Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described. >

Adaptive polynomial filters

Volterra series have been in the engineering literature for some time now, and yet there have been few attempts to measure Volterra kernels. This paper discusses techniques for measuring the Volterra kernels of weakly nonlinear systems. We introduce a new quick method for measuring the second Volterra kernel which is analogous to pseudonoise testing of a linear device. To illustrate the discussion we present an experimental example, an electro-acoustic transducer. Throughout the paper we emphasize the practical aspects of kernel measurement.

/pdf/measuring-volterra-kernels-3nbpzjptkf.pdf

Measuring Volterra kernels

https://www.dtc.umn.edu/s/resources/gg-se.pdf

A bibliography on nonlinear system identification

Representation, identification, and modeling are investigated for nonlinear biomedical systems. We begin by considering the conditions under which a nonlinear system can be represented or accurately approximated by a Volterra series (or functional expansion). Next, we examine system identification through estimating the kernels in a Volterra functional expansion approximation for the system. A recent kernel estimation technique that has proved to be effective in a number of biomedical applications is investigated as to running time and demonstrated on both clean and noisy data records, then it is used to illustrate identification of cascades of alternating dynamic linear and static nonlinear systems, both single-input single-output and multivariable cascades. During the presentation, we critically examine some interesting biological applications of kernel estimation techniques.

The identification of nonlinear biological systems: Volterra kernel approaches.

A brief overview of vortex-induced vibration (VIV) of bridge decks is presented, highlighting special VIV features concerning bridge decks. A popular VIV model (Van der Pol–type model) for bridge decks is examined in detail. Alternatively, a truncated Volterra series–based nonlinear oscillator is introduced to model the VIV system. Typical features of VIV such as the limit cycle oscillation (LCO), frequency shift, hysteresis, and beat phenomenon are parsimoniously and accurately captured in the proposed nonlinear model. As a functional expansion of a nonlinear system, the Volterra series is convenient for estimating the linear and nonlinear contributions to VIV. It is demonstrated that the relative contribution of nonlinear effects in VIV is around 50% of the total response for a range of bridge cross sections. The efficacy of the Volterra series as a reduced-order model (ROM) in capturing aerodynamic nonlinearities eliminates the need for reliance on conventional phenomenological models as it pro...

Vortex-induced Vibration of Bridge Decks: Volterra Series-Based Model

A new identification technique for a class of weakly nonlinear systems whose behavior is adequately characterized in terms of a finite Volterra functional series is presented. Application of the identification technique results in a complete specification of the nonlinear impulse responses which describe a weakly nonlinear system. The identification technique is a "black box" procedure in that only measurements at the system input and output terminals are required. A functional form for the second-order impulse response, h_2(t_1, t_2) , is derived for a nonlinear system with a finite number of power-law devices. The identification of h_2(t_1, t_2) is accomplished by exciting the system with a sum of exponentially decaying signals and appropriately processing the input and output signals using the pencil-of-functions system identification approach. This results in a complete set of linear equations involving all the parameters of h_2(t_1, t_2) . Solution of these equations uniquely determines h_2(t_l, t_2) . An example of the practical application of the technique to a common emitter amplifier is presented.

D. Weiner

Papers

Identification of weakly nonlinear systems using input and output measurements