As noted In Chapter 1, the general class of NARMAX models is extremely broad and includes almost all of the other discrete-time model classes discussed in this book, linear and nonlinear. Because of its enormity, little can be said about the qualitative behavior of the NARMAX family in general, but it is surprising how much can be said about the behavior of some important subclasses. In particular, there are sharp qualitative distinctions between nonlinear moving average models (NMAX models) and nonlinear autoregressive models (NARX models). Since these representations are equivalent for linear models (c.f. Chapter 2), this observation highlights one important difference between linear and nonlinear discrete-time dynamic models. Consequently, this chapter focuses primarily on the structure and qualitative behavior of various subclasses of the NARMAX family, particularly the NMAX and NARX classes. More specifically, Sec. 4.1 presents a brief discussion of the general NARMAX class, defining five important subclasses that are discussed in subsequent sections: the NMAX class (Sec. 4.2), the NARX class (Sec. 4.3), the class of (structurally) additive NARMAX models (Sec. 4.4), the class of polynomial NARMAX models (Sec. 4.5), and the class of rational NARMAX models (Sec. 4.6). More complex NARMAX models are then discussed briefly in Sec. 4.7 and Sec. 4.8 concludes the chapter with a brief summary of the NARMAX class. The general class of NARMAX models was defined and discussed in a series of papers by Billings and various co-authors (Billings and Voon, 1983, 1986a; Leontartis and Billings, 1985; Zhu and Billings, 1993) and is defined by the equation . . . y(k) = F(y(k – 1), . . . , y(k – p), u(k), . . . , u(k – q), e(k – 1), . . . , e(k – r)) + e(k). (4.1) . . . Because this book is primarily concerned with the qualitative input/output behavior of these models, it will be assumed that e(k) = 0 identically as in the class of linear ARMAX models discussed in Chapter 2.
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