Stabilizing and robust FOPI controller synthesis for first order plus time delay systems

Abstract

For all first order plus time delay (FOPTD) systems, a fractional order PI (FOPI) or a traditional integer order PID (IOPID) controller can be designed to fulfill three design specifications: gain crossover frequency, phase margin, and a flat phase constraint simultaneously. In this paper, a guideline for choosing feasible or achievable gain crossover frequency and phase margin specifications, and a new FOPI/IOPID controller synthesis are proposed for all FOPTD systems. Using this synthesis scheme, the complete feasible region of the gain crossover frequency and phase margin can be obtained and visualized in the plane. With this region as the prior knowledge, all combinations of the phase margin and gain crossover frequency can be verified before the controller design. Only if the combination is chosen from this achievable region, the existence of the stabilizing and desired FOPI/IOPID controller design can be guaranteed. Especially, it is interesting to compare the areas of these two feasible regions for the IOPID controller and the FOPI controller. This area comparison reveals, for the first time, the potential advantages of one controller over the other in terms of achievable performances. As a basic step, a scheme for finding the stabilizing region of the FOPI/IOPID controller is presented first, and then a new scheme for designing a stabilizing FOPI/IOPID controller satisfying the given gain crossover frequency, phase margin and flat phase constraint is proposed in details. Thereafter, the complete information about the feasible region of gain crossover frequency and phase margin is collected. This feasible region for the FOPI controller is compared with that for the traditional IOPID controller. This area comparison shows the advantage of the FOPI over the traditional IOPID clearly. Simulation illustration is presented to show the effectiveness and the performance of the designed FOPI controller comparing with the designed IOPID controller following the same synthesis in this paper.