Discretization of Duffing Oscillator Dynamics by Taylor-Lie Formulation Based Method Gagan Deep Meena 1 and S. Janardhanan 2 Abstract— This paper proposes a new discretization model for Duffing oscillator dynamics which is governed by a second order nonlinear differential equation. A Taylor-Lie formulation based method has been used for discretization. A detailed analysis and comparison of Taylor-Lie discretized model with Euler discretized model have been presented. I. INTRODUCTION Most physical systems have continuous time dynamics. Until a few decades ago, the control of such systems was also implemented in continuous time domain (i.e. using circuits and other conventional devices). With the advent of computers the replacement of continuous time systems with digital systems started. More sophisticated digital control and analysis algorithms came into existence. However, since the control and analysis algorithms are no longer continuous, it was not ideal to use a continuous time system representation. There the need of discrete time system became evident, and so did the need for discretization techniques. Discretization techniques were first proposed in 1960s by Kuo, Monroe, and Jury, [1]. Euler method is one of the simplest discretization techniques. This method provides a discrete model which matches closely to its continuous time counterpart at small sampling time. Though a variation in sampling time would result in changed discrete system. On the other hand, Taylor series expansion based discretization methods offer exactness in matching of continuous and discrete time state trajectories, irrespective of sampling time used for discretization [2]. In case of linear time invariant (LTI) systems an exact discrete model can be obtained, which has a closed form expression. A significant amount of research has been done in the area of discretization. Discretization of nonlinear systems has taken lead in the late 1980s. A number of attempts have already been made for the discretization of various nonlinear systems [3]–[14]. Duffing oscillator (DO) is an important family of nonlinear equations, which has been used to model several damped and driven oscillators [15]. It is one of the standard and classical examples to study relaxation oscillations, chaotic behaviors, jump phenomena 1 Gagan Deep Meena is with Department of Electrical Engi- neering, Indian Institute of technology Delhi, New Delhi, INDIA, er.gagandeepmeena@gmail.com 2 S. Janardhanan is is with Department of Electrical Engineering, Indian Institute of technology Delhi, New Delhi, INDIA, janas@ee.iitd.ac.in for frequency based analysis. The discretization of DO dynamics has received scant attention. Its study will open doors for other similar nonlinear oscillatory systems. In 1989, Reinhall, Caughey and Storti analyzed the error due to discretization techniques and presented the role of chaotic behavior for numerical integration over long periods and inherent difference between continuous and discrete models [16]. Following this a standard finite difference scheme also has been applied to obtain a discrete structure of DO dynamics [17]. Taylor-Lie discretization technique has been proposed by the authors in [18] with a class of nonlinear systems. It makes a comparison with Euler discretization technique. In this paper Taylor-Lie discretization technique has been applied on the DO dynamics. All its parameters have been tuned in such a way that DO is least reactive to sampling time variation and its exact discrete model can be obtained. Trial and error approach have been used for tuning all the parameters of DO. The simulation results shown in section 5 validate the claim of exact discretization of DO. A time and frequency response analysis also have been presented, the exactness of continuous and discrete time state trajectories have also been validated by the phase portrait analysis. Main Contributions of the paper are as follows • Implementation of Taylor-Lie Formulation based ap- proach of discretization on Duffing oscillator. • Validation of exact state trajectories by phase portrait, time response and frequency response analysis. Paper Organization: Section II discusses the preliminar- ies for the state of art techniques. The dynamics of the DO has been explained in the section III. DO has been discretized using by Taylor-Lie formulation based approach in section IV. All the simulation results validating all the claims for exactness of continuous and discrete models are presented in Section V. Section VI concludes the paper. II. PRELIMINARIES Before moving on to the discretization of DO dynamics let us first look at the state of art discretization techniques. For the discretization of continuous time systems Euler and Taylor methods are the most commonly used methods. Euler Discretization technique which can be used for all linear and nonlinear continuous time systems. It is capable of giving a close matching of continuous and discrete responses. For a continuous time system represented by ˙ x = f (x, u) (1) 2017 Australian and New Zealand Control Conference (ANZCC) December 17-20, 2017. Gold Coast Convention Centre, Australia 978-1-5386-2177-6/17/$31.00 ©2017 IEEE 23