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the frequency, magnitude, phase and offset of a biased sinusoidal signal. ... Index Terms—Adaptive control, continuous least-squares, sinusoidal pa- rameters ...
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for every I  m and J  m . Then, from Corollary 2, the saturated reset system is globally asymptotically stable and V (x) = xT P x yields a global Lyapunov function. VI. CONCLUSIONS In this paper we dealt with the problems of characterizing quadratic stability and computing ellipsoidal estimations of the domain of attraction for saturated hybrid systems. The results presented are based on a geometrical approach to the analysis of saturated functions, also in case of nested saturations, which permitted to formulate contractiveness conditions of ellipsoids for a rather generic class of saturated hybrid systems. An interesting forthcoming issue could be to exploit the hybrid loop to improve the performance of a controlled system in presence of exogenous signals. This could be achieved by designing the reset law and both the flow and jump sets.

REFERENCES [1] T. Alamo, A. Cepeda, and D. Limon, “Improved computation of ellipsoidal invariant sets for saturated control systems,” in Proc. 44th IEEE Conf. Decision and Control and European Control Conf, CDC-ECC 2005, Seville, Spain, Dec. 2005, pp. 6216–6221. [2] T. Alamo, A. Cepeda, D. Limon, and E. F. Camacho, “A new concept of invariance for saturated systems,” Automatica, vol. 42, pp. 1515–1521, 2006. [3] O. Beker, C. V. Hollot, Y. Chait, and H. Han, “Fundamental properties of reset control systems,” Automatica, vol. 40, pp. 905–915, 2004. [4] M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: Model and optimal control theory,” IEEE Trans. Autom. Control, vol. 43, no. 1, pp. 31–45, Jan. 1998. [5] M. Fiacchini, S. Tarbouriech, and C. Prieur, “Ellipsoidal invariant sets for saturated hybrid systems,” in American Control Conf. 2011 (ACC ’11), San Francisco, CA, Jun. 2011. [6] R. Goebel, J. P. Hespanha, A. R. Teel, C. Cai, and R. Sanfelice, “Hybrid systems: Generalized solutions and robust stability,” in Proc. IFAC: Symp. Nonlinear Control Systems, Stuttgart, Germany, 2004, pp. 1–12. [7] R. Goebel, R. Sanfelice, and A. R. Teel, “Hybrid dynamical systems,” IEEE Control Syst. Mag., vol. 29, no. 2, pp. 28–93, Apr. 2009. [8] J. M. Gomes da Silva, Jr and S. Tarbouriech, “Local stabilization of discrete-time linear systems with saturating controls: An LMI-based approach,” IEEE Trans. Autom. Control, vol. 46, no. 1, pp. 119–125, Jan. 2001. [9] R. L. Grossmann, A. Nerode, A. P. Ravn, and H. Rischel, “Hybrid systems,” in Lecture Notes in Computer Science. Berlin, Germany: Springer-Verlag, 1993, vol. 736. [10] J. P. Hespanha, D. Liberzon, and A. R. Teel, “Lyapunov conditions for input-to-state stability of impulsive systems,” Automatica, vol. 44, no. 11, pp. 2735–2744. [11] T. Hu and Z. Lin, “Exact characterization of invariant ellipsoids for single input linear systems subject to actuator saturation,” IEEE Trans. Autom. Control, vol. 47, no. 1, pp. 164–169, Jan. 2002. [12] T. Hu, Z. Lin, and B. M. Chen, “Analysis and design for discrete-time linear systems subject to actuator saturation,” Syst. & Control Lett., vol. 45, no. 2, pp. 97–112, 2002. [13] M. Lazar and W. P. M. H. Heemels, “Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions,” Automatica, vol. 45, no. 1, pp. 180–185, 2009. [14] D. Liberzon, Switching in Systems and Control. Boston, MA: Birkhauser, 2003. [15] D. Nesic, L. Zaccarian, and A. R. Teel, “Stability properties of reset systems,” Automatica, vol. 44, no. 8, pp. 2019–2026, 2008. [16] C. Prieur, R. Goebel, and A. R. Teel, “Hybrid feedback control and robust stabilization of nonlinear systems,” IEEE Trans. Autom. Control, vol. 52, no. 11, pp. 2103–2117, Nov. 2007. [17] R. T. Rockafellar, Convex Analysis. Princeton, MJ: Princeton University Press, 1970. [18] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, volume 44. Cambridge, U.K.: Cambridge University Press, 1993. [19] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design. London, U.K.: Springer-Verlag, 2005.

[20] S. Tarbouriech, T. Loquen, and C. Prieur, “Anti-windup strategy for reset control systems,” Int. J. Robust and Nonlin. Control, vol. 21, no. 10, pp. 1159–1177, 2011. [21] S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva, Jr, “Stability analysis and stabilization of systems presenting nested saturations,” IEEE Trans. Autom. Control, vol. 51, no. 8, pp. 1364–1371, Aug. 2006.

Non Adaptive Second-Order Generalized Integrator for Identification of a Biased Sinusoidal Signal Giuseppe Fedele, Member, IEEE, and Andrea Ferrise

Abstract—This note presents a new algorithm that is designed to identify the frequency, magnitude, phase and offset of a biased sinusoidal signal. The structure of the algorithm includes an orthogonal system generator based on a second-order generalized integrator. The proposed strategy has the advantages of a fast and accurate signal reconstruction capability and a good rejection to noise. Index Terms—Adaptive control, continuous least-squares, sinusoidal parameters estimation.

I. INTRODUCTION The estimation problem of the parameters set of a sinusoidal signal is a crucial task in a wide range of applications such as control theory [1], [2], signal processing [3], biomedical engineering [4], instrumentation and measurements, power systems [5]–[7], to name just a few. The problem is relevant in power systems area where grid-connected devices require an accurate and fast detection of the phase angle, amplitude and frequency of the utility voltage to assure the correct generation of the reference signals. Especially in this case, a relevant issue associated with grid-connected systems, is the presence of an offset in the measured grid voltage. This voltage offset is typically introduced by the measurements and data conversion processes and causes an error, for the estimated parameters, at the same frequency of the grid voltage. As a consequence, a filtering process can be performed at the expense of dynamic performance degradation [8]. Therefore, a very attractive challenge is the synthesis of an accurate method for the on line estimation of the entire parameters set of a biased sinusoidal signal. Several papers that address this problem can be found in the recent literature. Among them, a large part deals with the case of multi-sinusoidal estimation problem [9]–[14]. Although the estimation of a single biased sinusoid is a subcase of this general problem, however many of these methods fail in presence of a bias term since they require signals with strictly positive frequencies and they cannot be adapted to deal with the problem here discussed. For these reasons many techniques are proposed to deal with a biased sinusoidal signal [8], [15]–[17]. This paper belongs to this category and it provides an estimation algorithm for the entire set of parameters. Manuscript received December 14, 2010; revised June 14, 2011 and June 14, 2011; accepted November 08, 2011. Date of publication December 08, 2011; date of current version June 22, 2012. Recommended by Associate Editor E. Weyer. The authors are with the Department of Electronics, Computer and System Science, University of Calabria, 87036 Arcavacata di Rende (CS), Italy (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2011.2178877

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In [15] a seventh-order global estimator is presented for the recovery of frequency, amplitude and offset. However it is not able to reconstruct the input signal as it does not estimate the initial phase angle. Moreover the presence of the quadratic term of the input signal may heavily affect the estimation in case of noisy data. In [16] a new approach to the problem of globally convergent frequency estimator is proposed where the estimator is represented by a fourth-order system. In [17] a scheme is proposed to estimate offset, amplitude and frequency. The main advantage of this method, with respect to similar schemes, is to be small and simple, showing good performances in the simulations [18]. In fact, a frequency parameter adaptive law is proposed that is robust with regard to unaccounted disturbances. However, as remarked in [18], the tuning procedure should be done carefully. Amplitude and bias estimators have been also presented but bad transients could be obtained due to possible divisions by zero. Therefore, these estimators should not be taken into account while not close to steady-state conditions. The most commonly employed approach to estimate the entire set of parameters uses phase-locked-loop (PLL) topologies [19]–[21]. The structure of the main PLL topology is a feedback control system that automatically adjusts the phase of a locally generated signal to match the phase of the input signal. The main difference among single-phase PLL topologies consists in the orthogonal voltage system generation subsystem. In [19] an interesting PLL scheme is proposed to reconstruct an unbiased sinusoid by estimating in-phase and quadrature-phase amplitudes of the fundamental component of the input signal. In order to deal with the case of biased sinusoid signal, in [8] and references therein, an effective method is used to create orthogonal reference signals based on a second-order generalized integrator (OSG-SOGI) according to the information about the signal frequency that can be obtained by using a PLL system in a feedback closed-loop. The reliability of such a method has been tested in a wide range of practical scenarios but the dependence on initial conditions of the estimate may limit its use. Other applications of OSG-SOGI can be found in [22], [24]. Based on previous results in [25], an estimation scheme, relying on OSG-SOGI, is here presented. A continuous-time least-squares algorithm is used to estimate both bias and frequency. The remaining parameters are obtained via simple relationships between the OSG-SOGI output signals. The exponential convergence of this estimator is proved. The resulting estimator permits to reconstruct the unknown input signal with satisfactory accuracy. To properly test the proposed method, a comparison makes sense with techniques that directly deal with biased sinusoid. In the following the method presented in [17] and the PLL-based one presented in [8] are considered. In particular, as far as our knowledge is concerned, the method used in [8] is the most recently discussed one that is able to match the phase angle of a biased sinusoid. Unlike OSG-SOGI based approach, the method in [17] does not provide the estimation of the initial phase angle. Furthermore, the OSG-SOGI method is less sensitive to initial conditions choice than the method in [8] that requires an initial condition sufficiently close to the unknown frequency to be estimated. The paper is organized as follows: Section II presents OSG-SOGI scheme; in Section III the estimation method is discussed; Section IV contains simulation results. Finally, Section V is devoted to conclusions. II. OSG BASED ON SOGI The closed-loop diagram representing the OSG-SOGI is depicted in Fig. 1. In order to generate two orthogonal signals, the reference signal v (t) and the resonant frequency !s are needed in input. In the operating mode, if the resonant frequency is equal to the reference signal frequency, OSG-SOGI generates two sine waves (v1 (t) and v2 (t)) that have the same magnitude of v (t) and with a phase shift of =2 each

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Fig. 1. Block diagram of the orthogonal signals generator based on SOGI.

other. Moreover, v1 (t) is in phase with the fundamental of the input signal. The input signal v (t) is the biased sinusoid

v (t) = A0 + Ac sin(c (t))

(1)

where c (t) = !c t + c is the phase angle and fA0 ; Ac ; !c ; c g are the unknown parameters to be estimated. The OSG-SOGI system is governed by the following set of differential equations:

dq2 (t) = !s v1 (t) dt dv1 (t) = Ks !s (v (t) 0 v1 (t)) 0 !s q2 (t) dt v2 (t) = q2 (t) 0 Ks (v (t) 0 v1 (t)):

(2) (3) (4)

The gain Ks affects the bandwidth of the OSG-SOGI. Small values of the frequency !s result in a slowdown of the dynamic response. In such a case, the steady-state condition for the OSG-SOGI method is obtained for sufficiently long observation times. On the other hand, high values of the resonant frequency permit to quickly reach the steadystate condition obtaining estimations for the unknown parameters even for small observation times. As it is evident, the orthogonal component q2 (t) is directly affected by the presence of an offset, which does not appear in v1 (t) and v2 (t) because of the derivative actions on the input signal v (t). In order to simplify the estimation algorithm analysis, the response of the OSG-SOGI is approximated as its sinusoidal steady-state response. Even if the input signal parameters may be time-varying, they will be assumed to vary slowly enough that the approximation is valid. Moreover, due to the structure of the OSG-SOGI p poles p1;2 = 0(Ks 6 Ks2 0 4)!s =2, by choosing Ks 2 (0; 2), the transient response decays as e0K ! =2 . It is effortless to show that the output signals converge exponentially fast to the following steady-state signals:

v1 (t) = m1 Ac sin(!c t + c +  )

(5)

v2 (t) = 0 m3 Ac cos(!c t + c +  )

(6)

q2 (t) = A0 Ks 0 m2 Ac cos(!c t + c +  )

(7)

where

m1 =

Ks !s !c

(!s2 !s m2 = m1 !c

0 !c2 )2 + Ks2 !s2 !c2

(8) (9)

and

m3 = m1

!c !s

(10)

with

 = sgn[!s 0 !c ]

 2

0 arctan

Ks !s !c : !s2 0 !c2

(11)

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The sign function sgn(1) is defined as sgn(x) =

+1 01

By using the identity

iff x  0; iff x < 0.

(12)

Remark 1: Generally, the OSG-SOGI structure requires an adaptive tuning with respect to its resonant frequency. This can be achieved by adjusting the resonant frequency of the SOGI on-line using, for example, the frequency provided by the feedback control loop of a PLL structure as proposed in [8] or the frequency tracking method based on modulating functions as proposed in [22], [23]. In Section III, it will be shown that the unknown parameters can be estimated through the measurements of the OSG-SOGI output signals without any adaption of the resonant frequency.

dP (t) 01 dP 01 (t) d 01 (t)] = [P (t)P P (t) + P (t) =0 dt dt dt

we obtain dP (t) T = P (t) 0 tP (t)(t) (t)P (t) dt

y (t) = T (t) 3

(13)

(22)

with P (0) = Q001 . Differentiating (18) and using (22), we find that the parameters update satisfies d^(t) = dt

III. ESTIMATION METHOD In the nominal case, i.e., when the input signal is not affected by noise, (6) and (7) lead to the following linear relationship:

(21)

tP (t)(t)(T (t)^(t) 0 y (t)):

0

(23)

Therefore, (2)–(4) and the two matrix differential equations ((22), (23)) define the overall 7th-order estimation algorithm. In fact, as it can be noted, the algorithm consists of two differential equations ((2) and (3), two equations related to vector ^(t) ((23)) and the remaining three equations to update the components of the symmetric matrix P (t).

where (t) is the regressors vector A. Convergence Analysis

q2 (t) 0Ks

(t) =

(14)

3 is the vector of unknown parameters 3

 =

!c2 A0 !c2

2

0

e0(t0 )  (T ( )^( ) 0 y ( ))2 d + 1 0t ^ T + e ( (t) 0 ^0 ) Q0 (^(t) 0 ^0 ) 2

(17)

where Q0 = Q0T > 0 and   0 are design constants, ^0 = ^(0) is the initial parameters estimate and the linear term in the integrand function provides a zero-convergent estimation error in the nominal case, as explained in the following. The cost function (17) includes the forgetting factor  to possibly discount the past data and a penalty on the initial error between the estimate ^0 and 3 [26]. Setting @J=@ ^ = 0 and assuming ^( ) constant in [0; t], the estimate that minimizes (17) is ^(t) = P (t) e0t Q0 ^0 +

t 0

e0(t0 )  ( )y ( )d

P (t) =

e0t Q0 +

0

e0(t0 )  ( )T ( )d

P 01 (t)  e0

t t0

 ( )T ( )d

(26)

holds with  2 (0; t] because of the non-negative definition of the integrand function. Solving the integral in (26) with  = 2=!c , one has P 01 (t)  e02=!

t0

 !c

M1 +

Ac m2  sin((t))M2 !c2

(27) where

:

A20 Ks2 + 12 m22 A2c 0A0 Ks2 ; 2 0A0 Ks Ks2 !c 04A0 Ks + m2 Ac cos((t)) 2Ks M2 = 2Ks 0 M1 =

01

(19)

To achieve computational efficiency, it is desirable to compute P (t) recursively. This amounts to replace the above equation by the differential equation dP 01 (t) 01 T (t) + t(t) (t): = 0P dt

(25)

To investigate the convergence properties of the proposed estimator, a preliminary result is here introduced. The proof follows the one presented in [26] based on the concept of persistence of excitation. Proposition 1: The norm of the matrix P 01 (t) is unbounded as t ! 1. Proof: The inequality

(18)

with t

(t) = e0t P (t)Q0 (0):

(16)

To obtain the unknown vector 3 , an estimate ^(t) is computed by minimizing the cost function t

(24)

from which it is straightforward to derive the following expression for the estimation error:

y (t) = !s2 v2 (t):

1

d 01 01 (t)(t)] = 0P (t)(t) [P dt

(15)

and

J (^) =

Let (t) = ^(t) 0 3 be the estimation error and consider the term 0 P 1 (t)(t). From (20) and (23), one can obtain

(20)

2

(28) (29)

and (t) = !c t + c +  . Note that M1 is a constant symmetric positive definite matrix and each element of M2 is bounded. Therefore the Frobenius norm of P 01 (t) tends to infinity as O(t).

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As an immediate consequence of previous results, the norm of the absolute error, i.e., 0t e k(t)k  (30) kQ0 (0)k 0 kP 1 (t)k tends to zero exponentially. Given the estimate of A0 and !c , the amplitude Ac is estimated as ^c (t) = A

v12

(t) +

!^ (t) (t)2 !

m12

(31)

^0 (t): (t) 0 Ks A

(32)

where (t) = q2

Moreover the phase angle of the sinusoidal signal can be estimated as (t) ^ = arg (t) 0 

(33) Fig. 2. Example 1. Signal reconstruction.

with (t) =

0

2 ! ^ c (t) ! ^ c (t) (t) + | v1 2 ! !s

s

(t)

(34)

where | is the imaginary unit. If the signal v(t) is corrupted by additional bounded disturbance, namely w(t), the noise affects both the regressors and the signal y(t). However the noise present in the output signals is bounded due to the filtering characteristics of the OSG-SOGI system. In this case the steady-state error estimation is expressed as  (t)n (t) = P

where n depends on the noise w(t) as t 0(t0 )  (  )w( )d n = e 0

(35)

(36)

and the bar notation stands for noisy variables. By similar arguments, the norm of P 01 (t) tends to infinity as O(t). Moreover 0t 0 1 t + e kn k  kkkwk : (37) 2 

Therefore, kn k tends to infinity in O(t) time and, as a consequence, the boundedness of the estimation error norm can be guaranteed. Remark 2: Note that in (31) the signal term (^ !c (t)=!s )(t) where (t) is the auxiliary signal defined in (32), could be replaced with (!s =^ !c (t))v2 (t). However, the signal v2 (t) is more corrupted by the input noise because its dependence on the second-order derivative of the signal v(t). IV. SIMULATIONS In this section, some examples are given to illustrate the behavior of the proposed method, namely FF. Input signals, sampled with a period of Ts = 3 2 1004 s, are affected by a zero mean Gaussian noise with a signal-to-noise (SNR) ratio equal to 10. The SNR is measured in decibels as the logarithm of the average power of the reference signal samples and the noise ones over the experiment time as n01 2 y(kTs ) k=0 SN R 10 log10 (38) n01 w(kTs )2 k=0 where n is the number of samples.

1) Example 1: The aim of this example is to highlight the capability of the proposed method to identify all the parameters of a biased sinusoid and then to on-line reconstruct the input signal. The biased sinusoid v(t) = 1 + 5 sin (3t + =4) is considered as OSG-SOGI input signal. The parameters of FF are chosen as Ks = 1,  = 10, 2 2 T Q0 = (1=)I , ^0 = [!s ; !s ] and !s is chosen equal to twice the unknown angular frequency initial value. Fig. 2 depicts the noisy input signal, the nominal and the reconstructed one. As it can be noted FF seems to show a fast and accurate signal reconstruction capability and a good rejection to noise. 2) Example 2: In this example, a signal with two frequency steps is used to compare the proposed method with the OSG-SOGI based PLL (CTA) in [8] and the linear adaptive scheme (ABKNS) in [17]. Let us assume an input signal v(t) = 2 + 2 sin(!c t)

(39)

with 4; !c =

8; 2;

 t < 30  t < 60 60  t < 90. 0

30

(40)

All the methods are initialized with the same initial condition ! ^ c (0) = 3:2. The parameters of FF are chosen as Ks = 1, 2 2T !s = ! ^ c (0),  = 1, Q0 = (1=)I , ^0 = [!s ; !s ] . CTA method is ^ c (0) and the PI regulator gains Kp = 1:51 tuned with k = 1, ! = ! and KI = 0:89. As far as ABKNS method is concerned, the version

with a correction term is considered as proposed in [17]. This modification permits to reduce estimation error in case of noisy signal and it ensures boundedness of the closed system signals. The chosen parameters are k = 10, = 1 and 0 = 5. The results are depicted in Fig. 3. All the methods deal with the first frequency sweep in a satisfactory manner but CTA cannot perform the tracking of the final frequency that is 75% less than the intermediate frequency. As it can be noted, the proposed method performs a better filtering action on the estimated frequency with respect to other methods. Note however that the filtering characteristics of CTA and AKBNS can be improved by an accuate choice of the gain parameters at the expense of longer transients. This means that, in the case of heavily noisy signals, a tradeoff between filtering actions and transient responses is required making it difficult to correctly tune the free parameters.

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Fig. 3. Example 2. Frequency estimation.

V. CONCLUSIONS In this paper, an estimator for the on-line recovering of the frequency, amplitude, phase and offset of a biased sinusoidal signal has been proposed. The discussed method is based on a continuous least squares approach by considering a cost function with a forgetting factor and a regressors vector that is weighted by a linear term. Such regressors permit to have an exponentially zero-convergent estimation error when no noises are present and a bounded estimation error in the case of noisy input signals. Simulations have been conducted in order to illustrate the reconstruction capability of the method and its behavior dealing with abrupt changes in the input signal frequency.

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[11] R. Marino and P. Tomei, “Global estimation of n unknown frequencies,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1324–1328, Aug. 2002. [12] G. Obregón-Pulido, B. Castillo-Toledo, and A. A. Loukianov, “Globally convergent estimator for n—frequencies,” IEEE Trans. Autom. Control, vol. 47, no. 5, pp. 857–863, May 2002. [13] B. B. Sharma and I. N. Kar, “Design of asymptotically convergent frequency estimator using contraction theory,” IEEE Trans. Autom. Control, vol. 53, no. 8, pp. 1932–1937, Aug. 2008. [14] X. Chen, “Identification for a signal composed of multiple sinusoids,” IET Contr. Theor. Appl., vol. 2, no. 10, pp. 875–883, 2008. [15] M. Hou, “Amplitude and frequency estimator of a sinusoid,” IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 855–858, May 2005. [16] A. Bobtsov, “New approach to the problem of globally convergent frequency estimator,” Int. J. Adapt. Contr. and Sig. Proc., vol. 22, no. 3, pp. 306–317, 2008. [17] S. Aranovskiy, A. Bobtsov, A. Kremlev, N. Nikolaev, and O. Slita, “Identification of frequency of biased harmonic signal,” Eur. J. Contr., vol. 2, pp. 129–139, 2010. [18] G. Damm, S. Aranovskiy, A. Bobtsov, A. Kremlev, N. Nikolaev, and O. Slita, “Discussion on: Identification of frequency of biased harmonic signal,” Eur. J. Contr., vol. 2, pp. 140–143, 2010. [19] M. Karimi-Ghartemani, H. Karimi, and M. R. Iravani, “A magnitude/phase-locked loop system based on estimation of frequency and in-phase/quadrature-phase amplitudes,” IEEE Trans. Ind. Electron., vol. 51, no. 2, pp. 511–517, Apr. 2004. [20] B. Wu and M. Bodson, “A magnitude/phase-locked loop approach to parameter estimation of periodic signals,” IEEE Trans. Autom. Control, vol. 48, no. 4, pp. 612–618, Apr. 2003. [21] D. Jovcic, “Phase locked loop system for FACTS,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1116–1124, Aug. 2003. [22] G. Fedele, C. Picardi, and D. Sgrò, “A power electrical signal tracking strategy based on the modulating functions method,” IEEE Trans. Ind. Electron., vol. 56, no. 10, pp. 4079–4087, Oct. 2009. [23] G. Fedele and L. Coluccio, “A recursive scheme for frequency estimation using the modulating functions method,” Appl. Math. and Comp., vol. 216, no. 5, pp. 1393–1400, 2010. [24] P. Rodriguez, A. Luna, I. Candela, R. Mujal, R. Teodorescu, and F. Blaabjerg, “Multiresonant frequency-locked loop for grid synchronization of power converters under distorted grid conditions,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 127–138, Jan. 2011. [25] G. Fedele, A. Ferrise, and D. Frascino, “Structural properties of the SOGI system for parameters estimation of a biased sinusoid,” in Proc. 9th Int. Conf. Environment and Electrical Engineering (EEEIC), May 2010, pp. 438–441. [26] P. Ioannou and B. Fidan, Adaptive Control Tutorial. Philadelphia, PA: SIAM, 2006.