fuzzy, neural and neuro-fuzzy controllers for an inverted pendulum

Intituto Tecnológico y de Estudios Superiores de Monterrey, Intelligent Controll, Fall 2011, Monterrey, N.L. Mexico

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Abstract—This paper presents the design of a fuzzy logic,

artificial neural network (ANN) and neural-fuzzy (ANFIS)

controllers for the inverted pendulum-cart system. The dynamic

nonlinear equations are used to model the system. Fuzzy logic

controller is one of the most important applications using the

expert knowledge to control the system. Artificial Neural

Networks has similar properties to Fuzzy systems but it includes a

learning stage that allows to combine both strategies to implement

a Neuro-Fuzzy controller which results in the optimization of

fuzzy logic and ANN behavior. The simulation results of each

controller are compared.

Index Terms—Fuzzy Control; Artificial Neural Control;

ANFIS Control; Inverted Pendulum.

I. INTRODUCTION

HE inverted pendulum is a multivariable and nonlinear

system that is commonly used to test control strategies.

The inverted pendulum may simulate natural phenomena such

as the walking motion, missile guidance, biomechanical

devices, etc.

This paper contains the comparison of Fuzzy logic, artificial

neural network and a synergy of both strategies responses in

the control of the inverted pendulum using the non-linear

equations. The approach of Passino [1] is used for the fuzzy

rule database and then an ANN is trained with the Fuzzy

Controller dynamics to emulate it and control the pendulum

system. Finally a neuro-fuzzy controller is implemented.

II. MODELING THE SYSTEM

A. Non-Linear Dynamics.

The diagram of the inverted pendulum is shown in Fig. 1.

The dynamic non-linear equations of the inverted pendulum

system are as follows:

umLmLxbxmM sincos 2 (1)

cossin2 xmLmgLImL (2)

Where M is the mass of cart, m is the mass of the pendulum,

b is the friction of the cart resisting motion, L is the length of

the pendulum to its center of mass, I is the inertia of the

pendulum, u(t) is the force applied to the cart, x represents the

cart position coordinate and

is the angle of the pendulum

measured from the vertical.

Fig.1. Diagram of the inverted pendulum-cart system.

Using the non-linear equations the system uses the

parameters of Table I.

TABLE I

INVERTED PENDULUM PARAMETERS.

Parameters Value

M 0.5 kg

m 0.2 kg

b 0.1 N/(m/s)

L 0.3 m

I 0.006 kg

m 2

g 9.8 m/

s2

The simulation of the controller was performed in Simulink

MATLAB, this software allows to implement the model of the

system using the non-linear dynamics equations in the form of

block diagrams using a subsystem function. After substituting

the parameters of the Table I into the equations, the resulting

system is shown in Fig. 2.

Fuzzy, Neural and Neuro-Fuzzy Controllers for

an Inverted Pendulum

Esteban Salazar (353869), Ernesto León (808770)

T


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Fig.2. Inverted pendulum non-linear system.

This block represents the plant that the control system will

perform the manipulation.

B. Control Strategies

The strategies used in this paper to control the inverted

pendulum-cart system are Fuzzy logic, ANN and ANFIS.

The Fuzzy controller is designed to automate how a human

expert who is successful at this task would control the system.

First, the information of the expert is collected in linguistic

descriptions at the form of if… then rules. The statements that

quantifies the configuration of the inverted pendulum proposed

by Passino [1] are: (i) the statement “error is positive large”

can represent the situation where the pendulum is at a

significant angle to the left of the vertical, (ii) the statement

“error is negative small” can represent the situation where the

pendulum is just slightly to the right of the vertical but not

close to the vertical to justify quantifying it as zero and far

from negative large and (iii) the statement “error is zero”

represents the situation where the pendulum is near the vertical

position. For the change in the error the negative sentence

means that the pendulum direction is closing to the vertical

position and for a positive change in error represents that the

pendulum is moving away the vertical position.

Once the fuzzy controller is properly tuned, an artificial

neural network is trained to simulate the fuzzy controller

behavior using the back-propagation algorithm in MATLAB.

III. FUZZY LOGIC CONTROLLER

The fuzzy logic controller is known as knowledge based

system [2], where the knowledge of an expert about a system

behavior is pass on the form of if… then rules to a data base of

fuzzy rules that determines the manipulation of the fuzzy

controller to the plant.

The inverted pendulum system has two variables for

controlling: (i) the angle of the pendulum with respect of the

vertical position and (ii) the position of the cart. The cart

where the pendulum is mounting, it moves in straight direction

to compensate the changes in the angle variations. From [1],

the fuzzy rules for the error and the change of the error are in

the form of fuzzy sets that describes the relation of the actual

position with the desired. The fuzzy sets that describes the

membership functions of the error and the change in error for

the theta angle of the pendulum are the same form for both and

is shown in Fig. 3, where the manipulation fuzzy sets of the

controller is shown in Fig. 4.

Fig.3. Fuzzy sets for error and change in error

for theta.

Fig.4. Fuzzy sets for controller manipulation.

From Fig. 3 it is possible to observe that fuzzy sets for error

and chance in error is the range [-3 3] and from Fig. 4 the

range is [-5 5].

For the cart position controller the manipulation fuzzy sets

are the same from Fig. 4 meanwhile the error and the change in

error is shown in Fig. 5, where the range of values is [-10 10].

Fig.5. Fuzzy sets for error and change in error for

cart position.


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Using the approach from section II.B and the fuzzy sets

from above the rule bases can be defined. For the pendulum

problem there are two inputs (Error, change in error) and

five linguistic values (Negative Large, Negative Small,

Zero, Positive Small, Positive Large) for each case of these,

there are at most 52

= 25 possible rules. The rule table for

the inverted pendulum for controlling the angle and the cart-

position is shown in Table II.

TABLE II

THETA AND CART POSITION RULE DATABASE

de

e NL NS Z PS PL

NL PL PL PL PS Z

NS PL PL PS Z NS

Z PL PS Z NS NL

PS PS Z NS NL NL

PL Z NS NL NL NL

The fuzzy controller for the angle of the pendulum and the

cart position are implemented using a Simulink block, where

the designed controllers are embedded. The simulation process

is shown in Fig. 6, the block diagram consist in a desired

position using a constant, then calculation of the error and

change in error that are the inputs for the fuzzy controller, next

the manipulation for the plant or the subsystem and finally the

actual position is provided for the feedback.

Fig.6. Block diagram of the controller simulation.

To test the performance of the system two disturbances are

added to the system, a change in the desired position from 0 to

3 meters of the cart (see Fig. 14) and a disturbance of 10

degrees in the angle of the pendulum (see Fig 17).

IV. NEURAL NETWORK CONTROLLER

Controlling a dynamical system means forcing it to behave

in some desired way. Once the desired goal is specified, the

design of the controller is dictated by the knowledge about the

plant and the data available [3]. A neural network can learn

from the dynamics of other controllers by adding a training

process to ANN architecture.

The training process of the ANN in this paper used the data

of the manipulation or output of the Fuzzy Controller and the

two input vectors that correspond to the error and change in

error, to the train the ANN. The objective of the training is that

the ANN has to adjust the bias and its weights using the back-

propagation algorithm and the MSE (Mean Square Error)

method to calculate the error each iteration of the regression

process. It can be observed that the ANN is able to

approximate the behavior of the manipulation signal of the

Fuzzy Controller successfully (see Fig. 7).

Fig.7. Output data of the Fuzzy Controller versus the

learning ANN response.

The ANN controller is then incorporated to the block

diagram of the Fig. 6 to simulate the system response. Then

the response of the system to a change in the desired position

and angle disturbances is shown in Fig. 15 and Fig. 18

respectively.

V. ANFIS CONTROLLER

A. Basic Principles of ANFIS

An alternative of controlling is the ANFIS (Adaptive

Network Fuzzy Inference System) Controller that uses a

routine for Sugeno-type fuzzy inference systems with a hybrid

learning algorithm to identify parameters of the Sugeno-type

fuzzy inferences. The hybrid method is a combination of the

least-square method and the back-propagation gradient descent

method for training the FIS membership function parameters

to emulate a given training data set [4].

B. Training the ANFIS Controller

The training process of the ANFIS controller was performed

in the ANFIS MATLAB Toolbox. The training data (see Fig

8) is a matrix that consists in the Error, change in error and the

output of the fuzzy controller, using this data the toolbox

generates a Sugeno-type fuzzy inference system that fixes the

parameters of the fuzzy sets with an hybrid optimization

algorithm. A value of desired minimum error of 0.01 is set

before the training process.

The learning process finishes if the number of epochs


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selected for trained or while the desired error are reached (Fig.

9) the toolbox adjusts the Sugeno-type fuzzy inference system

with two inputs for the error and change in error and the

manipulation that consist in Sugeno functions. The ANFIS

controller training for the angle of the pendulum inputs are

shown in Fig.10 and Fig. 11 respectively.

Fig.8. ANFIS controller training data.

Fig. 9. ANFIS controller successfully training.

In Fig. 9 it is possible to see that the ANFIS training reaches

the desired error in a less number of epochs than ANN

learning.

Fig.10. Sugeno fuzzy sets resulting from ANFIS

Theta controller training for Error.

Fig.11. Sugeno fuzzy sets resulting from ANFIS training

for the change in Error.

Then for the cart position controller the process is the same

and the resulting fuzzy sets for Error and change in error is

shown in Fig. 12 and Fig. 13 respectively.

Fig.12. Sugeno fuzzy sets resulting from ANFIS cart position

controller training for the Error.

Fig.13. Sugeno fuzzy sets resulting from ANFIS cart position

controller training for the change in Error.

The ANFIS controller is then incorporated to the block

diagram of the Fig. 6 to simulate the system response. Then

the response of the system to a change in the desired position

and angle disturbances is shown in Fig. 16 and Fig. 19

respectively.

VI. SIMULATION RESULTS

The structure of the inverted pendulum system by Simulink

is shown in Fig.6, applying the dynamic model of the inverted

pendulum using the non-linear equation shown in Fig.2 and the

parameters given in Table I.

The simulations consist in two tests: (i) first a change in

reference is added to the desired of the cart position and the

(ii) second is a disturbance in the angle of the pendulum and

the cart reference position remains in a value of zero.


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Fig.14. Fuzzy controller response to a change in the desired

reference where the solid line represents the theta position and

the dotted represents the cart position.

Fig.15. ANN controller response to a change in the desired



Fig.16. ANFIS controller response to a change in the desired



The test of changes in reference of Fig.14,15,16 shows that

the ANN and Fuzzy controllers has similar responses, but the

Fuzzy has lower oscillation levels in the manipulation of the

cart position than the Fuzzy controller, meanwhile the ANFIS

manipulation of the cart position has no overshooting.

Furthermore, Fuzzy and ANN controlled cart position in

similar time, which is around 15 to 16, meanwhile the ANFIS

controlled in lower time, which is a value around 4 to 5.

Fig.17. Fuzzy controller response to a disturbance in the

pendulum where the solid line represents the theta position

and the dotted represents the cart position.

Fig.18. ANN controller response to a disturbance in the



Fig.19. Fuzzy controller response to a disturbance in the




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The test of disturbances in the angle of the inverted

pendulum of Fig.17,18,19 shows that the ANN and Fuzzy

controller has similar responses in the angle theta overshoot

value, which are certain of 2 and -2, meanwhile the ANFIS

controller has higher overshooting in the range of 10 and -10,

but this changes in the angle allows the ANFIS to control in

lower time than Fuzzy and ANN. The ANN and Fuzzy

controlled in time equal to 30 and the ANFIS controlled

approximately at time 15.

VII. CONCLUSION

This work presents the design of a Fuzzy, Neural and

ANFIS controllers for an inverted pendulum-cart system. A

non-linear model of the system has been considered to develop

the controllers. The response of the system with each

controller has been compared. These responses are shown in

section VI. From the simulation results, it is possible to

conclude that the three controllers can successfully stabilize

the inverted pendulum and also controls the cart at the desired

positions.

From the experimentation it is possible to conclude that the

ANFIS has the best performance due to the lower rate of

changes in the angle of the pendulum and the time that the

controller requires to stabilize both systems is smaller than the

Fuzzy and the ANN controllers.

Then it is possible to conclude that the ANFIS training

methodology obtain the best characteristics of both controllers

(Fuzzy and ANN) to optimize the system performance.

REFERENCES

[1] Passino K. M, “Fuzzy Control,” Menlo Park, Calif.:Addison-Wesley, pp

26-33, c1998.

[2] Driankov D.,”An introduction to fuzzy control,” Berlin: New York:

Springer, pp. 23-40, c1996.

[3] Hung T. Nguyen, “A first course in fuzzy and neural control,” Boca

Raton, FL: Chapman & Hall/CRC Press, c2003.

[4] Mathworks, “Advanced Fuzzy Inference Techniques: ANFIS,” [Online]

Matlab Documents [Access on date Dec 1, 2011].

http://www.mathworks.com/help/toolbox/fuzzy/anfis.html