Optimal sensor and actuator placement is an important unsolved problem in control theory. Nearly every downstream control decision is affected by these sensor and actuator locations, but determining optimal locations amounts to an intractable brute-force search among the combinatorial possibilities. Indeed, there are (np) = n!/((n-p)!p!) possible choices of p point sensors out of an n-dimensional state x. Determining optimal sensor and actuator placement in general, even for linear feedback control, is an open challenge. Instead, sensor and actuator locations are routinely chosen according to heuristics and intuition. For moderate-sized search spaces, the sensor placement problem has well-known model-based solutions using optimal experiment design [1], [2], and information theoretic and Bayesian criteria [3]-[7]. As discussed in "Summary," this article explores how to design optimal sensor locations for signal reconstruction in a framework that scales to arbitrarily large problems, leveraging modern techniques in machine learning and sparse sampling. Reducing the number of sensors through principled selection may be critically enabling when sensors are costly, and it may also enable faster state estimation for low-latency, high-bandwidth control.

Sparse principal component analysis (SPCA) has emerged as a powerful technique for modern data analysis, providing improved interpretation of low-rank structures by identifying localized spatial st...

Optimal sensor and actuator selection is a central challenge in high-dimensional estimation and control. Nearly all subsequent control decisions are affected by these sensor and actuator locations. In this article, we exploit balanced model reduction and greedy optimization to efficiently determine sensor and actuator selections that optimize observability and controllability. In particular, we determine locations that optimize scalar measures of observability and controllability using greedy matrix QR pivoting on the dominant modes of the direct and adjoint balancing transformations. Pivoting runtime scales linearly with the state dimension, making this method tractable for high-dimensional systems. The results are demonstrated on the linearized Ginzburg–Landau system, for which our algorithm approximates known optimal placements computed using costly gradient descent methods.

The characterization of intermittent, multiscale, and transient dynamics using data-driven analysis remains an open challenge. We demonstrate an application of the dynamic mode decomposition (DMD) with sparse sampling for the diagnostic analysis of multiscale physics. The DMD method is an ideal spatiotemporal matrix decomposition that correlates spatial features of computational or experimental data to periodic temporal behavior. DMD can be modified into a multiresolution analysis to separate complex dynamics into a hierarchy of multiresolution timescale components, where each level of the hierarchy divides dynamics into distinct background (slow) and foreground (fast) timescales. The multiresolution DMD is capable of characterizing nonlinear dynamical systems in an equation-free manner by recursively decomposing the state of the system into low-rank spatial modes and their temporal Fourier dynamics. Moreover, these multiresolution DMD modes can be used to determine sparse sampling locations which are nearly optimal for dynamic regime classification and full state reconstruction. Specifically, optimized sensors are efficiently chosen using QR column pivots of the DMD library, thus avoiding an NP-hard selection process. We demonstrate the efficacy of the method on several examples, including global sea-surface temperature data, and show that only a small number of sensors are needed for accurate global reconstructions and classification of El Nin͂o events.

The CANDECOMP/PARAFAC (CP) tensor decomposition is a popular dimensionality-reduction method for multiway data. Dimensionality reduction is often sought after since many high-dimensional tensors have low intrinsic rank relative to the dimension of the ambient measurement space. However, the emergence of ‘big data’ poses significant computational challenges for computing this fundamental tensor decomposition. By leveraging modern randomized algorithms, we demonstrate that coherent structures can be learned from a smaller representation of the tensor in a fraction of the time. Thus, this simple but powerful algorithm enables one to compute the approximate CP decomposition even for massive tensors. The approximation error can thereby be controlled via oversampling and the computation of power iterations. In addition to theoretical results, several empirical results demonstrate the performance of the proposed algorithm.