COMPUTATION OF SCALING MATRICES £>! AND Gx
The main advantage of the v upper bound above is its ease of implementation, so that we look for a simple method of computation of (sub) optimal scaling matrices D and G1, which minimize to some extent a(FfllbiMuD((l – jGi)Ff1/4), with Fi = I + G-
■ In the special but significant case of a full complex block Ді, the scaling matrices are simply D — I and Gi = 0. This is especially the case of a robust performance problem in the face of parametric uncertainties.
■ Concerning the computation of a suboptimal diagonal scaling matrix D, which minimizes to some extent o(DiMib(l), a first method uses the Perron eigenvector approach (Safonov, 1982). An associated routine is available in the Robust Control Toolbox of Matlab. The method is computationally efficient.
■ A second method for computing a suboptimal diagonal scaling matrix D is to minimize the Frobenius norm instead of the 2-norm of DiMnDf1: see the classical Osborne’s method and its variations: see especially (Beck and Doyle, 1992), which proposes an efficient implementation of the method, and included references. Here again, an algorithm is available in the Robust Control Toolbox of Matlab.
■ Concerning the computation of a not necessarily diagonal scaling matrix Du see also the routines in the p Analysis and Synthesis Toolbox.
■ Concerning the computation of a scaling matrix G i, a simple suboptimal method is proposed in (Young et al., 1995). Loosely speaking, the idea is simply to cancel with G the skewed hermitian part of the blocks of DMnDfl, which correspond to the real parametric uncertainties.
See also chapter 10 (subsection 5.2) for an other treatment of the same problem. See finally chapter 9 (section 4.) for a first evaluation of these methods on the missile example.