# REALIZATION OF UNCERTAIN SYSTEMS UNDER AN LFT FORM

1. INTRODUCTION

The issue is to transform a closed loop subject to model uncertainties (parametric uncertainties and neglected dynamics) into the standard in­terconnection structure M ( s ) — Д Д ) of Figure 3.1. As illustrated in chapter 1, the key issue is to take into account the parametric uncertain­ties entering the open loop plant model: the uncertain transfer matrix G(s, 6) (where S is a vector of uncertain parameters) is to be transformed into an LFT Fi(H(s), Д), where Д = diag(SiIqi) is a real model perturb­ation. We first consider the simple case of parametric uncertainties entering in an affine way the state-space model of the plant. A simple method (Morton and McAfoos, 1985; Morton, 1985) is indeed available for this special case, which is often encountered in practice. The general prob­lem is then considered in the third section. It is proved in e. g. (Belcastro and Chang, 1992; Lambrechts et al., 1993; Cheng and DeMoor, 1994)
that an LFT model can be obtained in the following very general case: the coefficients of the state space model or transfer matrix are rational functions of the parametric uncertainties. This covers most of the engin­eering examples. The problem is however the potential non minimality of the computed LFT model: see also (Font, 1995). This is an important problem from a practical point of view. Consider a simple example with two parametric uncertainties <5i and S?. Assume that an LFT model of the transfer matrix G ( s. 6 ) was computed with:

The LFT model is non minimal if an other LFT model could be found, which equivalently models G(s, 6) with a simpler structure for the real model perturbation, e. g. : Д = diag(Si, S2)

The model perturbation (3.2) is more attractive than the one of equation

(3.1) for two reasons. When applying the у tools to the interconnection structure M – A, the computational amount is an increasing function of the complexity of the model perturbation G. As a second reason, when computing e. g. the classical у upper bound of (Fan et al., 1991), the result is a priori more conservative with the model perturbation (3.1). It is indeed observed in practice that the more repeated a scalar, the more conservative the у upper bound (Packard and Doyle, 1988). Neverthe­less, note that an LFT model can be reduced a posteriori with various heuristic methods (Beck et al., 1996).

As a final point, a simple method is proposed for transforming an un­certain physical plant model into an LFT form (section 4.). The next chapter will apply this method, as well as the method by (Morton and McAfoos, 1985; Morton, 1985), to the two aeronautical examples.