Finite-parameter feedback control for stabilizing the complex Ginzburg-Landau equation

In this paper, we prove the exponential stabilization of solutions for complex Ginzburg-Landau equations using finite-parameter feedback control algorithms, which employ finitely many volume elements, Fourier modes or nodal observables (controllers). We also propose a feedback control for steering solutions of the Ginzburg-Landau equation to a desired solution of the non-controlled system. In this latter problem, the feedback controller also involves the measurement of the solution to the non-controlled system.Comment: 20 page

Performance of a linear robust control strategy on a nonlinear model of spatially developing flows

International audienceThis paper investigates the control of self-excited oscillations in spatially developing flow systems such as jets and wakes using H 8 control theory on a complex Ginzburg-Landau (CGL) model. The coefficients used in this one-dimensional equation, which serves as a simple model of the evolution of hydrodynamic instability waves, are those selected by Roussopoulos & Monkewitz (Physica D 1996, vol. 97, p. 264) to model the behaviour of the near-wake of a circular cylinder. Based on noisy measurements at a point sensor typically located inside the cylinder wake, the compensator uses a linear H 8 filter based on the CGL model to construct a state estimate. This estimate is then used to compute linear H 8 control feedback at a point actuator location, which is typically located upstream of the sensor. The goal of the control scheme is to stabilize the system by minimizing a weighted average of the 'system response' and the 'control effort' while rigorously bounding the response of the controlled linear system to external disturbances. The application of such modern control and estimation rules stabilizes the linear CGL system at Reynolds numbers far above the critical Reynolds number Rec ˜ 47 at which linear global instability appears in the uncontrolled system. In so doing, many unstable modes of the uncontrolled CGL system are linearly stabilized by the single actuator/sensor pair and the model-based feedback control strategy. Further, the linear performance of the closed-loop system, in terms of the relevant transfer function norms quantifying the linear response of the controlled system to external disturbances, is substantially improved beyond that possible with the simple proportional measurement feedback proposed in previous studies. Above Re ˜ 84, the control designs significantly outperform the corresponding control designs in terms of their ability to stabilize the CGL system in the presence of worst-case disturbances. The extension of these control and estimation rules to the nonlinear CGL system on its attractor (a simple limit cycle) stabilizes the full nonlinear system back to the stationary state at Reynolds numbers up to Re ˜ 97 using a single actuator/sensor pair, fixed-gain linear feedback and an extended Kalman filter incorporating the system nolinearity. © 2004 Cambridge University Press

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navier–Stokes equations obtained with the novel velocity–vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulse-like excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the so-called parallel-flow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the short-term behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotating-disc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigid-walled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive

Machine Learning Accelerated PDE Backstepping Observers

State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.Comment: Accepted to the 61st IEEE Conference on Decision and Control (CDC), 202

Three-dimensional instabilities of a stratified cylinder wake

International audienceThis paper describes experimentally, numerically and theoretically how the three-dimensional instabilities of a cylinder wake are modified by the presence of a linear density stratification. The first part is focused on the case of a cylinder with a small tilt angle between the cylinder's axis and the vertical. The classical mode A well-known for a homogeneous fluid is still present. It is more unstable for moderate stratifications but it is stabilized by a strong stratification. The second part treats the case of a moderate tilt angle. For moderate stratifications, a new unstable mode appears, mode S, characterized by undulated layers of strong density gradients and axial flow. These structures correspond to Kelvin–Helmholtz billows created by the strong shear present in the critical layer of each tilted von Kármán vortex. The last two parts deal with the case of a strongly tilted cylinder. For a weak stratification, an instability (mode RT) appears far from the cylinder, due to the overturning of the isopycnals by the von Kármán vortices. For a strong stratification, a short wavelength unstable mode (mode L) appears, even in the absence of von Kármán vortices. It is probably due to the strong shear created by the lee waves upstream of a secondary recirculation bubble. A map of the four different unstable modes is established in terms of the three parameters of the study: the Reynolds number, the Froude number (characterizing the stratification) and the tilt angle

The control transmutation method and the cost of fast controls

8 pages, a4paper, no figures, changed content. The final version will appear in the SIAM Journal of Control and Optimization.International audienceIn this paper, the null controllability in any positive time T of the first-order equation (1) x'(t)=e^{i\theta}Ax(t)+Bu(t) (|\theta|<\pi/2 fixed) is deduced from the null controllability in some positive time L of the second-order equation (2) z''(t)=Az(t)+Bv(t). The differential equations (1) and (2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method explicits the input function u of (1) in terms of the input function v of (2): u(t,x)=\int k(t,s)v(s)ds, where the compactly supported kernel k depends on T and L only. It proves that the norm of a u steering the system (1) from an initial state x_{0} to zero grows at most like ||x_{0}||\exp(\alpha_{*}L^{2}/T) as the control time T tends to zero. (The rate \alpha_{*} is characterized independently by a one-dimensional controllability problem.) In the applications to the cost of fast controls for the heat equation, L is the length of the longest ray of geometric optics which does not intersect the control region