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This paper investigates the efficacy of an optimized fuzzy logic controller for real-time swing-up control and stabilization to a rigidly coupled twin-arm inverted pendulum system. The proposed fuzzy controller utilizes Lyapunov criteria for controller design to ensure system stability. The membership functions are further optimized based on the entropy function. The controller design is based on the black-box approach, eliminating the need for an accurate mathematical model of the system. The experimental results shows an improvement in the transient and steady-state response of the controlled system as compared to other state-of-the-art controllers. The proposed controller exhibits a small settling time of 4.0 s and reaches the stable swing-up position within 5 oscillations. Various error indices are evaluated that validates an overall improvement in the performance of the system.
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Received January 27, 2021, accepted February 8, 2021, date of publication February 11, 2021, date of current version April 7, 2021.
Digital Object Identifier 10.1109/ACCESS.2021.3058645
Real-Time Swing-Up Control of Non-Linear
Inverted Pendulum Using Lyapunov Based
Optimized Fuzzy Logic Control
BRIAN AZZOPARDI 2, (Senior Member, IEEE),
1Department of Electrical and Electronics Engineering, University of Petroleum and Energy Studies, Dehradun 248007, India
2MCAST Energy Research Group, Institute of Engineering and Transport, Malta College of Arts, Science and Technology, 9032 Paola, Malta.
Corresponding author: Abhinav Sharma (
This work was supported in part by the European Commission H2020 TWINNING Networking for Excellence in Electric Mobility
Operations (NEEMO) Project under Grant 857484.
ABSTRACT This paper investigates the efficacy of an optimized fuzzy logic controller for real-time
swing-up control and stabilization to a rigidly coupled twin-arm inverted pendulum system. The proposed
fuzzy controller utilizes Lyapunov criteria for controller design to ensure system stability. The membership
functions are further optimized based on the entropy function. The controller design is based on the black-box
approach, eliminating the need for an accurate mathematical model of the system. The experimental results
shows an improvement in the transient and steady-state response of the controlled system as compared to
other state-of-the-art controllers. The proposed controller exhibits a small settling time of 4.0 s and reaches
the stable swing-up position within 5 oscillations. Various error indices are evaluated that validates an overall
improvement in the performance of the system.
INDEX TERMS Fuzzy entropy, real-time control, twin-arm inverted pendulum, fuzzy membership function
Inverted pendulum (IP) system has always attracted control
system engineers due to its wide range of applications. The
inverted pendulum finds direct application in segway and the
extended system is also applicable in designing and modeling
complex systems like, bipedal walking, robotic manipulator
systems, missile control among many others [1], [2]. Being
a non-linear underactuated system, the control of inverted
pendulum is typically considered as a benchmark to test the
efficacy of new control algorithms [3]. Researchers have
applied several control strategies for the control and stabi-
lization of the inverted pendulum system. In [4], the authors
developed a feedback linearization control to stabilize the
inverted pendulum system. The authors added adaptive fuzzy
control to ensure asymptotic stability. The control system
was applied on a real-time cart-position tracking by keeping
The associate editor coordinating the review of this manuscript and
approving it for publication was Xiaojie Su.
the pendulum angle at its equilibrium position. In [5], the
authors proposed a fuzzy logic controller for swing up con-
trol of a real-time pendulum. The authors designed a fuzzy
separate fuzzy controller for cart position control and the
pendulum angle stabilization was achieved in 10 seconds.
In [6], the authors proposed an optimized fuzzy controller
for an inverted pendulum system based on the minimization
of an objective function which is dependent on the mean
square error. The Gaussian membership function for the
fuzzy controller was optimized using an objective function
defined with the help of mean square error. The developed
controller is used to track the pendulum angle trajectory.
In [7], authors proposed a Takagi-Sugeno based fuzzy logic
controller for swing-up control of inverted pendulum. The
rule base for the controller is designed using Lyapunov’s
direct method, which ensures the stability of the system.
In [8], the authors proposed an artificial neural network based
controller to stabilize an inverted pendulum for a segway.
The authors developed the controller of this mobile inverted
VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see 50715
A. Jain et al.: Real-Time Swing-Up Control of Non-Linear IP
pendulum by using radial basis function to independently
determine the gains of the PID controller for both pendulum
angle and position control. The real-time implementation of
the controller on the mobile inverted pendulum showed the
movement of the platform while maintaining the pendulum
at an erected position. In [9], the authors used cascaded fuzzy
controllers for a flexible joint robot manipulator. The cascade
controller employs 3 fuzzy controllers out of which 2 con-
trollers are used to control the motor rotation, and deflection
angle. The output of these two fuzzy controllers is then fed to
the third controller which produces the final control signal.
The cascaded controller approach results in the reduction
of link vibrations and achieves faster tracking and reduces
the settling time. In [10], authors develoepd a vision based
feedback controller for stabilization of inverted pendulum.
The authors deployed vision based pendulum angle measure-
ment for feedback in place of encoders which are used in
a majority of applications that require angle measurement.
The vision based feedback control loop had a maximum time
delay of about 35 ms, and a resolution of 0.1o. The proposed
controller demonstrated a satisfactory stabilization and a high
disturbance rejection with an accuracy of ±0.2. In [11],
authors designed a linear quadratic regulator for swing-up
stabilization of a real-time inverted pendulum system. The
authors used the Lagrangian method for parameter identifi-
cation and the total energy at the upright position was forced
to zero, thereby obtaining the optimal control signal. In [12],
authors developed a swing up and stabilization controller for
a real-time rotatory inverted pendulum system. The control
action is obtained by switching the control objective between
two separately designed controllers for swing-up and sta-
bilization action, respectively.. In [13], authors deployed a
hybrid self-tuning Fuzzy based adaptive PID controller for
swing-up control for a real-time inverted pendulum system.
The authors designed two separate adaptive PID controllers
to control the cart position and to provide stabilization of
the pendulum angle.. In [14], the authors developed a fuzzy
based virtual model control (VMC) for stabilization of the
pendulum angle under parametric uncertainty. The proposed
controller is essentially divided into three steps: (a) imagine
and attach virtual components to the system followed by, (b)
obtain the virtual forces and torques and finally (c) feeding
these values to the real system to realize the virtual forces and
torques. The performance of VMC controller was found out
to be superior as compared to the linear quadratic regulator.
In [15], the authors developed a fuzzy-based linear quadratic
regulator to control a double link rotatory inverted pendulum.
The authors employed the Mamdani type fuzzy model to
adjust the linear state feedback controller gains. The con-
troller gain matrix was further optimized by adding Kalman
filter. In [16], authors developed a fuzzy controller based on
a guaranteed cost control objective function for swing up
control of the inverted pendulum system. The controller is
built around the linearized model of the inverted pendulum
system. The stability of the controller is analyzed using the
Lyapunov method. The proposed cost controller ensured the
stability of the system and aids in disturbance rejection..
In [17], the authors developed a self-tuning linear-quadratic
regulator for swing-up control and stabilization of an inverted
pendulum system. The authors developed a cognitive model
of the inverted pendulum system based on which the actuator
dynamics and the controller has been designed.
The fuzzy logic system is usually used to design controllers
for non-linear systems due to its inherent characteristic to
handle the system without a need for an accurate math-
ematical model [18], [19]. Fuzzy logic systems are also
advantageous where the control signal is to be generated in
presence of vague/noisy measurement data [20]. Fuzzy-logic
based systems are beneficial in a wide range of applica-
tions which has already been tested and proved by multiple
sources [21]–[24]. The selection of the correct membership
function has been amongst the most researched area for
optimizing the performance of fuzzy controllers. In [25],
the authors used s-function for defining membership func-
tion (MF) and maximizing fuzzy entropy corresponding to
the MF. The performance of the developed method had been
evaluated for image processing applications. In [26], the
authors proposed the tuning of the gains of a fuzzy type PID
controller by applying particle swarm optimization (PSO)
technique. The proposed algorithm was implemented to con-
trol an industrial DC drive. The simulation and experimental
results indicates an improved performance and robustness of
the controller. In [27], the authors proposed the optimiza-
tion of a fuzzy system using cross-mutated operation using
PSO. The robust performance of the proposed algorithm is
evaluated for: (a) the economic load dispatch system and (b)
self-provisioning system used in communication network ser-
vices. Results indicate an improved system efficiency and
better robustness as compared to the hybrid PSO technique.
In [28], the authors investigated the application of PD type
fuzzy logic controller in trajectory tracking of differential
drive mobile bot. The authors used a Takagi-Sugeno based
fuzzy controller having 7 sets in each variable. The per-
formance of the controller has been compared with PID
and PD controllers. The results indicate a superior perfor-
mance of PD type fuzzy logic controller as compared to
conventional controllers. In [18], the authors investigated a
statistical-based optimization approach for finding the opti-
mum support in a fuzzy logic system using fuzzy entropy
This paper proposes an algorithm to optimize the mem-
bership function for designing a fuzzy logic controller. The
proposed algorithm is tested for real-time swing-up control
and stabilization of the inverted pendulum system and per-
formance indices for the proposed controller are compared
with state-of-the-art controllers. The key novelty features of
the proposed work are:
1. Designed an optimal fuzzy controller based on a novel
objective function which comprises of fuzzy entropy.
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A. Jain et al.: Real-Time Swing-Up Control of Non-Linear IP
2. Developed a fuzzy controller based on Lyapunov stabil-
ity criteria to ensure asymptotic stability of the devel-
oped controller.
3. The developed controller is deployed for real-time
swing up stabilization of a rigidly coupled twin-arm
inverted pendulum system.
4. To compare the performance of the proposed controller
with other state-f-the-art controllers based on some
key parameters to validate the efficacy of the proposed
The rest of the paper layout is: Section 2 illustrates the digital
pendulum model. Section 3 describes the proposed fuzzy
logic controller (FLC) developed for the system. In section 4
the objective function and the optimization technique used to
optimise the fuzzy set (FS) is discussed in detail. Section 5
explains the experimental results of a swing-up control for
a real-time inverted pendulum system. In section 6, robust-
ness of the proposed algorithm is analyzed. In section 7,
the proposed controller is compared with state-of-the-art
controllers based on certain key performance parameters..
Finally, Section 7 discusses the salient findings of the study.
The figure of the rigidly coupled twin-arm digital pendulum
with the cart system is illustrated in Figure 1.
FIGURE 1. Cart driven twin-arm inverted pendulum [29].
Mathematically the forces acting on the system can be
summarized as:
θ2sinθcosθmal cosθF
where, a=1
Assuming the state variables: x1=θ,x2=˙
x4= ˙x. Then:
TABLE 1. Parameters for real-time model.
4l3mpla cos2x1
˙x4=mpag sin x1cos x1+4mpla
3mpa cos2x1
To design a PID controller a linearized model is obtained.
The equations are linearized around the inverted position,
i.e. θ=0 (operating point).
0 1 0 0
4l3mpla 0 0 0
0 0 1 0
43mpa0 0 0
3mpla 4l
Table 1 shows the parameters for the real-time model
used in the experiment setup, manufactured by ‘‘Feedback
instruments the digital pendulum system: 33-936S’’ [29].
By substituting the value from Table 1 in(8), the following
eigenvalues are obtained:
e1=0,e2=4.43,e3= −4.43,e4=1
As evident from the eigen values, we can conclude that
the system is unstable. The criteria for controller design is
to make the eigenvalues negative and in turn, stabilize the
Figure 2 illustrates the control block diagram for the same.
The computer is connected to a data acquisition (DAQ) card
which is an interface between the analog pendulum system
and the digital computer. The control signal is generated
by MATLAB – Simulinkrand is a digital signal, which is
then converted to an analog signal of ±5 volts by the DAQ
interface, which is converted to ±24 volts for motor operation
by DC motor interface. The position of the cart and the angle
of the pendulum are measured using encoders. The first is
attached to the DC motor and the latter to the cart-pendulum
for respective angular measurements. These encoders give
analog signals for real-time measurement which are again
converted to digital values via DAQ card.
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FIGURE 2. Digital control block diagram [29].
For designing an FLC precise mathematical model of the
system is not mandatory, however expert knowledge of the
system under study is required, as FLCs are primarily inspired
by the decision-making process of human beings. These can
infer linear or nonlinear complex relationships between the
input and output variable(s) [30]. The controller architecture
designed for IP consists of a PD (proportional-derivative)
type FLC designed to control the cart for driving the pendu-
lum to an inverted position. The architecture of the PD type
fuzzy logic controller used is depicted in Figure 3.
FIGURE 3. FLC architecture for the digital pendulum.
For control applications, FS are commonly named concerning
their relevant position with reference to error. As for any
control system, the desired error is always ‘zero’, hence FS
associated around ‘zero error’ is named as ‘zero’. Moving
on to the positive x-axis the FS which are associated with
the positive error is named ‘positive’ and for the negative
x-axis the FS which is associated with the negative error is
named ‘negative’, respectively [31]. Initially, the MFs are
distributed uniformly around the universe of discourse and
Gaussian fuzzy MFs are chosen. The proposed controller
is a two-input one output system; having (i) error in angle
(), and (ii) rate of change for error in angle (d( )dt)
as input variables and (iii) control signal (u) for DC motor,
acting as an output variable. Fuzzy sets for error in pendulum
angle are given in Figure 4. Here three overlapping normal
(i.e. µmax =1) Gaussian FS are defined which are distributed
uniformly across the universe of discourse. For example, the
equation for set zero ‘‘ZE’’ can be written as:
FIGURE 4. Predefined membership function.
The system dynamics can be represented as:
The linguistic control rules will be formed without know-
ing the terms f(x),g(x),d(x) and w(disturbance) consid-
ering the pendulum as a black box system. The linguistic
control rules are:
R1: State variables of the system: x1=θ,x2=˙
x3=x,x4= ˙x
R2: ˙x2is proportional to the control input u
R3: ˙x4is proportional to x3
The statements R1, R2 and R3 ensure the stability of the
system. The controller design objective is to find a u(control
force) at which the system is stable, considering the operating
point as [x1x2x3x4]T=0.
Let V=1
4] be the Lyapunov function,
hence as per the state variables defined for the system we can
Using the control rules R2 and R3
As per classical Lyapunov synthesis, the control input uthat
ensures ˙
V<0, can be formed using the following rule base:
If x1is negative AND x2is negative, THEN uis positive
If x1is negative AND x2is positive, THEN uis zero
If x1is positive AND x2is positive, THEN uis negative
If x1is positive AND x2is negative, THEN uis zero
Using the product operator for inference and centre of grav-
ity as the defuzzification process, the control signal can be
depicted as (13), as shown at the bottom of the next page.
To analyse control law and we assume µp(x)=
e(xax)2, µn(x)=e(x+ax)2, µz(x)=ex2, (14), as shown
at the bottom of the next page.
Simplifying equation (14) we have, u, as shown at the bottom
of the next page.
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u= au
2tanh 2ax1x1+tanh(2ax2x2)(15)
The fuzzy set design parameters ax1,ax2and auare computed
using the maximum entropy principle for the fuzzy system.
The following section discusses the optimization principle
The proposed optimization principle is based on the maxi-
mization of entropy of MF displaced by standard deviation
data obtained from the system [32].
The mathematical expression for probability measures of
FS according to which fuzzy entropy can be written as [29]:
H(A)= Z
{µilog µi+(1µi) log(1 µi)}(16)
For example, fuzzy entropy for positive MF can be calculated
Hµp(x)= Z
f(e(xax)2)dx (17)
As previously discussed in section III. The standard Gaus-
sian fuzzy sets is used for representing the fuzzy variables,
namely: error,change in error, and control output. The
control signal expressed in equation (15) depends on the
membership function parameters which are optimized using
the maximum entropy principle with stopping criteria of a
minimum optimal control cost function which is defined as:
Hence, for obtaining the optimized membership function the
optimization problem is defined as:
maximize:H(A)= − Z
{µilog µi+(1µi) log(1 µi)}
subject to:min (J) (19)
The optimization process is carried out on the membership
function by displacing the standard Gaussian membership
function using standard deviation for individual fuzzy sets
which are to be optimized. For instance, considering the input
variable error in pendulum angle 3 membership functions
are defined: zero, positive and negative. The membership
function zero can be defined as:
By displacing the set one can obtain the new fuzzy set µ
which can be written as:
where, σeis the standard deviation obtained for the error
in pendulum angle. The graphical depiction of a displaced
membership function is represented in Figure 5. The figure
depicts a few intermediate sets for the fuzzy set ‘‘zero’ when
it is displaced. During the optimization process, the algorithm
is supplied with the value of the standard deviation for each
of the fuzzy variables which is then optimized using (15),
(18) and (19). The standard deviation is obtained through
stabilization of the IP system using PID controller:
σe=1.6752 σ˙e=3.57 σc=0.3248
u=µn(x1)µn(x2) (au)+µp(x1)µp(x2) (au)
u= au
u= au
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Using the displaced fuzzy sets, the optimization problem can
now be defined as:
maximize:H(A)= Z
ilog µ
×log(1 µ
subject to:min (J) (22)
Here µ
idepicts the corresponding displaced fuzzy sets. The
objective function is optimized using genetic algorithm.
FIGURE 5. Displaced fuzzy set ‘‘zero’’.
TABLE 2. GA parameters.
The parameters used for genetic algorithm (GA) is given
in Table 2.. Using the GA based optimization of the objective
function, the resultant optimized FS is used to replace prede-
fined MFs and is thus used for designing optimized FLC. The
same method is utilized to obtain optimized MFs for the rate
of change of error in angle, and the control signal of the DC
Figure 6 depicts the real-time hardware in action and indicates
the stabilized inverted position achieved during experiments.
In swing up stabilization, the pendulum system is at an
initial angle of θ=180o(the natural equilibrium of a
simple pendulum). The control philosophy for the inverted
pendulum is fairly simple: here the controller’s target is to
swing upright and maintain the position of the pendulum to
FIGURE 6. Experimental setup indicating pendulum in a stable inverted
an inverted position by counteracting the earth’s gravitational
force. To generate this counteractive force, the cart is moved
back and forth, due to which the pendulum gains inertia
leading to an oscillatory motion. Once the pendulum reaches
the desired inverted position the cart tries to maintain the
inverted position [33].
The PID control algorithm consists of two controllers with
only one being active at a time. One is designed for swing-
ing up the pendulum pole and the other for stabilization of
pendulum as it reaches the inverted position. The control
algorithm for pendulum swing up is designed to regulate the
force applied to the cart in such a way that the pendulum
starts to oscillate with a successive increase in the oscilla-
tion magnitude. When the pendulum reaches the inverted
position the stabilization algorithm then tries to maintain the
inverted position with minimal control effort applied to the
cart.Here, the PID settings have been optimized for minimum
ISE (Integral square error)once the values are obtained using
the Ziegler-Nichols method [29]. Pendulum angle stabiliza-
tion using a PID controller is shown in Figure 7. The con-
troller performance parameters are observed as: (a) Settling
time – ts=18 seconds (b) Peak value – Mp=6.01 radians.
The PID control is now replaced by the fuzzy logic controller
as illustrated in Figure 3. Pendulum angle stabilization con-
trol using FLC is given in Figure 8. With this result, the con-
troller performance parameters are observed as (a) Settling
time – ts=8.4 seconds (b) Peak value – Mp=4.9 radians.
This indicates an improvement over PID control, with the set-
tling time being reduced by 53.33% and peak value by 18%.
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FIGURE 7. Pendulum angle for PID controller.
FIGURE 8. Pendulum angle for FLC.
The predefined MFs used in FLC are replaced by opti-
mized MFs obtained from the proposed algorithm and opti-
mized FLC is used to control the digital pendulum. The
pendulum angle stabilization using optimized FLC is given
in Figure 9.
With this result the controller performance parameters are
observed as: (a) Settling time – ts=4.0 seconds (b) Peak
value – Mp=5.52 radians. These values indicate an improve-
ment over PID control, with the settling time being reduced
by 77.78% and peak value by 8.2%. However, the comparison
of these parameters over FLC indicates a reduction of settling
time by 52.38% but an increase in peak value by 12.6%. One
of the biggest improvements exhibited by the proposed con-
troller is the reduction in oscillations, as the pendulum angle
gets stabilized in 5 oscillations, while it took 26 oscillations
for PID and 9 oscillations for FLC to stabilize the pendulum
FIGURE 9. Pendulum angle for optimized FLC.
Figure 10 delineates the cart position for PID, fuzzy, and
optimized fuzzy controller. The back and forth movement of
the cart provides inertia to the pendulum and is responsible for
maintaining the inverted position of the pendulum. The cart
movements are random until the pendulum angle is stabilized
and are periodic once the inverted position is obtained.
FIGURE 10. Cart position comparison for PID, FLC, novel FLC.
Figure 11 depicts the control forces (u) generated by the
respective controllers. The back and forth cart movements can
be associated with crisp control forces generated before the
sudden spike in the control force where the spike is generated
while the pendulum is being balanced to stabilize it to an
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inverted position. After which the motor continuously tries
to counter the effect of gravity on the pendulum.
FIGURE 11. Control force comparison for PID, FLC, novel FLC.
Comparison of performance (pendulum angle stabilization)
in swing up mode for PID control, FLC, and optimized FLC
are given in Figure 12. Pendulum angle stabilization compar-
ison indicates that steady-state error for all the controllers
is ‘‘0’’. It can, however, be concluded that the pendulum
angle is stabilized within a shorter duration for optimized
FLC as compared to PID control or FLC. Table 3 summarizes
‘‘Settling time (ts)’’, ‘‘Peak Value (Mp)’’ and the oscillations
exhibited by the pendulum to be stabilized by the three dis-
tinct controllers.
TABLE 3. Settling time and peak value comparison for pendulum angle
Table 4 summarizes error indices (error in pendulum angle)
for PID, FLC, and proposed controller; these indices include:
root mean square error (RMSE), integral square error (ISE),
integral time multiplied square error (ITSE), integral absolute
error (IAE), integral time multiplied square error (ITAE) [34]:
FIGURE 12. Comparison of performance for PID, FLC, novel FLC.
TABLE 4. Performance indices.
As the steady-state error for the (PID, FLC, and novel
FLC) controllers is: ‘‘0’’, the comparisons have been carried
out for: Settling time, Peak value, and error indices. The
proposed ‘optimized FLC’ exhibits minimum settling time
(fast convergence) among the three controllers. The peak
value is however only marginally different between the three
controllers, which is an unavoidable phenomenon in swing
up angle stabilization of an inverted pendulum. Further, the
minimum values of the performance indices are demonstrated
by optimized FLC. Hence it is innocuous to say that the pro-
posed optimized FLC is an efficient and effective controller,
and shows an improvement over benchmark PID control and
conventional FLC.
In this section, the authors have examined the robustness of
the proposed algorithm. The robust analysis is carried out
under small disturbances within: (a) the system parameters
and (b) the measurement unit by adding external noise. The
block diagram illustrating both internal and external distur-
bances that are added to the system is shown in Figure 13.
For robustness analysis the first step is to vary system parame-
ters to evaluate the robustness of the proposed controller. The
parameters chosen for this investigation are: (a) mass of cart,
(b) combined mass of pendulum. Both the parameter values
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FIGURE 13. System block diagram subjected to internal and external
are varied with a deviation of 10% and the proposed entropy
based fuzzy controller is tested to determine the robustness of
the system.
In the first case, the controller response is tested under cart
mass variation of 10% as compared to the default parameters
value. The result validates the robustness of the proposed con-
troller as the controller successfully stabilizes the system with
a slight increase in the settling time as shown in Figure 14.
FIGURE 14. System performance under variation in cart mass.
In the second case, the controller response is observed
under a 10% variation in pendulum mass when compared
with the default system parameters. The result obtain indi-
cates the robustness of the proposed controller as the con-
troller aids in stabilizing the system. The controller response
shows that the pendulum stabilizes at 360(equivalent to
inverted position) with a minor effect in the settling time as
depicted in Figure 15.
In the last case, the response of the controller is observed
when there is 10% variation in both pendulum and cart mass.
The response under twin parameter variation validates the
robustness of the proposed controller as the overall system
remain stable with a marginal increase in settling time as
shown in Figure 16.
To check the robustness under external disturbance, the
authors included a random noise generator which is added
to the measurement unit as shown in Figure 13. The peak-
to-peak amplitude of the external noise is kept at 10% of
FIGURE 15. System performance under variation in pendulum mass.
FIGURE 16. System performance under twin parameter variation.
the controller signal, which is 0.5 V. Figure 17 shows the
comparison of controller response to the system subjected
to the external noise condition. The results obtain indicate
the robustness of the proposed controller as it helps in sta-
bilizing the system and the performance indices observed
remains fairly unchanged with a small increase in the system
overshoot. It is important to point out that the noise signal
frequency is kept same as the transients observed by the
The performance parameters for the proposed novel
controller algorithm are compared with state-of-the-art con-
trollers. It is well-known that the steady-state error for effi-
cient controllers is 0. Hence, the comparison between the
controllers is based on transient performance parameters.
For performance evaluation the state-of-the-art work has
been selected based on the similarity of either or all of
the following parameters: (a), controller algorithm used,
(b) simulation and (c) real-time experimental deployment.
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FIGURE 17. System performance under noise.
For steady-state performance evaluation, the settling times
are compared and to determine the transient performance,
the number of oscillations exhibited by the system to reach
steady-state are observed.
In [13], the authors developed Fuzzy logic based adap-
tive PID controller for swing-up stabilization of pendulum
angle. The hardware platform used for validating the exper-
iment results by authors is the same as used to validate this
research work. The authors used the fuzzy logic system to
optimize the gains of the PID controller and hence it uti-
lizes the principle of adaptive PID controller. The authors
reported a settling time of 5 seconds. In [16], the authors
utilized a hybrid controller approach using fuzzy logic con-
trol for swing up controller, switching to state feedback
control for stabilization, and using LQR (guaranteed cost
control) for uncertainty handling. The authors reported a
settling time of 7.7 seconds. In [17], the authors developed a
self-tuning regulator based on a precise actuator model. The
controller achieved a settling time of 8 seconds. In [7], the
authors developed a FLC based on Lyapunov’s direct method
to achieve swing-up stabilization of an inverted pendulum
system. The authors didn’t compute the optimized fuzzy logic
system and the controller is designed using Lyapunov stabil-
ity criteria. In contrast, the algorithm proposed in this paper
utilizes the Lyapunov method and the fuzzy sets are further
optimized to compute the optimum controller. The authors
reported a settling time of 8.7 seconds. In [35], the authors
deployed a Takagi-Sugeno based fuzzy controller, state feed-
back controller, and sliding mode controller strategies for
swing-up stabilization of an inverted pendulum system. The
controller had four input variables. The authors deployed
Takagi-Sugeno based controller and didn’t compute the opti-
mum membership functions, although the authors compared
the performance of three different controllers. The proposed
controller in this paper is based on the Mamdani method and
hence optimization of fuzzy sets becomes an integral part of
controller implementation. The author reported a settling time
of 12.8 seconds. Table 5 summarizes the comparison of the
proposed controller with other benchmark control algorithms.
TABLE 5. Settling time and oscillation comparison of the proposed
controller with reference controller.
Figure 18 depicts the comparison of the settling time of the
proposed controller with few benchmarked controllers.
FIGURE 18. Comparison of settling time with state-of-the-art controllers.
It is observed that the proposed controller exhibits the
fastest settling time along with minimum oscillations when
compared with benchmarked controllers. This proves the
efficacy of the proposed optimization algorithm in finding
the optimal membership function for a fuzzy logic controller
design problem.
In this work, a novel optimization method has been proposed
to find the membership function for fuzzy controller based
on the ‘‘Fuzzy Entropy’’ function. The proposed method
uses predefined fuzzy sets and optimizes the support of the
set by evaluating the objective function. The fuzzy sets are
optimized using genetic algorithms with stopping criteria as
minimizing the optimal control cost function.
The proposed algorithm is applied for the swing-up sta-
bilization for a real-time inverted pendulum system. A PD
type fuzzy logic controller is designed based on the Lyapunov
method which ensures asymptotic stability. Furthermore, the
membership functions of the FLC are optimized based on
50724 VOLUME 9, 2021
A. Jain et al.: Real-Time Swing-Up Control of Non-Linear IP
the data obtained from the real-time PID swing-up control
of the pendulum. The control objective of the experiment is
to achieve swing-up stabilization in an inverted pendulum
system. The results depict an improvement in system per-
formance parameters (like ts, Mp,etc.) for optimized-FLC
as compared to FLC or PID. The performance of the pro-
posed controller is also compared with other state-of-the-
art controllers present in the literature that adopted similar
hardware/controller principle. The experimental results indi-
cate an improvement in the performance parameters of the
proposed controller.
In the current research work, the proposed methodology is
used to optimize FS having Gaussian MF. In the future the
authors intend to develop the same optimization technique
for fuzzy-logic based systems having different membership
functions like triangular, trapezoidal, s-function, etc. The
applicability of this technique is limited to systems having the
availability of reference data as a constrained requirement.
For the majority of practical systems data is available as a
reference set or can be determined with the help of simula-
tion/experimental analysis, therefore availability of data will
not be a major constraint for applicability of the proposed
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ARPIT JAIN received the B.Eng. degree from
SVITS, Indore, India, in 2007, the M.Eng. degree
from Thapar University, Patiala, India, in 2009,
and the Ph.D. degree from UPES, Dehradun, India,
in 2018. He is currently working as an Assistant
Professor (Selection Grade) with the Department
of Electrical and Electronics Engineering, Uni-
versity of Petroleum and Energy Studies (UPES).
He has over ten years of teaching and research
experience and a rich experience in curriculum
design. His research interests include real-time control systems, fuzzy logic,
machine learning, and neural networks.
VOLUME 9, 2021 50725
A. Jain et al.: Real-Time Swing-Up Control of Non-Linear IP
ABHINAV SHARMA received the B.Tech. degree
from H. N. B. Garhwal University, Srinagar, India,
in 2009, and the M.Tech. and Ph.D. degrees from
the Govind Ballabh Pant University of Agriculture
and Technology, Pantnagar, India, in 2011 and
2016, respectively. He is currently working as an
Assistant Professor (Senior Scale) with the Depart-
ment of Electrical and Electronics Engineering,
University of Petroleum and Energy Studies,
Dehradun. He has published number of research
papers in journals and conferences. His research interests include adaptive
array signal processing, smart antennas, artificial intelligence, and machine
VIBHU JATELY received the Ph.D. degree in elec-
trical engineering from the College of Technology,
G. B. Pant University, Pantnagar, India, in 2017.
Since July 2019, he has been working as a Postdoc-
toral Fellow with the MCAST Energy Research
Group, MCAST, Malta. He has over four years
of teaching experience during which he has also
worked under United Nations Development Pro-
gramme in Ethiopia. His research interests include
control in photovoltaic systems, control in power
electronics, and grid-connected photovoltaic systems.
received the B.Eng. degree from the University of
Malta, in 2002, and the Ph.D. degree from The
University of Manchester, U.K., in 2011. He is
currently a Senior Academic with the Malta Col-
lege of Arts, Science and Technology (MCAST),
where he leads the Energy Research Group. He is
also a Visiting Senior Lecturer with the University
of Malta. His research interests include photo-
voltaics and electric mobility network integration
and future urban low-carbon society.
degree from NIT Silchar, the M.Tech. degree
from Tezpur University, and the Ph.D. degree
from UPES, Dehradun. He has rich teaching and
industry experience. He is currently working as
a Professor and the Head of the Department of
Electrical and Electronics Engineering, School
of Engineering, UPES. He has published more
than 50 research papers in international/national
journals and conferences. He has filed 13 patents
and authored several books. He is selected as an outstanding scientist of
the twenty-first century by Cambridge Biographical Centre. His research
interests include wireless sensor networks, embedded systems, robotics,
automation and control, the IoT (Node-MCU, ESP8266), nanotechnology,
artificial intelligence, and machine learning.
50726 VOLUME 9, 2021
... There are many research studies dedicated to get a better control performance over the inverted pendulum. Jain et al. [5] proposed fuzzy control logic using the Lyapunov technique to ensure system stability. The design approach is based on blackbox mathematical modeling. ...
... The goal of the control system is to balance an inverted pendulum system by giving a control signal (force) to a cart that has been installed with a pendulum system [12] [13]. Some examples of inverted pendulum system applications are: balancing system in a rocket system when the rocket takes off [14], Missile Launcher [15] [16], Segway [17], balancing robots [18]- [20], humanoid robot [21]- [23], etc [24]. Some researches about inverted pendulum have been done by many. ...
Full-text available
The inverted pendulum system is highly popular in control system applications and has the characteristics of unstable, nonlinear, and fast dynamics. A nonlinear controller is needed to control a system with these characteristics. In addition, there are disturbances and parameter uncertainty issues to be solved in the inverted pendulum system. Therefore, this study uses a nonlinear controller, which is the backstepping sliding mode control. The controller is robust to parameter uncertainty and disturbances so that it is suitable for controlling an inverted pendulum system. Based on testing with step and sine reference signals without interference, the controller can stabilize the system well and has a fast response. In testing with disturbances and mass uncertainty, the backstepping sliding mode controller is robust against these changes and able to make the system reach the reference value. Compared with sliding mode control, backstepping sliding mode control has a better and more robust response to disturbances and parameter uncertainty.
In this article, the design and implementation of a cooperative multitasking control strategy for standalone photovoltaic (SPV) system is presented. A double-loop control strategy based on the fuzzy logic controller is developed to achieve high performance from the SPV without using the maximum power point technique, balance the power in the system, regulate the common dc-link voltage, and maintain system operation stability under disturbance. To ensure uninterruptible power supply to the critical loads during diminution of generated power from the SPV, turned on and off the diesel generator and the dc dump load, power management technique based on load scheduling technique is developed and implemented using the state-flow approach. Furthermore, the stability analysis in open and closed-loop using the average model of the dc–dc buck-boost converter is investigated. The performance of the dc off-grid configuration based on a single dc–dc converter and their control strategies are performed under load and solar irradiation change using MATLAB/Simulink and in real-time through hardware prototyping with DSP controller TMDS28379D.
Conference Paper
This paper presents several proposed motion models of high-order physical systems in three main concepts called macro-scale, regular-scale, and nano-scale. In fact, it is very difficult to find an exact numerical solution for the high-order differential equations because all numerical methods only yield the approximate solutions. In addition, loop gain is not widely used in many negative feedback systems because it is an approximation value. To overcome the limitations of the high-order differential equations and the loop gain, the waveforms of the physical periodic motions are expressed by helix functions at time variation, and the characteristics of complex functions are used to examine the behaviors of the transmission spaces and the transmission networks in the different motion models including the Earth's motion, the simple pendulum systems, and the electronic systems. Furthermore, the force of attraction and the friction or the resistance in the different scales obey the conservation law and the superposition principle; therefore, three superposition formulas are introduced to derive the transfer functions in high-order mechatronic systems. The operating regions, the effects of the overshoot phenomena, the breaking chemical bonds, and the difference between negative and positive feedbacks in these systems are also introduced. As a result, the use of complex functions, helix waves, and superposition principle leads to a complete control theory with which many behaviors of the physical systems can be explained and predicted easily.
Full-text available
There are two main approaches to safety-critical control. The first one relies on computation of control invariant sets and is presented in the first part of this work. The second approach draws from the topic of optimal control and relies on the ability to realize Model-Predictive-Controllers online to guarantee the safety of a system. In the second approach, safety is ensured at a planning stage by solving the control problem subject for some explicitly defined constraints on the state and control input. Both approaches have distinct advantages but also major drawbacks that hinder their practical effectiveness, namely scalability for the first one and computational complexity for the second. We therefore present an approach that draws from the advantages of both approaches to deliver efficient and scalable methods of ensuring safety for nonlinear dynamical systems. In particular, we show that identifying a backup control law that stabilizes the system is in fact sufficient to exploit some of the set-invariance conditions presented in the first part of this work. Indeed, one only needs to be able to numerically integrate the closed-loop dynamics of the system over a finite horizon under this backup law to compute all the information necessary for evaluating the regulation map and enforcing safety. The effect of relaxing the stabilization requirements of the backup law is also studied, and weaker but more practical safety guarantees are brought forward. We then explore the relationship between the optimality of the backup law and how conservative the resulting safety filter is. Finally, methods of selecting a safe input with varying levels of trade-off between conservatism and computational complexity are proposed and illustrated on multiple robotic systems, namely: a two-wheeled inverted pendulum (Segway), an industrial manipulator, a quadrotor, and a lower body exoskeleton.
Full-text available
In this paper, a Fuzzy based Linear Quadratic Regulator (FLQR) and Linear Quadratic Gaussian (FLQG) controllers are developed for stability control of a Double Link Rotary Inverted Pendulum (DLRIP) system. The aim of this work is to study dynamic performance analysis of both FLQR and FLQG controllers and to compare them with the classical LQR and LQG controllers, respectively. A dynamic mechanical simulation model of the DLRIP was obtained using both the numerically SimMechanics toolbox in MATLAB and the analytically dynamic nonlinear equations. To determine the control performance of the controllers, Settling Time (Ts), Peak Overshoot (PO), Steady-State Error (Ess), and the total Root Mean Squared Errors (RMSEs) of the joint positions are computed. Furthermore, the dynamic responses of the controllers were compared based on robustness analysis under internal and external disturbances. To show the control performance of the controllers, several simulations were conducted. Based on the comparative results, the dynamic responses of both FLQR and FLQG controllers produce much better results than the dynamic responses of the classical LQR and LQG controllers, respectively. Moreover, the robustness results indicate that the FLQR and FLQG controllers under the internal and external disturbances were effective.
Full-text available
A control problem of an inverted pendulum in the presence of parametric uncertainty has been investigated in this paper. In particular, synthesis and implementation of an automatic self–tuning regulator for a real inverted pendulum have been given. The main cores of the control system are a swing–up control method and a stabilisation regulator. The first one is based on the energy of an inverted pendulum, whereas the second one uses the linear–quadratic regulator (LQR). Because not all of the inverted pendulum parameter values are exactly known an automatic self–tuning mechanism for designed control system has been proposed. It bases on a devised procedure for identifying parameters. The entire derived control system enables effective a pendulum swing–up and its stabilisation at an upper position. The performance of the proposed control system has been validated by simulation in Matlab/Simulink environment with the use of the inverted pendulum model as well as through experimental works using the constructed inverted pendulum on a cart.
Full-text available
This paper presents the realisation of self-erecting inverted pendulum controls via two switched control approaches, a rule based fuzzy control for swing up inverted pendulum rod to pose upright position from downright position and an optimal state feedback control for stabilization as pendulum on upright position close to its equilibrium vertical line. The aim of this study is to solve two important problems on self-erecting inverted pendulum; swing up and stability in its upright balance position. Simulation and experimental results showed that control methods enabled the inverted pendulum swinging up and reaching its stable attitude in upright position even though small impulse and pulse disturbances were given.
Full-text available
Most of control strategies are based on complex mathematical equations and the success of them is directly related to the precise acquisition of the mathematical model of the system. Virtual Model Control (VMC) allows the system to be mechanically controlled like a puppet with intuitive approaches instead of complex mathematical equations. In this paper, a novel implementation of fuzzy VMC on inverted pendulum, which is an under actuated mechanism, is presented. The cart of the inverted pendulum is controlled by manipulating the vertical equilibrium point of the pendulum with two cascaded VMCs. A Takagi-Sugeno fuzzy parameter tuner is designed to improve the VMC performance. An LQR controller is also designed to compare the performances of the controllers. The complete control system is implemented on FPGA based embedded platform in real time. The simulation and experimental results indicated that the proposed VMC successfully controls the inverted pendulum. In addition, the VMC with fuzzy parameter tuner has lower rise time, settling time, IAE and ITAE when compared to the LQR which is widely used controller in the literature.
Purpose This paper aims to keep the pendulum on the linear moving car vertically balanced and to bring the car to the equilibrium position with the designed controllers. Design/methodology/approach As inverted pendulum systems are structurally unstable and nonlinear dynamic systems, they are important mechanisms used in engineering and technological developments to apply control techniques on these systems and to develop control algorithms, thus ensuring that the controllers designed for real-time balancing of these systems have certain performance criteria and the selection of each controller method according to performance criteria in the presence of destructive effects is very helpful in getting information about applying the methods to other systems. Findings As a result, the designed controllers are implemented on a real-time and real system, and the performance results of the system are obtained graphically, compared and analyzed. Originality/value In this study, motion equations of a linear inverted pendulum system are obtained, and classical and artificial intelligence adaptive control algorithms are designed and implemented for real-time control. Classic proportional-integral-derivative (PID) controller, fuzzy logic controller and PID-type Fuzzy adaptive controller methods are used to control the system. Self-tuning PID-type fuzzy adaptive controller was used first in the literature search and success results have been obtained. In this regard, the authors have the idea that this work is an innovative aspect of real-time with self-tuning PID-type fuzzy adaptive controller.
Purpose The purpose of this paper is to study the application of entropy based optimized fuzzy logic control for a real-time non-linear system. Optimization of the fuzzy membership function (MF) is one of the most explored areas for performance improvement of the fuzzy logic controllers (FLC). Conversely, majority of previous works are motivated on choosing an optimized shape for the MF, while on the other hand the support of fuzzy set is not accounted. Design/methodology/approach The proposed investigation provides the optimal support for predefined MFs by using genetic algorithms-based optimization of fuzzy entropy-based objective function. Findings The experimental results obtained indicate an improvement in the performance of the controller which includes improvement in error indices, transient and steady-state parameters. The applicability of proposed algorithm has been verified through real-time control of the twin rotor multiple-input, multiple-output system (TRMS). Research limitations/implications The proposed algorithm has been used for the optimization of triangular sets, and can also be used for the optimization of other fussy sets, such as Gaussian, s-function, etc. Practical implications The proposed optimization can be combined with other algorithms which optimize the mathematical function (shape), and a potent optimization tool for designing of the FLC can be formulated. Originality/value This paper presents the application of a new optimized FLC which is tested for control of pitch and yaw angles in a TRMS. The performance of the proposed optimized FLC shows significant improvement when compared with standard references.
In this paper, the swing up and stabilization control problems for a rotary inverted pendulum (RIP) system are solved via new control schemes, in which the swing up strategy is designed by using trajectory planning and inertia effect such that the pendulum can be swung to a desired position to trigger the stabilization controller, and the stabilization scheme is implemented by resorting to the nonlinear adaptive neural network (NN) control method and linear matrix inequation technique. By using Lyapunov stability theory, it can be proved that the target trajectory can be boundedly tracked by the arm section, and the pendulum section can be balanced in the upright position with a small error. Finally, two experiment results are respectively given to show the effectiveness of the proposed swing up and stabilization methods.
Purpose The aim of this paper is to propose a robust robot fuzzy logic proportional-derivative (PD) controller for trajectory tracking of autonomous nonholonomic differential drive wheeled mobile robot (WMR) of the type Quanser Qbot. Design/methodology/approach Fuzzy robot control approach is used for developing a robust fuzzy PD controller for trajectory tracking of a nonholonomic differential drive WMR. The linear/angular velocity of the differential drive mobile robot are formulated such that the tracking errors between the robot’s trajectory and the reference path converge asymptotically to zero. Here, a new controller zero-order Takagy–Sugeno trajectory tracking (ZTS-TT) controller is deduced for robot’s speed regulation based on the fuzzy PD controller. The WMR used for the experimental implementation is Quanser Qbot which has two differential drive wheels; therefore, the right/left wheel velocity of the differential wheels of the robot are worked out using inverse kinematics model. The controller is implemented using MATLAB Simulink with QUARC framework, downloaded and compiled into executable (.exe) on the robot based on the WIFI TCP/IP connection. Findings Compared to other fuzzy proportional-integral-derivative (PID) controllers, the proposed fuzzy PD controller was found to be robust, stable and consuming less resources on the robot. The comparative results of the proposed ZTS-TT controller and the conventional PD controller demonstrated clearly that the proposed ZTS-TT controller provides better tracking performances, flexibility, robustness and stability for the WMR. Practical implications The proposed fuzzy PD controller can be improved using hybrid techniques. The proposed approach can be developed for obstacle detection and collision avoidance in combination with trajectory tracking for use in environments with obstacles. Originality/value A robust fuzzy logic PD is developed and its performances are compared to the existing fuzzy PID controller. A ZTS-TT controller is deduced for trajectory tracking of an autonomous nonholonomic differential drive mobile robot (i.e. Quanser Qbot).
The future of mobile telecommunications is aimed at creating a user-centric wireless world which takes into account user's preferences, as well as communication context (e.g., network and terminal heterogeneity or user location). In order to enable context-aware personalization of communication services offered by next-generation mobile networks, we propose an agent-based approach in combination with semantic reasoning techniques from the Semantic Web. In particular, we use ontology-based user profiles to create an agent-based context-aware service which supports personalization according to user's preferences. An ontology is created which contains knowledge regarding terminal capabilities and user preferences, as well as a software agent which manages user personalization according to the context extracted from the user profile.