By Laurent El Ghaoui, Silviu-Iulian Niculescu
Linear matrix inequalities (LMIs) have lately emerged as priceless instruments for fixing a few keep watch over difficulties. This booklet offers an up to date account of the LMI process and covers themes comparable to fresh LMI algorithms, research and synthesis matters, nonconvex difficulties, and functions. It additionally emphasizes purposes of the tactic to parts except keep watch over. the elemental notion of the LMI approach up to speed is to approximate a given keep an eye on challenge through an optimization challenge with linear goal and so-called LMI constraints. The LMI approach results in an effective numerical resolution and is very fitted to issues of doubtful facts and a number of (possibly conflicting) requisites.
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Additional info for Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)
To find the new iterate, two steps are taken. In the predictor step, we seek to approach the optimum, that is, to satisfy ZF = 0. In the corrector step, we seek to return close to the central path by ensuring ZF = fj,I. The search directions 6x, 6Z for each step (predictor and corrector) are computed by linearization of the constraint ZF = 0 (predictor) or ZF = p,I (corrector) around the current value of (x,Z). Each step thus gives rise to a linear system in the elements of 6x, 6Z. Note that ZF can be linearized in a number of ways, depending on the specific method used.
Even this approach, however, is limited by the size of linear systems that can be solved. In what follows, we briefly describe primal-dual path-following methods, following Kojima et al. . 35) are both strictly feasible, then we may write the optimality conditions where £ is an affine subspace: We can interpret the above conditions as complementarity conditions over the positive semidefinite cone, similar to those arising in LP . The convexity of the original SDP is transported here into a property of monotonicity of £, which is crucial for the algorithm outlined below to converge globally.
3 El Ghaoui and Niculescu Examples Let us now show how the above tools can be used to describe compactly a wide array of uncertain matrices and dynamical systems. To simplify our description, we take the operator point of view mentioned previously. Matrices The above framework covers the case when some (matrix) data occurring in the robust decision problem is affected by a perturbation vector <5, in an algebraic manner, and the vector 6 is unknown-but-bounded. Consider, for example, an "uncertain matrix" M(£), where M is an algebraic function of vector <5, and 6 is unknown-but-bounded in the maximum-norm sense.