By Laurent El Ghaoui, Silviu-Iulian Niculescu

ISBN-10: 0898714389

ISBN-13: 9780898714388

Linear matrix inequalities (LMIs) have lately emerged as important instruments for fixing a few keep an eye on difficulties. This ebook offers an up to date account of the LMI approach and covers subject matters akin to contemporary LMI algorithms, research and synthesis matters, nonconvex difficulties, and purposes. It additionally emphasizes functions of the strategy to parts except regulate. the elemental thought of the LMI process on top of things is to approximate a given regulate challenge through an optimization challenge with linear goal and so-called LMI constraints. The LMI process results in a good numerical answer and is especially suited for issues of doubtful information and a number of (possibly conflicting) requirements.

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**Sample text**

Ben Tal and Nemirovski consider a truss topology design problem with uncertainty on the loading forces [41, 40]. In a more recent work [42], they introduce and study the complexity of robust optimality problems in the sense defined above in the context of convex optimization. Their approach is based on ellipsoidal bounds for the perturbation. 5) belong to the class of min-max problems. To attack them, we can thus use a versatile technique, called Lagrange relaxation, that enables us to approximate a set of complicated constraints by a "more tractable" set.

The above problem is convex; however, it is in general very difficult to solve (NP-hard; see [154, 323]). 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.

Another purpose of this chapter is to open the method to other areas. In our presentation, we have tried to give an "optimization point of view" of the method in an effort to put it in perspective with classical relaxation methods. This effort is mainly motivated by the belief that the approach could be useful in many other fields, especially regarding robustness issues. This topic of robustness is receiving renewed attention in the field of optimization, as demonstrated by a series of recent papers [42, 127, 39].

### Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control) by Laurent El Ghaoui, Silviu-Iulian Niculescu

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