A proper orthogonal decomposition (POD)-based model reduction technique is utilized to develop a closed-loop nonlinear flow control system. By using POD, the Navier-Stokes partial differential equations are recast as a set of nonlinear ordinary differential equations in terms of the unknown Galerkin coefficients. A sliding mode estimator is then employed to estimate, in finite time, the unknown coefficients in the reduced-order model for the actuated flow system. The estimated coefficients are utilized as feedback measurements in a robust nonlinear control law. A rigorous analysis is utilized to analyze the convergence of the sliding mode estimator, and a Lyapunov-based stability analysis is used to prove asymptotic regulation of the flow field velocity to a desired velocity profile. The control objective of tracking a desired velocity profile presented here is a proof of concept only; the proposed methodology could be applied to various flow control objectives. Numerical simulation results are provided to demonstrate the capability of the estimator/control system to regulate the velocity of the flow field to a desired state.