By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as worthy instruments for fixing a couple of keep an eye on difficulties. This publication presents an updated account of the LMI process and covers subject matters resembling contemporary LMI algorithms, research and synthesis matters, nonconvex difficulties, and functions. It additionally emphasizes purposes of the tactic to components except keep watch over. the fundamental concept of the LMI process on top of things is to approximate a given regulate challenge through an optimization challenge with linear aim and so-called LMI constraints. The LMI process ends up in an effective numerical answer and is very suited for issues of doubtful info and a number of (possibly conflicting) necessities.
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Additional resources for Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)
4. ) In turn, the SDP relaxations open up interesting perspectives for handling combinatorial problems with uncertain data, in view of the tools developed for the robust SDP. We illustrate this via a simple example. Consider the NP-hard problem where W is a given symmetric matrix. When W assumes some special structure, the above problem is known in the combinatorial optimization literature as "the maximum cut" (MAX-CUT) problem [161, 154]. 3. , <5j). 1. 47) minTrS subject to W < 5, 5 diagonal. , an upper bound) of the problem.
1. , Murray  for interesting examples in biology). A typical problem in control of delay system is to analyze stability as a function of delay, where the delay is viewed as a parameter of the system. The problem is still largely open ; even in the linear case, and with multiple (noncommensurable) delays, the problem is NP-hard (see Toker and Ozbay  for a precise statement and results). , no information on the delay size is used) and delay-dependent stability analysis (based on this information).
Introduce p = sx] our robustness analysis problem is equivalent to checking if Let us eliminate the "uncertain parameter" s by noting that Let us apply a Lagrange relaxation idea now. A sufficient condition for stability is that there exists a "multiplier" matrix S > 0 such that A simple computation leads to the classical Lyapunov condition which corresponds to the Lyapunov function candidate V(x) = xTSx. Let us give another example of the "Lagrange relaxation" method for dynamical systems. ) To analyze stability, the idea is to apply Lagrange relaxation twice.
Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control) by Laurent El Ghaoui, Silviu-Iulian Niculescu