Volterra series

Some attempts to exploit sensor dynamics for concentration prediction were carried out in the past. Davide et al. approached the problem using dynamic system theory, applying non-linear Volterra series to the modelling of Thickness Shear Mode Resonator (TSMR) sensors [4], This approach gave rise to non-linear models where the difficulty to discriminate the intrinsic sensor properties from those of the gas delivery systems limited the efficiency of the approach. 

The Volterra Series. For a time invariant system defined by equation 4.25, it is possible to form a Taylor series expansion of the non-linear function to give [Priestley, 1988] ... 

Genceli, H. and Nikolaou, M., Design of robust constrained nonlinear model predictive controllers with Volterra series, AIChE J. 41, 9, 2098-2107 (1995). 

The convolution defined in (4.2.1) is a linear operation applied to the input function x(t). Nonlinear systems transform the input signal into the output signal in a nonlinear fashion. A general nonlinear transformation can be described by the Volterra series. It forms the basis for the theory of weakly nonlinear and time-invariant systems [Marl, Schl] and for general analysis of time series [Kanl, Pril]. In quantum mechanics, the Volterra series corresponds to time-dependent perturbation theory, and in optics it leads to the definition of nonlinear susceptibilities [Bliil]. 

The Volterra series is an expansion of the response signal into multi-dimensional convolution integrals,... 

Convolution integrals can readily be calculated in Fourier space. To this end, the Volterra series is Fourier transformed [Blal, Bliil],... 

They differ from the kernels it (ti, ..., r ) of the Volterra series only by a faster signal decay with increasing time arguments [Bliil]. For coinciding time arguments the crosscorrelation function is the sum of the n-dimensional impulse-response function h with the impulse-response functions hm of lower orders m < n. The stochastic impulse-response functions h are the kernels of an expansion of the system response y(t) similar to the Volterra series (4.2.4) but with functionals orthogonalized for white-noise excitation x t) [Bliil, Marl, Leel, Schl], This expansion is known by the name Wiener series, and the h are referred to as Wiener kernels. 

D. Verotta, Volterra series in pharmacokinetics and pharmacodynamics. J Pharmacokinet Pharmacodyn 30 337-362 (2003). 

B Maner, FJ Doyle III, B Ogunnaike, and R Pearson. Nonlinear model predictive control of a multivariable polymerization reactor using second-order Volterra series. Automatica, 32 1285-1302, 1996. 

The inability to estimate the Volterra kernels in the general case of an infinite series prompted Wiener to suggest the orthogonalization of the Volterra series when a GWN test input is used. The functional terms of the Wiener series are constructed on the basis of a Gram-Schmidt orthogonalization procedure requiring that the covariance between any two Wiener functionals be zero. The resulting Wiener series expansion takes the form ... 

The nonlinear ffequaicy respixise (NLFR) method for investigation of adsorption systems has been developed as an extension of the classical FR method, by applying the mathematical tools of Volterra series and the concept of higher-order FRFs. For that reason, after a very brief survey of other methods for adsorption equiUbrium and kinetic measurements, we will give a short overview of the application of the classical FR method and a brief description of the concept of higher-order FRFs. 

NLFR is a quasi-stationary response of a nonlinear system to a periodic (sinusoidal or cosinusoidal) input, around a steady state. One of the most convenient tools for treating nonlinear FRs is the concept of higher-order FRFs [52], which is based on Volterra series and generalized Fourier transform. This concept will be briefly presented below. 

On the other hand, the response of a weakly nonlinear system, for which the system nonlinearity has a polynomial form (or can be developed in a Taylor series) can be represented in the form of a Volterra series ... 

The first element of the Volterra series yi corresponds to the linearized model, while y2, ys, are the correction functions of the 1st, 2nd,. .. orders. 

Similar to Taylor series expansion, a Volterra series of indefinite length is needed for exact representation of a nonlinear system, but for practical applications, finite series can be used. 

Step 3. Definition of the outputs. The particle outputs Q and Tf) are represented in the form of Volterra series [52] of the general form ... 

Another approach using nonlinear frequency respruise analysis with the help of the Volterra series expansirui and generalized Fourier transform was also proposed and applied to the study of methanol or ferrocyanide oxidation [666-668]. 

The traditional GMDH method (Farlow, 1984 Ivakhnenko, 1971) is based on anrm-derlying assumption that the data can be modeled by using an approximation of the Volterra Series or Kolmorgorov-Gabor polynomial (Madala and Ivakhnenko, 1994) as shown in equation (1). 

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