Performance Limitations of Observer-Based Feedback for Transient Energy Growth Suppression

Abstract

Transient energy growth suppression is a common control objective for feedback flow control aimed at delaying transition to turbulence. A prevailing control approach in this context is observer-based feedback, in which a full-state feedback controller is applied to state estimates from an observer. The present study identifies a fundamental performance limitation of observer-based feedback control: whenever the uncontrolled system exhibits transient energy growth in response to optimal disturbances, control by observer-based feedback will necessarily lead to transient energy growth in response to optimal disturbances for the closed-loop system as well. Indeed, this result establishes that observerbased feedback can be a poor candidate for controller synthesis in the context of transient energy growth suppression and transition delay: the performance objective of transient energy growth suppression can never be achieved by means of observer-based feedback. Further, an illustrative example is used to show that alternative forms of output feedback are not necessarily subject to these same performance limitations, and should also be considered in the context of transient energy growth suppression and transition control.

Advances in Output Feedback Control of Transient Energy Growth in a Linearized Channel Flow

“…Despite "optimality" of the LQG solution, the approach has inherent performance limitations with regards to TEG suppression. Indeed, as proven in [5], LQG controllers will always exhibit G > 1 whenever the uncontrolled system exhibits G > 1. Further, our previous work suggests that LQG can dramatically degrade TEG performance within the context of channel flows [6].…”

“…In the present study, we will always focus on the worst-case response of the system by considering the maxmimum TEG, G. For controlled systems, the worst-case response will be computed for the controlled system. For TEG analysis, this worst-case analysis is only fair; without performing an objective worst-case analysis, the performance limitations of certain control structures can remain hidden [5]-potentially giving a false impression that one controller outperforms another in terms of TEG.…”

“…To overcome the performance limitations of observer-based feedback structures, we previously proposed reconsidering the standard LQR problem above, but with a structural SOF constraint on the controller, u(t) = F y(t). The benefit of the SOF control structure is that the input is determined directly from the measured output, removing the need for an observer; indeed, SOF controllers constitute semi-proper compensators, and so satisfy the necessary conditions for overcoming the performance limitations of observerbased feedback with respect to TEG [5].…”

“…The standard approach for sensor-based output feedback design combines an optimal observer to estimate the flow state from available sensor measurements, and to then determine the control action based on the state estimate (i.e., observer-based feedback). Although observer-based feedback guarantees modal stability, it was recently established that such strategies have inherent performance limitations with regards to nonmodal mechanisms for TEG [5]. Indeed, observer-based feedback is incapable of completely eliminating TEG whenever the uncontrolled system exhibits TEG.…”

Suppressing subcritical transition in plane Poiseuille flow

“…In the present work, we focus on the initial optimal disturbance computed based on the linear model. The optimal disturbance is calculated for the uncontrolled and controlled systems, respectively, because an implementation of controller to the system can possibly induce new optimal disturbance that leads to a larger transient energy growth than for the uncontrolled system [25]. However, this point is rarely discussed in past direct numerical simulation studies [26][27][28] aiming to evaluate the overall performance of a feedback controller.…”