Stabilization of Rotary Inverted Pendulum Using … of Rotary Inverted Pendulum Using Fuzzy Logic...

Stabilization of Rotary Inverted Pendulum Using Fuzzy Logic

Abstract

The Rotary Inverted Pendulum (RIP) has sufficient suitable characteristics from control point of view to be famous for one of the best systems for evaluating control strategies and controllability tests. Having multivariable nonlinear unstable non minimum phase dynamics for a simple system puts it in gracious systems’ category to test performance and efficiency of different control methods. In this paper we use a fuzzy logic, and LQR method to deal with the challenging characteristic of the system, and evaluate the used control techniques as well as new methods. In this paper a new PID controller is made to swing up the pendulum, then it is shown that fuzzy controllers can be used instead of the conventional controllers such as PID and better performance can be achieved using the proposed method. Fuzzy logic may not be a fast controller, but it's very useful for most systems and also for complicated systems because of the design simplicity, so a suitable fuzzy controller is used in this paper, The results of simulations are also shown for comparison. First we model the rotary inverted pendulum, and then swing it up, at the end we stabilize the pendulum oscillations around its upright position.

Keywords: Modeling, Swing up, Linear Quadratic Regulator, Fuzzy Logic, Rotary Inverted Pendulum.

1. Introduction

Advantages of the rotary inverted pendulum system with unstable poles and non minimum phase

dynamics, nonlinear equations, and multivariable model with a simple structure lead to select RIP for testing new control techniques on as a benchmark, therefore engineers like to use it for validating and evaluating the effectiveness, robustness, and accuracy of our suggested control techniques. The RIP

structure includes a dc motor as an actuator, a rotary arm with angular position ' 1 ' and angular

velocity ' 1 ' which rotates horizontally based on the motion of the motor, the mounted pendulum on

the arm oscillates easily with an actuation. Two encoders detect 1 and 2 , which 2 are the angular position of pendulum and the arm respectively. In the following researches some control techniques like Swing up fuzzy controller for inverted pendulum based on a human control strategy [1], Nonlinear Control [2] , swing up a pendulum by energy control[3,4], etc are introduced. Sukontanakarn and Parnichkun implemented a real-time controller for self-erection of the pendulum from downward position and balanced the pendulum in vertical-upright position, an energy based PD controller was applied for self-erecting the pendulum while LQR controller was applied to balance the pendulum[5],Mary P.and Marimuthu N.S. swung up rotary inverted pendulum using a hybrid controller, using a PD position control to swing up the pendulum and a Fuzzy Logic Controller to balance and stabilize it at its upright position[6], Krishen and M.Becerra swung up the pendulum using Fuzzy Logic and LQR[7], M. Roman, E. Bobasu, D. Sendrescu did Modelling of the Rotary Inverted Pendulum System[8], Md. Akhtaruzzaman and A. A. Shafie modeled and Controled of a Rotary Inverted Pendulum Using Various Methods[9], Necdet Sinan and Mehmet Onder Efe Swinging up and Stabilized Control Experiment for a Rotary Inverted Pendulum[10].

In this paper we try to swing up the pendulum with fuzzy controller and then keep it at upright position and set the equilibrium point to zero with another controller.

For this RIP system because of its simple structure it's possible to linearize the system about zero as its operating point and can use conventional control methods such as LQR for various control purposes.

Arman Khashayar Islamic Azad University,

South Tehran Branch arman.khashayar@gmail.

com

Mohammad Ali Nekoui Islamic Azad University,

South Tehran Branch [email protected]

Hamed Ahangar-Asr Islamic Azad University,

South Tehran Branch [email protected]

Stabilization of Rotary Inverted Pendulum Using Fuzzy Logic Arman Khashayar, Mohammad Ali Nekoui, Hamed Ahangar-Asr

International Journal of Intelligent Information Processing(IJIIP) Volume2, Number4, December 2011 doi : 10.4156/ijiip.vol2.issue4.3

23

2 DESCRIBTION MODEL

The typical Rotary Inverted Pendulum system (RIP) includes a dc motor as an actuator, a rotary arm which rotates clockwise and anticlockwise based on dc motor's motion. Two rotary encoders as angular position detectors for detecting the angular position of arm and pendulum (

1 ,2 ) are applied. while

rotation, the pendulum mounted on the arm, falls immediately with each actuation which leads to displacement of the arm from its upright position as shown in figure 2, a controller should be applied to keep the pendulum in the upright position , a typical schematic of the system model in this paper is shown in figure1.

Figure 1. Typical schematic of RIP system

As Its seen in figure 1 the length of the rotary arm is 'L1≠0', but the length of the arm from its center

of mass that is in middle of its length is 'l1=0',however length of the pendulum is 'L1' and its length from its center of mass is 'l1'. J1 and J2 denote moments of inertia of the arm and pendulum about their centre of mass, and m1, m2 denote the masses of arm and pendulum respectively. We can see the relevant parameters of RIP system in figure 1.

According to figure1 parameters specify the angle of the pendulum from the upright position and the angular velocity of the pendulum. specify angle of the rotary arm and angular velocity of

the arm respectively. The upright position of pendulum is defined to be zero as the ideal position for the pendulum to stabilize. As we know an actuation of dc motor leads to rotation of the arm and causes an angular error for the pendulum with respect to its upright position which causes it to fall down. Therefore It’s necessary reduce the angular error to zero or keep the pendulum in the upright position. When the pendulum stands in the upright position the angular error is considered to be zero which shows that the used control techniques applied are working well .For evaluating the effectiveness of our controller we should first find the dynamical equations of this system[8]. We use the Euler-Lagrange equation to extract the equations of this system for applying the proposed control techniques to stabilizes the system. Euler-Lagrange equation is as follows:

( ) ii i

d L LQ

dt q q

In the above equation the energy loss is considered to be negligible. The Lagrangian equation is revealed by total potential energy ( V total ) subtracted from total kinetic energy ( T total ) as below:

( , ) total totalL q q T V

The total kinetic energy is considered as sum of kinetic energy of the rotary arm and the pendulum rod:

T total = T rotary arm + T pendulum rod

2 2,

1 1,

(1)

(2)

(3)


24

Figure 2. side view of moving the pendulum

The kinetic energy of the rotary arm is:

11Lvarm

Moving on x and y directions:

11222 cos Llvx

222 sinlv y

Then Total kinetic energy is as follows:

222

22

22

211 2

1

2

1

2

1

2

1 JvmvmJT yxtotal

by substitution of (5) and(6) into (7):

222

22222

212222

211 2

1)sin(

2

1)cos(

2

1

2

1 JlmLlmJTtotal

2 2 2 2 21 1 2 2 2 2 1 1 2 1 1 2 2 2 2 2

1 1 1 1cos

2 2 2 2totalT J m l m L m L l J

Total potential energy is equal to potential energy of the pendulum:

2 2 2costotal pendulumV V m gl

Therefore substitution of Eq. 8 and 9 into Eq. 2 yields:

2 2 2 2 21 1 2 1 1 2 2 2 2

22 1 1 2 2 2 2 2 2 2 2 2

1 1 1( )

2 2 2

cos ( sin ) cos

L J m L m l J

m L l l m gl

Hence the Euler-Lagrange equations become:

1 1

2 2

( )

( ) 0

e

d L L

dt

d L L

dt

Solving the equation substituting Eq. 10 into Eq. 11,12 yields:

(9)

(10)

(11)

(4)

(5)

(6)

(7)

(8)

(12)


25

2 21 2 1 1 2 1 2 2 2 2 1 2 2 2[ ] ( cos ) ( sin ) eJ m L m L l m L l

22 2 2 2 2 1 2 2 1 2 2 2[ ] ( cos ) ( sin ) 0J m l m L l m gl

Electromagnetic torque of the dc motor is given by:

1t a t a r t a t ae

a a a a

K V K V K V K V

R R R R

Substituting the Eq. 15 into Eq. 13 yields:

2 21 2 1 1 2 1 2 2 2 2 1 2 2 2

1

[ ] ( cos ) ( sin ) e

t a t a r t a t ae

a a a a

J m L m L l m L l

K V K V K V K V

R R R R

Table.1 represents the parameters of the RIP system according to [5]:

Table 1: Parameters of the Rotary Inverted Pendulum system

Parameter value Parameter Value m1(kg) 0.83 J1(kg.m-2) 0.00208 m2(kg) 0.1 J2(kg.m-2) 0.001 l1(m) 0.300 g(m.s-2) 9.81 L1(m) 0.600 Ra 28.6 L2(m) 0.300 Kv 0.168

C1, C2 (N.m.s) 0 Kt 1.68 Va 60.0

e 0.0098

because of the structure of the system it can be linearized around zero operating point, and linear controllers such as LQR can be used. To linearize the RIP model the following approximations are made:

cos 1,sin

The linearized RIP model is as follows:

2 21 2 1 1 2 1 2 2 2 1 2 2 2

1

[ ] ( ) ( )

t a t a r t a t ae

a a a a

J m L m L l m L l

K V K V K V K V

R R R R

22 2 2 2 2 1 2 1 2 2 2[ ] ( ) ( ) 0J m l m L l m gl

Using the data in table 1 , the steady state of linearized model becomes:

(13)

(15)

(16)

(17)

(19)

(18)

(14)


26

.

1 1

.. .

1 1

.2

2.

..2

2

1

.

1

2

.

2

0 1 0 0 0

0 0.4915 14.6564 0 2.93

0 0 0 1 0

0 1.4744 93.0191 1 8.78

1 0 0 0 0

0 0 1 0 0

a

a

V

y V

Eq.20 is defined by the following equations:

1 1 2 2

( ) ( ) ( )

( ) ( ) ( ) [ , , , ]Tx t Ax t Bu t

y t Cx t Where x t

3.USING FUZZY METHOD

Fundamentals of fuzzy logic control belong to Lotfizade's theorem which is based on experimental knowledge. Fuzzy logic control strategy is very useful even for systems with complicated structures that designers can hardly implement their intended ideas easily. Taking advantage of experiment based pieces of information helps them have a simple structure. As an example if you can't linearize a system, fuzzy logic controller can be a suitable suggestion. In this part application of a Fuzzy Logic Controller to the highly nonlinear and unstable RIP system is presented. First an initial voltage is applied to the dc motor to rotate the rotary arm and make the system unbalanced, which makes the angular error larger with respect to the upright position, then it is the controller's responsibility to swing the pendulum up and stabilized it. Therefore the controller should decrease the angular error to zero. The fuzzy controller should use the real state variables ( ) that are detected by the sensors. When RIP

outputs are measured we should observe the behavior of the outputs and use the suitable control input to manage the system’s behavior properly so that the pendulum can lead to our desirable point. We used output angular position and velocity ( ) as input of FLC to diagnose how we can use the

motor decrease the angular position error to zero with rotating the arm clockwise and anticlockwise repeatedly[10]. Our FLC inputs are angular position and velocity of pendulum that contain the necessary information which is then conveyed to the rule base. The fuzzy rules are experimental rules based on experts’ previous experience about the operation of the real system. Using these commands a voltage is applied to the motor to control the pendulum.FLC output which is the voltage that directly goes into the dc motor to implement fuzzy commands. Now, the commanded voltage causes the motor to rotate the arm to achieve the control objective. In the following sentences an example of fuzzy rule base is shown.

IF INPUT1(2 ) IS NL AND INPUT.2(

2 ) IS NL THEN OUTPUT(VOLTAGE) IS NL

As an example we explain the first rule, that reveals if the angular position of pendulum (input1 or

2 ) is negative large (NL) and the angular velocity of pendulum (input2 or 2 ) is NL , that means a

large force is required to displace the pendulum to its upright position in order to decrease 2 to zero.

Therefore a voltage applied to the dc motor to rotate the rotary arm exists according to the rule base output. As it is shown in steady state model of RIP, the state variable '

1 ' or ' angular velocity of rotary

arm ' is controllable, means the motor manages to swing the pendulum up with rotating the arm

2 2, 1 1,

2 2,

(20)

(21)


27

clockwise or anticlockwise regularly. As defined the inputs of FLC are the values of 2 and

2 . The

output of FLC is the dc motor’s input voltage to rotate the arm and control the pendulum. The FLC consists of two input variables with two sets made up of 5 triangular membership functions, and 52=25 rules as shown in the following figure:

Figure 4. membership functions of input variable (

2 )or(2 )

Membership functions of input variable (2 ) are all chosen to be triangular. Now we actuate the dc

motor with a pulse with an amplitude of 60(Volt).With the above stated input the pendulum falls until 1 second which is the time of actuating then using the FLC the pendulum swings up and is set to zero degree smoothly, so the FLC controller can balance the pendulum at the upright position after a few seconds.

Figure 8. Angular position of the pendulum (

2 ) using FLC with 2 ,

2 inputs for swing up and

for stabilize the pendulum.


2 ) by using FLC with 1 ،

1 inputs for swing up and

2 , 2 for stabilize the pendulum .

Using a similar FLC, with inputs instead of and considering the control of pendulum

using relevant parameters ( ) for swinging the pendulum up, we need to use negative feedback

strategy to set the 2 near the zero and stabilize it to zero with another FLC with as its inputs.As

shown in Fig.9 the angular position of pendulum (2 ) has a faster response in comparison with the

controller with previous fuzzy controller with inputs .

3. DESIGNING LQR CONTROLLER for THE ROTARY INVERTED

In this optimal control technique we try to minimize the defined error as a cost function and the Linear Quadratic Regulator (LQR) method minimizes the cost function (J)

1 1, 2 2,

1 1,

2 2,

2 2,


28

0[ ( ) ( ) ( ) ( )]T TJ x t Qx t u t Ru t dt

considering the state dynamics as:

0( ) ( ) ( ); (0)x t Ax t Bu t x x

The weighting matrices of the states x and input u are Q and R respectively. Here the cost function is

minimized with and the input u=-kx will be found to displace suitable minimum phase poles to stabilize the RIP system immediately[9].Weighting is required for the Q matrix and is defined in a way that the weight coefficients for the angular position of the pendulum and the arm position be bigger than the other coefficients according to the design objective :

Using MATLAB we have:

Q diag 60,.05,60,.05 ;

R 1; K,S,e LQR A, B,Q, R,0 ;Anew A B*K;

K,S,e LQR A, B,Q, R,0 ;Anew A B*K; v,ev eig Anew ;

So optimal gain K and the eigen values of the new system are defined as follow as :

7.7460 5.1440 36.6942 4.6414K ;

10.243 0 0 0

0 0 2.3427 2.0421 0

0 0 0 2.3427 2.0421

EigenValues i

i

As it was expected using this controller with the coefficients K =[k1,k2,k3,k4], all poles go to the left

side of imaginary axis. In Figure (10) the response of system with a LQR controller is shown.


2 ) using LQR.

As shown in the above picture

2 is set to zero after 4 seconds, and LQR makes it stable properly.

In the following figure 2 is set to zero by using a hybrid controller. Initially the fuzzy controller with

inputs is used for swinging the pendulum up by 1.52 seconds, then the LQR controller is used to

stabilize the pendulum and at the end another fuzzy controller with inputs is used for fine tuning. 1 1,

2 2,

(22)

(23)


29


2 ) using fuzzy and LQR.

As we see in figure 12 only by using a group of fuzzy controllers we can get the best response .


2 )

Now we want to get the best response for set

2 to zero by using the proposed hybrid controller

using fuzzy method. The new fuzzy controller with inputs initiate to swing up the pendulum like a

PID controller that works based on energy then the previous fuzzy controller with inputs is

implemented to stabilize the pendulum and set 2 to zero by switching on it.

The LQR method for non zero initial condition couldn't set the pendulum to zero, but fuzzy controller doesn't have this problem. In the following diagrams the stabilization the RIP system using FLC with an initial condition is presented.


2 ) by using FLC with inputs for swing up and

for stabilize the pendulum with an initial condition (2 =1 rad).

Figure 14. Angular position of the pendulum (2 ) using a few fuzzy controller to swing up and

stabilize the pendulum with an initial condition (2 =1 rad).

1 1,

2 2,

1 1,

2 2,


30

For increasing accuracy of the fuzzy controllers after few seconds each controllers were switched to low level controllers step by step.

4. CONCLUSION

The oscillatory response of the fuzzy control method is produced because of the highly nonlinear behavior of the system. At first only one Fuzzy Logic Controller was used but as it took quite a long time for the controller to swing up the pendulum and keep it in the upright position in comparison with the linearized techniques such as LQR we decided to use the hybrid controller instead. However as you see in this paper we get the best response with the fuzzy method easily with a little creativity, implementing the variable inputs to the system. As we see despite the existing of a fuzzy controller for the high nonlinear system, it has a little time delay which shows that the proposed fuzzy controller is so effective.

5. References [1] S. Yasunobu, M. Mori” Swing up fuzzy controller for inverted pendulum based on a human

control strategy.” Pp.1621,(1997) . [2] Torres-Pomales W. and Gonzalez O.R. “Nonlinear control of swing-up inverted pendulum”

pp.259-264 ,(1997) [3] Astrom.K.J. and K. Furuta.“Swinging up a Pendulum by Energy Control” pp. 1-6, (1996).

http://www.control.utoronto.ca/~broucke/ece1653s/Intro/ast_fur96.pdf [4] Astrom, K.J. and K. Furuta. “Swinging up a pendulum by energy control” pp.287-

295,(2000) .http://www.ece.ucsb.edu/~hespanha/ece229/references/AstromFurutaAUTOM00.pdf [5] Viroch Sukontanakarn and Manukid Parnichkun “Real-Time Optimal Control for Rotary Inverted

Pendulum” pp. 1106-1115,(2009) [6] Mary, M. And Marimuthu, “Design of intelligent hybrid controller for swing-up and stabilization

of rotary inverted pendulum.” ,(2008). [7] Jyoti Krishen and Victor M.Becerra.“Efficient fuzzy Control of a Rotary Inverted

Pendulum Based on LQR Mapping “,(2006.) [8] M. Roman, E. Bobasu, D. Sendrescu ," Modelling of the Rotary Inverted Pendulum System",

IEEE Control Systems,(2010) [9] Md. Akhtaruzzaman and A. A. Shafie," Modeling and Control of a Rotary Inverted Pendulum

Using Various Methods, Comparative Assessment and Result Analysis"Proceedings of the 2010 IEEE International Conference on Mechatronics and Automation August 4-7, 2010, Xi'an, China

[10] Necdet Sinan Ozbek, Mehmet Onder Efe" Swing up and Stabilization Control Experiment for aRotary Inverted Pendulum- An Educational Comparison"IEEE International Conference on Systems Man and Cybernetics (SMC), 2010


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Stabilization of Rotary Inverted Pendulum Using … of Rotary Inverted Pendulum Using Fuzzy Logic...

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