The Extension of the Gauss Approach for the Solution of an Overdetermined Set of Algebraic Non Linear Equations

In this brief is presented an extension of the Gauss approach for the solution of an overdetermined set of algebraic non linear equations. Further, it is shown that the measurement error, in the Gauss approach, is in the measurement direction and that error has a unique decomposition: 1) the first component, which is orthogonal to the Jacobian range space, the residual and 2) the other which is on the Jacobian range space. The latter is hidden in the Jacobian space when one minimizes the residual. The extension of the Gauss approach is then in the sense to minimize the norm of the error. In engineering, the measurements may have gross errors, and then detection, identification, and correction of those errors are necessary. The Largest Normalized Error Test will be developed for that purpose. Considering the cyber-attack possibility, modeled as a malicious data attack, the error correction step is paramount. Applications on power networks will be used to show the hidden error component when using the Gauss minimization, and also to illustrate all the steps of the presented procedure as well as comparison to the current Gauss approach.