# Advances in Linear Matrix Inequality Methods in Control by Laurent El Ghaoui, Silviu-Iulian Niculescu

By Laurent El Ghaoui, Silviu-Iulian Niculescu

Linear matrix inequalities (LMIs) have lately emerged as worthwhile instruments for fixing a couple of keep watch over difficulties. This booklet presents an up to date account of the LMI technique and covers issues comparable to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and functions. It additionally emphasizes purposes of the strategy to components except regulate. the fundamental proposal of the LMI technique up to the mark is to approximate a given keep watch over challenge through an optimization challenge with linear target and so-called LMI constraints. The LMI procedure ends up in a good numerical answer and is very suited for issues of doubtful information and a number of (possibly conflicting) necessities.

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Example text

Polytopic systems are considered in Chapter 9 of this book. Refined perturbation sets The above models are very rough in that they impose no restriction on the rate of change of the perturbed elements. 22) represents a specific operating mode for the system. In practice, this model will be very conservative, as it does not prevent the system from jumping arbitrarily fast from a mode to any other. Some refined models simply impose a bound on the rate of variation of the perturbation parameters and make use of LMI analysis, which takes into account this bound.

Therefore, with present technology, primal-dual methods are highly efficient if m is not too large (say, less than 1000), but if it is much larger, they are not practical unless sparsity can be exploited. 5. Algorithms and software for SDP 23 quite effectively exploits sparsity in SDPs arising in applications to graph theory. Even this approach, however, is limited by the size of linear systems that can be solved. In what follows, we briefly describe primal-dual path-following methods, following Kojima et al.

Their approach is based on ellipsoidal bounds for the perturbation. 5) belong to the class of min-max problems. To attack them, we can thus use a versatile technique, called Lagrange relaxation, that enables us to approximate a set of complicated constraints by a "more tractable" set. To understand the technique, let us assume that the uncertainty set A is a subset of R' that can be described by a finite number of (nonlinear) constraints where the g^s are given scalar-valued functions of the perturbation vector 6.

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