Intelligent Control and Automation, 2011, 2, 160-166
doi:10.4236/ica.2011.22019 Published Online May 2011 (http://www.SciRP.org/journal/ica)
LQG Control Design for Balancing an Inverted Pendulum
Mobile Robot
Ragnar Eide, Per Magne Egelid, Alexander Stamsø, Hamid Reza Karimi
Department o f En g i neer i n g , Faculty of Engineering and Science, University of Agder,
Grimstad, Norway
Received December 26, 2011; revised May 17, 2011; acce pted May 19, 2011
Abstract
The objective of this paper is to design linear quadratic controllers for a system with an inverted pendulum
on a mobile robot. To this goal, it has to be determined which control strategy delivers better performance
with respect to pendulum’s angle and the robot’s position. The inverted pendulum represents a challenging
control problem, since it continually moves toward an uncontrolled state. Simulation study has been done in
MATLAB Simulink environment shows that both LQR and LQG are capable to control this system success-
fully. The result shows, however, that LQR produced better response compared to a LQG strategy.
Keywords: LQG Control, Inverted Pendulum, Mobile Robot
1. Introduction
Inverted pendulum has been the subject of numerous
studies in automatic control since the forties. The in-
verted pendulum is a typical representative of a class of
high-order nonlinear and non-minimum phase systems
[1]. Since the system is inherently nonlinear, it is useful
to illustrate some of the ideas in nonlinear control.
Wheeled mobile robots have in the recent years
become increasingly important in industry, since they
provide a large degree of flexibility and efficiency with
respect to transportation, inspection, and operation. In
many cases, however, it is the control, or lack of
knowledge in control, which limit the area of applicatio n .
In this system, an inverted pendulum is attached to a
robot equipped with a motor that drives it along a
horizontal track (robot will in the following mean an
autonomous robot provided with the Robotics Starter Kit
from National Instruments). The user is able to dictate
the position and velocity of the robot through the motor.
The pendulum is characterized by an unstable equi-
librium point, and its behavior can be used in the analysis
and stability control of many similar systems.
Many different control methods are proposed for the
inverted pendulum problem. The Proportional-Integral-
Derivative (PID) and Proportional-Derivative (PD)
controllers [2] and [3], Model Predictive Control (MPC)
[4], and fuzzy contr ol [5] to mentio n a few. How ever one
of the obstacles by using the PID and PD controllers are
that they alone cannot effectively control all of the
pendulum state variables (modes) since they are of lower
order than the pendulum itself. Hence, they are usually
replaced by a full-order controller [3]. A linear state
feedback controller based on the linearized inverted
pendulum model can instead be used, and may also be
extended with a disturbance observer (Kalman filter) to
improve the disturbance rejection performance.
The proposed method is to balance an inverted pen-
dulum placed on top of a mobile robot by use of LQR/
LQG control methods. Our solution implements an LQG
controller with comparison to a simple LQR controller.
The controller found by means of a more analytical
approach will be tested with implementation of the
controller in the MATLAB/Simulink environment.
With varying input forces the goal is to design a
controller capable of meeting the following requirements:
 settling time,
s
T, less than 5 seconds
 maximum overshoot of 10 degrees (0.175 radians)
 rise time T less than 0.5 s
r
We will in part one describe a simplified mathematical
model of the problem. Then, when having a state space
model of the system, we can in part two go on and
determine the controllers and observers. Finally, the
results from simulations will then be presented and
compared in part three.

R. EIDE ET AL.161
2. Derivation of the Mathema tical Mode l
The main objective in this part is to come up with a valid
model that can be used as a basis for the control design.
The overall system can be described as Figure 1 depicts.
The mass of the robot and the pendulum is here denoted
as M and m, respectively, together with the applied force
F and the angle

referred to the vertical axis.
There are a lot of aspects to take into consideration
when modeling this system, of which some are more
crucial than others. W ithout loosing the generality of the
model, the authors have considered the following
assumptions as good approximations for the sake of
simplifying the model:
 no friction in the hinge between the pendulum and the
robot
 no friction between the wheels and the horizontal
plane
 small angle approximations, i.e. the pendulum does
not move more than a few degrees
 sensors for measuring all the states are when con-
sidering LQR control assumed to be available
By applying the Lagrange’s equations with respect to
the position of the robot,
x
, and the pendulum deflec-
tion angle

, and taking into account the moments
around the center of mass, the following nonlinear
dynamic equations of the pendulum system are obtained
[6]:

2=
I
mlmglsin mlxcos

  
 
 (1)
where I is moment of inertia of the pendulum,

is the
angle in counterclockwise direction with respect to the
vertical line (see Figure 1), and is the distance to th e
center of mass (equal to l
2L). Furthermore, the robot is
governed by a equation of motion relating the forces
Figure 1. The inverted pendulum placed on a simplified
mobile robot.
applied by the pendulum on the robot. Then, according to
Newtons law of motion, the equation of motion of the
robot in horizontal direction can be given as


2=
M
mx mlcosmlsinF    
 
 (2)
where F can be derived out from the physical properties
of the motor to be [7]
22
2
=g
tmtKm K
KK
F
V
Rr Rr

 x (3)
Here ,,, ,,
mttg
K
KKR

and are coefficients de-
pendent on th e physical properties of th e motor and gear .
The different parameters for (3) is not known for the
robot used in this project. However, [8] provide values
which will be used in the following. Based on the
previous derivation, we can now define four state vari-
ables, given as the state vector
r

T
T
1234
==xxxx xx





 x (4)
Equations (1), (2), and (3) need to be linearized in
order to represent the system on state space form
=
x
Ax BV

 (5)
=
y
Cx (6)
where
A
, , and are matrices needed for the
control design and is the (input) voltage applied on
the motor. The linearizing point of interest is the unstable
equilibrium point
BC
V

T
=0000x
The assumption of small angle approximation gives
sin =

 , cos =1

, and . This lead to the
following linearized equations of motion (note that (3)
will be the same)
=0 




2
=mgl mlx
Iml





 (7)



=
F
ml
x
M
m




 (8)
We want to make (7) and (8) more convenient for the
state space representation by expressing

 and
x
 in
the following way
 
22
2
=mg mg
t t
KK KK
g
x
V
Lml RrMLml RrM Lml
 
 
  (9)



22
2
=
mg
t t
mg
t
LK Kmlg
xx
LM ml Rr LMml
LK KV
Rr LMml


 

 
(10)

R. EIDE ET AL. 162
where for simplicity 2
=
I
ml
Lml
 and =
t
M
Mm

.
Based on these equations and the following relation
T
24
=xxx





x
we can now represenfirst
The system on state sen be
t (9) and (10) as a order system.
pace form th





22
2
22
2
01 00

00
00
0
0
mg
mg
t t
mg
t
mg
t
KK g
LK Kmlg
LM ml Rr LMml
LK K
Rr LMmlV
KK
RrMLml







 






 






(11)
(12)
Now, by using the values for the different constants
given in [8], the matrix equations becomes


When having the system on state space form, the
following step is to design a LQR controller (assuming
that all the states are known and can be measured). Then,
by assuming no sensor available for msuring the angle,
we will continue with an observer dsign. Finally, an
op
placement procedure
nkage
es in
covior. However, one disadvantage with this
ethod is that the placing of the poles at desired
=00 01
tLm
l RrM Lml 




xx
1000
=0010 y


x
01 000
0 15.14 3.040
00 010






xx
3.39
+

V
037.2331.61 08.33



1000
0010
y 

x
ea
e
timal LQR (LQG controller)will be proposed by im-
plementing a Kalman filter.
3. Linear Quadratic Regulator (LQR)
Design
There are different methods, or procedures, to control the
nverted pendulum. One is the pole i
having the advantage of giving a much clearer li
between adjusted parameters and the resulting ch ang
ntroller beha
m
locations can lead to high gains [8]. In this section a
linear quadratic regulator (LQR) is proposed as a
solution. The principles of a LQR controller is given in
Figure 2. Here is the state space system represented with
its matrices
A
, B, and C with the LQR controller
(shown with the

K
).
The LQR problem rests upon the following three
assumptions [9]:
1) All the states (x(t)) are available for feedback, i.e. it
can be measured by sensors etc.
2) The system a stabiliz ble which means that all of
its unstable modee cont
re a
s arrollable
bsv(A,C) and ctrb(A,B)
an the control area is
ca for a linear system with quadratic
pe
3) The system are detectable having all its unstable
modes observable
To check whether the system is controllable and
observable, we use the functions o
d find this to be true.
LQR design is a part of what in
lled optimal control. This regulator provides an
optimal control law
rformance index yielding a cost function on the form
[10]
T
 
T
0
=d
J
xt tttt
Qxu Ru (13)
where T
=QQ and T =RR are weighting parameters
that penalize the states and the control effort, respec-
tively. These matrices are therefore controller tuning
parame

ters.
nce to
se of certain
e the st
It is cru must be chosen in
accorda the want
responr word;
i
cial that
emphasize we
states, or in
Q
otheto give the how we will
penaliz ates. Likewise, the chosen value(s) of R
will penalize the control effort u. Hence, in an optimal
control problem the control system seeks to maximize
the return from the system with mnimum cost. In a LQR
design, because of the quadratic performance index of
the cost function, the system has a mathe matical solution
that yields an optimal control law


=txtuK (14)
where u is the control input and K is the gain given
as 1
=TS

K
RB . It can be shown (see [10]) that S can

Figure 2. Schematic representation of state space control
using a LQR controller.

R. EIDE ET AL.163
be found by solving the algebraic Riccati Equation
(15)
The process of minimizing the cost function therefore
involves to solve this equation, which will be done with
the use of MATLAB function lqr. In this project the
parameters in was initially chosen according to


1=0 TT
SAA SQPBRBS
Q
Bryson’s Rule (see [11] for details) to be
100

000
01 0 0


=0032.650


Q
00 0 1


and the control weight of the performance index R was
set to 1.
Here we can see that the chosen values in Q result in
a relatively large penalty in the states
(16)
1
x
and 3
x
. This
means that if 1
x
or 3
x
e effect
is large, the
will amplify ofe largvalues in Q th 1
x
and 3
x
in the opti-
n problem are to mization proble.he izat
minimize J, the optim contm
m Since t
al optim rol io ust forc
u e the states
1
x
and 3
x
to be small (which makesysically sene ph
since 1
x
and 3
x
represent the position of the robot and
the angle of the pendulum
ha
,ectively). Tha respis vlues
must be modified during subsequent iterations to achieve
as good response as possible (refer to thext section for
results). On the other nd, the small e n
R
relative to t
max values in Q involves vy lowenalty on the
control effort u in the minimization of
he
er p
J
, and the
optimal control u can then be rge. For this small R,
e gain Kcan then be large resulting in a faster response.
In thephysical world this might involve instability
problems, especially in systems with saturation [8].
After having specified the initial weighting factors,
one important task is then to simulate to check if the
results correspond with the specified dign goals given
in the introduction If not, an ad justment o f the weigh ting
factors to get a controller more in line with te specified
design goals must be performed. However, difficulty in
finding the right weighting factors limits the application
of the LQR based controller synthesis [?]. By an iterative
study when changing Q values and running the
la
th
es
.h
com-
mand K = lqr(A,B,Q,R)


10.0000 12.6123 25.8648 6.7915 K.
The simulation of time response with this controller
will be shown in the next section.
State Estimation
As mentioned for the case of the LQR controller, all
sensors for measuring the different states are assumed to
be available. This is not valid assumption in practice. A
are not immediately avai
a
void of sensors means that all states (full-order state
observers), or some of the states (reduced order observer),
lable for use in any control
schemes beyond just stabilization. Thus, an observer is
re e schematics of the
stem with the observer is shown in Figure 3 below.
from Figure 3, the observer state
eq
lied upon to supply accurate estimations of the states at
all robot-pendulum positions. Th
syAs can be seen
uations are g i v en by

ˆˆ ˆ =
x
AxBuL yCx 
 (17)
ˆˆ
=yCx (18)
where ˆ
x
is the estimate of the actual state
x
. Further-
more, Equations (17) and (18) can be re-written to
become


ˆˆ
=
x
ALCxBuLy


 (19)
This, in turn, is the governing equations for a full
order observer, having two inputs u and y and one
output, ˆ
x
. Since we already know and servers
of this ki is simple in design a
estimation of all the states around t
From 3 we can se
difference between the mea
estimauts and correct the mde
Th
A, B
nd
o
u, ob
nd
Figure
ted outp
provides accurate
he linearized point.
e that the observer is im-
plemented by using a duplicate of the linearized system
dynamics and adding in a correction term that is simply a
gain on the error in the estimates. Thus, we will feed
back the sured and the
l continuously.
e proportional observer gain matrix,
L
, can be found
by pole placement method by use of e placecommand
in MATLAB. The poles were determined to be ten times
faster than the system poles. These were found to be


T
*=18.0542, 5.4482, 3.4398, 2.4187eig ABK,
which yields the gain matrix
34.4 0.2
91.0 1.1
=30.4 52.5
1103.6 726.4
th












L (20)
When combining the control-law design with the esti-
mator design we can get the compensator (see Section 6
for results).
4. Kalman Filter Desig
n
In the previous design of the state observer, the mea-
surements =
y
Cx
is not usually the case in
were assumed to be
practical li
inputs yielding the state equations to be on the general
stochastic state space form
noise free. This
fe. Other unknown
=
=
x
AxBu Gd
yCxHdn



 (21)
where the matrices can be set to an identity matrix, G
H
can be set toro, is stationary, zero mean zed

R. EIDE ET AL. 164
Figure 3. Schematic of state space control using a observer
where L is the observer gain and K is the LQR gain matrix.
Gaussian white process noise, and is sensor noise of
the same kind.
In this section we will show how the Kalman filter can
be applied to estimate the state vector,
n
ˆ
x
uand the output vector by using the known inputs and the mea-
sureme . Block schematics of a Kalman filter
connected to the plant is depicted in Figure 4.
It can be shown that this will result in an optimal state
estimation. The Kalman filter is essentially a set of
inimizes the estimated error covariance when some
.
is given by
ˆ
y
nts
y
mathematical equations that implement a predictor-cor-
rector type estimator that is optimal in the sense that it
m
presumed conditions are met [12]The mean square
estimation error


 

ˆˆ =T

J
Ext xtxt xt




(22)
where
T

  
ˆ=0Ext xtyt


(23)
The optimal Kalman gain is given by
 
1
=T
e
LtS tCR
 (24)
where

e
St is the same as
J
given in (22). Further-
more, when inft, the alge
T
braic Riccati equation can
be written
1
e
(25)
and are the process and measurement
noise cnpectively. Tuning
red if these are not known. Note that when
coer will be used
for me the position. Since this device does
not involv ancan in the fo
mal Kalman gain for
as

=0
e
St

0= T
eenen
SAASQSCR CS


where n Q
ovaria
filter are requi
easurem
e
Finally, the sub-opti
n R
ces, res
nt of
y noise, we
of the Kalman
ntrolling the robot, a quadrature encod
llowing set =0
n
R.
a steady state
Kalman filter can be expressed as 1
=T
e
LSCR

.
5. LQG Controller
The LQG controller is simply the combi
ar feed
ot system. The sche-
atics a LQG is in essence similar to that depicted in
nce, the observer gain matrix
in this figure can now be defined as the Kalman gain.
nation of a
Kalman filter with a regular LQR controller. The se-
paration principle guarantees that these can be designed
and computed independently [13]. LQG controllers can
be used both in linear time-invariant systems as well as
in linear time-variant systems. The application to linear
time-variant systems enables the design of line-
back controllers for non-linear uncertain systems, which
is the case for the pendulum-rob
m
Figure 3 in Section 3.1. He
L
However, we also assume disturbances in form of noise,
such that the system in compressed form can be
described as in Figure 5.
From the system given by (21), the feedback control
law given by (14), and the full state observer equation
given by (19) defined previously, we can by combing
these have the following output-feedback co ntr oller to be


ˆˆ =
ˆ
=
x
ALCBKxLy
uKx
 

 (26)
which is the equations the software is based upon when
lculating the state estimates. Since the optimal LQR
controller for this system is already found to have the
feedback gain matrix
ca


10.0000 12.6123K 25.8648 6.7915 
be this will
used directly in the LQG design. The remaining is to
find the Kalman gain matrix. We first assume the system
as given in (21). After the measurement and disturbance
noise covariance matrices are determined, the MATLAB
Figure 4. Kalman filter used as an optimal observer.
Figure 5. LQG regulator.

R. EIDE ET AL.165
function kalman was used to find the optimal Kalman
gain L and the covariance matrix P. Measurement noise
covariance matrix is determined out from an expected
noise on each channel. The Kalman gain was found as
(27
6. Simulations
Figure 6 gives the time response of the system with a
step input simulated by use of lsim function in
MATLAB.
Note that a steady state error was reduced by a scalin
The reason for the difference between the plots is due
to the different simulation methods. When simulating the
LQG, we made it be that only the
subjected to a impulse force. This resulted in the Figure
8. We actually see here that the LQR gave better results.
4.95445.0879
0.7797 0.8612
=
25.9957 27.8975
L






5.0879 5.4689

)
g
factor N after twas calculated
by a rscale function. Figure 7 is the response when u sin g
a observer to estimate the states. Note that the simulation
in this case was done in the Simulink environment with a
impulse function as input.
he reference. This factor
pendulum was
. Conclusions 7
This report has shown that by manipulating the state/
control weightings and noise covariance matrices pro-
perly, both LQR and LQG will give satisfactory result.
Although the results has been somewhat limited, it has
provided us with some useful knowledge concerning
Figure 6. Time response of the system using a LQR con-
troller.
Figure 7. Time response of the system with LQR and ob-
server.
Figure 8. Time response of the system with LQG and im-
ulse force on pendulum.
differences between LQR and LQG control. It could
have been interesting to see how other cost functions
would have affected the results, and how the system
would have reacted if some of the input values had been
changed.
8. References
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p
[2] J. Ngamwiwit, N. Komine, S. Nundrakwang and T. Ben-
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Pendulum System,” Proceedings of the 44th IEEE Con-
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R. EIDE ET AL. 166
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Ikeda and H. O. Wang, “Fuzzy Regulators
servers: Relaxed Stability Conditions and
ber 2005.
[3
Pendulum,” Proceedings of the 5th International Con-
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[4] R. Balan and V. Maties, “A Solution of the Inverse Pen-
dulum on a Cart Problem Using Predictive Control,”
Techn
[5] K. Tanaka, T.
and Fuzzy Ob
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