Time series approach to the response of time-invariant pseudolinear differential equations with application in satellite simulation

Abstract

In this study, a novel time series-based approach is proposed for a specific class of nonlinear dynamical systems referred to as time-invariant pseudolinear systems. These systems are characterized, similar to linear dynamic systems, by system and control matrices that depend linearly on the state vector, while their second and higher-order derivatives with respect to the state variables are assumed to be zero. A methodology is developed to approximate the state transition matrix of such systems using a time series polynomial expansion, particularly for the autonomous case. For systems with external inputs, a complementary technique is introduced to accommodate input-dependent dynamics. To evaluate the effectiveness of the proposed methods, four case studies are presented. The first two are autonomous systems with known analytical solutions, allowing for direct validation. The third example involves a non-autonomous system with an input specifically designed to yield a known response. The final example demonstrates the application of the method to a real-world problem: satellite attitude dynamics, modeled as a pseudolinear system subject to a quaternion norm constraint. In all cases, the solutions obtained via the proposed approach are compared with those generated using the classical fourth-order Runge-Kutta method. Simulation results reveal that the proposed method outperforms the Runge-Kutta scheme in terms of accuracy, suggesting its potential utility for the discretization of pseudolinear systems commonly encountered in engineering applications.