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On the rank minimization problem over a positive semidefinite linear matrix inequality. (English) Zbl 0872.93029

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.

MSC:

93B50 Synthesis problems
15A39 Linear inequalities of matrices
93D15 Stabilization of systems by feedback
15A03 Vector spaces, linear dependence, rank, lineability
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