Performance of a linear robust control strategy on a nonlinear model of spatially developing flows

Abstract

This paper investigates the control of self-excited oscillations in spatially developing flow systems such as jets and wakes using ${\mathcal H}_{\infty}$ control theory on a complex Ginzburg–Landau (CGL) model. The coefficients used in this one-dimensional equation, which serves as a simple model of the evolution of hydrodynamic instability waves, are those selected by Roussopoulos & Monkewitz (Physica D 1996, vol. 97, p. 264) to model the behaviour of the near-wake of a circular cylinder. Based on noisy measurements at a point sensor typically located inside the cylinder wake, the compensator uses a linear ${\mathcal H}_{\infty}$ filter based on the CGL model to construct a state estimate. This estimate is then used to compute linear ${\mathcal H}_{\infty}$ control feedback at a point actuator location, which is typically located upstream of the sensor. The goal of the control scheme is to stabilize the system by minimizing a weighted average of the ‘system response’ and the ‘control effort’ while rigorously bounding the response of the controlled linear system to external disturbances. The application of such modern control and estimation rules stabilizes the linear CGL system at Reynolds numbers far above the critical Reynolds number $Re_c \,{\approx}\, 47$ at which linear global instability appears in the uncontrolled system. In so doing, many unstable modes of the uncontrolled CGL system are linearly stabilized by the single actuator/sensor pair and the model-based feedback control strategy. Further, the linear performance of the closed-loop system, in terms of the relevant transfer function norms quantifying the linear response of the controlled system to external disturbances, is substantially improved beyond that possible with the simple proportional measurement feedback proposed in previous studies. Above $Re\,{\approx}\, 84$, the ${\mathcal H}_{\infty}$ control designs significantly outperform the corresponding ${\mathcal H}_2$ control designs in terms of their ability to stabilize the CGL system in the presence of worst-case disturbances. The extension of these control and estimation rules to the nonlinear CGL system on its attractor (a simple limit cycle) stabilizes the full nonlinear system back to the stationary state at Reynolds numbers up to $Re\,{\approx}\, 97$ using a single actuator/sensor pair, fixed-gain linear feedback and an extended Kalman filter incorporating the system nolinearity.