We are interested in performing an observer-based state feedback stabilization of the Ginzburg-Landau equation. This equation is often considered as a good surrogate model for fluid flows. One particular constraint, in fluid systems, is the high number of degrees of freedom involved in the system dynamics. In the design process, modeling the intrinsic hydrodynamic instability by global modes allows to define low-order controllers, that, at moderate Reynolds numbers, can stabilize the flow with a minimal actuation cost (Raymond & Thevenet 2010). To be able to act in real-time, the computation time to update the control needs to be smaller than the characteristic time of the flow. The low dimension of the controller is then mandatory in practical applications. Moreover, the full state of the system is rarely known, but only some measurements, submitted to a certain level of noise, are available. Estimation of the reduced-order state may become a delicate task since consistency between the measurement and the low-order model is not guaranteed. Indeed, the measure does not distinguish the contribution of the part involved in the reduced-order model from the neglected part. Projection-based reduced-order methods, with various choices of basis, are then classically employed for minimizing the observation error. It is first clear that the finite number of unstable modes, if they exist, need to be modeled, since they drive the oscillatory behavior of the flow. For an accurate estimation, the infinite dimensional stable part needs to be incorporated in some way (Sipp & Schmid 2016). Adding a finite (even large) set of eigenmodes have demonstrated poor performances (Barbagallo et al. 2009, Ehrenstein et al. 2011, Barbagallo et al. 2011) and even sometimes a loss of stability of the compensated system. This failure has been understood to be caused, among other things, by the non-normality of the considered basis and its lack of input-output representativeness. Modeling the stable part can also be expressed with other bases such as POD modes, balanced truncation modes or balanced POD. These bases have the advantage of controlling the projection error and then to ensure a reduction of the observation inconsistency. Unfortunately, due to the truncation in the Galerkin projection, the dynamics of the estimator is corrupted. That certainly may affect slightly the computation of the controller and Kalman gain, but worst, it corrupts the estimator dynamics, that renders uncertain the compensated system performances.

We propose to model the stable part of the system, in the Kalman filter, by an integral term coming from the underlying linear dynamics of the full stable part excited by the actuation. An associated low-dimensional kernel can be pre-computed at a reasonable cost even for large scale systems. The structure of the proposed compensator guarantees stability of the compensated system.

In a second step, additional stable global modes can be incorporated in the model, that allows us to take into account the response of the stable part to external perturbations and thus to enhance the estimation. We give some guidelines for mode's selection in that context, in a perspective of robustness improvement of the control law.

The Ginzburg-Landau equation is used in this paper to demonstrate the necessity of having a special care of the estimation.
This equation is widely used for modeling typical features of non-parallel fluid flows (Roussopoulos & Monkewitz 1996), and it is commonly employed as a benchmark control problem (Bagheri et al. 2009, Chen & Rowley 2011).