Bifurcation Analysis and Sliding Mode Control of
Ziegler’s Pendulum
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
E. Naseri*, R. Ghaderi*,
A. Ranjbar N.*, S.H. Hosseinnia*, M. Mahmoudian, S. Momani
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* Babol (Noushirvani) University of Technology, Faculty of Electrical and
Computer Engineering, P.O. Box 47135-484, Babol, Iran
([email protected]) ,([email protected]) ,([email protected])
** Department of Mathematics, Mutah University, P.O. Box: 7,
Al-Karak, Jordan
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
Abstract: Chaos and bifurcation is studied in Ziegler’s Pendulum. The sliding mode
controller is proposed to stabilize and control a two-arm pendulum. This controller
provides the robustness together with a satisfactory time response. The simulation result
verifies the significance of the proposed controller.
Keywords: Bifurcation Analysis, Chaos, Sliding Mode Control, Ziegler’s Pendulum,
Stability.
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1. INTRODUCTION:

Chaotic systems have recently been much considered
due to their potential usage in different fields of
science and technology particularly in electronic
systems (Roy R et al, 2000), secure communication
(Yang. T, 2004), computer (Chen. L et al, 2003). To
develop the chaotic theory, a control approach of
chaotic systems has been considered as an important
problem (Liu Shu-Yong et al, 2008). It was initially
assumed that, control of chaotic systems is impossible
and they have uncontrollable and unpredictable
dynamic. The imagination was changed when three
researchers (Grebogi, Yorke, ott) in the subject of
chaos (Ott E et al, 1990) have shown other vice. The
endeavour has been proceeded to control the chaos
using different approaches, e. g. feedback
linearization (Gallegos. JA, 1994; Yassen. MT, 2005;
Abdous F. et. al. , 2008a), Delayed feedback control
(Abdous F. et. al. , 2008b), OPF (E. R Hunt, 1991)
and TDFC (Pyragas. k, 1991).
The main objective of this paper is to control a
Pendulum model as a base plate of nonlinear systems
via sliding mode control. The chaotic model, Chaos
and their effects have recently been investigated.
Sensitivity to initial conditions is a common factor
and a main cause of chaos occurrence. Meanwhile
regular nonlinear control techniques may be applied
to cope with the chaos.
Bifurcation refers to the phenomenon that when the
system parameter reaches the critical value, the
system behaviour will vary suddenly and generally, it
may result in the breakdown of the completely
engineering system and cause inestimable loss.
Bifurcation can usually be divided into static and
dynamic. Up till now, many positive results of the
researches in this field have been achieved (Chen G et
al, 2000; Wang X et al, 2000; Xuedi Wang, Lixin
Tian, 2006). Chaos controlling can be divided into
two categories: one is to suppress the chaotic
dynamical behaviour and other is to generate or
enhance chaos in a nonlinear system. Chaos is
generally believed to be harmful, so the research
mainly focuses on how to remove or lessen the
chaotic of the system. Meanwhile, the bifurcation and
chaos are closely related, because the changes of the
system parameter may lead to system bifurcation and
then cause chaos.
The paper is organized as follows:
Section 2 deals with the chaos and bifurcation
phenomenon in Ziegler’s Pendulum. The sliding
mode controller is briefly described in section 3. The
Sliding Mode Control is used to control and stable
this system in section 4. Finally the work will be
closed by a conclusion at section 5.
2. STABILITY ANALYSIS OF ZIEGLER'S
SYSTEM
A schematic diagram of Ziegler’s PENDULUM
(Ziegler H, 1952) is shown in Figure (1). This system
is consisted of two arms of length l. The mass is
assumed either negligible or concentrated together
with separate connected mass at the end of tips. The
bars are connected by the frictionless joint, and the
configuration of the system is completely specified by
the two angles, formed between the vertical and each
of the two bars, respectively.
the instability is of the divergence type (increasing
amplitude without oscillation), associated with a
positive real root 2 of (2). Secondly, for example, if
  0.2 the flutter region extends from p = 2.244 to p =
6.580; for higher loads, the equilibrium states (which
have very high magnitudes of 01 and 02) are stable
(until p =23.770). To stabilize the chaos and delete
the bifurcation, a sliding mode control will be used in
next section.
Fig(1): Double pendulum with eccentric follower
load.
A dimensionless movement equation is as follows
(Guran A and R. H. Plaut, 1993):
1   2 cos(1   2 )   2 2 sin(1   2 )
,

(21   2 )k ( p)  p sin(1   2 )

2
1 cos(1   2 )   2  1 sin(1   2 )
(1)

( 2  1 )k ( p)  p .
m
l
Where :   1  1 , p 
,
m2
k (0)

K (  K (0) / l )
, k ( p) 
l
K (0)
Linearization at an equilibrium point
1  0,2  0 finds the characteristics polynomial
(Guran A and R. H. Plaut, 1993) which is as follows:
2
4
2

(   )  ( p   p  2 k ( p)

2
2

(2)
 k ( p)  2k ( p))  k ( p)  0,
p
  cos(
)
k ( p)
For the standard model, Fig. 2 depicts the real and
imaginary parts of the roots of (2) as functions of p
for the perfect system (   0 ) and for
  0, 0.1, 0.15 and 0.2 .Two features in Fig. 3 are of
theoretical interest, even though they occur after the
system has become unstable. For   0 , flutter begins
at p = 2.086 and ends at p =4.914; for higher loads,

Fig(2): Root curves of characteristic polynomial
for various eccentricities. ( k ( p)  1,   3 )
3. SLIDING MODE CONTROLLER
DESIGNATION
The objective of this section is to design a sliding
mode controller (SMC) to achieve a closed loop
control. The designation procedure will be briefly
shown here. The controller has a duty to stabilize the
system to maintain a steady state for a specific class
of initial condition. Due to robustness of the SMC
against the variation of parameters of model (Slotine
and Sastry, 1983; Haeri and Emadzadeh, 2006), it is
of a primary goal to be used. On the other hand,
model uncertainty deteriorates the performance of
nonlinear process control.
In the sliding mode control terminology, a control
input u is designed such that states approach to zero
in finite time. Thereafter states have to stay in the
location for the rest of time. To gain the benefit of
sliding mode control, a sliding surface has to be
defined first. This surface introduces a desired
dynamic and the route of approaching the states
towards a stable point. This also defines the switching
of the sliding control (as a compliment of control law)
in the surface. Any outside state in a finite time
approaches to the surface. Therefore, the dynamics in
equation (1) will be presented in the state space
format by the following definition:
(3)
1  X1 , 1  X 2 ,2  X 3 ,2  X 4
This alters the model, which is as follows:
 X1  X 2

1



   cos( X  X ) ( p sin( X 1  X 3 ) 

1
3






k
(
p
)(
X

X
)
cos( X 1  X 3 )

1
3

 X   k ( p ) X  cos( X  X ) p
3
1
2
  f1 ( X )
 2 
2



 X sin( X 1  X 3 )  2k ( p) X 1 )
 4


(4)
  X 2 2 cos( X 1  X 3 )sin( X 1  X 3 ) 








X3  X4

1


2

   cos( X  X ) ( X 2 sin( X 1  X 3 ) 

1
3



  p cos( X 1  X 3 )sin( X 1  X 3 )




2
 X 4    X 4 cos( X 1  X 3 )sin( X 1  X 3 )
  f2 ( X )

  k ( p ) X cos( X  X )


1
1
3



  k ( p ) ( X 1  X 3 )  p





k
(
p
)
X
cos(
X

X
)
3
1
3



Two inputs of uSMC1 and uSMC2 are built to control the
pendulum which is used as follows:
X 2 k 1  X 2 k
X 2 k  f 1 (X 2 k 1 , X 2 k )  u SMC k , k  1, 2
(5)
The primary aim is to achieve the equilibrium point
e =(0,0,0,0) for the state. In one hand, the state has to
approach zero for any initial condition and also in
presence of uncertainty in parameters. The design
procedure of sliding mode control for a controllable
and observable system consists of two stages:
1. Designating the sliding surface, showing the
system dynamic
2. The control law must be completed to attract
the states to the surface
Therefore the sliding surface is defined as:
(6)
sk (t )  k X 2 k 1 (t )  X 2 k (t ), k  0, k  1, 2.
where  k is chosen according to the sliding dynamics.
When any state touches the surface, it is told the
sliding mode is taken place. At this time, the state
dynamics will be controlled via sliding mode
dynamics. After the touch, the state must stay in the
surface. The sliding mode control needs two stages
of:
1. Approaching phase to the surface S (t )  0
2.
A sliding phase to S (t )  0
1 2
sk
2
is candidate as a Lyapunov function where sk is the
To verify the stability requirements function V 
sliding surface. To guarantee the stability the
differentiation of Lyapunov function has to be
negative definite. A sufficient condition of transition
from the first phase to the second , will be defined by
the sliding condition as Differentiation of V when
approaches to:
(7)
V  s k s k  0, k  1, 2.
Differentiation of (6) yields:
(8)
sk   k X 2k 1 (t )  X 2k (t ),
  k X 2k (t )  f k ( X )  uSMCk , k  1, 2.
Replacement of equation (8) into (7) achieves:
V  sk sk   k X 2k 1 (t )  X 2 k (t ),
(9)
 sk [  k X 2k (t )  f k ( X )  uSMCk ], k  1, 2.
The selection sk   K S k sgn( sk ) meets the sliding
condition and yields the control law in the following
form:
(10)
uSMC   K S sgn(sk )  k X 2k (t )  f k ( X )
k
k
uSMCk  ueq  uc , k  1, 2.
where uc  KS k sgn(sk ) and ueq  k X 2k (t )  fk ( X ) are the
correcting control law and the equivalent control,
respectively, whereas KS k  0 is the switching
coefficient. An equivalent control ueq (t ) causes the
system dynamics to approach to the sliding surface
and uc is corrective control law which complete the
control law in ueq (t ) .
4. SIMULATION RESULTS
To verify the capability of the proposed controller
system (1) is parameterized as:
  0.2, k ( p)  1,   3,
( X1 (0), X 2 (0), X 3 (0), X 4 (0))  (0.2,0.2,0.3,0.4)
The flutter region will be occurred when p is change
from p  2.244 to p  6.58 [12].
The simulation is performed by MATLAB when
Runge-Kutta is used a solver for p  2.5 . To avoid
occurrence of chattering, saturation function is gained
instead of the signum function in (10). This is defined
as:
(11)
S (t )  
 1

sat ( S )   t    S (t )  
1
S (t )  

The simulation result is shown in Figure (3)-(5).
Figure (3) shows the controlled states, whereas
Figures (4) and (5) illustrate the control input and the
sliding surface, respectively. It should be noted that
the control is triggered at 150 seconds after the
simulation starts. As it can be seen as soon as the
input control is activated the states are approaching
towards the steady state. The inputs as shown in
Figure (4) are practically realizable. These verify the
performance of the controller to stabilize the chaos.
2
2
X1
S1
Control in
action
0
0
-2
-2
5
X2
1
0
S2
Control in
action
0
-5
-1
0
0
50
100
150
200
Fig 5: Sliding surfaces
250
4
X3
5. CONCLUSION:
Control in
action
2
In this work, bifurcation points are found in Ziegler’s
pendulum when the parameter p is changed. The
sliding mode controller has been designed to stabilize
the chaos and to vanish the bifurcation. The
simulation result verifies the significance of the
proposed controller.
0
-2
-4
4
Control in
action
X4
2
0
REFERENCE:
-2
-4
0
50
100
150
200
250
Figure (3): State response of the pendulum
considering different initial condition and parameters
in (11).
U
SMC1
0
-5
150
155
150
155
2
U
SMC2
-10
145
4
0
145
Figure (4): The control signals uSMC1 and uSMC2 .
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