Treating Noise Channels as Measured Inputs

A linear models is given by:

$$y=Gu+He$$

Where G is an operator that takes the measured inputs u to the outputs and captures the system dynamics. H is an operator that describes the properties of the additive output disturbance and takes the hypothetical (unmeasured) noise source inputs to the outputs. H represents the noise model. When you specify to estimate a noise model, the resulting model include one noise channel e at the input for each output in your system.

To study noise contributions in more detail, it might be useful to convert the noise channels to measured channels using noisecnv:

m_GH = noisecnv(m)

This operation creates a model m_GH that represents both measured inputs u and noise inputs e, treating both sources as measured signals. m_GH is a model from u and e to y, describing the transfer functions G and H.

Converting noise channels to measured inputs loses information about the variance of the innovations e. For example, step response due to the noise channels does not take into consideration the magnitude of the noise contributions. To include this variance information, normalize e such that v becomes white noise with an identity covariance matrix, where

$$e=Lv$$

To normalize e, use the following command:

m_GH = noisecnv(m,'Norm')

This command creates a model where u and v are treated as measured signals, as follows:

$$y(t)=Gu(t)+HLv=\left[\begin{array}{cc}G& HL\end{array}\right]\left[\begin{array}{c}u\\ v\end{array}\right]$$

For example, the scaling by L causes the step responses from v to y to reflect the size of the disturbance influence.

The converted noise sources are named in a way that relates the noise channel to the corresponding output. Unnormalized noise sources e are assigned names such as 'e@y1', 'e@y2', ..., 'e@yn', where 'e@yn' refers to the noise input associated with the output yn. Similarly, normalized noise sources v, are named 'v@y1', 'v@y2', ..., 'v@yn'.

If you want to create a model that has only the noise channels of an identified model as its measured inputs, use the noise2meas command. It results in a model with y(t) = He or y(t) = HLv, where e or v is treated as a measured input.

Note

When you plot models in the app that include noise sources, you can select to view the response of the noise model corresponding to specific outputs. For more information, see Selecting Measured and Noise Channels in Plots.