Krylov subspace approximation for quadratic-bilinear differential system

Many nonlinear systems with nonlinearities of the form $ {1}/({k+x}) $ 1/(k+x), $ {\rm e}^{x} $ ex, $ x^{\alpha } $ xα, ln $ (x) $ (x) can be converted into quadratic-bilinear differential algebraic equations (QBDAEs) by introducing new variables and operating some algebra computations. Previous researches claim that the first two generalised transfer functions are enough to capture the dynamical behaviours of the system. In this research, we attempt to use traditional Krylov subspace methods to obtain the reduced model of the original QBDAEs so that the input–output behaviours of the reduced system can approximate that of the initial system. Model order reduction method proposed in this paper can match higher moments of first two transfer functions. Moreover, it does not involve the computation of tensor, and the main subfunction of this method is Arnoldi process which is easy to realise. We also analyse the stability and matching property of the reduction process. In addition, the numerical cost is given. The theory result and numerical result show the validity of the proposed method and the small computation expense of this method.