By Laurent El Ghaoui, Silviu-Iulian Niculescu

Linear matrix inequalities (LMIs) have lately emerged as helpful instruments for fixing a couple of keep watch over difficulties. This publication offers an updated account of the LMI technique and covers themes akin to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and purposes. It additionally emphasizes purposes of the tactic to parts except keep watch over.

The simple proposal of the LMI procedure up to speed is to approximate a given keep watch over challenge through an optimization challenge with linear aim and so-called LMI constraints. The LMI strategy ends up in an effective numerical resolution and is especially suited for difficulties with doubtful facts and a number of (possibly conflicting) requirements.

Since the early Nineteen Nineties, with the improvement of interior-point equipment for fixing LMI difficulties, the LMI procedure has received elevated curiosity. One good thing about this system is its skill to deal with huge sessions of regulate difficulties through effective numerical instruments. This technique is commonly appropriate, not just up to speed yet additionally in different components the place uncertainty arises. LMI suggestions supply a universal language for lots of engineering difficulties. Notions now well known in regulate, comparable to uncertainty and robustness, are getting used in different parts by utilizing LMIs. this method is especially appealing for commercial purposes. it truly is like minded for the improvement of CAD instruments that aid engineers clear up research and synthesis difficulties.

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**Extra resources for Advances in linear matrix inequality methods in control**

**Example text**

23). Define Define also the convex set Let us fix the decision variable x again. 4. Robust SDP 17 Relaxing the rank constraint yields lower bound which, by virtue of the saddle-point theorem (recall X is convex), is equivalent to the SDP in variable X The above is the dual of the SDP obtained via Lagrange relaxation. 1 how this connection between Lagrange and rank relaxation has been recognized earlier in the special case of combinatorial optimization problems. For a related point of view on the geometrical properties of rank-constrained symmetric matrices, see [192] and references therein.

Good references on robust control include the book by Zhou, Doyle, and Glover [446]; Dahleh and Diaz-Bobillo [88]; and Green and Limebeer [175]. In other areas of engineering, uncertainty is generally dealt with using stochastic models for uncertainty. ; a good reference on this subject is the book by Dempster [97]. Models with deterministic uncertainty are less classical in engineering optimization. Ben Tal and Nemirovski consider a truss topology design problem with uncertainty on the loading forces [41, 40].

Note that we do not assume the continuous system to be stable, since the decay rate a may be negative; in some sense, we are requiring that the "degree of stability" of the discrete-time behavior dominates that of the continuous-time one. The above condition is an LMI in 5 for fixed A, a, which shows that the problem of checking stability (based on quadratic stability) is tractable, provided a plane search (on A, a) is used. The main advantage of the LMI formulation is that it allows for designing an appropriate impulse M: with the change of variables U = SM, the above conditions become convex in both S, U.