A systematic methodology for empirical modeling of non-linear state space systems

摘要

In this paper the authors formulate a theoretical framework for the empirical modelling of non-linear state space systems. The classification of non-linear system data, selection of model structure and order, system parameterisation, stationarity of the data, handling of outliers and noise in the data, parameter estimation and model validation can all be addressed with established, though loosely associated numerical techniques, often referred to as nonlinear process modelling. Relatively few researchers in system identification are comfortable with the application of these numerical techniques, such as time series embedding, surrogate data methods, non-linear stationarity, Lyapunov exponents for chaotic processes and nonlinear predictability. The authors reinterpret some of the above non-linear empirical concepts against the established background for linear state space system identification. Hereby we lay a basis for a systematic methodology to address empirical modelling of non-linear process dynamics, which can be implemented in a non-linear system identification toolbox. In particular, we apply surrogate data methods for the classification of data as stochastic or deterministic. For deterministic data, we embed the individual observations of the process and separate the embedding variables by non-linear factor analysis to arrive at a state space parameterisation of the system. The separation function makes no prior assumptions about the probability distributions of the observations and is robust against dynamic and measurement noise. An ensemble learning technique is used to estimate the parameters of the separation function. After parameterisation of the system a multiple-layer perceptron neural network maps the time evolution of the state vector onto the observations, one sample step ahead. In this manner, the dynamics of the process are captured. Model order is established against the Schwarz information criterion, formulated for multidimensional observations as a function of the model order and modelling error. Model validation is performed against the R~2 statistic, as well as in terms of free-run prediction performance.