Fuzzy logic control vs. conventional pid

International Conference on Intelligent and Advanced Systems 2007
Fuzzy Logic Control vs. Conventional PID Control
of an Inverted Pendulum Robot
M. I. H. Nour, J. Ooi, , and K. Y. Chan
University of Nottingham Malaysia Campus,
Jln Broga, 43500, Semenyih, Malaysia
mutasim.nour@nottingham.edu.my, jesamine_85@hotmail.com
Abstract - This paper addresses some of the potential benefits of
using fuzzy logic controllers to control an inverted pendulum
system. The stages of the development of a fuzzy logic controller
using a four input Takagi-Sugeno fuzzy model were presented.
The main idea of this paper is to implement and optimize fuzzy
logic control algorithms in order to balance the inverted
pendulum and at the same time reducing the computational time
of the controller. In this work, the inverted pendulum system was
modeled and constructed using Simulink and the performance of
the proposed fuzzy logic controller is compared to the more
commonly used PID controller through simulations using Matlab.
Simulation results show that the Fuzzy Logic Controllers are far
more superior compared to PID controllers in terms of overshoot,
settling time and response to parameter changes.
I. INTRODUCTION
It has been traditional for roboticists to mimic the human
body. The human body is so perfect in many ways that it
seems like a long way before a robot will ever get close to
exactly representing a human body.
One of the less thought about issues in robotics is the issue
of balance, which can be appropriately represented by the
balancing act of an inverted pendulum. This explains the fact
that although many investigations have been carried out on the
inverted pendulum problem [1]-[10], researchers are still
constantly experimenting and building it as the inverted
pendulum is a stepping stone to greater balancing control
systems such as balancing robots.
Therefore, in order to control the balancing act of the
inverted pendulum, a control system is needed. As known,
fuzzy logic control systems model the human decision making
process based on rules and have become popular elements as
they are inexpensive to implement, able to solve complicated
nonlinear control problems and display robust behavior
compared to the more commonly used conventional PID
control systems [1, 2, 11].
In general, there are numerous and various control problems
such as balancing control systems which involve phenomena
that are not amenable to simple mathematical modeling. As
known, conventional control system which relies on the
mathematical model of the underlying system has been
successfully implemented to various simple and non-linear
control systems. However, it has not been widely used with
complicated, non-linear and time varying systems [3, 4, 5, 11].
On the other hand, fuzzy logic is a powerful and excellent
analytical method with numerous applications in embedded
control and information processing. Fuzzy provides a
straightforward and easy path to describe or illustrate specific
outcomes or conclusions from vague, ambiguous or imprecise
information [1].
Review on existing conventional and fuzzy logic techniques
has highlight the significance and importance of control
systems. Researchers have proven that fuzzy logic control
systems are able to overcome nonlinear control problems
which may not be solved easily using conventional methods
and the delicate process in designing a fuzzy logic controller
that is able to mimic the human experience and knowledge in
controlling a system. Therefore, it will be interesting to show
that fuzzy logic controllers are able to control many of these
problems without having a clear understanding of the
underlying phenomena and are more favorable and superior
compared to conventional PID controllers.
This paper presents the systematic design of a fuzzy logic
controller using the Takagi-Sugeno model for a car-pendulum
mechanical system, well-known as the inverted pendulum
problem. Here, the inverted pendulum and control system is
first modeled before putting them into simulations using
Matlab where the control system is further tuned to increase its
performance. The control system is then further optimized in
order to reduce the computational time of the system by
reducing the number of rule bases. This is followed by the
implementation and comparison of the PID and fuzzy
controllers through simulations.
II. INVERTED PENDULUM MODEL
Fig. 1 shows the block diagram of an inverted pendulum
system with a feedback fuzzy control block. The output of the
plant ( xx,,,θθ ) is fed into the controller to produce the
subsequent force to balance the pendulum to its upright
position and at the same time maintaining the cart initial
position.
The inverted pendulum system consists of a moving cart and
a pivoted bar that is free to oscillate in the x-y plane. However,
the cart is constrained to move only in the x-plane as shown in
Fig. 2. In Fig. 2, mc is the mass of the cart, mp is the mass of the
pendulum, μ is the coefficient of friction, g is the acceleration
of gravity and I is the moment of inertia of the pendulum about
the pivot.
From Fig. 2 and the pendulum’s free body diagram, the state
equations in terms of the control force, F can be expressed as,

Fig. 1. Block diagram of inverted pendulum system with feedback fuzzy logic
controller.
Fig. 2. Inverted pendulum system.
θ
θθθθ
2
2
cos)(
3
4
cossinsin
3
4
3
4
lmmm
gmlmF
x
ppc
pp
−+
−−
= (1)
θ
θθθθθ
θ 2
2
cos)(
3
4
cossincossin)(
lmlmm
lmFgmm
ppc
ppc
−+
+−+
= (2)
where length, L =2l and I = 4/3 mp l 2
. θ is the falling angle,
θ is the angular velocity, θ is the angular acceleration of the
pendulum. x is the acceleration of the cart.
Since the control force, F in terms of the motor voltage, V
can be expressed as [6],
x
Rr
KK
V
Rr
KK
F
gmgm
2
22
−= (3)
where x is the velocity of the cart, Km is the motor torque
constant, Kg is the gearbox ratio, R is the motor armature
resistance and r is the motor pinion radius, the state equations
for the inverted pendulum in terms of the motor voltage, V has
been derived as,
ppc
p
gmgm
mmm
gmx
Rr
KK
V
Rr
KK
x
−+
−−
=
)(
3
4
)(
3
4
2
22
θ
(4)
lmlmm
x
Rr
KK
V
Rr
KK
gmm
ppc
gmgm
pc
−+
+−+
=
)(
3
4
)( 2
22
θ
θ (5)
The model of the inverted pendulum is then created using
Simulink. The input to the plant is the disturbance force
imparted to the cart. The current angle and angular velocity is
fed back to the system to calculate the angular acceleration and
acceleration of the cart and pendulum respectively. The
angular acceleration and the acceleration are then integrated to
obtain the outputs of angle, angular acceleration, position and
velocity.
As a whole, the balancing algorithm (controller) measures
two outputs from the plant (Inverted Pendulum) and calculates
the torque forces, F needed for balance. Fig. 3 shows how the
forces, F are determined from the angle, angular velocity,
position and velocity measured from their respective sensors.
Fig. 3 shows that the motor shaft encoder measures the
position of the cart’s wheel while the angle sensor measures the
tilt angle of the pendulum. The angular velocity and the
velocity of the cart are then derived respectively from the
measured tilt angle and the cart’s position. The four triangles
K1, K2, K3 and K4 are the “knobs” that apply gain to the four
feedback signals. They are summed together and fed back to
the system as the PWM motor voltage to drive the cart. This
can be expressed as,
+×+×= ),(),( 21 KocityAngularVelKAngleF θθ
),(),( 43 KxVelocityKxPosition ×+× (6)
The controller input gains, K1, K2, K3 and K4 are determined
using the Linear-Quadratic Regulator (LQR) method described
by Friedland [11]. This method finds the optimal K based on
the state feedback law and the state-space equation derived
earlier. It is found that the input gains, K1, K2, K3 and K4
respectively are approximately 40, 10, 3 and 4.
III. FUZZY LOGIC SYSTEM
Fuzzy controllers are very simple. They consist of an input
stage, a processing stage and an output stage. The input or
fuzzification stage maps sensor or other inputs to the
appropriate membership functions and truth values. The
processing or the rule evaluation stage invokes each
appropriate rule and generates a result for each, then combines
the results of the rules.
The output or the defuzzification stage then converts the
combined result back into a specific control output value using
the Centroid Method.
Fig. 4 shows the fuzzy inference process as discussed above
while Fig. 5 shows the shape and range of the membership
functions for the input angle, angular velocity, position and
velocity respectively.

Fig. 3. Plant and controller block diagram.
The input variables NL, NS, Z, PS and PL for the inputs
angle and angular velocity represent the membership functions
of Negative Large, Negative Small, Zero, Positive Small and
Positive Large respectively. While the input variables N, Z and
P for the inputs position and velocity represent the membership
functions Negative, Zero and Positive respectively.
Fig. 4. Fuzzy control process.
For this particular problem, an output window that consists
of 13 fuzzy singletons is used. Each fuzzy singleton is a linear
function that defines the output of the system as given in (7).
Output force, F = 54321 KxKxKKK ++++ θθ (7)
where K5 is a constant.
Fig. 6 shows the membership functions of the output force
while Table 1 represents the function of each membership
function defined in Fig. 6. The variables N1 to N6 represent a
negative force while variables P8 to P13 represent a positive
force and Z7 represents zero force
Values of gains K1 to K5 in Table 1 are obtained through
tuning of the input and output membership functions based on
the assumptions as follows:
• Output force is non linear. At larger angle, output
force is larger.
• Influence of inputθ towards output > Influence of θ
> Influence of x and x .
• Constant K5 is required for the fine tuning of the
output.
• A lower overshoot response and shorter settling time
is desired.
Fig. 5. Membership functions of inputs.
Fig. 6. Output membership functions.
Table 2 on the other hand defines the relationship between
the input variables and the output variable which is the required
force to balance the pendulum.
The rule base is constructed based on the assumptions as
follows. Considering in terms of the inputs angle or angular
velocity,
• The larger the –ve input, the larger the –ve force.
• The larger the +ve input, the larger the +ve force.
0.2
0.4
0.6
0.8
1
-3 -2 -1 0 1 2 3
N P
Membership Functions of Input Position and Velocity
Position (m), Velocity (m/s)
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
NL NS
Z
PS PL
Membership Functions of Input Angular Velocity
Angular Velocity (rad/s)
0.2
0.4
0.6
0.8
1
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
NL NS Z PS PL
Membership Functions of Input Angle
Angle (radians)

• At zero input, the force depends on the magnitude of
the other three inputs.
TABLE 1
OUTPUT FORCE & INPUT GAINS
TABLE 2
RULE BASE
Considering in terms of the inputs position or velocity,
• The larger the +ve input, the larger the –ve force.
• The larger the –ve input, the larger the +ve force.
• At zero input, the force depends on the magnitude if
the other three inputs.
Therefore, by assigning the weighting values of 1 to 5
respectively to NL, NS, Z, PS, and PL for inputs angle and
angular velocity and weighting values of 1 to 3 respectively to
N, Z and P for inputs position and velocity, the rule base is
determined using the formula,
)3()3()1( +−++−+−+= xxF θθ (8)
where F denotes the values 1 to 13 as represented in the output
membership functions in Fig. 6 while xx,,,θθ denotes the
weighting values of the respective inputs corresponding to the
membership functions.
Equation (8) is derived in such a way that:
• The addition of iθ and iθ as well as the subtraction of
ix and ix in the calculation of iF correspond to the
direction of the inputs with respect to the direction of
F as explained above.
• The subtraction of the value 1 from iθ shows that the
influence of iθ is greater than the influence of iθ
towards the required force.
• The addition of the value 3 to ix and ix is necessary
based on how the weighting values are assigned.
The weighting values area signed based (7) in such a way
that:
• The higher the weighting values of inputs angle and
angular velocity, the larger the force in the positive
direction.
• The higher the weighting values of inputs position and
velocity, the larger the force in the negative direction.
For example, given the conditions as follow:
• The falling angle of the pendulum is large in the
positive direction.
• The pendulum is falling slowly (small angular
velocity) in the negative direction.
• The cart is moving in the negative direction at the
negative position.
∴ )31()31()12(5 +−++−+−+=iF
10= (Refer to Table 2)
IV. SIMULATION
The inverted pendulum and fuzzy controller systems
modeled are then implemented and simulated in the Matlab
environment using Simulink and the Fuzzy Logic Toolbox.
The fuzzy logic controller modeled was first implemented
using the graphical user interface in the fuzzy logic toolbox.
As the fuzzy logic toolbox is designed to work flawlessly with
Simulink, the system is then embedded directly into simulation.
Fig. 7 shows the fuzzy logic control system of the inverted
pendulum system. In this system, a step response is used as the
reference position. The system works base on the concept that
at a smaller step reference, the force required to move the
pendulum to that reference position is smaller and vice versa.
This force is known as the disturbance force or the reference
force as mentioned earlier which will be imparted to the
inverted pendulum system.
Due to this disturbance force, the pendulum will be
displaced from its balance state. The fuzzy logic controller at
the feedback path will then try to balance the pendulum by
evaluating the current state of the angle and the angular
velocity of the pendulum and the current position and the
velocity of the cart to produce the appropriate force and
direction to balance the pendulum back to its upright position.

The fuzzy control system here was first implemented, tuned
and optimized using the fuzzy logic toolbox before being
implemented into the fuzzy logic controller.
The inverted pendulum plant on the other hand consists of
several masks layer that defines the whole inverted pendulum
system. Due to this, the physical specifications that define the
pole length, pole mass, cart mass and the acceleration due to
gravity, g can be modified to test the system and controller’s
performance at different conditions.
In order to reduce the system’s computational time, the 225
rule based designed previously are optimized and reduced to 16
rule base. The optimization process is simplified as the
Sugeno-type inference is used in this fuzzy logic controller.
The optimization process involves the reduction of the
number of membership functions of all the inputs to two
membership functions and also the reduction of the 225 rules to
16 only as discussed earlier. Table 3 shows the newly
optimized rule base.
TABLE 3
OPTIMIZED RULE BASE
Fig. 7. Simulink model of inverted pendulum problem with fuzzy logic control.
V. RESULTS AND DISCUSSION
This section discusses the simulation development results
and compares the fuzzy logic controller performance at
different physical conditions. The optimized simulation results
are also compared with the previous results to show that there
is only a very slight degradation of performance in the
controller when the rule base of the system is reduced and
tuned to reduce the computational time of the controller.
Besides that, the sensors are also tested to show the
relationship between the falling angle of the pendulum and the
speed of the motors.
A. Simulation Results
The Simulink model in Fig. 8 is simulated with relative
tolerance of 0.001s. Fig. 8 and 9 present the simulation for the
inputs angle and position and the output control force with pole
mass of 0.1kg, cart mass of 1.0kg, pole length of 1.0m, input
step reference of 1.0 and acceleration due to gravity of 9.8ms-2
.
Due to the step response, the actual simulation of the system
only starts at t=1s. In other words, the simulation of the system
only starts when the disturbance force is applied to the system.
Fig. 9 shows that the cart is able to reach its desired position
in about 6s while balancing the pendulum. The desired
position here represents the reference position of the system
which is 1 step in this case.
Fig. 8 on the other hand shows that with a step response of 1,
a disturbance force of about 3N in the opposite direction is
applied to the system. Therefore, in order for the cart to reach
its desired or reference position in the positive direction and at
the same time balancing the pendulum at the shortest time, a
negative disturbance force is applied to the system. When a
negative force is applied, the pendulum will be displaced to the
positive direction due to the inertia of the pendulum.
Due to this, the fuzzy logic controller will apply the
appropriate force in the positive direction to balance the
pendulum. This explains the force and the angle response at
the start of the simulation and the very small negative
displacement of the cart’s position as shown in Fig. 8. The
response also shows that the falling angle of the pendulum
requires 6.5s to return to its upright position of zero angle.
Fig. 10 shows the simulation of the optimized system. The
only difference between the previous and the optimized system
is that the optimized system takes an extra time of 0.5s to
balance the pole to its upright position.
B. Fuzzy Control vs. Conventional PID Control
Fig. 11 and Fig. 12 show two sets of results comparing the
application of fuzzy control and conventional control (PID
controller) techniques to the inverted pendulum problem
simulation. For the same system parameters here, the PID
controller proportional gain, Kp, derivative gain, Kd and
integral gain, Ki are found to be 9, 14, and 0.06 respectively.
The first two graph show that the fuzzy logic controller gives a
smaller overshoot and shorter settling time. In the second set,
the mass of the cart is changed without modifying the
controllers. Fig. 14 shows that the conventional controller
totally failed to balance the pendulum as it was designed for
the nominal value of cart mass. On the other hand, the fuzzy
logic controller exhibited small performance degradation due
to this parameter change as shown in Fig. 13. This proves that
fuzzy logic is not based on the mathematical model of the
inverted pendulum and more robust to mass variations.
Fig. 8. Falling angle versus force imparted response.

Fig. 9. Desired position versus cart position response.
Fig. 10. Optimized falling angle versus force imparted response.
Fig. 11. Falling angle response for fuzzy control.
Fig. 12. Falling angle response for conventional PID control.
Fig. 13. Falling angle response for fuzzy control (Mass changed).
Fig. 14. Falling angle response for conventional PID control (Mass changed).
VI. CONCLUSIONS
In this paper, an optimized fuzzy logic controller has been
implemented in the Matlab environment, using the Fuzzy
Logic Toolbox and Simulink. It has been used to control an
inverted pendulum system. The study has identified some of
the potential benefit of using fuzzy logic controllers. In
comparison with the modern control theory, fuzzy logic is
simpler to implement as it eliminates the complicated
mathematical modeling process and uses a set of control rules
instead. The achieved results showed that proposed fuzzy logic
controller is more robust to parameter variations when
compared to the PID controller.
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Fuzzy logic control vs. conventional pid

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