Solving quadratic distance problems: an LMI-based approach G. Chesi 1 , A. Garulli 1 , A. Tesi 2 , A. Vicino 1 1 Dipartimento di Ingegneria dell’Informazione, Universit`a di Siena Via Roma, 56 - 53100 Siena, Italia Tel.: +39 0577 233612; Fax: +39 0577 233602 E-mail: chesi,garulli,vicino@ing.unisi.it 2 Dipartimento di Sistemi e Informatica, Universit`a di Firenze Via di S. Marta, 3 - 50139 Firenze, Italia E-mail: atesi@dsi.unifi.it Abstract The computation of the minimum distance from a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems. This paper presents a general framework in which some classes of minimum distance problems are tackled via LMI techniques. Exploiting a suit- able representation of homogeneous forms, a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one-parameter family of LMI optimization problems. Several properties of the proposed tech- nique are discussed. In particular, tightness of the lower bound is investigated, providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures opti- mality a priori. Extensive numerical simulations are reported showing promising performance of the proposed method. Keywords: Optimization, distance problems, Linear Matrix Inequalities, homoge- neous forms. 1