An extended instrumental variable (EIV) method is considered for the stochastic Hammerstein system (ARMAX and general model structure). The EIV method provides consistent parameter estimates by eliminating noise-induced bias in the least square (LS) method. To estimate the parameters, the Hammerstein model is formulated using the bilinear parameterization. The bilinear model is identified by introducing the nonlinear instrumental variables obtained from transformed delayed outputs using nonlinear mapping and polynomial basis of delayed inputs. These instruments are analyzed in full generality by computing the bounds on expected relationship between instruments and noise for the general noise disturbance structure. Then, a specific case with hyperbolic tangent (tanh) transformation is considered. Comparative performance analysis of the proposed IV method with the existing IV method, the data filtering-based LS methods, and the extended LS method shows improvement in the statistical properties of parameters estimates.
AlbertAE (1972) Regression and the Moore–Penrose Pseudoinverse, New York: Academic Press.
2.
BaiEWChanKS (2008) Identification of an additive nonlinear system and its applications in generalized Hammerstein models. Automatica44(2): 430–436.
3.
BalestrinoALandiAOuld-ZmirliMet al. (2001) Automatic nonlinear auto-tuning method for Hammerstein modeling of electrical drives. IEEE Transactions on Industrial Electronics48(3): 645–655.
4.
BerkKN (1974) Consistent autoregressive spectral estimates. The Annals of Statistics2(3): 489–502.
5.
BoxGEPPierceDA (1970) Distribution of residual autocorrelations in Autoregressive-integrated moving average time series models. Journal of the American Statistical Association65(332): 1509–1526.
6.
BrulsJChouCTHaverkampBet al. (1999) Linear and nonlinear system identification using separable least squares. European Journal of Control5(11): 116–128.
7.
BuhlmannP (1995) Moving-average representation of autoregressive approximations. Stochastic Processes and their Applications60: 331–342.
8.
CainesPE (1976) On the asymptotic normality of instrumental variable and least squares estimators. IEEE Transactions on Automatic Control21(4): 598–600.
9.
DingFChenT (2005) Identification of Hammerstein nonlinear ARMAX systems. Automatica41(9): 1479–1489.
10.
DingFShiYChenT (2007) Auxiliary model-based least-squares identification methods for Hammerstein output-error systems. Systems & Control Letters56(5): 373–380.
11.
GrimmettGStirzakerD (2001) Probability and Random Processes, Oxford: Oxford University Press.
12.
HuPDingF (2013) Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principal. Nonlinear Dynamics73(1): 583–592.
13.
JuradoF (2006) A method for the identification of solid oxide fuel cells using a Hammerstein model. Journal of Power Sources154(1): 145–152.
14.
KuoPHoseinAFarmanbordaMS (2012) Nonlinear output feedback control of a flexible link using adaptive neural network: controller design. Journal of Vibration and Control19(11): 1690–1708.
15.
Laurain V, Gilson M, Garnier H, et al. (2008) Refined instrumental variable methods for identification of Hammerstein continuous time Box–Jenkins model. In: Proceedings of the 47th IEEE conference on decision and control, Cancun, Mexico, 9–11 December 2008, pp. 1386–1391. IEEE. DOI: 10.1109/CDC.2008.4738853.
16.
LiJDingFHuaL (2014) Maximum likelihood Newton recursive and the Newton iterative estimation algorithms for Hammerstein CARAR systems. Nonlinear Dynamics75(1): 235–245.
17.
LjungL (1999) System Identification: Theory for the User, Upper Saddle River, NJ: Prentice-Hall.
18.
NarendraKSGallmanPG (1966) An iterative method for the identification of nonlinear systems using Hammerstein model. IEEE Transactions on Automatic Control11(3): 546–550.
19.
SchoukensJWidanageWDGodfreyKRet al. (2007) Initial estimates for the dynamics of a Hammerstein system. Automatica43(7): 1296–1301.
20.
StoneMH (1948) The generalized Weierstrass approximation theorem. Mathematics Magazine21(5): 237–254.
21.
SoderstromTStoicaP (1981) Comparison of some instrumental variable methods-consistency and accuracy aspects. Automatica17(1): 101–115.
22.
SoderstromTStoicaP (1983) Instrumental Variable Methods for System Identification, Berlin: Springer.
23.
SoderstromTStoicaP (1989) System identification, Eaglewood Cliffs, NJ: Prentice-Hall.
24.
SrinivasanRRengaswamyRNarasimhanSet al. (2005) Control loop performance assessment. 2. Hammerstein model approach for stiction diagnosis. Industrial & Engineering Chemistry Research44(17): 6719–6728.
25.
StoicaPSoderstromT (1983) Optimal instrumental variable estimation and approximate implementations. IEEE Transactions on Automatic Control28(7): 757–771.
26.
TerrenceMAAndrewBN (2010) Uniform convergence of Vapnik–Chervonenkis classes under ergodic sampling. The Annals of Probability38(4): 1345–1367.
27.
VörösJ (1999) Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities. IEEE Transactions on Automatic Control44(11): 2145–2149.
28.
WangDQ (2016) Hierarchical parameter estimation for a class of MIMO Hammerstein systems based on the reframed models. Applied Mathematics Letters57: 13–19.
29.
WangDQDingF (2016a) Parameter estimation algorithms for multivariable Hammerstein CARMA systems. Information Sciences355-356: 237–248.
30.
WangDQDingFChuY (2013) Data filtering based recursive least square algorithm for Hammerstein systems using the key term separation principal. Information Sciences222: 203–212.
31.
WangDQLiuHBDingF (2015) Highly efficient identification methods for dual-rate Hammerstein systems. IEEE Transactions on Control System Technology23(5): 1952–1960.
32.
WangYDingF (2015) Iterative estimation for a non-linear IIR filter with moving average noise by means of the data filtering technique. IMA Journal of Mathematics Control and Information. Epub ahead of print 24 December 2015. DOI: 10.1093/imamci/dnv067.
33.
WangYDingF (2016b) Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model. Automatica71: 308–313.
34.
WangYDingF (2016c) Recursive least squares algorithm and gradient algorithm for Hammerstein–Wiener systems using the data filtering. Nonlinear Dynamics84(2): 1045–1053.
35.
WangZJiZ (2014) Data-filtering-based iterative identification methods for nonlinear FIR-MA systems. Journal of Vibration and Control20(14): 2193–2201.
36.
WestwickDKearneyR (2001) Separable least squares identification of nonlinear Hammerstein models: application to stretch reflex dynamics. Annals of Biomedical Engineering29(8): 707–718.
37.
XieLYangL (2010) Gradient-based iterative identification for nonuniform sampling output error systems. Journal of Vibration and Control17(3): 471–478.
38.
YoungPC (1970) An instrumental variable method for real time identification of a noisy process. Automatica6(2): 271–287.
