By Stephen Boyd, Laurent El Ghaoui, Eric Feron, Vendataramanan Balakrishnan
During this textual content, the authors decrease a wide selection of difficulties bobbing up in procedure and regulate conception to a handful of convex and quasiconvex optimization difficulties that contain linear matrix inequalities.
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Extra info for Linear Matrix Inequalities in System & Control Theory
Example text
See the Notes and References for more details. This electronic version is for personal use and may not be duplicated or distributed. 7 Some Matrix Problems Ellipsoidal Approximation The problem of approximating some subset of Rn with an ellipsoid arises in many fields and has a long history; see the Notes and References. ). In some cases the problem can be cast as a CP or an EVP, and hence solved exactly. As an example consider the problem of finding the ellipsoid centered around the origin of smallest volume that contains a polytope described by its vertices.
S-procedure example In Chapter 5 we will encounter the following constraint on the variable P : for all ξ = 0 and π satisfying π T π ≤ ξ T C T Cξ, ξ π T AT P + P A P B BT P 0 ξ π < 0. 34) is equivalent to the existence of τ ≥ 0 such that AT P + P A + τ C T C P B < 0. 34) holds can be expressed as an LMIP (in P and the scalar variable τ ). 7 Some LMI Problems with Analytic Solutions There are analytic solutions to several LMI problems of special form, often with important system and control theoretic interpretations.
In most of the problems encountered in this book, there are no implicit equality constraints or nontrivial common nullspace for F , so we can just take A = I, b = 0, and F˜ = F . Using this reduction we can, at least in principle, always deal with strictly feasible LMIs. For example we have inf cT x F (x) ≥ 0 = inf cT (Az + b) F˜ (z) ≥ 0 = inf cT (Az + b) F˜ (z) > 0 since the LMI F˜ (z) ≥ 0 is either infeasible or strictly feasible. 23) where A ∈ Rk×k is given, and the symmetric matrix P is the variable.