Reply to the comments and suggestions made in the
reviewers' reports
Comment 1. On page 5, why do the authors choose 12 as the sliding
parameter $C_s$? Is it chosen by authors or referred from other articles?
If from articles, please provide the reference paper.
Response: In this paper, the rising time for the real overload to track the
unit overload is required to be less than 0.3 seconds. From Theorem 1, we
know that the response speed is directly related to the sliding mode
parameter Cs . That is, the larger Cs is, the quicker the system response
is. On the other hand, if the sliding mode parameter Cs is increased, the
reaching parameter Cr also needs to be increased so as to ensure that the
system state is able to enter into the next sliding surface quickly.
According to (10), the increase in Cr will cause difficulty in reducing
the chattering. Thus, the choice of the sliding mode parameter Cs should
be small. However, the rising time is required to be less than 0.3 seconds.
Thus, the sliding mode parameter is chosen as 12.
We have changed
“To achieve ... of the sliding parameter.”
in original manuscript to
“From Theorem 1 ... of the sliding parameter.”
in the revised paper.
Comment 2. Please clarify how to derive equations (31) and (32) in
detail? In (32), should the exponent of the left expression be “$-C_s^f
t_1$” rather than “$-C_s^f t_2$”?
Response: Proof of (31). Since Csf  0 , the exponential function i eCs t
f
is monotonically decreasing. When t1  t2 , we have 1eCs t1  1eCs t2 .
f
f
Thus,
Cr
Cr
 Csf t1
 Csf t2
.


e



e
1
1
Csf
Csf
(31)
This completes the proof of (31).
Proof of (32). The exponential function e Cs t2 is nonnegative.
f
When 1  2 , we have
Cr
Cr
 Csf t2
 Csf t2
.


e



e
1
2
Csf
Csf
(32)
This completes the proof of (32).
There is a mistake in (30) in the original manuscript. Namely,
formula (30) was written as:
Csf xif,1  i e Cs ti 
f
Cr
,
Csf
i  1,2.
(30)
A “  ” sign was missed on the left hand of this equation. To be precise,
formula (30) should be written as:
Csf xif,1  i e Cs ti 
f
That is,
Cr
,
Csf
i  1,2.
Csf xif,1 
Cr
 Csf ti


e
,
i
Csf
i  1,2.
(30)
This point has been corrected in the revised paper. Apparently, the
confusion to (31) and (32) is caused by the error (i.e., a missing “  ” sign)
in formula (30). It is now being corrected and hence the confusion should
have been removed. We have given proofs of equations (31) and (32) in
this Reply. Since they are rather straightforward and hence are not
included in the revised paper.
Comment 3. The author should restate the design procedure of multiple
sliding mode recursive method in detail to show that it is guaranteed that
the above method can work as it.
Response. In the revised paper, we have added a paragraph towards the
end of Section 4.1, where the design procedure of multiple sliding mode
recursive method is restated with explanation.
Comment 4. Section 4.2 and 4.3 are not only straightforward but also
redundant. Suggest that the authors combine them together or just delete
them.
Response. The basic form of variable structure control law is
u  (GB)1GAx  (GB)1 Cr sgn( S ( x )) .
(10)
Since the existence of the sign function, chattering will appear in the
system if no other measures are taken. This is undesirable, and hence it is
necessary to make effort to reduce the chattering. To illustrate this point,
we include the simulated curve showing the results for the case when
chattering is present in the revised paper. In addition, Section 4.2 and
Section 4.3 are combined into one section.
Comment 5. Authors should provide detailed information of simulation.
Just the maximal allowable values for deflection angle are given and then
the author sketched some graphs. Please give parameters’ specifications.
Response. In our simulation study， the following controller
ui  (GB)1GAx ,

1
1
ui  (GB) GAx  (GB ) Cr
Si ( x )
Si ( x ) 
,
,
S ( x )  0,
S ( x )  0,
(38)
Is applied to the system
 0
x 1
 TM2
1 
 0 
x   vK M  u ,
2 TMM 
 gTM2 

(1)
so as to show the control effect, where G  [Cs 1] . Details on the
parameters used are: the time constant of the missile ( K M )，the flight
speed of the missile ( v ), the gravitational acceleration ( g ), sliding mode
parameter ( Cs ), reaching parameter ( Cr ), and reducing chattering factor
(  ). In addition, we also need the maximal elevator deflection angle
allowable to be used in evaluation and simulation step.
In the original manuscript, we forget to give detailed information on
these parameters. In the revised paper, the values of these parameters are
given.
We have changed
“For … is taken to be 0.001 second.”
in the original manuscript to
“For … in Figure 6.” and “In the … is taken to be 0.001 second.”
in the revised paper.
Comment 6. English expression should be further improved and there are
many typos in this paper. I just point out some obvious typos. For
example, on Page 1, Line 1, “by using of this specification” should be “by
using this specification”; on Page 1, Line 14, “the state will move
smoothing” should be “the state will move smoothly”; on Page 2, the 7th
last line, “T denotes the transport” should be “T denotes the transpose”,
on Page 2, the last line above Section 2, “We making some concluding
remarks” should be “We make some concluding remarks”; on Page 4, the
last line above Section 3, “it following from (10)” should be “it follows
from (10)”, and so on.
Responses. We have corrected the typos mentioned above in the revised
paper. In addition, the following errors have also been corrected.
1. In formula (8), (10) and (21), “ GB1 ” has been changed to “ (GB)1 .”
2. In the original manuscript, formula (9) is
1

u  GB GAx,
u
1
1

u  GB GAx  GB C rsgn( S ( x )),
S ( x ) 0 ,
S ( x ) 0 .
(9)
It has been changed to
u  (GB)1GAx,


1
1

u  (GB) GAx  (GB) C rsgn( S ( x )),
S ( x ) 0 ,
S ( x ) 0 .
(9)
in the revised paper.
3. In the original manuscript, formula (38) is
1

ui  GB GAx ,
u
u  GB 1GAx  GB 1C r SiS(ix( x) )  ,

 i
S ( x ) 0 ,
S ( x ) 0 .
(38)
It has been changed to
ui  (GB)1GAx ,

Si ( x )
1
1
ui  (GB) GAx  (GB ) C r Si ( x )  ,
in the revised paper.
4. On page 7 of the original paper, the second line is
0
0
 x1,1
“If x1,1
”
It has been changed to
0
0
 x2,1
“If x1,1
”.
S ( x ) 0 ,
S ( x ) 0 .
(38)