Iterative Feedback Tuning of the Proportional-Integral-Differential Control of Flow Over a Circular Cylinder

In this paper, we numerically perform a feedback gain optimization of the proportional-integral-differential (PID) control of flows over a circular cylinder at Re = 60 and 100. We measure the transverse velocity at a centerline location in the wake as a sensing variable and provide blowing and suction at the upper and lower slots on the cylinder surface as an actuation based on the PID control. The PID control gains are optimized by an iterative feedback tuning method that is one of the typical model-free gain optimization methods. For the feedback tuning, the cost function J is constructed such that the optimal control gains minimize the sensing velocity fluctuations. The control gains are iteratively updated using the gradient of the cost function until the control system satisfies a stopping criterion. For various sensing locations, the present control with optimal control gains successfully reduces the sensing velocity fluctuations and attenuates or annihilates vortex shedding in the wake, resulting in the reduction of the mean drag and lift fluctuations. It is also shown that the optimal actuation velocities obtained from different sensing locations have nearly the same phase in time. Finally, the occurrence of the integral windup and differential oscillations during PID controls is discussed.