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Estimating consistent parameters of a structured state-space representationrequires a reliable initialization when the vector of parameters is computed byusing a gradient-based algorithm. In the eponymous companion paper accepted forpublication in IEEE Transactions on Automatic Control, the problem of supplyinga reliable initial vector of parameters is tackled. More precisely, by assumingthat a reliable initial fully-parameterized state-space model of the system isavailable, the aforementioned paper addresses the challenging problem oftransforming this initial fully-parameterized model into the structuredstate-space parameterization satisfied by the system to be identified. Twosolutions to solve such a parameterization problem are more preciselyintroduced in the IEEE TAC paper. First, a solution based on a null-space-basedreformulation of a set of equations arising from the aforementioned similaritytransformation problem is considered. Second, an algorithm dedicated tonon-convex optimization is presented in order to transform the initialfully-parameterized model into the structured state-space parameterization ofthe system to be identified. In this technical report, a specific attention ispaid to the gradient computation required by the optimization algorithms usedin aforementioned to solve the aforementioned problem. These gradientformulations are indeed necessary to apply the quasi-NewtonBroyden-Fletcher-Goldfarb-Shanno (BFGS) methods used for the null-space basedas well as the least-squares-formulated optimization techniques introduced inthe IEEE TAC paper. For the sake of conciseness, we only focus on the smoothversion of the optimization problem introduced in aforementioned paper.Interested readers can easily extend the following results by using the chainrule as well as the sub-gradient computation techniques.

Identification of parameterized gray-box state-space systems: from a  black-box linear time-invariant representation to a structured one: detailed  derivation of the gradients involved in the cost functions