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It is shown that the state feedback matrix of a linear system optimal with respect to a quadratic performance index can be expanded in a MacLaurin series in parameters which change the order of the system. The first two terms of this series are employed in a near-optimum design for a high-order plant. The result of the near-optimum design is superior to that achieved by a conventional low-order design, while the amount of computation is considerably less than that required for a high-order design. An example of a second-order design for a fifth-order plant is given.