Identification of discrete time bilinear systems through equivalent linear models

This work discusses 3 methods for identification of discrete-time, time-invariant, single-input, multi-output, state-space bilinear models using the steady-state portion of a single input/output time-history recorded by exciting the system with a linear combination of sines and cosines of user-selected frequencies enriched by a subtle amount of random component. The proposed methods rely on conversion of the bilinear system into an equivalent linear model (ELM) by an accurate approximation of the state in the bilinear term using a set of sine and cosine basis functions whose frequencies are obtained using an Nth order perturbation theory approximation of the state.;In the first proposed method. Observer/Kalman Filter Identification (OKID), a linear time invariant (LTI) system identification algorithm, is used to identify the aforementioned ELM from which the original bilinear model is recovered.;In the second proposed method, the aforementioned ELM is converted to an overparameterized ELM by replacing its original state time-history with its past input/output time-history. The order of the new ELM is reduced by discarding its unobservable states. Since the state time-history of the ELM is (known and) common to the bilinear model, the original bilinear system is recovered via a least-square fit.;On the other hand, in the third proposed method, the aforementioned ELM is identified using a direct method of LTI system identification. Perturbation theory is used to show that the identified ELM can be used to identify the steady-state portion of the state time- history corresponding to (N + 1)th order perturbation theory approximation of the state. An iterative process is defined whereby the steady-state portion of the state time-history is identified exactly. The bilinear system matrices are identified via a least-square fit using the steady-state portion of the identified state-time history.;Numerical examples are also provided.