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 overdetermined set of linear equations AX ≈ 8 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 ||_F². However, this assumption is often impractical. Sampling errors, modeling errors, and, sometimes, human errors bring inaccuracies to A as well. In this note, 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.