In this paper, a practical and systematic tuning procedure combining both frequency-domain (FD) and time-domain (TD) specifications is proposed to obtain an optimal robust fractional order (FO) PI${}^{\lambda }$D (FOPI${}^{\lambda }$D) controller for the first order plus time delay (FOPTD) processes. The FD specifications (i.e. phase margin (PM), gain crossover frequency (${\omega }_{gc}$) and flat phase constrain (FPC)) guarantee the systemic stability and robustness to plant gain variations. Meanwhile, the TD specification (i.e. the smallest $J_{ITAE}$) achieves optimal dynamic performance. Furthermore, the entire feasible regions of two frequency-domain specifications ${\omega }_{gc}$ and PM have been obtained with a synthesis scheme and visualized in three-dimensional plots which can be used as prior knowledge before the controller design. The comparisons of feasible region with FOPI and integer order PID (IOPID) controllers clearly present the superiority of proposed FOPI${}^{\lambda }$D controller. Simulation illustration for delay dominant systems, lag dominant systems and high order system with one zero, using the proposed optimal robust FOPI${}^{\lambda }$D controller is presented to demonstrate the significant performance improvement over FOPI controller, three-parameter FOPID controller, Ziegler–Nichols FOPID controller, fractional filter-FOPID controller and SIMC-PI controller.

在本文中，提出了一种结合频域 (FD) 和时域 (TD) 规范的实用且系统的调谐程序，以获得最佳的稳健分数阶 (FO) PI${}^{\lambda }$D (FOPI)${}^{\lambda }$D) 用于一阶加时延 (FOPTD) 过程的控制器。FD 规格（即相位裕度（PM）、增益交叉频率（${\omega }_{GC}$) 和平坦相位约束 (FPC)) 保证系统稳定性和对植物增益变化的鲁棒性。同时，TD规范（即最小的$J_{\mathrm{一世}吨\mathrm{一种}乙}$) 实现最佳动态性能。此外，两个频域规范的整个可行区域${\omega }_{GC}$和 PM 已通过综合方案获得并在三维图中可视化，可用作控制器设计之前的先验知识。可行区域与 FOPI 和整数阶 PID (IOPID) 控制器的比较清楚地展示了所提出的 FOPI 的优越性${}^{\lambda }$D 控制器。使用所提出的最优鲁棒 FOPI 对延迟主导系统、滞后主导系统和具有一零的高阶系统的仿真说明${}^{\lambda }$提出 D 控制器是为了展示与 FOPI 控制器、三参数 FOPID 控制器、Ziegler-Nichols FOPID 控制器、分数滤波器-FOPID 控制器和 SIMC-PI 控制器相比的显着性能改进。