A paper on convex optimization in system identification (aka learning dynamical systems), written by Mark Tobenkin, myself, and Alex Megretski, has been published in IEEE Transactions on Automatic Control. This paper reports some of the key findings of the work we did when all three of us were MIT, but has taken a while to get into a final form for publishing.
It is available open-access here:
This paper provides methods to address two major challenges in nonlinear system identification: guaranteeing model stability, and identifying long-term dependence between inputs and outputs. In particular, we provide convex parameterizations of flexible sets of nonlinear models with guaranteed stability, and also convex upper bounds on simulation error. Taken together, these allow tools such as sum-of-squares programming to be used to identify highly accurate nonlinear models from data.