Contents -- 1 Motivations and Perspectives -- 1.1 Modeling complex systems -- 1.1.1 Fundamental models -- 1.1.2 Assumptions, approximations, and simplifications -- 1.1.3 Continuous- vs. discrete-time models -- 1.1.4 Empirical models -- 1.1.5 Gray-box models -- 1.1.6 Indirect empirical modeling -- 1.2 Inherently nonlinear behavior -- 1.2.1 Harmonic generation -- 1.2.2 Subharmonic generation -- 1.2.3 Chaotic response to simple inputs -- 1.2.4 Input-dependent stability -- 1.2.5 Asymmetric responses to symmetric inputs -- 1.2.6 Steady-state multiplicity -- 1.3 Example 1: distillation columns -- 1.3.1 Hammerstein models -- 1.3.2 Wiener models -- 1.3.3 A bilinear model -- 1.3.4 A polynomial NARMAX model -- 1.4 Example 2: chemical reactors -- 1.4.1 Hammerstein and Wiener models -- 1.4.2 Bilinear models -- 1.4.3 Polynomial NARMAX models -- 1.4.4 Linear multimodels -- 1.4.5 The Uryson model -- 1.5 Organization of this book -- 2 Linear Dynamic Models -- 2.1 Four second-order linear models -- 2.2 Realizations of linear models -- 2.2.1 Autoregressive moving average models -- 2.2.2 Moving average and autoregressive models -- 2.2.3 State-space models -- 2.2.4 Exponential and BIBO stability of linear models -- 2.3 Characterization of linear models -- 2.4 Infinite-dimensional linear models -- 2.4.1 Continuous-time examples -- 2.4.2 Discrete-time slow decay models -- 2.4.3 Fractional Brownian motion -- 2.5 Time-varying linear models -- 2.5.1 First-order systems -- 2.5.2 Periodically time-varying systems -- 2.6 Summary: the nature of linearity -- 3 Four Views of Nonlinearity -- 3.1 Bilinear models -- 3.1.1 Four examples and a general result -- 3.1.2 Completely bilinear models -- 3.1.3 Stability and steady-state behavior -- 3.2 Homogeneous models -- 3.2.1 Homogeneous functions and homogeneous models -- 3.2.2 Homomorphic systems.

3.2.3 Homogeneous ARMAX models of order zero -- 3.3 Positive homogeneous models -- 3.3.1 Positive-homogeneous functions and models -- 3.3.2 PH°-ARMAX models -- 3.3.3 TARMAX models -- 3.4 Static-linear models -- 3.4.1 Definition of the model class -- 3.4.2 A static-linear Uryson model -- 3.4.3 Bilinear models -- 3.4.4 Mallows' nonlinear data smoothers -- 3.5 Summary: the nature of nonlinearity -- 4 NARMAX Models -- 4.1 Classes of NARMAX models -- 4.2 Nonlinear moving average models -- 4.2.1 Two NMAX model examples -- 4.2.2 Qualitative behavior of NMAX models -- 4.3 Nonlinear autoregressive models -- 4.3.1 A simple example -- 4.3.2 Responses to periodic inputs -- 4.3.3 NARX model stability -- 4.3.4 Steady-state behavior of NARX models -- 4.3.5 Differences between NARX and NARX* models -- 4.4 Additive NARMAX Models -- 4.4.1 Wiener vs. Hammerstein models -- 4.4.2 EXPAR vs. modified EXPAR models -- 4.4.3 Stability of additive models -- 4.4.4 Steady-state behavior of NAARX models -- 4.5 Polynomial NARMAX models -- 4.5.1 Robinson's AR-Volterra model -- 4.5.2 A more general example -- 4.6 Rational NARMAX models -- 4.6.1 Zhu and Billings model -- 4.6.2 Another rational NARMAX example -- 4.7 More Complex NARMAX Models -- 4.7.1 Projection-pursuit models -- 4.7.2 Neural network models -- 4.7.3 Cybenko's approximation result -- 4.7.4 Radial basis functions -- 4.8 Summary: the nature of NARMAX models -- 5 Volterra Models -- 5.1 Definitions and basic results -- 5.1.1 The class V[sub(N,M)] and related model classes -- 5.1.2 Four simple examples -- 5.1.3 Stochastic characterizations -- 5.2 Four important subsets of V[sub(N,M)] -- 5.2.1 The class H[sub(N,M)] -- 5.2.2 The class U[sup(r)][sub(N,M)] -- 5.2.3 The class W[sub(N,M)] -- 5.2.4 The class P[sup(r)][sub(N,M)] -- 5.3 Block-oriented nonlinear models -- 5.3.1 Block-oriented model structures.

5.3.2 Equivalence with the class V[sub(N,M)] -- 5.4 Pruned Volterra models -- 5.4.1 The class of pruned Volterra models -- 5.4.2 An example: prunings of V[sub(2,2)] -- 5.4.3 The PPOD model structure -- 5.5 Infinite-dimensional Volterra models -- 5.5.1 Infinite-dimensional Hammerstein models -- 5.5.2 Robinson's AR-Volterra model -- 5.6 Bilinear models -- 5.6.1 Matching conditions -- 5.6.2 The completely bilinear case -- 5.6.3 A superdiagonal example -- 5.7 Summary: the nature of Volterra models -- 6 Linear Multimodels -- 6.1 A motivating example -- 6.2 Three classes of multimodels -- 6.2.1 Johansen-Foss discrete-time models -- 6.2.2 A modifed Johansen-Foss model -- 6.2.3 Tong's TARSO model class -- 6.3 Two important details -- 6.3.1 Local model selection criteria -- 6.3.2 Affine vs. linear models -- 6.4 Input-selected multimodels -- 6.4.1 Input-selected moving average models -- 6.4.2 Steady states of input-selected models -- 6.4.3 J-F vs. modified J-F models -- 6.4.4 Two input-selected examples -- 6.5 Output-selected multimodels -- 6.5.1 Output-selected autoregressive models -- 6.5.2 Steady-state behavior -- 6.5.3 Two output-selected examples -- 6.6 More general selection schemes -- 6.6.1 Johanssen-Foss and modified models -- 6.6.2 The isola model -- 6.6.3 Wiener and Hammerstein models -- 6.7 TARMAX models -- 6.7.1 The first-order model -- 6.7.2 The multimodel representation -- 6.7.3 Steady-state behavior -- 6.7.4 Some representation results -- 6.7.5 Positive systems -- 6.7.6 PHADD models -- 6.8 Summary: the nature of multimodels -- 7 Relations between Model Classes -- 7.1 Inclusions and exclusions -- 7.1.1 The basic inclusions -- 7.1.2 Some important exclusions -- 7.2 Basic notions of category theory -- 7.2.1 Definition of a category -- 7.2.2 Classes vs. sets -- 7.2.3 Some illuminating examples -- 7.2.4 Some simple "non-examples".

7.3 The discrete-time dynamic model category -- 7.3.1 Composition of morphisms -- 7.3.2 The category DTDM and its objects -- 7.3.3 The morphism sets in DTDM -- 7.4 Restricted model categories -- 7.4.1 Subcategories -- 7.4.2 Linear model categories -- 7.4.3 Structural model categories -- 7.4.4 Behavioral model categories -- 7.5 Empirical modeling and IO subcategories -- 7.5.1 IO subcategories -- 7.5.2 Example 1: the category Aff[sup(SS)] -- 7.5.3 Example 2: the category Gauss -- 7.5.4 Example 3: the category Median -- 7.6 Structure-behavior relations -- 7.6.1 Joint subcategories -- 7.6.2 Linear model characterizations -- 7.6.3 Volterra model characterizations -- 7.6.4 Homomorphic system characterizations -- 7.7 Functors, linearization and inversion -- 7.7.1 Basic notion of a functor -- 7.7.2 Linearization functors -- 7.7.3 Inverse NARX models -- 7.8 Isomorphic model categories -- 7.8.1 Autoregressive vs. moving average models -- 7.8.2 Discrete- vs. continuous-time -- 7.9 Homomorphic systems -- 7.9.1 Relation to linear models -- 7.9.2 Relation to homogeneous models -- 7.9.3 Constructing new model categories -- 7.10 Summary: the utility of category theory -- 8 The Art of Model Development -- 8.1 The model development process -- 8.1.1 Parameter estimation -- 8.1.2 Outliers, disturbances and data pretreatment -- 8.1.3 Goodness-of-fit is not enough -- 8.2 Case study 1-bilinear model identification -- 8.3 Model structure selction -- 8.3.1 Structural implications of behavior -- 8.3.2 Behavioral implications of structure -- 8.4 Input sequence design -- 8.4.1 Effectiveness criteria and practical constraints -- 8.4.2 Design parameters -- 8.4.3 Random step sequences -- 8.4.4 The sine-power sequences -- 8.5 Case study 2-structure and inputs -- 8.5.1 The Eaton-Rawlings reactor model -- 8.5.2 Exact discretization -- 8.5.3 Linear approximations.

8.5.4 Nonlinear approximations -- 8.6 Summary: the nature of empirical modeling -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.