- Email: [email protected]
Discussion on: “Subspace-based Identification Algorithms for Hammerstein and Wiener Models”
European Journal of Control (2005)11:137–149 # 2005 EUCA
Discussion on: ‘‘Subspace-based Identification Algorithms for Hammerstein and Wiener Models’’ Er-Wei Bai Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA
1. The Approach The paper by Gomez and Baeyens follows the two stage identification approach for Hammerstein and Wiener models. In general, if the nonlinearity is of polynomial type, identification of Hammerstein and to the certain extent, Wiener models is a bilinear problem. In the two stage approach, by treating the bilinear terms as independent ones, a Hammerstein model possesses a ‘‘linear representation’’. Then, any linear identification algorithm applies. In the paper, subspace algorithms are applied. In the second stage, the over-parameterized parameter vector is projected back to the original parameter space in some ways. The idea of the two stage identification was proposed during the seventies [2,3]. However, the optimal projection problem was not solved in the context of the two stage identification until the works of [1,4] where the singular vectors are shown to be the optimal ones. The two stage identification approach is by now well known and is one of the most common approaches for Hammerstein model identifications.
2. Discussion The paper shows that the two stage approach applies to Wiener models as well if the nonlinearity is invertible. This is a welcome addition to the literature of Hammerstein and Wiener model identifications. To make the approach practical, however, several tough
E-mail: [email protected]
issues have to be further investigated. Some are related to the two stage approach in particular and others are with the general frameworks of Hammerstein and Wiener model identifications. Dimension problem. This is one of the most serious problems with the two stage approach. A distinct advantage of the two stage approach lies in its simplicity of a ‘‘linear representation’’. Because of this ‘‘linear representation’’, any existing linear identification algorithm applies. However, parameterizing a bilinear system into a ‘‘linear presentation’’ is not free and the price is the increased dimension. Consider a SISO case with D ¼ 0. The number of unknown parameters in Bi, i ¼ 1, . . . , r, is n þ r. In the over-parameterization form of (6) and (7), the number of unknown parameters in Bi, becomes n r. In practices, the reason why the unknown nonlinearity may be assumed to be a polynomial or of polynomial type is that any continuous function can be approximated within arbitrary accuracy by a polynomial on a compact set. To have a good approximation, however, the order of the polynomial has to be reasonably high and this causes the problem. Even with modest orders n ¼ r ¼ 10, we have n þ r ¼ 20
and
n r ¼ 100:
Thus, by over-parameterizing a bilinear system into a ‘‘linear’’ one, dimension is increased by a factor of 10. It is well known in the adaptive community that the performance of an adaptive estimation algorithm is closely coupled with the number of parameters to be estimated. The performance
138
Discussion on: ‘‘Subspace-based Identification Algorithms for H. and W. Models’’
deteriorates quickly when the dimension increases. Thus, the two stage approach works better for a low dimensional system unless some modifications can be made. Polynomial representation of nonlinearities. The two stage approach has to assume that the unknown nonlinearity is linear in unknown parameters. If the nonlinearity is nonlinear in unknown parameters, over-parameterization would not make the entire system ‘‘linear’’ and the approach falls apart. How to extend the approach to a more general class of nonlinearities is a relevant issue. Invertibility of the output nonlinearity. It was assumed in the paper that the output nonlinearity N is invertible. Though this assumption is valid in some applications and becomes standard for Wiener model identifications, one has to realize that it also brings some unintended consequence, e.g., the sensitivity issue. Again, a polynomial representation of the output nonlinearity is an approximation and the inverse can be very sensitive to the approximation error as well as the noise if the order of a polynomial is high. Consider a SISO output nonlinearity with measurement noise k
Order. In the two stage approach, the order of the linear system as well as the order of polynomial representing the unknown nonlinearity are available a priori. This is unlikely in reality. In general, not the exact orders but the upper bounds are available. Developing an algorithm using only the upper bounds not the exact order information will be useful. Unknown nonlinearity structures. This issue has more to do with general identification of a block oriented system identification. The problem is not trivial but often encountered in reality. Several approaches have been proposed in the literature including stochastic method, graphical method, Fourier transform method, orthonormal basis method and etc. In a sense, orthonormal basis method is very similar to the polynomial approach discussed in the paper and the difference is that the order is not known a priori. It is conceivable to combine the order estimation into the two stage approach so some nonlinearities with unknown structures would be allowed.
3. Summary
yk ¼ Nðvk Þ þ k Suppose N1 exists and takes a polynomial form as assumed in the paper. Then, in the absence of noise X i gi ðyk Þ ¼ N1 ðyk Þ ¼ N1 Nðvk Þ ¼ vk : Now, in the presence of noise k, one obtains X i gi ðyk Þ ¼ vk þ N1 ðNðvk Þ þ k Þ vk : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Though there exist a number of well developed methods in the literature, the field of Hammerstein and Wiener model identification is still a very active area. There are many questions and problems that existing methods fail to answer. The hope is that the discussion will help to stimulate more discussions that will lead to further research in the area.
References
new noise term
The new noise term can be significantly larger than 1=3 the original k. For instance, let Nðvk Þ ¼ vk and N1 ðyk Þ ¼ y3k . For small noises k, N
1
ðNðvk Þ þ k Þ vk ¼ vk 1 þ
k 1=3
vk
!3 vk
2=3
3 k v k : 4=3
The energy of the noise is amplified by 9 vk .
1. Bai EW. An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems. Automatica 1998; 34(3): 333–338 2. Chang F, Luus R. A non-iterative method for identification using Hammerstein model. IEEE Trans. Autom Control 1971; 16: 464–468 3. Hsia T. A multi-stage least squares method for identifying Hammerstein model non-linear systems, Proceedings of the CDC, Clearwater Florida, pp 934–938 4. Rangan S, Wolodkin G, Poolla K. ‘‘Identification methods for Hammerstein systems’’, Proceedings of the CDC, New Orleans, pp 697–702
Discussion on: ‘‘Subspace-based Identification Algorithms for Hammerstein and Wiener Models’’ Er-Wei Bai Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA
1. The Approach The paper by Gomez and Baeyens follows the two stage identification approach for Hammerstein and Wiener models. In general, if the nonlinearity is of polynomial type, identification of Hammerstein and to the certain extent, Wiener models is a bilinear problem. In the two stage approach, by treating the bilinear terms as independent ones, a Hammerstein model possesses a ‘‘linear representation’’. Then, any linear identification algorithm applies. In the paper, subspace algorithms are applied. In the second stage, the over-parameterized parameter vector is projected back to the original parameter space in some ways. The idea of the two stage identification was proposed during the seventies [2,3]. However, the optimal projection problem was not solved in the context of the two stage identification until the works of [1,4] where the singular vectors are shown to be the optimal ones. The two stage identification approach is by now well known and is one of the most common approaches for Hammerstein model identifications.
2. Discussion The paper shows that the two stage approach applies to Wiener models as well if the nonlinearity is invertible. This is a welcome addition to the literature of Hammerstein and Wiener model identifications. To make the approach practical, however, several tough
E-mail: [email protected]
issues have to be further investigated. Some are related to the two stage approach in particular and others are with the general frameworks of Hammerstein and Wiener model identifications. Dimension problem. This is one of the most serious problems with the two stage approach. A distinct advantage of the two stage approach lies in its simplicity of a ‘‘linear representation’’. Because of this ‘‘linear representation’’, any existing linear identification algorithm applies. However, parameterizing a bilinear system into a ‘‘linear presentation’’ is not free and the price is the increased dimension. Consider a SISO case with D ¼ 0. The number of unknown parameters in Bi, i ¼ 1, . . . , r, is n þ r. In the over-parameterization form of (6) and (7), the number of unknown parameters in Bi, becomes n r. In practices, the reason why the unknown nonlinearity may be assumed to be a polynomial or of polynomial type is that any continuous function can be approximated within arbitrary accuracy by a polynomial on a compact set. To have a good approximation, however, the order of the polynomial has to be reasonably high and this causes the problem. Even with modest orders n ¼ r ¼ 10, we have n þ r ¼ 20
and
n r ¼ 100:
Thus, by over-parameterizing a bilinear system into a ‘‘linear’’ one, dimension is increased by a factor of 10. It is well known in the adaptive community that the performance of an adaptive estimation algorithm is closely coupled with the number of parameters to be estimated. The performance
138
Discussion on: ‘‘Subspace-based Identification Algorithms for H. and W. Models’’
deteriorates quickly when the dimension increases. Thus, the two stage approach works better for a low dimensional system unless some modifications can be made. Polynomial representation of nonlinearities. The two stage approach has to assume that the unknown nonlinearity is linear in unknown parameters. If the nonlinearity is nonlinear in unknown parameters, over-parameterization would not make the entire system ‘‘linear’’ and the approach falls apart. How to extend the approach to a more general class of nonlinearities is a relevant issue. Invertibility of the output nonlinearity. It was assumed in the paper that the output nonlinearity N is invertible. Though this assumption is valid in some applications and becomes standard for Wiener model identifications, one has to realize that it also brings some unintended consequence, e.g., the sensitivity issue. Again, a polynomial representation of the output nonlinearity is an approximation and the inverse can be very sensitive to the approximation error as well as the noise if the order of a polynomial is high. Consider a SISO output nonlinearity with measurement noise k
Order. In the two stage approach, the order of the linear system as well as the order of polynomial representing the unknown nonlinearity are available a priori. This is unlikely in reality. In general, not the exact orders but the upper bounds are available. Developing an algorithm using only the upper bounds not the exact order information will be useful. Unknown nonlinearity structures. This issue has more to do with general identification of a block oriented system identification. The problem is not trivial but often encountered in reality. Several approaches have been proposed in the literature including stochastic method, graphical method, Fourier transform method, orthonormal basis method and etc. In a sense, orthonormal basis method is very similar to the polynomial approach discussed in the paper and the difference is that the order is not known a priori. It is conceivable to combine the order estimation into the two stage approach so some nonlinearities with unknown structures would be allowed.
3. Summary
yk ¼ Nðvk Þ þ k Suppose N1 exists and takes a polynomial form as assumed in the paper. Then, in the absence of noise X i gi ðyk Þ ¼ N1 ðyk Þ ¼ N1 Nðvk Þ ¼ vk : Now, in the presence of noise k, one obtains X i gi ðyk Þ ¼ vk þ N1 ðNðvk Þ þ k Þ vk : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Though there exist a number of well developed methods in the literature, the field of Hammerstein and Wiener model identification is still a very active area. There are many questions and problems that existing methods fail to answer. The hope is that the discussion will help to stimulate more discussions that will lead to further research in the area.
References
new noise term
The new noise term can be significantly larger than 1=3 the original k. For instance, let Nðvk Þ ¼ vk and N1 ðyk Þ ¼ y3k . For small noises k, N
1
ðNðvk Þ þ k Þ vk ¼ vk 1 þ
k 1=3
vk
!3 vk
2=3
3 k v k : 4=3
The energy of the noise is amplified by 9 vk .
1. Bai EW. An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems. Automatica 1998; 34(3): 333–338 2. Chang F, Luus R. A non-iterative method for identification using Hammerstein model. IEEE Trans. Autom Control 1971; 16: 464–468 3. Hsia T. A multi-stage least squares method for identifying Hammerstein model non-linear systems, Proceedings of the CDC, Clearwater Florida, pp 934–938 4. Rangan S, Wolodkin G, Poolla K. ‘‘Identification methods for Hammerstein systems’’, Proceedings of the CDC, New Orleans, pp 697–702