# Autopilot Design for a Compound Control Small-Scale Solid Rocket in the Initial Stage of Launch

Authors: Tian Dong; Changjian Zhao; Zhiguo Song
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

In this paper, an autopilot design method for a compound control small-scale solid rocket is proposed. The rocket has multiple actuators, including a flexible nozzle for pitching and yawing channels, aerodynamic fins for rolling channel, and lateral thrusters which work in on-off mode for all three channels. In order to keep the aircraft steady in the initial stage of launch when the dynamic pressure is low, the autopilot is aimed at optimizing the cooperation among the actuators. Firstly, without considering the discontinuous lateral thrust, the control law for flexible nozzle and aerodynamic fins is achieved via the sliding mode control approach. On this basis, an object to be controlled with choiceness is obtained for the lateral thrusters controlled loop. Secondly, the operation logic of lateral thrusters is programmed, regarding rolling moment as priority. Thirdly, after a continuous controller is obtained, a discretization method for the lateral thrusters control law is designed combining the characteristics of sliding mode control and Lyapunov’s stableness theorem. Finally, the fundamental cause why compound control improves the system stability is given theoretically. Simulation results validate the improved response performance and robustness against uncertainties and disturbance of the autopilot.


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## Body

## 1. Introduction

Traditional aircraft controlled only by aerodynamic fins is hard to keep steady in the initial stage of launch, because the aerodynamic fin control efficiency is low for low-velocity aircraft, which leads to a difficulty in aircraft design [1]. Therefore, some aircrafts apply the flexible nozzles into pitching and yawing channels to make the control magnitude independent of velocity. However, the flexible nozzle cannot provide a control moment for the rolling channel. The instability of the rolling channel would lead to bad performance of the autopilot, because of aerodynamic couplings [2]. Therefore, it is necessary to provide sufficient control for the advanced aircrafts to keep steady for three channels.Another effective method to promote the performance of the control system is to employ both traditional aerodynamic fins and lateral thrusters at the same time. Recently, a lot of research on high-altitude interception has been done to improve the autopilot’s performance by this method [3]. The advantage of this compound control technology is to provide sufficient control through lateral thrust, especially when the dynamic pressure gets low, and make full use of the nonworking medium consuming air force. It will be meaningful to apply this compound control technology in the initial stage of launch to keep stability against disturbance. However, the lateral thrusters are usually used in the pitching and yawing channels instead of the rolling channel.In order to make full use of the aerodynamic fins, the flexible nozzles, and the lateral thrusters and avoid their disadvantages, in this paper, the flexible nozzle is used in the pitching and yawing channels, the aerodynamic fins are applied into the rolling channel, and the lateral thrusters are employed in all the three channels.A well-designed compound-control autopilot shall take advantage of all actuators. The main difficulty during design is that the aerodynamic fins and the flexible nozzle provide continuous control force while the lateral thrusters work in on-off mode, which makes the whole control system to be a discrete/continuous hybrid system. The design methods for the hybrid control system have been studied in many researches. In the early stage, Hirokawa et al. [4] decompose the MIMO control system into a feedback and feedforward controller design problem. The feedback controller is designed using the algebraic approach, and the feedforward controller is designed to achieve the minimum fuel consumption independently. The control allocation method is used to optimize the cooperation in many researches. Xu et al. [5] take advantage of the robust adaptive sliding sector to produce a virtual control effort signal and use a control allocator based on the L2 optimal control allocation strategy to distribute the virtual signal to the two pairs of actuators. Mathavaraj et al. [6] allocate the control command in the daisy chain mode, which chooses to use the aerodynamic fin system as much as possible in order to save fuel for the lateral thruster system. Bi et al. [7] provide an ADRC-based controller which obtains the required control moment according to the angle of attack tracking deviation then allocate the respective commands to the aerodynamic deflection and lateral thruster using the dynamic control allocation. Li and Zhou [8] propose a θ-D feedback controller to steer the aircraft to track reference acceleration commands based on the linear-like structure, which ensures the asymptotic stability based on Lyapunov’s theorem. Zhao et al. [9] complete the aerodynamic fin controlled loop based on the concept of finite time stability, then using a backstepping approach to design a lateral thruster control law. Generally, most researches focus on the control method and specific allocation of the two actuators, while few of them study the stability issue caused by discrete characteristics of lateral thrust.To promote the compound control performance for small-scale solid rockets, in this paper, the control method is proposed creatively. At first, a continuous controller is designed by the sliding mode control approach without considering the discrete characteristic of the lateral thrusters. Then, a operation logic of the lateral thrusters is optimized that provides a control moment for the rolling channel as a priority. Finally, according to the operation logic, the lateral thrust control law is discretized under the premise of the stable system based on Lyapunov’s theorem. Simulation results validate the effectiveness of the proposed approach and the improved performance compared to the traditional autopilot without lateral thrusters.


## 2. Dynamic Model of the Compound Controlled Rocket

Consider that the aircraft has a flexible nozzle which controls the yaw and pitch motions and aerodynamic fins which control the roll motion. Also, eight lateral thrusters are arranged in a lap with a 90-degree interval. Four of them provide lateral thrust parallel to thez axis of the body coordinate system, and the other four provide lateral thrust parallel to y axis of the body coordinate, as Figure 1 illustrates. x,y,z represent the axes of the body coordinate system to describe the force and moment provided by lateral thrusters.Figure 1
The diagram of the layout of lateral thrusters.The thrusters can provide control force for the yawing and pitching channels and work in pairs to generate a rolling moment for the rolling channel. Without loss of generality, suppose each thruster is able to generate a thrustP¯, with the arm of x2e relative to the x axis. The specific number, force, and moment of each thruster are shown in Table 1. For example, if the lateral thruster No. 1 is working, it will produce a moment of P¯⋅x2e parallel to the x axis, “0” force relative to the y axis, and force P¯ relative to the z axis.Table 1
Specific information of each thruster.
Thruster
No. 1
No. 2
No. 3
No. 4
No. 5
No. 6
No. 7
No. 8
x (moment)
P¯x2e
−P¯x2e
P¯x2e
−P¯x2e
P¯x2e
−P¯x2e
P¯x2e
−P¯x2e
y (force/N)
0
0
P¯
P¯
0
0
−P¯
−P¯
z (fore/N)
P¯
P¯
0
0
−P¯
−P¯
0
0Based on the sphere Earth model, the six-degree-of-freedom dynamic equations are established as follows, neglecting the rotation of the Earth.(1)ẋẏż=VxVyVz,dVxdtdVydtdVzdt=GBmPePeδφ−Peδψ+∂CA∂δγqSm∂CN∂δγqSm∂CZ∂δγqSmδγ+0+2ṁωz1x1e−2ṁωy1x1e+gxgygz,with
(2)GB=cosφcosψcosφsinψsinγ−sinφcosγcosφsinψcosγ+sinφsinγsinφcosψcosφcosγ+sinφsinψsinγsinφsinψcosγ−cosφsinγ−sinψcosψsinγcosψcosγγ̇ψ̇φ̇=ωx1+φ̇sinψωy1cosγ−ωz1sinγωy1sinγ+ωz1cosγcosψ,(3)Ix000Iy000Izdωx1dtdωy1dtdωz1dt+Iz−Iyωz1ωy1Ix−Izωx1ωz1Iy−Ixωy1ωx1=∂Cmx∂δγqSmlk∂Cmy∂δγqSmlk∂Cmz∂δγqSmlk+−∂Clp/∂δγqSmlk2ωx1V−∂Cnr/∂δγqSmlk2ωy1V−∂Cmq/∂δγqSmlk2ωz1Vδγ+0−Pex1eδψ−Pex1eδφ+pgx2epzx3e−pyx3e−İxωx1İyωy1İzωz1+ṁ0−x1e2ωy1−x1e2ωz1,where x,y,z are the position components in the launch coordinate system, Vx,Vy,Vz are the velocity components in the launch coordinate system, m is the current mess of the aircraft, Pe is the thrust of the main engine, δφ,δψ,δγ are the flexible nozzle deflection angles for pitching and yawing channels and aileron deflection angle, CA,CN,CZ are the coefficients of aerodynamic force of the aircraft, q means the dynamic pressure, Sm is the reference area of the aircraft, ωx1,ωy1,ωz1 are angular velocity components observed in the body coordinate system, x1e is the distance from the engine nozzle to the center of mess, gx,gy,gz are the gravitational coefficients in three axes, φ,ψ,γ are the three attitude angles, Ix,Iy,Iz are the components of the rotational inertia, Cmx,Cmy,Cmz are the coefficients of the aircraft’s aerodynamic moment, lk is the reference aerodynamic length of the aircraft, Clp,Cnr,Cmq are the coefficients of the aerodynamic damping, pg is the equivalent lateral thrust in the rolling channel, with an arm of x2e relative to the rolling axis, and py and pz are the equivalent lateral thrusts in the pitching channel and the yawing channel, respectively, with an arm of x3e relative to the pitching axis or yawing axis.


## 3. Compound Control System Design

The sliding mode method is a nonlinear control method which has great robustness against the model parameter variation and disturbance [10, 11]. It is applicable to the condition of large aerodynamic disturbance caused by the lateral thrusters. Accordingly, the aerodynamic surface controlled loop is designed at first via the sliding mode control approach. Suppose the lateral thrust is zero, the sliding mode surface is designed as
(4)s=c1φc−φ+φ̇c−φ̇c2ψc−ψ+ψ̇c−ψ̇c3γc−γ+γ̇c−γ̇=s1s2s3,where elements in c are the controlling parameters that are greater than zero, and φc,ψc,γc denote the control command of attitude angles. In order to improve the dynamic performance, the exponential reaching law has been chosen as
(5)ṡ=c1φ̇c−φ̇+φ¨c−φ¨c2ψ̇c−ψ̇+ψ¨c−ψ¨c3γ̇c−γ̇+γ¨c−γ¨=−ε1sgns1−k1s1−ε2sgns2−k2s2−ε3sgns3−k3s3,where ε,k are the control parameters with elements greater than zero. Substituting Equation (2) and Equation (3) into Equation (5) successively, the flexible nozzle deflection for the pitching channel, the flexible nozzle deflection for the yawing channel, and the aileron deflection angle δϕ,δψ,δγ are solved.
(6)δφδψδγ=cosγIzcosψPex1e0−Aφ⋅cosγIzcosψ0cosγIyPex1e−Aψ⋅cosγIy00−AγIx−1−ε1sgns1−k1s1−ε2sgns2−k2s2−ε3sgns3−k3s3+−c1φ̇c−ωzcosγcosψ−φ¨c+ωxωycosγcosψ−cosγcosψ1IzIy−Ixωyωx+Izωz+ṁxe2ωz−c2ψ̇c−ωycosγ−ψ¨c+ωxωzcosγ−cosγIyIx−Izωxωz+Iyωy+ṁxe2ωy−c3γ̇c−ωx−γ¨c−1IxIz−Iyωzωy+Ixωx,with
(7)AγAψAφ=∂Cmx1∂δγqSmlk∂Cmy1∂δγqSmlk∂Cmz1∂δγqSmlk+−∂Clp/∂δγqSmlk2ωx1V−∂Cnr/∂δγqSmlk2ωy1V−∂Cmq/∂δγqSmlk2ωz1V.On this basis, the inner loop is considered as an object to be controlled with choiceness in the lateral thruster controlled loop. A continuous controller is designed via the sliding mode control approach firstly, without the consideration of the discontinuous feature of the lateral thrust. Similar to the aerodynamic surface control loop, the sliding mode surface is designed as(8)s′=c1′φc−φ+φ̇c−φ̇c2′ψc−ψ+ψ̇c−ψ̇c3′γc−γ+γ̇c−γ̇=s1′s2′s3′,where c′ are controlling parameters with elements greater than zero. Then, the derivative of s′ is
(9)ṡ′=c1′φ̇c−φ̇+φ¨c−φ¨c2′ψ̇c−ψ̇+ψ¨c−ψ¨c3′γ̇c−γ̇+γ¨c−γ¨.In order to improve the dynamic performance, the exponential reaching law has been chosen as(10)ṡ′=−ε1′sgns1′−k1′s1′−ε2′sgns2′−k2′s2′−ε3′sgns3′−k3′s3′,where ε′,k′ are the controlling parameters with elements greater than zero. Substituting Equation (2), Equation (3), and Equation (10) into Equation (9) successively, the continuous lateral thrust py,pz,pg are
(11)pypzpg=−ε1′sgns1′−k1′s1′−c1′φ̇c−ωzcosγcosψ+φ¨c−ωxωycosγcosψ−cosγIzcosψIx−Iyωxωy+Cδγ−Pex3eδφ−İzωz−ṁx21eωzIzcosψx3ecosγ−ε2′sgns2′−k2′s2′−c2′ψ̇c−ωycosγ+ψ¨c−ωxωzcosγ−cosγIyIz−Ixωxωz+Bδγ−Pex3eδψ−İyωy−ṁx21eωy−Iyx3ecosγ−ε3′sgns3′−k3′s3′−c3′γ̇c−ωx+γ¨c−1Ix−Iz−Iyωzωy+Aδγ−İxωx−Ixx2e,where δφ,δψ,δγ are obtained by Equation (6).Finally, considering the lateral thruster work in on-off mode, the operation logic of lateral thrusters is designed to satisfy the rolling channel demand preferentially, because the flexible nozzle would provide relatively more control in the pitching and yawing channels, and the main purpose for the lateral thrusters is to provide sufficient control for the rolling channel. The additional effect of the operation logic design for lateral thrusters is to decrease the deflection of the flexible nozzle, which can save energy for the main engine. According to the considerations above and the layout of lateral thrusters, one possible control combination for the three channels is shown in Table2.Table 2
The possible control combination of lateral thrusters.
Modes
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
Mode 9
Pg/N
0
0
0
0
(±)2P¯
(±)2P¯
(±)2P¯
(±)2P¯
(±)4P¯
Py/N
(±)2P¯
0
(±)2P¯
(±)P¯
0
(±)2P¯
0
(±)P¯
0
Pz/N
0
(±)2P¯
(±)2P¯
(±)P¯
0
0
(±)2P¯
(±)P¯
0According to the operation logic of the lateral thrusters, the continuouspy,pz,pg are discretized according to the following rules which can guarantee the stability proved by Lyapunov’s stableness theorem.In order to describe the discretization method, denote the desired lateral thrust obtained by Equation (11) as
(12)ud=pypzpgT.Considering the limitation of the discrete operation logic, the lateral thrusters are hard to provide the desired control input precisely. The actual lateral thrustsu are described as
(13)u=ud−dc,where dc=dcy,dcz,dcxT denotes the thrust adjustment during the discretization. It is obvious that the most desired lateral thrust is between two practical values, neglecting the input saturation. The relationship among ud and those two practical thrust values ul and uu is
(14)uli<udi<uui,i=1,2,3.For example, ifP¯=50N and the desired lateral thrust for the rolling channel is 150 N, the practical values ul3=100N and uu3=200N, according to the operation logic.Therefore, the thrust adjustment has two possible values, which are defined as(15)dcli=udi−uli>0,i=1,2,3,dcui=udi−uui<0.Take advantage of Lyapunov’s stableness theorem, to determine the actual thrust adjustmentsdc. Consider a Lyapunov function candidate as
(16)V=12s′Ts′.So the derivative ofV is
(17)V̇=s′Ts′̇.It is obvious that ifs′Ts′̇<0, the system is asymptotically stable. Substituting Equation (2), Equation (3), Equation (6), and Equation (11) into Equation (9) successively, one can obtain that
(18)ṡ′=−ε1′sgns1′−k1′s1′−bydcy−ε2′sgns2′−k2′s2′+bzdcz−ε3′sgns3′−k3′s3′+bxdcx,with
(19)bybzbx=cosγIzcosψx3ecosγIyx3e1Ixx3e.by,bz,bx are greater than zero, according to the physical truth. When ṡi′<0 for i=1,2,3, to make sure that s′Ts′̇<0, let si′>0 for i=1,2,3. Therefore, the thrust adjustment dc is solved as
(20)dc=dcy=dcl,ifsgns1′=1dcu,ifsgns1′=−1dcz=dcu,ifsgns2′=1dcl,ifsgns2′=−1dcx=dcu,ifsgns3′=1dcl,ifsgns3′=−1.Substituting Equation (20) into Equation (13), the actual control inputs of lateral thrusters are obtained.Similar to the sliding-mode control, the discretization method makes the lateral thrusters turn on and off frequently whens′ is close to 0, which wastes a lot of fuel and may cause the undesired vibration. The decreasing chattering approach for sliding-mode control can also be used when the thrust adjustment is determined.


## 4. Stability Analysis

The aircraft is a MIMO nonlinear system with input saturation. Define the control input saturation as(21)satui=umax,ui≥umax,i=1,2,3,ui,umin<ui<umax,umin,ui≤umin,where satu is the actual control input, and umax>0 and umin<0 are saturation amplitudes. The part beyond saturation Δ can be described as
(22)Δ=u−satu.ConsiderΔ as perturbations of the system. Substituting δφδψδγ and pypzpg obtained from Equation (6) and Equation (11) into Equation (21), the practical control input satuδ and satup are got. Then substituting satuδ and satup into Equation (9), one can obtain
(23)s′̇=−ε1′sgns1′−k1′s1′−ε2′sgns2′−k2′s2′−ε3′sgns3′−k3′s3′+BδΔδ+BpΔp,where
(24)Bδ=cosγIzcosψPex1e0−Aφ⋅cosγIzcosψ0cosγIyPex1e−Aψ⋅cosγIy00−AγIx,Bp=cosγIzcosψx3e000−cosγIyx3e000−x2eIx.Consider a Lyapunov function candidate as(25)V=12s′Ts′.The derivative ofV is
(26)V̇=s′Ts′̇=∑i=13−εi′si′−ki′si′2+BδΔδ+BpΔpTs′≤∑i=13−εi′si′−ki′si′2+∑i=13BδΔδ+BpΔpi⋅si′.During the realistic mission,BδΔδ+BpΔpi⋅si′ is unknown but actually bounded. Thus, there exists a positive constant D that satisfies ∑i=13BδΔδ+BpΔpi⋅si′<D. And Equation (26) can be rewritten as
(27)V̇≤∑i=13−εi′si′−ki′si′2+D.Further,(28)V̇≤−∑i=13ki′si′2+D≤−∑i=13kmin′⋅si′2+D=−2kmin′V+D,with
(29)kmin′=mink1′k2′k3′.According to Equation (28),
(30)V̇≤−2kmin′V−D2kmin′.Sincekmin′ and D are both constants,
(31)dV−D/2kmin′dt≤−2kmin′V−D2kmin′.WhenV>D/2kmin′, let λ≜lnV−D/2kmin′, then
(32)λ̇≤−2kmin′,so
(33)λ≤λ0−2kmin′t.Substituting λ=lnV−D/2kmin′ into Equation (33),
(34)lnV−D2kmin′≤lnV0−D2kmin′−2kmin′t.Then,
(35)V−D2kmin′≤V0−D2kmin′e−2kmin′t,so
(36)V≤e−2kmin′tV0+D2kmin′1−e−2kmin′t.As is known to all, 0<e−2kmin′t≤1,
(37)V≤V0+D2kmin′.WhenV<D/2kmin′, let λ≜ln−V+D/2kmin′; according to Equation (31),
(38)λ̇≥−2kmin′,so
(39)λ≥λ0−2kmin′t.Substituting λ=ln−V+D/2kmin′ into Equation (39),
(40)ln−V+D2kmin′≥ln−V0+D2kmin′−2kmin′t,so
(41)V≤e−2kmin′tV0+D2kmin′1−e−2kmin′t,which also leads to Equation (37).WhenV=D/2kmin′, Equation (37) is correct, as well.Therefore, the system is uniformly ultimately bounded. All the signals in the closed-loop systems converge to a compact setΩ which is defined as
(42)Ω=s1′,s2′,s3′,φ,ψ,γ,φ̇,ψ̇,γ̇∣V≤V0+D2kmin′.According to Equation (27), if D is not too great, there exists ε′ and k′ with positive elements that guarantee V̇<0, and the system will be asymptotically stable. The compound control improves the total capacity of the autopilot, which increases the saturation amplitudes and therefore decreases the perturbation. This is the reason why this compound control autopilot insures the stability while improving the dynamic performance for the aircraft.


## 5. Simulations and Analysis

Numerical results are obtained by using a six-degree-of-freedom aircraft model which is launched from the ground with the 90° angle of pitch. The aircraft tracks a predefined schema trajectory given by a period of attitude control commandsϕc,ψc,γc, where ψc,γc keep zero, and φc decreases from 90° to shape the trajectory. Steady wind and wind shear are considered in the simulation to evaluate the performance of the compound controller.The numerical results of the initial stage of launch are shown in Figures2–10. The red lines represent the results of compound control. The blue lines show the results of the flexible nozzle and aerodynamic surface control, without lateral thrusters. The green lines describe the nominal condition where there is no disturbance.Figure 2
Tracking deviation of angle of pitch in the initial stage of launch.Figure 3
Tracking deviation of angle of yaw in the initial stage of launch.Figure 4
Tracking deviation of angle of roll in the initial stage of launch.Figure 5
Variation of nozzle deflection for pitching channel in the initial stage of launch.Figure 6
Variation of nozzle deflection for yawing channel in the initial stage of launch.Figure 7
Variation of aileron deflection in the initial stage of launch.Figure 8
Variation of lateral thrustPy in the initial stage of launch.Figure 9
Variation of lateral thrustPz in the initial stage of launch.Figure 10
Variation of lateral thrustPg in the initial stage of launch.As can be seen from Figures2–4, the wind disturbance increases the tracking deviation, but the compound controller can still hold the system steady. The tracking deviation of the angle of pitch does not converge to zero, which is necessary, because the control demand is varying to shape the trajectory. The wind disturbance influences more at the beginning of the launch when the velocity is low. So the deflection angles of the nozzle and the aileron get great and the lateral thrusters keep working, as is presented from Figures 5–10.Furthermore, the comparison between compound control and control without lateral thrusters shows that the tracking deviations and actuator deflection under compound control are both smaller. In each channel, the actuators cooperate to generate a control moment in the same direction. And the direction of control moment is defined by Equation (3).Specifically, In Figure2, the tracking deviation of the angle of pitch under compound control is smaller than that under control without lateral thrusters, because of the control moment provided by lateral thrusters in Figure 8 and the decreased couplings when the angle of roll is near zero in Figure 4. As is shown in Figure 5, the deflection of the flexible nozzle under compound control is smaller than that under control without lateral thrusters which saves energy for the main engine. The flexible nozzle deflection for the pitching channel is negative which provides positive control moment. The corresponding lateral thrust for the pitching channel is negative which generates a positive control moment. It demonstrates that under the compound control law, the flexible nozzle and lateral thrusters cooperated with each other and there is no undesirable counteracting. The same conclusion can be seen in the simulation results of the yawing channel in Figures 3, 6, and 9. The tracking deviation of the angle of roll and the usage of actuators for the rolling channel are shown in Figures 4, 7, and 10. The tracking deviation of the angle of roll is apparently smaller, and aileron saturation time decreases under compound control. This is because the control efficiency of ailerons is low when the velocity is low in the initial stage of launch, and the control moment provided by lateral thrusters is of the same direction as that induced by ailerons. This increases the actuators’ saturation amplitudes and therefore decreases the tracking deviation.It is demonstrated that the compound controller improves the performance of the autopilot by realizing the cooperation among actuators. That validates this compound control method is which effective to use the flexible nozzle, aerodynamic fins, and lateral thrusters to achieve a better response against disturbance and improve the stability of the aircraft.


## 6. Conclusions

This paper proposes a control scheme for aircrafts in the initial stage of launch. An autopilot design method is described for an aircraft with a flexible nozzle, aerodynamic fins, and lateral thrusters. This method is based on a nonlinear model, and the stability is guaranteed in each step of the design process. Simulation shows that the actuators are cooperated to improve the control which enhances the stability of the aircraft. Further, the disturbance from the lateral thrusters and other realistic impacts should be taken into consideration to improve the accuracy of modelling.



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*Source: 4749109-2019-03-06.xml*