By Laurent El Ghaoui, Silviu-Iulian Niculescu

Linear matrix inequalities (LMIs) have lately emerged as important instruments for fixing a few keep watch over difficulties. This e-book presents an up to date account of the LMI strategy and covers themes comparable to contemporary LMI algorithms, research and synthesis concerns, nonconvex difficulties, and functions. It additionally emphasizes functions of the strategy to components except regulate. the elemental thought of the LMI procedure up to the mark is to approximate a given keep an eye on challenge through an optimization challenge with linear goal and so-called LMI constraints. The LMI procedure ends up in an effective numerical answer and is very fitted to issues of doubtful information and a number of (possibly conflicting) standards.

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**Sample text**

Checking if a given candidate solution x is robustly feasible is already very difficult in general. Our objective is to find lower bounds on this problem in the form of SDP. 4. 24). The variables in this SDP are x (the original decision variable) and 5 (the "multiplier" matrix). 14)), the approximation is exact. 16 El Ghaoui and Niculescu A special case: Robust linear programming (LP) Consider the LP Assume that the a^'s and 6i's are subject to unstructured perturbations. , L, and p > 0 is a given measure of the perturbation level.

The corresponding characterization is Euclidean-norm bounds. } 2 The above representation is usually referred to as the "linear-fractional transformation" (LFT) [446]. We believe the term representation is more appropriate here. 3. Uncertainty models 11 Maximum-norm bounds. ) Sector bounds. Assume A = {s • I \ s G C, characterization is s + s* > 0}. The corresponding Of course, it is possible to mix different bounds to cover more complicated cases. 2 Operator point of view The above uncertainty models for data matrices can be interpreted as input-output maps, as follows.

18 El Ghaoui and Niculescu Robust synthesis We now seek a sufficient condition, A being fixed, so that holds. Now x becomes a variable. Due to possible cross products between x and P in the previous sufficient condition, the problem is not convex, contrary to what happens in the robust SDP case. 27) is true if and only if the optimum is A 2 n. 5). In general, the above problem is not convex; however, several efficient techniques can be used (see Part V). One of these techniques, described in Chapters 16 and 17, relies on a simple linearization of the quadratic objective and leads to a sequence of SDPs.