# Full-order optimal compensators for flow control: the multi-input case

## TL;DR

This paper develops an iterative algorithm for designing full-order optimal flow controllers with multiple actuators and sensors, avoiding complex Riccati equations, and demonstrates its effectiveness and scalability in large dynamical systems.

## Contribution

It extends the ADA algorithm to multi-input, coupled, and robust H-infinity controllers, enabling direct design without model reduction for large flow systems.

## Key findings

- Algorithm outperforms existing methods in convergence speed.
- Scalable with the number of actuators in large systems.
- Effective in complex flow control scenarios like Kuramoto-Sivashinsky.

## Abstract

Flow control has been the subject of numerous experimental and theoretical works. In this numerical study, we analyse full-order, optimal controllers for large dynamical systems in presence of multiple actuators and sensors. We start from the original technique proposed by Bewley et al. (2016), the adjoint of the direct-adjoint (ADA) algorithm. The algorithm is iterative and allows bypassing the solution of the algebraic Riccati equation associated with the optimal control problems, typically unfeasible for large systems. We extend ADA into a more generalized framework that includes the design of multi-input, coupled controllers and robust controllers based on the H_{\infty} framework. The full-order controllers do not require any preliminary step of model reduction or low-order approximation: this feature allows to pre-assess the optimal performances of an actuated flow without relying on any estimation process or further hypothesis. We show that the algorithm outperforms analogous technique, in terms of convergence performances considering two numerical cases: a distributed system and the linearized Kuramoto-Sivashinsky equation, mimicking a full three-dimensional control setup. For the ADA algorithm we find excellent scalability with the number of inputs (actuators) in terms of convergence to the solution, making the method a viable way for full-order controller design in complex settings.