TomTat luanan NCS dangvocong english

MINISTRY OF EDUCATION AND TRAINING

MINISTRY OF NATIONAL DEFENSE

ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY

DANG VO CONG

RESEARCHING AND BUILDING
THE CONTROL ALGORITH FOR A GROUND-TO-GROUND
MISSILES CLASS WITH VERTICAL LAUNCH

Specialization: Control Engineering and Automation
Code:

9 52 02 16

SUMMARY OF PhD THESIS IN ENGINEERING

HANOI – 2019


The thesis has been completed at
ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY

Scientific supervisors:
1. Prof. Dr.Sc Nguyen Duc Cuong
2. Dr. Nguyen Duc Thanh

Reviewer 1: Assoc.Prof. Dr Pham Trung Dung
Military Technical Academy

Reviewer 2: Assoc.Prof. Dr Tran Duc Thuan
Academy Of Military Science And Technology
Reviewer 3: Dr. Nguyen Van Nam
Air Defence- Air Force Academy
The thesis will be defended in front of the Doctoral Evaluating Committee at
Academy level held at Academy of Military Science and Technology at 8:30 AM,
date…month…, 2019.

This thesis can be found at:
- The Library of Academy of Military Science and Technology
- Vietnam National library


1
INTRODUCTION
1. Urgency of the thesis
In modern combat at the tactical level, the use of ground-toground missiles with high precision will prevail when these only
need to be equipped with aerodynamic controlling surfaces, without a
complex thrust vectoring system.
On the other hand, the vertical launch when applied to ground-toground missiles in general has shown many advantages, one of which
is the ease to change the launch direction even if the missile has
already been launched. Moreover, it does not require a large space,
which is necessary for launch with an angle of decline. The
combination of vertical launching into the ground-to-ground missiles
class equipped with just aerodynamic controlling surfaces, will create
the ground-to-ground missiles with very high combat effectiveness at
the tactical level.
However, with the ground-to-ground missiles in general and the
ground-to-ground missiles class with vertical launch direction using
aerodynamic controlling surfaces in particular, when the missile just

leaves the launch device, the velocity is still low, so the control
effectiveness is poor, and it is strongly influenced by manufacturing
errors, especially, thrust vector error compared with the symmetry
axis of the missile (later referred to as the thrust vector errors). These
errors are the main cause of the deviation of flight trajectory, which
can cause the the missile to deviate from the desired trajectory or
even become unstable right from the launching . If the missile is
equipped with the thrust vector control system, these errors will be
easily compensated, however, with the ground-to-ground missiles
class only equipped with aerodynamic controlling surfaces, the
problem will be much more difficult. Therefore, it is necessary to
study the control problem of the ground-to-ground missiles class with
vertical launch and only equipped with aerodynamic controlling
surfaces in the first phase after launching


2
On the other hand, the old generation of ground-to-ground
missiles are the ones that have no final phase control and most are
only equipped with inertial navigation system (INS) in order to
ensure stable attitude and flight with a pre-defined program of angle
as a function of time during the first phase. The absence of control of
the missiles at the final phase results in very low impact accuracy of
the old generation of missiles.
In recent decades, in the world, ground-to-ground missiles have
been developing in the direction of adding final phase control in by a
combined navigation method, so the accuracy is very high, that is the
inertial navigation system (INS) combined with Global Navigation
Satellite System (GNSS). Therefore, studying and clarifying the final
phase control problem for ground-to-ground missiles is also an

urgent issue.
Stemming from the above problems, the problem “Researching
and building the control algorithm for a ground-to-ground missiles
class with the vertical launching” was set and solved in the thesis is
of high practicality and scientific significanty. The thesis will focus
on researching and proposing control algorithms in the first phase
(vertical launch, acceleration and direction change) and trajectory
control solutions in the final phase for a ground-to-ground missiles
class taking the diversification of attack trajectories into account,
making it difficult for the enemy's airforce firepower, increasing the
efficiency in destroying targets.
2. The objective of the dissertation
Proposed a trajectory control algorithm for a ground-to-ground
missiles class, this algorithm ensure the necessary accuracy when there
are significant manufacturing errors and high enough uncertainty of
aerodynamic characteristics.
3. The subject and scope of the study
This thesis only considers the ground-to-ground missiles class
shooting predetermined fixed targets. The subject is the ground-toground missiles class only equipped with aerodynamic controlling


3
surfaces with duck-type aerodynamic diagram, vertical launching with
final-phase control.
The scope of this study is to build control algorithms in vertical
plane with assumptions that ground-to-surface missiles were equipped
with a inertial navigation system in combination with Global
Navigation Satellite System and ideal high pressure measuring devices
to determine parameters about coordinates (longitude, latitude),
velocity, altitude, attitude, angular velocities, normal factor, later

referred to as “combined navigation system”.
4. Research Methodology
This thesis applied theoretical methods combined with numerical
methods.
5. Scientific significance and practical meaning of the thesis
The dissertation will propose an algorithm to control trajectories
for the ground-to-ground missiles class which are only equipped with
aerodynamic controlling surfaces, vertical launching with final-phase
control.
It is possible to apply the built-in control algorithm in the thesis to
design control systems for short-range and mid-range ground-toground missiles to improve and create new domestic ground-toground missiles.
6. Structure of dissertation
The entire thesis is 148 pages long, presented in 4 chapters along
with the Introduction, Conclusion, List of published scientific works,
References and Appendixes
Chapter 1. OVERVIEW OF GROUND-TO-GROUND MISSLES
AND PROPOSAL TO THE PROBLEM OF TRAJECTORY
CONTROL FOR A GROUND-TO-GROUND MISSLES CLASS
1.1. Overview of ground-to-ground missiles
Ground-to-ground missiles are the missiles launched from the
ground or the sea to attack targets on land or at sea.


4
The ground-to-ground missiles can be divided into two types, the
first one is called cruise missile. With this type of missiles,
trajectory is maintained by the lifting force of the controlling surfaces.
The second type is the missiles whose trajectory is mostly "ballistic"
trajectory, also known as Ballistic Missile. This type of missiles apply
propulsion to launch in a already-calculated trajectory, then fly

inertially in the very dilute atmosphere, at the last phase, the missiles
plunged into the target either not being controlled (old generation) or
controlled (new generation).
Old generation ground-to-ground missiles often use inertial
navigation system, with angle control and only controlling in positive
phase (when propulsion is working) method, so the accuracy is not
high. The new generation of missiles is supplemented by a navigation
system based on Global Navigation Satellite System technology and
additional last-phase control so the accuracy is very high.
To shoot on land or sea surface targets at short range, ground-toground missiles can only be equipped with aerodynamic controlling
surfaces, with this missiles class, the structure will be simpler but still
give very high accuracy using combined navigation system. Extra
missiles designed by Israel are a typical example.
1.2. Research situation in Vietnam and abroad
1.2.1. Research situation abroad
Methodical overseas studies on the issue of control of ground-toground missiles are not published because of military confidentiality.
Some studies have been published but can only be applied to the old
generation of ground-to-ground missiles or only applied to short-range
ground-to-ground missiles with the general method of adjusting the
trajectory to reduce the dispersion of the falling point.
1.2.2. Research situation in Vietnam
In-country studies are mainly about methods of navigating air-toair missiles and ground-to-air missiles with the general trend of
adopting modern control theory to improve traditional navigating


5
methods. There are a number of studies that built control algorithms
for maritime cruise missiles and unmanned aerial vehicles (UAVs),
but all have common limitations which are the inventors have
solidified the coefficients in differential equations describing their

movement with seeing the velocity and altitude as small variable
quantities. The domestic research works on control of landslide in the
final phase are not yet available.
1.3. Proposing the problem of trajectory control for a ground-toground missiles class
With a pre-determined fixed target on the ground (sea surface), the
trajectory of new generation ballistic ground-to-ground missiles is in
the form of a rainbow and most of them go through 3 phases (figure
1.4): launch and change direction, with control; inertial flying without
control and controlling the final phase of approaching the target.
Launch and direction
change

Y(H)

1

Missile is
controled in the
final phase

Balistic phase,
uncontroled

2

3

A

B

Position of missile at the
first time of final phase

h0
Vertical launch phase

O

X(L)

Figure 1.4. A typical trajectory for new generation of ground -toground missiles

1.3.1. Control problem in the first phase
In the first phase, missiles is controlled so that its trajectory
follows the reference trajectory with the error at the end of the
control within the pre-calculated allowable limit. Reference


6
trajectories are orbits when assuming that missiles maintain a
constant overload during the intended control time ( T0  t  Tdk ) to hit
the target ignoring errors and disturbances from the external
environment.
 = 1
Real
trajectory
The end of the first phase

Reference trajectory

Vertical launch phase

O
L
Figure 1.7. Initial phase trajectory, vertical launch and direction change

The first phase of missiles’ trajectory will be influenced by the
errors in the manufacturing and assembly process such as: the errors
between the thrust vector versus the symmetry axis of the missiles,
the central errors compared with the symmetry axis of missiles, the
assembling missiles’ lifting surfaces errors compared to the
longitudinal axis of missiles, the assembling missiles’ controlling
surfaces errors, assembling errors of missile sections.
In the above 5 types of errors, the following 3 types of errors are
aerodynamic in nature (gas flow interaction with a solid of an
asymmetrical shape), so these errors can be easily compensated by
aerodynamic controlling surfaces. Particularly, the error of thrust
vector against the symmetry axis of the missile has the strongest
influence and is the main cause of the trajectory deviation in the first
phase. If the missiles are equipped with a thrust vector control
system, the thrust vector error problem will be compensated by the
controlling surfaces itself.


7
However, with the ground-to-ground missiles class in the thesis,
only equipped with aerodynamic controlling surfaces, the thrust vector
error is a big challenge because it causes aerodynamic imbalance of
missiles, leading to the possibly large deviations of movement
trajectory of missiles with reference trajectories, which could even

destabilize missiles right at the beginning because when the missile
just leaves the launch device, the velocity is still very small, the control
effectiveness of the aerodynamic controlling surfaces does not exit.
Therefore, in the thesis, the first phase control algorithm needs to work
towards overcoming the effect of the thrust vector error on the vertical
axis of the missiles. This is an unknown factor for each missile, which
means that the built control algorithm must ensure the necessary
accuracy with the uncertainty of the thrust vector error.
1.3.2. Control problem in the final phase
The problem of controlling the trajectory of the missiles in the
final phase, apart from the purpose of achieving high accuracy when
attacking a target, also takes into account the plan to diversify the
trajectories in order to achieve a tactical advantage to cause
difficulties for enemy’s airfoce firepower to increase the
effectiveness of destroying targets.
Y(H)

The missile is
controled in the
final phase

The position of A
missile at the first
time of the first phase

(

h0

2)

(

1)

O
0

X(L)

Figure 1.11. Controlled trajectory in the final phase of the ground-toground missiles


8
Accordingly, in the final phase, missiles can be controlled to attack
the target in many different trajectory, such as (figure 1.11): low attack
trajectory, intermediate trajectory, vertical attack trajectory. To achieve
these trajectories, missiles are controlled to follow predetermined
directional trajectories according to the tactical scenario.
1.4. Conclusion of chapter 1
Chapter 1 presents general issues on ground-to-ground missiles,
and analysis of national and foreign studies related to the problem of
controlling ground-to-ground missiles. On that basis, the thesis has
proposed the problem of trajectory control for a ground-to-ground
missiles class under three phases.
In the first phase, TL the missiles are controlled to fly according
to the reference trajectory built in advance, the middle phase, the
missiles fly according to inertia without being controlled at high
altitude, in the final phase, the thesis proposes the plan to diversify
attack trajectories to make it difficult for enemy's airforce firepower

to effectively destroy the targets.
Chapter 2. THE METHOD OF BUILDING
REFERENCE TRAJECTORY FOR A GROUND-TO-GROUND
MISSILES CLASS
2.1. The motion of ground-to-ground missiles takes into account
the curvature and rotation of the Earth around its axis
2.1.1. Coordinate frames
When considering the movement of the ground-to-ground
missiles, due to missile flying with quite a large distance, it must take
into account the curvature and rotation of the Earth around its axis.
At this time, in addition to the basic coordinate systems, it must also
use Earth fixed coordinate frame and local geographic coordinate
frame. These coordinate frames fully describe the rotation and
spherical shape of the Earth.
2.1.2. The equations of motion of the ground-to-ground missile
when taking into account the curvature and rotation of the Earth
around its axis


9

1) d V = P cos  − X a − g sin 
 dt
m
 d  ( P sin  + Y ) g cos  V

 2)
=
−
+ cos  − 2 cos  sin ;

mV
V
r
 dt
 d V
d
V sin  cos 
dr
= cos  cos  ; 4)
=−
;5)
= V sin  ;
3)
r
dt
r cos 
dt
 dt

 d 
6) J zz  z  = M z + M z cly + M zz ;
 dt 

 d
dm
P
7)
=  z ;8)  =  −  ; 9)
=−
dt

Je
 dt

(2.11)

2.2. Trajectory plane is compensated when taking into account
affect of the Coriolis force
The trajectory plane is the plane that passes through center of the
Earth, the starting point and end point. If missile is launched in the
trajectory plane (  0 ) that is calculated when ignoring the rotation of
the Earth around the axis, then the missile will touch the ground at
point M’ differs from M (target), so it is necessary to build a new
trajectory plane (  0   ) with correction amount  (figure 2.5):

  sin  =

MM '
OM

(2.12)

Figure 2.5. Trajectory plane of groundto-ground missiles is compensated

In order to implement the compensation for trajectory plane, with
the missiles that is vertical launch, it is possible to perform rotation


10
missile in the vertical launch phase by the cren stabilizing system
without having to have ground complex hydraulic-mechanical systems.

2.3. Method of building reference trajectories for a ground-toground missiles class
The reference trajectory of the ground-to-ground missiles class
which the thesis is aiming at is built according to the following
criteria: there is a vertical phase then descends towards the target in
the trajectory plane and ends at the position of the target at the
distance L given.
2.3.1. Mathematical basis of method of building reference
trajectories
Since the second phase has exhausted the thrust, the missile only
flies with the “ballistic” trajectory so the range will be determined by
the velocity (V1) and the trajectory angle ( 1 ) at the end of control
time (figure 2.7). To achieve a given range, we need to adjust 1 . The
method is: assuming that the missile maintains a constant normal
load factor ( nyc ) during the control time ( T0  t  Tdk ), depending on
the normal load factor, the 1 will be different, then the achieved
range is also different (expression 2.13)

0,
0  t  T0

n y = n yc , T0  t  Tdk (2.13)

t  Tdk
0,

Figure 2.7. The range depending on constant
normal load factor

2.3.2. Numerical method to build reference trajectories
The ultimate purpose of the building a reference trajectory is that

with a given range ( L) of the target, it is necessary to determine the
corresponding value of constant normal load factor to be imposed


11

L [km]

during the control of the first phase. To solve this problem, first of
all, we must solve the problem that is: with each of the imposed
normal load factor values (n yc ) , find the corresponding range ( L) .
This problem is solved by numerical simulation method (test shot
on computer) using automation tool of Matlab/Simulink software.
The end result is to build a curve representing the relationship
L = f (n yc ) , (figure 2.8). From the curve L = f (n yc ) , it is possible to
deduce the normal load factor value when giving the range ( L) .
120
110
100
90
80
70
60
50
40
30
20
0

-10 -9.5 -9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3.0 -2.5 -2 -1.5 -1 -0.5

0

nyc

Figure 2.8. Building curve L = f (n yc ) by numerical method

2.4. Conclusion of chapter 2
The main content of chapter 2 presented the method of building
reference trajectory for ground-to-ground missiles class taking into
account the influence of curvature and rotation of the Earth around
the axis. The method of building reference trajectory has shown how
to build the curve L = f (nyc ) by linear interpolation based on the
results of the test shot on the computer. From the curve L = f (nyc ) , it
is possible to determine the constant normal load factor to be imposed
( nyc ) with any range.
Chương 3. BUILDING THE TRAJECTORY
CONTROL ALGORITHM FOR A GROUND-TO-GROUND
MISSILES CLASS
3.1. Disadvantages of analytic approach
In order to approach analytical methods, the research works on
automatic flight vehicle often use the coagulation method to simplify
the system of equations describing the movement of the center and


12
rotational motion around their center of mass. However, the groundto-ground is the control object with complex mathematical models,
with the parameters that change greatly in the first phase. At this
time, hardening the coefficients will do wrong the mathematical
model of the control object.

Therefore, the problem in the thesis will be approached in the
direction of approaching the theoretical model with the real model with
modern simulation tools on Matlab/Simulink software. Algorithms are
built on the basis of flight dynamics theory. Using numerical methods
to assess the stability and sustainability of control algorithms.
3.2. Building a reference tracking trajectory algorith for ground-toground missiles class
In principle, in order to traking to reference trajectory, it is only
necessary to use the algorithm to control the trajectory angle (  )
which today with the strong development of Micro-ElectroMechanical-System and Global-Navigation-Satellite-System (GNSS)
allow to determine quite accurately (signed “PI” algorithm):
(3.5)
u = K p ( −  M ) + Ki  ( −  M )dt + K zz
However, in the problem of the thesis, the control object is the
ground-to-ground missile with special characteristics: in the early
times error of thrust vector can make quite trajectory error. So, after
that, “PI” controller can only make angle errors (  ) small but
doesn’t make deviation (  ) small. Therefore, the real trajectory
doesn’t the approach reference trajectory.
3.2.1. Basis of forming algorith to track referrence trajectory
Definition: Trajectory deviation (also known as linear deviation,
signed  ), is the algebraic distance between the center of the missle
and its projection on the velocity vector located on the reference
trajectory (figure 3.4).
The real trajectory is traked the reference trajectory when t → Tdk
then  → 0 .


13

Figure 3.4. Basis of forming algorith to track reference trajectory

If control basis on only angle of trajectory, it means making vector
V’ gradually parallels the vector VM (  →  M ), however the linear
deviation (  ) uncertainly be disappeared. To  is gradually
eliminated we need to correct the direction of the current velocity
vector V’ by forming a correction angle  , this angle will gradually
decrease when  → 0, so we can consider  proportional to Δ:
 = −Ks 

(3.6)

When  is calculated, we will calculate the desired trajectory
angle (  mm ), after that we will calculate the desired perpendicular
acceleration ( Wy _ mm ), and finally normal load factor ( n y _ mm ).
To linear deviation (  ) is gradually eliminated, algorithm (3.5)
needs add control signal according to the normal load factor value to
real normal load factor approaches to desired normal load factor
( ny → ny _ mm ). The expression of the control algorithm is (signed
“PI+ ss ” algorithm):
t

u = K p ( −  M ) + Ki  ( −  M )dt + K zz + K ny (ny − nymm )
0

(3.16)


14
By transforming, the expression (3.16) can be written by another
way:

t
d ( )
(3.23)
u = K p ( ) + K i  ( )dt + K d
+ K zz + K  
0

dt

3.2.2. Evaluation criteria
Limit of errors at the end of control time ( t = Tdk ):
  0.10 and   5 m

Limit of angle of attack and normal load factor:
ny  10 and   150

(3.24)
(3.25)

3.2.3. Determine the input parameters of the control algorithm
In the expression (3.16) there are 2 parameters (  (t ) and n y (t ) ) that
cannot be directly measured, they must be determined through
intermediate calculations with parameters that provided directly from
Combined Navigation System.
3.2.4. Selection method of coefficients in control algorithm
The coefficients ( K p , Ki , K z , K ny ) in expression (3.16) are
linearized in each piece of time, their values at the end of each
segment are selected using the “Simulink Design Optimization” tool
in SIMULINK software [12], [38].
3.3. Building trajectory control solutions in the final phase for a

ground-to-ground missiles class
3.3.1. Determine the height at the first time of the final phase ( h0 )
The h0 is determined based on the dynamic pressure ( V 2 / 2 )
when it is enough for the controlling surfaces work effectively. We
can determine h0 by numerical simulation.
3.3.2. Problems about attitude of missile at the first time of final phase
It may be concerned that in second phase, missile flies and not
controled for a long time with high-altitude (diluted air density) will
not guaranteed about required attitude at the first time of final phase.
The large angle of attack and large angle of cren maybe are causes.


15
However, due to the static stability of the missile, so at the end of
time of second phase angle of attack will drop quickly to 0 (  → 0 ).
Moreover, cren stability system will ensure angle of cren be
decreased fastly because the missiles with aerodynamic diagrams of
the “+” sign or the “×” sign has moment of inertia around axis of
symmetry is very smaller moment of inertia around the other axis
( J xx  J yy , J zz ). Such as, missile is guaranteed about required attitude
at the first time of final phase.
3.3.3. Desired trajectory tracking control algorith
In the final phase, missile will be controlled to track to
predetermined trajectories. Predetermined trajectories that consists of
lines and they depend on tactical scenarios. The lines in the
predetermined trajectory are called “desired lines”. Desired trajectory
tracking algorithm is the same as the reference trajectory tracking
algorith in the first phase, however the reference trajectory is now
replaced by the line. Expression of algorithm is the same as (3.16).
3.3.4. Solutions to diversify attack plan of ground-to-ground missiles

in the final phase
We can diversify the attack plan of ground-to-ground missiles in
the final phase in 3 different trajectories (Figure 3.14): intermediate
attack trajectory (1), low attack trajectory (3), high and vertical attack
trajectory (2). Extend the range by high and vertical trajectory (4).

Figure 3.14. Trajectories in final phase for ground-to-ground missiles


16
3.3.4.1. Intermediate attack trajectory
Intermediate attack trajectory is the traditional trajectory, the
missile is controlled to trak the line of sight (BM) to the target.
H

B

 BM


h0
O

L0

Position of the missile at the first time
of final phase


V

Position of target

M

L

Figure 3.17. Intermediate attack trajectory in the final phase
3.3.4.2. High and vertical attack trajectory
The missile will be controled to track to predetermined trajectory
BB1B2M that consists of desired lines BB1, B1B2, B2M.

Figure 3.19. High and vertical attack trajectory in the final phase

B- position of missile at the first time of the final phase (when
altitude drops to h0 ). B1 is intersection of the lines (d1) and (d2).
Where (d1) is a line passing through point B and its slope coefficient
is tg 0 ,  0 is angle of trajectory at the time that altitude drops to h0 .
(d2) is a horizontal line at a predetermined altitude ( h1 ), ( h1  h0 ). B2


17
is intersection of the line (d2) and the line (d3), where (d3) is a vertical
line and passes through point M (target).
3.3.4.3. Low attack trajectory
The missile will be controled to track to predetermined trajectory
BCM (figure 3.20).

Figure 3.20. Low attack trajectory in the final phase

B is defined as the position of missile at the first time of final
phase. C is intersection of the lines (d1) and (d2), where: (d1) is a line
passing through B and matching the line of sight BM at an angle 1
and (d2) is a line that passes through M and matching the horizontal
an angle  2 . The angles ( 1 and  2 ) are predetermined before
according to criteria that corresponding the low attack trajectory. In
hypothetical missile model: 1 = 300 , 2 = 200 .
3.4. Conclusion of chapter 3
Chapter 3 presents the steps to build the trajectory tracking
algorithm in the first phase. Control algorithm that author proposed, in
addition to angle of trajectory component, normal factor component is
also added to eliminate linear deviation (  ), therefore, the missile is
controled to track to the reference trajectory with high accuracy even in
the condition of a quite error between the thrust vector and the
symmetry axis of missile. In chapter 3, the thesis also developed a
control solution in the final phase taking into account the diversifying


18
the trajectories. To realize diversification of trajectories, in the final
phase, missile will be not controlled to track to reference trajectory but
track to predetermined trajectories to attack target.
Chapter 4. SIMULATION TO VERIFY
CONTROL ALGORITHM OF A GROUND-TO-GROUND
MISSILES CLASS
4.1. Simulation to verify control algorithm of ground-to-ground
missile affected by thrust vector error
Numerical simulation is done with the increasing level of moment
caused by thrust vector error. The criteria to verify is the limit of angle
error and linear deviation at the end of control (   0.10 ,   5m ).

- When M z  300 Nm ( Δ y  0.10 ):
4

50

M = -300 Nm

M = -300 Nm

z

z

M = -100 Nm
z

M = -100 Nm

40

M = 0 Nm
z

z

M = 0 Nm

M = 100 Nm
z

30

z

0

 [m]

 []

z

M = 300 Nm

1

-1
-0.04
-0.06

-5
0

-0.08
14

5

15

10

16

16

Time [s]

M = 100 Nm
z

M = 300 Nm
z

20
10
5
0
5
-10
0

5

10

16

Time [s]

Hình 4.6. Angle error  when

Hình 4.7. Linear deviation 

M z  ≤300 Nm

when M z  ≤300 Nm

Comment: The angle error and linear deviation when the missile
has just been launched has increased sharply but it decreased rapidly
and error in steady state is small (figure 4.6 and figure 4.7). It means
that in the early time, the real trajectory is different from the
reference trajectory but then it will be closed the reference trajectory
with limit of the angle of attack and normal factor are guaranteed.
- When M z = 500 Nm ( Δ y  0.50 ):


19
5

Mz = - 500 Nm

Mz = + 500 Nm

100

1
0
-1

Mz = + 500 Nm

 [m]

 []

Mz = - 300  + 300 Nm

-0.02

-5

50
Mz = - 300  + 300 Nm

-0.05

5
-5
-0.1
14

-10
0

5

15

16

10

Mz = - 500 Nm

16

0

Time [s]

5

10

16

Time [s]

Hình 4.13. Angle error  when

Hình 4.14. Linear deviation 

M z  =500 Nm

when M z  =500 Nm

Comment: The angle error at the early times is quite large (up to
10 ), however at the end of time it is still within the required range
(< Δθ max =0.10 , figure 4.13). Linear deviation at the end of control

0

is also within the required range (< Δ max =5 m , figure 4.14).
When the error of thrust vector is small, the component corrected
linear deviation in the algorithm (3.16) is not clear effective (Figure
4.11, 4.12), but when the error of thrust vector is large then its
effective is clear (Figure 4.15, 4.16).
4

60

PI+ss
PI

PI+ss
PI

50

2

-2

 [m]

 []

40

0

0.1

30
20

0.05
0

-4

-0.05
-0.1
15

-6
0

5

15.5

16

10

16

Time [s]

10
5
0
-5
-10
0

5

10

16

Time [s]

Figure 4.11. Compare  (“PI”
algorithm and PI + ssΔ ”

Figure 4.12. Compare  (“PI”
algorithm and “ PI + ssΔ ”

algorithm) when M z  =300 Nm

algorithm) when M z  =300 Nm


20
120

5

PI+ss
PI

PI+ss
PI

100
80
60

 []

 [m]

0

40

0.4

-5

-10
0

5
-5

0.2
0.1

0
-0.1
-0.2

5

-40
0

16

10

5

10

16

16

Time [s]

Time [s]

Figure 4.15. Compare  (“PI”
algorithm and PI + ssΔ ”

Figure 4.16. Compare  (“PI”
algorithm and “ PI + ssΔ ”

algorithm) when M z  =500 Nm

algorithm) when M z  =500 Nm

When increasing the moment caused by error of thrust vector (>
500 Nm), the control algorithm (3.16) is not guaranteed. However,
this problem is not practical because the moment caused by error of
thrust vector due to the 500 Nm is too large with the current
manufacturing technology.
4.2. Simulation to verify control algorithm of ground-to-ground
missile in the first phase with uncertainty of the parameters
4.2.1. Uncertainty of mz
4

50

k1=0.7
k2=1.0
k3=1.3

2

k1=0.7
k2=1.0
k3=1.3

40

 [m]

 []

30

0
0.1

-2

0.05

-4

-0.05

0

-0.1

-6
0

5

14

10

15

5
0
-5

16

16

Time [s]

Figure 4.21. Angle error  with


uncertainty of mz

20

0

5

10

Time [s]

Figure 4.22. Linear deviation 
with uncertainty of mz

16

21
4.2.2. Uncertainty of mz
4

80

k1=0.7
k2=1.0
k3=1.3

2

60

1

 [m]

 []

0
-2

k1=0.7
k2=1.0
k3=1.3

0.05

40
20

0
-4

5
-5

-0.1
14

-6
-7
0

5

15

16

10

16

-20
0

5

10

16

Time [s]

Time [s]

Figure 4.23. Angle error  with

Figure 4.24. Linear deviation 
with uncertainty of mz



uncertainty of mz

Comment: According to the simulation results, it can be affirmed
that the control algorithm (3.16) ensures the required criteria with the
uncertainty of the parameters about 30%
4.3. Simulation to verify control solutions of ground-to-ground
missile in the final phase
4.3.1. Intermediate attack trajectory
30

300

BM

Real trajectory

5

250

25

0

20
15

10

10

-5

150

 [m]

H [km]

200

15

B

100

150

155

160

165

50
0

5

-50

5
0
0

0
85
20

90
40

95

60

100
80

-100

M
100

-150

Figure 4.25. Trajectory of missile in
the final phase

135

140

145

150

155

160

165

Time [s]

L [km]

Figure 4.26. Linear deviation in
the final phase

4.3.2. High and vertical attack trajectory


22
100

25

0

20

15

B

 [m]

H [km]

-100

15

B2
10

10

B1

-200
20

-300

5
0
-5

-400

5

5
-500

0
75 80

0
0

20

90

100

40

M

60

80

170

-600
165

100

170

175

180

180

185

190

190

200

195

200

Time [s]

L [km]

Figure 4.31. Vertical attack
trajectory in the final phase

Figure 4.32. Linear deviation in the
final phase

4.3.3. Low attack trajectory
70
15

40
30

10

0

C

-5

5
0

5

200

70

80

 [m]

H [km]

50

10

250

Reference trajectory
Real trajectory

B

60

M
100

90

-10

100

140 160 180 200 220

50
0

15
10
0
0

-50
-100

20

40

60

80

100
120

140

L [km]

160

180

200

220

Time [s]

Figure 4.37. Low attack trajectory in Figure 4.38. Linear deviation in
the final phase
the final phase

4.3.4. Hight trajectory, reach out to extend the range
30

500
Reference trajectory
Real trajectory

0

20

15

15

10

B
B2

B1

 [m]

H [km]

25

-500

5

-1000

10
0

-1500

5
0
0

0
85

100

-5

120

M
20

40

60

80

100

120

L [km]

Figure 4.43. Hight trajectory, reach
out to extend the range

-2000

220

230

240

240

250
250

260
260

Time [s]

Figure 4.44. Linear deviation in
the final phase


23
Comment:
The test results showed that, with all four plans, the missiles hit
the target with satisfaction of the approach angle (tactical
requirement), limit of normal factor and angle of attack. Therefore,

the solution to diversify the trajectory of ground-to-ground missiles
in the final phase is very feasible.
4.4. Conclusion of chapter 4
The simulation results with the hypothetical missile model assume
that the control algorithm in the first phase is capable of
compensating for the thrust vector error compared with the symmetry
axis of the missile in a fairly wide range. This result is a scientific
basis for making recommendations on the required limit of the thrust
vector error compared with the symmetry axis of the missile in
fabrication and assembly with missiles using solid fuel engine.
The simulation results also showed that the control algorithm in
the first phase still ensures the necessary accuracy with a rather large
uncertainty of the parameters belong to control object. This problem
helps to reduce the rigor of calculation in design and fabrication assembly to satisfy the current domestic technology conditions.
In chapter 4, trajectory control solutions in the final phase
proposed by the author was also verified to show its feasibility. This
partly explains why the new generation of ground-to-groud missiles
that added Global Navigation Satellite System shot very accurately.
CONCLUSION
1. The main results of thesis
The contents of the thesis have solved the problem of controling
trajectory for a ground-to-ground missiles class that have only
aerodynamic controlling surfaces with vertical launch and controled
final phase. Accordingly, there are 3 basic issues were solved in the
thesis:
Firstly, a method of building reference trajectory for ground to
ground missiles was proposed with the considerations of curvature
and rotation of the Earth around its axis;