On Regularization of Least Square Problems via Quadratic Constraints

We consider uncertainty reduction in least square problems raised in system identification with unknown state space. We assume existence of some prior information obtained through a finite series of measurements. This data is modeled in the form of a finite collection of quadratic constraints enclosing the state space. A simple closed form expression is derived for the optimal solution featuring geometric insights and intuitions that reveal a two-fold effort in reducing uncertainty: by correcting the observation error and by improving the condition number of the data matrix. To deal with the dual problem of finding the optimal Lagrange multipliers, we introduce an approximate, positive semidefinite program that can be easily solved using the standard numerical techniques.