Academic literature on the topic 'Threshold autoregressive process'

This chapter briefly discusses the class of linear multimodels, which are globally nonlinear dynamic models, obtained by “piecing together” several local, linear dynamic models. The motivation behind this approach to model development is the observation that an approximate model’s complexity generally increases with the range of operation over which it must be valid. In particular, it has been noted repeatedly that linear models are often adequate approximations of process dynamics over a sufficiently narrow operating range. Thus, if the total operating range can be decomposed into small enough subsets, it is reasonable to expect that linear models will provide reasonable characterizations of process dynamics over these local regimes. The process of piecing these local models together may be approached in a number of different ways, and the details of this process are important, as subsequent examples illustrate. The principal issues that must be addressed in developing linear multimodels are illustrated with the example of a batch fermentation reactor, described in Sec. 6.1. Three possible definitions of discrete-time linear multimodels are presented in Secs. 6.2.1 through 6.2.3. The first of these definitions is based on Johansen and Foss (1993), whereas the second definition represents an apparently slight variation on the first that can lead to fundamentally different qualitative behavior. The third definition of linear multimodels is a special case of the first, described in Tong (1990) under the name open-loop threshold autoregressive models. The primary difference between Tong’s definition and that of Johansen and Foss is whether the regions of local model validity can overlap: they can in the models of Johansen and Foss, but they cannot in Tong’s model. An extremely important practical issue is the criterion by which local models are selected. The primary focus here is on Tong’s class of linear multimodels where each local model completely describes the global model dynamics over some specified operating range. If this operating range is defined entirely in terms of the input sequence {u(k)} , these models will be designated input-selected, whereas if the operating range is defined entirely in terms of the output sequence {y(k)}, they will be called output-selected; if both inputs and outputs are involved, the term generally selected will be used.