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Engineering, Technology & Applied Science Research Vol. 12, No. 5, 2022, 9388-9394 9388 
 

www.etasr.com Duong et al.: Settling Time Optimization of a Critically Damped System with Input Shaping for … 

 

Settling Time Optimization of a Critically Damped 

System with Input Shaping for Vibration Suppression 

Control 
 

Minh-Duc Duong 

School of Electrical and Electronic Engineering 

Hanoi University of Science and Technology 

Hanoi, Vietnam 

duc.duongminh@hust.edu.vn 

Quy-Thinh Dao 

School of Electrical and Electronic Engineering 

Hanoi University of Science and Technology 

Hanoi, Vietnam 

thinh.daoquy@hust.edu.vn 

Trong-Hieu Do 

School of Electrical and Electronic Engineering 

Hanoi University of Science and Technology 

Hanoi, Vietnam 

hieu.dotrong@hust.edu.vn 
 

Received: 5 August 2022 | Revised: 23 August 2022 | Accepted: 24 August 2022 

 

Abstract-The input shaping technique is widely used as 

feedforward control for vibration suppression of flexible dynamic 

systems. The main disadvantage of the input shaping technique is 

the increasing system time response since the input shaper 

contains time delay parts. However, with the same reference 

input, the actuator effort in the case of using an input shaper is 

smaller than the one in the case without an input shaper. Thus, it 

is possible to decrease the system response time by designing the 

feedback controller to maximize the actuator effort. This paper 

proposes a design approach to design the Proportional-Derivative 

(PD) controller for position control of the actuator so that the 

settling time of the flexible system with input shaper is 

minimized. The actuator system with a PD controller is 

equivalent to a critically damped system, and the condition for 

the controller gains is established. In addition, the settling time 

and actuator effort with shaped step input are calculated. The 

controller gains can be determined by solving the settling time 

optimization problem with the actuator effort constraint. The 

effectiveness of the proposed approach is verified via experiments 

with an overhead crane model. 

Keywords-flexible system; input shaping;PD controller; settling 

time optimization; overhead crane    

I. INTRODUCTION  

The vibration of flexible dynamic systems such as live load 
and flexible beam systems [1-3] and flexible robot manipulator 
and cranes [4-12] often causes a decrease in operation speed 
and accuracy. Due to sensor noise and unmodeled flexible 
dynamic problems, vibration suppression control using 
feedback control has often substantial limitations [4-7]. Open-
loop control is effective and widely used for the vibration 
suppression control of flexible machines. If the vibration 
dynamics are known with some confidence, then several 

techniques for modifying commands can suppress the system's 
vibration [8-13]. Among them, input shaping [13], which 
convolves a sequence of impulses with the command signal, is 
one of the most attractive techniques. Various improvements 
and applications of input shaping have been reported [14, 15]. 
The input shaping technique has also been used along with 
feedback control to optimize the system performance [16-24]. 
In [16-18], the concurrent design of the Proportional-Derivative 
(PD) controller and input shaping was considered. The PD 
controller parameters were chosen to fasten the feedback 
system response, and the input shaping was designed to 
eliminate the natural vibration frequency of the PD feedback 
control system. The same idea was deployed in [19] for the 
combination of input shaping technique and a Linear Quadratic 
Regulator (LQR) feedback control system. These works only 
considered the single mode vibration. The design technique 
was proposed for a feedback control system with multi-mode 
vibration in [20-21]. The vibration model uncertainty was also 
considered when combining input shaping and feedback 
controller [22-24].  

The above mentioned studies used input shaping that 
eliminates the vibration of the feedback system to optimize the 
system performance, not to suppress the vibration of a flexible 
system. In general, the vibration suppression control of a 
flexible dynamic system includes a feedback control part for 
actuator position control and a feed-forward control part for 
vibration control, as shown in Figure 1. When using an Input 
Shaper (IS) as feed-forward control, the IS only depends on the 
vibration model, not the feedback system. The feedback control 
is usually designed independently with the feed-forward 
controller. This design method is simple but does not optimize 
the system performance.  

Corresponding author: Minh-Duc Duong



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www.etasr.com Duong et al.: Settling Time Optimization of a Critically Damped System with Input Shaping for … 

 

 

Fig. 1.  Feed-forward control structure for vibration suppression control. 

Since the IS includes time delay parts that slow down the 
system, various efforts have been made to shorten the delay, 
see [14] for more details. However, most of them have not 
considered combining a feedback controller to reduce the 
system response. In this paper, a design technique to optimize 
the time performance of the system with the IS is proposed. At 
first, the IS is designed using the vibration’s parameters. Next, 
the PD controller is chosen to minimize settling time while 
keeping the actuator effort within the acceptable range. For this 
purpose, design techniques such as the one in [16] can be 
applied. However, that technique can only be applied to an 
under-damped feedback system. In many applications, the 
under-damped system is not expected since the existence of 
overshoot may cause limit excess. Thus, in this paper, we 
consider a design technique for a critically damped system. The 
PD feedback controller is designed to optimize the settling 
time, in consideration that the IS has been added to shape the 
input. The actuator effort is used as a design constraint. The 
settling time and actuator effort calculation are established such 
that the optimization problem can be easily solved by an 
optimization toolbox.  

II. INPUT SHAPING METHOD 

Input preshaping [13] is a feed-forward control technique 
for vibration suppression. The idea of input preshaping method 
is to cancel the vibration of an impulse input by generating 
another impulse that causes the inverse phase vibration with the 
first one. Let’s consider a simple vibratory system that can be 
expressed as second order system as following: 

�� � �������� � �02�2�2.�0
0.���02    (1) 
where ��is the undamped natural frequency, 
� is the damping 
ratio of the system, Y(s) and V(s) are the Laplace transforms of 
output y(t) and input v(t) respectively. If an impulse input with 
amplitude ��  is put into the system at time �� , then the output 
response y(t) is calculated as: 

���� � �� . sin��. � � �� �    (2) 
where �� � �� . ������� �  ������!�!"�, � � ���1 $ 
� % , and 
�� � ���1 $ 
� %��.  

In order to suppress the vibration caused by the impulse, we 
consider applying a second impulse to the system. Then the 
response of two impulses is calculated as: 

� � ��. sin��. � � ��� � �%. sin��. � � �%�    (3) 
Using the trigonometric relation, we can obtain: 

� � �. sin��. � � �!& �    (4) 

where: 

'� � (����)*�� � �%�)*�%�% � ��� +,��� � �%+,��%�%�!& � tan�� / 012�34150�2�34�016724150�6724�8   
By setting the response of two impulses equal to zero after 

applying the last impulse, we can easily obtain (supposing that 
the time of applying the first impulse is �� � 0 and the impulse 
amplitude are normalized �� � �% � 1): 

9���� : � ;
��5< <�5<0 ∆� >    (5) 

where ' ? �   $A.B(1$B2∆� �  A�0.√1$B2 . 
The two-impulse shaper is called the Zero Vibration (ZV) 

shaper. In general, if we apply N impulses with amplitude ��  
and at time ��  (i = 1, …, N), then the response of N impulses is 
calculated as: 

� � ∑ �� ���E�F� � ∑ �� . sin��. � � �� �E�     (6) 
where G� � (�∑ �� �)*��E�F� �% � �∑ �� +,���E�F� �%�!& � tan�� H∑ 0"2�34"I"J1∑ 0"6724"I"J1 K   

By setting the amplitude and its derivative equal to zero 
after applying the last impulse, we can obtain the result for 3 
impulses (N=3) as follows: 

9���� : � ;
���5<�� %<��5<�� <

�
��5<��0 ∆� 2Δ� >    (7) 

The three-impulse shaper or Zero Vibration Derivative 
(ZVD) shaper is more robust than the ZV shaper. More results 
can be seen in [14, 15]. By convolving the reference input and 
the above impulse series, the vibration can be suppressed. This 
paper will apply vibration suppression control using input 
preshaping with two and three impulses. It is noted that the 
response of the system with input preshaping is slower than 
that without input preshaping. The delay time is ∆t with ZV 
and 2∆t with ZVD input preshaping. 

III. CONTROLLER DESIGN 

In a flexible system, the actuator, such as an electrical 
motor, is controlled by a motor driver under velocity control 
mode. The actuator then can be modeled as follows: 

M�2�N�2� � GP��� � <Q�RQ25��2    (8) 
where ?6  is the gain and S6  is time constant of the system, U(s) 
and X(s) are the Laplace transforms of control voltage input 
u(t) and actuator position x(t) respectively. 

The system structure is shown in Figure 2, where GPD(s) is 
the transfer function of the feedback controller, Gp(s) is the 
transfer function of the plant (actuator and driver). The IS 
includes N impulses designed based on the flexible system’s 
vibration frequency and damping. We consider the system step 



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input to design the PD controller for position control. System 
response and settling time are calculated. In addition, the 
actuator effort is also determined. Then the settling time 
optimization with actuator effort constraint can be established, 
and the selection of PD controller parameters can be made by 
solving this optimization problem. 

 

 

Fig. 2.  Input shaper – PD feedback control structure. 

A. System Output Response 

The PD controller can be described by the following 
transfer function: 

�TU��� � ?U � � ?T     (9) 
where ?T V 0  is the proportional gain and ?U V 0  is the 
derivative gain. Then, the transfer function of the closed-loop 
system is: 

�WX � �Y�ZB�Y�ZB�1 �
?+?BS+ ��?+?ZS+�2�?+?Z�1S+ ��?+?ZS+

     (10) 

By setting: 
?+?BS+ � [  and <Q<\RQ � ] we have: 

�WX � [��]�2�[S+�1S+ ��] �
^���_���    (11) 

Because the underdamped system is not expected, the 
denominator of GCL(s) must have negative real roots. In 
addition, we require minimum response time. Thus, the 
denominator should have double negative roots. The condition 
for being double root is: 

/[S+�12S+ 82 � N ⇔ /?+?B�12S+ 82 � ?+?ZS+      (12) 
Now, we introduce an input step with magnitude L into the 

system at time t0, b��� � cd ef�g2 , the response of the feedback 
system is: 

^��� � cd ef�g2 h25E2�5ijQk1jQ 25E    (13) 
Thus, the response of the system with single step input can 

be expressed as: 

l��� � c.hm� n� $ � �m�!�!�� � o�o $ ���� $ ��� �m�!�!��p (14) 
where o � h.RQ5�%RQ  and � � Eh. 

Since the overshoot is not expected to appear, x(t) must be a 
non-decreasing or non-increasing function in the region � V ��. 
Thus, the root �q  of equation lr ��� � 0  must be outside the 
region � V �� , or �q s �� . Solving the equation lr ��� � 0  we 
obtain �q � �� � mm�t. Thus: 

�q s �� ⟺ mm�t s 0 ⇔ o s � ⟺ <v�RQ s ?T     (15) 
From the system response with single step input (14), the 

response of the system with convolved step command after step 
k is: 

���� � chm� w� ∑ X�xyF� $ � ∑ X�  �mz!�!{|xyF� �o�o $ �� ∑ X� z� $ �y | �mz!�!{|xyF� }    (16) 
Since 

hm� � � 1, the response y(t) can be rewritten as: 
���� � L ∑ XyxyF� $ L ∑ Xy  �mz!�!{|xyF� �  
X�[ $ o� ∑ Xy z� $ �y | �mz!�!{|xyF�     (17) 

B. Settling Time 

Settling time is a primary concern for most industrial 
applications. The faster the settling time, the higher throughput 
and accuracy are. The system is defined to be settled when the 
output response falls below a certain percentage (usually 5% or 
2%) of the step magnitude. The settling time is the required 
time for the output curve to reach and stay within a range of a 
certain percentage (usually 5% or 2%) of the final value. In this 
paper, we use the range of 5% to calculate settling time. We 
can describe the definition of settling time ts as the following 
conditions: 

1 $ ��!�c � 0.05, ∀� V �2    (18) 
1 $ ��!g�c � 0.05    (19) 

It is obvious that settling time is determined after the final 
step. Thus, the output y(t) to calculate settling time is the output 
with N steps. Since ∑ X�3yF� � 1 , and substituting (17) into 
(19), we have: 

1 $ ��!g�c � ∑ Xy  �mz!g�!{|EyF� $  
�[ $ o� ∑ Xy z�2 $ �y | �mz!g�!{|EyF� � 0.05    (20a) 

Moreover, since �2 � �3 , it can make the approximation 
 �mz!g�!{| ≅ 0  and z�2 $ �y | �mz!g�!{| ≅ 0  with j < N. Then 
(20a) becomes: 

X3 n �m(!g�!�) $ ([ $ o)(�2 $ �3) �m(!g�!�)p � 0.05   
(20b) 

Using the Lambert W function W(z) (i.e.  

� � �(�) �(�)) [25], the settling time ts can be determined as: 

�2 � �3 � �h�m $
�
m �(

�.��m.d
�

ie�
m(h�m) )    (21) 

C. Actuator Effort 

With reference input as step input, we are also interested in 
the actuator effort when designing the controller. According to 
Figure 2, with the input R(s), the actuator effort can be 
determined as: 

�(s) � ������.��5� . R(s) �
2(RQ25�)(<Q<v25<Q<\)
RQ.2�5(<Q<v5�)25<Q<\

. b(�)    (22) 



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If the input r(t) is the step input with magnitude L at the 

time t0, b(�) � cd
ef�g
2 , then:  

�(s) � X �!�2 (RQ25�)(<Q<v25<Q<\)RQ.2�5(<Q<v5�)25<Q<\    (23) 
The actuator effort can be calculated as: 

�(�) � L <\RQ�<Q<v�RQ n 
�m(!�!�) � (�� $ o)(� $ ��) �m(!�!�)p    (24) 

where �� � <\�<Q<\.<vRQ<\�<Q<v� . Then, the actuator effort with 
convolved step command after step k is: 

�(�) � <\RQ�<Q<v
�

RQ
 × 

∑ Xy w �mz!�!�| � (�� $ o)z� $ ��| �mz!�!�|}xyF�     (25) 
The actuator effort u(t) can be represented as the form:  

�(�) � Zx  �m! � �x � �m!     (26) 
where:  

Zx � <\RQ�<Q<v
�

RQ
∑ Xy n1 $ (�� $ o)�y p m!{xyF�   

�x � <\RQ�<Q<v
�

RQ
∑ Xy (�� $ o) m!{xyF�   

To find the maximum value of u(t), we solve the equation: 

�r (�) � �($oZx � �x )$o�x �� �m! � 0    (27) 
Thus, the maximum value of u(t) can be obtained at time: 

�x� � ���mT�m��     (28) 
It is noted that we can only consider u(t) after step k, and 

u(t) is decreased when t > tku. Therefore, the maximum value of 
actuator effort Ukmax is calculated as: 

�x�m� � � Zx  
�m!� � �x �x  �m!�   if �x� � �x

Zx  �m!�� � �x �x� �m!��   if �x� � �x    (29) 
To guarantee that u(t) is within the allowable actuator effort 

range, the following constraint is required after the step 
corresponding to each impulse: 

�x�m� � ��m�  for all k = 1, …, N    (30) 
where ��m�  is the maximum allowable actuator effort. 
D. Controller Designing  

The designing process of actuator position control using a 
PD controller and vibration suppression control using input 
shaping for a flexible dynamic system is: 

Step 1: Determine the system vibration parameters, 
including natural vibration frequency and damping factor. 

Step 2: Choose the appropriate input preshaping techniques 
such as ZV, ZVD, etc. See [14, 15] for more details about IS 
techniques. 

Step 3: Choose the PD controller parameters ?T  and ?U  to 
minimize the system settling time by solving the following 

optimization problem: minimizing (21) with constraints (12), 
(15), and (30).  

IV. APPLICATION TO OVERHEAD CRANE 

Figure 3 describes the overhead crane. In this figure, x is 
the cart’s position, l is the rope length, m is the load mass, M is 
the cart mass, and    is the angle between the rope and the 
vertical axis (Y axis). According to [26], the overhead crane 
linearized model can be described as: 

¡� � ([ � ¢)l£ � ¤� lr � ¢¥ £     (31) 
$l£ � ¥ £ � ¤¦  r � §     (32) 

where ¤�  and ¤¦  are the equivalent viscous damping of the cart 
and the load respectively. It can be seen that (31) describes the 
relation between the input force and the cart’s position. The 
load angle plays the role of disturbance. Equation (32) 
describes the effect of the cart’s motion to the vibration of the 
load.   

 

 

Fig. 3.  Overhead crane model. 

In practice, the cart is controlled by a motor with a driver. 
That can allow us to control the velocity of the cart. Therefore, 
to control the cart’s position, we use the motor model with a 
driver instead of (31). The cart plays the role of an actuator in a 
flexible system and has (8) as the transfer function. The control 
of the overhead crane is to move the load to the desired 
position while suppressing the load vibration. To control the 
cart’s position precisely, a PD controller is used. In addition, to 
suppress the load vibration, input preshaping is applied. 

The actual experiments are conducted with the overhead 
crane model shown in Figure 4. The vibration model can be 
calculated from (32), however, the model’s parameters may not 
be precise. Thus, we measured the sway angle and identified 
the natural vibration frequency and damping factor. The 
experiment parameters are shown in Table I. 

TABLE I.  EXPERIMENT PARAMETERS 

Parameter Values 

?6 4 
S6 0.2 
�� 4.45 rad/s 

� 0.007 
g 9.81 m/s

2
 

Umax 10 V 



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Fig. 4.  Experimental crane system. 

Then, two types of input shaping including ZV and ZVD 
can be designed using (5) for ZV and (7) for ZVD. Using the 
designed ISs, the PD controllers that optimize the settling time 
were calculated. We call them optimized PD-ZV and optimized 
PD-ZVD. In addition, the PD controller that optimizes the 
settling time of the feedback system using step input only and 
with the same constraint is also calculated. We call it optimized 
PD without IS. The ISs (5) and (7) are then applied to the 
optimized PD without IS. We call them ZV-optimized PD and 
ZVD-optimized PD without IS. The results are shown in Table 
II.  

 

 
Fig. 5.  Cart’s position in three cases: optimized PD without IS, ZV-
optimized PD without IS, and optimized PD-ZV. 

 

Fig. 6.  Actuator effort in three cases: optimized PD without IS, ZV-
optimized PD without IS, and optimized PD-ZV. 

 

Fig. 7.  Sway angle in three cases: optimized PD without IS, ZV-optimized 
PD without IS, and optimized PD-ZV. 

 
Fig. 8.  Cart’s position in three cases: optimized PD without IS, ZVD-
optimized PD without IS, and optimized PD-ZVD. 

 
Fig. 9.  Actuator effort in three cases: optimized PD without IS, ZVD-
optimized PD without IS, and optimized PD-ZVD. 

 
Fig. 10.  Sway angle in three cases: optimized PD without IS, ZVD-
optimized PD without IS, and optimized PD-ZVD. 

0 1 2 3 4 5 6 7 8 9

Time (s)

0

5

10

15

20

25

30

35

C
a

r
t'

s 
p

o
si

ti
o

n
 (

c
m

)

Optimized PD without IS

ZV- Optimized PD without IS

Optimized PD-ZV

A
c
tu

a
to

r
 E

ff
o

r
t 

(V
)

0 1 2 3 4 5 6 7 8 9

Time (s)

-10

-5

0

5

10

S
w

a
y

 a
n

g
le

 (
d

e
g
r
e
e
)

Optimized PD without IS

Optimized PD-ZV

ZV-Optimized PD without IS

0 1 2 3 4 5 6 7 8 9

Time (s)

0

5

10

15

20

25

30

35

C
a

r
t'

s 
p

o
si

ti
o
n

 (
c
m

)

Optimized PD without IS

ZVD-Optimized PD without IS

Optimized PD-ZVD
A

c
tu

a
to

r
 E

ff
o

r
t 

(V
)

S
w

a
y

 a
n

g
le

 (
d

e
g

r
e
e
)



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TABLE II.  CONTROLLER PARAMETERS AND SETTLING TIME 

Parameter 

Values 

Optimized PD 

without IS 

Optimized 

PD-ZV 

Optimized 

PD-ZVD 

?T 0.325 0.52 0.59 
?U 0.01 0.07 0.09 

Settling time 

(s) 

2.66 

Using ZV: 3.18 

Using ZVD: 3.67 

2.27 2.64 

 

The experimental results are shown in Figure 5-9. It is clear 
that ZV and ZVD can suppress the payload vibration 
significantly. In the case of optimized PD without IS, the 
settling time is 2.66s. But when the IS is applied to the 
vibration suppression control, the settling time increases to 
3.18s for ZV shaper and 3.67s for ZVD shaper. In addition, the 
actuator effort also reduces from 10V in the case of optimized 
PD without IS to 6.80V and 5.90V when applying ZV and 
ZVD shapers respectively. Therefore, we should design the PD 
controller inconsideration of using IS at the system input. At a 
result, the actuator effort is larger but still in the required limit, 
and the system moves faster with settling time only 2.27s for 
ZV and 2.64s for ZVD shapers.   

It is also found that in the case of optimized PD-ZV the 
settling time is smaller than the one in the case of ZV-
optimized PD without IS, but the residual vibration magnitude 
is larger. The reason for this is the difference between the 
actual vibration frequency and the designed vibration 
frequency and the effect of the feedback system to the response 
of the IS shaper. This can be improved by using ZVD shaper. It 
is clear that in the case of optimized PD-ZVD the settling time 
is smaller than the one in the case of ZVD-optimized PD 
without IS, and the residual vibration magnitude is almost the 
same. 

V. CONCLUSIONS 

The current paper proposes a design process for the PD 
controller of the actuator in a flexible system that uses input 
shaper for vibration suppression control. The controller gains 
are chosen such that the feedback system is of the critically 
damped type and system’s settling time is minimized while 
keeping the actuator effort constraint. The experiments with 
overhead crane model show the effectiveness of the proposed 
designing process. The settling time is reduced, in comparison 
to the case of the optimizing PD controller without considering 
the use of the input shaper. In the future, a designing tool will 
be developed to apply the proposed designing process for a real 
flexible system. In addition, the designing process for the 
system with time varying vibration frequency is also 
considered.  

ACKNOWLEDGMENT 

The paper is supported by the Hanoi University of Science 
and Technology project T2021-PC-002. 

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