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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: 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: Prof Dr.Sc Nguyen Duc Cuong 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 INTRODUCTION 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 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 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 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 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” Research Methodology This thesis applied theoretical methods combined with numerical methods 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 Structure of dissertation The entire thesis is 148 pages long, presented in chapters along with the Introduction, Conclusion, List of published scientific works, References and Appendixes Chapter 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 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 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 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) Missile is controled in the final phase Balistic phase, uncontroled 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 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 types of errors, the following 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 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 X(L) Figure 1.11 Controlled trajectory in the final phase of the ground-toground missiles 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 Chapter 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 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 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 -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 nyc Figure 2.8 Building curve L = f (n yc ) by numerical method 2.4 Conclusion of chapter The main content of chapter 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 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 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  → 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 ) (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   dt 3.2.2 Evaluation criteria Limit of errors at the end of control time ( t = Tdk ):   0.10 and   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 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 / ) 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 (  → ) 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 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 ,  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  The angles ( 1 and  ) 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 Chapter 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 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 ): 50 M = -300 Nm M = -300 Nm z z M = -100 Nm z M = -100 Nm 40 M = Nm z z M = Nm M = 100 Nm z 30 z  [m]  [] z M = 300 Nm -1 -0.04 -0.06 -5 -0.08 14 15 10 16 16 Time [s] M = 100 Nm z M = 300 Nm z 20 10 5 -10 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 Mz = - 500 Nm Mz = + 500 Nm 100 -1 Mz = + 500 Nm  [m]  [] Mz = - 300  + 300 Nm -0.02 -5 50 Mz = - 300  + 300 Nm -0.05 -5 -0.1 14 -10 15 16 10 Mz = - 500 Nm 16 Time [s] 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 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) 60 PI+ss PI PI+ss PI 50 -2  [m]  [] 40 0.1 30 20 0.05 -4 -0.05 -0.1 15 -6 15.5 16 10 16 Time [s] 10 -5 -10 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 PI+ss PI PI+ss PI 100 80 60  []  [m] 40 0.4 -5 -10 -5 0.2 0.1 -0.1 -0.2 -40 16 10 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 50 k1=0.7 k2=1.0 k3=1.3 k1=0.7 k2=1.0 k3=1.3 40  [m]  [] 30 0.1 -2 0.05 -4 -0.05 -0.1 -6 14 10 15 -5 16 16 Time [s] Figure 4.21 Angle error  with  uncertainty of mz 20 10 Time [s] Figure 4.22 Linear deviation  with uncertainty of mz 16 21 4.2.2 Uncertainty of mz 80 k1=0.7 k2=1.0 k3=1.3 60  [m]  [] -2 k1=0.7 k2=1.0 k3=1.3 0.05 40 20 -4 -5 -0.1 14 -6 -7 15 16 10 16 -20 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 250 25 20 15 10 10 -5 150  [m] H [km] 200 15 B 100 150 155 160 165 50 -50 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 20 15 B  [m] H [km] -100 15 B2 10 10 B1 -200 20 -300 -5 -400 5 -500 75 80 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 C -5 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 15 10 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 20 15 15 10 B B2 B1  [m] H [km] 25 -500 -1000 10 -1500 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 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 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 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; 24 The second problem, a reference trajectory tracking algorithm in the first phase was synthesized when the missile is vertical launch and change direction for a ground-to-ground missiles class In the first phase, missile will be controlled to track to reference trajectory under conditions strongly influenced by fabrication - assembly errors, particularily the thrust vector error compared with the symmetry axis of the missile; The third problem, building solutions to control missile’s trajectory in the final phase for a ground-to-ground missiles class In the final phase, missile will be controlled to track to predetermined trajectories depend on tactical scenario to attack target, thereby make difficult to air defense of the enemy and increase the effectiveness of destroying the target New contribution of the thesis Proposed a method of building flight trajectory in the first stage taking into account the rotation effect of the Earth around the axis and the solution to form the desired trajectories in the final stage in accordance with the combat conditions of the ground-to-ground missiles 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 Further research and development direction In theory: continue to study and consider other factors affecting the trajectory of missiles such as wind, sensor error, etc It is possible to expand the problem of diversifying the trajectories in the horizontal plane to complete the various attack scenarios for groundto-ground missiles In practice: the thesis’s control algorithms can be used to design or improve the control system for some kinds of short-range groundto-ground missiles in our Army THE SCIENTIFIC PUBLICATIONS Dang Vo Cong, Nguyen Duc Cuong, Nguyen Duc Thanh, Dang Cong Vu, Nguyen Sy Hieu, 2016, “Effect of manufacturing errors on the flight trajectory of missile with autonomous control system”, Journal of Science and Technique (No.180, 10/2016), Military Technical Academy, pp 91-103 Dang Vo Cong, Nguyen Duc Cuong, Nguyen Duc Thanh, Dang Cong Vu, Nguyen Sy Hieu, 2017, “Development of a control algorithm to track a preset trajectory on autonomous stage for sounding rocket”, Journal of Science and Technique (No.182, 2/2017), Military Technical Academy, pp 70-80 Dang Vo Cong, Nguyen Duc Cuong, Nguyen Duc Thanh, Pham Tuan Hung, 2018, “Trajectory tracking problem and reference trajectory constructing method for surface-to-surface missiles”, Journal of Military Science and Technology (No.53, 2/2018), Academy of Military Science and Technology, pp 3-11 Dang Vo Cong, Nguyen Duc Cuong, Nguyen Duc Thanh, 2018, “Development of trajectory control algorithms for surface-to-surface missiles”, Journal of Military Science and Technology (No.54, 4/2018), Academy of Military Science and Technology, pp 3-12