Lionel Mathelin

A dynamic mode decomposition approach for large and arbitrarily sampled systems

Detection of coherent structures is of crucial importance for understanding the dynamics of a fluid flow. In this regard, the recently introduced Dynamic Mode Decomposition (DMD) has raised an increasing interest in the community. It allows to efficiently determine the dominant spatial modes, and their associated growth rate and frequency in time, responsible for describing the time-evolution of an observation of the physical system at hand. However, the underlying algorithm requires uniformly sampled and time-resolved data, which may limit its usability in practical situations. Further, the computational cost associated with the DMD analysis of a large dataset is high, both in terms of central processing unit and memory. In this contribution, we present an alternative algorithm to achieve this decomposition, overcoming the above-mentioned limitations. A synthetic case, a two-dimensional restriction of an experimental flow over an open cavity, and a large-scale three-dimensional simulation, provide exampl...

A machine learning approach for constrained sensor placement

Sensor placement is of pivotal importance in closed-loop control as measurements are key to design the control laws. In this article, a novel machine learning-based sensor placement algorithm is proposed in order to recover a high-dimensional field from a limited amount of local measurements with a linear estimator. Unlike many other methods, our algorithm does not rely on a reduced order model and achieves good results even with a small number of sensors. In many situations, sensors cannot be placed arbitrarily, either because of their geometry or because of the environment they are in. Our algorithm naturally accounts for these constraints as well as being robust to noise. Its performance is illustrated on a fluid flow example and compared to two state of the art methods, Effective Independence and FrameSense, on the recovery of the pressure field from limited noisy pressure measurements.

A statistical learning strategy for closed-loop control of fluid flows

This work discusses a closed-loop control strategy for complex systems utilizing scarce and streaming data. A discrete embedding space is first built using hash functions applied to the sensor measurements from which a Markov process model is derived, approximating the complex system’s dynamics. A control strategy is then learned using reinforcement learning once rewards relevant with respect to the control objective are identified. This method is designed for experimental configurations, requiring no computations nor prior knowledge of the system, and enjoys intrinsic robustness. It is illustrated on two systems: the control of the transitions of a Lorenz’63 dynamical system, and the control of the drag of a cylinder flow. The method is shown to perform well.

Closed-loop fluid flow control using a low dimensional model

This paper is devoted to the problem of defining a control strategy to minimize the drag of a bluff body in a 2-D cross-flow. A reduced model is obtained using a robust statistical reduction approach and an optimal orbit in the phase space is determined using an open-loop control strategy. This open-loop control law is inexpensive to derive as it relies on a reduced order model. Since the deviations from the optimal orbit are meant to remain small, the non-linear flow model can be linearized around the orbit at each time. To compensate for the deviations, a closed-loop control is applied. The design of a robust controller is difficult due to the large number of state space variables, conflicting specifications and parameter uncertainties. Further, the model is a time-varying process, so that Linear Time Invariant design methods cannot be directly applied.

Identification-Based Closed-Loop Control Strategies for a Cylinder Wake Flow

Four closed-loop control strategies are discussed to reduce the drag of a cylinder wake flow: Linear Quadratic Gaussian (LQG) control, gain-scheduled LQG (GS-LQG) control, gain-scheduled PI control, and multimodel predictive control (M-MPC). The control models are obtained in an input–output framework through ARMAX (for LQG control), multi-ARMAX (for GS-LQG control and M-MPC), and multi-ARX (for gain-scheduled PI control). The use of system identification for the underlying flow control problem gets rid of the difficult task of developing accurate and robust reduced-order models for the Navier–Stokes (NS) equations. The control is introduced through sucking of fluid through the cylinder surface. The drag on the cylinder is reduced for all control methods. The robustness of all control strategies is tested against unmodeled disturbances and/or dynamics through a detailed simulation of an NS equation-based model by varying in time the Reynolds number around its nominal value 200. For the considered cylinder wake, the M-MPC approach is the best solution. The application of the presented closed-loop control algorithms for the cylinder drag control as a benchmark problem constitutes promising solutions for other related flow control problems in industries.

A continuous reinforcement learning strategy for closed-loop control in fluid dynamics

This article presents the application of a continuous Reinforcement Learning (RL) framework to the closed-loop control of the drag of a cylinder wake flow. RL is a pure-driven approach that can rely on scarce and streaming data to find an optimal control policy for a task in an unknown environment. An implicit model of the system at hand is learned from sensor measurements, allowing for gradients evaluation and leading to quick convergence to an efficient control policy. This data-driven method is well suited for experimental configurations, requiring no computations nor prior knowledge of the system, and enjoys intrinsic robustness. These properties are highly desirable in flow control where real-time constraints, limited embedded hardware computational power and severe noise prevent the use of model-based strategies. Specific issues of reinforcement learning in the context of closed-loop flow control are briefly discussed.

An adaptive compressed-sensing equation-free approach for closed-loop nonlinear control

We present a method which seeks to combine the efficiency of an optimal control command with the robustness of a closed-loop controller. While the performance of optimal control is excellent, it cannot account for perturbations to the system under control which make the resulting performance to drop. On the other hand, a closed-loop control makes use of the actual state of the system and is thus robust, at the price of limited performance. We here present a control method which is both robust (closed-loop) and achieves a nearoptimal performance. The controller relies on an approximation of the optimal control command in the state space where the system lies upon reduction. The command can then be updated when a new observation of the system becomes available, hence achieving a closed-loop control. The approximation of the command in the state space is a computationally costly step but is achieved off-line and only once. To minimize its cost while achieving a good accuracy in the approximation, an adaptive multi-wavelets approach, combined with compressed-sensing acceleration exploiting compressibility, is here proposed. The control algorithm is demonstrated on the control of a cylinder wake in a 2-D flow.