» Download Article

Authors
Ase, Hajime; Katayama, Tohru; Tanaka, Hideyuki

Identifier
10.3182/20090706-3-FR-2004.00181

Index Terms
Nonlinear System Identification

Abstract
We develop a state-space method of identifying a Wiener-Hammerstein system, where a nonlinearity is sandwiched by two linear systems. By dividing it into the linear system and Hammerstein system composed of the nonlinearity and the second linear system, we propose an iterative method of identifying the Hammerstein system by the orthogonal decomposition subspace method (ORT) and the linear system by minimizing the square norm of output prediction error, for which the identified Hammerstein model plays a role of instrument of measuring the output of the linear system. The data driven local coordinate (DDLC)-based gradient method is applied to this minimization. Numerical results for the benchmark problem are included to show the applicability of the present method.

References

1) Ase, H. and T. Katayama (2008). Identification of Wiener models by separable
least-squares. In: Proc. 50th Stochastic Systems Symposium (SSS08), Kyoto, Japan.
2) Bai, E.W.(2002). A blind approach to the Hammerstein-Wiener model identification.
Automatica 38(3), 967-979. 
2) Bai, E.W. and J. Reyland, Jr.(2008). Towards identification of Wiener systems with
the least amount of a priori information on the nonlinearity. Automatica 44(4), 910-919.
3) Bauer, D. and B. Ninness (2002). Asymptotic properties of least-squares estimation
of Hammerstein-Wiener models. Int. J. Control 75(1), 35-51.
4) Billings, S.A. and S.Y. Fakhouri (1982). Identifications of systems containing linear 
dynamic and static nonlinear elements. Automatica 18(1), 15-26.
5) Boutayeb, M. and M. Darouach(1995). Recursive identification method for MISO Wiener-
Hammerstein model. IEEE Trans. Automatic Control 40(2), 287-291.
6) Bruls, J., C.T. Chou, B.R.J. Haverkamp and M. Verhaegen(1999). Linear and nonlinear
system identification using separable least-squares. European J. Control 5(1), 116-128.
7) Crama, P. and J. Schoukens (2005). Computing an initial estimate of a Wiener-
Hammerstein system with a random phase multisine excitation. IEEE Trans. Instrumentation and Measurement 54(1), 117-122.
8) Ding, F. and T. Chen (2005). Identification of Hammerstein nonlinear ARMAX systems.
Automatica 41(9), 1479-1489.
9) Gomez, J.C.and E.Baeyens(2002). Subspace identification of multivariable Hammerstein
and Wiener models. In: Proc. IFAC 15th World Congress. Barcelona.
10) Haber, R.and H. Unbehauen(1990). Structure identification of nonlinear dynamic systems- a survey on input/output approaches. Automatica 26(4), 651-677.
11) Hunter, I.W. and M.J. Korenberg (1986). The identification of nonlinear biological systems: Wiener and Hammerstein cascade models. Biological Cybernetics 55, 135-144.
12) Kalafatis, A.D., L. Wang and W.R. Cluett(1997). Identification of Wiener-type 
nonlinear systems in a noisy environment. Int. J. Control 66(6), 923-941.
13) Katayama, T.(2005).Subspace Methods for System Identification. Springer.
14) Ljung, L.(1999). System Identification - Theory for the User. 2nd ed. Prentice-Hall.
15) Ljung, L.(2006). Some aspects of nonlinear system identification. In: Proc. 14th IFAC
Symposium on System Identification (SYSID2006). Newcastle, Australia. pp.110-121.
16) McKelvey, T., A. Helmersson and T. Ribarits(2004). Data driven local coordinates 
for multivariable linear systems and their applications to system identification. 
Automatica 40(9),1629-1635.
17) Palanthandalam-Madapusi, H.J., D.S. Bernstein and A.J. Ridley(2007). Subspace
identification of periodically switching Hammerstein-Wiener models for magnetosphere
dynamics. IEEE Control Systems Magazine 27(5),109-123.
18) Schoukens, J., J. Suykens and L. Ljung 2008a). Wiener-Hammerstein Benchmark.
SYSID2009 Webpage.
19) Schoukens, J., R. Pintelon and M. Enqvist (2008b). Study of the LTI relations between
the outputs of the coupled Wiener systems and its application to the generation of 
initial estimates for Wiener-Hammerstein systems. Automatica 44(7), 1654-1665.
20) Schoukens, J., W.D. Widanage, K.R. Godfrey and R. Pintelon (2007). Initial estimates
for the dynamics of a Hammerstein system. Automatica 43(7), 1296-1301.
21) Tian, C. and T. Fujii (2005). Nonlinear system identification of rapid thermal 
processing. Control Engineering Practice 13(6), 681-687.
22) Vandersteen, G., Y. Rolain and J. Schoukens (1997). Non-parametric estimation of the
frequency response functions of the linear blocks of a Wiener-Hammerstein model.
Automatica 33(7), 1351-1355.
23) Verhaegen, M. and D. Westwick(1996). Identifying MIMO Hammerstein systems in the
context of subspace model identification methods. Int. J. Control 63(2), 331-349.
24) Westwick, D. and M.Verhaegen (1996). Identifying MIMO Wiener systems using subspace
model identification methods. Signal Processing 52(2), 235-258. 
25) Wigren, T.(1993). Recursive prediction error identification using the nonlinear
Wiener model. Automatica 29(4), 1011-1025.
26) Wills, A. and B. Ninness (2008). On gradient-based search for multivariable system
estimates. IEEE Trans. Automatic Control 53(1), 298-306.
27) Zhang, Q., A. Iouditski and L. Ljung (2006). Identification of Wiener system with
monotonous nonlinearity. In: Proc.14th IFAC Symposium on SystemI dentification (SYSID 
2006). Newcastle, Austria, pp. 166-171.