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This paper addresses the existence and design of reduced-order controllers for a rather general class of control design problems which can be characterized in the LMI framework. By incorporating certain matrix pencils into the LMI based formulation for solving the problems, this paper presents a unified controller degree bound for the problems in terms of the minimal rank of the matrix pencils with respect to their generalized eigenvalues in the unstable region and at infinity. When the full-order controller exists, this paper shows that there exists a reduced-order controller if one of the matrix pencils has unstable finite generalized eigenvalue(s) or infinite eigenstructure. Moreover, this paper demonstrates that the computational problem of finding the controllers with such new degree bound is convex by providing LMI based design methods for constructing the reduced-order controllers.