# Optimal Robust PID Control for First- and Second-Order Plus Dead-Time Processes
## 1 Introduction  
Proportional-integral-derivative (PID) control has been widely used because the control structure is simple and the role of tuning parameters is clear. See [1] as example of complex industrial application. The epoch-making rule of PID control has been investigated as the Ziegler and Nichols (ZN) method [2] and has been studied for more than half a century [3; 4; 5; 6].  
In the control system, stability is crucially important, and therefore, robust PID control has been studied [7; 8]. [PERSON] et al. have proposed a fast calculation method [9]. In this method, the stabilizing regions in the PID parameter space are decided for a fixed proportional gain, and the boundaries for stabilizing the proportional gain are also provided. Works [10; 11; 12] have proposed methods for the computation of all stabilizing PI(D) controllers, where the stability regions of PI(D) controllers are given. In [10], stabilizing PI controllers that achieve user-specified gain and phase margins are studied, and [11] gives the computation of stability boundaries of PI controller parameters that guarantee stability. [PERSON] and [PERSON] have proposed a method for the computation of all possible robustly stabilizing PID controllers for interval plants [10]. On the other hand, tracking performance also has to be improved. Since stability and tracking performance are in a trade-off relationship, a trade-off design method is required [13; 14; 15]. However, these trade-off design methods require complex calculations and so are unsuitable for industrial engineers.  
A robust PID design method has been proposed [16; 17] in which the tracking performance is optimized subject to the prescribed stability margin. Because of its usefulness, the unified design method has been extended from the first-order plus dead-time (FOPDT) system to the second-orderplus dead-time (SOPDT) system [18], and a two-degrees-of-freedom (2 DoF) design has also been proposed [19]. Since these unified design methods are designed based on normalized plant models, the unified approach does not need a different solution whenever a different control system is designed.  
The present method is inspired by the work presented in [18]. In this work, both FOPDT and SOPDT are considered. In a unified way, both process models' structures are considered and an optimal PID controller is adjusted for servo tracking and the disturbance rejection. However, in this approach, robustness is considered by simply adjusting the controller proportional gain. In the current work, a similar problem is faced, but the robustness is introduced in its full affectation of the controller parameters. Obviously, this affects the expression of all three controller parameters but allows for the improvement of performance. The rest of the paper is structured as follows. In Section 2, the controlled system model and the design objective are described. Section 3 gives the proposed PID tuning method, and the effectiveness of the proposed interpolation method is confirmed through numerical examples in Section 4. Finally, concluding remarks are presented in Section 5.