Volterra Series Based Nonlinear Model: a Comparison of Kernel Identification Schemes

Volterra series based model, which is comprised of uniand multi-convolutions in terms of various inputs, could provide an accurate description of nonlinearities while preserving memory effects missed in static transformations. The basic premise of the Volterra theory of nonlinear systems is that any nonlinear system can be modeled as an infinite sum of multidimensional convolution integrals of increasing order, whereas, if the considered nonlinear system possesses fading memory, then a truncated Volterra series can be utilized. The applications of Volterra theory in wind engineering are usually focused on the frequency domain in a statistic point of view (e.g., Tognarelli et al. 1997) while recently Volterra theory based model in time domain from a deterministic point of view has been developed (e.g., Wu and Kareem 2012). The identification of Volterra series kernels (especially higher-order kernels) is a critical issue to establish the Volterra series based nonlinear analysis framework. This research focuses on the systematical comparison of various identification schemes for the truncated Volterra system. The advantages and disadvantages of the considered identification schemes will be examined in detail through a numerical nonlinear system. There are various approaches to identify the Volterra kernels. The first attempt to obtain the Volttera kernels is to derive them from the analytical expression governing the nonlinear system under investigation. For example, Duhamel integral is one of the simplest analytical expressions for a linear system. However, for most of engineering issues of interest there are no such analytical expressions. As considered by several studies, an alternate approach to identify the Volterra kernels is based on system identification techniques, where the continuous Volterra series is discretized to corresponding discrete Volterre system. In discrete time/frequency domain, there are mainly four popular Volterra identification techniques, which are (1) general identification method (e.g., Rugh 1981), (2) harmonic probing method (e.g., Bedrosian and Rice 1971), (3) identification based on Wiener orthogonal kernels (e.g., Lee and Schetzen 1965) and (4) identification based on impulse function concept (e.g., Schetzen 1965). It is important to investigate the input sequence utilized in the Volterra kernels identification, especially as the identified kernels are higher-order ones. There is no special requirement of input sequence for the first identification technique. For the last three techniques, the input sequences are harmonic components, Gaussian white noise and impulse function, respectively. For the first general identification method, the finite Volterra model in the discrete form actually, as it is unfolded, belongs to a larger class of finite-dimensional nonlinear moving average (NMA) models, where there are abundant system identification techniques. Fig. 1 shows a identified second-order kernel of a nonlinear system using the general identification scheme. For the second harmonic probing method, the obtained Volterra series expression in frequency domain has an intuitive interpretation of the observed physical phenomenon, especially with respect to the superhamonic and intermodulation distortion (harmonic distortion). As is well known, the first-order kernel of linear system is easily identified due to the superposition principle. Unfortunately, this easy way is not valid any more for the kernel identification of nonlinear systems. As the frequency components of the output may consist of contributions from kernels of various orders, a tough task is to separate the contributions of different order kernels to the same output frequency component. For the third identification scheme based on Wiener orthogonal kernels, the crosscorrelation technique, which is successfully applied to the identification of a stationary linear system, is utilized. Fig. 2 shows a identified second-order kernel of a nonlinear system based on Wiener orthogonal kernels. The fourth identification scheme is based on impulse function concept. a generalized impulse-function-based kernel identification scheme is developed by the authors, where the kernels of the second-order Volterra system is expressed as (Wu and Kareem 2012)   2 1 2 1 ( ) [ ( )] [ ( )] h t y t y t               (1)   2 1 2 1 1 2 2 1 1 2 2 1 2 1 ( , ) [ ( ) ( )] [ ( )] [ ( )] 2 h t t y t t y t y t                           (2) where h1 represents the first-order kernel which describes the linear behavior of the system; hn the higher-order terms which indicates the nonlinear behavior existing in the system; δ(t) represents the Dirac delta function (unit-impulse function); y[δ(t)] indicates the unit-impulse response; α, 1 and 2 are selected constants. Fig. 3 shows the identified firstand second-order kernels of a nonlinear system based on impulse input. Figure 1: Identified second-order kernel using general identification technique. Figure 2: Identified second-order kernel based on Wiener orthogonal kernels. Figure 3: Identified firstand second-order kernels using impulse input. Reference [1] Bedrosian, E. and Rice, S.O., 1971. The output properties of Volterra systems (nonlinear systems with memory) driven by harmonic and Gaussian inputs. In: Proceedings IEEE, 59, 1688-1707. [2] Lee, Y. W. and Schetzen, M., 1965. Measurement of the Wiener kernels of a nonlinear system by crosscorrelation. Int. J. Control, 2, 237-254. [3] Rugh, W.J., 1981. Nonlinear System Theory, the John Hopkins University Press, Baltimore, MA. [4] Schetzen, M., 1965. Measurement of the kernels of a nonlinear system of finite order. Int. J. Control, 1 (3), 251–63. [5] Tognarelli, M. A., Zhao, J., Rao, K. B. and Kareem, A., 1997. Equivalent statistical quadratization and cubicization for nonlinear systems. J. Engrg. Mech., ASEC, 123 (5), 512-532. [6] Wu, T. and Kareem, A., 2012. Nonlinear aerodynamic and aeroelastic analysis framework for cablesupported bridges. In: 3 rd American Association for Wind Engineering Workshop, Hyannis, MA, USA. 0 10 20 30 40 0 10 20 30 40 -100 0 100 Time step [n] Time step [n] S e c o n d o rd e r k e rn e l h 2 [n ,n ]