Mesbahi, M.; Papavassilopoulos, G. P. On the rank minimization problem over a positive semidefinite linear matrix inequality. (English) Zbl 0872.93029 IEEE Trans. Autom. Control 42, No. 2, 239-243 (1997). Let the system \(\dot x= Ax+ Bu\), \(y= Cx\) be stabilized by the \(k\)th order controller \(\dot z= A_Kz+ B_Ky\), \(u= C_Kz+ D_Ky\). According to El Ghaudi and Gabinet, the existence of the stabilizing controller can be reduced to a MIN-RANK problem: find min rank \(X\), subject to \(Q+ M(X)\geq 0\), \(X\geq 0\); \(M\) symmetry preserving on the space of symmetric matrices, \(Q\) symmetric and the ordering is to be interpreted in the sense of Löwner: \(A\geq B\) iff \(A-B\) is positive definite. The authors’ method of solving the problem employs idea from the ordered linear complementarity theory and the notion of the least element in a vector lattice. Reviewer: A.Vaněček (Praha) Cited in 1 ReviewCited in 41 Documents MSC: 93B50 Synthesis problems 15A39 Linear inequalities of matrices 93D15 Stabilization of systems by feedback 15A03 Vector spaces, linear dependence, rank, lineability Keywords:output stabilization; linear matrix inequalities; rank minimization; ordered linear complementarity theory; least element; vector lattice PDFBibTeX XMLCite \textit{M. Mesbahi} and \textit{G. P. Papavassilopoulos}, IEEE Trans. Autom. Control 42, No. 2, 239--243 (1997; Zbl 0872.93029) Full Text: DOI