  We consider cost-constrained sparse sensor selection for full-state
reconstruction, applying a well-known greedy algorithm to dynamical systems for
which the usual singular value decomposition (SVD) basis may not be available
or preferred. We apply the cost-modified, column-pivoted QR decomposition to a
physically relevant basis -- the pivots correspond to sensor locations, and
these locations are penalized with a heterogeneous cost function. In
considering different bases, we are able to account for the dynamics of the
particular system, yielding sensor arrays that are nearly Pareto optimal in
sensor cost and performance in the chosen basis. This flexibility extends our
framework to include actuation and dynamic estimation, and to select sensors
without training data. We provide three examples from the physical and
engineering sciences and evaluate sensor selection in three dynamically
relevant bases: truncated balanced modes for control systems, dynamic mode
decomposition (DMD) modes, and a basis of analytic modes. We find that these
bases all yield effective sensor arrays and reconstructions for their
respective systems. When possible, we compare to results using an SVD basis and
evaluate tradeoffs between methods.
