1. IntroductionModern missiles have the characteristics of high speed, high control accuracy, stronger mobility and larger flight envelope [1]. However, the low cost makes the device performance not high, and there is a problem of insufficient error accuracy in the manufacturing process. Then, disturbance deviations such as large difference of aerodynamic parameters, deterioration of structural mass characteristics, and deterioration of inertial navigation drift performance occur. In addition, during the guidance process, the missile will encounter random wind, turbulence and other external disturbances. These factors bring great difficulties to the design of attitude control system. It is necessary to design an attitude control system with excellent performance to achieve high-precision and high stability attitude control.At present, missile control mostly adopts PID feedback control [2,3]. In PID control, it takes time to transmit control commands and the actuator responds to control commands. When the actuator responds to the control command, the characteristics of the controlled object change greatly. This leads to poor control quality in trajectory simulation or flight experiments. Therefore, for small missiles with small and complex moments of inertia, it is difficult to meet the requirements of high-performance attitude control using classical control theory PID.Compared with PID control method, active disturbance rejection control technology [4] is represented by active disturbance rejection controller (ADRC), including tracking differentiator, extended state observer (ESO), nonlinear feedback and other technologies. The controller designed by this technique has the characteristics of small overshoot, fast convergence, high precision, strong anti-disturbance ability and simple algorithm [5]. Extended state observer is the core part of ADRC, which expands the disturbance affecting the controlled output into a new state quantity. Using special feedback mechanism to establish the extended state observer, It does not depend on the mathematical model of the controlled object. It has a good inhibition effect on the changes of the parameters and structure of the controlled object, as well as internal and external disturbances. The algorithm is especially suitable for solving various problems existing in the actual flight of the missile [6]. In view of the above advantages, the extended state observer has been widely concerned since it was proposed. It has been successfully applied in many fields such as aviation, aerospace, weapons and industrial control [7,8,9].Research has been conducted in-depth on the field of the extended state observer. References [10,11] analyzed the convergence of uncertain nonlinear systems and multiple-input multiple-output systems, respectively. Literatures [12,13] discussed the relationship between the bandwidth of linear extended state observer and observer error for the controlled object with uncertain model. Reference [14] studied aircraft flight control based on second-order attitude observer and a nonlinear feedback. The active disturbance rejection controller based on extended state observer was designed in the literature [15]. The attitude stability control of a single channel spacecraft with flexible solar panels was analyzed. References [16,17] studied the control law and attitude stability of satellites based on ADRC technology.In the aspect of aircraft attitude control, extended state observer also played an important role and has achieved a lot of research results. Literature [18] took “Tihang-2” small solid rocket as the object, designed the three-channel attitude controller based on the extended state observer, and verified its performance. Literature [19] applied the third-order extended state observer to the design of ADRC, studied the attitude maneuver control problem with three-channel coupling, and completed the parameter setting of ADRC. References [20,21,22] proposed robust flight control method based on the extended state observer. The nonlinear model of aircraft pitch channel was linearized, and the disturbance was compensated to the extended model. Finally, the desired angle of attack was tracked and controlled. Furthermore, some scholars focused on the disturbance in attitude control. The variable structure control scheme was proposed in literatures [23,24], which made the system sensitive to parameters and external disturbances after entering the sliding mode region, but also caused high frequency flutter. The attitude control of flexible multi-body satellite non-gyro or gyro fault was studied in reference [25]. Active disturbance rejection attitude controller was used to realize high precision and high stability attitude control. However, other disturbance factors were not considered. Literatures [26,27,28] have proposed an adaptive control algorithm for spacecraft attitude. The algorithm is robust to parameter uncertainty, but did not consider the suppression effect of the distance. In the controller design mentioned above, some disturbance factors were considered, but the influence of internal and external disturbance on the attitude control were not considered comprehensively.

From previous analysis, the studies on the types of disturbances are not comprehensive enough. Besides, there are relatively few researches on micro air vehicles, whose roll control has the characteristics of relatively small moment of inertia and poor anti-disrurbance performance. Moreover, it directly affected the maneuverability and stability of the guidance process.

The research content in this paper is mainly reflected in the following points:

(1)

In the article, the internal and external disturbances, such as aerodynamic parameter perturbation, structural mass characteristic deviation, inertial navigation drift and wind disturbance, are regarded as the uncertainties of the control system. We compare with the existing research, and focuse on the influence of the uncertainties on the roll channel control for small air-to-surface missiles.

(2)

The system uncertainty is estimated through the extended state observer, and the estimated value is fed back to the control law through nonlinear feedback. It is compensated to the attitude control of the roll channel, and the roll angle is controlled according to the control command of the roll channel. Consequently, the stable tracking control of missile roll attitude is realized.

(3)

The internal and external disturbances are comprehensively considered, and different parameters are set for comparative analysis. The algorithm is compared with other adrc algorithms and PID to verify the effectiveness of the algorithm.

This paper is organized as follows: Section 2 mainly describes the dynamic model of small missile roll channel. Section 3 is mainly about the design of controller based on ESO. First, a description of the ADRC model is presented. Further, the schematic structure of the ESO based controller is also designed. In Section 4, the simulation results under different disturbances and the comprehensive discussion are carried out in detail. 2. Dynamic ModelIn the design of missile guidance and control system [29], the roll control of the missile body around its longitudinal axis is very important. However, the dynamic environment of the rolling channel is relatively poor under the influence of such interference factors as centroid deviation, processing and installation errors, and high altitude wind. They lead to coupling between pitch, yaw and roll channels and a decrease in guidance accuracy. Therefore, it is required that the roll control should have strong anti-disturbance ability, observe and compensate the above uncertain disturbance, and restrain its influence on attitude control.For small missiles facing the scale, under the assumption that gravity is ignored, the body disturbance motion can be divided into longitudinal disturbance motion and lateral disturbance motion. The lateral movement refers to the lateral centroid movement of the missile body, the angular movement of the missile body around the longitudinal axis and the angular movement of the missile body around the normal axis. The lateral movement parameters are mainly the control parameters [30]. Based on the small disturbance assumption, the lateral disturbance deviation of the body can be obtained according to the dynamic equation and kinematic equation of the body. As shown in Equation (1) below:

e1,e2|e1>r0,0≤e2≤βx1G2=e1,e2|e2>0,e2≥β2e1+r0,e2≥βe1G3=e1,e2|e1<−r0,0≥e2≥βe1G4=e1,e2|e2<0,e2≤β2e1−r0,e2≤βe1G0=e1,e2e2<r0,β2e1−r0≤e2≤β2e1+r0

(27)

For Gi(i∈1,2,3,4), respectively, construct the following discontinuous piecewise smooth Lyapunov positive definite functions Vi(e1,e2):

Vie1,e2=β2e1−r0,e1,e2∈G1e2−βe1,e1,e2∈G2−β2e1+r0,e1,e2∈G3−e2+βx1,e1,e2∈G4

(28)

If parameters r0,β1,β2,β,σ can be selected for fixed α, so that the derivative of function Vi(e1,e2) along the trajectory of system (26) is less than zero outside the region G0 [34]. It can be proved that the system (26) is stable according to the multiple Lyapunov function theorem. 4. Numerical Simulation

The main purpose of attitude controller design is to track the roll speed instruction quickly and accurately, and control the roll speed to achieve the expected roll angle. This section uses Matlab software platform to simulate the missile roll state control and verify the effectiveness of the proposed control method.

In this section, the mass and inertia loss of the aircraft, the deviation of the aircraft structure from the ideal situation, the absolute plus relative deviation of aerodynamic and aerodynamic moment coefficients, and wind disturbance are considered, respectively. In order to verify the adaptive ability of ADRC, a failure mode, namely rudder damage, is also considered.

4.1. Distence Characteristic Analysis

The influence of disturbance factors on rolling attitude control loop is analyzed.

(1)

Steerage deviation: The steering effect is the control ability of the steering gear to the course in the missile guidance process. When the missile needs to change the direction at the same angle, the shorter the time required, the smaller the angle the steering gear turns, and the better the steering effect.

(2)

Centroid deviation: The center of mass deviation refers to the deviation between the actual center of mass and the theoretical center of mass of the missile, including the longitudinal deviation and the transverse deviation of the center of mass. Since the center of mass does not coincide with the center of reference moment, the coupling of aerodynamic force and attitude system will inevitably affect the attitude system.

(3)

Stuck steering gear: In addition to the uncertainty disturbance, the complex and harsh environment is very easy to cause the aging of the structure and some components; thus, causing the failure in the flight process. The common fault of the control surface is that the steering gear is stuck. When the control surface deflects to a certain angle and cannot continue to deflect, it is stuck. After the jamming fault occurs, the control surface will no longer change according to the signal given by the controller. At this time, the rudder can only give a constant signal.

(4)

Wind disturbance: Describe the original quantity of wind disturbance factors for wind speed, wind speed is a random quantity, the size and direction are highly, season, climate, location, and the influence of such factors as trimming wind is pointed out that it must be the height of the horizontal wind, usually arrived at a certain height, the wind speed with height increase abruptly, then suddenly decreases, and routine as “wind shear”. Because the time for the aircraft to pass through the shear wind area is generally very short, the effect of the shear wind is similar to the impulse moment.

(5)

Drifts error: Because other information of the body comes from navigation and sensor measured data, there will be drifts error and other deviations in the process of long-term storage or multiple experiments, which will also affect the control of rolling accuracy.

4.2. Simulation Conditions

The initial conditions of the simulation experiment in this paper are as follows: the missile is 3000 m above the ground, the initial pitch angle is 0∘, the yaw angle is 0∘, the roll angle is 0∘, the roll angle speed is 3 rad/s, the yaw angle speed is 0 rad/s, and the pitch angle speed is 0 rad/s. The moment of inertia Jx, Jy and Jz of the three axes of the missile are 0.3 kg·m2, 8.58 kg·m2 and 8.58 kg·m2 respectively, and the sampling time is 0.05 s.

The parameter design of the controller in this paper is: b=mxδx·Q·Sref·Lref/Jx, b1=210, b2=38.

Based on the given initial simulation conditions, the proposed rolling attitude control method based on ESO is verified by simulation.

4.3. Effectiveness of the Control Method against Multiple Disturbances

Experiment 1: Deviation based on steerage

The rudder torque deflection was set as −10%, and the experimental verification was carried out in the three-axis direction and in combination to test the robustness and stability of the controller.

As can be observed from Figure 2 and Figure 3, when steerage deviation is applied, the attitude tracking error of the roll channel is almost 0, and the control precision of the controller is relatively high, with rapid response and no overshoot. The disturbance is accurately estimated by the extended attitude observer, which successfully compensates the influence of the rudder effect deviation in the model, and verifies the feasibility of the controller.As shown in Figure 4 and Figure 5, when a single or integrated steerage deviation is applied, the roll angle deflection of the roll channel can quickly become stable, which successfully illustrates the effectiveness of the extended attitude observer for rudder deflection disturbance estimation.

Experiment 2: Based on centroid deviation

Considering the variation range of the missile’s three-axis centroid deviation, the centroid deviation is 1% and 2% respectively for comparative experiments. The experimental results are shown in Figure 6 and Figure 7. The controller in this paper can suppress the disturbance of centroid deviation quickly and realize the stable control of diagonal rate.

Experiment 3: Based on the steering gear stuck

The effectiveness of the ADRC controller proposed in this paper is studied under the condition that the single actuator is stuck and fails. The experimental results are shown in Figure 8 and Figure 9. It can be observed from the figure that, in the case of steering gear stuck, although there are some fluctuations in the middle, the overall stable control of roll angle can be achieved.

Experiment 4: Based on wind disturbance

In this section, comparative experiments are carried out when the gust wind is 5 m/s, 10 m/s and 20 m/s, respectively, and the gust wind action height is 2650–2850 m. The experimental results are shown in Figure 10 and Figure 11. It can be observed from the figure that the larger the tangential wind speed is, the greater the amplitude change of the system at the early stage is. At 10 s, the system is near stable, meeting the performance indicators of fast response and effective response to trimming wind disturbance.

Experiment 5: Based on drifts error

In this section, the control effect of the system with drift error deviation in the range of 0.01∘/s and 0.1∘/s is verified by experiments. As can be observed from Figure 12 and Figure 13 below, the system rapidly changes to a stable state after the controller starts acting. It is further verified that the active disturbance rejection controller in this paper can effectively suppress the zero-drift disturbance of the model. 4.4. Comparison and Analysis

A variety of disturbance factors were applied to the system, including wind disturbance, centroid deviation, rudder effect deviation and dirfts error. Then, the ESO and PID control effects were verified.

The disturbance factors imposed are as follows: the disturbance wind speed is 10 m/s, the centroid deviation is 0.1∘/s, and the rudder effect deviation is 10.

As can be observed from Figure 14 and Figure 15, under the condition of comprehensive disturbance, the controller proposed in this paper is effective. The influence of disturbance, model deviation and noise is suppressed, and the control effect has no static error. The steady-state performance is obviously better than PID control, and the attitude angle error is smaller than PID control, which significantly improves the missile attitude maneuverability and stability. They satisfy the design requirements of the controller, and further verify the effectiveness of the controller.In order to verify the effectiveness of this algorithm and other adrc algorithms, the system is compared under the same experimental conditions. It can be observed from Figure 16 that compared with other ADRC, the ESO controller proposed in this paper has smaller amplitude of initial disturbance, tends to be stable faster, and has good control performance.