Time domain model order reduction of discrete-time bilinear systems by discrete Walsh functions

In this paper, the discrete Walsh functions are first used for time domain model order reduction (MOR) of discrete-time bilinear systems. Firstly, in terms of the discrete Walsh functions, the reduced order system is generated by means of the explicit solution formula for the discrete-time bilinear system. Meanwhile, we expand the discrete-time bilinear system under space spanned by the discrete Walsh functions. Then the reduced order system is produced by the orthogonal projection matrix that is defined by discrete Walsh coefficient matrix. Moreover, MOR method via discrete Walsh functions is investigated for discrete-time bilinear systems with nonzero initial conditions, where nonzero initial condition is also utilized to create the orthogonal projection matrix. Theoretical analyses show that the outputs of the resulting reduced order systems can match the first several outputs of the original system in finite intervals. Finally, an example is given to illustrate the effectiveness of the proposed methods.