By Laurent El Ghaoui, Silviu-Iulian Niculescu

Linear matrix inequalities (LMIs) have lately emerged as precious instruments for fixing a few keep an eye on difficulties. This ebook presents an updated account of the LMI approach and covers subject matters reminiscent of contemporary LMI algorithms, research and synthesis concerns, nonconvex difficulties, and purposes. It additionally emphasizes functions of the tactic to parts except keep watch over. the fundamental inspiration of the LMI technique on top of things is to approximate a given keep watch over challenge through an optimization challenge with linear goal and so-called LMI constraints. The LMI procedure results in an effective numerical resolution and is especially suited for issues of doubtful info and a number of (possibly conflicting) standards.

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**Sample text**

The above problem is an SDP, with decision variable S (linked to the quadratic Lyapunov function proving quadratic stability), and W. 14), the above condition is the well-known small gain theorem (see [64] and references therein). The above idea can be extended in various ways, as illustrated in some other chapters (Parts III and IV) of this book. For example, we may consider parameter-dependent, or frequency-dependent Lyapunov functions, based on so-called IQCs. Stochastic Lyapunov functions Stochastic Lyapunov functions have been introduced by Kushner [237], and their use in the context of LMI optimization has been introduced in [64, 120, 121].

In the multivariable case, the integers TI are greater than the degrees of the corresponding variable Si in M, and the issue of minimality is much more subtle. Finally, we note that when M is polynomial in its argument, an LFR can always be constructed such that D is strictly upper triangular, so that this LFR is everywhere well posed, as is M. 2 For other comments on the LFR formalism, see, for example, [108, 254]. The above lemma shows that we can handle almost arbitrary algebraic functions of perturbation parameters, provided we define the perturbation matrix A as a diagonal matrix, with repeated elements.

These algorithms have been implemented, and several interior-point software packages are now generally available. 1 Facts about SDP and related problems We briefly recall some important results on SDP and related problems. 37) is where Z is a symmetric NxN matrix and Ci is the ith coordinate of vector c. 1], and both problems have equal optimal objectives. Related convex problems We list several problems that are related to SDPs. Second-order cone programming problems are special cases of SDPs.