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**Sample text**

Blanchini F (1999) Set invariance in control—a survey. Automatica 35(11):1747–1767 18. Blanchini F, Miani S (2008) Set-theoretic methods in control. Birkh¨auser Boston, Boston, MA, USA 19. Blanchini F, Casagrande D, Miani S, Viaro U (2010) Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design. IEEE Trans Autom Contr 55(10):2271–2281 20. Blondel V, Tsitsiklis J (2000) The boundedness of all products of a pair of matrices is undecidable. Syst Contr Lett 41:135–140 21.

If So ∈ M but Go ∈ G, then Go (q) = G(q, θg∗ ). t. the particular identification approaches we consider in Sect. 5. 3 The [16]. ¯ notation E{x} = limN→∞ 1 N ∑Nk=1 E{x(k)} is adopted from the prediction-error framework of 38 R. T´oth et al. 4 Classical Model Structures Next we can investigate how the classical model structures can be formulated in the introduced LPV–PE setting. To follow the classical formulations, we will construct both the process and the noise components using an LPV–IO representation form.

1. 4) with P = [0, 1]. By recursive substitution for q−1 y, q−2 y, . 2 sin q−2p q−3 u + · · · , g3 p where the sequence of functions gi p converges to zero as i → ∞. 3) can be considered as a series expansion of S in terms of q−i which is convergent if S is asymptotically stable. 3) can be seen as the generalization of LPV–SS and LPV–IO representations with appropriate equivalence transformations available (see [23]). 3 An LPV Prediction-Error Framework By using the LPV impulse response representation, established in the previous section, it becomes possible to extend the classical PE framework to the LPV case allowing sophisticated analysis of the estimation of LPV–IO models.