Input and state estimation exploiting input sparsity

Motivated by cyber-physical security applications, we face the problem of estimating the state and the input of a linear system, where the input may represent the presence of adversarial attacks. We consider the case where classical filters cannot be used, because the number of measurements is too low, for example it is lower than the size of the input vector. If the input, although of large size, is known to be sparse, the problem can be tackled using techniques from compressed sensing theory. In this paper, we propose a recursive estimator, based on compressed sensing and Kalman-like filtering, which is able to reconstruct both the state and the input from noisy, compressed measurements. The proposed algorithm is proved to be feasible and numerically efficient, and simulations show a good recovery accuracy with respect to an oracle estimator.