A Blind Approach to Closed-Loop Identification of Hammerstein Systems (original) (raw)
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Blind maximum likelihood identification of Hammerstein systems
Automatica, 2008
This paper is about the identification of discrete-time Hammerstein systems from output measurements only (blind identification). Assuming that the unobserved input is white Gaussian noise, that the static nonlinearity is invertible, and that the output is observed without errors, a Gaussian maximum likelihood estimator is constructed. Its asymptotic properties are analyzed and the Cramér-Rao lower bound is calculated. In practice, the latter can be computed accurately without using the strong law of large numbers. A two-step procedure is described that allows to find high quality initial estimates to start up the iterative Gauss-Newton based optimization scheme. The paper includes the illustration of the method on a simulation example. A theoretical analysis demonstrates that additive output measurement noise introduces a bias that is proportional to the variance of that additive, unmodeled noise source. The simulations support this result, and show that this bias is insignificant beyond a certain Signal-to-Noise Ratio (40 dB in the example).
A new closed-loop identification method of a hammerstein-type system with a pure time delay
2007 Mediterranean Conference on Control & Automation, 2007
A procedure for the closed-loop identification of a class of Hammerstein type nonlinear plants with a pure time delay in the linear part, based on a generalization of the well-known Ziegler-Nichols' (ZN) experiment, is proposed. It provides a simultaneous estimation of the linear plant dynamic, input-output approximate model of the non-linear part and value of the time delay. As in the ZN method, only a suitable controller is needed for experiments. Tuning of PID controller based on the obtained plant model is compared with PID tuning recommended by Ziegler-Nichols method and clear advantage of knowledge of the plant model is demonstrated.
A Decoupling Derivative-Based Approach for Hammerstein System Identification
Proceedings of the 17th IFAC World Congress, 2008, 2008
This paper proposes a non iterative algorithm for the identification of Hammerstein model, using the sampled output data obtained from the step response, giving a continuoustime model for the linear part and a point-wise estimation of the nonlinear one. Key in the derivation of the results is the algebraic derivative method in the frequency domain yielding exact formula in terms of multiple integrals of the output signal, when placed in the time domain. By investigating the connection between such integrals and parameters to be estimated, a set of three linear regression equations is proposed. The first equation is used to estimate the structure of poles in the linear part, the second to estimate a point of the nonlinearity, the third to estimate the structure of zeros in the linear part. No a priori knowledge of the structure of the nonlinearity is required. The proposed algorithm is numerically robust, since it is based only on least squares estimation. Simulation results validate the proposed algorithm.
Identification of Hammerstein systems without explicit parameterisation of non-linearity
International Journal of Control, 2009
This article proposes a new approach to identification of Hammerstein systems, where a non-linearity precedes a linear dynamic system, driven by piece-wise constant inputs. The proposed approach does not require an explicit parameterisation of the non-linearity. Moreover, the non-linearity does not have to be static, but could be the one with finite memories like backlash. By exploiting input's piecewise constant property, the denominator of the linear system described by an ARX model is consistently identified from the information of the output only; next, a subspace direct equalisation method estimates the unmeasurable inner signal based on the resulted denominator estimate and output measurements. Contrary to the existing blind approaches, the numerator of the linear system is not required, which leads to a significant improvement of removing an error propagation. On the basis of the estimated inner signal, the measured input and output, the non-linearity and linear system are obtained separately. The proposed approach is validated and compared with two existing blind approaches through numerical and experimental examples.
Combined parametric-nonparametric identification of Hammerstein systems
Ieee Transactions on Automatic Control, 2004
A novel, parametric-nonparametric, methodology for Hammerstein system identification is proposed. Assuming random input and correlated output noise, the parameters of a nonlinear static characteristic and finite impulse-response system dynamics are estimated separately, each in two stages. First, the inner signal is recovered by a nonparametric regression function estimation method (Stage 1) and then system parameters are solved independently by the least squares (Stage 2). Convergence properties of the scheme are established and rates of convergence are given.
Automatica, 2009
The Two-Stage Algorithm (TSA) has been extensively used and adapted for the identification of Hammerstein systems. It is essentially based on a particular formulation of Hammerstein systems in the form of bilinearly parameterized linear regressions. This paper has been motivated by a somewhat contradictory fact: though the optimality of the TSA has been established by Bai in 1998 only in the case of some special weighting matrices, the unweighted TSA is usually used in practice. It is shown in this paper that the unweighted TSA indeed gives the optimal solution of the weighted nonlinear least-squares problem formulated with a particular weighting matrix. This provides a theoretical justification of the unweighted TSA, and also leads to a generalization of the obtained result to the case of colored noise with noise whitening. Numerical examples of identification of Hammerstein systems are presented to validate the theoretical analysis.
Sensor-to-sensor identification of Hammerstein systems
2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012
Traditional system identification uses measurements of the inputs, but when these measurements are not available, alternative methods, such as blind identification, output-only identification, or operational modal analysis, must be used. Yet another method is sensor-to-sensor identification (S2SID), which estimates pseudo transfer functions whose inputs are outputs of the original system. A special case of S2SID is transmissibility identification. Since S2SID depends on cancellation of the input, this approach does not extend to nonlinear systems. However, in the present paper we show that, for the case of a two-output Hammerstein system, the leastsquares estimate of the PTF is consistent, that is, asymptotically correct, despite the presence of the nonlinearities.
Conference proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference, 2013
This paper describes a novel method for the identification of Hammerstein systems with time-varying (TV) static nonlinearities and time invariant (TI) linear elements. This paper develops a linear parameter varying (LPV) state-space representation for such systems and presents a subspace identification technique that gives individual estimates of the Hammerstein components. The identification method is validated using simulated data of a TV model of ankle joint reflex stiffness where the threshold and gain of the model change as nonlinear functions of an exogenous signal. Pilot experiment of TV reflex EMG response identification in normal ankle joint during an imposed walking task demonstrate systematic changes in the reflex nonlinearity with the trajectory of joint position.
Hammerstein System Identification by a Semi-Parametric Method
2000
A semi-parametric algorithm for identification of Hammerstein systems in the presence of correlated noise is proposed. The procedure is based on the non-parametric kernel regression estimator and the standard least squares. The advantages of the method in comparison with the standard non-parametric approach are discussed. Limit properties of the proposed estimator are studied, and the simulation results are presented.
Mixed parametric-nonparametric identification of Hammerstein and Wiener systems - a survey
IFAC Proceedings Volumes, 2012
The paper surveys the ideas of cooperation between parametric and nonparametric (kernel-based) algorithms of nonlinear block-oriented system identification. Various strategies are proposed, dependently on the system structure, number of data and the prior knowledge. The estimates are consistent and their rates of convergence are presented. The aim of the paper is to show some recent results in the field in a systematic, ordered way.