Closed-loop linear model validation and order estimation using polyspectral analysis

Suppose that we perform closed-loop linear system identification using polyspectral analysis given noisy time-domain input-output measurements. In this setup, it is assumed that various disturbances affecting the system are zero-mean stationary Gaussian, whereas the closed-loop system operates under an external (possibly noisy) non-Gaussian input. The closed-loop system must be stable, but it is allowed to be unstable in the open loop. Various techniques have been proposed for system identification using polyspectral analysis. Having obtained a model, how do we know if the fitted model is “good?” This paper is devoted to the problem of statistical model validation using polyspectral analysis. We propose simple statistical tests based on the estimated polyspectrum (integrated bispectrum and/or integrated trispectrum) of an output error signal or the estimated cross-polyspectrum between the external reference and the output error signal. Model order estimation is performed by repeatedly using the model validation procedure. Computer simulation examples are presented in support of the proposed approaches