Tutorial (System Identification Toolbox)

System Identification Toolbox

Assessing the Model Uncertainty

The estimated model is always uncertain, due to disturbances in the observed data, and due to the lack of an absolutely correct model structure. The variability of the model that is due to the random disturbances in the output is estimated by most of the estimation procedures, and it can be displayed and illuminated in a number of ways. This variability answers the question of how different can the model be if the identification procedure is repeated, using the same model structure, but with a different data set that uses the same input sequence. It does not account for systematic errors due to an inadequate choice of model structure. There is no guarantee that the "true system" lies in the confidence interval.

The uncertainty in the different model views is displayed if the argument 'sd' is included in the argument list

```
command(Model,'sd',sd)
```

as explained in Graphs of Model Properties.

The uncertainty in the time response is displayed by

```
simsd(Model,u)
```

Ten possible models are drawn from the asymptotic distribution of the model Model. The response of each of them to the input u is graphed on the same diagram.

The uncertainty of these responses concerns the external, input-output properties of the model. It reflects the effects of inadequate excitation and the presence of disturbances.

You can also directly get the standard deviation of the simulated result by

```
[ysim,ysimsd] = sim(Model,u)
```

The uncertainty in internal representations is manifested in the covariance matrix of the estimated parameters

```
Model.CovarianceMatrix
```

which is used to give the standard deviations of all model characteristics. The parametric uncertainty is directly available as

Model.da for the standard deviations of Model.a

Note that state-space models, estimated in a free parameterization do not have well defined standard deviations of the matrix elements. The model still has stored the uncertainty of the input-output behavior, so other model representations and graphs will show the uncertainty. For a state-space model in a free parameterization, it is possible to first transform it to a canonical parameterization and then display the matrix parameter uncertainties.

Modelc = Model
Modelc.ss = 'canon'
Modelc.da

All routines for computing and displaying model characteristics also offer to calculate and show the uncertainties. See Transformations to Other Model Representations.

Large uncertainties in these representations are caused by excessively high model orders, inadequate excitation, or bad signal-to-noise ratios.

Noise-Free Simulations Comparing Different Models