Two mixed v upper bounds and a mixed slower bound were proposed in this chapter. All these v bounds can be computed in polynomial time.
The exact value of v can be obtained by computing recursively the exact value of /x: in the same way, the first mixed v upper bound can be computed either directly, or recursively using the mixed и upper bound of (Fan et al., 1991). A first solution for computing this mixed supper bound is thus to recursively compute the classical mixed supper bound of (Fan et al., 1991): this one is available in e. g. the ц Analysis and Synthesis Toolbox or in the LMI Control Toolbox. A more computationally efficient solution is to solve the quasi-convex LMI problem associated to this и upper bound (using e. g. the LMI Control Toolbox). In this context, when comparing the LMI problems associated to the mixed //and supper bounds, it is worth pointing out that the computational complexity is the same.
Concerning the second v upper bound, the idea is to transform the v problem into an augmented fi problem. When splitting the model perturbation as Д = diag(Ai, Д2), remember as apreliminary that Ді is to be maintained inside its unit ball while the size of Д2 is free. The main advantage of this second n upper bound is that it is easier to implement than the first one. Its computation is indeed done in two steps. The first one consists in computing D and G scaling matrices for the model perturbation Ді: in the special but practically important case of a full complex block Ді, D = 1 and G = 0 are simply chosen. Otherwise, simple suboptimal methods for computing D and G were proposed. The second step consists in a single computation of the ц upper bound of (Fan et al., 1991), applied to an augmented д problem: this second step is straightforward, since the /x upper bound of (Fan et al., 1991) is directly available in standard Matlab softwares.
The main drawback of this second v upper bound is that Д2 can only contain real (possibly repeated) scalars (unlike the first mixed v upper bound, which can be applied to a generic problem of robustness analysis) . Skewed n problems with such a specific structure are nevertheless encountered in practice. Chapter 7 especially illustrated the practical interest of the problem of checking a small gain condition despite parametric uncertainties (i. e. Дi is a full complex block while Д2 contains the parametric uncertainties). Such a problem can be especially encountered when analyzing the robust performance properties of a closed loop in the presence of parametric uncertainties. As an other physical example, in a linear closed loop containing nonlinearities (e. g. saturations), a sufficient condition for the absence of limit-cycles corresponds to a small gain condition, which is to be satisfied despite LTI parametric uncertainties in the linear part of the closed loop.
A power algorithm in the spirit of (Packard et al., 1988; Young and Doyle, 1990) is finally proposed for the computation of a mixed Д lower bound.
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