By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as important instruments for fixing a couple of keep an eye on difficulties. This booklet offers an up to date account of the LMI approach and covers themes equivalent to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and purposes. It additionally emphasizes functions of the tactic to components except keep an eye on.
The uncomplicated notion of the LMI technique up to speed is to approximate a given keep an eye on challenge through an optimization challenge with linear aim and so-called LMI constraints. The LMI technique ends up in an effective numerical resolution and is especially fitted to difficulties with doubtful information and a number of (possibly conflicting) requisites.
Since the early Nineties, with the improvement of interior-point equipment for fixing LMI difficulties, the LMI method has received elevated curiosity. One benefit of this method is its skill to deal with huge periods of keep watch over difficulties through effective numerical instruments. This technique is broadly appropriate, not just on top of things yet additionally in different parts the place uncertainty arises. LMI suggestions offer a universal language for plenty of engineering difficulties. Notions now well known in keep an eye on, equivalent to uncertainty and robustness, are getting used in different parts by utilizing LMIs. this system is very beautiful for business purposes. it truly is like minded for the improvement of CAD instruments that aid engineers resolve research and synthesis difficulties.
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Additional info for Advances in linear matrix inequality methods in control
The uncertainty set assumption made on the set A can appear also very specialized. In fact, it can handle a wide array of uncertainty bounds. The following is a short list of examples. Unstructured case. Assume This case is referred to as the "unstructured perturbations" case and is a classic since the development of HQQ control. The corresponding characterization is Euclidean-norm bounds. } 2 The above representation is usually referred to as the "linear-fractional transformation" (LFT) .
1 Decision problems with uncertain data Decision problems Many engineering analysis and design problems can be seen as decision problems. In control engineering, one must decide which controller gains to choose in order to satisfy the desired specifications. This decision involves several trade-offs. ). 3) minimize /o(x) subject to x € #, fi(x) < 0, i = 1,... , fp are given scalar-valued functions of the decision vector x G Rm, and X is a subset of R m . In some problems, X is infinite dimensional, and the decision vector x is a function.
Motivations The above uncertainty models for the data seem very specialized. However, they can cover a wide variety of uncertain matrices, as we now show. , [56, 446]. 2. Every (matrix-valued) rational function M(<5) of 6 € Rp that is well defined for 6 = 0 admits an LFR of the form valid for whenever det(/ — DA) ^ 0 for appropriate matrices M, L, R, D and integers We note that, when I = 1 (that is, for a monovariable rational matrix function), the matrices M, L, R, D are simply a state-space realization of the transfer matrix M+L(sI— D)~1R, where s = 1/6.
Advances in linear matrix inequality methods in control by Laurent El Ghaoui, Silviu-Iulian Niculescu