Identification of Frequency of Biased Harmonic ... - Semantic Scholar
European Journal of Control (2010)2:129–139 # 2010 EUCA DOI:10.3166/EJC.16.129–139
Identification of Frequency of Biased Harmonic Signal Stanislav Aranovskiy1, Alexey Bobtsov1,, Artem Kremlev1,1, Nikolay Nikolaev1, Olga Slita2 1
Department of Control Systems and Informatics, Saint-Petersburg State University of Information Technologies Mechanics and Optics, Kronverkski av. 49, 197101, Saint-Petersburg, Russia; 2 Department of Mechatronics and Robotics, Baltic State Technical University, 1-st Krasnoarmeiskaya st. 1, 190008, Saint-Petersburg, Russia
We consider a problem of identification of unknown frequency of a biased sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ with bounded disturbance or harmonic noise ðtÞ. A new proposed approach to estimation of frequency of biased sinusoidal signal is robust with regard to unaccounted disturbances, which are present in measurement of effective signal. Unlike known analogs, this approach allows to regulate time of estimation of unknown frequency !. The proposed approach also allows online amplitude and bias estimation. The proposed identification algorithm has smaller dimension than other known analogs. Keywords: Identification, Harmonic signal, Robustness, Estimation, Disturbance
1. Introduction We consider the problem of frequency identification of sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ for any unknown constant values 0 , , and bounded disturbance ðtÞ. Problem of frequency identification of a sinusoidal signal is a very important basic problem, which has different applications in theoretical and engineering disciplines, for instance, in rotational mechanical processes like induction motor, in active noise and vibration control, in helicopters, disk drivers and magnetic bearings [4, 8, 16, 26]. The estimation problem is an important problem in systems theory with applications in diverse fields. Most *Correspondence to: A. Bobtsov, E-mail: [email protected]
of the existing solutions have been sought from the perspective of signal processing and/or telecommunication: line enhancers [23], finite impulse response filters [21], infinite impulse response filters or notch filters [14, 18, 19] and frequency locket loop [11]. The reading of data in magnetic and optical disk drivers requires a periodic movement of the reading device, due to the eccentricity of the tracks. The effect of disturbances originating from external sources must be reduced in active noise and vibration control [5]. These sources are often rotating machines producing periodic noise and/or vibration [6]. The problem of active noise and vibration control in system, described by equation yðsÞ ¼ PðsÞðuðsÞ þ dðsÞÞ, where uðsÞ and dðsÞ are Laplace transform of the controller output and disturbance signals, is of particular interest. In these systems, dðtÞ is an offending noise or vibration source and PðsÞ is the transfer function of outputactuator-to error-sensor propagation path [18]. Often, the noise source consists mainly of periodic components due to rotating machinery generating the undesired noise signal. Examples of such noises include engine noise in turboprop aircraft, engine noise in automobiles and ventilation noise in HVAC system. There are few approaches how to solve the problem of disturbance attenuation – indirect approach (see Fig. 1) and direct approach. In order to use indirect approach, it is necessary to know frequency of disturbing influence. In practice, the frequency of disturbance is usually not known and may even vary during operation. It is often impossible Received 2 October 2008; Accepted 9 September 2009 Recommended by S.M. Savaresi, A.J. van der Schaft
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online amplitude and bias estimation was not discussed. 6. Algorithm proposed in this paper has robust properties with regard to unaccounted disturbances. In the papers [2, 7, 8, 10, 16, 24], the problem of robustness and noise rejection was not widely discussed.
2. Problem Statement Fig. 1. Indirect approach.
Consider measured signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ;
to use sensors measuring frequency of disturbing influence, so the problem of frequency identification of unknown harmonic signal arises. There are many different approaches dedicated to identification of unknown frequency of a sinusoidal function, see [1–3, 7, 8, 10, 12, 13, 15, 16, 20, 22, 24, 25]. Let us note that modern approaches to identification of parameter ! > 0 are not limited with studying the case of a single sinusoid, see [1, 13, 15, 19, 20]. In particular, papers [18, 21, 24] consider problem of frequency identification of a biased sinusoidal signal, and papers [2, 14, 16, 17] present common case of harmonic signal, which is a sum of n sinusoidal functions with different frequencies. This approach has the following advantages: 1. Algorithm proposed in this paper for bias sinusoidal signal has dynamic order equal to three, which is less than the dynamic order in most results, published in [2, 14, 16–18, 24]. In [2, 14, 17, 18, 24] minimal dimension of dynamic order of the algorithm is four, and in [16] dimension of the algorithm amounts to nine. 2. It requires fewer multiplications than the algorithms in [7, 24]. 3. Algorithm proposed in this paper has simpler structure than [7, 15, 17] and requires smaller number of parameters for the realization of the frequency estimator. 4. Algorithm of identification, proposed in the given paper allows to regulate rate of convergence of tuned parameter (estimation of frequency of signal yðtÞ ¼ 0 þ sinð! t þ Þ). In the papers [1–3, 7, 8, 10, 12, 13, 16, 20, 22, 24, 25], the problem of regulated rate of convergence of tuned parameter was not discussed. 5. As in [7], the approach proposed in this paper may be extended for the task of online amplitude and bias estimation (see remark 4). In the papers [1–3, 8, 10, 12, 13, 16, 20, 22, 24, 25], the problem of
ð1Þ
which is a biased sinusoid with unknown bias 0 and amplitude , unknown frequency ! and unknown phase , ðtÞ is bounded disturbance. The purpose is a design of identification algorithm, which would ensure realization of conditions ^ j ¼ 0 for ðtÞ ¼ 0; lim j! !ðtÞ
1Þ
t!1
^ j 0 < 1 for ðtÞ 6¼ 0; lim j! !ðtÞ
2Þ
t!1
ð2Þ ð2aÞ
^ is a current estimation of parameter ! for where !ðtÞ any 0, , , ! > 0 and the number 0 represents a certain domain depending on the disturbance amplitude ðtÞ. Assumption: In the identification of the parameter ! under the effect of the disturbance ðtÞ, we assume that the amplitude of the signal ðtÞ is less than the amplitude of the useful signal yðtÞ ¼ 0 þ sinð! t þ Þ.
3. Main Result It is known that for generating signal (1) it is possible to use differential equation :::
:
:
y ðtÞ ¼ !2 yðtÞ ¼ yðtÞ;
ð3Þ
where ¼ !2 is a constant parameter. Lemma: Consider an auxiliary second-order filter 8 < &_1 ðtÞ ¼ &2 ðtÞ; &_ ðtÞ ¼ 2&2 ðtÞ 2 &1 ðtÞ þ yðtÞ; ð4Þ : 2 &ðtÞ ¼ &1 ðtÞ or &ðtÞ ¼
1 ðp þ Þ2
yðtÞ;
ð5Þ
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where p is differentiation operator and number > 0. Then differential equation (3) can be rewritten in the form _ ¼ 2€ _ þ &ðtÞ _ þ "y ðtÞ; yðtÞ & ðtÞ þ &ðtÞ 2
ð6Þ
where "y ðtÞ is exponentially decaying function of time caused by nonzero initial conditions. Proof: After Laplace transform of equation (3) we obtain sYðsÞ ¼
s
YðsÞ þ 2
ðs þ Þ DðsÞ þ ; ðs þ Þ2
2s2 þ 2 s ðs þ Þ2
YðsÞ ð7Þ
_ ¼ yðtÞ
ðp þ Þ
yðtÞ þ 2
Proposition 1: Let algorithm of identification of unknown parameter have the view _^ ^ ðtÞ ¼ k&_2 ðtÞð ðtÞÞ where number k > 0, and function &ðtÞ is solution of differential equation (4). Then purpose of (2) is achieved. Proof of the proposition 1. Consider estimation error of parameter of the following form ~ ¼ ðtÞ: ^ ðtÞ
ð10Þ
After differentiation of equation (10) we have
where s is complex variable, YðsÞ ¼ LfyðtÞg is Laplace image of signal y(t), and polynomial DðsÞ denotes sum of all terms, containing nonzero initial conditions. From equation (7) we find p
The following statement proves efficiency of ideal identification algorithm for achieving purpose (2).
2p2 þ 2 p ðp þ Þ2
yðtÞ þ "y ðtÞ; ð8Þ
where exponentially decaying function of time "y ðtÞ ¼ L1 fDðsÞ=ðs þ Þ2 g is determined by nonzero initial conditions. Substituting (5) into equation (8) we obtain _ ¼ 2€ _ þ &ðtÞ _ þ "y ðtÞ; yðtÞ & ðtÞ þ 2 &ðtÞ
_^ _ ~ ¼ k&_2 ðtÞðtÞ: ~ ~ ¼ _ ðtÞ ¼ 0 k&_2 ðtÞðtÞ ð11Þ Solving differential equation (11) we obtain ~ ¼ ðt ~ 0 Þekðt;t0 Þ ðtÞ
ð12Þ
where Zt &_2 ðÞd:
ðt; t0 Þ ¼
ð13Þ
t0
It is obvious that as polynomial ðp þ Þ2 is Hurwitz, function &ðtÞ takes the form þ &ðtÞ ¼ 0 þ sinð! t þ Þ
which was to be proved. Remark 1: As exponentially decaying function "y ðtÞ ¼ L1 fDðsÞ=ðs þ Þ2 g depends on parameter , it is possible to accelerate convergence of "y ðtÞ to zero by increasing . Now, based on lemma results we can formulate scheme of unknown parameter identification. First, _ let us suppose that function yðtÞ is measured. Then, neglecting exponentially decaying item "y ðtÞ, ideal identification algorithm can be written in the following way _^ ^ ^ _ zðtÞ k&_2 ðtÞðtÞ; ¼ k&ðtÞ ðtÞ ¼ k&_2 ðtÞð ðtÞÞ rffiffiffiffiffiffiffiffiffiffiffi ^ ^ ¼ ðtÞ !ðtÞ ; ð9Þ _ 2€ _ and where function zðtÞ ¼ yðtÞ & ðtÞ 2 &ðtÞ number k > 0.
ð14Þ
where 0 , and are constant coefficients depending on parameters of signal yðtÞ ¼ 0 þ sinð! t þ Þ and number , and is an exponentially decaying item, caused by transients. Neglecting and differentiating (14) we obtain _ ¼ ! cosð! t þ Þ &ðtÞ into (13) we have _ ¼ ! cosð! t þ Þ Substituting &ðtÞ Zt
Zt &_ ðÞd ¼ !
ðt; t0 Þ ¼
2
2 2
t0
2 d ðcosð! þ ÞÞ
t0
! t ! t0 2 !2 sinð2! t þ 2Þ þ 2 4! 2 2 !2 sinð2! t0 þ 2Þ 4! ¼ 0 t þ 1 ðt; t0 Þ ¼
2 2
2 2
ð15Þ
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8 < &_1 ðtÞ ¼ &2 ðtÞ; &_ ðtÞ ¼ 2&2 ðtÞ 2 &1 ðtÞ þ yðtÞ; : 2 &ðtÞ ¼ &1 ðtÞ:
where function 2 !2 t0 2 !2 sinð2! t þ 2Þ 1 ðt; t0 Þ ¼ þ 4! 2 2 !2 sinð2! t0 þ 2Þ 4!
Proposition 2: Algorithm (19)–(21) is stable and ~ ¼ 0 for ðtÞ ¼ 0. lim ðtÞ t!1 ~ ¼ Proof of the proposition 2. Consider function ðtÞ ^ ðtÞ (see equation (10)). From (20) we have
2 ! 2 . 2 Let us substitute (15) into (12)
is bounded for any t, and 0 ¼
~ ¼ ðt ~ 0 Þek0 t ek1 ðt;t0 Þ : ðtÞ
ð16Þ
It follows from equation (16) that lim ~ ¼ 0, and t!1 rffiffiffiffiffiffiffiffiffiffiffi ^ ^ ¼ ðtÞ hence !ðtÞ ! !ðtÞ for t ! 1. Proposition 1 is proven. Remark 2: It follows from equation (16) that function ^ ðtÞ converges faster to parameter by increasing coefficient k. It means that it is possible to decrease or increase rate of convergence of the tuned parameter to its real value in identification algorithm (9) by changing coefficient k. Remark 3: It follows from equation (16) that system (11) is exponentially stable. In its turn it ensures robustness of identification algorithm with respect to external disturbances. However, in our case signal y(t) is only measured but not its derivatives. To derive realizable scheme of identification algorithm let us consider the following variable ^ k&ðtÞyðtÞ: _ ðtÞ ¼ ðtÞ
ð21Þ
ð17Þ
~ ¼ ðtÞ k&ðtÞyðtÞ: _ ðtÞ Differentiating last equation we obtain _~ _ _ k€ _ yðtÞ ðtÞ ¼ ðtÞ & ðtÞyðtÞ k&ðtÞ _ _ þ k&_2 ðtÞð ðtÞ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞÞ _ _ _ yðtÞ þ k&ðtÞyðtÞ þ k€ & ðtÞyðtÞ k€ & ðtÞyðtÞ k&ðtÞ _ _ _ &ðtÞ ðtÞ _ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞ k&_2 ðtÞyðtÞ _ þ yðtÞÞ; _ 2€ _ ¼ zðtÞ ¼ &ðtÞ. _ where yðtÞ & ðtÞ 2 &ðtÞ So we obtain _~ _ _ &ðtÞ ðtÞ _ ðtÞ ¼ k&ðtÞð &ðtÞ k&_2 ðtÞyðtÞÞ ~ ¼ k&_2 ðtÞðtÞ; where equilibrium position ~ ¼ 0 is exponentially stable (see proof proposition 1). Proposition 2 is proven. Remark 4: Let us notice that proposed approach also allows online amplitude and bias estimation. From equations (4) and (6) we obtain ^ &_1 ðtÞ þ 2&_2 ðtÞ þ 2 &_1 ðtÞ;
1 ðtÞ ¼ ðtÞ
Differentiating equation (17) we obtain _^ _ _ _ yðtÞ ðtÞ ¼ ðtÞ k€ & ðtÞyðtÞ k&ðtÞ ^ _ 2€ _ _ ¼ k&ðtÞð yðtÞ & ðtÞ 2 &ðtÞÞ k&_2 ðtÞðtÞ _ _ yðtÞ k€ & ðtÞyðtÞ k&ðtÞ 2 ^ _ _ k&_2 ðtÞðtÞ ¼ k&ðtÞð2€ & ðtÞ &ðtÞÞ k€ & ðtÞyðtÞ: ð18Þ From equations (17) and (18) we receive realizable identification algorithm ^ _ _ _ k&_2 ðtÞðtÞ ðtÞ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞÞ k€ & ðtÞyðtÞ; ð19Þ
ð22Þ
_ where 1 ðtÞ is a current estimation of yðtÞ. Differentiating equation (6) for case "y ðtÞ ¼ 0 we obtain &1 ðtÞ y€ðtÞ ¼ 2€ &2 ðtÞ þ 2 &€1 ðtÞ þ € _ þ 2 &€1 ðtÞ þ € &1 ðtÞ: ¼ 2ð2&_2 ðtÞ 2 &_1 ðtÞ þ yðtÞÞ Substituting (6) into last equation we have y€ðtÞ ¼ 2ð2&_2 ðtÞ 2 &_1 ðtÞ þ 2&_2 ðtÞ þ 2 &_1 ðtÞ &1 ðtÞ þ &_1 ðtÞÞ þ 2 &€1 ðtÞ þ € ¼ 2 &_1 ðtÞ þ 2 &_2 ðtÞ þ &_2 ðtÞ: From last equation, we obtain
^ ¼ ðtÞ þ k&ðtÞyðtÞ; _ ðtÞ
ð20Þ
^ &_1 ðtÞ þ 2 &_2 ðtÞ þ ðtÞ ^ &_2 ðtÞ;
1 ðtÞðtÞ ¼ 2ðtÞ
ð23Þ
133
IFBHS
where 2 ðtÞ is a current estimation of y€ðtÞ. Let us consider the following variables y€ðtÞ ¼ sinð! t þ ’Þ; ^ 6¼ 0: ^ ¼ 2 ðtÞ for ðtÞ ðtÞ ^ ðtÞ
Consider scaled normalized adaptive notch filter (ANF) see for instance [8, 10, 12, 13, 24]
ðtÞ ¼
ð24Þ
From equations (22) and (24) we can receive realizable online amplitude and bias estimator sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðtÞ ^2 ^ðtÞ ¼ ð25Þ þ ðtÞ; ^ ðtÞ ^ for ðtÞ ^ 6¼ 0: ^0 ðtÞ ¼ yðtÞ ðtÞ
ð26Þ
Remark 5: In the papers [2,8,10,12,16], the problem of robustness and noise rejection was not widely discussed. However, in the paper [10] robustness according to [9] is considered. If the signal yðtÞ ¼ 0 þ sinð! t þ Þ is corrupted by additional bounded disturbance, parameter adaptation algorithm in (19) may be modified according to [1,9,17], in order to prevent parameter estimates drift ^ _ _ _ ðtÞ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞÞ k&_2 ðtÞðtÞ ^ k€ & ðtÞyðtÞ ka ðtÞ; ^ ¼ ðtÞ þ k&ðtÞyðtÞ; _ ðtÞ where ka 0. The modification of identification algorithm (19) assumes that outside bounded region in the statespace the derivative of Lyapunov function becomes negative. The main drawback is that a bias term is added to the parameter update equation, therefore zero residual error cannot be guaranteed when the disturbances are removed. In this case we can use the following approach [1] 8 ^ > > 0; < 0 ; > > > 0 1 > > < ^ ^ ka ¼ ka ðÞ ¼ @ 1A; 0 ^ 20 ; > > > 0 > > > ^ > : 1; > 20 ; where 0 is strictly positive constant.
4. Example Consider identification algorithm proposed in this paper and compare it with adaptive notch filter.
^x_ þ ! ^2 x ¼ ! ^2 y; x€ þ 2& !
ð27Þ
^ !; ^ !^_ ¼ ð2& x_ !yÞx
ð28Þ
¼(
" ; 2 #) _ x ^j g 1 þ N x2 þ f1 þ j! ^ ! "
ð29Þ
where – estimation of the frequency, y – measurable harmonic signal, & > 0 – is the damping coefficient, "; N; > 0 and 1. Let us compare ANF algorithm with algorithm proposed in this paper: – dynamic order of algorithm proposed in the given paper and ANF algorithm (27)–(29) are equal to three; – algorithm (19)–(21) has two unknown parameters in comparison with five unknown parameters of ANF algorithm, so tuning of proposed identification algorithm is much simpler; – proposed identification algorithm allows to estimate not only frequency of unknown harmonic signal but also gives estimation of amplitude and bias of identified signal, see equations (25) and (26). Let us simulate comparable algorithms. Simulation of algorithm (27)–(29) was done for the following values of constant parameters & ¼ 0:4, " ¼ 1:35, N ¼ 1, > 0:5 and ¼ 2, initial conditions _ xð0Þ ¼ 1, xð0Þ ¼ 1, ð0Þ ¼ 10, see [1]. Simulation of algorithm (27)–(29) was done for ¼ 1, k ¼ 1 and zero initial conditions. Let us consider problem of frequency identification signal yðtÞ ¼ sinð! tÞ for ¼ 10, ! ¼ 10. Fig. 2 shows simulation results of ANF and proposed simulation algorithms for yðtÞ ¼ 10 sinð10 tÞ. Let us consider problem of frequency identification signal yðtÞ ¼ sinð! tÞ for ¼ 50, ! ¼ 50. Fig. 3 shows simulation results of ANF and proposed algorithm for yðtÞ ¼ 50 sinð50 tÞ. As Figs. 2 and 3 show, both algorithms are efficient for different values of amplitude and frequency of harmonic signal. Varying of identification algorithm parameters can change rate of estimation of unknown frequency. Let us consider problem of frequency identification of biased harmonic signal yðtÞ ¼ 0 þ sinð! tÞ for 0 ¼ 10, ¼ 10, ! ¼ 10. Fig. 4 shows simulation results of ANF and proposed algorithm for yðtÞ ¼ 10 þ 10 sinð10 tÞ.
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Fig. 2. (a) ANF (27)–(29), (b) identification algorithm (19)–(21).
Fig. 3. (a) ANF (27)–(29), (b) identification algorithm (19)–(21).
Fig. 4. (a) ANF (27)–(29), (b) identification algorithm (19)–(21).
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As Fig. 4 shows, ANF algorithm is not efficient for biased harmonic signal. Figs. 5–10 show simulation results of identification algorithm (19)–(26) proposed in the paper. Figs. 5 and 6 show simulation results of the proposed algorithm for yðtÞ ¼ 10 þ 10 sinð10 tÞ. Let us consider problem of frequency identification of biased harmonic signal yðtÞ ¼ 0 þ sinð! tÞ for 0 ¼ 3, ¼ 2, ! ¼ 4. Figs. 7 and 8 show simulation results of proposed algorithm for yðtÞ ¼ 3þ 2 sinð4 tÞ. Let us consider problem of frequency identification of harmonic signal yðtÞ ¼ sinð! tÞ for ¼ 10, ^ and ! ¼ 10. Fig. 9 shows graphs of parameters ðtÞ ^ tuning for the sinusoidal signal yðtÞ ¼ 10 sin 10t !ðtÞ disturbed by the function ðtÞ ¼ 0:3 sinð6t þ 0:5Þ
^ ^ (b) function . Fig. 5. (a) frequency estimation, !;
Fig. 6. (a) bias estimation, ^0 ; (b) amplitude estimation ^.
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þ0:2 sinð40t 0:8Þ þ 0:03wðtÞ, where wðtÞ is a white noise. Computer simulation illustrates that robustness properties keep safe with respect to unaccounted disturbances (see Remark 3). Let us compare approaches [7,16] with algorithm proposed in this paper. Consider the sinusoidal signal yðtÞ ¼ sin t disturbed by function ðtÞ ¼ 0:05 sinð10tÞ and simulate comparable algorithms for different values of parameter . The results of computer simulation of approach [16] for parameters ¼ 10, ¼ k1 ¼ k2 ¼ & ¼ 1 (see page 863 and equations (3)–(6) from [16]) are presented in Fig. 11. We can see that for additional bounded disturbance the approach [16] ensures boundedness of all variables of the algorithm but parametrical convergence is not good.
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^ ^ (b) function . Fig. 7. (a) frequency estimation, !;
Fig. 8. (a) bias estimation, ^0 ; (b) amplitude estimation ^.
^ ^ (b) function . Fig. 9. (a) frequency estimation, !;
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The results of computer simulation approach [7] for parameters ¼ ¼ 1 (see page 856 and equation (24)–(26) from [7]) are presented in Fig. 12. We can see that for additional bounded disturbance the approach [7] ensures boundedness of all variables of the algorithm but parametrical convergence is bad. The results of computer simulation of this approach for parameters k ¼ ¼ 1 are presented in Fig. 13. We can see that parametrical convergence for this approach is better than in [7,16].
Fig. 10. The sinusoidal signal yðtÞ ¼ 10 sin 10t disturbed by function ðtÞ ¼ 0:3 sinð6t þ 0:5Þ þ 0:2 sinð40t 0:8Þ þ 0:03wðtÞ, where wðtÞ is a white noise.
^ for the case: (a) ¼ 5, (b) ¼ 10. Fig. 11. Transient for variable !
^ for the case: (a) ¼ 5, (b) ¼ 10. Fig. 12. Transient for variable !
Remark 6: The proposed estimation algorithm has an exponentially stable equilibrium point if one considers the undisturbed system. We have good simulation results for periodical disturbance (see Fig. 13). But unfortunately that is not the case if there is a periodic disturbance. In that case, there is a new equilibrium point that appears in the system. The attraction region of this equilibrium point is proportional to its amplitude
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^ for the case: (a) ¼ 5, (b) ¼ 10. Fig. 13. Transient for variable !
and its frequency. For this reason, even if its amplitude is small, it still will have an attraction region, and as a consequence, the desired equilibrium point is no more global. It will have an attraction region, and convergence will depend on initial conditions (see for example [8]). We leave this case as future works.
5. Conclusion We considered the problem of identification of frequency of a sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ for any unknown constant values 0 , , , ! > 0 and bounded disturbance or harmonic noise ðtÞ. Designed algorithm of identification (19)–(21): – is stable with regard to unaccounted disturbances presenting in measurement of effective biased sinusoidal signal; – allows accelerating rate of convergence of estimate ^ to because of increasing coefficient k (see ðtÞ Remarks 1 and 2); – has an ability of extension (22)–(26) for online estimation of amplitude and bias with no increase of dynamic order (see Remark 4 and example); – has the least dynamic order in comparison with works [1,7,10,12,16,24].
Acknowledgment This work was supported in part by RFBR Grant No. 09-08-00139.
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Identification of Frequency of Biased Harmonic Signal Stanislav Aranovskiy1, Alexey Bobtsov1,, Artem Kremlev1,1, Nikolay Nikolaev1, Olga Slita2 1
Department of Control Systems and Informatics, Saint-Petersburg State University of Information Technologies Mechanics and Optics, Kronverkski av. 49, 197101, Saint-Petersburg, Russia; 2 Department of Mechatronics and Robotics, Baltic State Technical University, 1-st Krasnoarmeiskaya st. 1, 190008, Saint-Petersburg, Russia
We consider a problem of identification of unknown frequency of a biased sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ with bounded disturbance or harmonic noise ðtÞ. A new proposed approach to estimation of frequency of biased sinusoidal signal is robust with regard to unaccounted disturbances, which are present in measurement of effective signal. Unlike known analogs, this approach allows to regulate time of estimation of unknown frequency !. The proposed approach also allows online amplitude and bias estimation. The proposed identification algorithm has smaller dimension than other known analogs. Keywords: Identification, Harmonic signal, Robustness, Estimation, Disturbance
1. Introduction We consider the problem of frequency identification of sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ for any unknown constant values 0 , , and bounded disturbance ðtÞ. Problem of frequency identification of a sinusoidal signal is a very important basic problem, which has different applications in theoretical and engineering disciplines, for instance, in rotational mechanical processes like induction motor, in active noise and vibration control, in helicopters, disk drivers and magnetic bearings [4, 8, 16, 26]. The estimation problem is an important problem in systems theory with applications in diverse fields. Most *Correspondence to: A. Bobtsov, E-mail: [email protected]
of the existing solutions have been sought from the perspective of signal processing and/or telecommunication: line enhancers [23], finite impulse response filters [21], infinite impulse response filters or notch filters [14, 18, 19] and frequency locket loop [11]. The reading of data in magnetic and optical disk drivers requires a periodic movement of the reading device, due to the eccentricity of the tracks. The effect of disturbances originating from external sources must be reduced in active noise and vibration control [5]. These sources are often rotating machines producing periodic noise and/or vibration [6]. The problem of active noise and vibration control in system, described by equation yðsÞ ¼ PðsÞðuðsÞ þ dðsÞÞ, where uðsÞ and dðsÞ are Laplace transform of the controller output and disturbance signals, is of particular interest. In these systems, dðtÞ is an offending noise or vibration source and PðsÞ is the transfer function of outputactuator-to error-sensor propagation path [18]. Often, the noise source consists mainly of periodic components due to rotating machinery generating the undesired noise signal. Examples of such noises include engine noise in turboprop aircraft, engine noise in automobiles and ventilation noise in HVAC system. There are few approaches how to solve the problem of disturbance attenuation – indirect approach (see Fig. 1) and direct approach. In order to use indirect approach, it is necessary to know frequency of disturbing influence. In practice, the frequency of disturbance is usually not known and may even vary during operation. It is often impossible Received 2 October 2008; Accepted 9 September 2009 Recommended by S.M. Savaresi, A.J. van der Schaft
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online amplitude and bias estimation was not discussed. 6. Algorithm proposed in this paper has robust properties with regard to unaccounted disturbances. In the papers [2, 7, 8, 10, 16, 24], the problem of robustness and noise rejection was not widely discussed.
2. Problem Statement Fig. 1. Indirect approach.
Consider measured signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ;
to use sensors measuring frequency of disturbing influence, so the problem of frequency identification of unknown harmonic signal arises. There are many different approaches dedicated to identification of unknown frequency of a sinusoidal function, see [1–3, 7, 8, 10, 12, 13, 15, 16, 20, 22, 24, 25]. Let us note that modern approaches to identification of parameter ! > 0 are not limited with studying the case of a single sinusoid, see [1, 13, 15, 19, 20]. In particular, papers [18, 21, 24] consider problem of frequency identification of a biased sinusoidal signal, and papers [2, 14, 16, 17] present common case of harmonic signal, which is a sum of n sinusoidal functions with different frequencies. This approach has the following advantages: 1. Algorithm proposed in this paper for bias sinusoidal signal has dynamic order equal to three, which is less than the dynamic order in most results, published in [2, 14, 16–18, 24]. In [2, 14, 17, 18, 24] minimal dimension of dynamic order of the algorithm is four, and in [16] dimension of the algorithm amounts to nine. 2. It requires fewer multiplications than the algorithms in [7, 24]. 3. Algorithm proposed in this paper has simpler structure than [7, 15, 17] and requires smaller number of parameters for the realization of the frequency estimator. 4. Algorithm of identification, proposed in the given paper allows to regulate rate of convergence of tuned parameter (estimation of frequency of signal yðtÞ ¼ 0 þ sinð! t þ Þ). In the papers [1–3, 7, 8, 10, 12, 13, 16, 20, 22, 24, 25], the problem of regulated rate of convergence of tuned parameter was not discussed. 5. As in [7], the approach proposed in this paper may be extended for the task of online amplitude and bias estimation (see remark 4). In the papers [1–3, 8, 10, 12, 13, 16, 20, 22, 24, 25], the problem of
ð1Þ
which is a biased sinusoid with unknown bias 0 and amplitude , unknown frequency ! and unknown phase , ðtÞ is bounded disturbance. The purpose is a design of identification algorithm, which would ensure realization of conditions ^ j ¼ 0 for ðtÞ ¼ 0; lim j! !ðtÞ
1Þ
t!1
^ j 0 < 1 for ðtÞ 6¼ 0; lim j! !ðtÞ
2Þ
t!1
ð2Þ ð2aÞ
^ is a current estimation of parameter ! for where !ðtÞ any 0, , , ! > 0 and the number 0 represents a certain domain depending on the disturbance amplitude ðtÞ. Assumption: In the identification of the parameter ! under the effect of the disturbance ðtÞ, we assume that the amplitude of the signal ðtÞ is less than the amplitude of the useful signal yðtÞ ¼ 0 þ sinð! t þ Þ.
3. Main Result It is known that for generating signal (1) it is possible to use differential equation :::
:
:
y ðtÞ ¼ !2 yðtÞ ¼ yðtÞ;
ð3Þ
where ¼ !2 is a constant parameter. Lemma: Consider an auxiliary second-order filter 8 < &_1 ðtÞ ¼ &2 ðtÞ; &_ ðtÞ ¼ 2&2 ðtÞ 2 &1 ðtÞ þ yðtÞ; ð4Þ : 2 &ðtÞ ¼ &1 ðtÞ or &ðtÞ ¼
1 ðp þ Þ2
yðtÞ;
ð5Þ
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where p is differentiation operator and number > 0. Then differential equation (3) can be rewritten in the form _ ¼ 2€ _ þ &ðtÞ _ þ "y ðtÞ; yðtÞ & ðtÞ þ &ðtÞ 2
ð6Þ
where "y ðtÞ is exponentially decaying function of time caused by nonzero initial conditions. Proof: After Laplace transform of equation (3) we obtain sYðsÞ ¼
s
YðsÞ þ 2
ðs þ Þ DðsÞ þ ; ðs þ Þ2
2s2 þ 2 s ðs þ Þ2
YðsÞ ð7Þ
_ ¼ yðtÞ
ðp þ Þ
yðtÞ þ 2
Proposition 1: Let algorithm of identification of unknown parameter have the view _^ ^ ðtÞ ¼ k&_2 ðtÞð ðtÞÞ where number k > 0, and function &ðtÞ is solution of differential equation (4). Then purpose of (2) is achieved. Proof of the proposition 1. Consider estimation error of parameter of the following form ~ ¼ ðtÞ: ^ ðtÞ
ð10Þ
After differentiation of equation (10) we have
where s is complex variable, YðsÞ ¼ LfyðtÞg is Laplace image of signal y(t), and polynomial DðsÞ denotes sum of all terms, containing nonzero initial conditions. From equation (7) we find p
The following statement proves efficiency of ideal identification algorithm for achieving purpose (2).
2p2 þ 2 p ðp þ Þ2
yðtÞ þ "y ðtÞ; ð8Þ
where exponentially decaying function of time "y ðtÞ ¼ L1 fDðsÞ=ðs þ Þ2 g is determined by nonzero initial conditions. Substituting (5) into equation (8) we obtain _ ¼ 2€ _ þ &ðtÞ _ þ "y ðtÞ; yðtÞ & ðtÞ þ 2 &ðtÞ
_^ _ ~ ¼ k&_2 ðtÞðtÞ: ~ ~ ¼ _ ðtÞ ¼ 0 k&_2 ðtÞðtÞ ð11Þ Solving differential equation (11) we obtain ~ ¼ ðt ~ 0 Þekðt;t0 Þ ðtÞ
ð12Þ
where Zt &_2 ðÞd:
ðt; t0 Þ ¼
ð13Þ
t0
It is obvious that as polynomial ðp þ Þ2 is Hurwitz, function &ðtÞ takes the form þ &ðtÞ ¼ 0 þ sinð! t þ Þ
which was to be proved. Remark 1: As exponentially decaying function "y ðtÞ ¼ L1 fDðsÞ=ðs þ Þ2 g depends on parameter , it is possible to accelerate convergence of "y ðtÞ to zero by increasing . Now, based on lemma results we can formulate scheme of unknown parameter identification. First, _ let us suppose that function yðtÞ is measured. Then, neglecting exponentially decaying item "y ðtÞ, ideal identification algorithm can be written in the following way _^ ^ ^ _ zðtÞ k&_2 ðtÞðtÞ; ¼ k&ðtÞ ðtÞ ¼ k&_2 ðtÞð ðtÞÞ rffiffiffiffiffiffiffiffiffiffiffi ^ ^ ¼ ðtÞ !ðtÞ ; ð9Þ _ 2€ _ and where function zðtÞ ¼ yðtÞ & ðtÞ 2 &ðtÞ number k > 0.
ð14Þ
where 0 , and are constant coefficients depending on parameters of signal yðtÞ ¼ 0 þ sinð! t þ Þ and number , and is an exponentially decaying item, caused by transients. Neglecting and differentiating (14) we obtain _ ¼ ! cosð! t þ Þ &ðtÞ into (13) we have _ ¼ ! cosð! t þ Þ Substituting &ðtÞ Zt
Zt &_ ðÞd ¼ !
ðt; t0 Þ ¼
2
2 2
t0
2 d ðcosð! þ ÞÞ
t0
! t ! t0 2 !2 sinð2! t þ 2Þ þ 2 4! 2 2 !2 sinð2! t0 þ 2Þ 4! ¼ 0 t þ 1 ðt; t0 Þ ¼
2 2
2 2
ð15Þ
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8 < &_1 ðtÞ ¼ &2 ðtÞ; &_ ðtÞ ¼ 2&2 ðtÞ 2 &1 ðtÞ þ yðtÞ; : 2 &ðtÞ ¼ &1 ðtÞ:
where function 2 !2 t0 2 !2 sinð2! t þ 2Þ 1 ðt; t0 Þ ¼ þ 4! 2 2 !2 sinð2! t0 þ 2Þ 4!
Proposition 2: Algorithm (19)–(21) is stable and ~ ¼ 0 for ðtÞ ¼ 0. lim ðtÞ t!1 ~ ¼ Proof of the proposition 2. Consider function ðtÞ ^ ðtÞ (see equation (10)). From (20) we have
2 ! 2 . 2 Let us substitute (15) into (12)
is bounded for any t, and 0 ¼
~ ¼ ðt ~ 0 Þek0 t ek1 ðt;t0 Þ : ðtÞ
ð16Þ
It follows from equation (16) that lim ~ ¼ 0, and t!1 rffiffiffiffiffiffiffiffiffiffiffi ^ ^ ¼ ðtÞ hence !ðtÞ ! !ðtÞ for t ! 1. Proposition 1 is proven. Remark 2: It follows from equation (16) that function ^ ðtÞ converges faster to parameter by increasing coefficient k. It means that it is possible to decrease or increase rate of convergence of the tuned parameter to its real value in identification algorithm (9) by changing coefficient k. Remark 3: It follows from equation (16) that system (11) is exponentially stable. In its turn it ensures robustness of identification algorithm with respect to external disturbances. However, in our case signal y(t) is only measured but not its derivatives. To derive realizable scheme of identification algorithm let us consider the following variable ^ k&ðtÞyðtÞ: _ ðtÞ ¼ ðtÞ
ð21Þ
ð17Þ
~ ¼ ðtÞ k&ðtÞyðtÞ: _ ðtÞ Differentiating last equation we obtain _~ _ _ k€ _ yðtÞ ðtÞ ¼ ðtÞ & ðtÞyðtÞ k&ðtÞ _ _ þ k&_2 ðtÞð ðtÞ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞÞ _ _ _ yðtÞ þ k&ðtÞyðtÞ þ k€ & ðtÞyðtÞ k€ & ðtÞyðtÞ k&ðtÞ _ _ _ &ðtÞ ðtÞ _ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞ k&_2 ðtÞyðtÞ _ þ yðtÞÞ; _ 2€ _ ¼ zðtÞ ¼ &ðtÞ. _ where yðtÞ & ðtÞ 2 &ðtÞ So we obtain _~ _ _ &ðtÞ ðtÞ _ ðtÞ ¼ k&ðtÞð &ðtÞ k&_2 ðtÞyðtÞÞ ~ ¼ k&_2 ðtÞðtÞ; where equilibrium position ~ ¼ 0 is exponentially stable (see proof proposition 1). Proposition 2 is proven. Remark 4: Let us notice that proposed approach also allows online amplitude and bias estimation. From equations (4) and (6) we obtain ^ &_1 ðtÞ þ 2&_2 ðtÞ þ 2 &_1 ðtÞ;
1 ðtÞ ¼ ðtÞ
Differentiating equation (17) we obtain _^ _ _ _ yðtÞ ðtÞ ¼ ðtÞ k€ & ðtÞyðtÞ k&ðtÞ ^ _ 2€ _ _ ¼ k&ðtÞð yðtÞ & ðtÞ 2 &ðtÞÞ k&_2 ðtÞðtÞ _ _ yðtÞ k€ & ðtÞyðtÞ k&ðtÞ 2 ^ _ _ k&_2 ðtÞðtÞ ¼ k&ðtÞð2€ & ðtÞ &ðtÞÞ k€ & ðtÞyðtÞ: ð18Þ From equations (17) and (18) we receive realizable identification algorithm ^ _ _ _ k&_2 ðtÞðtÞ ðtÞ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞÞ k€ & ðtÞyðtÞ; ð19Þ
ð22Þ
_ where 1 ðtÞ is a current estimation of yðtÞ. Differentiating equation (6) for case "y ðtÞ ¼ 0 we obtain &1 ðtÞ y€ðtÞ ¼ 2€ &2 ðtÞ þ 2 &€1 ðtÞ þ € _ þ 2 &€1 ðtÞ þ € &1 ðtÞ: ¼ 2ð2&_2 ðtÞ 2 &_1 ðtÞ þ yðtÞÞ Substituting (6) into last equation we have y€ðtÞ ¼ 2ð2&_2 ðtÞ 2 &_1 ðtÞ þ 2&_2 ðtÞ þ 2 &_1 ðtÞ &1 ðtÞ þ &_1 ðtÞÞ þ 2 &€1 ðtÞ þ € ¼ 2 &_1 ðtÞ þ 2 &_2 ðtÞ þ &_2 ðtÞ: From last equation, we obtain
^ ¼ ðtÞ þ k&ðtÞyðtÞ; _ ðtÞ
ð20Þ
^ &_1 ðtÞ þ 2 &_2 ðtÞ þ ðtÞ ^ &_2 ðtÞ;
1 ðtÞðtÞ ¼ 2ðtÞ
ð23Þ
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where 2 ðtÞ is a current estimation of y€ðtÞ. Let us consider the following variables y€ðtÞ ¼ sinð! t þ ’Þ; ^ 6¼ 0: ^ ¼ 2 ðtÞ for ðtÞ ðtÞ ^ ðtÞ
Consider scaled normalized adaptive notch filter (ANF) see for instance [8, 10, 12, 13, 24]
ðtÞ ¼
ð24Þ
From equations (22) and (24) we can receive realizable online amplitude and bias estimator sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðtÞ ^2 ^ðtÞ ¼ ð25Þ þ ðtÞ; ^ ðtÞ ^ for ðtÞ ^ 6¼ 0: ^0 ðtÞ ¼ yðtÞ ðtÞ
ð26Þ
Remark 5: In the papers [2,8,10,12,16], the problem of robustness and noise rejection was not widely discussed. However, in the paper [10] robustness according to [9] is considered. If the signal yðtÞ ¼ 0 þ sinð! t þ Þ is corrupted by additional bounded disturbance, parameter adaptation algorithm in (19) may be modified according to [1,9,17], in order to prevent parameter estimates drift ^ _ _ _ ðtÞ ¼ k&ðtÞð2€ & ðtÞ 2 &ðtÞÞ k&_2 ðtÞðtÞ ^ k€ & ðtÞyðtÞ ka ðtÞ; ^ ¼ ðtÞ þ k&ðtÞyðtÞ; _ ðtÞ where ka 0. The modification of identification algorithm (19) assumes that outside bounded region in the statespace the derivative of Lyapunov function becomes negative. The main drawback is that a bias term is added to the parameter update equation, therefore zero residual error cannot be guaranteed when the disturbances are removed. In this case we can use the following approach [1] 8 ^ > > 0; < 0 ; > > > 0 1 > > < ^ ^ ka ¼ ka ðÞ ¼ @ 1A; 0 ^ 20 ; > > > 0 > > > ^ > : 1; > 20 ; where 0 is strictly positive constant.
4. Example Consider identification algorithm proposed in this paper and compare it with adaptive notch filter.
^x_ þ ! ^2 x ¼ ! ^2 y; x€ þ 2& !
ð27Þ
^ !; ^ !^_ ¼ ð2& x_ !yÞx
ð28Þ
¼(
" ; 2 #) _ x ^j g 1 þ N x2 þ f1 þ j! ^ ! "
ð29Þ
where – estimation of the frequency, y – measurable harmonic signal, & > 0 – is the damping coefficient, "; N; > 0 and 1. Let us compare ANF algorithm with algorithm proposed in this paper: – dynamic order of algorithm proposed in the given paper and ANF algorithm (27)–(29) are equal to three; – algorithm (19)–(21) has two unknown parameters in comparison with five unknown parameters of ANF algorithm, so tuning of proposed identification algorithm is much simpler; – proposed identification algorithm allows to estimate not only frequency of unknown harmonic signal but also gives estimation of amplitude and bias of identified signal, see equations (25) and (26). Let us simulate comparable algorithms. Simulation of algorithm (27)–(29) was done for the following values of constant parameters & ¼ 0:4, " ¼ 1:35, N ¼ 1, > 0:5 and ¼ 2, initial conditions _ xð0Þ ¼ 1, xð0Þ ¼ 1, ð0Þ ¼ 10, see [1]. Simulation of algorithm (27)–(29) was done for ¼ 1, k ¼ 1 and zero initial conditions. Let us consider problem of frequency identification signal yðtÞ ¼ sinð! tÞ for ¼ 10, ! ¼ 10. Fig. 2 shows simulation results of ANF and proposed simulation algorithms for yðtÞ ¼ 10 sinð10 tÞ. Let us consider problem of frequency identification signal yðtÞ ¼ sinð! tÞ for ¼ 50, ! ¼ 50. Fig. 3 shows simulation results of ANF and proposed algorithm for yðtÞ ¼ 50 sinð50 tÞ. As Figs. 2 and 3 show, both algorithms are efficient for different values of amplitude and frequency of harmonic signal. Varying of identification algorithm parameters can change rate of estimation of unknown frequency. Let us consider problem of frequency identification of biased harmonic signal yðtÞ ¼ 0 þ sinð! tÞ for 0 ¼ 10, ¼ 10, ! ¼ 10. Fig. 4 shows simulation results of ANF and proposed algorithm for yðtÞ ¼ 10 þ 10 sinð10 tÞ.
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Fig. 2. (a) ANF (27)–(29), (b) identification algorithm (19)–(21).
Fig. 3. (a) ANF (27)–(29), (b) identification algorithm (19)–(21).
Fig. 4. (a) ANF (27)–(29), (b) identification algorithm (19)–(21).
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As Fig. 4 shows, ANF algorithm is not efficient for biased harmonic signal. Figs. 5–10 show simulation results of identification algorithm (19)–(26) proposed in the paper. Figs. 5 and 6 show simulation results of the proposed algorithm for yðtÞ ¼ 10 þ 10 sinð10 tÞ. Let us consider problem of frequency identification of biased harmonic signal yðtÞ ¼ 0 þ sinð! tÞ for 0 ¼ 3, ¼ 2, ! ¼ 4. Figs. 7 and 8 show simulation results of proposed algorithm for yðtÞ ¼ 3þ 2 sinð4 tÞ. Let us consider problem of frequency identification of harmonic signal yðtÞ ¼ sinð! tÞ for ¼ 10, ^ and ! ¼ 10. Fig. 9 shows graphs of parameters ðtÞ ^ tuning for the sinusoidal signal yðtÞ ¼ 10 sin 10t !ðtÞ disturbed by the function ðtÞ ¼ 0:3 sinð6t þ 0:5Þ
^ ^ (b) function . Fig. 5. (a) frequency estimation, !;
Fig. 6. (a) bias estimation, ^0 ; (b) amplitude estimation ^.
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þ0:2 sinð40t 0:8Þ þ 0:03wðtÞ, where wðtÞ is a white noise. Computer simulation illustrates that robustness properties keep safe with respect to unaccounted disturbances (see Remark 3). Let us compare approaches [7,16] with algorithm proposed in this paper. Consider the sinusoidal signal yðtÞ ¼ sin t disturbed by function ðtÞ ¼ 0:05 sinð10tÞ and simulate comparable algorithms for different values of parameter . The results of computer simulation of approach [16] for parameters ¼ 10, ¼ k1 ¼ k2 ¼ & ¼ 1 (see page 863 and equations (3)–(6) from [16]) are presented in Fig. 11. We can see that for additional bounded disturbance the approach [16] ensures boundedness of all variables of the algorithm but parametrical convergence is not good.
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^ ^ (b) function . Fig. 7. (a) frequency estimation, !;
Fig. 8. (a) bias estimation, ^0 ; (b) amplitude estimation ^.
^ ^ (b) function . Fig. 9. (a) frequency estimation, !;
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The results of computer simulation approach [7] for parameters ¼ ¼ 1 (see page 856 and equation (24)–(26) from [7]) are presented in Fig. 12. We can see that for additional bounded disturbance the approach [7] ensures boundedness of all variables of the algorithm but parametrical convergence is bad. The results of computer simulation of this approach for parameters k ¼ ¼ 1 are presented in Fig. 13. We can see that parametrical convergence for this approach is better than in [7,16].
Fig. 10. The sinusoidal signal yðtÞ ¼ 10 sin 10t disturbed by function ðtÞ ¼ 0:3 sinð6t þ 0:5Þ þ 0:2 sinð40t 0:8Þ þ 0:03wðtÞ, where wðtÞ is a white noise.
^ for the case: (a) ¼ 5, (b) ¼ 10. Fig. 11. Transient for variable !
^ for the case: (a) ¼ 5, (b) ¼ 10. Fig. 12. Transient for variable !
Remark 6: The proposed estimation algorithm has an exponentially stable equilibrium point if one considers the undisturbed system. We have good simulation results for periodical disturbance (see Fig. 13). But unfortunately that is not the case if there is a periodic disturbance. In that case, there is a new equilibrium point that appears in the system. The attraction region of this equilibrium point is proportional to its amplitude
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^ for the case: (a) ¼ 5, (b) ¼ 10. Fig. 13. Transient for variable !
and its frequency. For this reason, even if its amplitude is small, it still will have an attraction region, and as a consequence, the desired equilibrium point is no more global. It will have an attraction region, and convergence will depend on initial conditions (see for example [8]). We leave this case as future works.
5. Conclusion We considered the problem of identification of frequency of a sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ þ ðtÞ for any unknown constant values 0 , , , ! > 0 and bounded disturbance or harmonic noise ðtÞ. Designed algorithm of identification (19)–(21): – is stable with regard to unaccounted disturbances presenting in measurement of effective biased sinusoidal signal; – allows accelerating rate of convergence of estimate ^ to because of increasing coefficient k (see ðtÞ Remarks 1 and 2); – has an ability of extension (22)–(26) for online estimation of amplitude and bias with no increase of dynamic order (see Remark 4 and example); – has the least dynamic order in comparison with works [1,7,10,12,16,24].
Acknowledgment This work was supported in part by RFBR Grant No. 09-08-00139.
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