Indirect Closed-Loop Identification of Input-Output LPV Models: A Pre-filtering Approach Ahsan Ali * Mukhtar Ali * Herbert Werner * * Institute of Control Systems, Hamburg University of Technology, 21073 Hamburg, Germany (e-mail: ahsan.ali,mukhtar,h.werner@tuhh.de). Abstract: In this paper, a method is proposed to identify input-output models for Linear Parameter Varying systems in closed loop. In such situations, a formidable problem is the issue of dynamic dependence on the scheduling parameters. Because of this the coefficients of the system depend on time shifted versions of the scheduling parameter vector which leads to a significant increase in the complexity of the model. An approach to avoid this issue is proposed that is at the same time simple and computationally less demanding. The closed loop system is identified like an open loop system and plant parameters are later extracted. The effectiveness of the method is illustrated with a simulation example. 1. INTRODUCTION Control of Linear Parameter Varying (LPV) systems is a well established framework that has a wide range of applications. In this framework, nonlinear systems can be treated like linear ones without compromising much on the benefits that are often lost when only linearized models around a certain operating point are used. In the LPV methodology, nonlinear systems are modelled as gain scheduled linear systems and as a result, linear system techniques can be applied on them. Since linear systems theory is a well developed field, LPV methods have high potential for applications that require high performance over a wide range of operation. Controller synthesis tech- niques for such systems have been well developed and are available as commercial packages [Apkarian and Gahinet, 1995] [Apkarian et al., 1993]. One difficulty with this approach is that it is not trivial to obtain first principle models for such systems. Experi- mental methods based on identification from input-output measurements have therefore received attention in the recent past. Methods taking this approach can be classified based on the model structure: state space methods like [Verdult and Verhaegen, 2002]; truncated series expan- sions based methods like [T´oth et al., 2009] and input- output representation based approaches [Bamieh and Gi- arr´ e, 2002]. The last mentioned have been successfully applied to applications, see for example [Wei, 2006] and [Bamieh et al., 2001]. The input-output(I/O) model based LPV methods can be seen as an extension of LTI predic- tion error methods. By using suitable parameterizations in this setting, the identification problem can be recast as a linear regression problem allowing least squares (LS) based algorithms to be used for estimating system parameters [Bamieh and Giarr´ e, 2002]. Many systems are inherently unstable and it is very dif- ficult or virtually impossible for them to be identified in open loop. Such systems have then to be identified in closed-loop [Ljung, 1999]. Closed-loop system identi- fication techniques for LTI systems is a well developed area, see e.g. [Forssell and Ljung, 1999]. For the case of I/O LPV models, closed loop identification has been at- tempted in [Abbas and Werner, 2009],[Boonto and Werner, 2010],[Abbas et al., 2010]. In [Abbas and Werner, 2009], an instrumental variable technique for LTI systems has been extended to LPV systems. In [Boonto and Werner, 2010], cubic splines are used to identify the system coeffi- cients and in [Abbas et al., 2010], linear recurrent neural networks (LRNN) are used to identify the LPV system. For the state space case, see [Wingerden and Verhaegen, 2009]. An important issue in the indirect identification of LPV systems is that of the dynamic dependence on schedul- ing parameters. This issue arises when the predictor is being derived from the input-output relationship of the closed loop system. Due to the presence of time vary- ing coefficients, the multiplications do not commute and it becomes necessary that the coefficients be allowed to depend on finitely many, time shifted instances of ρ(k), i.e. {...ρ k-1 ,ρ k ,ρ k+1 ,... }. This issue further complicates the situation when the controller separation problem is considered. The case obviously does not remain as simple as that of LTI one. An elaborate framework to deal with the issues arising in I/O-LPV identification has been developed in [T´oth, 2008]. However using this framework for closed-loop systems yields complex equations. The issue has been addressed in [Abbas et al., 2010] by using LRNN. However, due to the nature of computations involving LRNN, this method may take hours before producing results. It is worth mentioning here that this issue of dynamic dependence also arises when input-output LPV models are being converted to state-spaceand vice versa,see e.g. [T´oth, 2008] and [Toth et al., 2011] for details. In this paper, a method to identify LPV systems is pre- sented that somehow ‘evades’ the issue of dynamic depen- dence of scheduling parameter and thus we can represent the predictor in linear regression form. The method is Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 4186