- Email: [email protected]
Final Comments by the Authors
142
Discussion on: ‘‘IFBHS’’
main reason for the works of multiple sinusoids estimations as [3] scheme.
3. Conclusion The work of Aranovskiy, Bobtsov, Kremlev, Nikolaev, and Slita deals with the classical problem of designing an algorithm for the identification of the unknown frequency of a measured sinusoidal signal corrupted by bias and bounded non-periodic disturbances. They propose interesting contributions in this important and difficult field that has been visited by a large number of authors in the last decades. The result’s main advantage with respect to similar schemes is to be small and simple, showing good performance in the simulations. The tuning procedure should be done carefully and could be greatly enhanced if based on prior approximative knowledge of the unknown frequency.
References 1. Hsu L, Ortega R, Damm G. A globally convergent frequency estimator. IEEE Trans Autom Control 1999; 44: 698–713 2. Damm G, Hsu L, Ortega R. Measurement noise applied to a globally convergent frequency estimator. In: Mathematical Theory of Networks and Systems, Beghi A, Finesso L, Picci G (eds), Il Poligrafo, pp. 735–738, 1998 3. Marino R, Tomei P. Global estimation of unknown frequencies. IEEE Trans Autom Control 2002; 47: 1324–1328 4. Netto MS, Annaswamy AM, Ortega R, Moya P. Adaptive control of a class of non-linearly parametrized systems using convexification. Int J Control 2000; 73: 1312–1321 5. Regalia P. An improved lattice-based adaptive IIR notch filter. IEEE Trans Signal Process 1991; 39: 2124–2128 6. Bodson M, Douglas S. Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency. Automatica 1997; 33
Final Comments by the Authors S. Aranovskiy, A. Bobtsov, A. Kremlev, N. Nikolaev and O. Slita We agree with most of remarks done by Gileny Damm [1]: a roughly initial estimation of the unknown frequency can simplify procedure of parameter tuning and improve estimator’s behavior; before using bias and amplitude one should ensure that transactions are finished. Also very important discussion is about periodic noise. And for this problem we want to present an extended result.
Extended result In this paper, we extend approach [2] for estimation of signal which is the sum of n sinusoids with unknown amplitudes, frequencies and phases yðtÞ ¼
n X
i sinð!i t þ ’i Þ
ð1Þ
i¼1
It is known that for generating signal (1) it is possible to use differential equation [3] yð2nÞ ¼ n yð2n2Þ þ n1 yð2n4Þ þ ::: þ 2 y€ þ 1 y: ð2Þ
It can be verified that n Y
ðs2 þ !2i Þ ¼ s2n n s2n2 ::: 2 s2 1
i¼1
¼ s2n ðsÞ: Let us rewrite (2) in s-domain by taking Laplace transform of both sides of the equation, ignoring terms depending on initial conditions s2n YðsÞ ¼ ðsÞYðsÞ;
ð3Þ
sðs þ Þ2n1 YðsÞ ¼ aðsÞYðsÞ þ ðsÞYðsÞ;
ð4Þ
where aðsÞ ¼ sðs þ Þ2n1 s2n . From (4) we obtain _ ¼ yðtÞ
aðpÞ
ðp þ Þ2n1 ðpÞ þ yðtÞ ðp þ Þ2n1
yðtÞ ;
ð5Þ
143
Discussion on: ‘‘IFBHS’’
where p ¼ dtd .
From equations (6), (10) we obtain realizable identification algorithm of the following form (see Fig. 1)
Consider n-1-order filter &ðtÞ ¼
1 ðp þ Þ2n1
ð6Þ
yðtÞ:
Substituting (6) into (5) we obtain
k ðtÞ
_ ¼ aðpÞ yðtÞ & ðtÞ þ ðpÞ & ðtÞ ¼ zðtÞ þ
T
ðtÞe
ð7Þ ;
where ðtÞ ¼ colf & ð2n2Þ ; &ð2n4Þ ; &ð2n6Þ ; :::; €&_; €&; &g, zðtÞ ¼ aðpÞ & ðtÞ and e ¼ colfn ; n1 ; :::; 2 ; 1 g. _ is measured. First let us suppose that function yðtÞ Then ideal identification algorithm can be written as _ ^e ðtÞ ¼ k ðtÞ
T
ðtÞðe ^e Þ
_ zðtÞÞ k ðtÞ ¼ k ðtÞðyðtÞ
T
ð8Þ
ðtÞ^e ;
where number k > 0. However, in our case signal yðtÞ is only measured but not its derivatives. To derive realizable scheme of identification algorithm let us consider the following variable e ðtÞ ¼ ^e ðtÞ k ðtÞyðtÞ:
ð9Þ
Differentiating equation (9) we obtain _ k ðtÞ zðtÞ k ðtÞ _ e ðtÞ ¼ k ðtÞyðtÞ _ _ k ðtÞyðtÞ k ðtÞyðtÞ ¼ k _ ðtÞyðtÞ k ðtÞ zðtÞ T k ðtÞ ðtÞ^e
^e ðtÞ ¼ k ðtÞyðtÞ þ e ðtÞ; zðtÞ _ e ðtÞ ¼ k _ ðtÞyðtÞ k ðtÞ
T
ðtÞ^e :
ð10Þ
Fig. 1. Structural chart of the identification algorithm (6), (11), (12).
T
ðtÞ^e :
ð11Þ
ð12Þ
Algorithm (6), (11), (12) has ð3n 1Þ-order.
Conclusion Problem of frequency identification of sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ has been considered in [2] for any unknown constant values 0 ; ; ; ! > 0. Obtained result has been extended for estimation of the signal which is the sum of n sinusoids with unknown amplitudes, frequencies and phases. Algorithm (6), (11), (12) has ð3n 1Þ-order.
References 1. Damm G. Discussion on: Identification of frequency of biased harmonic signal. European J. Control 2010; 16 2. Aranovskiy S, Bobtsov A, Kremlev A, Nikolaev N, Slita O. Identification of frequency of biased harmonic signal Eur J. Control 2010; 16. 3. Xia X. Global frequency estimation using adaptive identifiers, IEEE Trans Autom Control 2002; 47: 1188 –1193.
Discussion on: ‘‘IFBHS’’
main reason for the works of multiple sinusoids estimations as [3] scheme.
3. Conclusion The work of Aranovskiy, Bobtsov, Kremlev, Nikolaev, and Slita deals with the classical problem of designing an algorithm for the identification of the unknown frequency of a measured sinusoidal signal corrupted by bias and bounded non-periodic disturbances. They propose interesting contributions in this important and difficult field that has been visited by a large number of authors in the last decades. The result’s main advantage with respect to similar schemes is to be small and simple, showing good performance in the simulations. The tuning procedure should be done carefully and could be greatly enhanced if based on prior approximative knowledge of the unknown frequency.
References 1. Hsu L, Ortega R, Damm G. A globally convergent frequency estimator. IEEE Trans Autom Control 1999; 44: 698–713 2. Damm G, Hsu L, Ortega R. Measurement noise applied to a globally convergent frequency estimator. In: Mathematical Theory of Networks and Systems, Beghi A, Finesso L, Picci G (eds), Il Poligrafo, pp. 735–738, 1998 3. Marino R, Tomei P. Global estimation of unknown frequencies. IEEE Trans Autom Control 2002; 47: 1324–1328 4. Netto MS, Annaswamy AM, Ortega R, Moya P. Adaptive control of a class of non-linearly parametrized systems using convexification. Int J Control 2000; 73: 1312–1321 5. Regalia P. An improved lattice-based adaptive IIR notch filter. IEEE Trans Signal Process 1991; 39: 2124–2128 6. Bodson M, Douglas S. Adaptive algorithms for the rejection of sinusoidal disturbances with unknown frequency. Automatica 1997; 33
Final Comments by the Authors S. Aranovskiy, A. Bobtsov, A. Kremlev, N. Nikolaev and O. Slita We agree with most of remarks done by Gileny Damm [1]: a roughly initial estimation of the unknown frequency can simplify procedure of parameter tuning and improve estimator’s behavior; before using bias and amplitude one should ensure that transactions are finished. Also very important discussion is about periodic noise. And for this problem we want to present an extended result.
Extended result In this paper, we extend approach [2] for estimation of signal which is the sum of n sinusoids with unknown amplitudes, frequencies and phases yðtÞ ¼
n X
i sinð!i t þ ’i Þ
ð1Þ
i¼1
It is known that for generating signal (1) it is possible to use differential equation [3] yð2nÞ ¼ n yð2n2Þ þ n1 yð2n4Þ þ ::: þ 2 y€ þ 1 y: ð2Þ
It can be verified that n Y
ðs2 þ !2i Þ ¼ s2n n s2n2 ::: 2 s2 1
i¼1
¼ s2n ðsÞ: Let us rewrite (2) in s-domain by taking Laplace transform of both sides of the equation, ignoring terms depending on initial conditions s2n YðsÞ ¼ ðsÞYðsÞ;
ð3Þ
sðs þ Þ2n1 YðsÞ ¼ aðsÞYðsÞ þ ðsÞYðsÞ;
ð4Þ
where aðsÞ ¼ sðs þ Þ2n1 s2n . From (4) we obtain _ ¼ yðtÞ
aðpÞ
ðp þ Þ2n1 ðpÞ þ yðtÞ ðp þ Þ2n1
yðtÞ ;
ð5Þ
143
Discussion on: ‘‘IFBHS’’
where p ¼ dtd .
From equations (6), (10) we obtain realizable identification algorithm of the following form (see Fig. 1)
Consider n-1-order filter &ðtÞ ¼
1 ðp þ Þ2n1
ð6Þ
yðtÞ:
Substituting (6) into (5) we obtain
k ðtÞ
_ ¼ aðpÞ yðtÞ & ðtÞ þ ðpÞ & ðtÞ ¼ zðtÞ þ
T
ðtÞe
ð7Þ ;
where ðtÞ ¼ colf & ð2n2Þ ; &ð2n4Þ ; &ð2n6Þ ; :::; €&_; €&; &g, zðtÞ ¼ aðpÞ & ðtÞ and e ¼ colfn ; n1 ; :::; 2 ; 1 g. _ is measured. First let us suppose that function yðtÞ Then ideal identification algorithm can be written as _ ^e ðtÞ ¼ k ðtÞ
T
ðtÞðe ^e Þ
_ zðtÞÞ k ðtÞ ¼ k ðtÞðyðtÞ
T
ð8Þ
ðtÞ^e ;
where number k > 0. However, in our case signal yðtÞ is only measured but not its derivatives. To derive realizable scheme of identification algorithm let us consider the following variable e ðtÞ ¼ ^e ðtÞ k ðtÞyðtÞ:
ð9Þ
Differentiating equation (9) we obtain _ k ðtÞ zðtÞ k ðtÞ _ e ðtÞ ¼ k ðtÞyðtÞ _ _ k ðtÞyðtÞ k ðtÞyðtÞ ¼ k _ ðtÞyðtÞ k ðtÞ zðtÞ T k ðtÞ ðtÞ^e
^e ðtÞ ¼ k ðtÞyðtÞ þ e ðtÞ; zðtÞ _ e ðtÞ ¼ k _ ðtÞyðtÞ k ðtÞ
T
ðtÞ^e :
ð10Þ
Fig. 1. Structural chart of the identification algorithm (6), (11), (12).
T
ðtÞ^e :
ð11Þ
ð12Þ
Algorithm (6), (11), (12) has ð3n 1Þ-order.
Conclusion Problem of frequency identification of sinusoidal signal yðtÞ ¼ 0 þ sinð! t þ Þ has been considered in [2] for any unknown constant values 0 ; ; ; ! > 0. Obtained result has been extended for estimation of the signal which is the sum of n sinusoids with unknown amplitudes, frequencies and phases. Algorithm (6), (11), (12) has ð3n 1Þ-order.
References 1. Damm G. Discussion on: Identification of frequency of biased harmonic signal. European J. Control 2010; 16 2. Aranovskiy S, Bobtsov A, Kremlev A, Nikolaev N, Slita O. Identification of frequency of biased harmonic signal Eur J. Control 2010; 16. 3. Xia X. Global frequency estimation using adaptive identifiers, IEEE Trans Autom Control 2002; 47: 1188 –1193.