1. D’Almeida 1
Sami D'Almeida
August 04, 2015
Article Assignment 14
Stability Control of an Inverted Pendulum
Abstract
In this article a novel method of designing a controller for an inverted pendulum is
developed. Laplace transform has been used to solve the differential equations that carried out
the factors influencing the dynamic of the inverted pendulum as a continuation of the work
proposed by Domenico, Fabio, and Carmine (2010) and then an output-feedback controller has
been designed using Single Input Single Output (SISO) tools from MATLAB. Additionally, a
stochastic method of initializing the weighting matrices to minimize the performance index of
the Linear Quadratic Regulator (LQR) is examined. The stochastic method is based on binomial
distribution to initialize the populations in order to minimize the number of iteration used in the
traditional PSO introduced by Kavey and Won-sook (2014). The purpose of my research is to
investigate a more optimal and low cost method of designing a control system for an inverted
pendulum. Upon simulation with MATLAB Simulink it was shown that the new output-feedback
controller easily and more effectively controls the stability of the pendulum in its vertical
unstable position without the need of any integral gain.

2. D’Almeida 2
Introduction
A statistical analysis conducted by David and Roger (2003) shows that between 1990 and
2000 26% of plane crashes were due to mechanical failure especially stability control issue. It is
therefore important to investigate more effective ways of controlling the stability of aircrafts and
robotic devices. An inverted pendulum have been chosen as a bench mark prototype to carry out
the control because aircraft motion is mainly influenced by different kind of frictions that cause
vibration and the whole system acts like a pendulum placed on top of a moving cart which
constantly sends disturbance to the pendulum. It was shown by previous authors such as Siebert
(1986) that if one can control the stability of the pendulum while the cart if moving then he/she
can definitely control the stability of the entire aircraft or any rocket or robotic device.
One of the most important step in designing a control system is the proper modeling of
the dynamic of the system. The two fundamentals law of dynamic (Newton and Lagrange law)
have proved that the dynamic of an inverted pendulum is carried out by a system of two second
order differential equations. Most research show that by performing a suitable change of
variable, these equations become a linear time-invariant (LTI) system also known as a state-
feedback system; such a controller is effectively optimized by a method called a linear quadratic
regulator (LQR) as shown in “Controller Design of an Inverted Pendulum Using Pole Placement
and LQR” by Kumar & Mehrota (2012). The (LQR) had been identified as one of the most
optimal technique to design a controller for an inverted pendulum. However the reliability of this
method is questionable because it does not offer a systematic method of determining the
weighting matrices that can minimize the performance index; the selection of the matrices was
based on the designer’s experience and usually performed by trial and error. The following
studies conducted in the past show that not only very few of the articles in the past have actually

3. D’Almeida 3
solved the dynamic equation in order to design the control system for an inverted pendulum, but
also very few articles have extensively covered output-feedback control system design for the
inverted pendulum. Moreover most articles that have used state-feedback designs have not
prescribed a systematic way of finding the weighting matrices for the LQR optimization. All
these shortcomings in the previous studies will therefore be eliminated in my summer research.
First a solution of the differential equations will be provided using Laplace transform. Second
the ratio between the output and the input obtained from the Laplace transform will be used as
transfer function to design an output-feedback controller using SISO tools from MATLAB.
Finally a stochastic method will be used along with the traditional PSO algorithm in order to
propose a systematic and more efficient way of choosing the weighting matrices.
Literature Review
An inverted pendulum is a non-linear system which can be controlled by an output-
feedback or state-feedback controller. While the state-feedback assumes that the designer knows
all state variables, the output-feedback consider the fact that not all state variables are always
available for measurement. In the state-feedback which most authors explored, there is no
systematic way of determining the gain matrices. The following authors investigate the factors
that control the dynamic of an inverted pendulum and various evolutionary algorithms of
selecting the weighting matrices through either statistical analysis or computer simulation.
The mathematical modeling of the system was first introduced by Siebert (1986) who
used a geometrical approach to investigate the factors that influence the dynamic of the inverted
pendulum. Guida, Fabio, and Carmine (2010) also used similar method to confirm that the main
factors that influence the control system of an inverted pendulum are the dry friction in slide
bearing and dry friction in the joint bearing. Jeremic (2012) recently used energy approach to

4. D’Almeida 4
derive the same dynamic equations that the previous authors introduced with a dynamic
approach based on Newton or Lagrange law. Three (3) steps were followed: First they
introduced a quick overview of static and dynamic friction models; then they assumed the
existence of dry friction in slide bearing and no friction in joint bearing and then no friction in
slide bearing and friction in the joint bearing; finally they interpreted each case and proposed
results in picture form. Their articles were the starting point of the design and implementation of
the controllers for an inverted pendulum as it carried out the dynamic equations that the
following authors will use as a tool to implement their control system.
Now that the dynamic equations have been carried out, most of the designers such as
Dotoli et al. (2001), Wait and Lee (2008), Tahir and Jawad (2012), and Reddy, Kumar, and Tao
(2014) introduced an output-feedback control design for an inverted pendulum using a Sliding
Mode Controller (SMC). They assert that the SMC is one of the best robust control device for the
inverted pendulum. They used simulation in MATLAB to conduct their investigation. They
introduced three Lyapunov’s functions and for each function a control device was designed: The
proportional Integral derivative (PID), the proportional derivative (PD), and the SMC; then they
simulated the performance of the pendulum using each of the control device; finally the analysis
and comparison of each response confirmed their claim. These authors intended to analyze the
performance of an inverted pendulum using different models in order to select the best design
that can efficiently control the dynamic of the inverted pendulum but they did not include
friction in their model.
Due to the vast application of an inverted pendulum in robotic devices, control designers
are now interested in optimal control. Nguyen et al. (2011) and Hi-Le, Duc-Trung, & Nhu-Lan
(2012) are one of the recent designers who introduced the concept of optimization in control

5. D’Almeida 5
system engineering in order to obtain a high quality control. They posit that by using a suitable
change of variable in the traditional dynamic equations, an LTI problem is obtained and by using
hedge algebra to solve the problem a high quality control can be designed. In order to prove their
claim, they used an empirical method by designing three different control devices: the
conventional fuzzy control (CFC), the fuzzy control using hedge algebras (FCHA) derived from
the CFC by adding the concept of hedge algebra, and the optimal fuzzy control using hedge
algebras (OFCHA) built from the FCHA by adding an optimization parameter. Upon simulation
of each controller, it was determined that the OFCHA and FCHA are simpler, more effective,
offer higher control quality, and more understandable in comparison with the method based on
CFC.
One of the problems with the optimization method is the initialization of the weighting
matrices that can minimize the performance index. The LQR method was therefore introduced to
design a state-feedback control system for the inverted pendulum. While Siebert (1986) was one
of the first to introduce the concept by using pole placement method to initialize the matrices,
Deris and Omatu (1996) assert that the Genetic Algorithm (GA), an evolutionary algorithm that
take sample possible individual (elements of the matrices) and employ coding (conversion from
decimal to 8 bits), selection, crossover, and mutation is even more efficient. The authors support
their claim by conducting two experiments. They designed two Proportional Integral Derivative
(PID) state- feedback controller: one using the GA algorithm to select the weighting matrices and
the other using the conventional manual pole placement method; after simulating each controller
on the inverted pendulum, the authors found that the two controllers offer approximately the
same gain but the time response of the controller based on the GA is small compare to the
conventional controller.

6. D’Almeida 6
Recently the PSO has been identified as the most effective algorithm of selecting the
weighting matrices. Mobayen, Rabiei, Moradi, and Mohammady (2011), Zara et al. (2012), and
Kumar and Pandey (2015) posit that in comparison with the GA, the Particle Swarm
Optimization (PSO) based on social swarm behavior is very efficient and robust in designing of
optimal LQR controller. To support their claim, they used empirical method by designing three
different control devices: trial and error, GA, and PSO; then compare their robustness by
simulation on aircraft landing system using MATLAB as software. High promising results
demonstrate that the proposed method is very flexible, efficient and robust against changes in
parameters, and can obtain higher quality solution with better computational efficiency and fast
convergence. The simulation results are very satisfactory in comparison with previous
experiments, that is, GA and trial and error. The authors write this article to explore a new
efficient and optimal possible algorithm for selecting the weighting matrices in LQR control
design.
From the PSO, more efficient and optimal method have been introduced. While the PSO
was based on Newton classic mechanic theory, the Quantum Particle Swarm Optimization
(QDPSO) introduced by Kavey and Won-sook (2014) is based on quantum mechanics principle.
Vinodh and Jovitha (2014) add an Adaptive Inertia Weight Factor (AIWF) to the traditional PSO
to design a more efficient PSO named Adaptive Particle Swarm Optimization (APSO). The
simulation suggests that the new PSOs carried out other techniques in terms of rising time,
settling time and quadratic performance index. Additionally, they are competitive in terms of
maximum overshoot percentage and steady-state error.
From the LQR method, the concept of hybridization has been introduced by some
designers. Asa et al. (2008), Benjanarasuth and Nudrakwang (2008), Singh, Bhatotia, & Mitra

7. D’Almeida 7
(2012) assert that by assembling two or more different design of controllers one will surely get a
hybrid controller with better performance. They started their research by designing 3 models of
controllers: the linear quadratic regulator (LQR) obtained from the CFC by minimizing the
performance index; the fuzzy logic controller (FLC) also known as a state controller which
actual output is a result from both input and prior output of the system; and the adaptive neuro-
fuzzy inference system (ANFIS) controller, which is a combination of the first two. A
comparison of simulation results shows that each controller can give stability for the inverted
pendulum but that the settling time is less in the case of hybrid controller. Further investigation
also confirmed that hybrid controller is more robust to parameter variation compare to the other
controllers. These authors intended to inform control engineers that hybridization is another
efficient and easy way to obtain a high quality control without using mathematical model. But
the drawback with this method is the high cost of the controller.
The research presented above investigate different methods of designing a control
system for the inverted pendulum. In my summer research, first I will use Laplace Transform to
solve the dynamic differential equations. Second a PID output-feedback control system will be
implemented using the results from Laplace transform as a transfer function. Finally a stochastic
method of selecting the weighting matrices will be examined.

8. D’Almeida 8
Methods
1- Mathematical Model of the Cart Pendulum
The free boby diagram of the system is
shown on the left (figure 1). By
applying Newton’s law we get the
following equations in which x
represents the position of the cart , ϴ is
the pendulum angle with respect to the
vertical stable position, F is the motrice
force applied to the cart as a disturbance
to the pendulum and Lfric and τfric are
respectively the friction force between cart and the bench and the friction torque in the
joint or pivot point of the pendulum.
{
(𝑚 𝑝 + 𝑚 𝑐)𝑥̈ + 𝑙𝑚 𝑝 sin(𝜃) 𝜃̈ − 𝑙𝑚 𝑝 sin(𝜃) 𝜃̇2
= 𝐹 + Lfric (1)
4
3
𝑙2
𝑚 𝑝 𝜃̈ + 𝑔𝑙𝑚 𝑝 𝑠𝑖𝑛(𝜃) + 𝑙𝑚 𝑝 𝑐𝑜𝑠(𝜃)𝑥̈ = τfric (2)
With the approximation that the unstable position of the inverted pendulum corresponds to
(𝑥, 𝜃, 𝑥̇, 𝜃̇) = (0, 𝜋, 0, 0), the mathematical model linearized around the unstable position of the
pendulum is given by the system below:
{
𝑥̈ =
1
𝑀
(𝐹 + 𝑙𝑚 𝑝 𝛼̈ + Lfric) (3)
𝛼̈ =
3
4𝑙2 𝑚 𝑝
(𝑔𝑙𝑚 𝑝 𝛼 + 𝑙𝑚 𝑝 𝑥̈ + τfric) (4)
Were 𝛼 = 𝜃 − 𝜋 𝑎𝑛𝑑 𝑀 = 𝑚 𝑝 + 𝑚 𝑐

9. D’Almeida 9
By using the friction model proposed by Guida, Fabrio, and Carmine (2010) where (Lfric =
−bẋ and τfric = −σα̇ ), equations (3) and (4) become:
{
𝑥̈ =
1
𝑀
(𝐹 + 𝑙𝑚 𝑝 𝛼̈ − bẋ) (5)
𝛼̈ =
3
4𝑙2 𝑚 𝑝
(𝑔𝑙𝑚 𝑝 𝛼 + 𝑙𝑚 𝑝 𝑥̈ − σα̇ ) (6)
By applying Laplace transform to the above time domain equations (5) and (6) we obtain
frequency domain equations that give us a relationship between the over all input force F and the
output representing the angular position α with the initial condition α(0) = 0,
α̇(0) = 0, and F(0) = 0. We will call 𝛼̃(𝑠), 𝐹̃(𝑠), and 𝑋̃(𝑠) the laplace transform of
𝛼(𝑡), 𝐹(𝑡), 𝑎𝑛𝑑 𝑥(𝑡) respectively.
From (5) it follows:
𝑠2
𝑋̃(𝑠) =
1
𝑀
(𝐹̃(𝑠) + 𝑙𝑚 𝑝 𝑠2
𝛼̃(𝑠) − 𝑏𝑠𝑋̃(𝑠))
By solving for 𝑋̃(𝑠) we get:
𝑋̃(𝑠) =
𝐹̃(𝑠) + 𝑙𝑚 𝑝 𝑠2
𝛼̃(𝑠)
𝑏𝑠 + 𝑀𝑠2
(7)
From (6) it follows:
𝑠2
𝛼̃(𝑠) =
3
4𝑙2 𝑚 𝑝
(𝑔𝑙𝑚 𝑝 𝛼̃(𝑠) + 𝑙𝑚 𝑝 𝑠2
𝑋̃(𝑠) − σs𝛼̃(𝑠) )
By solving for 𝛼̃(𝑠) we get:
𝛼̃(𝑠) =
3𝑙𝑚 𝑝 𝑠2
𝑋̃(𝑠)
4𝑚 𝑝 𝑙2 𝑠2 − 3𝑔𝑙𝑚 𝑝 + 3σs
(8)

10. D’Almeida 10
Now by substituing (7) in (8) we have:
𝜶̃(𝒔) =
𝟑
𝟒𝑴𝒍
𝒔
𝒔 𝟑 + (
𝟑𝝈
𝟒𝒎 𝒑 𝒍 𝟐 +
𝒃
𝑴
) 𝒔 𝟐 + (
𝟑𝒃𝝈
𝟒𝑴𝒎 𝒑 𝒍 𝟐 −
𝟑𝒈
𝟒𝒍
) 𝒔 −
𝟑𝒃𝒈
𝟒𝑴𝒍
𝑭̃(𝒔) (9)
In this study we are not interested in the motion of the cart but only the stability of the pendulum
in it unstable position. So we will not find an explicit formula for 𝑋̃(𝑠).
Equation (9) gives us the transfer function that will be usefull for the design of the PID
controller:
𝑻(𝒔) =
𝜶̃(𝒔)
𝑭̃(𝒔)
=
𝟑
𝟒𝑴𝒍
𝒔
𝒔 𝟑 + (
𝟑𝝈
𝟒𝒎 𝒑 𝒍 𝟐 +
𝒃
𝑴
) 𝒔 𝟐 + (
𝟑𝒃𝝈
𝟒𝑴𝒎 𝒑 𝒍 𝟐 −
𝟑𝒈
𝟒𝒍
) 𝒔 −
𝟑𝒃𝒈
𝟒𝑴𝒍
2- PID Controller Design with SISO Tools from MATLAB
Refering to a MATLAB tutorial for a PID controller design, we create the m-file in appendix A
that design the plant model. The plant structure is shown below:

11. D’Almeida 11
In our study since we are attempting to control the pendulum’s position, which should return to
the vertical after the initial disturbance, the reference signal (r) we are tracking should be zero.
Moreover ∅ = 𝛼̃(𝑠) and:
𝑇(𝑠) =
𝛼̃(𝑠)
𝐹̃(𝑠)
=
𝑃𝑝𝑒𝑛𝑑(𝑠)
1 + 𝐶(𝑠)
=
3
4𝑀𝑙
𝑠
𝑠3 + (
3𝜎
4𝑚 𝑝 𝑙2 +
𝑏
𝑀
) 𝑠2 + (
3𝑏𝜎
4𝑀𝑚 𝑝 𝑙2 −
3𝑔
4𝑙
) 𝑠 −
3𝑏𝑔
4𝑀𝑙
From the above transfer function and the built-in functions C = pid(KP , KI , Kd) and
T = feedback(P_pend , C) in MATLAB where KP , Ki , Kd represents the proportional gain, the
integral gain, and the derivative gain respectively, we can stimulate the dynamic of the pendulum
under the close-loop PID (Proportional Integral Derivative) feedback control. Before getting into
the next section, let’s clarify the practical meaning of each of the three gains KP , Ki , Kd.
- The proportional gain KP is the amount the error signal (e) is multiplied by directly so
that the proportional part of the controller acts o the present value of the error.

12. D’Almeida 12
- The integral gain Ki is the inverse of the time applied to the error signal so that the
integral part of the controller represents the average of past errors. It’s can also be seen as
the acceleration of the error signal.
- The derivative gain Kd is the speed in which the error signal changes so that the
derivative part of the controller represents a prediction of future errors based on linear
extrapolation.
From the above definition, the control function in time domain can be formulated as:
𝐶(𝑡) = 𝐾𝑝 𝑒(𝑡) + 𝐾𝑑
𝑑
𝑑𝑡
𝑒(𝑡) + 𝐾𝑖 ∫ 𝑒(𝑡)𝑑𝑡 (10). This formula has a very mportant
application developed in the next section.
3- Analog Design of the Controller
From formula (10) above it can be seen that the controller can be easily designed by combining
three (3) operational amplifier:
- An inverted amplifier with gain Kp;
- A derivative amplifier with gain Kd;
- An integration amplifier with gain Ki
All the three amplifiers should be mounted together in a summation mode with a fourth amplifier
as shown on the block diagram below.

13. D’Almeida 13
The schematic of each section is shown below.
- Inverted amplifier
-−𝐾 𝑝 𝑒 𝑡
−𝐾𝑖 𝑒 𝑡 𝑑𝑡
−𝐾 𝑑
𝑑
𝑑𝑡
𝑒 𝑡

14. D’Almeida 14
- Integration amplifier
- Derivative amplifier
- Summation amplifier
𝑅1 = 𝑅2 = 𝑅3 = 𝑅
𝑘𝑖 =
1
𝑅 𝐵 𝐶𝑖
𝐾 𝑑 = 𝑅𝐶

15. D’Almeida 15
Results
Before we move forward let us remind the reader that our goal is to stabilize the pendulum in its
unstable position as quickly as possible. In practice, this means we want a design requirements of
α ≤ 0.05 radian and t < 3 seconds.
1- Initial Simulation
The MATLAB code in appendix B is obtained by modifying the code in appendix A that
we used to define the plant. The initial simulation consists in tuning the controller by
sending an impulse disturbance in order to find out how it affects the initial gains
KP = 1 , KI = 1 , and Kd = 1. From there we will know how to adjust the gains in order
to minimize the angle 𝛼. By running the code in appendix B we get the following graph
representing the response of the pendulum angular position to the above disturbance.

16. D’Almeida 16
2- Influence of the gains on the Stability
By changing the set up for the gain so that KP = 100 , KI = 1 , and Kd = 1 we obtain the
response below:
By changing the set up for the gain so that KP = 100 , KI = 1 , and Kd = 20 we obtain
the response below:

17. D’Almeida 17
By changing the set up for the gain so that KP = 100 , KI = 0 , and Kd = 20 we obtain the
response below:

18. D’Almeida 18
The sumary of the reults is shown in the folowing table.
Gain
Maximum
angle (rad)
Final
Position
(rad)
Settle
time (s)
𝐾𝑝 𝐾𝑖 𝐾𝑑 𝛼 𝑚𝑎𝑥 𝛼 𝑓𝑖𝑛𝑎𝑙 𝑡𝑓𝑖𝑛𝑎𝑙
Simulation 1 1 1 1 0.654 -0.0143 5.04
Simulation 2 100 1 1 0.162 0 3
Simulation 3 100 1 20 0.0427 0 1.015
Simulation 4 100 0 20 0.0427 0 1.015

19. D’Almeida 19
Discussion
From the previous results the reader can see that during simulation 1, the pendulum is
subject to an aperiodic motion where its swinged up to an amplitude of 0.654 before stabilizing
at an angle of 0.0143 radian counterclock wise after 5.04 seconds. In the real world this means,
for example in case of a rocket, that we lunch a rocket toward a target located at 0 radian from
the north meridian but it ends up impacting another target located at 0.0143 radian. This
unaccuracy will certainly results in the death of thousands of people. Since the design
requirements are not met, a second simulation is needed.
From the second simulation one can see that the pendulum has a pseudo-periodic motion
with a maximal amplitude of 0.162 radian and then stabilized at its vertical unstable position as
expected. But the problem here is that the amplitude and the settling time are above the
maximum specifications. Also in the real word the pseudo periodic motion can be interpreted as
a vibration of an airplane for 3 seconds. A third simulation is therefore an obligation.
The third simulation is one of the best so ever because not only the pendulum is no
longer subject to vibration (pseudo-periodic motion) mut also the one time maximum amplitude
is less than 0.05 randian and it takes only 1.015 second for the pendulum to be stabilized at the
vertical unstable position. We therefore meet the design requirements and a further simulation is
no longer needed. But as one can see, we have never tuned the integral gain during the previous
simulation; Maybe we don’t realy need an integral gain. In order order to verify this assumption
another simulation with 𝐾𝑖 = 0 is needed.

20. D’Almeida 20
The fourth simulation is a strong evidense that an integral gain is not necessary for the
stabilization of the inverted pendulum. All we need is Kp = 100 and Kd = 20 for the pendulum
to stabilize at its unstable position in approximately one second.
Conclusion
In this research paper we were able to solve the dynamic equations using Laplace
transform and from the solution found, a PID controller with 𝐾𝑝 = 100 , 𝐾𝑖 = 1, and 𝐾𝑑 = 20
has been identified to be above and beyond the design requirements. However the same result is
obtained by setting 𝐾𝑝 = 100 , 𝐾𝑖 = 0, and 𝐾𝑑 = 20 . Therefore a PD design might be used in
order to reduce the cost of design. But should one chose to use a PID design, he/she must make
sure that the integral gain does not exceed 7 otherwise the pendulum will stabilize at a nonzeero
angle about its unstable position. We did not deeply cover the stochastic method of selecting the
weighting matrix because as my mentor suggested, this will be the main topic of my PhD thesis
that I will be working on for the nex four years.

21. D’Almeida 21
Appendix A: MATLAB Code that defined the Plant
Appendix B
% this code defines the pid
mp = 0.2;
mc = 0.5;
b = 0.1;
sigma = 0.01;
g = 9.8;
l = 0.3;
M = 0.7;
A = 3/(4*M*l);
B = 3*sigma/(4*mp*l^2) + (b/M);
D = 3*b*sigma/(4*M*mp*l^2-3*g/(4*l));
E = 3*b*g/(4*M*l);
s = tf('s');
p_pend = (A*s)/(s^3+B*s^2+D*s-E);

22. D’Almeida 22
Appendix B: This Code performed the simulations
Acknowledgement
Moysey Brio, PhD
Andrew Huerta, PhD
Nura Dualey, MA
% this code performs the simulation for the PID
controller
mp = 0.2;
mc = 0.5;
b = 0.1;
sigma = 0.01;
g = 9.8;
l = 0.3;
M = 0.7;
A = 3/(4*M*l);
B = 3*sigma/(4*mp*l^2) + (b/M);
D = 3*b*sigma/(4*M*mp*l^2-3*g/(4*l));
E = 3*b*g/(4*M*l);
s = tf('s');
p_pend = (A*s)/(s^3+B*s^2+D*s-E);
kp = 1;
ki = 1;
kd = 1;
C = pid(kp,ki,kd);
T = feedback(p_pend,C);
t = 0:0.01:10;
impulse(T,t)
title('Response of Pendulum Angle to an Impulse
Disturbance Under PID Control: kp = 1, ki = 1, kd
= 1')

23. D’Almeida 23
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