Synthesizes Optimal Conduction Law for Radio Self-Guided Missile Class

The paper considers a method to build optimal conduction law solve the problem of local optimization of the omniscient form based on Letov-Kalman approach. The obtained optimum lead law is similar to the traditional proportional lead law, but the ratio coefficient varies with missile-target distance. The law of conductivity can be realized for the active radio self-guided missile class. DOI: 10.35840/2631-5009/7547 Introduction In the field of rocket guidance, number of studies suggests different laws of conduction. The law of proportional approach with the fixed coefficients (traditional proportional conduction law) has been widely applied since its high realization ability. However, the paper [1,2] has shown that this lead law has many limitations in the case of self-guided missiles with maneuverable targets. Therefore, the law of conduction has been continuously studied to find a new law of conduction that can improve the quality of missile guidance in the case of mobile targets [3]. Optimized Control Algorithm According to Local Standards Consider a linear control system with a defined structure as follows: ( ) ( ) ( ) ( ) ( ) ( ) = + + y y y y y t t t t t t x F B  x u ξ (1) Where: y x the output state vector of the system y ξ the systemic noise vector of the Gaussian white noise with the mathematical expectation of 0; u(t)control signal vector Fy, By -Control efficiency matrix. The index function of the local quality: ( ) ( ) ( ) ( ) ( ) ( ) 0 = t T T T y T y I x t x t Q x t x t U t Ku t dt         ∫ (2) Which: XT the request state vector • Page 2 of 6 • Trung et al. Int J Astronaut Aeronautical Eng 2021, 6:047 ISSN: 2631-5009 | Citation: Trung VA (2021) Synthesizes Optimal Conduction Law for Radio Self-Guided Missile Class. Int J Astronaut Aeronautical Eng 6:047 Qmatrix of penalty coefficients according to accuracy at time t Kmatrix of the penalty coefficients according to the magnitude of the control signal Solving the problem of finding optimal control signals for system (1) and slab (2) by Bellman dynamic planning method. 1 y T y u K B Q x x −   =   (3) Equations of State Describe Self-Conduction Kinetics in Space The spatial self-conducting geometry is described by the vector equation 2 1 1 = + + mt R R a a R R R R λ λ λ     − − −                  (4) Which: mt a  , a is the total acceleration vector of the target and missile λ  is the unit vector of missile-target distance vector with = R R λ ⋅   To describe the coordinates of the vectors , , , , mt a a λ λ λ        in the fixed ground coordinate system (inertial coordinate system) we use vectors. The corresponding state silk is as follows: = ; = ; = = ; = ; T T T q xq yq zq q xq yq zq q xq yq zq T T mt mt mt mt q xq yq zq q xq yq zq a a a a a a a a λ λ λ λ λ λ λ λ λ λ λ λ                                       (5) The state vector describes the instantaneous state of the line of sight: = T T T y q q X λ λ′     (6) Put: 3 3 3 3 3 3 3 3 O O O 1 1 = ; = ; u = ; = ; = 2 I I mt y y mt mt q I F aq a R R R R I I R R         −       − −         B B   ξ (7) In which: O3is a square matrix of order 3, including only zeros; I3 is a 3 × 3 unit matrix. From equations (5) to (7) we have self-conducting geometry described in terms of equations of state = + + y y y y mt mt x x u ξ F B B  (8) Select the Optimal Slab and Determine the Optimal Lead Law Select the optimal slab as follows: ( ) ( ) 1 2 2 0 h u I q t K a t dt λ ω = ⋅ + ⋅ ∫ (9) In which: qh > 0 is the penalty coefficient according to the instantaneous slip; ku > 0 is the penalty factor for the total acceleration (or overload) required up to time t. Vector depicts the desired state of sight T T T q x O T   =   (10) Put: 3 3 u u 3 3 3 = ; = k .I h h O O Q q O I       K (11) Then the (9) equivalent of the slab ( ) ( ) ( ) ( ) ( ) ( ) 1 0 = T T T y h T y u I x t x t Q x t x t u t K u t dt         ∫ (12) When synthesizing the conduction law in the absence of information about the target acceleration, • Page 3 of 6 • Trung et al. Int J Astronaut Aeronautical Eng 2021, 6:047 ISSN: 2631-5009 | Citation: Trung VA (2021) Synthesizes Optimal Conduction Law for Radio Self-Guided Missile Class. Int J Astronaut Aeronautical Eng 6:047 the target acceleration can be considered as an unknown random effect. The state equation without the impact of the target acceleration has the form y y y y x F x B u = +  (13) Applying optimal control algorithm for system (13) and local criteria blade (12) we have 1 h q q u q u a R k λ   = = ⋅      (14) From (5) and (14) we have optimal lead law corresponding to the criterion (9) or (12) 1 h u q a R k λ   = ⋅     


Introduction
In the field of rocket guidance, number of studies suggests different laws of conduction. The law of proportional approach with the fixed coefficients (traditional proportional conduction law) has been widely applied since its high realization ability. However, the paper [1,2] has shown that this lead law has many limitations in the case of self-guided missiles with maneuverable targets. Therefore, the law of conduction has been continuously studied to find a new law of conduction that can improve the quality of missile guidance in the case of mobile targets [3].

Optimized Control Algorithm According to Local Standards
Consider a linear control system with a defined structure as follows: Where: y x -the output state vector of the system y ξ -the systemic noise vector of the Gaussian white noise with the mathematical expectation of 0;

u(t)-control signal vector
Fy, By -Control efficiency matrix.
The index function of the local quality: Which: X T -the request state vector Q-matrix of penalty coefficients according to accuracy at time t K-matrix of the penalty coefficients according to the magnitude of the control signal Solving the problem of finding optimal control signals for system (1) and slab (2) by Bellman dynamic planning method.

Equations of State Describe Self-Conduction Kinetics in Space
The spatial self-conducting geometry is described by the vector equation The state vector describes the instantaneous state of the line of sight: Put: In which: O 3 -is a square matrix of order 3, including only zeros; I 3 is a 3 × 3 unit matrix.
From equations (5) to (7) we have self-conducting geometry described in terms of equations of state

Select the Optimal Slab and Determine the Optimal Lead Law
Select the optimal slab as follows: In which: q h > 0 is the penalty coefficient according to the instantaneous slip; k u > 0 is the penalty factor for the total acceleration (or overload) required up to time t.
Vector depicts the desired state of sight Put: Then the (9) equivalent of the slab When synthesizing the conduction law in the absence of information about the target acceleration, the target acceleration can be considered as an unknown random effect. The state equation without the impact of the target acceleration has the form y y y y Applying optimal control algorithm for system (13) and local criteria blade (12) we have From (5) and (14) we have optimal lead law corresponding to the criterion (9) or (12) The According to [4] the line of sight kinetics is described by the equation In which:  In which: V R = −  is the speed at which the missile approaches the target According to [5], the instantaneous slip h of the self-conductive process is determined by

Replace (18) into (17) and transform we have
Where ΔV is the magnitude of the vector ( ) The instantaneous slip (h) of the self-conduction process is proportional to the rotation speed of the line of sight so allowing the self-guided control ring to ensure that the instantaneous slip h → 0 will corre-spond to the control of the rocket maneuver so that ω λ → 0. Equation (19) shows that when the missile is self-conducting according to the law (15), the self-conducting control ring corresponds to the first order of inertia stage with the time constant: When a missile attacks a mobile target, the total active self-driving time is usually very short. There-fore, to ensure the required slip at the meeting point, we set a requirement to limit the transient time of the self-conducting control ring right from the start of self-conduction: Tcp-allowable limit of self-conductive control loop time constant.
From (21) and (22) we have the boundary condition of (22) being equivalent For (qh, ku) choose according to (23) then the law of leading (15) becomes The law of conductivity (24) ensures that the transient time constant of the self-conductive control loop is equal to the limit of the permissible value. Unlike the traditional proportional conduction law, this law of conduction has a proportional coefficient that decreases with distance and depends on the time constant limit [6].

The Simulation Results Evaluate the Optimal Lead Law
Simulations are performed with traditional proportional conduction law (coefficient K = 3) and optimal conduction law (24) under the same conditions:

Comment
• When the target is strongly mobile, the slip at the meeting point of the orbit corresponding to the law of conduction (24) is significantly smaller than the slip at the meeting point of the conduction trajec-tory corresponding to the traditional proportional conduction law.
• With traditional proportional conduction law, at the beginning of self-conductivity, the required missile overload is of small value so that the initial slip is not quickly eliminated from the start thanks to the maximum effective overload of the name fire.
• With the law of conductivity (24), because slip is always required to reduce quickly with limited transition time, the day from the start of self-conductivity, according to the requirements of the law of conductivity, the missile using the overload is maximized to quickly reduce the initial slip. Therefore the initial slip is reduced faster and the slip at meeting point is guaranteed to be small enough as required. The trajectory of target

Conclusion
The survey simulation results show that the obtained optimal conduction law meets the quality requirements of the self-conductive control loop and is more effective than the traditional ratio approach law when the target is maneuverable. This law of conductivity can be applied to the active radio self-guided missile class, which helps to improve the conductivity of these missiles in aerial combat with highly maneuverable flying targets.