By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as important instruments for fixing a few keep an eye on difficulties. This ebook presents an up to date account of the LMI technique and covers subject matters corresponding to contemporary LMI algorithms, research and synthesis matters, nonconvex difficulties, and purposes. It additionally emphasizes purposes of the tactic to components except regulate. the elemental concept of the LMI approach on top of things is to approximate a given keep watch over challenge through an optimization challenge with linear target and so-called LMI constraints. The LMI process results in an effective numerical answer and is very fitted to issues of doubtful information 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)
The following condition on po, • • • > 9p holds if there exist scalars T» > 0, i = 1, . . ,p, such that When p=l, the converse holds, provided that there is some 60 such that gi(6o) > 0. 3 Uncertainty models A typical decision problem is defined via some data, which we collect for convenience in a matrix M. For example, if the problem is defined in terms of the transfer function of an LTI system M(s) — D + C(sl — A)~1B, the matrix M contains the four matrices A, B, C, D. 3. Uncertainty models 9 set A.
Crl . mcmaster . 6 Illustrations in control This book abounds with examples of applications of the LMI framework to control problems. In this section, we seek to provide complementary examples. , have usually interesting interpretations for the analysis of (optimal) control problems. For example, the perturbation theory for the general SDP recently developed by Shapiro [372, 373] and Bonnans, Cominetti, and Shapiro  can be used for the analysis of Lyapunov and Riccati inequalities and equations encountered in optimal control.
Let us give another example of the "Lagrange relaxation" method for dynamical systems. ) To analyze stability, the idea is to apply Lagrange relaxation twice. In a first step, we "forget" uncertainty and apply Lagrange relaxation to the "frozen" system. This results in a sufficient condition for stability: There exists a matrix 5 such that This condition is known as a quadratic stability condition, since it requires the existence of a quadratic Lyapunov function proving stability of the uncertain system.