Sliding mode estimation and closed‐loop active flow control under actuator uncertainty

Abstract

This article presents a nonlinear closed-loop active flow control (AFC) method, which achieves asymptotic regulation of a fluid flow velocity field in the presence of actuator uncertainty and sensor measurement limitations. To achieve the result, a reduced-order model of the flow dynamics is derived, which utilizes proper orthogonal decomposition (POD) to express the Navier-Stokes equations as a set of nonlinear ordinary differential equations. The reduced-order model formally incorporates the actuation effects of synthetic jet actuators (SJA). Challenges inherent in the resulting POD-based reduced-order model include (1) the states are not directly measurable, (2) the measurement equation is in a nonstandard mathematical form, and (3) the SJA model contains parametric uncertainty. To address these challenges, a sliding mode observer (SMO) is designed to estimate the unmeasurable states in the reduced-order model of the actuated flow field dynamics. A salient feature of the proposed SMO is that it formally compensates for the parametric uncertainty inherent in the SJA model. The SMO is rigorously proven to achieve local finite-time estimation of the unmeasurable state in the presence of the parametric uncertainty in the SJA. The state estimates are then utilized in a nonlinear control law, which regulates the flow field velocity to a desired state. A Lyapunov-based stability analysis is provided to prove local asymptotic regulation of the flow field velocity. To illustrate the performance of the proposed estimation and AFC method, comparative numerical simulation results are provided, which demonstrate the improved performance that is achieved by incorporating the uncertainty compensator. K E Y W O R D S robust control, sliding mode estimation, uncertainty List of Symbols , flow field velocity p , flow field pressure (s) , POD mode (s) , actuation mode x(t) , time-varying Galerkin coefficient