By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as worthy instruments for fixing a few keep an eye on difficulties. This booklet offers an up to date account of the LMI process and covers subject matters similar to fresh LMI algorithms, research and synthesis concerns, nonconvex difficulties, and functions. It additionally emphasizes functions of the strategy to parts except keep an eye on.
The simple inspiration of the LMI strategy up to speed is to approximate a given regulate challenge through an optimization challenge with linear target and so-called LMI constraints. The LMI strategy results in an effective numerical answer and is especially fitted to difficulties with doubtful info and a number of (possibly conflicting) requisites.
Since the early Nineteen Nineties, with the advance of interior-point tools for fixing LMI difficulties, the LMI procedure has won elevated curiosity. One benefit of this method is its skill to deal with huge sessions of keep an eye on difficulties through effective numerical instruments. This method is commonly appropriate, not just up to speed yet additionally in different parts the place uncertainty arises. LMI innovations offer a universal language for plenty of engineering difficulties. Notions now renowned in keep watch over, resembling uncertainty and robustness, are getting used in different components by using LMIs. this system is very appealing for commercial functions. it's compatible for the improvement of CAD instruments that aid engineers remedy research and synthesis difficulties.
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Additional resources for Advances in linear matrix inequality methods in control
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]. 2 defines robust solutions to problems with uncertain data and outlines the general ideas involved in Lagrange relaxations for these problems.
The above approach can be extended to some other classes of nonstandard Riccati equations, involving, for example, jump linear systems , problems with indefinite control weighting matrix R , etc. 2 Impulse differential systems Impulse differential systems are systems subject to state discontinuities (or jumps). These systems are hybrid, in the sense that they obey to a mixture of discrete- and continuoustime behaviors. See [27, 358] for references and also  for an application in the context of telecommunications.
Generalized eigenvalue problems (GEVPs) are generalizations of SDPs, of the form minimize A subject to Here, A, B, and C are symmetric matrices that depend affinely on x e Rm. GEVPs are not convex but are quasi convex and amenable to interior-point methods similar to those used for SDPs; see [63, 295]. 2 Interior-point methods One way to view interior-point methods is as a reduction of the original problem to a sequence of differentiable, unconstrained minimization subproblems. Each subproblem can then be efficiently solved, perhaps approximately.