8]. Motivated by the significant development of the more popular time domain counterparts, ... of Ac, Bc, Cc, Dc, given the noise-corrupted samples of the.
Electrical Engineering in Japan, Vol. 132, No. 1, 2000
Translated from Denki Gakkai Ronbunshi, Vol. 119-C, No. 3, March 1999, pp. 326334
Frequency Domain Subspace Identification with the Aid of the w-Operator ZI-JIANG YANG and SHINGO SANADA Kyushu Institute of Technology, Japan
parametric model from frequency response data require nonlinear optimization techniques and may not ensure global convergence [4]. In recent years, subspace-based identification methods have attracted much attention, because of their excellent numerical reliability owing to singular value decomposition (SVD) and LQ factorization techniques [5 8]. Motivated by the significant development of the more popular time domain counterparts, frequency domain subspace identification algorithms have also been studied recently by several researchers [911]. In Refs. 9 and 10, the subspace identification algorithms are performed on discrete-time models. The corresponding continuous-time model can be obtained by transforming the identified discrete-time model through the bilinear transformation z 2 sT / 2 sT (where z, s, T are the z-operator, Laplace operator, and the sampling period, respectively). To preserve the invariance of the frequency response, it is required to prewarp the frequency scale [10]. Direct identification of continuous-time models is also considerable. However, the data matrices often become ill-conditioned [11, 12], if we simply rewrite the Laplace operator s as s jZ, where Z denotes the frequency. In Ref. 11, the Forsythe polynomials [12] are utilized to improve the conditioning of the data matrices involved in the frequency domain subspace identification algorithms, and some interesting results are shown through theoretical analysis. However, the recursive computing procedure using the Forsythe polynomials seems to be complicated and is not easy to understand. Recently, the w-operator approach has been proposed by several researchers for time domain subspace identification of continuous-time models [1315]. The w-operator w s D / s D , where D > 0, is introduced to avoid direct numerical differentiations. The subspace identification algorithms can then be applied to the derived w-operator state-space model. Since the w-operator corresponds to the well-known Laguerre filter, which is frequently used for nonparametric model identification, the condition number of the matrix consisting of filtered input signals does not
SUMMARY Frequency domain subspace identification algorithms have been studied recently by several researchers in the literature, motivated by the significant development of the more popular time domain counterparts. Usually, this class of methods are focused on discrete-time models, since in the case of continuous-time models, the data matrices often become ill-conditioned if we simply rewrite the Laplace operator s as s jZ, where Z denotes the frequency. This paper proposes an efficient and convenient approach to frequency domain subspace identification for continuous-time systems. The operator w s D / s D is introduced to avoid the ill-conditioned problem. Hence, the system can be identified based on a state-space model in the w-operator. Then the estimated w-operator state-space model can be transformed back to the common continuous-time state-space model. An instrumental variable matrix in the frequency domain is also proposed to obtain consistent estimates of the equivalent system matrices in the presence of measurement noise. Simulation results are included to verify the efficiency of the proposed algorithms. © 2000 Scripta Technica, Electr Eng Jpn, 132(1): 4656, 2000
Key words: Subspace model identification; continuous-time system; w-operator; frequency domain; instrumental variable.
1. Introduction Due to recent development of electronic technologies, it often occurs in practice that the frequency response samples of a dynamic system can be obtained by sophisticated spectrum analyzers. Methods of synthesizing a parametric model based on the measured frequency response have been studied extensively for a long time [13]. However, most of the conventional methods of constructing a
© 2000 Scripta Technica 46
where Imu1 denotes an m u 1 column vector, whose elements are all equal to unity. If we rewrite s as s jZ and substitute it into Eqs. (3) and (4), then it is clear that the frequency response of the system is expressed as Gc jZ . It is assumed here that M points of frequency response samples measured at frequencies Zk k 1, . . . , M are given as
increase drastically as the system order increases [13 16]. In this paper, we propose a new approach to freq u e n cy d o m a i n s u b s p a c e i d e n t i fi c a t i o n o f continuous-time models, by extending the time domain w-operator to its frequency counterpart. First, we substitute w jZ jZ D / jZ D into the derived w-operator state-space model so that a subspace identification algorithm can be applied. The identified w-operator state-space model can be transformed back to the common continuous-time state-space model in the Laplace operator. The w-operator is solely a bilinear transformation of the Laplace operator and hence no discretization error arises, so that frequency scale prewarping is not necessary. In the frequency domain subspace identification methods reported so far in the literature, consistent identification results can be obtained by utilizing a weighting matrix based on the covariance matrix of the measurement noise [10, 11]. However, in many practical situations, it is not easy to obtain information concerning the measurement noise in advance. In this paper, an instrumental variable (IV) matrix in the frequency domain is also proposed in order to obtain consistent estimates in the presence of measurement noise. Simulation results are included to verify the efficiency of the proposed algorithms.
(5) where N jZk is a stochastic noise with zero mean. Our task here is to estimate the equivalent matrices of Ac, Bc, Cc, Dc, given the noise-corrupted samples of the frequency response. 3. w-Operator Model 3.1 Introduction of the w-operator model As claimed in the literature, if we identify the Laplace operator model (4) directly with s jZ, the condition numbers of the data matrices involved in the identification algorithms increase drastically as the system order increases, such that the identification problem becomes illconditioned [11]. Instead of direct use of the Laplace operator s, we introduce here the w-operator [1315]:
2. Statement of the Problem
(6)
Consider a continuous-time system represented by the following continuous-time state-space model:
where D > 0, and in the frequency domain, we have
(1)
(7)
where u t Rm, y t Rl, x t Rn are the system input, output, and state variable vectors, respectively, and Ac Rnun, Bc Rnum, Cc Rlun, Dc Rlum. Using the Laplace operator s, we have the following state-space model and transfer function model, respectively:
According to Eq. (6), we can express the Laplace operator s as (8)
(2)
Substituting it into Eq. (3), we have the following transfer function model in the w-operator:
(3) (9)
where In denotes an n u n unity matrix. When the input is given as U s Imu1, the output becomes Gc s , and Eq. (2) can be rewritten as
The equivalent state-space model is expressed as (10)
(4)
where
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Lemma 3 Let q t n. If the Laplace operator statespace model (2) is controllable, then (11) has full row rank. Proof: Since the system is controllable, for all O C, the matrix [Ac OIn Bc], has full row rank [18]. Using Eq. (11), we have
(12) Inspection of Eq. (11) indicates that it is necessary that D O Ac , where O() denotes the eigenvalues of a matrix, to ensure the existence of the w-operator models shown in Eqs. (9) and (10) [1315]. That is, D should not coincide with one of the real poles of the system. In the sequel, it is assumed that D O Ac . For design policy of a suitable D, the reader is referred to Ref. 14 and 15.
and hence the matrix [Aw OInBw] has full row rank, that is,
3.2 Some properties of the Z-operator model Before providing concrete identification algorithms, some important properties of the w-operator model are summarized here. Lemma 1 For the Laplace operator transfer function model (3) and the w-operator transfer function model (9), the following are equivalent:
has full row rank [18]. Based on Lemmas 1 to 3, we can prove the following lemma, along the same lines as Lemmas 1 and 2 in Ref. 10. Lemma 4 Let M t n. If the Laplace operator statespace model (2) is controllable, then for Zk k 1, . . . , M such that jZk O Ac ,
Proof: If jZ O Ac , then we have
has full row rank. Similarly, if the Laplace operator state-space model (2) is observable, we can prove the following lemmas. Lemma 5 Let q t n. If the Laplace operator statespace model (2) is observable, then
From Eq. (7), we have jZ D 1 w jZ / 1 w jZ . Substituting it into jZIn Ac , we can verify that
that is, w jZ O Aw . In a similar manner, it can be shown that if w jZ O Aw , then jZ O Ac . Lemma 2 For the Laplace operator transfer function model (3) and the w-operator transfer function model (9), if jZ O Ac , then
has full column rank. Lemma 6 Let M t n l. If the Laplace operator state-space model (2) is observable, then for Zk k 1, . . . , M such that jZk O Ac ,
Proof: Lemma 1 ensures that both Gw w jZ and Gc jZ exist. Substituting Eqs. (7) and (11) into Eq. (9), we can verify that Gw w jZ Gc jZ . " Lemmas 1 and 2 imply that if the frequency response of the Laplace operator model exists, the frequency response of the w-operator model also exists and is equivalent to the former.
has full column rank.
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3.3 Construction of data matrices In the frequency domain, the w-operator model can be expressed as (13) (20) Using the measured frequency response Gm jZ at Z, we have the following result:
(21)
(14)
(22) The operations involved in the above data matrices where are the frequency counterparts of the following Laguerre filters which have been extensively applied in system identification problems:
(15)
A j 1 -th-order Laguerre filter consists of a first-order low-pass filter and a j-th-order all-pass filter, and its impulse response is the classical Laguerre function [16]. So far, the Laguerre filters have been studied mainly for approximation of linear systems [16]. In this study, however, we introduce the Laguerre filters to improve the conditioning of the corresponding data matrices. It is pointed out in the literature that the condition number of matrix I fi,M in Eq. (20) may influence the identification results significantly [8]. Inspection of I fi,M indicates that the power of jZ D / jZ D in each row vector differs from that of the others, that is, although the amplitude of each row vector is the same as that of the others, the phase of it can be shifted according to the power of jZ D / jZ D . It is not difficult to understand that we can shift the phase of each row vector of I fi,M by the w-operators with a suitable D, while keeping the amplitude unchanged. Therefore, it is easy to make the matrix have full row rank without large condition numbers [16]. On the other hand, for the Laplace operator model (2), the corresponding matrix becomes
(16)
(17)
Furthermore, with the samples of the frequency response at Zk k 1, . . . , M , we have (18) where
(19) 49
Multiplying both sides of Eq. (23) from the right by P AI , we have the following equation:
Especially for comparatively large bandwidth, as the system order increases, the differences of magnitudes among the row vectors may be very large, such that the condition number increases drastically [11]. This may make the identification results degenerate significantly. Since we are interested only in real-valued matrices of the system, we will rewrite the complex matrices with separated real and imaginary parts:
(26) f It is assumed here that the effects of the noise term N i,M on the right-hand side of Eq. (23) can be negligible. How to treat the noise term will be discussed in the next subsection. Inspection of Eq. (26) indicates that if rank G fi,MP AI rank Gic fwMP AI n, then t h e colu m n space of G fi,MP AI is equal to that of Gi such that we can recover the column space of Gi from the matrix product G fi,MP AI [5, 6]. This will be explained in detail as follows. The LQ factorizations of the input and output data matrix pair, and of the input and state variable matrix pair are given as the following, respectively:
(23) where
(24)
(27) 4. Identification Algorithms where R11 Rmiumi, R21 Rliumi, R22 Rliuli, Rx1 Rnumi, Rx2 Rnun. We can derive the following equation from Eqs. (23) and (27), when the noise term in Eq. (23) is neglected:
4.1 MOESP method When the noise effects are negligible, we can apply the MOESP method (MIMO Output-Error State Space model identification approach) [68] to identify the w-operator state-space model. Before discussing the MOESP method in detail, we first state the following theorem. Theorem 1 Let M t i n. If the Laplace operator state-space model (2) is controllable, then for Zk k 1, . . . , M such that jZk O Ac ,
(28) Hence, since Q2QT1 tain
0, QxQT1
0, and Q1QT1
Imi, we ob(29)
That is, (30)
Proof: According to the conditions stated in Theorem 1 and the result of Lemma 2, we can conclude that Gw w jZk Gc jZk exists for Zk k 1, . . . , M . By using the results of Lemmas 3 and 4, we can prove the following result along the same lines of Lemma 5 in Ref. 10:
where (31)
Furthermore, the result of the theorem holds according to Lemma 3 in Ref. 10. E f as Define the projection onto the null space of I i,M
According to the result of Theorem 1, Rx2 has full rank. Hence, we have rank Rx2Qx n. Furthermore, we have rank Gi n accor ding to Lemma 5, and hence we have rank GiRx2Qx r ank R22Q2 n. This implies that the column space of Gi is equal to that of R22. The basis vectors of the column space of R22 can be obtained by the following SVD:
(25)
(32)
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where Un Rliun, UAn Rliu li n , Sn Rnun, S2 R lin u lin , VTn Rnuli, V An T R lin uli, and n is the estimated system order, that is, the number of the relatively large ones in the li singular values of R22. Since the column space of R22 is spanned by the orthogonal matrix Un, there exists a nonsingular matrix T such that
Case 2: Dc 0. Dc 0 corresponds to the case where the system under study is strictly proper. In practical situations, it is often the case that the system under study is strictly proper. Therefore, it is recommended to take the condition Dc 0 into account. Based on Eqs. (9) and (12), we can compute BwT and DwT by solving the following equations:
(33)
(39)
Based on the above discussions, we can identify the w-operator state-space model and hence the original Laplace operator state-space model by the following MOESP method [5, 6]: Step 1 Construct the data matrices G fi,M and I fi,M , where i ! n.
(40) where
Step 2 Perform the LQ factorization of the matrix pair of G fi,M and I fi,M . Step 3 Perform the SVD of matrix R22. Step 4 Compute AwT and CwT by solving the following equations:
(41)
(34)
It is obvious that M2 has full column rank if M1 has full column rank. Step 6 Compute the matrices AcT, BcT, CcT, Dc using Eq. (12).
(35) and U 2 are the submatrices composed of the where U 1 n n first l i 1 rows and last l i 1 rows of Un, respectively. Notice that Un 1 : l, : is a common notation in Matlab, which denotes the submatr ix composed of the fir st l columns of Un. Step 5 Compute BwT and Dw. Case 1: No prior information of Dc is available. Compute BwT and DwT by solving the following equation:
4.2 Instrumental variable method When the noise effects cannot be neglected, with the MOESP method, the estimate of the range space of Gi is usually biased, except for the case where the covariance of the noise term N fi,M in Eq. (23) is a multiple of the unit matrix [8, 10, 11]. In Refs. 10 and 11, consistent identification results can be obtained by utilizing a weighting matrix based on the covariance matrix of the measurement noise [10, 11]. However, in many practical situations, it is not easy to obtain information concerning the measurement noise in advance. To yield consistent estimates of the range space of Gi, we propose here an IV-MOESP (Instrumental Variable MOESP) method with the IV matrix P fE,M defined as the following:
(36) where (37)
(42) where
(38) (43) Notice that the result of Lemma 6 guarantees that M1 has full column rank.
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To remove I fi,M and N fi,M in Eq. (23), we multiply both T sides of Eq. (23) from the right by 1 / M P AI P fE,M . This leads to
(47).
Step 2 Perform the LQ factorization as shown in Eq. Step 3 Perform the SVD of matrix R22RT32.
(49) (44)
Step 4 Compute AwT and CwT. Step 5 Compute BwT and Dw. Step 6 Compute AcT, BcT, CcT, Dc .
Then consistent estimate of the range space of Gi can be achieved if the following two conditions are satisfied [8, 10, 11]:
5. Numerical Example (45)
Consider a sixth-order SISO continuous-time system studied in Ref. 11, whose poles are located at 0.1 r 0.995i, 0.25 r 4.9937i, 0.06 r 2.9994i, respectively:
(46) The first condition is introduced to remove the second term on the right-hand side of Eq. (44). This is guaranteed if the noise N jZk k 1, . . . , M is a stochastic noise with zero mean. The second condition is the rank condition to recover the column space of Gi. Although it is not an easy task to prove the second condition theoretically, it is considered to be satisfied in generic cases [8, 19]. We next perform LQ factorization of the matrix composed of the inputoutput data and IV matrix as follows:
(50) (47)
Usually, the variance estimation error due to the measurement noise decreases when the number of frequency response samples increases. In this study, however, for the sake of comparison with the results in Ref. 11, the data are sampled under the same condition as in Ref. 11. The frequency response of the system is sampled at Zk = 0.05, 0.10, 0.15, . . . , 9 rad/s with a spacing interval of 0.05 rad/s, that is, M = 180. Monte Carlo simulations are performed based on 100 independent data sets with different realizations of the measurement noise. As in Ref. 11, two types of measurements noise are considered here: 1: Relative noise
where R11 Rmiumi, R21 Rliumi, R22 Rliuli, R31 RmEumi, R32 RmEuli, R33 RmEumE. fT Equations (31) and (47) lead to P AI P E, M T T T f T Q 2 Q 2P E,M Q2 R32. We then have the following result based on Eqs. (44), (45), and (47):
(48)
(51)
It is obvious that when rank condition (46) is satisfied, the column space of Gi is equal to that of R22RT32. Additionally, since the size of matrix R22RT32 is li u mE, to recover the column space of Gi, it is necessary that mE t n. The concrete algorithm of the IV-MOESP method is described as follows. Step 1 Construct the data matrices G fi,M, I fi,M , and f P E,M , where i ! n and E t n / m.
2: Absolute noise (52) where eR jZk and eI jZk are unit variance, zero-mean, white Gaussian noises.
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the averaged model obtained through 100 experiments is shown in (c) of Fig. 1. In (b) and (c), the dashed and solid lines denote, respectively, the true and estimated frequency responses. It can be verified in Fig. 1 that due to the noise effects, the third resonance mode is not identified by the MOESP method. On the other hand, the results of the IV-MOESP method shown in Fig. 2 indicate that when the IV-MOESP method is applied, the noise effects are removed and the identification results are very satisfactory.
5.1 Identification results In each case of measurement noise, the proposed identification algorithms are performed on 100 data sets with different realizations of the measurement noise. For the sake of comparison, the order of the system model is fixed to 6, no matter what the exact singular values are. 5.1.1 The case of relative noise The identification results of the MOESP method are shown in Fig. 1. The nonsingular values and identified frequency responses of 100 experiments are shown, respectively, in (a) and (b) of Fig. 1. The frequency response of
The results using the MOESP method are shown in Fig. 3. In this case, since the noise level is not so significant,
Fig. 1. Results using the MOESP method in the case of relative noise (i = 15, D = 5).
Fig. 2. Results using the IV-MOESP method in the case of relative noise (i = 15, D = 5, E = 30).
5.1.2 The case of absolute noise
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the MOESP method still yields acceptable results. In contrast, in Ref. 11, the system model cannot be correctly identified except when a weighting matrix based on the covariance matrix of the measurement noise is utilized. The results obtained by the IV-MOESP method are shown in Fig. 4. As expected, the IV-MOESP method yields very satisfactory identification results. Moreover, a comparison of (a) of Fig. 3 with (a) of Fig. 4 indicates that it is easier for the IV-MOESP method to determine the system order based on the singular values.
In the proposed identification algorithms, the parameter D in the w-operator, and the positive integers i and
E are design parameters that should be chosen appropriately by the users. Here we present guidelines for these design parameters. Considering the sensitivity to errors of the estimated system parameters when we transform the identified w-operator model back to the Laplace operator model, the value of D should not be too far from the absolute values of the dominant poles of the system. For detailed analysis, the reader is referred to Refs. 14 and 15. Also, as mentioned in Section 2, D should not coincide with one of the real poles of the system. It should be emphasized here that the results are not sensitive to the value of D, according to our numerical experience. In the simulation studies in this paper, we used D = 5 and D = 10, respectively, for the two types of noise, to show that the results are not sensitive to the value of D. Although in the MOESP method the results
Fig. 3. Results using the MOESP method in the case of absolute noise (i = 15, D = 10).
Fig. 4. Results using the IV-MOESP method in the case of absolute noise (i = 15, D = 10, E = 30).
5.1.3 Guidelines for the design parameters of the identification algorithms
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may be influenced by the value of D, in the IV-MOESP method we have verified that the identification results are very satisfactory for a wide range of D. In the subspace identification methods, the roles of i and E are still not clarified, and no method of choosing optimal i and E has yet been established [20]. Usually, it is recommended to choose a large value of i. In the simulation studies in this paper, we chose i = 15, as is done in Ref. 11. We did not find a significant improvement when we increased i further. Although it is theoretically sufficient to let E t n / m, and E is usually chosen such that E i, we have verified that better identification results can be achieved when E = 20 to 40.
7. Verhaegen M, Dewilde P. Subspace model identification Part 2: Analysis of the elementary output-error state-space model identification algorithm. Int J Control 1992;56:12111241. 8. Verhaegen M. Subspace model identification Part 3: Analysis of the ordinary output-error state-space model identification algorithm. Int J Control 1993;58:555586. 9. Liu K, Jacques RN, Miller DW. Frequency domain structural system identification by observability range space extraction. J Dyn Syst Meas Control 1996;118:211220. 10. McKelvey T, Akcay H, Ljung L. Subspace-based multivariable system identification from frequency response data. IEEE Trans Autom Control 1996;AC41:960978. 11. Van Overschee P, Moor BD. Continuous-time frequency domain subspace system identification. Signal Process 1996;52:179194. 12. Rolain Y, Pintelon R, Xu KQ, Vold H. On the use of orthogonal polynomials in high order frequency domain system identification and its application to modal parameter estimation. Proc IEEE Conference on Decision and Control, p 33653373, 1994. 13. Haverkamp BRJ, Verhaegen M, Chou CT, Johansson R. Continuous-time subspace model identification using Laguerre filtering. 11th IFAC/IFORS Symp on Identification and System Parameter Estimation, p 11431148, 1997. 14. Yang ZJ. Subspace model identification of continuous-time systems with the aid of the w-operator. Trans Soc Instrum Control Eng 1998;34:546554. (in Japanese) 15. Yang ZJ, Sagara S, Wada K. Subspace identification of continuous-time model systems. 11th IFAC/IFORS Symp on Identification and System Parameter Estimation, p 16711676, 1997. 16. Wahlberg B. System identification using Laguerre models. IEEE Trans Autom Control 1991;AC36:551562. 17. Ljung L. System identification: Theory for the user. PrenticeHall; 1987. 18. Zhou K, Doyle JC, Glover K. Robust and optimal control. PrenticeHall; 1996. 19. Viberg M, Wahlberg B, Ottersten B. Analysis of state space system identification methods based on instrumental variables and subspace fitting. Automatica 1997;33:16031616. 20. Wahlberg B, Jansson M. 4SISD linear regression. Proc IEEE Conference on Decision and Control, p 28582863, 1994.
6. Conclusions In this paper, we have proposed an efficient and convenient approach to frequency domain subspace identification for continuous-time systems, by using the w-operator. Furthermore, we have proposed an IV method by constructing an IV matrix in the frequency domain based on the w-operator. The consistency of the IV method was shown theoretically. The efficiency of the proposed identification algorithms was verified through numerical simulations, and, finally, the guidelines for the design parameters of the identification algorithms were described briefly. REFERENCES 1. Schoukens J, Pintelon R. Identification of linear systemsA practical guideline to accurate modeling. Pergamon Press; 1991. 2. Sanathanan CK, Koerner J. Transfer function synthesis as a ratio of two complex polynomials. IEEE Trans Autom Control 1963;AC-8:5658. 3. Pintelon R, Guillaume P, Rolain Y, Schoukens J, Van Hamme H. Parametric identification of transfer functions in the frequency domainA survey. IEEE Trans Autom Control 1994;AC-39:557566. 4. Friedman JH, Khargonekar PP. A comparative applications study of frequency domain identification techniques. Proc American Control Conference, p 30553059, 1995. 5. Viberg M. Subspace-based methods for the identification of linear time-invariant systems. Automatica 1995;31:18351851. 6. Verhaegen M, Dewilde P. Subspace model identification Part 1: The output error state-space model identification class of algorithms. Int J Control 1992;56:11871210.
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AUTHORS (from left to right)
Zi-Jiang Yang (member) received his D.Eng. degree from Kyushu University in 1992. From 1992 to 1996, he was a research associate in the Department of Electrical Engineering, Kyushu Institute of Technology. Since 1996, he has been an associate professor in the Faculty of Computer Engineering and System Science, Kyushu Institute of Technology. His research interests include system identification, and related signal processing, genetic algorithms, neural networks, wavelet analysis, and motion control. He is a member of the Institute of Systems, Control and Information Engineers, IEICE, the Society of Instrument and Control Engineers, and the Robotics Society of Japan. Shingo Sanada (nonmember) graduated from the Faculty of Computer Engineering and System Science, Kyushu Institute of Technology, in 1997. He is currently in the masters course, Department of Information System Engineering, Graduate School of Computer Engineering and System Science, Kyushu Institute of Technology. He is engaged in research on system identification.
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