# Control-oriented model reduction for minimizing transient energy growth in shear flows

## TL;DR

This paper develops a control-oriented reduced-order modeling approach to design feedback controllers that effectively minimize transient energy growth in shear flows, enabling more efficient flow control strategies.

## Contribution

It introduces a combined model reduction technique using proper orthogonal decomposition and balanced truncation tailored for control design in shear flows.

## Key findings

- Reduced-order models enable tractable controller synthesis.
- Controllers based on reduced models outperform traditional LQ strategies.
- The approach effectively minimizes transient energy growth in linearized channel flow.

## Abstract

A linear non-modal mechanism for transient amplification of perturbation energy is known to trigger sub-critical transition to turbulence in many shear flows. Feedback control strategies for minimizing this transient energy growth can be formulated as convex optimization problems based on linear matrix inequalities. Unfortunately, solving the requisite linear matrix inequality problem can be computationally prohibitive within the context of high-dimensional fluid flows. In this work, we investigate the utility of control-oriented reduced-order models to facilitate the design of feedback flow control strategies that minimize the maximum transient energy growth. An output projection onto proper orthogonal decomposition modes is used to faithfully capture the system energy. Subsequently, a balanced truncation is performed to reduce the state dimension, while preserving the system's input-output properties. The model reduction and control approaches are studied within the context of a linearized channel flow with blowing and suction actuation at the walls. Controller synthesis for this linearized channel flow system becomes tractable through the use of the proposed control-oriented reduced-order models. Further, the resulting controllers are found to reduce the maximum transient energy growth compared with more conventional linear quadratic optimal control strategies.