  Estimating consistent parameters of a structured state-space representation
requires a reliable initialization when the vector of parameters is computed by
using a gradient-based algorithm. In the eponymous companion paper accepted for
publication in IEEE Transactions on Automatic Control, the problem of supplying
a reliable initial vector of parameters is tackled. More precisely, by assuming
that a reliable initial fully-parameterized state-space model of the system is
available, the aforementioned paper addresses the challenging problem of
transforming this initial fully-parameterized model into the structured
state-space parameterization satisfied by the system to be identified. Two
solutions to solve such a parameterization problem are more precisely
introduced in the IEEE TAC paper. First, a solution based on a null-space-based
reformulation of a set of equations arising from the aforementioned similarity
transformation problem is considered. Second, an algorithm dedicated to
non-convex optimization is presented in order to transform the initial
fully-parameterized model into the structured state-space parameterization of
the system to be identified. In this technical report, a specific attention is
paid to the gradient computation required by the optimization algorithms used
in aforementioned to solve the aforementioned problem. These gradient
formulations are indeed necessary to apply the quasi-Newton
Broyden-Fletcher-Goldfarb-Shanno (BFGS) methods used for the null-space based
as well as the least-squares-formulated optimization techniques introduced in
the IEEE TAC paper. For the sake of conciseness, we only focus on the smooth
version of the optimization problem introduced in aforementioned paper.
Interested readers can easily extend the following results by using the chain
rule as well as the sub-gradient computation techniques.
