By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as beneficial instruments for fixing a few regulate difficulties. This booklet offers an up to date account of the LMI process and covers subject matters resembling contemporary LMI algorithms, research and synthesis concerns, nonconvex difficulties, and functions. It additionally emphasizes functions of the tactic to components except regulate. the elemental concept of the LMI process up to speed is to approximate a given keep watch over challenge through an optimization challenge with linear goal and so-called LMI constraints. The LMI technique results in a good numerical resolution and is very fitted to issues of doubtful facts and a number of (possibly conflicting) requisites.
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Additional resources for Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)
6 Link with Lyapunov functions There is more than one connection between the approach proposed here and Lyapunov theory; we will briefly hint at these connections here. 4. Robust SDP 19 Lyapunov functions and Lagrange multipliers Lyapunov functions can be interpreted in the context of Lagrange relaxations, which establish an interesting link between two "central" methods in control and optimization. For the sake of simplicity, we first consider the problem of establishing stability of the LTI system where x 6 Rn is the state, and A 6 R n x n is a (constant) square matrix.
27) is true if and only if the optimum is A 2 n. 5). In general, the above problem is not convex; however, several efficient techniques can be used (see Part V). One of these techniques, described in Chapters 16 and 17, relies on a simple linearization of the quadratic objective and leads to a sequence of SDPs. In some cases, this approximation technique is guaranteed to yield the global optimum of the original problem (see ). 5 On quality of relaxations Associated with the relaxation methods comes the need to evaluate the quality of the approximations involved.
Robustness) seem to play a role that goes sometimes unrecognized yet, such as interval linear algebra. 1 LMIs in combinatorial optimization SDPs are now recognized as yielding very efficient relaxations for (hard) combinatorial problems . 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.
Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control) by Laurent El Ghaoui, Silviu-Iulian Niculescu