Identification using Convexification and Recursion

System identification studies how to construct mathematical models for dynamical systems from the input and output data, which finds applications in many scenarios, such as predicting future output of the system or building model based controllers for regulating the output the system. Among many other methods, convex optimization is becoming an increasingly useful tool for solving system identification problems. The reason is that many identification problems can be formulated as, or transformed into convex optimization problems. This transformation is commonly referred to as the convexification technique. The first theme of the thesis is to understand the efficacy of the convexification idea by examining two specific examples. We first establish that a l1 norm based approach can indeed help in exploiting the sparsity information of the underlying parameter vector under certain persistent excitation assumptions. After that, we analyze how the nuclear norm minimization heuristic performs on a low-rank Hankel matrix completion problem. The underlying key is to construct the dual certificate based on the structure information that is available in the problem setting. Recursive algorithms are ubiquitous in system identification. The second theme of the thesis is the study of some existing recursive algorithms, by establishing new connections, giving new insights or interpretations to them. We first establish a connection between a basic property of the convolution operator and the score function estimation. Based on this relationship, we show how certain recursive Bayesian algorithms can be exploited to estimate the score function for systems with intractable transition densities. We also provide a new derivation and interpretation of the recursive direct weight optimization method, by exploiting certain structural information that is present in the algorithm. Finally, we study how an improved randomization strategy can be found for the randomized Kaczmarz algorithm, and how the convergence rate of the classical Kaczmarz algorithm can be studied by the stability analysis of a related time varying linear dynamical system.