A Fuzzy logic controller for stabilization and control of Double Inverted Pendulum ( DIP ) using different Membership functions ( MF ' s )

Double Inverted Pendulum (DIP) on cart is a highly non-linear system widely used as a testing bed for verification of newly designed control laws and controllers. In this study control of DIP is obtained using fuzzy logic controllers having different Membership functions(MF's) i.e. triangular, trapezoidal and gbell. The effects of shape of MF's on various controlling parameters i.e. stabilization time, maximum degree of overshoot and steady state error is also illustrated. Simulation results are shown with the help of graphs and tables which proves the validity of proposed method. KeywordsDouble Inverted Pendulum, Fuzzy Logic, Membership function, Matlab-Simulink.  The DIP is a multi-variable, unstable system which is difficult to stabilize in upright position . It represents a kinematic joint for robotic knee and arm. It can also be considered as a model of human and of other animal postural control [2]. In this paper fuzzy logic reasoning is used for stabilization and control of DIP. Fuzzy logic controller is able to stabilize the non-linear systems effectively and increases their flexibility to a great extent . This study shows a comparison between three different Membership functions (MF's) namely triangular, trapezoidal and gbell in terms of stabilization time, maximum degree of overshoot and steady state error. The affect of a particular MF's on performance of Inverted Pendulum system is illustrated in this study. There are several studies which have been done recently for the stabilization and control of DIP. Jianqiang Yi, Naoyoshi Yubazaki and Kaoru Hizota  proposed a new fuzzy controller AMO Advanced Modeling and Optimization.  1841-4311 547 Ashwani Kharola, Dr Pravin Patil, Punit Gupta having six inputs and one output based Single Input Rules Modules (SIRM). Ding Chengjun, Duan Ping, Zhang Ming-lu and Zhang Yan-fang [5] used genetic algorithm to optimize the weighting coefficient and MF's parameters of fuzzy controller. Sandeep Kumar Yadav, Sachin Sharma and Mr. Narinder Singh  performed the analysis of DIP using Linear Quadratic Regulator (LQR) controller. I.Zamani and M.H.Zarif  designed a fuzzy controller based on Lyapunov theorem. Atabak Nejadfard, M.J. Yazdanpanah and Iraj Hassanzadeh [7] proposed a friction compensation of DIP using neuro-fuzzy model. Ehsan kiankhah, Mohammad Teshnelab and Mahdi Aliyari Shoorehdeli  designed a neuro fuzzy controller using feedback-error learning for control of DIP etc. 2.0 Double Inverted Pendulum The classical DIP system consist of a pair of rigid pendulum rods i.e. Bottom pendulum and top pendulum which are interconnected with a joint. The bottom pendulum is attached to a cart which moves in the horizontal direction . Sensors are used to measure angle of each pendulum from vertical i.e. Ө1 and Ө2 and displacement of cart x. The final objective is to control the system such that Ө1, Ө2 and x should be equal to zero . A view of DIP on cart is shown in figure 1.0 [6]. Figure 1.0 A view of DIP on cart 2.1 Mathematical model of DIP The mathematical equations for control of DIP are derived separately for each subsystem i.e. Cart, First Pendulum and Second Pendulum. The equations are as follows: A. Cart ẍ = 1/M[F-N1-bẋ] where ẍ is acceleration of Cart, M is mass of Cart, F is the applied Force to Cart, N1 is the interaction force between Cart and First Pendulum and b is friction coefficient. 548 A Fuzzy logic controller for stabilization and control of Double Inverted Pendulum (DIP) using different Membership functions (MF's)