Identification of Cascade Dynamic Nonlinear Systems: A Bargaining-Game-Theory-Based Approach

Cascade dynamic nonlinear systems can describe a wide class of engineering problems, but little efforts have been devoted to the identification of such systems so far. One of the difficulties comes from its non-convex characteristic. In this paper, the identification of a general cascade dynamic nonlinear system is rearranged and transformed into a convex problem involving a double-input single-output nonlinear system. In order to limit the estimate error at the frequencies of interest and to overcome the singularity problem incurred in the least-square-based methods, the identification problem is, thereafter, decomposed into a multi-objective optimization problem, in which the objective functions are defined in terms of the spectra of the unbiased error function at the frequencies of interest and are expressed as a first-order polynomial of the model parameters to be identified. The coefficients of the first-order polynomial are derived in an explicit expression involving the system input and the measured noised output. To tackle the convergence performance of the multi-objective optimization problem, the bargaining game theory is used to model the interactions and the competitions among multiple objectives defined at the frequencies of interest. Using the game-theory-based approach, both the global information and the local information are taken into account in the optimization, which leads to an obvious improvement of the convergence performance. Numerical studies demonstrate that the proposed bargaining-game-theory-based algorithm is effective and efficient for the multi-objective optimization problem, and so is the identification of the cascade dynamic nonlinear systems.