LQR control for a rotary double inverted pendulum - Nguyen Dang ...

LQR control for a rotary inverted pendulum
C. Mira 1
1 Program of Physical Engineering, School of Science and Humanities, EAFIT University, Medellín, Colombia
Abstract - This paper presents the procedure to obtain a
Linear Quadratic Regulator (LQR) control for a rotary
double inverted pendulum. The theoretical description of
the movement is developed using the Euler-Lagrange
equations and it is implemented using Maple. Analysis of
stability, controllability and observability are carried
out, also, using Maple tools. Finally the obtained control
is presented as a linear combination of the dynamic
variables of the system. Also is shown the evolution of
these variables from a set of initial conditions to a stable
state and is exposed a short analysis about the ability of
response of the control respect the initial conditions.
Keywords: Rotary double inverted pendulum, control
LQR, Euler-Lagrange.
1 Introduction
The inverted pendulum is a typical nonlinear
mechanical system used for testing control algorithms
[1]. Generally, these control system are based in
feedback control methods. They are different kinds of
inverted pendulum, but the common goal is to balance a
link on end using feedback control [2].
In this paper the case of study is a multiple input
single output control for a rotary double inverted
pendulum. The rotational is a rather challenging
inverted pendulum [2].
2 Problem
The problem consist in describing the movement of
the system composed by three arms of large L,L1,L2 and
three masses M, m1,m2.
y
z
θ
L
Ф1
Fig 1. Schematic rotary double inverted pendulum
M
m1
L1
m2
Ф
2
L2
x
The movement of the first arm of large L is
constrained to the plane xy and it rotates around the z
axis. The movements of the other two arms are
constrained to a vertical plane normal to the first arm.
The first arm is driven by a DC motor, whose
voltage input is defined by a multiple input single output
control system. The inputs of the control system are the
angular positions θ, Φ1 y Φ2 of the three arms. The
objective is to maintain the arms, L1, L2 in vertical
position.
The differential equations that describe the
dynamics of the system are:
Where,
(1)

In this expression u(t) is the output of the control
system.
Where,
Where,
(2)
(3)
3 Method
The equations that describe the problem are
developed using Euler-Lagrange mechanics. The
lagrangian is the difference between the kinetic energy
and the potential energy. For the system under study it
could be written as:
Where,
The equations of motion are obtained by means of
the minimum action principle. Where the action is
defined as:
Functional derivatives of the action respect each
movement parameter (θ, Ф1, Ф2) are computed. The
functional derivative of the action respect the angle θ is
equal to u(t), which is the only external action in the
system. The functional derivatives of the action respect
Ф1 and Ф2 are equal to zero.
Subsequently the movement equations are solved
for the angular velocities. Controllability and
observability matrices are constructed to verify that the
(4)
(5)

system with particular parameter values (M, m1, m2, L,
L1, and L2) is controllable. Finally the LQR control is
computed and simulations of it behaviour are made.
4 Results
A controlled is designed for a system with the
parameters listed in the table 1.
Table 1 Parameters values
Parameter Value
g 9.8 m/s 2
M 10 kg
m1 2 kg
m2 1.5 kg
L 3 m
L1
1 m
0.5 m
L2
The LQR controller is founded as a linear
combination of the dynamic variables of the system.
For implementation purposes the output of the
control is converted in a voltage signal. The input
variables are sensed by linear potentiometers and are
analogical differentiated. A summation of the input
variables regulated by resistances, proportional to the
coefficients in the linear combination, is made to obtain
the output voltage. A scheme of the analogical control
system is shown in the figure 2.
Fig 2. Analogical circuit for the control system
Also simulations of the behaviour of the control are
made. By means of these simulations is found that the
initial values of the angles Ф1 and Ф2 are limited to a
maximum value 36 degrees.
(6)
In the following graphics some results are shown
for a particular case of solution where the initial
conditions were those listed in the table 2.
Table 2 Initial conditions for a particular case of study
Value
Angle
(degrees)
θ 0
Ф1 18
32.7
Ф2
The evolution of position of the arm control driven
from the initial condition to the equilibrium condition is
shown in the figure 3.
Fig 3 Evolution of θ for the particular case of study
(angle in radians, time in seconds).
The evolutions of the angular positions of the
inverted arms are shown in figures 4 and 5.
Fig 4 Evolution of Ф1 for the particular case of study
(angle in radians, time in seconds).

Fig 5 Evolution of Ф2 for the particular case of study
(angle in radians, time in seconds).
5 Conclusions
A LQR control for a double inverted pendulum was
developed. This control is represented by a linear
combination of the dynamic variables of the system, then
is feasible its implementation with analogical
electronics.
Animations made show that the angles that
describe the positions of the elements evolve to
equilibrium conditions.
The control found has a lower robustness because it
only allows initial angles lower than 36 degrees. A more
robust control system would imply more advanced
techniques.
The use of advanced computation tools like Maple
has great advantages. Otherwise long mathematic
operations should be made involving a lot of time and
limiting the possibility of changing parameters.
The formulation of Euler-Lagrange mechanics
allows finding the movement equations of the system in
consistent way. These procedure is also facilitate by the
use ·Physics Maple library.
6 References
[1] S.A: Reshmin and F. L. Chernous’ko. “A Time-
Optimal Control Synthesis for a Nonlinear Pendulum”.
Journal of Computer and Systems Sciences
International, 2007, Vol. 46, No. 1, pp. 9–18. ISSN
1064-2307.
[2] S. Awtar, N. King, T. Allen, I. Bang, M. Hagan,
D. Skidmore, K. Craig. “Inverted Pendulum Systems:
Rotary and Arm-Driven: A Mechatronic System Design
Case Study”. Mechatronics. Department of Mechanical
Engineering, Aeronautical Engineering and Mechanics
Rensselaer Polytechnic Institute
[3] K. D. Pham, M. K. Sain, and S. R. LIBERTY
“Cost Cumulant Control: State-Feedback, Finite-
Horizon Paradigm with Application to Seismic
Protection”. Journal of Optimization Theory and
Applications: Vol. 115, No. 3, pp. 685–710, December
2002.