We consider the problem of jointly estimating the states and sparse inputs of a linear dynamical system using noisy low-dimensional observations. We exploit the underlying sparsity in the inputs using fictitious sparsity-promoting Gaussian priors with unknown variances (as hyperparameters). We develop two Bayesian learning-based techniques to estimate states and inputs: sparse Bayesian learning and variational Bayesian inference. Through numerical simulations, we illustrate that our algorithms outperform the conventional Kalman filtering based algorithm and other state-of-the-art sparsity-driven algorithms, especially in the low-dimensional measurement regime.

The stabilizability of a linear dynamical system (LDS) refers to the existence of control inputs that drive the system state to zero. In this article, we analyze both the theoretical and algorithmic aspects of the stabilizability of an LDS using sparse control inputs with potentially time-varying supports. We show that an LDS is stabilizable using sparse control inputs if and only if it is stabilizable (using unconstrained inputs). For a stabilizable LDS, we present an algorithm to determine the sparse control inputs that steer the system state to zero. We show that all stabilizable LDSs are also sparse mean square stabilizable when the process noise has zero mean and bounded second moment. For such an LDS, we devise a method to sequentially estimate the sparse control inputs to stabilize the LDS in the mean square sense. We prove that a detectable and stabilizable LDS is sparse stabilizable through output feedback and develop an algorithm for finding the corresponding sparse control inputs. Finally, we analyze the stabilizability of an LDS using sparse control inputs with common support. Our results shed light on the conditions under which a given LDS is stabilizable using sparse control inputs and the design of the corresponding control inputs.

The emergence of compressive sensing and the associated ℓ₁ recovery algorithms and theory have generated considerable excitement and interest in their applications. This chapter will examine recent developments and a complementary set of tools based on a Bayesian framework to address the general problem of sparse signal recovery and the challenges associated with it. Bayesian methods offer superior performance compared to convex optimization-based methods and are parameter tuning-free. They also have the flexibility necessary to deal with a diverse range of measurement modalities and structured sparsity in signals than hitherto possible. Parsimonious signal representation using overcomplete dictionaries for compression, estimation of sparse communication channels with large delay spread as in underwater acoustics, low-dimensional representation of MIMO wireless channels, brain imaging techniques, such as MEG and EEG, are a few examples. We provide a mathematically rigorous and in-depth overview of this fascinating area within sparse signal recovery. We highlight the generality and flexibility of Bayesian approaches and show how it greatly facilitates their deployment in communications-related applications, even though they generally lead to nonconvex optimization problems. Further, we show that, by reinterpreting the Bayesian cost function as a technique to perform covariance matching, one can develop new, ultrafast Bayesian algorithms for sparse signal recovery. As an example application, we discuss the utility of these algorithms in the context of 5G communications with several case studies including wideband time-varying channel estimation and low-resolution analog-to-digital conversion-based signal recovery.