# SLS algorithm for full network identification

Identification of a full network is performed on the basis of measurements of all node signals and possibly a number of external excitation signals. A sequential least-squares (SLS) algorithm is implemented and is applied to an ARMAX model structure for the network model:

$A(q) w(t) = B(q) r(t) + C(q) e(t)$

with polynomial matrices A, B, C.

While a standard ARMAX model estimate would require non-convex optimization, the SLS algorithm performs consecutive convex steps to solve the problem. The several steps involve:

- Estimate a (high-order) ARX model, and use that model to construct an estimate $\hat{e}$ of the innovation process $e$.
- Construct a parametric model by solving a linear regression problem with $r$ and $\hat{e}$ as inputs and $w$ as output.
- Iterate on the noise model that acts as a weighting in the linear regression problem of step 2.

Full network identification with SLS algorithm can be performed at the command line using the `nwidfullSLS`

function. Check the reference pages for `nwdata`

and `nwmodel`

to learn how to create the data and model objects required to run the identification.

Implementation limitations:

- All polynomials have a scalar model order variable, meaning that it is not possible to specify different orders for the different matrix entries in the polynomials. This also applies to the high order ARX model in step 1.
- All modules in the network are assumed to be strictly proper; i.e. they have at least one delay.
- The situation of rank reduced noise (dim($e$) < dim($w$)), is implemented as an (unconstrained) weighted LS method.
- The implemented algorithm requires the CVX toolbox to be installed (http://cvxr.com).
- An automated check is included that tests whether the model structure is globally network identifiable.

### References

- H.H.M. Weerts, M. Galrinho, G. Bottegal, H. Hjalmarsson and P.M.J. Van den Hof (2018). A sequential least squares algorithm for ARMAX dynamic network identification. IFAC PapersOnLine, Vol. 51-15, pp. 844-849.