Controller Synthesis for Periodic, Linear-Distributed Parameter Systems: A Channel Flow Application

Abstract

A framework is presented in which temporally periodic, linear, distributed parameter systems can be converted to a time-invariant system. This conversion is key for the control of the secondary instabilities in three-dimensional channel flow induced by an upstream traveling wave of zero-net mass flux of wall transpiration. Linearized dynamical equations derived from Floquet analysis have shown that the instabilities are a direct result of the primary disturbance of the traveling wave but do not provide an analytical framework upon which to design a feedback controller. The necessary observation, although simple but subtle, is that the dynamics of the steady-state flow induced by a traveling wave must be linearized and decomposed in a frame of reference moving with the traveling wave. The resulting linear time-invariant equations are appropriate for system theoretic feedback control synthesis, i.e., H 2 and H ∞ methods. Although the linearization method produces a time-invariant linear system in the moving frame, the controller is periodic from a fixed reference frame. This approach for constructing a time-invariant system with periodic inputs is applicable to any system in which the dynamics are described as a combination of a static base and a periodic primary disturbance.