Design of Nonlinear State Feedback Control Law for Underactuated TORA System: A Block Backstepping Approach Shubhobrata Rudra, Ranjit Kumar Barai, Madhubanti Maitra, Dharmadas Mandai, Shimul Dam, Somsubhra Ghosh, Prarthna Bhattacharyya and Arka Dutta Electrical Engineering Department, Jadavpur University, Kolkata-700032, India Abstract-This paper presents the formulation of a novel block-backstepping based control algorithm to address the stabilization problem for the well-known nonlinear benchmark TOA system. The ideas behind the method are as follows. At first, state model of the TOA system has been converted into block-strict feedback form. Then the control Lyapunov function has been desined for each cascaded ynamic block to derive the expression of the control input for the overall nonlinear system. The overall asymptotic stabiliy of the TOA system has been analyzed using Lyapunov Stabiliy Criteria. Finally, the efectiveness of the proposed control algorithm has been verfied in the simulation environment. Keywords: Underactuated System, TORA System, Block Backstepping Control, Lyapunov Stability Criteria. I. INTRODUCTION Stabilization of a TORA system (Translational Oscillator with a Rotational Actuator) has been considered as an active research area for control system engineers [1-4]. The TORA system has been considered as a benchmark nonlinear system for the class of underactuated mechanical systems. Nowadays, TORA system has received a conspicuous amount of research attention [1]-[4]. Early research activities on TORA were initiated by the need of the design of the controllers to address the complex control problem of a dual-spin aircrat, where the interaction between spin nd nutation complicates the algorithm design task for the conrol system engineers. Out-of-the-way, controlling the TORA is of independent interest as a benchmark problem in nonlinear control design [4]. Consequently, devising an eicient control algorithm for a TORA system remains as an active area of research. However, the control design of a TORA system is more complicated than that of a ully actuated system. Moreover, complicated coupling action between translational motion of the cart and angular rotation of the eccentric mass make the control design task more complicated. Backstepping is Lyapunov method based versatile control design approach for nonlinear systems that ensures the convergence of the regulated variables to zero [5-7]. Backstepping relies on the fact that the system under 978-1-4673-4603-0/12/$31.00 © 2012 IEEE consideration should be in a strict feedback form. Generally, single input single ouput systems satisy this condition under some simpliying assumptions [5-9]. Nevertheless, in case of multivariable control problem, quite oten the system sucture is not in he lower riangulr fom (srict feedback form or semi strict feedback form). In fact, ordinary backstepping eventually fails to generate a conrol algorithm for MIMO systems. However, if it is possible to represent the state space structure of such system in block strict feedback form by means of some state transformation, it is possible to address the conrol problems of MIMO system using backstepping technique. This technique is also known as block backstepping [8-10]. The seminal contribution of Reza Olfati Saber in the ield of underactuated system's reserch mkes it possible to fomulate different types of control algorithm for stabilization problem of the underactuated system [11-19]. Moreover, it is an inevitable fact that the proposed transformation [21] of an underactuated system's state model in the normal form simpliies the design task for the conrol system engineers. However, his proposed backstepping based conrol law [described in 11-13] is somewhat complicated and design of conrol algorithm for a TORA system using nomal form is not a very convenient way to design a backstepping conroller for the TORA system. The main objective of the paper is to fomulate a block backstepping based conrol algorithm for the TORA system. To accomplish the mentioned objective the paper is organized as follows: section II describes the mathematical model of the TORA system and precisely formulates the conrol objective in an analytic manner. Section III describes the derivation of the proposed algorithm. Section IV Analyzes the stability of the proposed conrol law. Section V describes the implantation procedure of the proposed algorithm on TORA system and analyzes the results. Finally, section V concludes the work. II. PROBLEM FORMULATION The nonlinear TORA system considered in this section was introduced by Wan, Benstein and Coppola in [4]. The picture in Fig. 1 illustrates the top view of this nonlinear benchmark mechanical system in which the rotational