The aim of the book is to present the state-of-the-art of the Linear Parameter-Varying (LPV) systems theory, to explore the lacking details, and to establish a framework for the modeling and identification of such systems. To this end, in the first part of the book (Chapters 3–6), a unified system theoretical background for the LPV case is provided to enable comparison of various model structures. In the second part (Chapters 7–9), a theory concerning LPV behaviours, and modeling and identification methods are presented. In particular, the classical prediction-error identification method is extended to LPV systems, and an LPV identification mechanism based on the Orthonormal Basis Function (OBF) expansion models is developed.
In the introductory Chapter 1 an overview of the origin of LPV systems is given, and in Chapter 2 basic concepts of the prediction-error identification method and orthonormal series expansion models are collected. In Chapter 3, a behavioral framework for the LPV case is developed yielding the basis for obtaining proper representations of LPV systems in a well-founded system theoretical sense. In Chapter 4, canonical forms and equivalence transformations between different descriptions of LPV systems, particularly the state-space and the input-output representation, are investigated. In Chapter 5, the OBF-based series-expansion representations of LPV systems are developed and finite truncated series models are studied. In turn, in Chapter 6, a discretization problem of LPV systems in a zero-order hold setting is presented, and criteria for choosing the sampling time along with the exact and approximate discretization methods are given. In Chapter 7, modeling of nonlinear systems described by nonlinear differential equations in an LPV form is discussed exploiting the behavioral framework of Chapter 3. The way is investigated in which a nonlinear system can be realized, or approximated, by an LPV system and an algorithmic technique is presented enabling errorless conversion of nonlinear differential equations into LPV representations. In Chapter 8, the optimal basis selection problem for OBF-based series expansion model structures of LPV systems is considered for the case when only measured data records are available. A selection method founded on fuzzy clustering of estimated pole locations, and robust against the measurement noise is provided. Finally, in Chapter 9, the prediction-error identification framework for LPV systems, exploiting truncated series expansion models and relying on the results from earlier chapters, is established. Local and global identification approaches are presented, based respectively on the interpolation and linear regression concepts. Specifically, profitable properties of Wiener and Hammerstein OBF models in the LPV systems set-up are pointed out, and identification of LPV systems is discussed in this context. Both approaches are analysed in terms of bias, variance, convergence and consistency of the obtained model estimates, as well as applicability of the methods.
The book is completed by an Appendix containing the proofs of basic facts and a vast bibliography.