Missile channel simulation based on PID control

1 Introduction
Modern high-performance combat aircraft generally adopt thrust vectoring technology, and the threat of various high-altitude, high-speed, and high-maneuverability re-entry warheads has become more prominent, which puts forward new requirements for missile systems controlled by traditional pneumatic rudders. Modern missiles are required to be able to select attack targets, have certain anti-interference capabilities, and achieve all-weather operations, which makes missiles develop towards high precision, high intelligence, lightness and miniaturization; at the same time, the improvement of missile guidance and control accuracy has shifted from guidance to control. The range of missile targets is constantly expanding, from anti-aircraft to anti-cruise missiles, anti-ballistic missiles and other anti-missile tasks. High altitude, high speed, and high maneuverability have become important characteristics of today's missile targets. The high-speed and high-maneuverability characteristics of the target lead to the intensification of the relative motion of the missile, which puts forward high requirements on the terminal overload of the missile; on the other hand, the high-altitude characteristics of the target lead to a significant reduction in the efficiency of the missile system, and the available overload decreases significantly with the increase of altitude. In order to solve these contradictions, the PID control method is used here to control the three channels of the missile's pitch, yaw, and roll.

2 Model establishment
The study of missile guidance problems must be based on a certain mathematical model. Therefore, after selecting an appropriate coordinate system, the ideal control kinematic model of the missile's sub-channel is analyzed and derived, and the servo model is established.
2.1 Ideal control dynamic equation of sub-channel
Due to the existence of the roll angle of the missile, coupling will occur, which will increase the control difficulty and reduce the control accuracy. Therefore, coupling should be minimized and sub-channel control should be adopted. Due to the symmetry of the missile, when the roll angle is zero or small, the coupling between pitch and yaw is ignored, that is, a single-input single-output system. Therefore, the classical control theory can be used to study, analyze and design the missile control system by sub-channel.
The longitudinal motion is the longitudinal dynamic equation of the missile is:

In the formula, is the tangential force, is the normal force, is the pitch moment, m is the missile mass, V is the missile's flight velocity vector, α is the angle of attack, θ is the inclination angle of the trajectory, δz is the pitch rudder deflection angle, ωz is the angular velocity of the missile around the oz1 axis of the missile body coordinate system, X, Y are the drag and lift of the total aerodynamic force on the missile decomposed along the velocity coordinate system, Jz is the moment of inertia of the missile around the oz1 axis of the missile body coordinate system, and Mz is the pitch moment.
The lateral motion is the cross-linking coupling of the heading and the lateral direction, so the lateral dynamic equation of the missile is:

In the formula, -mVcosθ(dψv／dt) is the horizontal component of the missile's center of mass acceleration, "-" indicates that the centripetal force is positive, and the corresponding ψv is negative, and vice versa. It is determined by the definition of the positive and negative signs of the angles, dωx／dt, dωy／dt are the components of the missile's rotational angular acceleration vector on the missile body coordinate system axis, Jx, Jy, Jz are the missile's rotational inertia around the missile body coordinate system ox1, oy1, oz1 axis, Mx, My are the rolling moment and yaw moment, Y, Z are the lift and lateral force of the total aerodynamic force on the missile decomposed along the velocity coordinate system, ωx, ωy, ωz are the angular velocities of the missile around the missile body coordinate system ox1, oy1, oz1 axis.
2.2 Servo model
2.2.1 Establishment of motor model
The motor control schematic diagram is shown in Figure 1.

Assume the reduction ratio i, total moment of inertia J, torque M, input voltage u, current I, inductance L, resistance R, angular velocity and rotation angle of the drum ω and δk respectively, rudder deflection angle δ, torque characteristic of the electric servo is approximately A, mechanical characteristic is approximately -B, Mj is hinge torque, which is the hinge torque generated by unit rudder deflection angle, TM=L/R is the electrical time constant of the motor, then the transfer function of the servo under load is:

2.2.2 Rudder loop
The hinge torque of the rudder surface has a great influence on the servo, and the flight control system adopts a closed loop design to eliminate its influence. The rudder loop generally adopts two feedback compensation methods, position and speed, to eliminate the influence of the hinge torque on it.
The transfer function of position feedback is:

when Therefore, a strong feedback is introduced, and the motor output transfer is proportional to the input voltage, proportional to the feedback amount, and has nothing to do with the size of the hinge torque.
The transfer function of speed feedback is:

According to the above analysis, when a stronger speed feedback is introduced, the motor output angular velocity is proportional to the input voltage, and has nothing to do with the flight state, that is, the size of the hinge torque.
Therefore, the system structure of the servo position control system is shown in Figure 2.

3 Channel PID Control
The flight attitude of the missile is achieved by controlling the deflection of the three control surfaces of the missile (i.e., elevator, rudder, and roll rudder), changing the aerodynamic characteristics of the control surfaces, forming a rotational torque around the center of mass of the missile, and changing the flight attitude. Angular position control is divided into three channels, the pitch channel (controlling the pitch angle), the yaw channel (controlling the yaw angle), and the roll channel (controlling the roll angle).
3.1 PID Control of the Servo
According to the structural block diagram of the servo position control system shown in Figure 2, the current link uses an ammeter feedback, the speed feedback uses a speed measurement generator, and the position feedback uses a photoelectric encoder. The servo adopts a three-closed-loop control design, namely, the current loop, the speed loop, and the position loop. The PID parameters can be preliminarily determined by the "critical proportionality method". This method is suitable for occasions where the object transfer function is known, and the regulator is placed under pure proportional action in a closed control system. Gradually change the proportionality of the regulator from large to small to obtain a transition process of equal amplitude oscillation. The proportionality at this time becomes the critical proportionality δk, and the time interval between two adjacent wave peaks is called the critical oscillation period Tk, from which the various parameters, namely the values ​​of Kp, Ti, and Td, are calculated.
3.2 Longitudinal channel control
The traditional control scheme is to simplify the servo into an amplification link. The system only has angular velocity feedback, and its longitudinal channel transfer function is:

Where KM is the transfer coefficient, TM is the time constant, ξM is the relative damping coefficient, and T1 is the aerodynamic constant.
When designing the longitudinal channel of the servo link accurately, it is necessary to add a PID correction link and analyze the system to meet the design requirements. Figure 3 is the structural block diagram of its control system.

3.3 Lateral channel control
When the input command of the rolling channel is zero, that is, the rolling angle and angular velocity are kept at zero, the coupling effect of the pitch channel and the yaw channel is eliminated, and the three channels can be controlled separately. At this time, the control of the pitch channel and the yaw channel of the symmetrical structure missile is basically the same.
3.4 Roll channel control
The servo link is introduced into the rolling channel, similar to the longitudinal channel and the heading channel, and the PID correction link is introduced. The system is analyzed, and its angular velocity transfer function is:

where KMx is the transfer coefficient and TMx is the tilt time constant.

4 Simulation results
To verify the correctness and control effect of the control scheme, the following missile parameters are given: KM=0.171 7(1/s), TM=0.085 0(s), ξM=0.111 2, T1=6.521 7(s), KMx=170.778 9, TMx=1.006 3(s). The unit step signal is added to the servo system, longitudinal channel system, lateral channel system, and rolling channel system for digital simulation, and the traditional control system is simulated to compare the control results. Figure 4 is the time domain step response curve of the servo system. From the simulation curve in Figure 4, it can be seen that the overshoot is 9.5%, the rise time is 41.9 ms, the adjustment time (2% error band) is 88.8 ms, and the steady-state error is 0.

Figure 5 is the time domain step response curve of the longitudinal channel. It can be seen from the simulation curve in Figure 5 that when the servo link is accurately considered, the time domain step response curve of the longitudinal channel of the PID correction link responds well, with an overshoot of 11.4%, a rise time of 170.6 ms, an adjustment time (2% error band) of 356.3 ms, and a steady-state error of 0.

Figure 6 is the time domain step response curve of the lateral channel. From the simulation curve in Figure 6, it can be seen that when the servo link is accurately considered, the time domain step response curve of the lateral channel of the PID correction link responds well, with an overshoot of 11.4%, a rise time of 168.3 ms, a regulation time (2% error band) of 347.1 ms, and a steady-state error of 0.

Figure 7 is the time domain step response curve of the roll channel. From the simulation curve of Figure 7, it can be seen that when the servo link is accurately considered, the time domain step response curve of the roll channel of the PID correction link responds well, with an overshoot of 9.81%, a rise time of 178.6 ms, a regulation time (2% error band) of 397.1 ms, and a steady-state error of 0.

5 Conclusion
This paper uses the critical proportionality method to obtain PID parameters and uses MATLAB/Simulink for time domain simulation. From the simulation results, the PID sub-channel control method can improve the accuracy, rapidity and stability of the traditional pneumatic rudder missile control system. Of course, this is only a result compared with the traditional control scheme. The actual parameters must be continuously debugged in the physical simulation, and the control system must be corrected and improved to obtain a satisfactory control effect. The simulation results show that each channel system reflects well and can achieve real-time control requirements.