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Triple stage inverted pendulum belongs to a category of highly unstable and underactuated system which exhibits highly nonlinear behavior due to its complex dynamics (Gluck et al., 2013). These systems are testing bed platforms for comparing various control algorithms and strategies (Kizir et al., 2010; Henmi et al., 2015). The system resembles dynamics of humanoid robot, flexible space structures and therefore attracts interest of many researchers (Mao et al., 2013; Eltohamy & Kuo, 2010). (Farwig & Unbehauen, 1990) examined stabilisation of triple link inverted pendulum based on state space approach. The authors utilised a computer-aided design package called
'KEDDC' for building overall control structure. Experiments were performed which proved validity of proposed technique. (Li et al., 2002) proposed an adaptive fuzzy control of quadruple inverted pendulum. A mathematical model of proposed system has been derived and verified through simulations. The study also considered brief description of single, double and triple inverted pendulum systems. According to (Awrejcewicz & Kudra, 2007) a model of flat triple pendulum having obstacles at its position can be used to represent a model of piston-connecting rod-crankshaft assembly of four-stroke combustion engine. The proposed model can be successfully applied to analyse noise generated by impacts of piston and cylinder barrel.
A PID-neural adaptive controller for stabilisation of triple inverted pendulum was proposed by (Liang et al., 2010). The control law for PID was obtained using back-propagation algorithm and Lyapunov method was applied for learning of adaptive controllers. In a study by (Ling et al., 2011) comparison of single neuron adaptive proportional differential (SNPD) control and Linear quadratic regulator (LQR) control for stabilisation of triple inverted pendulum was performed. The results showed that SNPD strategy provides better stability and enhances anti-jamming capacity of the system. (Zhang et al., 2012) designed a variable gain LQR controller for control of three stage inverted pendulum. The authors proposed a new algorithm based on Schur method for solution of Riccati equation. Real time experiments were performed to demonstrate the proposed control. According to (Huang et al., 2012) an intelligent control based on combination of neural network and genetic algorithm can be applied to simulate a triple inverted pendulum system. The results showed that proposed control provides faster convergence and accompanies less iterations. (Molazadeh et al., 2014) designed a LQR controller for control of triple inverted pendulum. The tuning of LQR gains was achieved using fuzzy logic, GA and GA-particle swarm optimisation (PSO) algorithm. The results showed superiority of fuzzy controller over other two controllers.
In a study by (Zhang et al., 2015) a PID neural network (PIDNN) controller was optimised using cloud genetic algorithm (CGA) for control of three-stage inverted pendulum. It was observed that CGA provides faster convergence and avoids premature convergence phenomenon of genetic algorithms (GAs). The study further presented a three dimensional animated simulation to verify proposed control. (Chen & Theodomile, 2016) designed a feedback weight matrix of linear quadratic regulator (LQR) optimal control and feedback parameters of linear optimal control for determining configuration of fuzzy controller. The fuzzy controller was further applied to control a three-stage inverted pendulum. The results showed that the proposed controller selected simple parameters and exhibited good performance. This paper presents a new approach of designing PID based Adaptive neuro fuzzy inference system (ANFIS) and neural controllers for stabilisation of triple link inverted pendulum on cart. The paper describes in detail designing procedure of proposed controllers. The study compares linear and constant type MFs of different shapes. The performance of MFs were judge based on their respective training error tolerances. The relation between number of MFs and training error of ANFIS has also been demonstrated. The performances of controllers were compared in terms of three performance parameters namely settling time, maximum overshoot and steady state error.