By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as necessary instruments for fixing a few regulate difficulties. This e-book offers an updated account of the LMI procedure and covers subject matters equivalent to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and purposes. It additionally emphasizes purposes of the tactic to parts except regulate. the fundamental suggestion of the LMI process 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 results in an effective numerical resolution and is very fitted to issues of doubtful facts and a number of (possibly conflicting) necessities.
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These methods can be described in the context of self-concordant barrier and potential functions; this is the point of view taken in [296, 404]. Primal-dual interior-point methods Without doubt, the most efficient interior-point methods for LP are the primal-dual interior-point methods . When a primal-dual interior-point method is used, each subproblem can be described by a linear least squares problem or, equivalently, a system of linear equations. These methods are very efficient for four reasons: (1) they enjoy robust convergence from infeasible starting points, with guaranteed complexity when implemented accordingly (though typically this is not done in practice); (2) they converge rapidly, even if the solutions are not unique; (3) the linear systems to be solved are sparse (or can be made sparse by, for example, using standard techniques to handle dense columns in the constraint matrix); consequently, very large problems can be solved; (4) the inherent ill conditioning in the linear systems to be solved does not, in practice, greatly limit achievable accuracy.
A survey of results and further comments are given in . 6, we have given a delay-independent stability result, based on Lagrange relaxations, which is related to a particular choice of Lyapunov-Krasovskii functional candidate. This approach can be extended to a delay-dependent result, using an embedding technique (comparison principle) as follows. 30). 30) (in fact, the above is an integration of the original equations over one delay interval). 6) and obtain the following delay-dependent relaxed LMI condition: Computing a lower bound on the largest allowable delay is a generalized eigenvalue optimization problem (GEVP).
3. There are other motivations for us to use such models. ). To cope with analysis and design problems for such systems, we almost always have to use (finite-dimensional, linear, stationary) approximations. We believe that a finer way to handle complexity is to use approximations with uncertainty. We thus make the distinction between "physical" and "engineering" uncertainties. The first are due to ignorance of the precise physical behavior; the second come from a voluntary simplifying assumption.
Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control) by Laurent El Ghaoui, Silviu-Iulian Niculescu