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On the rank minimization problem over a positive semidefinite linear matrix inequality. (English) Zbl 0872.93029
Let the system x ˙=Ax+Bu, y=Cx be stabilized by the kth order controller z ˙=A K z+B K y, u=C K z+D K y. 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)0, X0; M symmetry preserving on the space of symmetric matrices, Q symmetric and the ordering is to be interpreted in the sense of Löwner: AB 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.
93B50Synthesis problems
15A39Linear inequalities of matrices
93D15Stabilization of systems by feedback
15A03Vector spaces, linear dependence, rank