Discretization of nonlinear input-driven dynamical systems using the Adomian Decomposition Method

A numerical decomposition method proposed by Adomian provides solutions to nonlinear, or stochastic, continuous time systems without the usual restrictive restraints. It is applicable to differential, delay differential, integro-differential, and partial differential equations without the need for linearization or other restrictions. It also works through both uncoupled boundary conditions as well as delay systems. In the following paper, a new time discretization method for the development of a sample-data representation of a nonlinear continuous-time input-driven dynamical system is proposed. The proposed method is based on both the zero-order hold (ZOH) assumption as well as the Adomian Decomposition Method which exhibit unique algorithmic and computational advantages. To take advantage of this method, the following steps must be followed. First, the method is applied to a linear input-driven dynamical system to explicitly derive an exact sample-data representation, producing proper results. Second, the method is then applied to a nonlinear input-driven dynamical system, which thereby derives exact and approximate sample-data representations, the latter being most suited for practical applications. To evaluate the performance, the proposed discretization procedure was tested using simulations in a case study which involved an illustrative two-degree-of-freedom mechanical system that exhibited nonlinear behavior considering various control and input variable profiles. In conclusion, the suggested algorithm, in comparison to the results of a Taylor-Lie series expansion method, demonstrated increased performance and efficiency.