Estimation and Characterization of Nonlinear Behavior of Nonlinear Spatio-Temporal Systems

1. Motivation Spatio-temporal systems (STSs) described by partial differential equations (PDEs) are widely used in physical and engineering systems [1]. Traditional methods for the analysis of a PDE system rely on an analytical solution of the system, which is difficult to obtain for most nonlinear PDEs. Alternative methods such as qualitative analysis of solutions or numerical methods for an approximation are studied using functional analysis and generalized function theory [2], or finite element methods and difference methods [3]. Identification of STSs has also been studied recently using finite dimensional parametric MIMO models to approximate infinite dimensional systems [4]. The estimation of spatio-temporal systems is formulated into a traditional identification problem of an MIMO system. Because multiple input/output variables are involved including their nonlinear combinations, existing methods are usually computationally intensive and not applicable for online problems. Therefore, an effective and powerful method for the estimation of nonlinear behavior of a STS is deserved to be further investigated. Importantly, this study will also aim to estimate physical characteristics characterized by some important parameters in the system PDE model. This will provide an important insight into the analysis and design of physical and structural properties of the dynamical system under study. 2. The method Nonlinear STSs are firstly transformed into a class of MIMO partially linear systems (PLSs), and a new online identification algorithm for this class of PLSs is proposed using a pruning error minimization principle and least squares support vector machines (LS-SVM). Many benchmark physical and engineering systems can be transformed into an MIMO-PLS which keeps important physical spatiotemporal relationships that are very helpful in system identification and also in analysis of the underlying system. Compared with existing methods, the proposed algorithm can make full use of prior structural information on system physical models, can realize online estimation of system dynamics and achieve online characterization of some important nonlinear physical characteristics of the system. Given a nonlinear STS with appropriate boundary conditions