Estimating the disturbance or clutter covariance is a centrally important problem in radar space time adaptive processing (STAP). The disturbance covariance matrix should be inferred from training sample observations in practice. Large number of homogeneous training samples are generally not available because of difficulty of guaranteeing target free disturbance observation, practical limitations imposed by the spatio-temporal nonstationarity, and system considerations. In this dissertation, we look to address the aforementioned challenges by exploiting physically inspired constraints into ML estimation. While adding constraints is beneficial to achieve satisfactory performance in the practical regime of limited training, it leads to a challenging problem. We focus on breaking this classical trade-off between computational tractability and desirable performance measures, particularly in training starved regimes. In particular, we exploit both the structure of the disturbance covariance and importantly the knowledge of the clutter rank to yield a new rank constrained maximum likelihood (RCML) estimator. In addition, we derive a new covariance estimator for STAP that jointly considers a Toeplitz structure and a rank constraint on the clutter component. Finally, we address the problem of working with inexact physical radar parameters under a practical radar environment. We propose a robust covariance estimation method via an expected likelihood (EL) approach. We analyze covariance estimation algorithms under three different cases of imperfect constraints: 1) only rank constraint, 2) both rank and noise power constraint, and 3) condition number constraint. For each case, we formulate estimation of the constraint as an optimization problem with the EL criterion and formally derive and prove a significant analytical result such as uniqueness of the solution.