# Fitting a Nonlinear, Dynamic System

This section describes the fitting process and the main challenges associated with it for a nonlinear, dynamic system.

## Identification Problem for Fitting

The bushing identification is formulated as follows:

$Cost\left(b\right)=\sum _{i=1}^{ncase}\left\{{\left(1-\frac{{k}_{d_\text{model}}^{i}}{{k}_{d_test}^{i}}\right)}^{2}+{\left(1-\frac{{\phi }_{\text{model}}^{i}}{{\phi }_{test}^{i}}\right)}^{2}\right\}$ , ncase = the number of measurements made

such that the constraints:

$H\left({x}_{0}+{x}_{m}{e}^{j\left({\omega }_{0}t+{\phi }_{x}\right)},j{\omega }_{0}{x}_{m}{e}^{j\left({\omega }_{0}t+{\phi }_{x}\right)},{y}_{0}+{k}_{d}{u}_{m}{e}^{j\left({\omega }_{0}t+\phi \right)},{u}_{0}+{u}_{m}{e}^{j{\omega }_{0}t},p\right)=0$

Subject to limits on the parameters p:

${L}_{i}\le {p}_{i}\le {U}_{i},i=1\dots {N}_{p}$