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A new approach for robust fixed-order $H_\infty$ controller design by convex optimization is proposed. Linear time-invariant single-input single-output systems represented by a finite set of complex values in the frequency domain are considered. It is shown that the H infinity robust performance condition can be approximated by a set of linear or convex constraints with respect to the parameters of a linearly parameterized controller in the Nyquist diagram. Multimodel and frequency-domain uncertainty can be directly considered in the proposed approach by increasing the number of constraints. The proposed method is compared with the standard H infinity control problem. It is shown by an example that for an unstable uncertain model, a PID controller can be designed with the proposed approach which gives better H infinity performance than a 7th order unstable controller obtained by the standard H infinity solution.