1. No category

Paper
Int’l J. of Aeronautical & Space Sci. 16(1), 28–40 (2015)
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.28
Experimental Study of Adaptive Sliding Mode Control for Vibration of
a Flexible Rectangular Plate
Jingyu Yang*, Zhiqi Liu, Xuanming Cui, Shiying Qu and Chu Wang
Joint Laboratory of Spacecraft Dynamics and Control, Faculty of Aerospace Engineering, Shenyang Aerospace
University,Shenyang, China, 110136
Zhou Lanwei and Guoping Chen
The State Key Laboratory of Mechanics and Control for Mechanical Structures, Nanjing University of Aeronautics and
Astronautics, Nanjing, China 210016
Abstract
This paper aims to address the intelligent active vibration control problem of a flexible rectangular plate vibration involving
parameter variation and external disturbance. An adaptive sliding mode (ASM) MIMO control strategy and smart piezoelectric
materials are proposed as a solution, where the controller design can deal with problems of an external disturbance and
parametric uncertainty in system. Compared with the current ‘classical’ control design, the proposed ASM MIMO control
strategy design has two advantages. First, unlike existing classical control algorithms, where only low intelligence of the
vibration control system is achieved, this paper shows that high intelligent of the vibration control system can be realized by
the ASM MIMO control strategy and smart piezoelectric materials. Second, the system performance is improved due to two
additional terms obtained in the active vibration control system. Detailed design principle and rigorous stability analysis are
provided. Finally, experiments and simulations were used to verify the effectiveness of the proposed strategy using a hardware
prototype based on NI instruments, a MATLAB/SIMULINK platform, and smart piezoelectric materials.
Key words: Plate Vibration, Active Control, ASM MIMO Controller, Intelligent Experimental ASM MIMO Control System, Smart
Piezoelectric Materials
1. Introduction
structures have been presented in several investigations. The
development of active vibration control experienced two
stages: ‘classical’ active vibration control and intelligent active
vibration control.
With respect to classical active vibration control, Dixiong
Yang (1) revealed the essential connections among several
popular chaos feedback control approaches, investigated the
intrinsic relationship between the stability transformation
method and speed feedback control method for controlling
the equilibrium of continuous autonomous systems. Danfeng
(2) applied a harmonic balance method to investigate the
amplitude of the self-excited vibration, and designed a
PI controller to control the amplitude of the vibration at a
Typically, modern spacecraft include some flexible
structures, such as solar arrays and antennas. Attitude
maneuvers are prone to excite vibrations in these flexible
structures because of their typically low damping. Such
vibrations have characteristics of large amplitudes and long
periods of time. Moreover, the coupling between elastic and
rigid-body motions can be aroused during the orientation of
a spacecraft. As a result, the issues of dynamic modeling and
vibration control of flexible spacecraft structures have been
studied intensively recently.
Literature reviews on active vibration control of flexible
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/bync/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: November 5, 2014 Revised : February 26, 2015 Accepted: March 2, 2015
Copyright ⓒ The Korean Society for Aeronautical & Space Sciences
(28~40)14-070.indd 28
28
* Associate professor, Corresponding author: [email protected]
http://ijass.org pISSN: 2093-274x eISSN: 2093-2480
2015-03-30 오후 3:39:41
Jingyu Yang Experimental Study of Adaptive Sliding Mode Control for Vibration of a Flexible Rectangular Plate
for suppressing the vibrations of the bending and torsional
modes, based on two low-order models of the bending and
torsional motion.
Using adaptive control theory and fuzzy control theory,
Zolfagharian (12) presented a mechatronic approach
integrating both passive and active controllers to deal with
unwanted noise and vibration produced in an automobile
wiper system operation; a bi-level adaptive-fuzzy controller
was used where the parameters were tuned simultaneously
by a multi-objective genetic algorithm (MOGA) to deal
with the conflicting interests in the wiper control problem.
Jaroslaw (13) presented a fuzzy logic-based robust feedback
anti-sway control system that could be applicable with or
without a sensor of sway angle of a payload, and proposed an
iterative procedure combining a pole placement method and
interval analysis of closed-loop characteristic polynomial
coefficients to design a robust control scheme. Nemanja (14)
dealt with active free vibration control of smart composite
beams using a particle-swarm optimized self-tuning fuzzy
logic controller, and proposed integration of a self-tuning
method with membership function optimization using the
PSO algorithm to improve FLC performance and robustness.
Young presented the concept of sliding mode control
in 1978, since when it has been used by many researchers
in the field of engineering control. However, few papers
talked about AVC of flexible structures using the sliding
mode control algorithm. A. El et al. (15) presented a nonlinear controller design for biaxial feed drive systems for
reducing the control input variance while maintaining
motion accuracy. Experimental results demonstrated a
significant performance improvement in terms of control
input variance while maintaining motion accuracy. Oliveira
et al. (16) presented an automatic tuning method for the
discontinuous component of the Sliding Mode Generalized
Predictive Controller (SMGPC), subject to constraints.
Simulations and performance indexes for common process
models in industry, such as non-minimum phase and timedelayed systems, resulted in better performance, improving
robustness and tracking accuracy. Chen et al. (17) proposed
an intelligent self-repairing control scheme for a class of nonlinear MIMO systems; the control method was applied to a
helicopter flight control system with loss-in-effectiveness
faults. Some simulation results illustrated the effectiveness
and feasibility of the proposed control scheme in the paper.
Ghasemi et al. (18) used the notion of finite-time stability to
apply it to the problem of coordinated motion in multi-agent
systems, obtained non-Lipschitzian closed-loop dynamics
restricted to the surface, which leads to the convergence of
the system trajectories to the origin in finite time. Acosta
et al. (19) introduced a sliding mode control approach for
given level. The effectiveness of this method showed good
prospects for its application to commercial maglev systems.
Bin (3) proposed a non linear dynamic modeling method for
a rigid-flexible coupling satellite antenna system composed
of laminated shell reflector undergoing a large overall
motion, and used a PD with a vibration force feedback
control strategy to eliminate the system vibration. Previdi
(4) presented a full analysis and the complete development
of a system for mechanical vibration reduction in a
kitchen hood using piezoelectric actuators, and provided a
minimum variance (MV) controller to get the theoretically
‘best’ performance in terms of noise reduction. Xiuting
(5) investigated the potential beneficial performance of
a quasi-zero-stiffness vibration isolator (QZS-VI) with a
simple linear time-delayed active control strategy, provide
a time-delayed active control which can not only further
strengthen the robustness of the system in stability, but also
noticeably improved system transmissibility performance
both in force and base excitations and obviously decrease
the settling time of system transient response subject to
an impact load. Dafang (6) investigated active vibration
control of a flexible beam with piezoelectric pieces on the
surface using an independent modal space control method,
which was able to control the first three modes of the beam
independently. Cazzulani (7) proposed exploiting the
measurements of Fiber Bragg Grating sensors as feedback
for active vibration control applications. The experimental
tests confirmed that smart structures with embedded FBG
sensors could be profitably designed to suppress vibrations.
Hu (8) presented a new and effective approach for vibration
suppression of large space structures, and developed the
equations of motion of a flexible structure with a set of
arbitrarily distributed CMGs. Xianglong (9) presented an
analytical study of active structural acoustic control of an
elastic cylindrical shell coupled to a two-stage vibration
isolation system, and developed an analytical activepassive model to attenuate sound radiating from the base
shell structure, which consisted of a rigid-body machine,
an intermediate rigid mass, and a supporting cylindrical
shell, all connected by a combination of passive and active
isolators.
Regarding intelligent active vibration control research,
using robust control theory, Zhang (10) investigated the
robust finite frequency H∞ passive fault-tolerant staticoutput-feedback controller design problem, used the
finite frequency H∞ control to attenuate the effect from the
external disturbance to the controlled output. Zhicheng
(11) developed an active vibration control for a two-hinged
plate, obtained state space representations for bending and
torsional vibrations, and designed two H∞ robust controllers
29
(28~40)14-070.indd 29
http://ijass.org
2015-03-30 오후 3:39:41
controller design with MIMO features, and the effectiveness of the vibration control
c
m,1
m, i
m  1, i  1
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theory.
Int’l J. of Aeronautical & Space Sci. 16(1), 28–40 (2015)
m
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system can maintain the vibration reduction performance based on intelligent
control
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plate structure will be considered. The proposedc method
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c
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achieves an accurate dynamics model and its structure is
control effort overcame only uncertainties, disturbances or
simple,1.thereby
providing
MIMO
unstable dynamics. Two illustrative examples were given to
Figure
Cantilever
plate controller design with
Figure
2. The discrete cantilever
m
m
m
m 2. The discrete cantilever plate
k
kFigure
1. proposed
Cantilever
plate
k
features,
and the effectiveness
of the vibration control system
show the feasibility of the method. ZhangFigure
et al. (20)
c
c
c
can maintain
the vibration
reduction performance based on
a fractional order sliding mode control (FROSMC) scheme
Considering
the
continuous
cantilever plate (Fig. 1) and discrete state of the
intelligent
controlplate
theory.
for the velocity control of permanent magnet
synchronous
Considering
the continuous
cantilever
(Fig. 1) and discrete state of the plate
c
c continuous
k
c
c
k
k
k
Considering
the
cantilever
plate (Fig. 1) and
motor (PMSM), and used a fuzzy logic inference
scheme
(Fig.
2)
(22),
we
can
establish
a
model
of thea rectangular plate
m
m
m
m
k
k
discretek state of the plate
(Fig. 2) dynamic
(22), we can
establish
(FLIS) to obtain the gain of switching control. Simulations
(Fig. 2) (22), we can establish a dynamic model of the rectangular plate
dynamicc model of the rectangular
plate
and experiments demonstrated that the proposed FROSMC
c
c

[
M
][
Y
]

[ P  P( L0  Y Y  Y )][Y ]  [ K  K ( L0  Y Y  Y )][Y ] 
[ L][ F ]
not only achieved better control performance with smaller
[ M ][Y]  [ P  P( L0  Y Y  Y )][Y ]  [ K  K ( L0  Y Y  Y )][Y ] 
[ L][ F ]
(1)
(1)
chatting
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Figure
Cantilever
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Figure 2. The discrete cantilever plate

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
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parameter variations. Zhang et al. (21) exploited the  robust
 
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 13 
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Consideringcontroller
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plate
of the plate
 





 (Fig. 1) and discrete state
H∞ sliding-mode
design cantilever
problem for
discrete
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y21 
time Markovian jump linear systems, and provided
two  where
 inertia

 21 
mij
matrices;
[Y ]   y22  ; [ F ]   F22  ;
matrices;
where [ M ]   
 inertia




 


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F  ;
m
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M
[
]
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matrices;
[
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[
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
where
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 22 
y 
F 

 22 

numerical
examples
illustrate
the aadvantages
and
theof the rectangular
(Fig. 2)
(22), wetocan
establish
dynamic model
 

 23 
 23 



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 
 y 

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
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F 

 
 23 
y 
F 


 23 
efficacy of the proposed method when simultaneously
 
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 31 
 31 
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 
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
31
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
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F 
y



 32 
 32 
 [ L ][ F ]an accurate dynamics

 and

considering intermittent observations,
system
model
its
structure
is
simple,
the
[ M ][Y] 
[ P  P(uncertainty,
L0  Y Y Y )][Y ]  [ K  K ( L0  Y Y  Y achieves
)][
Y ] 
(1)
 y 

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


F 

 
m jj  32 
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 32 
 33 
 33 
 





and external disturbance. In 2014, Yang et al. (22) presented


m




F
y
jj 
 33 
 33 





 

an experimental study
about flexible plate using
adaptive
y11 controller
K11
 m11

 11
 F11 
K

design
with
MIMO
features,
and
the
effectiveness
of the





 
 3
1
1



 11  K11
F 
 y 

K



fuzzy sliding mode
(AFSM)
control
strategy;
however,
the








12
12

 3


1
1









1
3
1
1








F

 y 






  13 
  13 

 1
3 1
1
limitations of above
control
strategy were  more
complex





1
1 2

 

  the vibration reduction

maintain
performance
based
on

 F can
K
  y  system




K




21
 21  21
  21 

1
2
1





and it was difficult to be applied in control engineering,
so

1
4
1



 
  1

K
K



F  ;
 y  ;

21
21
  inertia matrices;

mij
where [M ]  
[
]

[
F
]

Y
K


[
]
 22 
  22 
1 
4 1

  1
1
1 4 1
1




 

there is still s need
to investigate vibration

[ K ]   control using

F 
 y 




 23 
1
1 4 1
  23  theory.
 1


1
1 3















other intelligent control
methods.
Then,
the
advantages







F
y


K 31  

1K 32
1 3
1
 31 
  31 



3 1
1

 K  K

 



31


32
 y 

F 



of different control
strategies could be compared.
Surely,
m
m
k1 3m
1
3


 32 
  32 
  1
k


1




 




m jj 
F 
  y33 


 33 
 

1 33  K331  3 1c
K
the feasibility of control
algorithms in engineering
practice


1 c 1


K 33  K 33  


1
1 2 

would be confirmed.


 K11  K11

c
c
k
k
k

 3
1
1


The proposed
ASM
 MIMO control scheme can be used   stiffness matrices; K ij 

K ij ( L0  Yij Y ij Yij ) where i j  1 2 3

 1
3
1
1


stiffness
K ij ( L0  Yij Y ij Yij ) where i j  1 2 3

in active  vibration control
systems in
which matrices;
unknown  Kij 
m
m
m
k

k
1 2
1




K
K




21
21increase the complexity of 

disturbances
and
perturbations
1
4 1

  1

c
c



[ K ] 
  model and its structure is simple,
achieves
accurate
dynamics
thereby providing

a non-linear
complex vibration
system. an
Thus,
the intelligent
1
1 4 1
1







1
1 3
1
controller design becomes more challenging.
Moreover,  
K 31  K 32
c
k
c

k
k


1
3
1



 features, and
controller
design
with
MIMO
the
effectiveness

 of the vibration control
the proposed
scheme incorporates an adaptive

 control 

m
m
m


1
1
3
1



k
k



 K 33  
law mechanism
to deal with uncertainty. To achieveK 33better


1
1 2 

c
c
system can
the vibration
reduction performance based on intelligent control
tracking vibration control performance
in amaintain
limited time
and
to attenuate efficiently the effects of both disturbances and
stiffness matrices; Kij Kij ( L0  Yij Y ij Yij ) where i j 1 2 3 Fig. 1. Cantilever plate
Figure 1. Cantilever plate
Figure 2. The discrete c
theory.
uncertainties to an expectation
level, an ASM MIMO control
scheme was adopted in the control design for a structure
m
m
m
m
k
k
k
vibration
control system with parameter uncertainties.
c
c
c
Considering the continuous cantilever plate (Fig. 1) and discrete sta
Finally, experiments and numerical simulation examples
c
c
c
c
k
k
k
k
of a flexible plate structure vibration system are provided
(Fig. 2) (22), we can establish a dynamic model of the rectangular p
to confirm the effectiveness of the proposed intelligent
m
m
m
m
k
k
k
vibration control scheme.
1, i  1
1,2
n  1,2
1, k
n  1, i  1
n  1, k
n  1, i  1
n  1,2
n  1, i  1
n  1,2
2,1
2,1
m,1
2, i  1
2, i  1
m  1,2
2, i  2
m, i
m  1, i  1
m  1,2
n,1
n,1
n  1,2
m, i  1
n, i  1
n  1, i  1
n, i  2
n, k  1
n, k  1
n, i  1
n, k
n  1, k
n  1, i  1
n  1,2
m, k
m, k
m, k
n, i  2
n, i
n
n  1, k
2, k  1
2, k  1
m  1, i  1
n, i  1
n,1
2, i  2
n  1, k
1,1
1, i
1,2
1, i  1
1,2
2,1
2,1
m,1
2, i  1
2, i  1
m, i
m  1,2
n, i  1
n, i  1
n, i  2
n, i
n  1,2
1, i
1,2
n  1, i  1
2,1
m,1
m, i
m  1,2
n,1
c
In this section, the present discretization and reducedorder modeling method (DROMM) for modeling of a flexible
Figure 1. Cantilever plate
Considering the
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.28
n, i  1
k
n,1
n  1,2
c
n  1,2
m
n, i


























c
m11
[ M ][Y]c [ P  P( L0  Y Y  Y )][Y ]  [ K  K ( L0  Y Y  Y )][Y ] 
[
m, k
c
k
n, i  1
n  1, i  1
c
k
n, i  1
n  1, k
c
n  1, i  1
[M ] 
Fig. 2. The where
discrete cantilever
plate
n  1, k

mij
c
n, k  1
m
n, k
inertia matrices;
Figure 2. The discrete cantilever
plate



continuous cantilever
plate (Fig. 1) and discrete state of the plate
m
30














[ M ][Y]  [ P  P( L0  Y Y  Y )][Y ]  [ K  K ( L0  Y Y  Y )][Y ] 
[ L][ F ]


























jj 
n, k  1
m
K11 rectangular plate
(Fig. 2) (22), we can establish a dynamic model Kof
11  the
(28~40)14-070.indd 30
k
n, i  2
n, i  2
k
m, k
m, k
m  1, i  1
k
n,1
m
2, k  1
2, k  1
m, i  1
c
m  1,2
k
2, i  2
2, i  2
m  1, i  1
c
2. Dynamics Model of Flexible Plate
1, k
2, i  1
2, i  1
1, k
1, k
1, i  1
1,2
2,1
1, i  1
1, i  1

[K ]

K 21  K 21



n, i  1
n  1, i  1
n  1,2
1,1
m, i  1
m  1, i  1
n,1
n,1
2, i  2
m  1, i  1
m  1,2
n,1
1, i
1, i  1















2015-03-30




























y11 

y12 

y13 

y21 

[Y ]  y22 

y23 
;
[F ] 

y31 

y32 

y33 
1
 3 1
 1 3 1
1


1 2

(1)



1
4
1

 오후13:39:41 1 4 
where
]  we can establish m
matrices;
[Y ]  y ; [ F ]  F ;
(Fig. 2)[ M
(22),
a dynamic model
of inertia
the
rectangular
plate
q for
t  for
tracks
the
desired
trajectories
after an law
initial
rectangular
plate
system
output
the
the
actuator
actuator
forces
forces
andinitial
and
an an
estimation
estimation
law
forfor
thethe
unknown
unknown
parameter
parame
rectangular plate
q  t  and
tracks
the desired
trajectories
after
an
 output
ysome
for system
the actuator
forces
an estimation
law
forFall
thethe
unknown
parameters
such that
the
or
rectangular
plate system
parameters
being unknown, d
adaptation
process.
[ M ][Yall
]  [ P the
P( L Yrectangular
 Y  Y )][Y ]  [ K  K ( L  Y Y plate
Y )][Y ] 
[ L][ Fsystem
]
(1) F being unknown, derive a control law
somerectangular
or
y tracks
plate system
output
q  t parameters
the desired trajectories after an initial

F plate
y
rectangular
rectangular
plate
system
system
output
outputq  t q  ttracks
tracks
thethe
desired
desired
trajectories
trajectories
after
af
adaptation
process.
m
F
y
q  t  tracks
theactuator
trajectories
an initial
adaptation process.
m systemy output
Fdesiredforces
yfor the
and anafter
estimation
law for the unknown parame
 rectangular plate
F













11

























11
















jj 
ij
0

























jj 




 22 




 23 




 31 




 32 




 33 
0



























11 


12 


13 


21 


22 


23 


31 


32 


33 



























11 


12 


13 


21 



22 


23 


31 


32 


33 

















22 


23 


31 


32 


33 
steady-state
position
errors can
be eliminated
if wethe
restrict them to lie on a
:
for the
actuator forcesUndesirable
and an estimation
law for
the unknown
parameters
such that
 adaptation process.
K [M]K
m
inertia matrices;
; position
;
where
adaptation
adaptation
process.
process.
Undesirable
steady-state
eliminated
if we
restrict
to lie
ondesired
a
1 can be
 3 1errors
system
rectangular
platefor
output
tracks
thePlate
trajectories aft
Undesirable steady-state
position
errors
can
be
eliminated
if we
restrict
to
lie
on qa tthem
adaptation
 Rectangular
1
1process.
Yang
Experimental
Study
of
Adaptive
Sliding
Mode
Control
Vibration
of
a Flexible
 3 Jingyu


 them

1
3
1
1



sliding
surface
rectangular
plate
system
output
q
t
tracks
the
desired
trajectories
after
an
initial





  1 Undesirable
1 steady-state
3 1
can1be eliminated
if we restrict them to lie on a

 position


2
1 errors

1 K2 m surface1
 K



Undesirable
Undesirable
steady-state
steady-state
position
position
errors
errors
cancan
be be
eliminated
eliminated
if we
if we
restrict
restrit
sliding



1
4
1
1
C  C



 if we restrict them to lie on a
surfaceUndesirable
position
errorsadaptation
can
besystem
eliminated
process.
K]
4 steady-state
[sliding
  unknown,
1
1
plate
parameters
being
derive a control

1



q
q
0
1
1
4
1
1





adaptation
process.
 sliding



surface
1
1  4 1
1
1 law
1for3 the actuator
1 forces and an estimation law for the





K
K




q


q

0
(3)
1
1
3
1

sliding
sliding
surface
surface




3 steady-state
1unknown parameters
1
0 Undesirable
(3)be
the rectangular
plate
system
C  C
sliding
surface
position
errors
can
eliminated
if we restric


q  q 
 such that

1
3
1


is aerrors
constant
matrix,
eigenvalues
are
strictly
in
the
right-half complex
where


3 1if
1 eliminated
1whose
Undesirable
steady-state
position
can
be
we
restrict
them
to
lie
on
a
Assumed
that:


output
q(t)
tracks
the
desired
trajectories
after
an
initial


q


q

0
(3)


1
1 3 K 1  K 
1
1 2 
C  C 


q  
q q 
0q 
0
adaptation
process.
 is a1constant
whose
eigenvalues
are
strictly in the right-half complex
where
1 2matrix,



C
C


q surface
 in
q 
0 right-half complex
where  isa3 constant
areachieve
strictly
the
3 1
1
1
1 matrix, whose
 eigenvalues
1 Undesirable
 3 1we sliding

q
t
in (3)
the above
plane.
Formally,
this
by
replacing
the
desired
trajectory


steady-state
position errors can be eliminated

 1 3 1surface

sliding
 1 whose
strictly in the right-half complex
1 3 1
1
1

where
is
a
constant
matrix,
eigenvalues
are
3
1
1

2,
stiffness
matrices;
matrices;

 i, j=1,
stiffness
K 
K ( L  Y  Y  Y ) where
where
i j 
132 3

if we restrict
to lietrajectory
on a sliding surface
matrices;
j i  11 plane.
 vector;this
rices;
C ( L 1Y  Y  Ystiffness
) where
2 3,2 U Formally,
is where
the1 s 1 we
input
1 
2C 
q  t  eigenvalues
ineigenvalues
the above
achieve
desired
 X 1by

 1
where
where
isthe
aisthem
constant
a constant
matrix,
matrix,
whose
whose
areare
strictly
strictly
in the
in the
righ
r
2replacing
Assumed
C  C
Assumed
 that: that:
matrix, whose
q t  in in
the
above
plane.
Formally,
this
by replacing
the desired
trajectory
Cwe
C
 achieve
q  q 
0
is
a
constant
eigenvalues
are
strictly
the
right-half complex
where
4 1
1
1
1
4
1
1




1
4
1
1
derivation
by
the
virtual
“reference
trajectory.”




q  q 
0
(3)
(3)




trajectory q  t  in the above
this1 by replacing

1
1 4 1 [ P]1
1 plane.
1 Formally,
1we achieve
 4 1
1 4 1 the desired
1
X is the n  1 nodal velocity vector;


odal displacement
vector;
X is the

plane.
plane.
Formally,
we1we
achieve
thisthis
by by
replacing
replacing
thethe
desired
desired
trajectory
trajectoryq  tq
1
1 3
1
1
1 by
1
3 the virtual

derivation
“reference
1trajectory.”
1 Formally,
3
 by replacing
achieve
C  Cderivation
q eigenvalues
t  in the
above
plane.
Formally,
achieve
this
desired
by
virtual
“reference
trajectory.”
a constant
matrix,
whose
are strictly are strictly in the ri
wherethe
C we

 the

 isis
where
a1 constant
matrix,
eigenvalues

 trajectory
1
1
3C1
3 1

q
q    whose
qdt
1
3




 is a constant matrix,



where
whose
eigenvalues
are strictly
inplane.
the right-half

in the
Formally, wecomplex
achieve this by
derivation
“reference
trajectory.”

1
1  damping

cceleration vector;
[1P] 3isCthe
matrix, 1n by
n ; 1the
[ M]3virtual
is1 the
mass
1 right-half
1 3 complex
1
 C 
C
C





(4)
q

q


qdt
derivation
derivation
by
by
the
the
virtual
virtual
“reference
“reference
trajectory.”
trajectory.”
1
1 2 
1
1 2 
replacing
the
desired
trajectory
q
(t)
in
the
above
derivation


1
1
2

d
plane. Formally, we
(4)
q q  trajectory.”
qdt
 achieve this by replacing
q
the
desired
trajectory
derivation by the virtual
“reference




Accordingly,
and by
are
replaced
by trajectory.”
qreplacing
q the
virtual
“reference
q  t  in the above
plane.
Formally,
we
achieve
this
by
the
desired
trajectory

(4)
q

q


qdt
; F is a concentrated force that is applied to m ; [ K ] is the stiffness

s;
2 3 ,CU 
is
C the
( L  Y sY 1 Y input
)damping
where
vector;
j matrices;
i  1X2 3 ,Accordingly,
U Cis
the
vector;
  qdt

q qq q  
qdt
matrices;
C ( LsY1  Yinput
where
ji=1,
X
i  1replaced
2 3 3,
, U X is by
the s 1 input
vector; X
damping
where
2,
q  Y )and
q j, are
damping
matrices;
where
,
is
the
input
vector;
the virtual “reference trajectory.” (4)
(4)
q q  
qdt
Accordingly, q and q are replaced by
derivation
by











q




q
q
q
q
q
is the s×1derivation
input
vector;
X is the nd×1 nodal
displacement
“reference
trajectory.”
; and [ L] is influence isUcoefficient
matrix,
n by
 s . the virtual



Accordingly,
and
are
replaced
by
q
q
the n nodal
displacement
vector;velocity
X is the n
nodal velocity
vector;
X is the
vector;
the
×1
nodal
vector;
the
n
aln displacement
 1 nodal velocity
vector;
nthe
 1nddisplacement
nodal
velocity
vector;
the

isvector;
theX nisXthe
1 isis
nodal
vector;
X Xisisis
the
n d×1
1 Accordingly,
nodal
the
Accordingly,
are
replaced
replaced
 q  qvector;
 and
qX q is
(5)
q  qqand
qare
qvelocity
and
are
replaced
by by by
Accordingly,







q


 q
(5)

qareqreplaced
qd; q by
 damping
 matrix,
q q    qdt
Accordingly,
and
q
q
nodal
acceleration
vector;
[P]
is
the
n
×n
d
n
nodal acceleration vector; P is the damping
n n ; M
is thesliding
mass
Ifmatrix,
we define
the
surface

(4) (5)
q q    qdt
q  q  q q q  q
[M]
is
the
mass
matrix,
ndn×n
gferation
matrix,vector;
nPlate
 n ; [Modeling
P[]M ]is
is
damping
mass
matrix,
dn; F;ij is
[ Ma[]Pconcentrated
r is and
Note
that
the
matrix
(25)
is force
nowthat
a function
q r , rather than qd and q(5)
n matrix,
the
1 the
nodal
vector;
]Yisisthe
themass
damping
matrix,
n  n ;of[ M ]q
the mass
Flexible
d .
; acceleration
F is a concentrated force that is applied to m ; K is the stiffness
q qq qq q
q qq qq q
If
we
define
thematrix,
sliding
is applied
to the
mij; [K]
is the
stiffness
nd×nsurface
d ; and [L] is
(5)
q  q  q q q q qand q are replaced
Accordingly,
by
If we
define
sliding
surface

s surface
If we define the sliding
q  q  q  q  q
Accordingly,
replaced
by
qd×s. and
q n sare
influence
matrix,
n
matrix,
n n coefficient
;
and
L is influence
coefficient
matrix,
.


Note
that
the
matrix
Y
(25)
is
now
a
function
of
and
rathert


q
Note
that
the
matrix
Y
(25)
is
now
a
function
of
and
q
q q , ,rather
Fplied
is a concentrated
to m ; [ K ] isforce
the
stiffness
that is
m define
; [ K ] isthe
theforce
stiffness
sliding
n applied
n ; F If
matrix,
istowe
a concentrated
thatsurface
is applied to m ; [ K ] is the stiffness

Weiscan
demonstrate
convergence
of
the
tracking
by
now
using
the

 10 m,
the elasticityglobal
modulus
E If we
s used for illustration. The plate
10 magain
s define
 qthe
 sliding
q sliding
q surface
(6)
q  qthe
(6)
If we
define
surface
s  q  q  q  q  q
 q  q q q  q
q (6)
If we define the sliding surface
3. Example of Flexible Plate Modeling

The
control
law
and
adaptation
law
become









q




q
(5)
qagain
q demonstrate
and
trix, [ Ln] is
 s influence
.
coefficient
We
global
convergenceofofthe
thetracking
tracking
bynow
now
global
convergence
matrix, n matrix,
 n ; andn [ Ls ].is influence coefficient matrix, nq We
sq
. can
scan
 qcontrol
again
q  q law
 demonstrate
q and
 q adaptation
(6) by
The
law become
Plate
Modeling
oisson’s ratio = 0.3, 3.=Example
7900 kg/mof, Flexible
the elasticity
shear
modulus












s

s


q


q


q



q
q


q
q
q
q
Lyapunov
function
control
law
and
adaptation
become
Note that the qmatrix
Y (25) is now a functio
An control
example
is used
illustration.
The plate
is 10
m become
10
m, the elasticity law
modulus
 q define
 q  q  the
q  sliding
q
(6)
The
lawforThe
and
adaptation
law
IfE swe
surface

 Mˆ  q  q  Cˆ  q q  q  Kˆ  q   K s
aˆ   Y  q q q  q  s
If
we
define
the
sliding
surface
Lyapunov
function
Lyapunov
function
An
example
is
used
for
illustration.
The
plate
is
10
m
×
10
The
control
law
and
adaptation
law
become
= 206 GPa,
ratio
=dynamics
0.3,
 = 7900
kg/m
, thethat
elasticity
modulus
and thePlate
thickness
is 3.
0.001
m. Poisson’s
Using
the
modeling
method
above,
(7) ofq .q and q , rath
Note
the shear
matrix
Y (25) is
nowthat
a function
of
rather
than
q Yand
q ,is
q and
lexible
Modeling
matrix
now
a demonstrate
function
Example
of Flexible
Plate
ˆ  qqThe
T
qnow
can
again
convergenc
 Note
the
Y ˆ Modeling
(25)
isPoisson’s
now
a
function
,
rather
than
and
 matrix
ratio
Mˆ  q= q0.3,
of
CThe
q Note
now
Kˆ that
law
Kthe
s the
aˆadaptation
Y (25)
qqWe
.,qlaw
 q
sbecome
qq a
qYq(25)
qand

control
control
law
and
and
adaptation
law
become

 and
m, theNote
elasticity
=
206
GPa,
matrix
is
a
function
of
rather
than q

V  t that12 smodulus
Msmatrix
 12 a TEMa
q
q , (7)


ˆ
Note
that
the
Y
(25)
is
function
of
and
rather
than
and
.global
ˆ
q
q
  M  q and
 K sbecomeaˆ   Y  q q q  q  s
q  C  q q  q  K  q law
s (7)
 q  q  q  qq 
q
The
control
=79.4 GPa, and the
thickness
is 0.001law
m. Using the adaptation
dynamics modeling method
above,




ρ = 7900 kg/m , the elasticity shear modulus G =79.4
GPa,

s


q


q


q
(6)
q

q



V
t
s
Ms

Ma

V
t
s
Ms

Ma
  Y  q q q  q  s
q   Kˆ  q   Ka as
  demonstrate
Mˆ  q  q  Cˆ  q qglobal
(7)
s of the model ([M],[K],[P],[L]) can be obtained.
We
again
convergence
ofaˆisthe
tracking
by now
using
theq ,q rather
becan
Lyapunov
function
ˆand
and
the
thickness
0.001
m.
the
dynamics
modeling
Note
Note
that
that
the
the
matrix
matrix
Ydemonstrate
Y(25)
(25)
is
now
now
aˆ qfunction
aqglobal
function
of
of
and
, rather
than
qY
  than
by
We
can
again
demonstrate
convergence
of
the
tracking
ˆby
ˆnow
ˆis
ˆq qqK
the
parameters
ofisthe
([M],[K],[P],[L])
can
obtained.
Weis
can
again
demonstrate
global
convergence
of
the
tracking
by
now
using
the
 10
is10
sed
m for
illustration.
m, the elasticity
The
plate
modulus
10
m
Emodel
m,illustration.
theUsing
elasticity
modulus
E m
Note
that
the
matrix
Υ(25)
now
a
function
of
and
,


ˆ
ˆ
1



and
We
can
again
global
convergence
of
the
tracking



M

q
M
C

q
C

q
q

q

K

q
K

s
K
s
a

a
 
Ynow
qqqand
using
qqby
qqqn
q
q
q

An
example
used
for
The
plate
is
10
10
m,
the
elasticity
modulus
E









T
T
We
can
again
demonstrate
global
convergence
of
the
tracking
using
the
T
q tq  C
 
ˆ control
(8)
sˆ q qThe
C
 aˆ s 
Ms
qqrKG s M
qr  and
 sK
  Mˆ V
Y a qaq q  qlaw
s become
(7)
q



law
adaptation
method
above,
the
parameters
of
the
model
([M],[K],[P],[L])
Note
Notethat
thatthe
thematrix
matrix Y Y (25)
(25)isisnow
nowa afunction
functionofof q q and
and q q, rather
, rather
than
than
and2 q q.. We
. can again
1
1
rather
thanq q and
and
demonstrate
global
 a a
Ms
The the
control
and
adaptation
law become
(23)
confirmed
validity oflaw
theby
new
model
obtainedessential
by
comparing
essential
VVtt s s CC s sq q 
a a
 GGMMqq   s sMs
nfirmed the validity ofPaper
thebe
new
model
obtained
comparing
Lyapunov
function
can
obtained.
 tracking
2 2using

V  t  using
s Ms
Ma
a the
convergence
offunction
the global
tracking
by
now
Lyapunov
Lyapunov
function
son’s
, theratio
elasticity
= 0.3,shear
 ==modulus
7900
kg/m
, the
elasticity
shear
We
We
can
can
again
again
demonstrate
demonstrate
global
convergence
convergence
ofthe
of
the
tracking
byby
now
now
using
thethe
206
GPa,
Poisson’s
ratio
=
0.3,
modulus
=
7900
kg/m
,
the
elasticity
shear
modulus
Lyapunov
function
Lyapunov
Lyapunov
function
element model
features
between
a finite
obtained of
model,
as natural
Paper
(23)
confirmed
the and
validity
thesuch
model
 tracking
function
  Mˆ  q  q  Cˆ  q q  q  Kˆ  q   K s
aˆ   Y  q q q
We
Wecan
can
again
again
demonstrate
demonstrate
global
global
convergence
convergence
thetracking
by
now
using
the
the
  d by
using
Considering
the
equation
M of
qof
K
qnow

Ya
(26).
qthe
new
d  C ˆ q q  q
K
ˆ q  K s
een a finite element model
and by
obtained
model,
such
as

  q Considering
M
ˆ equation







V  t natural
s between
MsMˆ  qa Ma



C
q

q
a


Y
q

q

s
(7)

q
q
q










q

C
q

q

K
q

Ya
(26).
Considering
the
equation


q
q
the
M
q

C
q

q

K
q

Ya
(26).
obtained
comparing
essential
features
a
finite
q
q






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GPa,
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Lyapunov
Lyapunov
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2 the dynamics

V  tfunction
dthe
thedynamics
thicknessmodeling
is 0.001
m.
method
Using
modeling
above,
 Ma

1 T 2 a
1 T

Vis t 0.001
smethod
Msm.
 aUsing
G frequencies.
 Ma
=79.4
and 2the
the
dynamics
method
above,

V modeling
t 2 s Ms
 Ma
2 s Ms  2 a
element model and obtained model, such as natural
Lyapunov
Lyapunovfunction
function
1
T
  a TaT T T
V  t   a T Y T s 1 T1 aT  11sVTT1TKt DTssT TsTCs
(9)(8)
C s qs r  K
GDMCqrs 0sTMs
frequencies.
T
T
TKqq
1Cs
00q 
 
s
sTsK
KMsD s
 TsCs
VVVttt 

a a 
q
sd TsTC
K
C
 aTTaYYs(8)
dssK
the
4.
The
DESIGN
MIMO CONTROLLER V  t   sT   C  s  
equation
sqT Ms
 qC

s 2Ms
V2ats2Ma
qsr Ms
Ta
Vt G t 2 M
asMa
fbethe
obtained.
model ([M],[K],[P],[L])
can ofbeASM
qrs M
a D1aD(8)
Ms
 a Tsa   C  Ts  q r   G D(8)
2  Ms
the parameters
ofobtained.
the
model ([M],[K],[P],[L]) can beqVr obtained.
 Ca  s  q r   G  M qr   12 s2TConsidering
  a Ta
2


V
t

s


C
s

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G
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M

sT Ms
q
q


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
r
r
1 1T T
1 1T T  

V
V
 Ma
 t  t  2 s2 sMsMs 2 a2 aMa
2
Considering
Ya (26). T T
An n degree of freedom plate system (24) model
is described as the equation M  q  qd  C  q q  q d  K  q  
1 T1 T  T T V t  T Y T s  a 
 V
 t position
  q q  q not
  q(8)
  qcontain

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a
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V
t
s

s
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s
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s
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G
M
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M

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s
Ms
s
Ms



a

a

q
q
q
q


Considering
the
equation
M

C

K

Ya
(26).








q
q
is
Note
that
control
law
does
a
term
in,
because
the
error




a
a


that
ra
 the
rmed
l obtained
validity
byMIMO
comparing
of the
essential
model
obtained
comparing
rterm
Considering
equation
(26).
d essential
d
the
equation
C  qcontain
qcontain
Ya
law
(8)Mdoes
does
not
in,because
because the
posi
that
law
(8)
term
 q  qd not
rq d r K  q a
 (26).
Considering
equation
MNote
qConsidering
Ya
Paper
(23)
confirmed
thebyvalidity
of
the
newthe
model
obtained
comparing
qcontrol
GN
ofthe
ASM
CONTROLLER
Note
 qby
 q q control

4.new
The
Design
of Asm
Mimo
Controller
2 (26).
2 in,
d C
d  K  q essential
1 1T T  T T Considering

T
T



ˆ
ˆ
ˆ






the
equation
M
q

C
q

q

K
q

Ya
(26).

q
q


M
q

C
q

q

K
q


(2)
q
q







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(8)
(8) d
VV
M
qr q
t  t s s  C C
 s s q rq r G G M
 r 2 s2 sMsMs a a aa
d
V Tt   a T Y T s  a   sT K D s  sT Cs   sT  K D  C  s  0
(9)
(9)
T
T
  t   a T Y T s  a   sConsidering






TYa
T(8) does
Tthat
Note
control
not
at
V
K
s

s
Cs


s
K

C
s

0
(9)





T
T
T
T




Considering
the
the
equation
equation
M
M
q
q

C

C
q

q
q

q

K

K
q

q

Ya
(26).
(26).


T
q
q
q
q












  t   aas
D
D Y s  a
Y(10)
s  ashows
 sT K law
sthat
Cs the
the
sT  Koutput
 C  contain
s (9)
0errors
ned
a finite
model,
element
such as
model
natural
and
obtained
such
as natural
d   s VKd t 
dCa
 s
 shows
   output
Vsuch
sin
K
0(10)
D s that
D
 qsd.Cs
features
between
aincluded
finite
element
model
and
natural

An n
degree model,
of
freedom
plate
(24)obtained
model(10)
ismodel,
system
already
included
in
.
Expression
output
errors
Ds 
D converge
Expression
c
already
included
q
already
in
.
Expression
shows
that
the
errors
to
the
q
r
r
r  
of freedom plateConsidering
system
(24)
model
is
described
as
T
T
T
T
T






Ya (26).
Considering
the
the
equationMM qqqdqd C Cq qqqdqd K K qq

(26).
 Ya
 a  in,
s K D s  s Cs   s  K D  C  s 
 t   a Ya sterm
Note that control law (8) does notVcontain
described
asequation
Note that control law (8) does not contain a term Vin,
because
the
position
q is
T T
T T error
T Talready
 V
included
.is
Expression
(10) show
 s
t does
t aqT ais
Yis
sYalready
T sacontain

K
sincluded
sthe
D sa
s
Cs
s nCs
in

sTin,
vector
sof
 s0applied
0the position
sKqqTrDbecause
Note that control law (8)Note
does that
not contain
aqterm
in,
because
the
position
error
not
 aposition
because
the
position
error
.K
DK
DC C
where law
is(8)the
nNote
 1not
vector
of
nodal
displacements,
is
1in
that
control
law
(8)
term
er
frequencies.
control
does
contain
a
term
in,
because
the
error
ˆ
ˆ
ˆ
sliding
surface
(2)
sliding
surface
  q r  K  q  
T T
T T
T T
M  q  q

C
q

q

(2)
T T T T





r
VV
t  t a aYYs s
ais
a sthe
sK DKsDsns Cs
s1 Cs
vector
 s s KDK
 C
sExpression
s 0 0
(9)
(9)
sliding
surface


(10)
shows
that
the
output
errors
converge
to
the
DC
where q is the n 1 vector
of nodal
displacements,
of
applied
where q is the already
n  1 vector of nodal displacements,
 is
the nthat
 1law
vector
of applied
Note that
control
does
not contain
a term
included in q r . Expression
(10)
shows
the(8)
output
errors
converge
to thein, because the p
sliding
surface
sliding
ˆ the
included
in nodal
(10)
thatMthat
output
errors
converge
to
the
Note
Note
(8)
(8)
does
does
not
contain
contain
a (10)
term
a surface
term
in,
in,
because
because
thethe
position
position
error
error
q qi
q r . Expression
qalready
iscontrol
the
nlaw
law
nshows
symmetric
positive
definite
manipulator
inertia
matrix,
nodal
forces,
included
inthat
.the
Expression
shows
thattothe
output
errors
conve
 that
 control
q r not
already
included
in n×1
(10)
output
errors
converge
the
wherealready
q is the n×1
vector of
displacements,
τ isshows
the
q r . Expression






s

q


q

0
s

q


q

0
Rofnodal
ASMforces,
MIMO
Note
Note
that
control
control
law
(8)
(8)
does
does
not
notcontain
athe
aterm
term
in,because
because
the
the
position
position
error
error
q q isis
The
DESIGN
of
ASM
MIMO
MˆCONTROLLER
is the
 n law
symmetric
positive
manipulator
inertia
matrix,
 q4.
 that
vector
ofn applied
nodal
forces,
isisthe
n×n
nodal
forces,
Mˆcontain
n in,
nsymmetric
symmetric
positive
inertia matrix,
 q definite
CONTROLLER
s  q definite
q included
0 manipulator
(10) that the(10)
already
in q r . Expression
(10) shows
output erro
sliding
surface


ˆ
ˆshows
already
already
included
included
in
in
.
.
Expression
(10)
(10)
shows
that
that
the
the
output
output
errors
errors
converge
converge
toto
ths
q r q rExpression
sliding
surface


positive
definite
manipulator inertia
matrix,
is
the
n

1
n

1
C
q

q
q
is
the
vector
of
damping
forces,
and
K
q
is
the
vector
of elastic




sliding
surface
sliding surface


ˆ qshows
already
included
included
Expression
(10)
(10)
that
that
the
output
output
errors
errors
converge
converge
tothe
to
the
the
ˆr q.qr Expression
qC
This,
in
turn,
implies
that
as t 
t .. .Thus,
Thus,the
theadaptive
adaptivecontrolle
controll
vector
ofinin
and
isisthe
n×1
offorces,
n  1 n×1
n vector
1the
Cˆ  q q  q is thealready
vector
of damping
and
the
vector
of
elastic
This,
in
turn,
implies
q q
0 0 elastic
as
the
as
Thus,
This,
implies
that
 shows
n  1K  vector
n  1that
isasthe
of
damping
and
Kˆin
is
vector
of
 .q forces,
 q forces,
 q turn,
sreedom
described
plate
assystem
(24)
isofdamping
described
An
n model
degree
freedom
system
(24)qmodel
is described
as
s  q  qcontroller
0
(10)
 0 as
t


.
Thus,
the
adaptive
defined
by
(8)
is
This,
in
turn, plate
implies
that
sliding
surface


q

0
as
t
  . Thu
This,
in
turn,
implies
that
elastic forces.
adaptive
controller
defined
by
(8)
is
globally
asymptotically


qsliding
 q 0surface
(10)s  q  q 0
surface
forces. s sliding
s  q  q 0
(10)
The
controller ˆ design problem is as follows. Given the
stable and guarantees zero steady-state error for node
sliding
sliding
surface
surface
asymptoticallystable
stableand
and
guarantees
zero
steady-state
errorf
globally
asymptotically
zero
error
Kˆ  q  

Mˆ  q  qr  Cˆ  q q  q r forces.
K q 
 (2)
ˆ  q  q  Cˆ  q q  q  Kˆ  q  
q r forces.
M
globally
(2)guarantees
This,
in
turn,
implies
that(2)
qpositions.
 0 as t   . Thus, the adaptive controller
defined
by steady-state
(8)
is
r
r
desired
trajectory
q
(t),
and
with
some
or
all
the
rectangular
d
globally
asymptotically
s  qstable


This, in turn, implies that q  0 as The
t  controller
. Thus, thedesign
adaptive
controller
isthes desired
sq
qqq0
0
problem
is as defined
follows.by
Given
trajectory
q q t 0,and
andguarantees
with
 (8)
T
T
r
r
D
r
r
T
r d d
r
int turn,
implies
q  0 controller
as
  . Thus,
thebyadaptive
This, instable
turn, implies
that q This,
0 as
 steady-state
. Thus,
thethat
adaptive
defined
(8) is controller defi
globally asymptotically
and guarantees
zero
error
fort node
positions.
 q 

 00
s

s
q


q
q
(10)
(10)
The controller design problem is as
follows.
Given
the
desired
trajectory
q
t
,
and
with


The controller design problem is as follows. Given the desired trajectory q  t  , and with
globally asymptotically stable and guarantees zero
steady-state error for node positions.
This,
This,
in
in
turn,
turn,
implies
implies
that
that
qnode

q 
0parameters
0stable
as
ast t and

q.
Thus,
.error
the
adaptive
adaptive
controller
controller
defined
defined
byno
by
(
globally asymptotically stable
andasymptotically
guarantees
zero
steady-state
error
for
positions.
some or all
the
rectangular
plate
system
being
unknown,
derive
law
globally
asymptotically
guarantees
zero
steady-state
error
for
This,
in
turn,
implies
that

0Thus,
asthe
t

. Thus,
thea control
adaptive
contr
globally
stable
and
guarantees
zero
steady-state
for
node
positions.
Numerical
example
5.5.
Numerical
example
http://ijass.org
5. Numerical
example
31

This,
This,
ininturn,
turn,implies
implies
that
that
qparameters
q 
0 0 asas t 
t

.
Thus,
.
Thus,
the
the
adaptive
adaptive
controller
controller
defined
defined
by
by
(8)
(8)
is
is
some or all the
rectangular
plate
system
being
unknown,
derive
a
control
law
some or all the rectangular plate system parameters being unknown, derive a control law
5. Numerical
example
globally
globally
asymptotically
asymptotically
stable
stable
and
and
guarantees
guarantees
zero
zero
steady-state
steady-state
error
error
for
for
node
node
positi
pos
for the
actuator
forces
and an asymptotically
estimation
law
for
the unknown
parameters
such
that
the
5. Numerical
example
globally
stable
and
guarantees
zero
steady-state
err
5. an
Numerical
example
ˆ
ˆ M
ˆ
ˆMnode
5.
Numerical
example
globally
globally
asymptotically
stable
stable
and
and
guarantees
guarantees
zero
zero
steady-state
steady-state
error
error
for
for
node
positions.
positions.
ˆ
ˆ






Letting
M
,
C
q

q

P
,
K
Y

K
q
,


L
F
,

q


q


[
L
]
,


q
q
5.
Numerical
example
Letting
M
,
C
q

q

P
,
K
Y

K
q
,


L
F
,

q


q


[
L
]
,
q
q
for the actuator
forcesasymptotically
and
estimation
law
for
the
unknown
parameters
such
that
the
















    such
  M  ,  C q q  
 parameters
Letting
 that Mthe
 P  ,  Kr r Y rr K  q    
for the actuator forces and an estimation law for the unknown
rectangular plate system output q  t  tracks the desired trajectories after an initial
ˆ   M  , Cˆ  q q  
Lettingt  Mtracks
ˆ K
q 
, P ,  K LY Fexample
,  q ,  q qrL qFqr , [ L q],q  q q  5[ L] ,
Numerical
 P  ,Mˆ  K MY, after
Numerical
Letting
C
q q5.
K
5.
example
the
desired
trajectories
an
rectangular plate system Letting
output qMrectangular
ˆ
ˆ   M  , C  q qplate
Y1  K  q  ,    L  F  , q  q  q q 1[ L] ,
P  , an
qˆ tL tracks
the
after
 
P  system
,  K Y Mˆoutput

K M
q  ,, C
 Letting
qKq55
Ydesired
q q
ˆF q,q  
ˆ ],,trajectories
Letting
,P  initial

K1initial
(28~40)14-070.indd 31
2015-03-30
q q F
,

K
q[ML
  ,L Cexample
오후13:39:44


 M





 
5.
Numerical

q q  q q[L] ,5

5.5.Numerical
Numericalexample
example
d
d
d
r
r
r
r
r
r
r
r



KD  











100

100


100


100

100 
100
100


(10)
.
ssqqqq00
(10)
(10)
This, in
in turn,
turn, implies
implies that
that qq00 as
as tt..Thus,
Thus, the
the adaptive
adaptive controller
controller defined
defined by
by (8)
(8) isis
This,
 0 as t   . Thus, the adaptive controller defined by (8) is
globally asymptotically
asymptotically stable
stable and guarantees
guarantees zero
zero steady-state
steady-state error
error for
for node
node positions.
positions.
globally
Int’l J. of Aeronautical
& Space Sci. 16(1),and
28–40 (2015)
The
desired
joint
trajectory
is given by
e and guarantees zero steady-state error for node positions.
s  q  q 0
q1_d=sin(2  t); q2_d=sin(2  t); q3_d=sin(2  t); q4_d=sin(2  t); q5_d=sin(2  t);
The desired joint trajectory is given by
q1_d=sin(2Π
t); q2_d=sin(2Πt); q3_d=sin(2Πt);
 t); q7_d=sin(2  t); q8_d=sin(2  t); q9_d=sin(2  t);
q6_d=sin(2
q4_d=sin(2Πt); q5_d=sin(2Πt); q6_d=sin(2Πt);
Letting MM
M ,, CˆCˆ qq qq 

Letting
ˆˆ M
Letting

[[LL]],,
  

PP,, KKYYKKqq,,  LLFF,, qq qqrrqq qqr rThe
q7_d=sin(2
Πt); q8_d=sin(2
Πt); q9_d=sin(2
Πt);
displacements
and velocities
are chosen
as
K Y   K  q  ,    L  F  ,   q q r q qr  [ L] ,
The displacements and velocities are chosen as
5. Numerical
Numerical example
example
5.
5.
Numerical
example
5
1
5

1





5
1
1


 5

 1

 1

5
1




5
1











1
55
11





  
 ,  
5
1


,,
1
,

5
1

 





5
1

,
1
5
1










5
1
1





5
1









5
1
1


5 
1 







1 
55
11




1

100

y2
y3
y 2
y 3
y4
y 4
y5
y 5
y6
y 6
y7
y 7
y8
y 8
y9
y 9 
0
0
basis of the intelligent control scheme shown in Figures 3 and 4. When conducting
2
0active
velocity-tracking-23 velocity-tracking-43 velocity-tracking-39 velocity-tracking-59
system.
The desired joint 2trajectory is given10by
0
y 1
The initial displacements and velocities are chosen as
The
initial displacements and velocities are chosen as
x0=[1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0];
Using control law (8), Fig. 3 shows the ASM MIMO control
x =[1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0];
of the 57th, 59th, 61st, 39th, 41st, 43rd, 21st, 23rd, and 25th nodes.
Fig. 4 shows PD position tracking control of 57th, 59th, 61st,
Using control law (8), Figure 3 shows the ASM MIMO control of the 57th, 59th, 6
39th, 41st, 43rd, 21st, 23rd, and 25th nodes.
In this section, the effectiveness of ASM MIMO control
39th, 41st , 43rd , 21st, 23rd, and 25th nodes. Figure 4 shows PD position tracking contr
is examined with the new plant model. Comparison of
vibration
tracking
th
st
of 57th, response
59th, 61st, 39
, 41st ,control
43rd , 21during
, 23rd,the
and simulation
25th nodes.
time, and vibration control for the flexible plate vibration
system
is discussed
oneffectiveness
the basis of of
the
intelligent
In this
section, the
ASM
MIMO control
control is examined with the new
scheme shown in Figs. 3 and 4. When conducting active
suppression
ofComparison
the vibration,
nine points
are tracking
preferably
plant model.
of vibration
response
control during the simulatio
chosen as the tracking signal. From Figs. 3 and 4, the
time, and
vibrationtracking
control for
the flexible
plate
vibration
vibration
response
control
at each
point
is notsystem is discussed on the
position-tracking-23 position-tracking-43 position-tracking-39 position-tracking-59



100

100

.
100

100


100
there are some disturbances in the

100

100 
velocity-tracking-25 velocity-tracking-21 velocity-tracking-41 velocity-tracking-61 velocity-tracking-57
100
position-tracking-25 position-tracking-21 position-tracking-41 position-tracking-61 position-tracking-57





KD  







x   y1
10
suppression of the
0 vibration, nine points are preferably chosen as the tracking
q1_d=sin(2  t); q2_d=sin(2  t); q3_d=sin(2  t); q4_d=sin(2  t); q5_d=sin(2  t);
-2
0
5
time(s)
10
-10
0
5
time(s)
10
-2signal. From Figures -10
3 and
0
5
10
0 4, the
5 vibration
10
time(s)
time(s)
q6_d=sin(2  t); q7_d=sin(2  t); q8_d=sin(2  t); q9_d=sin(2  t);
2
10
The displacements0 and velocities are0 chosen as
x   y1
y 1
y2
y 2
-2
y3
0
y 3
5
10
y 4 y5 y 5
time(s)
y4
2
-10
y6
0
y 6
5
10
y 7 y8 y 8
time(s)
y7
10
-10
0
of the proposed ASM MIMO
control algorithm behavior are very good, especially w
-2
0
time(s)
2
time(s)
10
5
y 
9  time(s)
10
2
0
-2
x =[1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0,1.0,0];
0
5
10
0
5
10
0
0
when the PD control
method is adopted. In contrast, numerical simulation resu
10
0
y9
The initial displacements
and velocities
are chosen as
0
0
-2
2good
response tracking control at each point is
5
time(s)
10
2
-10
0
5
time(s)
10
0
5
time(s)
10
10
0
-10
10
Using control law (8), Figure 3 shows the ASM MIMO control of the 57th, 59th, 61st,
0
0
0
0
st
rd
th
-2 , 23 , and 25 -10
-2
-10
39th, 41st , 43rd , 21
nodes.
Figure
4 shows
PD 5position
tracking
control
0
5
10
0
5
10
0
10
0
5
10
th
th
of 57 , 59 ,
61st, 239th,
time(s)
st
rd
41 , 43 ,
0
2110st,
time(s)
rd
th
time(s)
time(s)
23 , and 25 nodes.
0
In this section,-2the effectiveness -10
of ASM MIMO control is examined with the new
0
5
time(s)
10
0
5
time(s)
10
plant model. Comparison of vibration response tracking control during the simulation
th
th st
Fig. 3. Position tracking
of the3.57Position
, 59th, 61sttracking
, 39th, 41st, of
43rdthe
, 21st57
, 23thrd,, 59
andth,2561
nodes.
Figure
, 39thUnit:
, 41stm., 43rd
, 21st, 23rd, and 25th
time, and vibration control for the flexible plate vibration system is discussed on the
nodes. Unit: m.
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.28
32
basis of the intelligent control scheme shown in Figures 3 and 4. When conducting
active suppression of the vibration, nine points are preferably chosen as the tracking
signal. From Figures 3 and 4, the vibration response tracking control at each point is not
(28~40)14-070.indd 32
2015-03-30 오후 3:39:44
Jingyu Yang Experimental Study of Adaptive Sliding Mode Control for Vibration of a Flexible Rectangular Plate
Table 1. Contrastive analysis results
Natural
1
2
3
4
5
6
7
Frequency
FEA result
(Regular
0.8359 2.027 5.0545 6.3662 7.2835 12.459 14.451
plate)
FEA result
(Piezoelectric 0.8392 2.0547 5.1419 6.5629 7.4702 13.05 14.792
plate)
Absolute
0.0033 0.0277 0.0874 0.1967 0.1867 0.591 0.341
error
Model test results and contrast analysis
This section presents the results obtained on the test-flexible plate. The plate is
composed of several segments, as shown in Figures 8-10. Table 2 shows the
experimental
th
th
st results
th ofstnatural
rd frequency,
st
rd obtained from different model test m
th
st
st
PDth, position
tracking
error
of 57
Fig. 4. PD position tracking error of 57Figure
, 59th, 614.
, 39
41st, 43rd, 21
, 23rd, and
25th nodes.
, 59 , 61 , 39 , 41 , 43 , 21 , 23 , and
We can see that data of Groups 7 and 8 are better than the other groups. In parti
25th nodes.
flexible plate. The plate is composed of several segments, as
see8-10.
that Table
the data
of Groups
and 8 are in agreement
shown incan
Figs.
2 shows
the 7experimental
results with the results of the
of natural frequency, obtained from different model test
numerical simulation. Additionally, the results of flexible cantilever plate struct
methods. We can see that data of Groups 7 and 8 are better
than the other groups. In particular, we can see that the data
6. Experimental study with smart
piezoelectric
nine PZT materials
patches are shown in Table 3.
good when the PD control method is adopted. In contrast,
numerical simulation results of the proposed ASM MIMO
control algorithm behavior are very good, especially when
there are some disturbances in the system.
6. Experimental study with smart piezoelectric
materials
FEA calculation and contrast analysis
FEA calculation and contrast analysis
These results are shown in Table
1 and
Figs. 5-7.
These
results
are shown in Table 1 and Figures 5-7.
It is easy to see that there are small difference between the
frequency of the PZT plate and the frequency of the regular
It is easy to see that there are small difference between the frequency of the PZT plate
plate.
and the frequency of the regular plate.
Model test results and contrast analysis
This section presents the results obtained on the test-
Figure 5. PZT plate
Fig. 5. PZT plate
Table 1. Contrastive
analysis
results
Table
1. Contrastive
analysis results
Natural
Table
1
2
3
4
5
6 2. Modal
7 test value
8 of natural
9 frequency before installed actuators/sensor
Frequency
FEA result
(Regular
0.8359 2.027 5.0545 6.3662 7.2835 12.459 14.451 14.669 16.525
plate)
FEA result
(Piezoelectric 0.8392 2.0547 5.1419 6.5629 7.4702 13.05 14.792 15.453 17.111
plate)
Absolute
0.0033 0.0277 0.0874 0.1967 0.1867 0.591 0.341 0.784 0.586
error
Model test results and contrast analysis
33
http://ijass.org
This section presents the results obtained on the test-flexible plate. The plate is
composed of several segments, as shown in Figures 8-10. Table 2 shows the
(28~40)14-070.indd 33
2015-03-30 오후 3:39:45
locations. This is illustrated in the example.
Mechanical-electric coupled model
The procedure of a new modeling method is discussed in this section, and the
Int’l J. of Aeronautical & Space Sci. 16(1), 28–40 (2015)
parameters of dynamic Eq. (3) are shown in Tables 4-10.
of Groups 7 and 8 are in agreement with the results of the
numerical simulation. Additionally, the results of flexible
cantilever plate structure with nine PZT patches are shown
in Table 3.
Optimal piezoelectric actuator locations and orientations
An intelligent optimal algorithm design method is
presented for the orientation, placement, and numbers
of actuators and sensors in closed-loop intelligent
vibration control systems (Fig. 10). The method can be
used to develop the non-linear programming problem;
it is efficient and can handle complex optimization
6. PZT plate FEA model
Fig. 6. PZT plateFigure
FEA model
Table 2. Modal test value of natural frequency before installed actuators/sensors
Location
A
Method
Group
1
Group
2
Random
Group
3
Group
4
Group
1
Group
2
Group
3
Group
4
Sine
Sweep Group
5
Group
6
Group
7
Group
8
Hammer
Shot
Group
1
Group
Random
2
Group
3
B
Sine
Sweep
Hammer
Shot
1
Table 3. Model test values of natural frequency
2
3
4
2.8
6.9
7.4
2.8
7
10.5
2.6
7.1
5
6
7
8
9
10.6 15.2 17.3 21.6 25.2
15
17.3 21.8 25.3 28.3
10.5 15.1 17.3 21.8 25.3 26.4
7.2 10.6 15.2 16.9 21.7 25.3 28.3 34.1
0.8 2.6
4.1
7.4
10.7 13.1 15.2 17.3 21.8
0.8 2.6
4.9
7.7
10.7 13.1 15.2 17.3 21.8
0.8 2.6
4.1
7.4
10.7 13.1 15.2 17.2 21.8
0.8 2.6
4.1
7.4
10.7 13.1 15.2 17.2 21.8
0.8 2.6
4.4
5.2
6.9
7.5
10.8 12.9 15.2
0.8 2.8
4.4
5.2
6.9
7.8
10.7 13.0 15.2
0.7 2.8
4.0
5.3
6.9
7.6
8.8
10.6 13.2
0.7 2.8
4.0
5.3
6.9
7.6
8.8
10.6 13.2
Failure
2.6
3.9
6.8
7.4
10.8 12.9 15.3 17.1
2.8
3.9
6.8
7.4
10.6 12.9 16.6 17.1
2.8
4
6.9
7.5
10.8 12.9 15.3 17.1
Group
0.7 2.7
1
3.9
6.7
7.4
8.6
10.5 13.0 15.2
Group
0.7 2.7
2
3.9
6.7
7.4
8.6
10.5 13.0 15.2
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.28
Failure
34
Optimal piezoelectric actuator locations and orientations
An intelligent optimal algorithm design method is presented for the orientation,
(28~40)14-070.indd 34
2015-03-30 오후 3:39:45
Jingyu Yang Experimental Study of Adaptive Sliding Mode Control for Vibration of a Flexible Rectangular Plate
objective optimization problem, and generated a set
of solutions that can be used to configure the number
and the actuators’s orientations and locations. This is
illustrated in the example.
problems and deal with many optimization variables.
Several numerical analysis results (27,28) were performed
to confirm the effectiveness of the method. Multiple
procedures for the scheme are possible for the multiTable 3. Model test values of natural frequency
Frequency
Frequency
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
Experiment Group 1 0.7 2.8 4.0 5.3 6.9
7.6 8.8
10.6
13.2
Experiment Group
1
0.7
2.8
4.0
5.3
6.9
7.6
8.8
10.6
13.2
Group 2 0.7 2.7 3.9
6.7
7.4 8.6
10.5
13.0
Group
2
0.7
2.7
3.9
6.7
7.4
8.6
10.5
13.0
Average 0.7 2.75 3.95 5.30 6.8
7.5 8.7
10.55 13.1
Average
0.7
2.75
3.95
5.30
6.8
7.5
8.7
10.55
13.114.67
FEM
Group 1 0.84 2.03
5.05 6.37 7.28 12.46 14.451
FEM
Group
1 0.840.14
2.030.72
Error
Error
0.14 0.72
5.050.25
6.370.437.280.22
12.46
3.7614.451
3.90114.67
1.57
0.25 0.43 0.22 3.76 3.901 1.57
(a)
(b)
Figure
7. first
Contrastive
analysis
of common
the first FEA
mode
shape
FEA
model and
Fig. 7. Contrastive analysis
of the
mode shape
between
model
andbetween
PZT platecommon
FEA model.
(a) Common
plate FEA model (b) PZT
Figure 7. Contrastive analysis of the first mode shape between common FEA model and
plate FEA model
35
(28~40)14-070.indd 35
http://ijass.org
2015-03-30 오후 3:39:45
Int’l J. of Aeronautical & Space Sci. 16(1), 28–40 (2015)
shown in Tables 4-10.
Mechanical-electric coupled model
To establish the vibration control model, in this section,
The procedure
of a new
modeling method
is discussed
PZT
PZTplate
plateFEA
FEAmodel.
model.(a)
(a)Common
Commonplate
plateFEA
FEAmodel
model(b)
(b)PZT
PZTplate
plateFEA
FEAmodel
model
experimental tests of the plate are used to determine
in this section, and the parameters of dynamic Eq. (3) are
parameters
the model.andThe
reason
for using
Figureof8. Experiment
control
instruments
(1) an
experiment for the input matrix L, instead of the finite
element method (FEM) is because the base displacement
of the controlled system can easily reflect the real situation
of an intelligent vibration control system. This would make
the control system (Figs. 8, 9) for vibration more effective.
Moreover, the other system parametric values as presented
in Tables 6-10.
Establish the experimental system
The experimental test results are shown in Figures 11-15.
Some experiments must be performed if the ASM MIMO
Figure 9. Experiment and control instruments (2)
Figure
8.8.Experiment
and
(1)
Fig.
8. Experiment
and control
instruments
(1)
Figure
Experiment
andcontrol
controlinstruments
instruments
(1)
Figure
9.9.Experiment
and
(2)
Fig.
9. Experiment
and control
instruments
(2)
Figure
Experiment
andcontrol
controlinstruments
instruments
(2)
Figure
10. Five
locations of PZT
Fig. 10. Five
locations
of PZT
Table 4. Experiment test
values
strain rangetest values of strain range
Table
4. of
Experiment
Location
Static
Figure
Figure10.
10.Five
Fivelocations
locationsofofPZT
PZT
Deformed
1
2
3
4
5
6
7
8
9
Group 1
995
977
131
140
131
968
131
140
1012
Group 2
986
968
140
123
140
986
123
149
995
Group 3
977
977
123
131
123
997
114
131
1004
Average
986
974
131
137
137
983
123
140
1003
Group 1
995
986
123
131
105
123
123
123
986
Group 2
986
1012
114
114
114
114
114
114
995
Group 3
977
995
131
123
131
105
140
131
977
Average
986
992
123
123
117
114
126
123
986
0
18
-8
-14
-20
3
-17
-17
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.28
Table 5. Model test value of natural frequency
36 after installed actuators and sensors
Frequency
(28~40)14-070.indd 36
Location Type
Sine
A
Sweep
1
2
3
4
5
6
0.68
2.72
3.91
6.54 7.14 10.40
7
8
9
13.03 14.96 17.17
2015-03-30 오후 3:39:45
Static
Group 1
995
977
131
140
131
968
131
140
1012
Group 2
986
968
140
123
140
986
123
149
995
Group 3
977
977
123
131
123
997
114
131
1004
Average
986
974
131
137
137
983
123
140
1003
Group 1
995
986
123
131
105
123
123
123
986
Jingyu Yang Experimental Study of Adaptive Sliding Mode Control for Vibration of a Flexible Rectangular Plate
control design is to be accepted by people in the research
Group 2 986 1012 114
field. From the experimental set-up photograph in Figure
Deformed
8, a steel cantilever plate bonded Group
with nine
piezoelectric
3 977
995 131
actuators is shown. Moreover, near the fixed end, a PCB
sensor is used to acquire the displacement
plate. 123
Averageof flexible
986 992
Then, the acquired displacement signal is transmitted to
0
18 is -8
the NI data acquisition (DAQ). Next, the NI compactrio
used to process experiment data and generate the actuating
control voltage signals for piezoelectric patches to actuate
the cantilevered plate. The computer loads the LABVIEW
114 114 114 114 114 995
control programs to the NI compactrio and achieves current
collect,
process,
123 131
105 store,
140 and
131display
977 of experimental data.
Finally, the experimental results above are presented to
confirm
the capacity
of the
123 117
114 126
123intelligent
986 vibration controller.
Additionally, in the experiments, two different control
-14 -20
3 to-17
-17the effectiveness of the
schemes
are presented
confirm
intelligent control system described. The validity of vibration
suppression with the ASM MIMO was tested effectively.
Table 5. Model test value
of natural
frequency
after installed
actuators
and sensors
Table
5. Model
test value
of natural
frequency
after installed actuators and sensors
Frequency
Location Type
Sine
A
Sweep
Sine
B
Sweep
Average
1
2
3
4
5
6
7
8
9
0.68
2.72
3.91
6.54 7.14 10.40
13.03 14.96 17.17
0.67
2.75
3.86
6.54 7.14 10.53
13.03 14.96 17.17
0.675 2.735 3.885 6.54 7.14 10.465 13.03 14.96 17.17
Table 6. Experimental
test values
of displacement
voltages
Table 6. Experimental
test values
of displacement
andand
voltage
Voltage (V) To establish the vibration control model, in this section, experimental tests of the plate
CM
C
O
L
U
M
N
1
2
3
4
5
6
7
8
0
×
0
0
0
0
0
0
0
0
0
9 0
4
4
6
5
9
6
9
8
7
9
10
10
9
9
are
parameters
for
× used×to determine
×
×
× of the
× model.
× The reason
×
× using×an experiment
×
×for ×
×
8
10
8
10
6
10
7
8
10
8
8
8
9
10
the
input
matrix
L,
instead
of
the
finite
element
method
(FEM)
is
because
the
base
-1.5 -2.1
-2
-2
-1
-2
-0.5 -1.9 -2.2
-2
-2
-2
-2.5 -1.9
-0.5 -0.8 -0.5 -0.5 -0.5 -0.5
-1
-0.2 -1.2 -1.5 -0.5 -0.9
0
-1
-1
-0.7 -0.8
-1
0
-0.7 -1.2
-1
-1
-1.1 -1.2 -1.2
-1
-1
3
2.8
2.5
3
1.5
2.8
2
2.8
2.5
2.5
3.5
2
2.5
3
2.5
2.2
2
2
1
1.2
1.3
2.5
1.5
2.5
2
2.3
2
2.2
2.3
1.7
2.5
2.4
1.5
2.3
1.3
2.3
2
2
1
2.3
1.3
1.6
-0.45 -0.5 -0.55 -0.45 -0.25 -0.45 -0.27 -0.57 -0.45 -0.25 -0.58 -0.32 -0.27 -0.28
-0.55 -0.35 -0.4 -0.17 -0.05 -0.27 -0.17 -0.37 -0.25 -0.15 -0.4 -0.23 -0.2 -0.25
2.8
2.5
2.5
2.2
2
3
2.1
2.8
2.1
2.5
2
2.3
2
2.1
Table 7. Experimental
test values
of input
matrix
Table 7. Experimental
test values
of input
matrix
LL
Row
C
O
L
U
M
N
1
2
3
4
5
6
7
8
9
1
0.000189
0.000237
0.00028
0.000355
0.00037
0.00041
0.00047
0.00048
0.00053
2
0.00016
0.00020
0.00024
0.00030
0.00032
0.00035
0.00040
0.00041
0.00045
3
0.00021
0.00026
0.00031
0.000398
0.00042
0.00046
0.00053
0.00053
0.00059
4
0.00022
0.000286
0.00034
0.000429
0.00045
0.0005
0.00057
0.00057
0.00064
5
0.00020
0.00026
0.00031
0.00039
0.00041
0.00045
0.00052
0.00052
0.00058
37
(28~40)14-070.indd 37
6
0.00020
0.00025
0.00031
0.000388
0.000414
0.00045
0.00051
0.00052
0.00058
7
0.00016
0.000204
0.000245
0.000306
0.000326
0.00035
0.00040
0.00041
0.00045
8
0.00020
0.000251
0.000301
0.000377
0.000402
0.00044
0.00050
0.00050
0.00056
9
0.00018
0.000232
0.000279
0.000349
0.000372
0.00040
0.00046
0.00047
0.00052
http://ijass.org
2015-03-30 오후 3:39:46
Int’l J. of Aeronautical & Space Sci. 16(1), 28–40 (2015)
7. Conclusions
In this paper, we studied the vibration tracking control
problem of a flexible plate structure system and proposed
an ASM MIMO control method. ASM MIMO control
is a combination of adaptive control and sliding mode
control, having the advantages of intelligent control,
MIMO control, and non-linear control in the sense of
robustness of the control system. The proposed sliding
mode control law is an intelligent feedback control law
that involves tracking positions and tracking velocities.
One advantage of the proposed ASM MIMO controller
is that it is a MIMO control law that differs from a
SISO controller; it can be used to deal with a complex
system with features of non linearity and parameter
coupling. The robust design of the ASM MIMO control
is another advantage compared with a conventional PD
control method. Simulation results demonstrated that
Table 8. ExperimentalTable
test values
of mass matrixtest
M values of mass matrix M
8. Experimental
Table 8. Experimental test values of mass matrix M
Row
1Row 2
3
4
5
6
7
1
02
03
04
05
06
07
C 1 0.88
C 21 00.88 0.88
0
O
00
00
00
00
00
0
LO 32 00
00.88 0.88
00
00
00
00
L 3 0
0
U
4 0
00
00.88
0.88
00
00
00
U 4 0
M
0
5 0
00
00
00.88
0.88
00
00
M 5 0
N
0
00
00
00
00.88
0.88
00
N 6 0
6
0
0
0
0
0
0.88
0
7 0
0
0
0
0
0
0.88
7
0
0
0
0
0
0
8 0
0
0
0
0
0
00.88
8
0
0
0
0
0
0
9 0
0
0
0
0
0
00
9 0
0
0
0
0
0
0
Table 9. Stiffness matrix K value
Table 9. Stiffness matrix
K value
Table
9. Stiffness matrix K value
Row
N/M
Row
1
2
3
4
5
6
7
N/M
1
2
3
4
1 202.7 -67.6
0
-67.6
05
06
07
202.7 654.6
-67.6 -218.2
0
-67.6
0
21 -218.2
0
-218.2
00
00
C 2 -218.2 654.6 -218.2
-218.2
0
0
-10.4
20.8
00
0
-10.4
00
C 3
O
3
0
-10.4
20.8
0
0
-10.4
0
4 -262.5
0
0
1050 -262.5
0
-262.5
LO 4 -262.5
0
0
1050
-262.5
0
-262.5
5
0
-138.1
0
-138.1 552.4 -138.1
0
UL 5
0
-138.1
0
-138.1
552.4 846.6
-138.1
6
0
0
-282.2
0
-282.2
00
U
M
6
-282.2
0
-282.2
846.6
0
00
00
0
-67.6
0
0
202.7
M 7
N
7
0
0
0
-67.6
0
0
202.7
0
0
0
0
-218.2
0
-218.2
N 8
-218.2
0
-218.2
98
00
00
00
00
0
-10.4
0
9
0
0
0
0
0
-10.4
0
8
08
00
00
00
00
00
00
0
0.88
0.88
0
0
9
09
00
00
00
00
00
00
00
0
0.88
0.88
8
9
08
09
00
00
00
00
0
0
00
0
-138.1
00
-138.1
0
0
-282.2
0
-282.2
-67.6
0
-67.6
0
654.6 -218.2
654.6
-218.2
-10.4
20.8
-10.4
20.8
Table 10. Damping matrix
c value
Table
10. Damping matrix c value
1
2
3
4
Row
5
6
7
8
9
1
202.7
-67.6
0
-67.6
0
0
0
0
0
2
3
4
5
6
7
8
9
-218.2 654.6 -218.2
0
-218.2
0
0
0
0
0
-10.4
20.8
0
0
-10.4
0
0
0
-262.5
0
0
1050 -262.5
0
-262.5
0
0
0
-138.1
0
-138.1 552.4 -138.1
0
-138.1
0
0
0
-282.2
0
-282.2 846.6
0
0
-282.2
0
0
0
-67.6
0
0
202.7 -67.6
0
0
0
0
0
-218.2
0
-218.2 654.6 -218.2
0
0
0
0
0
-10.4
0
-10.4
20.8
N·S/
M
C
O
L
U
M
N
7. Conclusions
DOI: http://dx.doi.org/10.5139/IJASS.2015.16.1.28
38
In this paper, we studied the vibration tracking control problem of a flexible plate
structure system and proposed an ASM MIMO control method. ASM MIMO control is
(28~40)14-070.indd 38
a combination of adaptive control and sliding mode control, having the advantages of
2015-03-30 오후 3:39:46
Figure 11. Location 1’s forced vibration response (controlled with 10 V voltage)
Jingyu Yang Experimental Study of Adaptive Sliding Mode Control for Vibration of a Flexible Rectangular Plate
uncontrolled controlled uncontrolled uncontrolled Figure
11.
Location1’s
1’sforced
forced vibration
response
(controlled
with 10with
V voltage)
Fig. 11. Location
vibration
response
(controlled
10 V
Figure
vibration
response
(controlled
with
voltage)
Figure
13.Location
Location
3’sforced
forced
vibrationresponse
responseofof
(controlled
with
2020VV20
voltage)
Fig.
12. L12.
ocation
2’s2’s
forced
vibration
(controlled
with
V
voltage)
voltage)
uncontrolled controlled uncontrolled controlled controlled controlled uncontrolled controlled Fig.
13. Location
3’s 3’s
forced
vibration
(controlled
V
Figure
13. Location
forced
vibration response
response (controlled
withwith
20 V 20
voltage)
voltage)
Figure 12. Location 2’s forced vibration response of (controlled with 20 V voltage)
uncontrolled uncontrolled controlled controlled Figure
14. Location
forced
vibration response
response (controlled
withwith
10 V 10
voltage)
Fig.
14. Location
4’s 4’s
forced
vibration
(controlled
V
voltage)
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