Tuning of Pid Controller of Inverted Pendulum Using Genetic Algorithm

The paper presents two different ways of mathematical modeling of Inverted Pendulum. A Proportional-Integral-Derative (PID) controller is designed for its stabilization. Some reference stable system is selected after designing of PID controller to optimize different types of error using Genetic Algorithms. The proposed system extends classical inverted pendulum by incorporating two moving masses. A tuning mechanism is implemented by genetic algorithm for optimizing different gains of controller parameter. Also, different performance indices are calculated in MATLAB environment. This paper exhibits to demonstrate the capability of genetic algorithm to solve complex and constraint optimization problems and as a general purpose optimization tool to solve control system design problems. Keyword-Inverted pendulum,Mathematical modelling,swing up control ,PID controller,Tuning,Genetic Algorithm,Performance Indeces,Error minimization. --------------------------------------------------------------------*****-------------------------------------------------------------------1.INTRODUCTION The inverted pendulum may be viewed as a classical problem in dynamics and control theory[1,9] and is widely used as a benchmark[16] for testing control algorithms(PID controllers, Linear Quadratic Regulator (LQR), neural networks, fuzzy logic control, genetic algorithms, etc)[7,8].The inverted pendulum is unstable[11] in the sense that it may fall any time in any direction unless a suitable control force is applied. The control objective of the inverted pendulum is to swing up[4] the pendulum hinged on the moving cart by a linear motor[12] from stable position (vertically down state) to the zero state(vertically upward state) and to keep the pendulum in vertically upward state in spite of the disturbance[5,13]. In the field of engineering and technology the importance of benchmark [14,7] needs no explanation. They make it easy to check whether a particular algorithm [6] yields the requisite results. Several work has been reported on the inverted pendulum for its stabilization. Attempts have been made in the past to control it using classical control [3]. In this paper, the controller is proposed to be tuned using Genetic Algorithm(GA)[15] technique.. Genetic Algorithm[2,10] have been shown to be capable of locating high performance areas in complex domains without experiencing difficulties, even associated with high dimensionality or false optima as many occurring with some other optimization method. 2. MECHANICAL CONSTRUCTION The system comprises of a horizontal plate connected to two wheels through a connecting rod. The wheels are independent of each other and are placed in the centre of the rail. The platform thus can move on a horizontal surface and is able to rotate about the axis of wheels. There are two masses on top of the system that can slide along the horizontal rail, the masses being on both sides of the rail. The system is shown in fig (1) 3. MATHEMATICAL MODEL OF THE PLANT Defining the angle of the rod from the vertical (reference) line as θ , displacement of the cart as x, assuming the force applied to the system be F , centre of gravity of the pendulum rod is at its geometric centre and l be the half length of the pendulum rod, the physical model of the system is shown in fig (2). IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163 __________________________________________________________________________________________ Volume: 01 Issue: 03 | Nov-2012, Available @ http://www.ijret.org 360 The Lagrangian of the entire system is given as, L= 1 2 (mx 2 +2mlx cos�+mlθ 2+Mx 2)+ 1 2 Iθ 2 ]-mglcos� The Euler-Lagrange’s equation for the system is : d dt δL δx − δL δx + bx = F d dt δL δθ − δL δθ + dθ = 0 The dynamics of the entire system using above equation is I + ml θ + ml cos θ x − mgl sin θ + dθ = 0 ..........(1) M + m x + ml cos θθ − ml sin θ θ 2 + bx = F ........(2) x1 = x + l sin θ