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For model-free time-delay systems, an analytical method is proposed to characterize the stabilizing PID region based on the frequency response data. Such characterization uses linear programming which is computationally efficient. The characteristic parameters of the controller are first extracted from the frequency response data. Subsequently, by employing an extended Hermite-Biehler theorem on quasipolynomials, the stabilizing polygon region with respect to the integral and derivative gains (ki    and kd) is described for a given proportional gain (kp) in term of the frequency response data. Simultaneously, the allowable stabilizing range of kp is derived such that the complete stabilizing set of the PID controller can be obtained easily. The proposed method avoids the complexity and inaccuracy of the model identification and thus provides a convenient approach for the design and tuning of the PID controller in practice. The advantage of the proposed algorithm lies in that the boundaries of the stabilizing region consist of several simple straight lines, the complete stabilizing set can be obtained efficiently, and it can be implemented automatically in computers

Characterization of the Stabilizing PID Controller Region for the Model-Free Time-Delay System