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

TL;DR

This paper derives the gradients needed for optimization algorithms to transform a fully-parameterized state-space model into a structured form, facilitating parameter estimation in gray-box systems.

Contribution

It provides detailed gradient formulations for non-convex optimization methods used in structured state-space system identification, extending previous work on initial parameter transformation.

Findings

Gradient formulas for BFGS optimization methods are derived.

The approach enables reliable parameter initialization for structured system identification.

Focus on smooth optimization problems facilitates practical implementation.

Abstract

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.