Numerical Discretization Estimation for Ordinary Differential Equation via Hybrid Discretization

Abstract

Simulation of control system is mostly developed based on the use of ordinary differential equation (ODE). With the advancement of technologies especially in term of real time computation, the traditional numerical approaches seem outdated to fit in to the current real-time discrete systems. Numerical method such as Euler’s approach has inaccurate approximation as compared to other methods such as Heun’s (RK2), Runge-Kutta (RK4), and Adams-Bashforth (AB2) methods, and these methods on the other hands suffers from high calculation time. In this work, Hybrid Discretization (HD) method is proposed to solve both approximation accuracy and calculation speed of the discretization. HD adapts RK2 method to correct the approximation error for one-to-ten step depending on the sampling time. Later, the system will return to Forward Euler’s method to maintain the calculation speed. The HD is applied to two first order ODE test functions and the result of this work shows a significance improvement in terms of the accuracy of the approximation and slight improvement in term of the calculation time. The accuracy of about 9% is obtained as compared to a similar step time in Euler’s, and comparable calculation time is maintained. In conclusion, it is shown that this new technique of discretization has better approximation than its counterparts and the method can serve as an important simulation tool in modeling and control of dynamic system.

Keywords

• hybrid discretization
• numerical methods
• ordinary differential equation
• simulation