 Open Access
 Total Downloads : 490
 Authors : Neeraj Jain, Rajeev Gupta, Girish Parmar
 Paper ID : IJERTV2IS121106
 Volume & Issue : Volume 02, Issue 12 (December 2013)
 Published (First Online): 01012014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Intelligent Controlling of an Inverted Pendulum Using PSOPID Controller
Neeraj Jain 
Rajeev Gupta 
GirishParmar 
Department of Electronics Engg. 
Department of Electronics Engg. 
Department of ElectronicsEngg. 
Rajasthan Technical University 
Rajasthan Technical University 
Rajasthan Technical University 
Kota, India 
Kota , India 
Kota, India 
Abstract Stabilizing the inverted pendulum is a standard problem in the field of control system. When force is applied to cart its position and pendulum angle deviate from its position .Many researchers have been applying different control algorithm and design techniques such as Neural
technique.

Modeling an Inverted Pendulum
Network, Genetic Algorithm (GA),Fuzzy logic, Particle swarm optimization on to a PID controller for stabilization of cart position and pendulum angle. The particle swarm optimization is a new evolutionary computation technique and has been introduced to solve several industrial problems [16]. Particle swarm optimization has better computational efficiency and it requires less number of parameter to adjust [13]. In this paper Particle Swarm Optimization (PSO) technique has been discussed to control the inverted pendulum problem and result is compared with conventional PID controller.
Keywords: Inverted pendulum; PID controller; PSO; System stability

Introduction
For control engineers inverted pendulum is a very good platform to verify different problems in the field of control theory. It is an excellent test benchmark for testing various complicated control problems.Normally inverted pendulum is stable when put on a cart, if a force is applied to the cart pendulum becomes unbalance until a suitable control strategy is applied. It is a Single input multiple output problembecause the system has one input the force applied to the cart, and two outputsposition of the cart and the angle of the pendulum.
The standard linear techniques cannot solve thesuch type of nonlinear dynamics of the system. This system is challenging for analysis. Due to the good features of particle swarm optimization (PSO) algorithm, presently it has been used as a new optimizer and applied to various types of research problems.PSO was developed through simulation of simplified social systems, and is robust in solving nonlinear optimization problems[8]. Objective of this paper is to design simulink model of inverted pendulum system, stabilize it with PID controller using particle swarm optimization technique (PSO) and compare the results with conventional PID
The inverted pendulum model is shown in Fig.1. Its modeling is done for analysis the pendulum position when force is applied to cart and pendulum is stabilize using PSOPID controller.
2.1 Inverted Pendulum on a Cart
The cart with an inverted pendulum, is shownin Fig. 1. An impulse force F Newton is applied to the cartSomeassumptions are madefor modeling of an inverted pendulum which is given below in table 1.
Table I.Assumption for Inverted Pendulum
Symbol
Parameter
Value
M
Mass of the cart
0.5 Kg
m
Mass of Pendulum
0.2 Kg
b
Friction of the cart
0.1/N/m/sec
l
Length to pendulum centre of mass
0.3 m
I
Inertia of pendulum
0.006Kg*m^2
F
Force applied to the cart
1 Newton
Pendulum angle from vertical
radian
Below are the two Free Body Diagrams of the system. Summing the forces in the Free Body Diagram of the cart in the horizontal direction, we get the following equation of motion:
+ + = (1)
Since we will be looking at the angle, as the output of interest, solve the equation (9) for X(s),
= + 2 ()(11)
2
Substitute value of X(s) from equation (11) to (10) and rearrange. The transfer function is:
2
() =
()
4
+ 2 3
+ 2
(12)
+
where,
= + + 2 ()2 (13)
Fig.1.Free Body Diagram of Inverted Pendulum
From the transfer function above it can be seen that there is both a pole and a zero at the origins. These can be canceled and the transfer function becomes:
Note that the forces can be sum in the vertical direction,
() =
()
3
+ 2 2
+
(14) From
but no useful information would be gained. Summing
the forces in the Free BodyDiagram of thependulumin the horizontal direction, we can get an equation for N
+
table I by putting the values of M, m, b, l, I in equation
(14) and get the transfer function of Fig. I
() = 4.545
(15)
= + 2 (2)
By substituting (2) equation into the (1) equation, we get the equation of motion for this system
()
3+0.1818 231.184.455


Particle Swarm Optimization
+ + + 2 = (3) To get the second equation of motion, sum the forces perpendicular to the pendulum
+ = + (4)
To get rid of the P and N terms in the equation above, sum the moments around the centroid of the pendulum to get the following equation
= (5) Combining equation (4) & (5), we get the second dynamic equation:
+ 2 + = (6)
These set of equations (3) & (6) should be linearized about = ,Assume that theta = + (represents a small angle from the vertical upward direction). Therefore, cos = 1, sin = 1 and (d /dt) 2 = 0. After linearization the two equations of motion become (where u represents the input):
+ 2 = (7) + +
= (8)
The PSO was originally designed by Kennedy and Eberhart and is one of the modern heuristic algorithm[10]. This technique involves simulating social behavior among individuals (particles) flying through a multidimensional search space, each particle representing a single intersection of all search dimensions. The particles evaluate their positions relative to a goal (fitness) at every iteration, and particles in local neighborhood share memories to adjust their own velocities and thus subsequent positions.
The various steps involved in Particle Swarm Optimization Algorithm are as follows:
Step 1: The velocity and position of all particles are randomly set to within predefined ranges.
Step 2: Velocity updating At each iteration, the velocities of all particles are updated according to,
vi = vi + c1R1 (pi,best pi ) + c2 R2 (gi,best pi ) where,
2.2. Transfer Function of Pendulum Model
piand vi – are the position and velocity of particle i, respectively;
To obtain the transfer function of the linearized sypsit,ebmestand gi,best – is the position with the best objective value
equations analytically, we must first take the Laplace transform of the system equations (7)& (8). The Laplace transforms are:
+ 2 2 = 2 (9)
+ 2 + 2 = ()(10)
found so far by Particle i and the entire population respectively;
w – is a parameter controlling the dynamics of flying;
R1 and R2 – are random variables in the range [0, 1];
c1and c2 – are factors controlling the related weighting of corresponding terms. The random variables help the PSO with the ability of stochastic searching.
Step 3: Position updating The positions of all particles are updated according to,
pi = pi + vi
After updating, pishould be checked and limited to the allowed range.
Step 4: Memory updating Update pi,bestand gi,bestwhen condition is met,
pi,best = pi if f(pi ) > f(pi,best )
gi,best = giif f(gi ) > f(gi,best )
where ,f(x)is the objective function to be optimized. Step 5: Stopping Condition The algorithm repeats steps 2 to 4 until certain stopping conditions are met, such as a predefined number iterations. Once stopped, the algorithm reports the values of gbestand f (gbest)as

Simulation and Results
Simulink model has been developed in MATLAB. It is tried to tune inverted pendulum system using particle swarm optimization by nonlinear equations. Simulink model with impulse disturbance has been shown in Fig.3.
Fig.4 depicted the response of inverted pendulum with conventional (Zeigler Nichols) PID controller and with impulse disturbance .It is observed from Fig 4.that response is not stable before 20 sec.
The PID values for this PID controller are Kp = 2, Ki = 5 and Kd = 2
its solution
Start
Impulse Input
PID
PID Controller
u y
Pendulum
Output
simout 2
simout 2
1 To Workspace 2 Out 1
Impulse Disturbance
Initialize Particles with Random Position andVelocity
Fig.3. PSOPID Model in Simulink
Evaluate Particles
Evaluate Particles
10
Compare and update, Pbest
Compare and update, Pbest
Theta (Degree)
Theta (Degree)
8
6
Compare and update, gbest
Compare and update, gbest
4
2
0
0 10 20 30 40 50
Time (Sec)
Meet stoppingcriteria
End
Fig.4 Impulse response of Pendulum with Conventional PID Controller
Update velocity andPosition
Update velocity andPosition
When PSO PID controller is applied with following settings
Initial population size (N) = 200 Step size = 40
c1 = c2 = 2.0 w = 0.8
The response is shown in Fig.5
Fig 2: Flowchart of PSO Algorithm
10
Theta (Degree)
Theta (Degree)
8 VI. References
6
4
2
0
0 10 20 30 40 50
Time (Sec)
Fig.5 Impulse response of pendulum with PSO PID controller
It is observed from Fig.5that by using PSO technique to tune PID controller system is stable and response is less oscillatory.
The value of integral error performance indices is also obtained. The integral of the absolute value of the error (IAE) is an appropriate measure of control performance when the effect on control performance is linear with the deviation. The integral of square of the error (ISE) is appropriate when large deviations cause greater performance degradation than small deviation.
Table1. Comparison of PID Techniques
S.No.
Type of Controller
IAE
ISE
1
Conventional (Zeigler Nichols)
4.58
7.14
2
PSO PID
1.61
2.37
It is observed from table 1 that on IAE & ISE error performance indices the result is much better with PSO PID Controller.

Conclusion
The results of the simulation have been shown. The proposed PSO improves the performance of inverted pendulum and can be easily introduced to any other nonlinear control problem. It is observed that the inverted pendulum position is stable very soon using PSOPID controller compare to conventional PID controller.Future work can be done on tuning PID controller with Genetic algorithm and type two fuzzy cascade controller.
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[14]. AnulaKhare, SarojRangnekar, A Review of Particle Swarm Optimization and its Applications in Solar Photovoltaic System,Elsevier,pp.2997 3006,2013.Neeraj Jain has received B.E. degree in Electronic Instrumentation and Control Engineering from the Engineering College, Kota, and the M.E. (2009) control & Instrumentation from NITTTR, Chandigarh. Presently he is pursuing Ph.D. from Electronics department in Rajasthan
Technical University, Kota. He has 15 years of experience in process industry and teaching. He is a life member of Institution of Engineers (IE), Kolkata. He has presented 23 research papers in conferences and journals. His research areas are transducers, process control, modern control system, Industrial automation, artificial intelligence.
Rajeev Gupta received the
B.E. (1986) degree in Electrical Engineering from the Engineering College, Kota, and the M.Tech. (1995) Control & Instrumentation Engg and Ph.D. (2004) degrees in System & Control
Engg.from the Indian Institute of Technology, Bombay, Powai, Mumbai India. His current research interests include: Multirate Periodic output feedback control, Robust control, Sliding mode control, Fuzzy logic control, Genetic Algorithms and Artificial Neural network based control. Heis the author/coauthor of 34 Research Papers; Seven Books of Electrical & Electronics Engineering.Heis a consultant to several Government organizations and private industries in India and abroad.Dr. Guptahasguided 8 M.Tech. Thesis and is guiding 13 Ph.D. students on various areas. He s life member of ISTE, New Delhi, Institution of Engineers, India, and IEEE, USA.
Dr. GirishParmar was born in Bikaner (Rajasthan), India, in 1975.He received B.Tech. in Instrumentation and Control Engineering from National Institute of Technology, Jalandhar (Punjab), India in 1997 and M.E. Electrical (Gold Medalist)
withspecialization inMeasurement and Instrumentation from Indian Institute of Technology, Roorkee, India in 1999. He obtained his Ph.D. in Electrical Engineering. in 2007 under Quality Improvement Programme from Indian Institute of Technology, Roorkee, India. He is life member of Systems Society of India (LMSSI) and Associate member of Institution of Engineers, India (AMIE). He has published 59 research papers in various International/National Journals and Conferences. He is author of several technical books. He has worked as aAssistant Professor in Department of Electronics Engineering at Rajasthan Technical University, from 1999 to 2011.He has worked as a Principal of Modi Institute of Technology, Kota from December, 2011 to 2013. Presently he is working as an Associate professor in Rajasthan Technical University,Kota. His research interests are in the area of Process Instrumentation & Control, Optimization, Signal Processing, System Engineering and Model Order Reduction of Large scale systems.