# 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.

## Leave a reply