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On the rank minimization problem over a positive semidefinite linear matrix inequality. (English) Zbl 0872.93029
Let the system $\stackrel{˙}{x}=Ax+Bu$, $y=Cx$ be stabilized by the $k$th order controller $\stackrel{˙}{z}={A}_{K}z+{B}_{K}y$, $u={C}_{K}z+{D}_{K}y$. 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 $Q+M\left(X\right)\ge 0$, $X\ge 0$; $M$ symmetry preserving on the space of symmetric matrices, $Q$ symmetric and the ordering is to be interpreted in the sense of Löwner: $A\ge B$ 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