Nonlinear system identification
System identification is a method of identifying or measuring the mathematical model of a system from measurements of the system inputs and outputs. The applications of system identification include any system where the inputs and outputs can be measured and include industrial processes, control systems, economic data, biology and the life sciences, medicine, social systems and many more.
A nonlinear system is defined as any system that is not linear, that is any system that does not satisfy the superposition principle. This negative definition tends to obscure that there are very many different types of nonlinear systems. Historically, system identification for nonlinear systems has developed by focusing on specific classes of system and can be broadly categorised into four basic approaches, each defined by a model class:
There are four steps to be followed for system identification: data gathering, model postulate, parameter identification and model validation. Data gathering is considered as the first and essential part in identification terminology, used as the input for the model which is prepared later. It consists of selecting an appropriate data set, pre-processing and processing. It involves the implementation of the known algorithms together with the transcription of flight tapes, data storage and data management, calibration, processing, analysis and presentation. Moreover, model validation is necessary to gain confidence in, or reject, a particular model. In particular, the parameter estimation and the model validation are integral parts of the system identification. Validation refers to the process of confirming the conceptual model and demonstrating an adequate correspondence between the computational results of the model and the actual data.
The early work was dominated by methods based on the Volterra series, which in the discrete time case can be expressed as
where u(k), y(k); k = 1, 2, 3, … are the measured input and output respectively and is the lth-order Volterra kernel, or lth-order nonlinear impulse response. The Volterra series is an extension of the linear convolution integral. Most of the earlier identification algorithms assumed that just the first two, linear and quadratic, Volterra kernels are present and used special inputs such as Gaussian white noise and correlation methods to identify the two Volterra kernels. In most of these methods the input has to be Gaussian and white which is a severe restriction for many real processes. These results were later extended to include the first three Volterra kernels, to allow different inputs, and other related developments including the Wiener series. A very important body of work was developed by Wiener, Lee, Bose and colleagues at MIT from the 1940s to the 1960s including the famous Lee and Schetzen method,. While these methods are still actively studied today there are several basic restrictions. These include the necessity of knowing the number of Volterra series terms a priori, the use of special inputs, and the large number of estimates that have to be identified. For example for a system where the first order Volterra kernel is described by say 30 samples, 30x30 points will be required for the second order kernel, 30x30x30 for the third order and so on and hence the amount of data required to provide good estimates becomes excessively large. These numbers can be reduced by exploiting certain symmetries but the requirements are still excessive irrespective of what algorithm is used for the identification.
...
Wikipedia