Greatly enhancing the modeling accuracy for distributed parameter systems by nonlinear time/space separation

Abstract

An effective modeling method for nonlinear distributed parameter systems (DPSs) is critical for both physical system analysis and industrial engineering. In this paper, we propose a novel DPS modeling approach, in which a high-order nonlinear Volterra series is used to separate the time/space variables. With almost no additional computational complexity, the modeling accuracy is improved more than 20 times in average comparing with the traditional method. Introduction -Most of the physical processes (e.g. thermal diffusion process [1,2,3,4,5,6,7], thermal radiation process [8], distributed quantum systems [9,10], concentration distribution process [11,12,13], crystal growth process [1, 6], etc.) are nonlinear distributed parameter systems (DPSs) with boundary conditions determined by the system structure. Thus, it is an urgent task to design an effective modeling method for nonlinear DPSs. The key problem in the design of nonlinear-DSP modeling method is how to separate the time/space variables. Some modeling approaches are previously proposed: These include the Karhunen-Loève (KL) approach [1,4,14,15], the spectrum analysis [16], the singular value decomposition (SVD) combined with the Galerkin's method [1,17], and so on. Among them, the KL approach is the most extensively studied and the most widely applied one. In this approach, the output T (z, t) is expanded as

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“…Boyd and Chua (1985) proved the superiority of this model structure compared with some other empirical models such as the non-linear auto-regressive moving average with exogenous inputs (NARMAX) model and the non-linear auto-regressive with exogenous inputs (NARX) model (Henson, 1997), etc., when capturing the dynamics of fading memory non-linear systems (FMNSs). Additionally, recently this kind of model has even shown its superior capability in approximating distributed parameter systems compared with the routine KL technique (Zhang et al, 2007a). Thereafter, more and more researchers had recognized the potential of Laguerre-Volterra model in non-linear process modelling and control, and begun to seek help from this promising model (Dumont and Fu, 1993;Parker, 2002;Parker and Doyle, 1998).…”