Linearisation by finite difference

  • Control vector: \(\underline x = (x_1, x_2, ..., x_N)\)

  • Observation vector: \(\underline{y} = \left[..., {\cal G}_i(\underline{x}), ..., {\cal G}_M(\underline{x})\right]^T\)

  • Linearization: \({\cal G}(\underline{x}^b + {\underline {\delta x}}) \; \approx\; {\cal G}(\underline{x}^b) + \underline{\underline G} \; {\underline {\delta x}})\) with \(\underline{\underline G}^T= \left( ..., \underline G_i^T\, ..., \underline G_M^T\right)\)

  • Approximation of columns of \(\underline{\underline G}\) by finite difference

    \(\displaystyle \underline G_i= \left[{\partial {\cal G}_i\over \partial x_1} (\underline{x}^b), \;{\partial {\cal G}_i\over \partial x_2} (\underline{x}^b), ...,{\partial {\cal G}_i\over \partial x_N} (\underline{x}^b)\right]^T\)