A Single-residual Partial Mutual Information (SPMI) approach to learning discrete-time inputs of stable nonlinear dynamic systems

The selection of input variables and their discrete time delays are fundamentally important in developing robust data-driven dynamic models for use in applied engineering settings, such as for controller design or system level calibration/optimisation. This work is not trivial, especially in the case of complex multivariate and non-linear dynamic systems. There are an array of model-free approaches to input selection explored in the literature, including Multivariate Mutual Information (MMI), Gamma Tests (GT), Self-Organising Maps (SOM) and Partial Mutual Information (PMI). Such a filter-based approach has advantages in exploring feature correlations and their associated information content in a data set directly, agnostic to the constraint of any specific model structure or form.

This paper investigates and expands upon the application of a PMI-based Input Selection (PMI-IS) methodology for resulting in a modified version of the algorithm. The modifications are: (1) Selection of input Dead Time (DT) using Mutual Information of First and Second Difference terms of input delays with the output. (2) The Number of Delayed Outputs (NDO) is selected based on the PMI incorporating the previously selected time delays; (3) The Number of Delayed Inputs (NDI) is selected based on the PMI incorporating the identified delay times and NDO; (4) The established Dual-residual PMI (DPMI) algorithm for input selection is simplified to a Single-residual PMI (SPMI) algorithm.

Three benchmark discrete-time non-linear dynamic systems and one practical demonstration are used in the case study to demonstrate the effectiveness of this learning algorithm for data-driven identification of time delays, in addition to the implementation details of this modified SPMI-IS methodology.