Analysis & Control of Inverted Pendulum System Using PID Controller

Vivek Kumar pathak. Int. Journal of Engineering Research and Application www.ijera.com
ISSN : 2248-9622, Vol. 7, Issue 5, ( Part -4) May 2017, pp.01-04
www.ijera.com DOI: 10.9790/9622-0705040104 1 | P a g e
Analysis & Control of Inverted Pendulum System Using PID
Controller
Vivek Kumar pathak *, Sankalp paliwal**
*(PG Student [PS&C], Dept. of EE, Babu Banarasi Das University, Lucknow, and Uttar Pradesh, India)
** (Department of EE, Babu Banarasi Das University, Lucknow, and Uttar Pradesh, India
ABSTRACT
This Analysis designs a two-loop proportional–integral–derivative (PID) controller for an inverted cart–
pendulum system via pole placement technique, where the (dominant) closed-loop poles to be placed at the
desired locations are obtained from an Linear quadratic regulator (LQR) design. It is seen that in addition to
yielding better responses (because of additional integral action) than this LQR (equivalent to two-loop PD
controller) design, the proposed PID controller is robust enough. The performance and of the PID compensation
are verified through simulations as well as experiments.
Keywords: PID, inverted pendulum, pole placement, LQR design, robust
I. INTRODUCTION
The inverted cart–pendulum is an example of
under actuated, non-minimum phase and highly
unstable system. Therefore a controller design is
difficult for such a system. The design becomes more
difficult because of the physical constraints on track
length, applied voltage and the pendulum angle.
It is well known that proportional–integral–
derivative (PID) controllers are widely used in control
systems The design of these controllers, however, are
generally carried out using some tuning approaches
such as trail error method, [1]. For the inverted cart–
pendulum system, it is however seen that a design of
LQ controller, which is equivalent to a two-loop PD
controller here, exhibits poor real-time cart responses
containing sustained oscillations. To achieve better
response, this paper designs a two-loop PID controller
using pole placement technique where the (dominant)
closed-loop poles are placed at the same locations as
obtained from the above LQR design. It is seen that
while the robustness of LQR design is almost retained
in the two-loop PID design, the real-time cart
response is superior to LQR because of additional
integral action. The performance and robustness of the
proposed two loop PID design have been verified
using simulations. In [2], a two-loop PID controller
was designed for an inverted cart–pendulum system
based on simultaneous tuning of the controllers using
pole–placement technique This controller was,
however, tested on SIMULINK environment only. In
[3], a PD controller was designed and tested on
experimental set-up, but, as expected, the real-time
pendulum angle and cart responses exhibit
oscillations. In [4], a linear state feedback controller
was used to ensure infinite gain margin (GM). In this
paper pole placement technique has been used to
stabilize the MIP system. The stabilization of MIP
system using pole placement method has been
discussed in the literature [8-9].
In this paper PID controllers have been used
to stabilize the MIP system. The tuning of the PID
controllers has been done using pole placement
technique. In this technique the dominant closed loop
poles are placed at desired locations which are
obtained from LQR technique. By placing these poles
in the characteristic equation of the MIP system the
parameters of the PID controllers are obtained.
II. STRUCTURE AND
MATHEMATICAL MODEL OF
INVERTED PENDULUM
The x inverted pendulum on a pivot driven
by horizontal control force is shown in fig 1(a) the
control based on the horizontal displacements x
inverted pendulum are the total kinetic energy and
potential energy . K=1/2MX2+1/2(XP
2
+Z2
), Z= mgzp
Let l is the distance from the pivot to the maas center
of the pendulum M,m are the piot and the pendulum
(x,z) respectively(x,z) is the position of the pivot in
the xoz coordinate is the speed in the xoz coordinate
(xp,zp) and angle ϴ and Fx is the horizontal control
force.
RESEARCH ARTICLE OPEN ACCESS

Figure -1
Inverted pendulum system free-body diagrams of the
two elements
Figure -2
free-body diagram of the cart in the horizontal
direction Summing the forces you get the following
equation of motion Mx+bx+N=F (1)
forces in the free-body diagram of the pendulum in
the horizontal direction, you get the following
expression for the reaction force .
N= mx + mlϴcosϴ-ml ϴ2
sinϴ (2)
One of the two governing equations for this system is
obtained if this equation is substituted into the first
equation
(M+m)x +bx+ mlϴcosϴ-ml ϴ sinϴ = F (3)
To get the second equation of motion for this system,
sum the forces perpendicular to the pendulu
PSINϴ+NCOSϴ-mg Sinϴ = ml2
ϴ+mxcosϴ (4)
To get rid of the and terms in the equation
above, sum the moments about the centroid of the
pendulum to get the following equation
-PlSINϴ -Nlcosϴ =Iϴ (5)
Combining these last two expressions, the second
governing equation is obtained
( I+ml2
) ϴ+mglsinϴ = - mlxcosϴ (6)
Let represent the deviation of the pendulum‟s
position from equilibrium, that is, = + . Again
presuming a small deviation ( ) from equilibrium, we
can use the following small angle non linear function
cosϴ = cos(Π+φ) = - 1 (7)
Sinϴ =Sin (Π+φ) = - φ (8)
ϴ2
= φ2
= 0 (9)
After substiting the above approximations into our
nonlinear governing equations, the two linearized
equations of
Motion are obtained. Note has been substituted for
the input
(I+ml2
) φ- mglsin φ = - mlx (10)
(M+m)x +bx-mlφ = u (11)
underline any of the headings, or add dashes, colons,
etc.
III. CONTROLLER DESIGN
Stabilizing the IP a two loop control scheme
has been used for stabilizing the IP as shown in figure
2. PID1 has been used for controlling the position x
while PID2 for controlling angle PID1 has been used
for controlling the position x while PID2 for
controlling angle .
The basic equation of a PID controller is given as
(12)
Where e (t) is the error between the reference and the
feedback of the system. In order to control the system
two PID controllers are required one for controlling
the position of the robot and the other for controlling
the pendulum angle. The two loop PID controller is
shown in figure 2. As shown in figure PID1 is used
for position control while PID2 is used for stabilizing
angle of the MIP. P1=X(s)/U(s) and P2=
represents the two transfer functions. The tuning of
the PID controllers has been done using pole
placement technique. In this technique dominant
closed loop poles are placed at desired locations
which are obtained from LQR technique.
The characteristic equation for the control
(13)
Where and are given
(14)
(15)
Let the desired characteristic equation be
(16)
The dominant poles of the two loop PID controller are
obtained using LQR design.
The performance index is given as
(17)
Here Q is the state weighted matrix and R is the
control weighted matrix the performance index j is
minimized using Riccati equation given by
(18)

State feedback gain vector K [K1,K2,K3,K4] is
obtained by
(19)
For simplicity the state weighted matrix is chosen as
Q=diag {q1,q2,q3,q4} and the control weighted
matrix is chosen as a scalar vector R=r. In this paper
Q vector is q1=500q, q2=q3=20q, q4=q and r= 10n
.
By trial and error It is found that for optimal results
q=100 and n=4.
.
Figure 3 two loop PID controller for IP system
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IV. SIMULATION RESULTS
The state space obtained after putting these values is
(16)
The closed loop poles are obtained by the
Eigen values of A-BK. The four poles obtained are -
5.7391,-5.451,-1.3209±1.209. The fifth pole is taken
as six times the real part of the most dominant pole -
7.92
The matrix has 2 rows because both the cart's
position and the pendulum's position are part of the
output. Specifically, the cart's position is the first
element of the output and the pendulum's deviation
from its equilibrium position is the second element
of
Figure 4 Angle vs. time curve
Figure 5 positions vs. time curve
Controller Kp Ki Kd
PID1 52.35 102.93 10
PID2 18.27 13.91 3.224
Table 1 PID Parameter Using Pole Placement
V. CONCLUSION
A two loop PID control scheme has been for
stabilization of IP system. The tuning of the PID
controllers has been done using pole placement.
Simulation results show effective stabilization of IP
system with low rise time tr and overshoot in both
position and angle curves.
REFERENCE
[1]. Naidu, D.S.: „Optimal control systems‟
(CRC Press, FL, 2003)
[2]. Nasir, A.: „Modelling and controller design
for an inverted pendulum system‟. Master of
Technology Degree, Faculty of Electrical
Engineering, Universiti Teknologi Malaysia,
Malaysia, 2007
[3]. Hung, C.C., Fernandez, B.R.: „Comparative
analysis of control design techniques for a
cart-inverted-pendulum in real-time
implementation‟. Proc. American Control
Conf., June 1993
[4]. Lin, Z., Saberi, A., Gutmann, M., Shamash,
Y.A.: „Linear controller for an inverted
pendulum having restricted travel: a high-

and-low gain approach‟, Automatica, 1996,
32, pp. 933–937
[5]. „Digital pendulum control experiments
manual‟ (Feedback Instruments Ltd, UK,
2002)
[6]. Wolovich, W.A.: „Automatic control
systems‟ (Saunders College Publishing,
Rochester, NY, 1994)
[7]. Skogestad, S., Postlethwaite, I.:
„Multivariable feedback control, analysis and
design‟ (John Wiley, Chichester, 2005)
[8]. [8]. A. Ghosh T.R. Krishnan B. Subudhi
“Robust proportional–integral–
derivativecompensation of an inverted cart–
pendulumsystem: an experimental study”
IET Control Theory and Applications
pp.1145-1152, 2012
[9]. [9] N.Rajak “Stabilization of Cart-Inverted
Pendulum Using Pole-Placement Method”
International Journal of Advanced
Engineering Research and Studies 308-312,
2015

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