Research on Sliding Mode and Backstepping Control of Missile Pitch Channal System

: A new type of proportional plus sliding mode inversion controller is constructed and applied to solve the attack angle tracking problem of missile pitch channel; At the same time, in order to solve the differential explosion problem of backstepping control, a method combining an integral type sliding surface with an accurate differentiator is proposed to solve the derivative of the desired pitch angular rate signal; At last, a kind of integral type Lyapunov function is constructed to prove the stability and boundness of the closed-loop signal of the whole system; Finally, a digital simulation experiment is done to verify the correctness of the method.


Introduction
The control problem of the missile pitch channel is the main problem of the whole missile control system design, because the yaw channel is identical or similar to the pitch channel, while the control of the roll channel is relatively simple. Generally, PID control can meet the control task [1][2][3][4][5]. In addition to the traditional PID control, the current popular variable structure control has the advantage of good rapidity; The disadvantage is that its dynamic performance often appears flutter [6][7][8][9]. At the same time, inversion control is also loved by the majority of researchers, because its precise design idea and backward design idea are more suitable for combining with Lyapunov function method, and can rigorously analyze stability; However, its disadvantage is that the theoretical analysis of the differential solution of the desired signal is relatively complex, which will lead to the problem of differential explosion; In order to solve this problem, many scholars try to use the accurate differentiator method to calculate; We also propose a method based on the combination of sliding mode and accurate differentiator to solve the differential problem. At the same time, on the basis of inversion, we use a method combining two-layer integral sliding mode and error proportional feedback to solve the problem of angle of attack tracking, which is also very innovative. The digital simulation results also show the stability of the whole system.

Problem Description
A kind of second order system can be used to describe the linear model of supersonic missile pitch channel as following: can also be estimated by the control law z  .

Backstepping controller Design with Tracking Differentiator
First step, we define a new error variable as d e    , then the first subsystem of missile system can be written as And it derivative can be solved as So according to the Lyapunov stability theory, all signal of the the whole system is bounded and stable.

Numerical simulation experiment for example
According to the air wind hole experiments, all constant air coefficients of a type of missiles are as follows Then simulation results can be shown by figures 1-5.
3 Figure 1 The attack angle of missile system It can be seen from Figure 1 to Figure 5 that the system oscillates severely at this time. We select the following parameters: It can be seen from Figure 6 to Figure 10 that the dynamic performance of the system has been greatly improved; The main manifestation is that the flutter is greatly weakened; The overshoot is reduced, and the flutter frequency is less. The rise time has decreased, but the overall rise time is still less than 0.15 seconds; Moreover, the adjustment time is greatly reduced from 0.6 seconds to 0.3 seconds. In general, the simulation results show that the proposed method is stable and valuable.

Conclusions
According to the angle of attack tracking task of the missile pitch channel control system, a kind of angle of attack stability controller combining backstepping and sliding mode method is designed. The error of sliding mode surface is designed according to the attack angle subsystem and the pitch angular speed subsystem respectively. In addition, according to the calculation requirements of the inversion differential, the sliding model accurate differentiator is introduced to solve the derivatives of the required variables, avoiding the differential explosion problem, so as to improve the dynamic performance of the entire system. Finally, the digital simulation experiment also shows the stability of the whole system.