Maximum likelihood identification of stable linear dynamical systems

This paper concerns maximum likelihood identification of linear time invariant state space models, subject to model stability constraints. We combine Expectation Maximization (EM) and Lagrangian relaxation to build tight bounds on the likelihood that can be optimized over a convex parametrization of all stable linear models using semidefinite programming. In particular, we propose two new algorithms: EM with latent States & Lagrangian relaxation (EMSL), and EM with latent Disturbances & Lagrangian relaxation (EMDL). We show that EMSL provides tighter bounds on the likelihood when the effect of disturbances is more significant than the effect of measurement noise, and EMDL provides tighter bounds when the situation is reversed. We also show that EMDL gives the most broadly applicable formulation of EM for identification of models with singular disturbance covariance. The two new algorithms are validated with extensive numerical simulations.