By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as beneficial instruments for fixing a couple of regulate difficulties. This ebook offers an updated account of the LMI technique and covers issues comparable to contemporary LMI algorithms, research and synthesis concerns, nonconvex difficulties, and functions. It additionally emphasizes purposes of the tactic to components except keep an eye on.
The simple proposal of the LMI procedure up to the mark 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 very fitted to difficulties with doubtful information and a number of (possibly conflicting) necessities.
Since the early Nineties, with the advance of interior-point equipment for fixing LMI difficulties, the LMI procedure has won elevated curiosity. One good thing about this system is its skill to deal with huge periods of regulate difficulties through effective numerical instruments. This procedure is commonly appropriate, not just up to the mark 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, akin to uncertainty and robustness, are getting used in different components by utilizing LMIs. this method is very beautiful for business functions. it truly is like minded for the improvement of CAD instruments that aid engineers resolve research and synthesis difficulties.
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Additional resources for Advances in linear matrix inequality methods in control
One of these techniques, described in Chapters 16 and 17, relies on a simple linearization of the quadratic objective and leads to a sequence of SDPs. In some cases, this approximation technique is guaranteed to yield the global optimum of the original problem (see ). 5 On quality of relaxations Associated with the relaxation methods comes the need to evaluate the quality of the approximations involved. In the context of a robust SDP, the first results are due to Nemirovski . , F/, and a scalar p, check if This seemingly simple problem is NP-hard.
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. In the language of control theory, we can interpret the above as a multispecification control problem, where several (possibly conflicting) constraints have to be satisfied. Let us comment on the limitations of the above mathematical framework.
Decision problems with uncertain data 7 In control theory, we interpret the above problems as robust multispecification control problems, where several (possibly conflicting) constraints have to be satisfied in a robust manner. We note that these problems are in general very hard to solve (noncomputationally tractable). References on complexity of such problems in control include [86, 291, 72, 52, 393, 144]. Robust decision problems first arose (to our knowledge) in the field of automatic control.