By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as precious instruments for fixing a couple of keep an eye on difficulties. This booklet presents an up to date account of the LMI technique and covers subject matters similar to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and purposes. It additionally emphasizes purposes of the tactic to parts except keep watch over.
The easy notion of the LMI strategy 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 process results in an effective numerical answer and is especially suited for difficulties with doubtful facts and a number of (possibly conflicting) necessities.
Since the early Nineteen Nineties, with the advance of interior-point tools for fixing LMI difficulties, the LMI process has won elevated curiosity. One benefit of this system is its skill to deal with huge sessions of regulate difficulties through effective numerical instruments. This strategy is commonly appropriate, not just up to speed yet additionally in different components the place uncertainty arises. LMI innovations supply a universal language for lots of engineering difficulties. Notions now well known in regulate, similar to uncertainty and robustness, are getting used in different parts by utilizing LMIs. this system is especially appealing for commercial purposes. it truly is well matched for the improvement of CAD instruments that aid engineers remedy research and synthesis difficulties.
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Linear matrix inequalities (LMIs) have lately emerged as beneficial instruments for fixing a couple of keep an eye on difficulties. This ebook presents an updated account of the LMI strategy and covers subject matters reminiscent of fresh LMI algorithms, research and synthesis concerns, nonconvex difficulties, and functions. It additionally emphasizes functions of the tactic to parts except regulate.
Integer recommendations for platforms of linear inequalities, equations, and congruences are thought of besides the development and theoretical research of integer programming algorithms. The complexity of algorithms is analyzed established upon parameters: the size, and the maximal modulus of the coefficients describing the stipulations of the matter.
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Extra resources for Advances in linear matrix inequality methods in control
The above idea can be extended in various ways, as illustrated in some other chapters (Parts III and IV) of this book. For example, we may consider parameter-dependent, or frequency-dependent Lyapunov functions, based on so-called IQCs. Stochastic Lyapunov functions Stochastic Lyapunov functions have been introduced by Kushner , and their use in the context of LMI optimization has been introduced in [64, 120, 121]. Consider stochastic system of the form where w is a scalar standard Wiener process (the "derivative" of this process can be interpreted as a random perturbation on the state-space matrix, hence the name multiplicative noise given to such perturbations).
Each subproblem can then be efficiently solved, perhaps approximately. These methods can be described in the context of self-concordant barrier and potential functions; this is the point of view taken in [296, 404]. Primal-dual interior-point methods Without doubt, the most efficient interior-point methods for LP are the primal-dual interior-point methods . When a primal-dual interior-point method is used, each subproblem can be described by a linear least squares problem or, equivalently, a system of linear equations.
3 4 El Ghaoui and Niculescu Multicriteria problems. The approach enables us to impose many different (possibly conflicting) specifications in the design process, allowing us to explore trade-offs and analyze limits of performance and feasibility. This offers a drastic advantage over design methods that rely on a single criterion deemed to reflect all design constraints; the choice of a relevant criterium is sometimes a nontrivial task. Wide applicability. The techniques used in the approach are relevant far beyond control and estimation.