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System Identification Toolbox™ software provides several mapping functions
*F*(*x*) for nonlinear ARX models. When used in a
model, these mapping functions collectively make up the *output
function* of the nonlinear ARX model architecture. For more information
about *F*(*x*), see Structure of Nonlinear ARX Models.

Each mapping function corresponds to an object class in this toolbox. When you estimate nonlinear ARX models in the app, System Identification Toolbox creates and configures objects based on these classes. You can also create and configure mapping functions at the command line.

Most mapping functions represent the nonlinear function as a summed series of nonlinear units,
such as wavelet networks or sigmoid functions, and also contain a linear component. You
can configure the number of nonlinear units *n* for estimation. For a
detailed description of each mapping function, see the corresponding reference
page.

Nonlinearity | Mapping Object | Structure | Comments |
---|---|---|---|

Wavelet network (default) | `idWaveletNetwork` |
$$g(x)={\displaystyle \sum _{k=1}^{n}{\alpha}_{k}}\kappa \left({\beta}_{k}\left(x-{\gamma}_{k}\right)\right)$$ where $$\kappa (s)$$ is the wavelet function. | By default, the estimation algorithm determines the number
of units n automatically. |

One layer sigmoid network | `idSigmoidNetwork` |
$$g(x)={\displaystyle \sum _{k=1}^{n}{\alpha}_{k}}\kappa \left({\beta}_{k}\left(x-{\gamma}_{k}\right)\right)$$ where $$\kappa (s)={\left({e}^{s}+1\right)}^{-1}$$ is the sigmoid function. $${\beta}_{k}$$ is a row vector such that $${\beta}_{k}(x-{\gamma}_{k})$$ is a scalar. | Default number of units n is 10. |

Tree partition | `idTreePartition` | Piecewise linear function over partitions of the regressor space defined by a binary tree. | The estimation algorithm determines the number of units automatically. Try using tree partitions for modeling data collected over a range of operating conditions. |

F is linear in x | `idLinear` | This estimator produces a model that is similar to the linear ARX model, but offers the additional flexibility of specifying custom regressors. | Use to specify custom regressors as the mapping function rather than a nonlinear mapping object . |

Multilayered neural network | `idFeedforwardNetwork` | Uses as a `network` object created using
the Deep Learning Toolbox™ software. | |

Custom network (user-defined) | `idCustomNetwork` | Similar to sigmoid network but you specify $$\kappa (s)$$. | (For advanced use) Uses the unit function that you specify. |