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 /spl ap/ 8 with the constraint that the matrix X be symmetric and positive definite...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2002-10, Vol.47 (10), p.1721-1725
Main Authors: Yixin Chen, McInroy, J.E.
Format: Article
Language:English
Subjects:
Applied sciences
Computer science
control theory
systems
Control theory. Systems
Covariance matrix
Equations
Error analysis
Errors
Exact sciences and technology
Human error
Intelligent control
Intelligent robots
Intelligent structures
Interpolation
curve fitting
Least squares method
Mathematical analysis
Mathematical methods in physics
Mathematical models
Modelling and identification
Numerical approximation and analysis
Optimization
Physics
Robot control
Robot dynamics
Sampling methods
Smart structures
Stiffness matrices
Studies
Symmetric matrices
Testing
Weight control
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Summary: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 /spl ap/ 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 /spl par/AX /spl par//sub F//sup 2/. 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.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2002.803545