Volterra series

About: Volterra series is a research topic. Over the lifetime, 2731 publications have been published within this topic receiving 46199 citations.

Random and pseudorandom inputs for Volterra filter identification

Abstract: This paper studies input signals for the identification of nonlinear discrete-time systems modeled via a truncated Volterra series representation. A Kronecker product representation of the truncated Volterra series is used to study the persistence of excitation (PE) conditions for this model. It is shown that i.i.d. sequences and deterministic pseudorandom multilevel sequences (PRMS's) are PE for a truncated Volterra series with nonlinearities of polynomial degree N if and only if the sequences take on N+1 or more distinct levels. It is well known that polynomial regression models, such as the Volterra series, suffer from severe ill-conditioning if the degree of the polynomial is large. The condition number of the data matrix corresponding to the truncated Volterra series, for both PRMS and i.i.d. inputs, is characterized in terms of the system memory length and order of nonlinearity. Hence, the trade-off between model complexity and ill-conditioning is described mathematically. A computationally efficient least squares identification algorithm based on PRMS or i.i.d. inputs is developed that avoids directly computing the inverse of the correlation-matrix. In many applications, short data records are used in which case it is demonstrated that Volterra filter identification is much more accurate using PRMS inputs rather than Gaussian white noise inputs. >

Application of higher order spectral analysis to cubically nonlinear system identification

Abstract: In this study, digital higher-order spectral analysis and frequency-domain Volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time-invariant systems up to third order. The primary focus is on consideration of random excitation of nonlinear systems and, thus, the approach makes extensive use of higher-order spectral analysis to determine the frequency-domain Volterra kernels, which correspond to linear, quadratic, and cubic transfer functions. Although the Volterra model is nonlinear in terms of its input, it is linear in terms of its unknown transfer functions. Thus, a least squares approach is used to determine the optimal (in a least squares sense) set of linear, quadratic, and cubic transfer functions. Of particular practical note, is the fact that the approach of this paper is valid for non-Gaussian, as well as Gaussian, random excitation. It may also be utilized for multitone inputs. The complexity of the problem addressed in this paper arises from two principal causes: (1) the necessity to work in a 3D frequency space to describe cubically nonlinear systems, and (2) the necessity to characterize the non-Gaussian random excitation by computing higher-order spectral moments up to sixth order. A detailed description of the approach used to determine the nonlinear transfer functions, including considerations necessary for digital implementation, is presented. >

The existence and uniqueness of Volterra series for nonlinear systems

Abstract: Given an input-output map described by a nonlinear control system \dot{x}=f(x,u) and nonlinear output y=h(x) , we present a simple straightforward means for obtaining a series representation of the output y(t) in terms of the input u(t) . When the control enters linearly, \dot{x} =f(x)+ ug(x) , the method yields the existence of a Volterra series representation. The proof is constructive and explicitly exhibits the kernels. It depends on standard mathematical tools such as the Fundamental Theorem of Calculus and the Cauchy estimates for the Taylor series coefficients of analytic functions. In addition, the uniqueness of Volterra series representations is discussed.

Orthogonal approaches to time-series analysis and system identification

Abstract: Some recent, efficient approaches to nonlinear system identification, ARMA modeling, and time-series analysis are described and illustrated. Sufficient detail and references are furnished to enable ready implementation, and examples are provided to demonstrate superiority over established classical techniques. The ARMA identification algorithm presented does not require a priori knowledge of, or assumptions about, the order of the system to be identified or signal to be modeled. A suboptimal, recursive, pairwise search of the orthogonal candidate data records is conducted, until a given least-squares criterion is satisfied. In the case of nonlinear systems modeling, discrete-time Volterra series is stressed, or rather a more efficient parallel-cascade approach. The model is constructed by adding parallel paths (each consisting of the cascade of dynamic linear and static nonlinear systems). In the case of time-series analysis, a non-Fourier sinusoidal series approach is stressed. The relevant frequencies are estimated by an orthogonal search procedure. A search of the candidate sinusoids is conducted until a given mean-square criterion is satisfied. >