IDENTIFICATION OF A CLASS OF LINEAR MODELS WITH NONLINEARLY VARYING PARAMETERS Fabio Previdi, Marco Lovera Dipartimento di Elettronica e Informazione, Politecnico di Milano via G. Ponzio 34/5 - 20133 Milano, Italy. Tel. +39-02-23993650, Fax +39-2-23993412 e-mail: previdi@elet.polimi.it; lovera@elet.polimi.it Abstract A novel class of hybrid linear/nonlinear models is proposed for nonlinear identification purposes and a specific parameter tuning algorithm is presented. The connections between this model class and the Local Model Network family are outlined. Finally, the capabilities of this model in identification problems is outlined by means of simulations. 1 Introduction Nonlinear models of the NARMAX class have been the subject of considerable attention in recent years [2, 4, 8] and a number of different paradigms for their representation and parametrisation have been proposed in the [8]. Although the generality of this kinds of nonlinear models makes them very useful in a number of applications in which little or no a priori information about the system to be studied is available, in many practical situations it would be possible and fruitful to take into account such priors in the structure of the model. A typical example of this kind of situation is given by a nonlinear plant which is led to operate in a number of different equilibria. In such a case it would be useful to separate the role of conventional input variables from the role of scheduling variables (i.e., variables defining the operating point of the plant), by letting them enter the model in distinct ways [9]. As a particular case of this approach, Linear Parametrically Varying (LPV) models have been recently proposed. These models are linear ones, in which a vector of scheduling variables enters the system matrices in an affine or linear fractional way; the advantage of such a representation for general nonlinear models is easily recognized in the connection with modern robust control theory [6, 10]. In this paper an attempt is made to combine the clear advantages of the parametrically varying approach with the generality of the NARX/NARMAX model class. The result is a hybrid linear/nonlinear model family for which a very efficient parameter tuning procedure has been developed and implemented. The paper is organized as follows: in Section 2 the proposed model class is defined and characterized w.r.t. the so-called Local Model Network family [4]; Section 3 is devoted to a detailed presentation of the identification algorithm for the model class, while in Section 4 some simulation results are presented. 2 The model family We consider the following model structure: () ()( ) ()( ) a n n t y t a t y t a t y a 1 1 ()( ) ()( ) () t e n k t u t b k t u t b b n b 0 (1) where () ( ) 2 , 0 WN t e ; a n is the maximum output lag, b n is the maximum input lag and k is the delay between model input and output. It is worth noting that Eq. (1) can be also written in the following general form: () ( ) ( ) () t e t f t y 1 (2.a) where ( ) ( ) ( )( ) ( ) [ ] b a n k t u k t u n t y t y t ,..., , ,..., 1 1 is the model regressor. Moreover, the previous Eq. (1) and (2.a) can be written as a linear regression with time varying parameters: () () ( ) () t e t t t y T 1 (2.b) where () () () () () [ ] 0 1 ,..., , ,..., t b t b t a t a t b a n n . The parameters in Eq. (1) are supposed to be time-varying according to the following affine transformations: () () t z a a t a i i i 2 1 for a n i ,..., 1 (3.a) () () t z b b t b i i i 2 1 for b n i ,..., 0 (3.b) and  zt is the output of a feed-forward neural network with a single hidden [7, 8]. () ( ) å å h i n h h n i i ih h b t w t z 1 1 1 ~ (4) where: h n is the number of unit in the hidden layer; n i is the number of inputs; h , ih w , h b are the network parameters and in particular they are respectively the weights connecting