Determining sensor locations tailored to important dynamical behavior
Finding an optimal location for a sensor is a problem of tremendous importance, particularly if these are expensive and only a limited number is available, e.g. when releasing surface drifters to monitor ocean currents, temperature sensors in buildings, and cameras for surveillance. In the paper “Sparsity enabled cluster reduced-order models for control” (open access) we discuss sensor placement in the context of fluid flows, i.e. finding few locations that measure the velocity in the flow. However, these sensors are not placed at random locations but their position is optimized to capture the dominant flow behavior by combining ideas from compressed sensing and machine learning. You can find the code for the paper on github.
The goal of this work is to optimize sensor locations tailored to a particular low-dimensional model that has been identified from data and captures the main physical mechanism. The modeling of dynamical processes can be a tremendously difficult endeavor (due to high-dimensionality, nonlinearities, multi-scale behavior, uncertainty, etc.), but is critical to obtain a better understanding and to make predictions into the future. Tailoring sensor locations to a dynamical model serves several needs:
- A model is typically (in fluids but also in other fields) learned from previously collected, high-dimensional data. However, the high-dimensionality strains computational ressources for exploratory parameter studies.
- Moreover, often the model shall be used in real-time applications for prediction and control. Here, in addition to the computational ressources, the high-dimensionality also strains data acquisition capabilities.
Thus, finding few sensor locations enables in some sense a compressed model for faster execution and analysis. There is a growing effort to find transformations that embed nonlinear dynamics in global linear representation, which allows one to apply powerful linear methods for the estimation, prediction and control to strongly nonlinear systems. This has motivated significant work on the Koopman and Perron–Frobenius operators. The cluster-based reduced-order model (CROM) framework (somewhat dull youtube video, but check out links in description) provides an efficient low-dimensional representation of the Perron–Frobenius operator using a data-driven discretization of phase space into clusters, on which probabilistic dynamics evolve. Although CROM is fundamentally low-dimensional, it often relies on access to full-state data for training and evaluation.
In this work, we demonstrate the sparse sensor selection for efficient operator-theoretic modeling of nonlinear systems, the so-called sparsity-enabled CROM.
We show that
- a sufficient, but small number of random measurements embed the cluster geometry and preserve the probabilistic dynamics, relying on compressed sensing and the restricted isometry property.
- Further, we demonstrate the ability to learn a reduced set of optimized sensors that are tailored to the specific CROM model and provide performance on par with the full high-dimensional CROM. These sparsity-enabled innovations are demonstrated on several high-dimensional nonlinear fluid systems of increasing complexity, and in all cases optimized sensors outperform randomly chosen sensors.
- We also show that the sparsity-enabled CROM can be used for feedback control, resulting in control performance that is similar to that of the high-dimensional CROM.
There is a tremendous potential using sparsity-promoting methods for efficient sensing, actuator placement, model identification, etc. In particular, the combination of sparsity-promoting techniques with linear embeddings of nonlinear systems will become a key enabler for real-time estimation and control.
