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This paper considers the problem of stabilizing linear time-invariant plants by a PID controller. If one of the tuning parameters reaches the extreme value, the closed-loop system contains a double pole on the positive imaginary axis or the integral gain is equal to zero. Using this property, the admissible ranges of the proportional gain, the integral gain and the derivative gain are derived respectively. If the proportional gain is fixed, the stability region in the plane with respect to the integral gain and the derivative gain is determined by plotting the stability boundary locus. The effectiveness of the method presented is illustrated a numerical example. Introduction The PID controller is widely used in industrial processes. It has simple structure and gives a satisfactory performance. The PID controller can be tuned in both practical and theoretical ways. In practical applications, the most famous tuning method is the Ziegler-Nichols tuning formula. To tune a PID controller in theoretical way, its basis is to find the stabilizing regions in the parameters space. For linear time-invariant plants, an important method based on the generalized version of the Hermite-Biehler theorem is proposed in [1] to obtain all stabilizing PID controllers. A linear programming approach is adopted to solve the same problem [2]. However, the computation time of these two methods will increase exponentially as the order of the plant varies. Using the Nyquist plot, a fast solution to this problem is given. But if the proportional gain reaches the extreme value, the corresponding integral gain and derivative gain are not derived [4]. By plotting the stability boundary locus, the stability region in the ) , ( d i K K plane for a given p K is determined [3, 5], while the admissible range of the proportional gain is not given. In terms of Kronecker summation operation, the stability regions of PID controllers are given in the ) , ( i p K K , ) , ( d p K K and ) , ( d i K K plane respectively [6]. Based on the inverse Nyquist plot, the possible range of the proportional gain for a given plant is estimated [7]. The admissible range of the proportional gain in stabilizing PID region is derived in [8]. This paper is arranged as follows. Firstly, a theorem is given to determine the stabilizing PID region is closed or not. Secondly, analytical expressions are derived to calculate the admissible ranges of the tuning parameters; finally, a numerical example is used to illustrate the proposed method. System Model The structure of the PID control system is shown in Fig. 1.

Extreme Value Condition of Stabilizing PID Region