Realizations of Linear Systems with Prescribed Sparsity Patterns
This paper solves the following problem: given a transfer function and desired sparsity patterns for its state-space realization, how can we find a realization of the function that adheres to them? While sparse controllers are studied in distributed control, their sparsity patterns are generally designed to match those of the plant. In contrast, we are interested in \textit{arbitrary} sparsity patterns, which are relevant to modeling problems in sensorimotor neuroscience. Though this problem is highly nonconvex, we solve it exactly. We first show that the problem reduces to finding an appropriate similarity transform from the modal realization, which in turn reduces to solving a system of multivariate polynomial equations. We then leverage tools from algebraic geometry (Gröbner basis, moment method) to solve the system. Algorithms are provided for both real- and complex-valued realization problems, and their efficacy is demonstrated on several examples.