Stochastic linearization of nonlinear point dissipative systems

Stochastic linearization produces a linear system with the same covariance kernel as the original nonlinear system. The method passes from factorization of finite-dimensional co-variance kernels through convergence results to the final input/output operator representation of the linear system. 1. Introduction. Linearization and hence stochastic linearization of a nonlinear system is about local behavior of the system in time and space. Since the system functions of monitoring and control are concerned with local behavior, they are usually based on linearizations of the underlying nonlinear system [19]. The nonparametric methods of linearization which are the subject of this investigation are based on the covariances of the input and output processes for the system. The data typically looks like Figure 1.1. Existence of the covariance is implied by the assumption that the underlying nonlinear system is point dissipative [3], that is, there is a compact set which each trajectory of the system without stochastic excitation enters and remains within. Nonparametric methods of linearization which only require observations of inputs and outputs rather than models of the nonlinear system are potentially useful in two situations [5]: first, when the system is evolving in time or is frequently reconfigured and model updates are difficult or expensive to obtain; second, when the monitoring or control functions are to be exercised at a low level by smart devices without the high level logic required for choosing or changing the system model. The covariance function R of a zero-mean output process determines a reproducing kernel Hilbert (RKH) space with kernel R. This RKH space is said to represent the output process [21] and has been exploited in signal analysis [28]. In a reasonable sense, the RKH space representation of the process contains all of the information on the process available from observations. Starting with a known linear system excited by a Wiener process [24] provides an explicit representation of the RKH space as a space of Hellinger integrable functions. Further, the linear input/output operator for the system provides a factorization of the nonnegative Hermitian operator on the space of Hellinger integrable functions with matrix representation R. When the underlying system is nonlinear, we show, in Section 3, that factoring discrete versions of R yields in the limit the matrix representation of a linearization of the nonlinear system. This stochastic linearization is the best possible in that when excited