Abstract
The primary difficulty in the identification of Hammerstein nonlinear systems (a static memoryless nonlinear system in series with a dynamic linear system) is that the output of the nonlinear system (input to the linear system) is unknown. By employing the theory of affine projection, we propose a gradient-based adaptive Hammerstein algorithm with variable step-size which estimates the Hammerstein nonlinear system parameters. The adaptive Hammerstein nonlinear system parameter estimation algorithm proposed is accomplished without linearizing the systems nonlinearity. To reduce the effects of eigenvalue spread as a result of the Hammerstein system nonlinearity, a new criterion that provides a measure of how close the Hammerstein filter is to optimum performance was used to update the step-size. Experimental results are presented to validate our proposed variable step-size adaptive Hammerstein algorithm given a real life system and a hypothetical case.
Highlights
Nonlinear system identification has been an area of active research for decades
Hammerstein systems are applied in the area of Neural Network since it provides a convenient way to deal with nonlinearity [10]
In block based adaptive Hammerstein algorithms, the Hammerstein system is overparameterized in such a way that the Hammerstein system is linear in the unknown parameters
Summary
Nonlinear system identification has been an area of active research for decades. Nonlinear systems research has led to the discovery of numerous types of nonlinear systems such as Volterra, Hammerstein, and Weiner nonlinear systems [1,2,3,4]. In block based adaptive Hammerstein algorithms, the Hammerstein system is overparameterized in such a way that the Hammerstein system is linear in the unknown parameters This allows the use of any linear estimation algorithm in solving the Hammerstein nonlinear system identification problem. Jeraj and Mathews derived an adaptive Hammerstein system identification algorithm by linearizing the system nonlinearity using a Gram-Schmidt orthogonalizer at the input to the linear subsystem (forming an MISO system) [17] This method suffers the same limitations as the block-based adaptive Hammerstein algorithms. Input signals with high eigen value spread, ill-conditioned tap input autocorrelation matrix can lead to divergence or poor performance of a fixed step-size adaptive algorithm To mitigate this problem, a number of variable step-size update algorithms have been proposed.
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