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Sparsity constraints on the control inputs of a linear dynamical systemnaturally arise in several practical applications such as networked control,computer vision, seismic signal processing, and cyber-physical systems. In thiswork, we consider the problem of jointly estimating the states and sparseinputs of such systems from low-dimensional (compressive) measurements. Due tothe low-dimensional measurements, conventional Kalman filtering and smoothingalgorithms fail to accurately estimate the states and inputs. We present aBayesian approach that exploits the input sparsity to significantly improveestimation accuracy. Sparsity in the input estimates is promoted by usingdifferent prior distributions on the input. We investigate two main approaches:regularizer-based MAP, and {Bayesian learning-based estimation}. We also extendthe approaches to handle control inputs with common support and analyze thetime and memory complexities of the presented algorithms. Finally, usingnumerical simulations, we show that our algorithms outperform thestate-of-the-art methods in terms of accuracy and time/memory complexities,especially in the low-dimensional measurement regime.

Joint State and Input Estimation for Linear Dynamical Systems with Sparse Control