By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as worthy instruments for fixing a few keep watch over difficulties. This booklet offers an up to date account of the LMI procedure and covers issues akin to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and functions. It additionally emphasizes functions of the tactic to parts except regulate.
The easy inspiration of the LMI strategy on top of things is to approximate a given keep an eye on challenge through an optimization challenge with linear target and so-called LMI constraints. The LMI process results in an effective numerical resolution and is especially suited for difficulties with doubtful facts and a number of (possibly conflicting) requirements.
Since the early Nineties, with the improvement of interior-point tools for fixing LMI difficulties, the LMI process has won elevated curiosity. One good thing about this method is its skill to deal with huge periods of keep an eye on difficulties through effective numerical instruments. This method is commonly appropriate, not just up to speed yet additionally in different components the place uncertainty arises. LMI thoughts supply a universal language for plenty of engineering difficulties. Notions now renowned in keep watch over, equivalent to uncertainty and robustness, are getting used in different parts by utilizing LMIs. this method is very appealing for commercial functions. it really is like minded for the improvement of CAD instruments that support engineers remedy research and synthesis difficulties.
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Extra info for Advances in linear matrix inequality methods in control
Http : //www . ensta . f r/~gropco/ • The code SDPpack, outlined in Chapter 3 of this book, implements a primal-dual algorithm due to Alizadeh, Haeberly, and Overton  and Alizadeh et al. . http : //www . cs . nyu . edu/cs/f aculty/overton/sdppack/sdppack . html • The code SDPA: developed by Fujisawa, Kojima, and Nakata; the underlying algorithm is described in . ftp : //ftp . is . t itech . ac . jp/pub/OpRes/sof tware/SDPA • The code SDPHA: developed by Brixius, Sheng, and Potra; for a description of the underlying algorithm, see the web page.
3. Uncertainty models 13 with Hnd free of delay and 6n as the nominal delay. The uncertainty 6 is of the form where A is an unknown-but-bounded real scalar (— 1 < A < 1). A way to handle this class of systems is to aproximate the delay uncertainty term using a first-order real Fade approximant. An LFR of the approximation is Using this LFR, we may obtain an LFR for the delayed system. ) Analysis and comparisons between several approximations of the delay uncertainty can be found in . 4 Toward other models We now briefly mention other possible models.
In control, this kind of problem occurs when only some parameters are available for measurement. 1 We take the uncertainty to be a matrix instead of a vector for reasons to be made clear later. 2. 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).