko, “An estimate for the fundamental matrix of singularlyperturbed differential-functional equations and some applications,”Diff.Equat., vol. 33, pp. 281–284, 1997.

[19] M. C. Delfour and S. K. Mitter, “Controllability, observability and op-timal feedback control of affine hereditary differential systems,”SIAMJ. Control, vol. 10, pp. 298–328, 1972.

[20] P. V. Kokotovic, “Applications of singular perturbation techniques tocontrol problems,”SIAM Rev., vol. 26, pp. 501–550, 1984.

[21] H. K. Khalil, “Feedback control of implicit singularly perturbed sys-tems,” inProc. 23rd Conf. Decision Control, Las Vegas, NV, 1984, pp.1219–1223.

Closed-Form Unbiased Frequency Estimation of a NoisySinusoid Using Notch Filters

Sergio M. Savaresi, S. Bittanti, and H. C. So

Abstract—In this note, the problem of the frequency estimation of a sinu-soid embedded in white noise is considered. The approach used herein is theminimization of the sample variance of the output of constrained notch fil-ters fed by the noisy sinusoid. In particular, this note focuses on closed-formexpressions of the frequency estimate, which can be obtained using notchfilters having an all-zeros finite-inpulse response (FIR) structure. The re-sults presented in this note are as follows. 1) It is shown that the FIR notchfilters obtained from standard second-order infinite-impulse response (IIR)filters are inadequate. 2) A new second-order IIR notch filter is proposed,which provides an unbiased estimate of the frequency. 3) The FIR filter ob-tained from the new IIR filter provides a closed-form unbiased frequencyestimate. 4) The closed-form frequency estimate obtained using the newFIR notch filter asymptotically converges toward the Pisarenko HarmonicDecomposition estimator and the Yule–Walker estimator.

Index Terms—Frequency estimation, harmonic analysis, notch filters,unbiased parameter identification.

I. INTRODUCTION AND PROBLEM STATEMENT

This note deals with the problem of estimating the frequency0 of aharmonic signals(t) = A cos(0t+'), given its noisy measurementy(t) = s(t)+n(t), t = 1; 2; . . . ; N , wheren(t) is a zero-mean whiteGaussian noise(n � WGN(0; �2)). This problem is frequently en-countered in real-world applications, especially in the fields of adaptivecontrol and signal processing, and numerous techniques have been de-veloped for its treatment (see, e.g., [4]–[7], [10]–[13], [15], [18], [21]).This note focuses on the class of estimation methods based on con-strained notch filters (see, e.g., [8] and the references cited therein).

The basic idea underlying notch-filters-based estimation techniquesis the minimization, with respect to, of the loss function

J() =

N

t=1

"(t;)2 (1)

where"(t;) = G(z�1;)y(t) is the output of a notch filter withtransfer functionG(z�1;), fed by the measured signaly(t). Thenotch ofG(z�1;) is centered around the frequency. In general,the dependence ofJ() on is nonlinear and nonconvex; hence, iter-ative quasi-Newton minimization methods must be used.

Manuscript received November 5, 2002; revised January 31, 2002. Recom-mended by Associate Editor A. Garulli. This work was supported by MIURproject “New Methods for Identification and Adaptive Control for IndustrialSystems,” and by the EU project “Nonlinear and Adaptive Control.”

S. M. Savaresi and S. Bittanti are with the Dipartimento di Elet-tronica e Informazione, Politecnico di Milano, 20133 Milan, Italy (e-mail:savaresi@elet.polimi.it).

H. C. So is with the Department of Computer Engineering and InformationTechnology, City University of Hong Kong, Hong Kong.

Digital Object Identifier 10.1109/TAC.2003.814278

0018-9286/03$17.00 © 2003 IEEE

1286 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003

Obviously, the most crucial design choice in a notch-basedestimation technique is the selection of the structure and of theparameterization of the filterG(z�1;). Usually, second-order IIRfilter with a strongly constrained parameterization are used. Startingfrom second-order filters, simple finite-impulse response (FIR) filtersor more sophisticated higher order infinite-impulse response (IIR)filters have been developed and proposed ([2], [9], [17]).

Two slightly different second-order IIR notch filters are typicallyused in practice. They have the following expressions:

G1(z�1

;; �) =1� 2 cos()z�1 + z�2

1� 2� cos()z�1 + �2z�2(2)

G2(z�1

;; �) =1� 2 cos()z�1 + z�2

1� (1 + �2) cos()z�1 + �2z�2: (3)

In (2) and (3), the parameter� (0 � � < 1) is known as thede-bi-asing parameteror thepoles-contraction factor(note that� only affectsthe position of the poles). In the literature, filters of this type are alsoknown asconstrained notch filter, where the termconstrainedrefersto the fact that their structure is strongly under parameterized: the fiveparameters of a fully-parameterized second-order digital IIR filter arereduced to one parameter only. As a matter of fact, since� is regardedas a design parameter, the only unknown parameter of (2) and (3) is theangular frequency. The main difference between (2) and (3) is that(3) provides arigorously unbiased estimation of the frequency of a puretone embedded in white noise, whereas (2) provides a biased estimate.It is easy to see that such bias is negligible if� � 1; the problem of thebias becomes severe if�� 1. The properties of such filters have beendiscussed and analyzed in a large number of works (see, e.g., [3], [14],[25], [26], and the references cited therein).

Note that if the unknown frequency0 is time-varying, and the min-imization of (1) is made recursively, the estimation algorithm usuallyis calledfrequency tracker(see e.g. [22]). Notch filters are frequentlyused for real-time recursive frequency estimation: in the literature thisproblem is referred to as adaptive notch filtering (ANF). This work doesnot focus on ANF but the results presented herein can be straightfor-wardly extended to ANF as well.

The goal of this note is to develop closed-form frequency estimatorsbased on notch filters. The starting point of this work can be summa-rized in the following simple observations. First, note that closed-formexpressions of the frequency estimator cannot be obtained if the notchfilter has a IIR structure, due to the autoregressive part of the filter.Moreover, observe that a constrained FIR notch filter can be easily ob-tained fromG1(z

�1;; �) by setting� = 0; unfortunately, this is notpossible usingG2(z

�1;; �) (note thatG2(z�1;; 0) is not a FIR).

Finally, note that the closed-form frequency estimate obtained fromG1(z

�1;; 0) is severely affected by a bias error.Starting from these observations, the main results and original con-

tributions of this note are the following.

• Section II: A new second-order IIR unbiasing constrainednotch filterG3(z

�1;; �) is developed and analyzed.• Section III: It is shown that a closed-form frequency esti-

mate can be obtained using the FIR filtersG1(z�1;; 0) and

G3(z�1;; 0); the major advantage ofG3(z

�1;; 0) over

G1(z�1;; 0) is that it provides a rigorously unbiased esti-

mate of0.• Section IV: It is shown that the closed-form frequency es-

timate provided by the FIR notch filterG3(z�1;; 0), if the

numberN of data snapshots is large, tends to the frequencyestimators provided by the Pisarenko Harmonic Decompo-sition (PHD) approach, and by the Yule–Walker (YW) ap-proach.

II. NEW UNBIASING SECOND-ORDERCONSTRAINEDNOTCH FILTER

As already remarked in Section I, one of the major drawbacks of thenotch filter (2) (the most widely used in practice) is that it provides abiased estimation of0. This bias is particularly severe when�� 1.

Starting from the cost function (1), a new unbiasing second-orderIIR notch filter can be obtained as follows.

• Consider the long-run (asymptotic) version of the cost function(1), namely

J() = limN!1

1

N

N

t=1

"(t;)2

where

"(t;) =G(z�1;)y(t):

It is easy to see thatJ() can be given the following expression(see [3]):

J() =1

2�

+�

��

G(ej!;)2

Sy(!)d! (4)

whereSy(!) is the power spectrum ofy(t), which can be splitinto the power spectra ofs(t) andn(t), namely

Sy(!) =Ss(!) + Sn(!) Sn(!) = �2

and

Ss(!) =A2

2

1

2�(! + 0) +

1

2�(! � 0) :

• Compute the asymptotic cost functionJ1() associated with thenotch filterG1(z

�1;; �), by plugging in (4) the expression ofthe notch filter (2) and the expressions ofSs(!) andSn(!)

J1() =J1(s)

() + J(n)1 ();

J(n)1 () =

1

2�

+�

��

G1(ej!;; �)

2

�2d!

J(s)1 () =

1

2�

A2

2G1(e

j!;0; �) :

For the computation ofJ(n)1 () (the contribution toJ1()

due to the noise) we have resorted to theRugizkaalgorithm (thisalgorithm is based on theory of residues; see [2]). The calculusof J

(s)1 () calls for cumbersome but easier computations. The

expressions obtained forJ(s)1 () andJ

(n)1 () are shown in (5a)

and (b) at the bottom of the page. It is interesting to note that an

J(s)1 () =

2A2 (cos()� cos(0))2

1 + �4 + 4�2 cos2(0) + 4�2 cos2()� 4�3 cos() cos(0)� 4� cos() cos(0)� 2�2(5a)

J(n)1 () =

�2

�

�3 + �2 � 6� cos2() + �+ 2cos2() + 1

(1 + �) (�2 + 2� cos() + 1) (�2 � 2� cos() + 1): (5b)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 1287

Fig. 1. Shape of the function�(; �) in the ranges 2 [0; �] and� 2 [0; 1].

expression very similar to (5b) is proposed in [25] (10). The twoexpressions (both correctly derived) do not coincide since theyhave a slightly different meaning.

• Observe that the bias in the frequency estimate obtained usingG1(z

�1;; �) is due to the fact thatJ(n)1 () is a function of.

This dependence ofJ(n)1 () on has the effect of moving the

minimum ofJ1() away from0 (note, instead, that0 is theminimum ofJ

(s)1 ()).

Now, observe that the minimum ofJ(s)1 () does not change

if J(s)1 () is multiplied by a strictly positive function of and

�, say�(; �) (obviously for 2 [0; �) and� 2 [0; 1)). This isdue by the presence of the factor(cos()�cos(0)) in J

(s)1 (),

which is null if = 0. Consider then the following function�(; �):

�(; �)

=(1+�) (�2+2� cos()+1)(�2�2� cos()+1)

(1+�2) (�3+�2�6� cos2()+�+1+2cos2()): (6)

Note that such function is the square-root of the inverse ofJ(n)1 () (but for the coefficient�2=�), multiplied by(1 + �2)).

A new unbiasing filter G3(z�1;; �) can be obtained from

G1(z�1;; �) and (6) as follows:

G3(z�1;; �) = �(; �)G1(z

�1;; �)

= �(; �)1� 2 cos()z�1 + z�2

1� 2� cos()z�1 + �2z�2: (7)

Due to the fact thatG3(z�1;; �) is simply obtained by multiplying

G1(z�1;; �) by �(; �), some remarks on the shape of�(; �) are

due (see Fig. 1 where�(; �) is plotted in the ranges 2 [0; �] and� 2 [0; 1]).

• �(; �) is not null in the ranges 2 [0; �] and � 2

[0; 1]; this can be easily seen from (6). This guarantees thewell posedness of the optimization problem based on the costfunction (1).• Note that�(; 1) = 1, whereas�(; 0) strongly differs

from 1; this is expected since�(; �) is a sort of “de-bi-asing factor” ofG1(z

�1;; �). Therefore,�(; �) leavesG1(z

�1;; �) almost unchanged if� is close to 1, whereas�(; �) provides a strong correction toG1(z

�1;; �) forsmall values of�.• Note that�(�=2; �) = 1 8� 2 [0; 1], and that�(; �) is

symmetric with respect to = �=2 in the range 2 [0; �].This is consistent with a peculiar feature ofG1(z

�1;; �):it provides an unbiased estimate� 2 [0; 1] if and only if0 = �=2 ([3]).

In order to get a complete understanding of the differences betweenthe three second-order constrained notch filtersG1(z

�1;; �),G2(z

�1;; �), andG3(z�1;; �), it is interesting to compare the

corresponding asymptotic cost functionsJ1(), J2(), andJ3(),respectively.

The closed-form expressions ofJ1() has already been computedin (5). Following the same procedure,J2() = J

(s)2 () + J

(n)2 ()

andJ3() = J(s)3 () + J

(n)3 () can be obtained as (8a), (8b), (9a),

and (9b), shown at the bottom of the next pageBy comparingJ1(), J2(), andJ3() it is apparent thatJ2()

andJ3() have the minimum exactly at0 (unbiased estimate), since

1288 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003

Fig. 2. Shape of the cost functionsJ (), J (), andJ () (A = 1, � = 1, = �=6, � = 0:5).

J(n)2 () andJ

(n)3 () do not depend on. On the contrary, as already

observed,J1() provides a biased estimate sinceJ(n)1 () is depen-

dent.The shapes of these three different cost functions can be better ap-

preciated from Fig. 2, whereJ1(), J2(), andJ3() are displayedfor A = 1, �2 = 1 (signal-to-noise ratio (SNR)= 0:5), 0 = �=6,and� = 0:5.

To conclude this section, it is worth remarking that the newfilter G3(z

�1;; �) merges the two main appealing features ofG1(z

�1;; �) and G2(z�1;; �): similarly to G1(z

�1;; �), anFIR filter can be obtained fromG3(z

�1;; �) by simply using� = 0;similarly to G2(z

�1;; �), G3(z�1;; �) provides an unbiased

estimate of0 8� 2 [0; 1). These features will be fully exploitedin the following section, in order to obtain closed-form frequencyestimates based on FIR notch filters.

III. CLOSED-FORM FREQUENCYESTIMATION VIA NOTCH FILTERS

A closed-form notch-based frequency estimate cannot be obtained ifthe filter has a IIR structure. Consider the FIR filters obtained by simplysetting� = 0 in (2) and in (7) (it has been already observed that setting� = 0 in G2(z

�1;; �) does not yield a FIR filter), namely

G1(z�1;; 0) =1� 2 cos()z�1+z�2

G3(z�1;; 0) =

1

2 cos2()+11� 2 cos()z�1+z�2 :

Using such filters, closed-form frequency estimators from the datacan be obtained as follows.

J(s)2 () =

2A2 (cos()� cos(0))2

1 + �4 cos2()� 2�4 cos() cos(0)� 2�2 + �4 + cos2() + 2�2 cos2()� 4�2 cos() cos(0)� 2 cos() cos(0) + 4�2 cos2(0)

(8a)

J(n)2 () =

�2

(1 + �2)�(8b)

J(s)3 () =

2A2 (cos()� cos(0))2 �2(; �)

1 + �4 + 4�2 cos2(0) + 4�2 cos2()� 4�3 cos() cos(0)� 4� cos() cos(0)� 2�2(9a)

J(n)3 () =

�2

(1 + �2)�: (9b)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 1289

Fig. 3. Asymptotic bias error of (11) (A = 1, � = 1).

A. Closed-Form Frequency Estimator Obtained UsingG1(z�1;; 0)

Consider the following cost function, obtained by plugging in (1) theFIR notch filterG1(z

�1;; 0):

J1() =

N

t=1

(y(t)� 2 cos ()y(t� 1) + y(t� 2))2

and differentiateJ1() with respect to:

dJ1()

d= 2

N

t=1

[(y(t)� 2 cos()y(t� 1) + y(t� 2))

� (2 sin()y(t� 1))] : (10)

Note that (10) is quadratic with respect tocos(); hence, by solvingdJ1()=d = 0 with respect tocos(), it is easy to see that the fol-lowing holds:

cos() =N

t=1y(t� 1) (y(t) + y(t� 2))

N

t=1y(t� 1)2

:

The closed-form frequency estimator, therefore, is given by

1 = arccosN

t=1y(t� 1) (y(t) + y(t� 2))

N

t=1y(t� 1)2

: (11)

As the numberN of data grows,1 tends to the minimum of theasymptotic cost function (5) (in the special case of� = 0). After somecumbersome computation, the following asymptotic expression of (11)

is obtained:

1 = !N!1

arccos�A2

�A2 + �2cos(0) : (12)

From (12), it is apparent that the frequency estimate is affected bya severe bias; note that the bias is null in the (trivial and unrealistic)case of zero noise(�2 = 0); it grows as the SNR decreases. To get aquantitative idea of this bias error, in Fig. 3 the asymptotic bias error inthe case of SNR= 0:5 (A = 1, �2 = 1) is displayed. As expected, thebias is null if0 = �=2; it is maximum for0 = 0 or 0 = �. Notethat, in average, the bias error is huge; hence, the frequency estimate(11) is of no use in practice.

B. Closed-Form Frequency Estimator Obtained UsingG3(z�1;; 0)

Consider the following cost function, obtained by plugging in (1) theFIR notch filterG3(z

�1;; 0):

J3() =

N

t=1

(y(t)� 2 cos()y(t� 1) + y(t� 2))2

2 cos2() + 1

and differentiateJ3() with respect to; (13), as shown at the bottomof the next page holds.

Consider now the problem of solvingdJ3()=d = 0 with respectto . After some manipulation the following expression is obtained:

N

t=1

(sin() (y(t)� 2 cos()y(t� 1) + y(t� 2))

� (y(t� 1) + cos()y(t) + cos()y(t� 2))) = 0: (14)

1290 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003

Equation (14) admits a trivial solution:sin() = 0. Assuming that0 6= f0; �g, the following quadratic form (with respect tocos())can be obtained from (14):

2

N

t=1

y(t�1) (y(t)+y(t�2)) cos2()+

N

t=1

2y(t�1)2�(y(t)+y(t�2))2 cos()�

N

t=1

y(t�1) (y(t)+y(t�2)) = 0: (15)

From (15), a closed-form frequency estimator can be computed. Ithas the expression shown in (16) at the bottom of the page.

As the numberN of data grows,3 tends to the minimum of theasymptotic cost function (8) (in the special case of� = 0)

3 !N!1

0:

Thus, the new filterG3(z�1;; 0) provides a simple closed-form

unbiased estimate of0. Interestingly, (16) is closely related to themethod given in [19] and [20], even if it the derivation of this result iscompletely different.

We conclude this section by briefly discussing the problem of usinga finite number of data. This note mainly deals with asymptotic re-sults, even if (16) is used forN finite. To get a rough and preliminaryindication on its behavior whenN is small, in Fig. 4 the average fre-quency estimate and error variance obtained using (16) forN = 100,N = 1:000, N = 10:000, andN = 100:000 are displayed. The re-sults in Fig. 4 have been obtained using 100 different uncorrelated (ob-viously for the noisy part only) realizations of the signaly. It is apparentthat both the bias and the variance errors rapidly decrease whenN getslarge. A comparison—for finite (and low) values ofN—between thisestimation algorithm and other estimation algorithms goes out of thescope of this note and might be the subject of future work.

IV. RELATED FREQUENCY-ESTIMATION METHODS

In the literature, other closed-form frequency estimators for har-monic signals in white noise have been proposed and analyzed. Twocelebrated estimators are the YW estimator, and the “ PHD estimator(see, e.g., [1], [7], [16], [19], [20], [23], and [24]). In this section, theywill be briefly recalled and compared with the asymptotic version ofthe notch-based estimator (16).

A. YW Approach

Given a signaly(t) = s(t) + n(t), wheres(t) = A cos(0t+ '),n � WGN(0; �2), the autocorrelation coefficients of order 1 and 2,sayr1 andr2, respectively, are given by

r1 = E [y(t)y(t� 1)] = A

2cos(0)

r2 = E [y(t)y(t� 2)] = A

2cos(20)

: (17)

By eliminating the parameterA in (17), the following equation isobtained:

2r1 cos2(0)� r2 cos(0)� r1 = 0:

Its solution with respect to0 provides the YW frequency estimator,given by

YW = arccosr2 + r22 + 8r21

4r1: (18)

B. PHD Approach

Given a zero-mean stationary signaly(t), its autocorrelation matrixof order three is given by

R =

r0 r1 r2

r1 r0 r1

r2 r1 r0

r0 = E y(t)2

r1 =E [y(t)y(t� 1)] r2 = E [y(t)y(t� 2)] :

The eigenvector associated with the smallest eigenvalue ofR has thefollowing form:

1 � r +pr +8r

2r1

T

: (19)

Pisarenko (see [16] and the analysis proposed in [7]) has proven that,if y(t) = s(t)+n(t) (s(t) = A cos(0t+'),n �WGN(0; �2), thesmallest eigenvalue ofR must have the following simple expression:

[ 1 �2 cos(0) 1 ]T : (20)

By comparing (19) and (20), the PHD frequency estimator is ob-tained

PHD = arccosr2 + r22 + 8r21

4r1: (21)

dJ3()

d=

N

t=1

2 (y(t)� 2 cos()y(t� 1) + y(t� 2)) ((2 + cos(2))2 sin()y(t� 1) + sin(2) (y(t)� 2 cos()y(t� 1) + y(t� 2)))

(2 + cos(2))2:

(13)

3 = arccos

�� N

t=12y(t�1)2�(y(t)+y(t�2))2 + N

t=12y(t�1)2�(y(t)+y(t� 2))2

2

+8 N

t=1y(t�1) (y(t)+y(t�2))

2

4 N

t=1y(t�1) (y(t)+y(t�2))

:

(16)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003 1291

Fig. 4. Average frequency estimate and error variance for different values ofN (A = 1, � = 1, = �=6).

Interestingly enough, the YW and PHD approaches provide exactlythe same results. This has been recently proven and discussed in [23]and [24].

Consider now the notch-based closed-form estimator (16). The fol-lowing result holds.

Proposition 1: Given a signaly(t) = s(t) + n(t), wheres(t) =A cos(0t + '), n � WGN(0; �2), the notch-filter based estimator(16) asymptotically converges towardYW andPHD, namely

limN!1

3 = arccosr2 + r2

2+ 8r2

1

4r1

r1 =E [y(t)y(t� 1)] r2 = E [y(t)y(t� 2)] :

Proof: If N is large, the following hold:

limN!1

1

N

N

t=1

y(t� 1) (y(t) + y(t� 2))

= limN!1

1

N

N

t=1

2y(t)y(t� 1)

= 2E [y(t)y(t� 1)] ; (22a)

limN!1

1

N

N

t=1

2y(t� 1)2 � (y(t) + y(t� 2))2

= �2E [y(t)y(t� 2)] : (22b)

By plugging in (16) the asymptotic expressions (22), it is easy to seethat

limN!1

3 = arccosr2 + r2

2+ 8r2

1

4r1

r1 =E [y(t)y(t� 1)] r2 = E [y(t)y(t� 2)] :

From a theoretical point of view the fact that (asymptotically)YW,PHD and3 are exactly the same, is particularly interesting: it showsthe equivalence of three classical approaches which have been indepen-dently conceived and developed following three completely differentpaths.

V. CONCLUSION AND FUTURE WORK

In this work, a notch FIR filter which provides and unbiasedclosed-form frequency estimate of harmonic signals in white noisehas been proposed. This estimator has been proven to convergeasymptotically to the well-known YW and PHD estimators.

These three equivalent estimators are very appealing since theyadmit a very simple closed-form expression starting from a set ofmeasured data. However, their main flaw is the sensitivity to the noise:they guarantee an unbiased estimate if the noise is white; when thenoise is colored, the bias error can be large.

In order to try to overcome this pitfall, probably the most interestingrationale is that proposed by Quinn and Fernandes in [15]: the idea is

1292 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 7, JULY 2003

to prefilter the data with a frequency enhancer, in an iterative fashion;at each step the frequency enhancer is centered around the frequencyestimated at the previous step. This approach seems to fit perfectly tosimple closed-form estimators.

The reduction of the noise-sensitivity of FIR-based frequency esti-mators using frequency enhancers is currently the subject of furtherinvestigation.

REFERENCES

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[2] K. J. Åström,Introduction to Stochastic Control Theory. New York:Academic, 1970.

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[4] S. Bittanti and G. Picci, Eds.,Identification, Adaptation, Learning—TheScience of Learning Models from Data. Berlin, Germany: Springer-Verlag, 1996, Computer and Systems Sciences.

[5] S. Bittanti and S. M. Savaresi, “On the parametrization and design of anextended Kalman filter frequency tracker,”IEEE Trans. Automat. Contr.,vol. 45, pp. 1718–1724, Sept. 2000.

[6] B. Boashash, “Estimating and interpreting the instantaneous frequencyof a signal,”Proc. IEEE, vol. 80, pp. 520–568, Apr. 1992.

[7] A. Eriksson and P. Stoica, “On statistical analysis of Pisarenko tone fre-quency estimator,”Signal Processing, vol. 31, pp. 349–353, 1993.

[8] P. Händel and A. Nehorai, “Tracking analysis of an adaptive notch filterwith constrained poles and zeros,”IEEE Trans. Signal Processing, vol.42, pp. 281–291, Feb. 1994.

[9] P. Händel, P. Tichavsky, and S. M. Savaresi, “Large error recovery fora class of frequency tracking algorithms,”Int. J. Adapt. Control SignalProcessing, vol. 12, pp. 417–436, 1998.

[10] L. Hsu, R. Ortega, and G. Damm, “A globally convergent frequency es-timator,” IEEE Trans. Automat. Contr., vol. 44, pp. 698–713, Apr. 1999.

[11] S. M. Kay, Modern Spectral Estimation: Theory and Applica-tions. Upper Saddle River, NJ: Prentice-Hall, 1988.

[12] B. La Scala and R. Bitmead, “Design of an extended Kalman filter fre-quency tracker,”IEEE Trans. Signal Processing, vol. 44, pp. 739–742,Mar. 1996.

[13] B. La Scala, R. Bitmead, and B. G. Quinn, “An extended Kalman filterfrequency tracker for high-noise environments,”IEEE Trans. SignalProcessing, vol. 44, pp. 431–434, Feb. 1996.

[14] G. Li, “A stable and efficient adaptive notch filter for direct frequencyestimation,”IEEE Trans. Signal Processing, vol. 45, pp. 2001–2009,Aug. 1997.

[15] B. G. Quinn and J. M. Fernandes, “A fast efficient technique for theestimation of frequency,”Biometrika, vol. 78, pp. 489–497, 1991.

[16] V. F. Pisarenko, “The retrieval of harmonics from a covariance function,”J. Roy. Astr., vol. 33, pp. 374–376, 1973.

[17] S. M. Savaresi, “Funnel filters: a new class of filters for frequency esti-mation of harmonic signals,”Automatica, vol. 33, no. 9, pp. 1711–1718,1997.

[18] S. M. Savaresi, R. Bitmead, and W. Dunstan, “Nonlinear system identi-fication using closed-loop data with no external excitation: the case of alean combustion process,”Int. J. Control, vol. 74, no. 18, pp. 1796–1806,2001.

[19] H. C. So, “A closed form frequency estimator for a noisy sinusoid,” in45th IEEE Midwest Symp. Circuits Systems, Tulsa, OK, 2002.

[20] H. C. So and S. K. Ip, “A novel frequency estimator and its comparativeperformances for short record lengths,” in11th Eur. Signal ProcessingConf., Toulouse, France, 2002.

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Comments on “A Robust State Observer Scheme”

M. Boutayeb and M. Darouach

Abstract—In this note, we point out that the sufficient conditions given ina recent paper to assure convergence of the proposed robust state observerare incomplete.

Index Terms—Robust observer design, stability analysis, uncertain sys-tems.

I. INTRODUCTION

In [2], a robust state observer scheme was proposed for uncertainlinear systems. The main result and theorem may be summarized asfollows. Consider the linear map

�1

_x = A0 +�A x+Bu

y = Cx+Du

where�A represents the model uncertainty, assumed to be boundedby � > 0 with k�Ak2 < �.

The authors propose the following observer dynamics:

�2

_x = A0x +Bu+H(y � y) + �

y = Cx+Du:

Theorem: We are given the uncertain system�1 and the state ob-server�2. If we set

� =�2xT x

2rT rP�1CTr; r = y � y

whereP > 0 satisfies the following matrix inequality:

(A0 �HC)TP + P (A0 �HC) + 2P 2 + �2I < 0

andH , such that(A0 �HC) is stable, then the state error vectore =x� x converges to zero. <

Manuscript received June 7, 2002; revised September 27, 2002, January 10,2003, and January 17, 2003. Recommended by Associate Editor M. E. Valcher.

M. Boutayeb is with the LSIIT-CNRS UMR 7005, University of LouisPasteur, 67400 Illkirch, France (e-mail: Mohamed.Boutayeb@ipst.ulp.u-strasbg.fr).

M. Darouach is with the CRAN-CNRS UMR 7039, Universityof Henri Poincaré, IUT of Longwy 54400, France (e-mail: Mo-hamed.Darouach@iut-longwy.uhp-nancy.fr).

Digital Object Identifier 10.1109/TAC.2003.812795

0018-9286/03$17.00 © 2003 IEEE