. NOTATIONS AND DEFINITIONS

The u> dependence is left out in the following: in the new interconnec­tion structure of Figure 1.11, the complex matrix M represents the value of the transfer matrix M(s) at s = ju>, while the model perturbation Д is an uncertain complex matrix, which also represents the value of the uncertain transfer matrix A(.s) at s = ju>. Remember especially that each block Az(s) of neglected dynamics becomes a full complex block Дг at s = ju).

A mixed structured perturbation Д is a free complex matrix with the following specific structure:

лС *C

, іЛ I, • • • , Д rri£

With classical notations (Fan et al., 1991), Д contains real scalars (which represent the parametric uncertainties), complex scalars Щ and full complex blocks ДJj7 Є Сктг+тс+‘>’ктг+гпс+ч (which represent the neg­lected dynamics). The integers mr, mc, me and kx define the structure of the perturbation. A real scalar (resp. a complex scalar 8flkmr+) is said to be repeated if the integer fc, (resp. kmr +l) is strictly greater than unity.

Д is said to be a complex model perturbation if it only contains com­plex scalars and full complex blocks. Conversely, Д is a real model per­turbation if it only contains real scalars. Д is finally a mixed model perturbation when it simultaneously contains real and complex uncer­tainties.

Remark: for the sake of completeness, the model perturbation Д above can also contain repeated complex scalars. Nevertheless, in many con­trol problems, Д only contains real (possibly repeated) scalars and full complex blocks. Specific problems however require the introduction of repeated complex scalars: see e. g. (Doyle and Packard, 1987) [5]. Note finally that a non repeated complex scalar can also be considered as a one dimensional full complex block.

The unit ball В Д is introduced in the space of the structured perturba­tion Д:

Note that the relation ег(Д) < 1 can be rewritten as and аД) В 1 . The s. s.v. is defined as:

liA(M) = l/min(k / ЗД Є кВА with det(I — MA) = 0) (1.28) = 0 if no (к, Д) exists

Remark: the notation дхд(М) emphasizes that this value simultan­eously depends on the complex matrix M and on the structure of the model perturbation Д. For the sake of simplicity, we will often drop out the Д dependence, i. e. simply note ц(М).