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A characterization of fixed order stabilizing controllers is provided for single-input single-output plants. The controller parameters are classified into two types: tractable parameters and intractable parameters. When the values of intractable parameters are fixed, the set of tractable parameter values corresponding to stabilizing controllers is shown to be a union of polyhedral sets. These polyhedral sets can be handled by linear programming. On the other hand, for an appropriate choice of intractable parameter values a probabilistic approach is employed. A similar characterization is also provided for H∞ control problems involving sensitivity and complementary sensitivity functions.