Report on State-of-the-art techniques for generation of LFT models

Abstract

Many physical dynamical systems can be described by differential equations depending on parameters, which are either not exactly known (i.e., uncertain) or are time-varying. The analysis and design of controllers for such systems ensuring the stability and performance requirements for all allowable parameter variations and over the whole range of operating conditions is a highly complex task but it may be efficiently addressed by employing advanced techniques like μ [32]. The key aspect of these linear system based analysis and synthesis approaches is employing special model uncertainty descriptions based on Linear Fractional Transformation (LFT) [32]. In principle, it is straightforward to transform a linear plant model including unstructured and structured uncertainties into a Linear Fractional Representation (LFR). However, this transformation is not unique and the blindfold application of ad-hoc methods [26] generally leads to LFRs of high complexity. LFT based robust control methods are numerically highly demanding and of high computational complexity, e.g., many methods are based on solving a large system of Linear Matrix Inequalities (LMIs) [9]. Therefore the application of these methods is restricted to LFRs of reasonable complexity, otherwise the computation time will be unacceptable, the numerical errors will increase or the methods may even fail. The report is structured as follows. In section 2 some ideas are presented to generate a linear, rational parametric model, which is required to allow the transformation into linear fractional form. This will be the first step in the LFR realization process as especially the COFCLUO models are either given as a nonlinear flight mechanics model or as a set of linear aeroelastic models, which both cannot be directly used for LFR generation. In section 3 some direct LFR realization approaches and the object-oriented LFR realization technique are briefly described. As one of the main aspects in LFR realization is to generate LFRs of least possible order, a method to calculate a lower bound for the LFR order is also presented, which may be used to quantify the order of the resulting LFRs. The object-oriented LFR realization approach will probably be the method of choice during the COFCLUO project as a very efficient software implementation exists, which is briefly described in section 4 [17, 18].