By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as helpful instruments for fixing a couple of keep watch over difficulties. This e-book offers an up to date account of the LMI procedure and covers themes comparable to contemporary LMI algorithms, research and synthesis matters, nonconvex difficulties, and functions. It additionally emphasizes functions of the tactic to parts except regulate. the elemental suggestion of the LMI technique up to the mark is to approximate a given keep an eye on challenge through an optimization challenge with linear target and so-called LMI constraints. The LMI strategy ends up in a good numerical resolution and is very fitted to issues of doubtful info and a number of (possibly conflicting) standards.
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Additional resources for Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)
Sample text
We may distinguish the following: • A robust feasibility problem, where we seek a vector x that belongs to • A robust optimality problem, defined as Here, we seek to minimize the worst-case value of the objective fo(x, A) over the set of robustly feasible solutions. ) Parameter-scheduled synthesis. In such a problem, we allow the decision vector to be a function of the perturbation. In this case, we assume that the decision vector is not finite dimensional but evolves in a set X of functions (of the perturbation A).
Robustness out of control 29 The problem is to find a u e U, if any, such that for every XQ € AQ, the resulting output satisfies y £ y. If the state-space matrices are exactly known, the problem is equivalent to a simple linear program. In the case when the state-space matrices are uncertain, the robust counterpart can be approximated using the robust decision methodology. In terms of optimization, this versatile framework gives rise to challenging large-scale problems. One way to avoid the curse of dimensionality is to construct ellipsoids of confidence for the future state and optimize (over a few time steps) the control vector.
1 Basic idea The basic idea of the LMI method is to formulate a given problem as an optimization problem with linear objective and linear matrix inequality (LMI) constraints. An LMI constraint on a vector x e Rm is one of the form where the symmetric matrices Fj = F? G R JVxJV , i = 0,... ,m, are given. The minimization problem where c € R m , and F > 0 means the matrix F is symmetric and positive semidefinite, is called a semidefinite program (SDP). The above framework is particularly attractive for the following reasons.