 Open Access
 Total Downloads : 773
 Authors : Ashwani Kharola, Punit Gupta
 Paper ID : IJERTV2IS90261
 Volume & Issue : Volume 02, Issue 09 (September 2013)
 Published (First Online): 10092013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Stabilization of Inverted Pendulum (IP) Using Two Fuzzylogic Controllers (FLC’s) Having Nine Linguistic Variables
Stabilization of Inverted Pendulum (IP) Using Two Fuzzylogic Controllers (FLCs) Having Nine Linguistic Variables
Ashwani Kharola *1, Punit Gupta *2
*Department of Mechanical Engineering, Graphic Era University, Dehradun, India.
ABATRACT This paper presents a Modelling and Simulation study of a control strategy for an Inverted Pendulum system. Inverted Pendulum is well known as a testing bed for various controllers. For independent control of Cart and Pendulum two FLCs have been defined separately using nine linguistic variables. Modelling and Simulation has been done in Matlab and Simulink respectively. In addition to Stepresponses and Pulse responses, an Openloop and Closedloop response using PID controllers are also shown. The proposed fuzzy control scheme successfully fulfils the control objectives and also has an excellent stabilizing ability to overcome the external impact acting on the pendulum system.
KEYWORDS Inverted Pendulum, Fuzzy logic Controller, UnitStep response, Pulse response.
the major concerns in industry machineries. In conventional control theory, most of control problems are usually solved by mathematical tools based on system models. But in true sense, there are many complex systems whose accurate mathematical models are not available or difficult to formulate. As an alternative to conventional control approach, the fuzzy control techniques can provide a good solution for these problems by introducing linguistic information
[2]. The control of inverted pendulum is fundamentally same as those involved in rocket or missile propulsion, walking robot, flying objects in space etc [3]. Figure 1.0 shows a view of inverted pendulum [3].

INTRODUCTION
The inverted pendulum is a classical problem in control system [1]. It is a system that is inherently unstable. The pendulum is mounted onto a nonstationary cart in the vertical position or at equilibrium position. The pendulum is unstable and free to fall over if there are any disturbances. On the other hand, if the pendulum is too thin and the cart is move by a force, it can flex and cause vibrations. These vibrations are one of the
(1)
Figure 1.0 view of Inverted Pendulum An Inverted Pendulum has two
equilibrium points: vertical upright equilibrium point and downward equilibrium point [4]. When the pendulum is standing at vertical upright position and the resultant of forces acting from all sides is
zero it is said to be in vertical upright equilibrium point. The vertical upright equilibrium point is inherently unstable, as any small disturbance may cause the pendulum to fall on the either side when the cart is at rest. When there is no external force acting on pendulum it will come into rest in downward position. This equilibrium position is stable.

MATHEMATICAL MODELLING OF INVERTED PENDULUM
The system consists of a pole with mass, m , hinged by an angle from vertical axis on a cart with mass, M, which is free to move in the x direction as shown in figure 1.1[2] . A force, F is required to push the cart horizontally, the friction coefficient of cart b, the length between axle centre and the centre of pendulum L, the inertia of pendulum I [6].
Figure 1.1 Diagram of Inverted Pendulum
For simulation the dynamic equations of the actual inverted pendulum system are derived as follows consider figure 1.2[2,7].
Figure 1.2 FBD of Cart & Pendulum
By Newtons Equation: For Cart:
Where, N and P are the interaction forces between the cart and pendulum.
For Pendulum:
It is necessary, however, to include the interaction forces N and P between the cart and the pendulum in order to model the dynamics. The inclusion of these forces requires modelling the x and y dynamics of the pendulum in addition to its theta dynamics. Therefore, the additional x and y equations for the pendulum are modelled as given by the equations below:
However xp and Yp are exact functions of theta. Therefore, their derivatives are
represented in terms of the derivatives of theta
(2)
After substituting these equations into eq. 7 and 9, we get:
Using the above nonlinear equations, a Matlab Simulink model has been developed [8]. This Simulink model is used in this work for stabilizing the upright position of the pendulum and to move the cart at desired position.

DESIGN OF FUZZY LOGIC CONTROLLER FOR INVERTED PENDULUM SYSTEM
In order to implement four inputs to the controller, the FLC were divided into two parts as can be seen in the figure
1.3[2,4]. The FLC 1 is used for controlling the carts position, where as the FLC 2 controls the pendulums angle. The FLC 1 receives Position (x) and Del Position (x dot) as the inputs while the FLC 2 receives Angle () and Del Angle (dot) as the inputs. The output variable of both the FLCs is force.
(3)
Figure1.3 FLC Overview IV.MEMBERSHIP FUNCTIONS:FLCs
According to the complexity of this inverted pendulum system, we have taken nine fuzzy subsets to quantize each fuzzy variable for both FLC as shown in table.
LINGUISTIC TERM 
LABEL 
Negative Extreme 
NE 
Negative Big 
NB 
Negative Medium 
NM 
Negative Small 
NS 
Zero 
ZE 
Positive Small 
PS 
Positive Medium 
PM 
Positive Big 
PB 
Positive Extreme 
PE 
Table 1.3 Standard labels of quantization.
Figure 1.4 till figure 1.6 show the membership functions of FLCs.
Figure 1.4 input 1 for FLC 1
Figure 1.5 input 2 of FLC 1
Figure 1.6 output of FLC 1

RULE BASE:
The following rule bases are applied for simulation study to control the Inverted Pendulum.
Table1.4 Fuzzy Rule Matrix (9×9) :FLC 1
Table 1.5 Fuzzy Rule Matrix (9×9):FLC 2
(4)

SURFACE VIEWERS
Figure 1.7 and Figure 1.8 shows surface viewer for FLC 1 and FLC 2 respectively.
Figure 1.7 Surface Viewer for
FLC 1
Figure 1.8 Surface Viewer for
FLC 2

SIMULINK MODEL:INVERTED PENDULUM

Simulink model for Step response of Inverted Pendulum using two FLCs
Figure 1.9 Simulink model using two
FLCs

Simulink SubSystem model for Inverted Pendulum
Figure 2.1 Openloop pulse response
(d) Closedloop response : using PID controller
Figure 2.2 Closedloop pulse response
(5)



StepResponses
Figure 2.0 IP: subsystem
model

Openloop response for Inverted Pendulum


Figure 2.3 output step response :Position x
Fig 2.4 output step response Del Position x
Figure 2.5 output step response for Angle
Fig 2.6 output step response for del_ angle

Pulseresponses
Fig 3.0 Output pulseresponse Del_angle

Openloop responses
Fig3.1 openloop impulse response Position x
Figure 3.2 open loop impulse response for angle
(6)

Closedloop response
Fig2. Oputut pulseresponse Del position
Figure 3.3 closedloop response for Position implementing PID control
Figure 3.4 closedloop response for Angle implementing PID control
CONCLUSION
As a conclusion, the objective in stabilizing the inverted pendulum has been achieved by using two block of Fuzzy Logic Controllers[5]. This can be verified through output responses of the system which satisfy the design criteria. The Output Stepresponse for both the FLCs shows that after 20sec all the transient behaviour stops and the Pendulum is stabilized. The Output Pulseresponse for both the FLCs shows that there is no carry over of the previous pulses and the system need not to be reinitialized. As can be seen from the Openloop response that the Pendulum swings all the way around due to impact, and the cart travels along with a jerky motion due to pendulum. We can infer from the ClosedLoop response that the PID Controller handles the nonlinear system very well because the angle is very small (.04 radians).
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