Estimation of symmetric positive-definite matrices from imperfect measurements

In a number of contexts relevant to control problems, including estimation of robot dynamics, covariance, and smart structure mass and stiffness matrices, we need to solve an over-determined set of linear equations AX ≈ B with the constraint that the matrix X be symmetric and positive definite. In the classical least squares method the measurements of A are assumed to be free of error, hence, all errors are confined to B. Thus, the “optimal” solution is given by minimizing the optimization criterion ‖AX−B‖F . However, this assumption is often impractical. Sampling errors, modeling errors, and, sometimes, human errors bring inaccuracies to A as well. In this paper, we introduce a different optimization criterion, based on area, which takes the errors in both A and B into consideration. Under the condition that the data matrices A and B are full rank, which in practice is easy to satisfy, the analytic expression of the global optimizer is derived. A method to handle the case that A is full rank and B loses rank is also discussed. Experimental results indicate that the new approach is practical, and improves performance.