CN102880060A

CN102880060A - Self-adaptive index time varying slip form posture control method of reentry flight vehicle

Info

Publication number: CN102880060A
Application number: CN2012104150066A
Authority: CN
Inventors: 刘向东; 王亮; 盛永智
Original assignee: Beijing Institute of Technology BIT
Current assignee: Beijing Institute of Technology BIT
Priority date: 2012-10-25
Filing date: 2012-10-25
Publication date: 2013-01-16
Anticipated expiration: 2032-10-25
Also published as: CN102880060B

Abstract

The invention relates to an adaptive exponential time-varying sliding mode attitude control method for a reentry aircraft, belonging to the technical field of aircraft control. First, take the unpowered reentry vehicle model as the object to establish the attitude motion equation; then rewrite it into the form of MIMO affine nonlinear system, and then apply the feedback linearization theory for linearization processing to obtain the three-channel linearization of pitch, roll-off, and yaw model; for the linearized system obtained, a modified adaptive exponential time-varying sliding mode controller is designed; then the control torque command for the attitude control of the re-entry vehicle is obtained and input to the re-entry vehicle to control the attitude. The present invention solves the problem of over-adaptation of the switching gain of the existing adaptive sliding mode control to a certain extent by combining the exponential time-varying sliding mode control with an adaptive method; it can effectively suppress the influence of system parameter uncertainty and external disturbances , to achieve precise attitude control.

Description

Adaptive exponential time-varying sliding mode attitude control method for reentry vehicle

技术领域 technical field

本发明涉及一种再入飞行器自适应指数时变滑模姿态控制方法，属于飞行器控制技术领域。The invention relates to an adaptive exponential time-varying sliding mode attitude control method for a reentry aircraft, belonging to the technical field of aircraft control.

背景技术 Background technique

对于再入飞行器来讲，再入过程中飞行条件（空域、速域）大范围变化，各通道间耦合严重，呈现出强烈的非线性动态特性。另外，各种不确定性外部扰动的存在以及飞行器的气动特性不能精确获知，导致其姿态控制变得异常复杂。再入飞行器控制系统设计要解决的关键问题是抑制上述非线性、强耦合和不确定性对系统性能影响。For the re-entry vehicle, the flight conditions (airspace and speed domain) change in a large range during the re-entry process, and the coupling between channels is serious, showing strong nonlinear dynamic characteristics. In addition, the existence of various uncertain external disturbances and the inability to accurately know the aerodynamic characteristics of the aircraft make its attitude control extremely complicated. The key problem to be solved in the design of the reentry vehicle control system is to suppress the influence of the above nonlinear, strong coupling and uncertainty on the system performance.

尽管各种先进的非线性控制方法（如动态逆、反馈线性化、轨迹线性化、反步法、自适应控制方法等）已广泛应用于再入飞行器姿态控制系统设计中，滑模变结构控制方法仍然是处理系统模型中存在有界扰动/不确定性和未建模动态首要选择。滑模变结构作为一种非线性控制方法，对系统存在的匹配参数不确定性以及扰动具有强鲁棒性。然而普通滑模然控制不足之处在于：1）到达段不具备鲁棒性；2）抖振问题；3）控制律中切换增益的选取问题。Although various advanced nonlinear control methods (such as dynamic inverse, feedback linearization, trajectory linearization, backstepping, adaptive control methods, etc.) method remains the first choice for dealing with bounded perturbations/uncertainties and unmodeled dynamics in system models. As a nonlinear control method, sliding mode variable structure has strong robustness to the uncertainty of matching parameters and disturbances in the system. However, the disadvantages of ordinary sliding mode control are: 1) the arrival stage is not robust; 2) the chattering problem; 3) the selection of the switching gain in the control law.

为了解决到达段不具备鲁棒性的问题，实现全局鲁棒的目的，A.Bartoszewicz[A.Bartoszewicz,Time-varying sliding modes for second-ordersystems,IEE Proceedings of Control Theory Application,143(5),1996:455-462.]采用时变滑模面替代时不变滑模面，使系统状态在初始时刻就处于滑模面上，以旋转或者平移的方式随时间趋近事先确定的时不变滑模面，但仍存在滑模控制量不光滑的问题。滑模变结构控制的抖振问题作为其固有特性，只能设法削弱而不能完全消除，目前已经有许多方法可以对其进行处理，比如：边界层方法[J.J.Slotine,Sliding mode controller design for nonlinear system,International Journal ofControl,40(2),1984:421-434.]；滑动扇区方法[K.Furuta,Y.Pan,Variable structurecontrol with sliding sector,Automatica,36(2),2000:211-228.]；高阶滑模控制方法[A.Levant,Sliding order and sliding accuracy in sliding mode control,InternationalJournal of Control,58(6),1993:1247-1263.]。一般来讲，滑模控制中切换增益是基于事先已知的系统中不确定性上界来确定的。然而，对于再入飞行器来讲，再入过程复杂多变，不容易获得这些不确定性上界。若切换增益取值过大，系统鲁棒性强，但抖振严重，容易激发系统的高频未建模动态引起系统不稳定；若切换增益取值太小，抖振小，但是系统抗干扰能力弱，鲁棒性较差。为此，需要寻求一种自适应方法，在线计算滑模控制的切换增益。In order to solve the problem that the arrival segment does not have robustness and achieve global robustness, A.Bartoszewicz [A.Bartoszewicz, Time-varying sliding modes for second-ordersystems, IEE Proceedings of Control Theory Application, 143(5), 1996 :455-462.] The time-varying sliding surface is used instead of the time-invariant sliding surface, so that the system state is on the sliding surface at the initial moment, and approaches the predetermined time-invariant sliding surface with time in the way of rotation or translation Die surface, but there is still the problem that the amount of sliding mode control is not smooth. The chattering problem of sliding mode variable structure control, as its inherent characteristics, can only be weakened but not completely eliminated. At present, there are many methods to deal with it, such as: boundary layer method [J.J.Slotine, Sliding mode controller design for nonlinear system ,International Journal of Control,40(2),1984:421-434.]; sliding sector method [K.Furuta,Y.Pan,Variable structure control with sliding sector,Automatica,36(2),2000:211-228. ]; high-order sliding mode control method [A. Levant, Sliding order and sliding accuracy in sliding mode control, International Journal of Control, 58(6), 1993:1247-1263.]. Generally speaking, the switching gain in sliding mode control is determined based on the known upper bound of uncertainty in the system in advance. However, for reentry vehicles, the reentry process is complex and changeable, and it is not easy to obtain these upper bounds of uncertainty. If the value of the switching gain is too large, the robustness of the system is strong, but the chattering is serious, and it is easy to stimulate the high-frequency unmodeled dynamics of the system to cause system instability; if the value of the switching gain is too small, the chattering is small, but the system is anti-interference Weak ability and poor robustness. Therefore, it is necessary to find an adaptive method to calculate the switching gain of sliding mode control online.

发明内容 Contents of the invention

本发明的目的是针对再入飞行器快时变、强耦合以及高度非线性的特点，通过将指数时变滑模与自适应控制方法结合，对于存在气动参数不确定性以及外部干扰力矩的再入飞行器，提出了一种高精度全局鲁棒姿态控制方法。The purpose of the present invention is to aim at the characteristics of fast time-varying, strong coupling and highly nonlinear re-entry aircraft, by combining the exponential time-varying sliding mode with the adaptive control method, for the re-entry aircraft with aerodynamic parameter uncertainty and external disturbance torque Aircraft, a high-precision global robust attitude control method is proposed.

本发明的目的是通过如下技术方案实现的：The purpose of the present invention is achieved through the following technical solutions:

步骤1，以关于机体坐标系（坐标系原点O取在飞行器质心，Ox轴与机体纵轴重合，指向头部为正；Oy轴位于机体纵对称面内与Ox轴垂直，指向上为正；Oz轴垂直于Oxy平面，方向按右手直角坐标系确定）x-O-y平面对称的无动力再入飞行器模型为对象，建立姿态运动方程：Step 1, with respect to the body coordinate system (the origin O of the coordinate system is taken at the center of mass of the aircraft, the Ox axis coincides with the longitudinal axis of the body, and it is positive when it points to the head; the Oy axis is located in the longitudinal symmetry plane of the body and is perpendicular to the Ox axis, and it is positive when it points upward; The Oz axis is perpendicular to the Oxy plane, and the direction is determined according to the right-hand rectangular coordinate system) The model of the unpowered reentry vehicle symmetrical to the x-O-y plane is used as the object, and the attitude motion equation is established:

$\overset{\cdot &Center Dot;}{α α} = = {ω ω}_{z z}$

$\overset{\cdot &Center Dot;}{β β} = = {ω ω}_{x x} sin sin α α + + {ω ω}_{y the y} cos cos α α$

$\overset{\cdot &Center Dot;}{μ μ} = = {ω ω}_{x x} cos cos α α - - {ω ω}_{y the y} sin sin α α$

${\overset{\cdot \cdot}{ω ω}}_{x x} = = \frac{{I I}_{yy yy}}{{I I}^{* *}} {M m}_{x x} + + \frac{{I I}_{xy xy}}{{I I}^{* *}} {M m}_{y the y} - - \frac{{I I}_{yy yy} (({I I}_{zz zz} - - {I I}_{yy yy})) - - {I I}_{xy xy}^{22}}{{I I}^{* *}} {ω ω}_{y the y} {ω ω}_{z z} - - \frac{{I I}_{xy xy} (({I I}_{yy yy} + + {I I}_{xx xxx} - - {I I}_{zz zz}))}{{I I}^{* *}} {ω ω}_{x x} {ω ω}_{z z} - - - - - - ((11))$

${\overset{\cdot &Center Dot;}{ω ω}}_{y the y} = = \frac{{I I}_{xy xy}}{{I I}^{* *}} {M m}_{x x} + + \frac{{I I}_{xx xxx}}{{I I}^{* *}} {M m}_{y the y} - - \frac{{I I}_{xx xxx} (({I I}_{xx xxx} - - {I I}_{zz zz})) + + {I I}_{xy xy}^{22}}{{I I}^{* *}} {ω ω}_{x x} {ω ω}_{z z} + + \frac{{I I}_{xy xy} (({I I}_{xx xxx} + + {I I}_{yy yy} - - {I I}_{zz zz}))}{{I I}^{* *}} {ω ω}_{y the y} {ω ω}_{z z}$

${\overset{\cdot &Center Dot;}{ω ω}}_{z z} = = \frac{11}{{I I}_{zz zz}} {M m}_{z z} - - \frac{{I I}_{yy yy} - - {I I}_{xx xxx}}{{I I}_{zz zz}} {ω ω}_{x x} {ω ω}_{y the y} - - \frac{{I I}_{xy xy}}{{I I}_{zz zz}} (({ω ω}_{y the y}^{22} - - {ω ω}_{x x}^{22}))$

式中，α,β,μ分别为攻角，侧滑角和倾侧角；ω_x,ω_y，ω_z分别为滚转、偏航和俯仰角速度；I_xx,I_yy,I_zz,I_xy分别为机体坐标系下关于x,y,z轴的转动惯量和惯量积，I_xz=I_yz=0，

M_x,M_y,M_z分别为机体坐标系下的气动力矩。其中，气动力矩为：In the formula, α, β, μ are attack angle, sideslip angle and roll angle respectively; ω _x , ω _y , ω _z are roll, yaw and pitch angular velocity respectively; I _xx , I _yy , I _zz , I _xy Respectively, the moment of inertia and product of inertia about the x, y, and z axes in the body coordinate system, I _xz =I _yz =0,

M _x , M _y , M _z are the aerodynamic moments in the body coordinate system, respectively. Among them, the aerodynamic moment is:

${M m}_{i i} = = \overset{^^}{q q} Sl Sl {C C}_{mi mi} ((α α,, β β,, Ma Ma,, {δ δ}_{x x},, {δ δ}_{y the y},, {δ δ}_{z z})),, i i = = x x,, y the y,, z z - - - - - - ((22))$

式中：

为动压，ρ为大气密度，V为飞行器飞行速度；S,l分别为飞行器的参考面积和参考长度；δ_x，δ_y，δ_z分别为副翼、方向舵和升降舵；C_mx,C_my,C_mz分别为滚转、偏航和俯仰力矩系数，为关于α,β,δ_x，δ_y,δ_z和马赫数Ma的函数。In the formula:

is the dynamic pressure, ρ is the atmospheric density, V is the flight speed of the aircraft; S, l are the reference area and reference length of the aircraft respectively; δ _x , δ _y , δ _z are the aileron, rudder and elevator respectively; C _mx , C _my , C _mz are roll, yaw and pitch moment coefficients respectively, which are functions of α, β, δ _x , δ _y , δ _z and the Mach number Ma.

由于地球自转角速度相比于飞行器的旋转运动慢得多，以及飞行器的旋转运动比位移运动快得多，因此忽略地球自转角速度和飞行器的位移运动在旋转运动方程中的作用。再入过程中采用BTT控制，侧滑角维持在零值附近，sin β≈0,tan β≈0,cos β≈1。Since the angular velocity of the earth's rotation is much slower than the rotational motion of the aircraft, and the rotational motion of the aircraft is much faster than the displacement motion, the effects of the angular velocity of the earth's rotation and the displacement motion of the aircraft in the rotational motion equation are ignored. BTT control is adopted during the reentry process, and the sideslip angle is maintained near zero, sin β≈0, tan β≈0, cos β≈1.

步骤2，将步骤1建立的再入飞行器模型改写成MIMO仿射非线性系统形式：Step 2, rewrite the reentry vehicle model established in step 1 into the form of MIMO affine nonlinear system:

$\overset{\cdot \cdot}{x x} = = f f ((x x)) + + g g ((x x)) u u - - - - - - ((33))$

Ω=h(x)Ω=h(x)

式中，x=[α β μ ω_x ω_y ω_z]^T是状态向量，Ω=[α β μ]^T是系统输出变量，u=[M_x M_y M_z]^T是计算所得的气动力矩，舵面偏转角指令[δ_x,δ_y,δ_z]^T通过对式(2)求逆计算得到。f(x)=[f₁(x)…f₆(x)]^T为6×1维矩阵，g(x)=[g₁(x) g₂(x) g₃(x)]^T为6×3维矩阵，h(x)=[h₁(x) h₂(x) h₃(x)]^T为3×1维矩阵。其中，In the formula, x=[α β μ ω _x ω _y ω _z ] ^T is the state vector, Ω=[α β μ] ^T is the system output variable, u=[M _x M _y M _z ] ^T is the calculated aerodynamic Moment, rudder surface deflection angle command [δ _x , δ _y , δ _z ] ^T is obtained by inverse calculation of formula (2). f(x)=[f ₁ (x)…f ₆ (x)] ^T is a 6×1-dimensional matrix, g(x)=[g ₁ (x) g ₂ (x) g ₃ (x)] ^T is 6×3-dimensional matrix, h(x)=[h ₁ (x) h ₂ (x) h ₃ (x)] ^T is a 3×1-dimensional matrix. in,

$\{\begin{matrix} {f f}_{11} ((x x)) = = {ω ω}_{z z} \\ {f f}_{22} ((x x)) = = {ω ω}_{x x} sin sin α α + + {ω ω}_{y the y} cos cos α α \\ {f f}_{33} ((x x)) = = {ω ω}_{x x} cos cos α α - - {ω ω}_{y the y} sin sin α α \\ {f f}_{44} ((x x)) = = - - \frac{{I I}_{yy yy} (({I I}_{zz zz} - - {I I}_{yy yy})) - - {I I}_{xy xy}^{22}}{{I I}^{* *}} {ω ω}_{y the y} {ω ω}_{z z} - - \frac{{I I}_{xy xy} (({I I}_{yy yy} + + {I I}_{xx xxx} - - {I I}_{zz zz}))}{{I I}^{* *}} {ω ω}_{x x} {ω ω}_{z z} \\ {f f}_{55} ((x x)) = = - - \frac{{I I}_{xx xxx} (({I I}_{xx xxx} - - {I I}_{yy yy})) + + {I I}_{xy xy}^{22}}{{I I}^{* *}} {ω ω}_{x x} {ω ω}_{z z} + + \frac{{I I}_{xy xy} (({I I}_{xx xxx} + + {I I}_{yy yy} - - {I I}_{zz zz}))}{{I I}^{* *}} {ω ω}_{y the y} {ω ω}_{z z} \\ {f f}_{66} ((x x)) = = - - \frac{{I I}_{yy yy} - - {I I}_{xx xxx}}{{I I}_{zz zz}} {ω ω}_{x x} {ω ω}_{y the y} - - \frac{{I I}_{xy xy}}{{I I}_{zz zz}} (({ω ω}_{y the y}^{22} - - {ω ω}_{x x}^{22})) \end{matrix},,$

$\{\begin{matrix} {g g}_{11} ((x x)) = = {[\begin{matrix} 00 & 00 & 00 & \frac{{I I}_{yy yy}}{{I I}^{* *}} & \frac{{I I}_{xy xy}}{{I I}^{* *}} & 00 \end{matrix}]}^{T T} \\ {g g}_{22} ((x x)) = = {[\begin{matrix} 00 & 00 & 00 & \frac{{I I}_{xy xy}}{{I I}^{* *}} & \frac{{I I}_{xx xxx}}{{I I}^{* *}} & 00 \end{matrix}]}^{T T} \\ {g g}_{33} ((x x)) = = {[\begin{matrix} 00 & 00 & 00 & 00 & 00 & \frac{11}{{I I}_{zz zz}} \end{matrix}]}^{T T} \end{matrix} . .$

步骤3，针对步骤2得到的仿射非线性系统，应用反馈线性化理论进行线性化处理，得到俯仰、滚装、偏航三通道线性化模型为：Step 3, for the affine nonlinear system obtained in step 2, apply the feedback linearization theory to perform linearization processing, and obtain the three-channel linearization model of pitch, roll-off and yaw as:

$\overset{\cdot &Center Dot; \cdot \cdot}{Ω Ω} = = F f ((x x)) + + E E. ((x x)) U u - - - - - - ((44))$

式中，In the formula,

$F f ((x x)) = = [\begin{matrix} {f f}_{33} ((x x)) \\ sin sin α α \cdot &Center Dot; {f f}_{11} ((x x)) + + cos cos α α \cdot \cdot {f f}_{22} ((x x)) + + (({ω ω}_{x x} cos cos α α - - {ω ω}_{y the y} sin sin α α)) \cdot &Center Dot; {ω ω}_{z z} \\ cos cos α α \cdot \cdot {f f}_{11} ((x x)) - - sin sin α α \cdot \cdot {f f}_{22} ((x x)) - - (({ω ω}_{x x} sin sin α α + + {ω ω}_{y the y} cos cos α α)) \cdot &Center Dot; {ω ω}_{z z} \end{matrix}],,$

$E E. ((x x)) = = [\begin{matrix} 00 & 00 & \frac{11}{{I I}_{zz zz}} \\ \frac{{I I}_{yy yy} sin sin α α}{{I I}^{* *}} + + \frac{{I I}_{xy xy} cos cos α α}{{I I}^{* *}} & \frac{{I I}_{xy xy} sin sin α α}{{I I}^{* *}} + + \frac{{I I}_{xx xxx} cos cos α α}{{I I}^{* *}} & 00 \\ \frac{{I I}_{yy yy} cos cos α α}{{I I}^{* *}} - - \frac{{I I}_{xy xy} sin sin α α}{{I I}^{* *}} & \frac{{I I}_{xy xy} cos cos α α}{{I I}^{* *}} - - \frac{{I I}_{xx xxx} sin sin α α}{{I I}^{* *}} & 00 \end{matrix}],,$

U＝[u₁ u₂ u₃]^T=[M_x M_y M_z]^T。U = [u ₁ u ₂ u ₃ ] ^T = [M _x M _y M _z ] ^T .

计算得：det(E(x))＝-1/(I^*I_zz)≠0，因此E(x)可逆。因此选择控制律形式为：Calculated: det(E(x))=-1/(I ^* I _zz )≠0, so E(x) is reversible. Therefore, the form of the control law is chosen as:

U＝E^-1(v-F)（5）U=E ^-1 (vF) (5)

其中，v=[v₁ v₂ v₃]^T是辅助控制量。Among them, v=[v ₁ v ₂ v ₃ ] ^T is the auxiliary control quantity.

将控制律代入线性化模型中，得如下解耦的积分器形式：Substituting the control law into the linearized model, the following decoupled integrator form is obtained:

$\overset{\cdot &Center Dot; \cdot \cdot}{Ω Ω} = = v v - - - - - - ((66))$

当再入飞行器模型的仿射非线性系统中存在参数不确定性及外部扰动时，经反馈线性化的系统模型表示为：When there are parameter uncertainties and external disturbances in the affine nonlinear system of the reentry vehicle model, the system model after feedback linearization is expressed as:

$\overset{\cdot &Center Dot; \cdot \cdot}{Ω Ω} = = F f + + ΔF ΔF + + ((E E. + + ΔE ΔE)) U u - - - - - - ((77))$

用Δv=[Δv₁ Δv₂ Δv₃]^T表示上式中的聚合扰动：Δv=ΔF+ΔEU。且设此不确定性扰动是有界的，即存在Δv_1max,Δv_2max,Δv_3max，使得|Δv₁|≤Δv_1max，|Δv₂|≤Δv_2max，|Δv₃|≤Δv_3max。Use Δv=[Δv ₁ Δv ₂ Δv ₃ ] ^T to represent the aggregation disturbance in the above formula: Δv=ΔF+ΔEU. And assume that the uncertainty disturbance is bounded, that is, there are Δv _1max , Δv _2max , Δv _3max , such that |Δv ₁ |≤Δv _1max , |Δv ₂ |≤Δv _2max , |Δv ₃ |≤Δv _3max .

将选择的控制律形式（5）和聚合扰动表达式Δv，代入经反馈线性化的系统模型式(7)，则考虑了参数不确定性以及扰动的再入飞行器反馈线性化系统为：Substituting the selected control law form (5) and the aggregated disturbance expression Δv into the feedback linearized system model formula (7), the feedback linearization system of the reentry vehicle considering the parameter uncertainty and disturbance is:

$\overset{\cdot &Center Dot; \cdot &Center Dot;}{Ω Ω} = = v v + + Δv Δv - - - - - - ((88))$

步骤4，针对步骤3得到的线性化系统，进行自适应指数时变滑模控制器设计。Step 4, for the linearized system obtained in step 3, design an adaptive exponential time-varying sliding mode controller.

首先，选择指数时变滑模面：First, choose an exponential time-varying sliding surface:

$S S ((t t)) = = \overset{\cdot &Center Dot;}{\overset{~ ~}{Ω Ω}} + + Λ Λ \overset{~ ~}{Ω Ω} + + A A {e e}^{- - at at} - - - - - - ((99))$

式中，

是系统跟踪误差，Ω_c=[α_c β_c μ_c]^T是制导环给出的姿态指令，S(t)=[s_α(t) s_β(t) s_μ(t)]^T∈R³是滑模面函数向量，A∈R³是跟系统状态初值相关的参数矩阵，Λ=diag{λ₁,λ₂,λ₃}∈R^3×3表示滑模面斜率，a∈R⁺决定了时变滑模面向时不变滑模面的趋近速度，令λ₁=λ₂=λ₃=a=λ。根据时变滑模理论，系统状态要求从初始时刻就要处于滑模面上，即满足：S(0)=0_3×1，则A的取值为：In the formula,

is the system tracking error, Ω _c =[α _c β _c μ _c ] ^T is the attitude command given by the guidance loop, S(t)=[s _α (t) s _β (t) s _μ (t)] ^T ∈ R ³ is the sliding mode surface function vector, A∈R ³ is the parameter matrix related to the initial value of the system state, Λ=diag{λ ₁ ,λ ₂ ,λ ₃ }∈R ^3×3 represents the slope of the sliding mode surface, a∈ R ⁺ determines the approach speed of the time-varying sliding mode surface to the time-invariant sliding mode surface, let λ ₁ =λ ₂ =λ ₃ =a=λ. According to the time-varying sliding mode theory, the system state is required to be on the sliding mode surface from the initial moment, that is, to satisfy: S(0)=0 _3×1 , then the value of A is:

$A A = = - - \overset{\cdot \cdot}{\overset{~ ~}{Ω Ω}} ((00)) - - Λ Λ \overset{~ ~}{Ω Ω} ((00)) = = - - Λ Λ \overset{~ ~}{Ω Ω} ((00)) - - - - - - ((1010))$

然后，设计修正的自适应指数时变滑模控制器形式为：Then, the modified adaptive exponential time-varying sliding mode controller is designed as:

$v v = = {v v}_{eq eq} + + {v v}_{sw sw} = = {\overset{\cdot &Center Dot; \cdot &Center Dot;}{Ω Ω}}_{c c} - - Λ Λ \overset{\cdot &Center Dot;}{\overset{~ ~}{Ω Ω}} + + Aλ Aλ {e e}^{- - λt λt} - - ηsat ηsat ((S S ((t t)))) - - - - - - ((1111))$

式中，

表示等价控制，v_sw=-ηsat(S(t))表示切换控制（采用饱和函数sat(·)是为了减小抖振），η=diag{η_α,η_β,η_μ}为滑模控制的切换增益。饱和函数sat(·)以及切换增益自适应算法分别表示为：In the formula,

represents equivalent control, v _sw =-ηsat(S(t)) represents switching control (the saturation function sat( ) is used to reduce chattering), η=diag{η _α ,η _β ,η _μ } is sliding Mode controlled switching gain. The saturation function sat( ) and the switching gain adaptive algorithm are respectively expressed as:

${\overset{\cdot \cdot}{η η}}_{j j} = = \frac{11}{{k k}_{j j}} ((- - {σ σ}_{j j} {η η}_{j j} + + | | {s the s}_{j j} ((t t)) | |)),, j j = = α α,, β β,, μ μ - - - - - - ((1313))$

其中，

表示边界层厚度，σ_j是一个较小的正常数，k_j>0为自适应率。η_j的自适应速度受k_j的控制，通过选择合适的k_j可以有效地避免到达阶段控制量的高频振动。不失一般性，本发明将k_j设置为常数，且k_α=k_β=k_μ=k。in,

Indicates the thickness of the boundary layer, σ _j is a small normal constant, and k _j >0 is the adaptive rate. The adaptive speed of η _j is controlled by k _j , and the high-frequency vibration of the control quantity at the arrival stage can be effectively avoided by choosing an appropriate k _j . Without loss of generality, the present invention sets k _j as a constant, and k _α =k _β =k _μ =k.

步骤5，根据步骤4，得到再入飞行器姿态控制的控制力矩指令：Step 5, according to step 4, obtain the control torque command of the attitude control of the reentry vehicle:

U＝E^-1(v-F)（14）U=E ^-1 (vF) (14)

然后根据气动力矩表达式（2），将控制力矩分配到气动舵面，计算得到姿态控制所需要舵面偏转角指令[δ_x δ_y δ_z]^T。Then according to the aerodynamic moment expression (2), the control torque is distributed to the aerodynamic rudder surface, and the rudder deflection angle command [δ _x δ _y δ _z ] ^T required for attitude control is calculated.

步骤6，将步骤5得到的舵面偏转角指令[δ_x δ_y δ_z]^T输入到再入飞行器，对姿态进行控制。同时，飞行器控制系统输出实时飞行状态（α,β,μ,ω_x,ω_y,ω_z），并作为反馈状态输入到姿态控制系统，重复步骤2-步骤6。Step 6: Input the rudder surface deflection angle command [δ _x δ _y δ _z ] ^T obtained in step 5 to the re-entry vehicle to control the attitude. At the same time, the aircraft control system outputs the real-time flight status (α, β, μ, ω _x , ω _y , ω _z ) and inputs it to the attitude control system as the feedback status, repeating steps 2-6.

从而在系统存在参数不确定性及外部扰动的情况下，通过控制舵面偏转角[δ_x δ_y δ_z]^T，实现对制导环给出的姿态指令Ω_c=[α_c β_c μ_c]^T的有效跟踪。Therefore, in the case of parameter uncertainties and external disturbances in the system, by controlling the steering surface deflection angle [δ _x δ _y δ _z ] ^T , the attitude command Ω _c =[α _c β _c μ _c ] ^T 's effective tracking.

有益效果Beneficial effect

本发明与现有技术相比的优点在于：The advantage of the present invention compared with prior art is:

1）本发明结合再入飞行器的特点，给出了模型简化的方法，分析了模型不确定性，应用反馈线性化方法对再入飞行器非线性动态方程进行了线性化处理。针对线性化后解耦的再入飞行器动态方程，提出了一种指数时变滑模姿态控制器设计方法，有效地解决了常规滑模控制到达段不具鲁棒性的问题，提高了控制系统的鲁棒性，并有效地改善了系统响应效果；1) The present invention combines the characteristics of the re-entry vehicle, provides a model simplification method, analyzes the model uncertainty, and uses the feedback linearization method to linearize the nonlinear dynamic equation of the re-entry vehicle. Aiming at the decoupled reentry vehicle dynamic equations after linearization, an exponential time-varying sliding mode attitude controller design method is proposed, which effectively solves the problem that the conventional sliding mode control is not robust in the arrival stage, and improves the performance of the control system. Robustness, and effectively improved the system response;

2）本发明引入的切换增益自适应调整算法，有效地解决了盲目调整滑模控制切换增益的问题，能够有效地提高系统的适应性；2) The self-adaptive adjustment algorithm of switching gain introduced by the present invention effectively solves the problem of blindly adjusting the switching gain of sliding mode control, and can effectively improve the adaptability of the system;

3）本发明通过将指数时变滑模控制与自适应方法结合，在一定程度上解决了现有自适应滑模控制的切换增益过度适应问题。3) The present invention solves the problem of over-adaptation of the switching gain of the existing adaptive sliding mode control to a certain extent by combining the exponential time-varying sliding mode control with the adaptive method.

本发明能够有效地抑制系统参数不确定性和外部扰动的影响，实现精确的姿态控制。The invention can effectively suppress the influence of system parameter uncertainty and external disturbance, and realize precise attitude control.

附图说明 Description of drawings

图1为本发明提出的自适应指数时变滑模控制方法的结构图；Fig. 1 is the structural diagram of the self-adaptive exponential time-varying sliding mode control method that the present invention proposes;

图2为具体实施中再入飞行器自适应指数时变滑模控制系统结构框图；Fig. 2 is the structural block diagram of reentry aircraft self-adaptive exponential time-varying sliding mode control system in concrete implementation;

图3为具体实施时再入飞行器姿态控制系统跟踪给定姿态角指令时自适应指数时变滑模控制与自适应普通滑模控制的响应曲线，其中，（a）为攻角响应曲线，（b）为侧滑角响应曲线；（c）为倾侧角响应曲线；Fig. 3 is the response curve of the adaptive exponential time-varying sliding mode control and the adaptive ordinary sliding mode control when the attitude control system of the reentry vehicle tracks a given attitude angle command during the specific implementation, where (a) is the response curve of the angle of attack, ( b) is the side slip angle response curve; (c) is the roll angle response curve;

图4为具体实施时再入飞行器姿态控制系统姿态角速度响应曲线。左图是采用自适应指数时变滑模控制器时的响应曲线，右图是采用自适应普通滑模控制器时的响应曲线；Fig. 4 is the attitude angular velocity response curve of the reentry aircraft attitude control system during the specific implementation. The left figure is the response curve when using the adaptive exponential time-varying sliding mode controller, and the right figure is the response curve when using the adaptive ordinary sliding mode controller;

图5为具体实施时再入飞行器姿态控制系统舵面偏转角响应曲线。左图是采用自适应指数时变滑模控制器时的响应曲线，右图是采用自适应普通滑模控制器时的响应曲线；Fig. 5 is the rudder surface deflection angle response curve of the reentry vehicle attitude control system during the specific implementation. The left figure is the response curve when using the adaptive exponential time-varying sliding mode controller, and the right figure is the response curve when using the adaptive ordinary sliding mode controller;

图6为具体实施时再入飞行器姿态控制系统自适应切换增益响应曲线。左图是采用自适应指数时变滑模控制器时的响应曲线，右图是采用自适应普通滑模控制器时的响应曲线；Fig. 6 is the adaptive switching gain response curve of the re-entry aircraft attitude control system during specific implementation. The left figure is the response curve when using the adaptive exponential time-varying sliding mode controller, and the right figure is the response curve when using the adaptive ordinary sliding mode controller;

图7为具体实施时再入飞行器姿态控制系统滑模面响应曲线。左图是采用自适应指数时变滑模控制器时的响应曲线，右图是采用自适应普通滑模控制器时的响应曲线。Fig. 7 is the sliding mode surface response curve of the attitude control system of the reentry vehicle during the specific implementation. The left figure is the response curve when using an adaptive exponential time-varying sliding mode controller, and the right figure is the response curve when using an adaptive ordinary sliding mode controller.

具体实施方式 Detailed ways

为了更好的说明本发明的目的和优点，下面结合附图和实施例加以进一步说明。In order to better illustrate the purpose and advantages of the present invention, further description will be given below in conjunction with the accompanying drawings and embodiments.

本发明实施的再入飞行器自适应指数时变滑模控制器结构图如图2所示，利用本发明提出的自适应指数时变滑模姿态控制系统实现对姿态角指令Ω_c=[α_c β_c μ_c]^T的有效跟踪。The structural diagram of the self-adaptive exponential time-varying sliding mode controller for the re-entry aircraft implemented by the present invention is shown in Figure 2. The self-adaptive exponential time-varying sliding mode attitude control system proposed by the present invention is used to realize the attitude angle command Ω _c =[α _c β _c μ _c ] ^T efficient tracking.

通常，设计自适应指数时变滑模控制器形式为：Usually, the form of designing an adaptive exponential time-varying sliding mode controller is:

$v v = = {v v}_{eq eq} + + {v v}_{sw sw} = = {\overset{\cdot \cdot \cdot \cdot}{Ω Ω}}_{c c} - - Λ Λ \overset{\cdot \cdot}{\overset{~ ~}{Ω Ω}} + + Aλ Aλ {e e}^{- - λt λt} - - ηsgn ηsgn ((S S ((t t)))) - - - - - - ((1515))$

式中，

表示等价控制，在标称模型的情况下，根据

推导可得；v_sw=-ηsgn(S(t))表示切换控制，目的是抵消模型中的不确定性及扰动。其中，η=diag{η_α,η_β,η_μ}为滑模控制的切换增益，其在线自适应更新算法为：In the formula,

denotes an equivalent control, in the case of the nominal model, according to

It can be derived; v _sw =-ηsgn(S(t)) means switching control, the purpose is to offset the uncertainty and disturbance in the model. Among them, η=diag{η _α ,η _β ,η _μ } is the switching gain of sliding mode control, and its online adaptive update algorithm is:

${\overset{\cdot \cdot}{η η}}_{j j} = = \frac{11}{{k k}_{j j}} | | {s the s}_{j j} ((t t)) | |,, j j = = α α,, β β,, μ μ - - - - - - ((1616))$

式中，k_j>0,j=α,β,μ为自适应率。η_j的自适应速度受k_j的控制，通过选择合适的k_j可以有效地避免到达阶段控制量的高频振动。不失一般性，本发明将k_j设置为常数，且k_α=k_β=k_μ=k。In the formula, k _j >0, j=α, β, μ is the adaptive rate. The adaptive speed of η _j is controlled by k _j , and the high-frequency vibration of the control quantity at the arrival stage can be effectively avoided by choosing an appropriate k _j . Without loss of generality, the present invention sets k _j as a constant, and k _α =k _β =k _μ =k.

对上述自适应指数时变滑模控制器进行修正。The above adaptive exponential time-varying sliding mode controller is modified.

由于控制律（15）中符号函数sgn(·)的存在，当系统状态穿越滑模面时，是非连续的。这就会带来不必要的抖振问题，严重影响飞行器舵机寿命及响应特性。另外，在实际应用时由于测量噪声、模型不匹配以及有限切换频率等因素的影响，S(t)不能精确限制到0。这时，根据式（16）可知自适应切换增益η会无限增长直至无界。为了克服上述不足，采用如下修正的连续化自适应指数时变滑模控制律：Due to the existence of the sign function sgn(·) in the control law (15), when the system state crosses the sliding mode surface, it is discontinuous. This will bring unnecessary chattering problems, which will seriously affect the service life and response characteristics of the aircraft steering gear. In addition, due to measurement noise, model mismatch, and limited switching frequency, S(t) cannot be precisely limited to 0 in practical applications. At this time, according to formula (16), it can be known that the adaptive switching gain η will grow infinitely until it is unbounded. In order to overcome the above shortcomings, the following modified continuous adaptive exponential time-varying sliding mode control law is adopted:

$v v = = {v v}_{eq eq} + + {v v}_{sw sw} = = {\overset{\cdot &Center Dot; \cdot \cdot}{Ω Ω}}_{c c} - - Λ Λ \overset{\cdot \cdot}{\overset{~ ~}{Ω Ω}} + + Aλ Aλ {e e}^{- - λt λt} - - ηsat ηsat ((S S ((t t)))) - - - - - - ((1717))$

式中，饱和函数sat(·)以及修正切换增益自适应算法分别表示为：In the formula, the saturation function sat( ) and the modified switching gain adaptive algorithm are respectively expressed as:

${\overset{\cdot &Center Dot;}{η η}}_{j j} = = \frac{11}{{k k}_{j j}} ((- - {σ σ}_{j j} {η η}_{j j} + + | | {s the s}_{j j} ((t t)) | |)),, j j = = α α,, β β,, μ μ - - - - - - ((1919))$

其中，

表示边界层厚度，σ_j是一个较小的正常数。in,

Indicates the thickness of the boundary layer, σ _j is a small normal constant.

稳定性分析：Stability Analysis:

对于（8）描述的考虑不确定性的再入飞行器非线性系统，采用式（15）所示的指数时变滑模控制律和相应的切换增益自适应算法（16），整个再入飞行器闭环姿态控制系统是渐进稳定的。For the nonlinear system of the reentry vehicle described in (8) considering the uncertainty, the exponential time-varying sliding mode control law shown in equation (15) and the corresponding switching gain adaptive algorithm (16) are used, and the closed loop of the entire reentry vehicle The attitude control system is asymptotically stable.

首先，定义自适应误差： $\tilde{η} = {[η_{1} - Δ v_{1 \max} η_{2} - Δ v_{2 \max} η_{3} - Δ v_{3 \max}]}^{T} .$ First, define the adaptive error: $\tilde{η} = {[η_{1} - Δ v_{1 \max} η_{2} - Δ v_{2 \max} η_{3} - Δ v_{3 \max}]}^{T} .$

选择正定Lyapunov函数为：Choose the positive definite Lyapunov function as:

$V V = = \frac{11}{22} {S S}^{T T} ((t t)) S S ((t t)) + + \frac{11}{22} k k {\overset{~ ~}{η η}}^{T T} \overset{~ ~}{η η} - - - - - - ((2020))$

对式（20）求其关于时间的导数，可得Calculate the derivative of equation (20) with respect to time, we can get

$\overset{\cdot &Center Dot;}{V V} = = {S S}^{T T} ((t t)) \overset{\cdot &Center Dot;}{S S} ((t t)) + + k k {\overset{~ ~}{η η}}^{T T} \overset{\cdot &Center Dot;}{\overset{~ ~}{η η}}$

$= = {S S}^{T T} ((t t)) Δv Δv + + [\begin{matrix} - - {Δv Δv}_{11 max max} & - - {Δv Δ v}_{22 max max} & - - {Δv Δ v}_{33 max max} \end{matrix}] | | S S ((t t)) | | \leq \leq 00 - - - - - - ((21 twenty one))$

从式（21）可知，

是半负定的，这意味着V是非增长且有界的，即V(t)≤V(0)。因此，可得S(t)和自适应增益η是有界的。进一步，令

且对其从时间0→t积分，可得From formula (21), we can see that,

is negative semi-definite, which means that V is non-growing and bounded, that is, V(t)≤V(0). Therefore, available S(t) and adaptive gain η are bounded. Further, make

And integrate it from time 0→t, we can get

${&Integral; &Integral;}_{00}^{t t} Ξ ξ ((t t)) dτ dτ = = V V ((00)) - - V V ((t t)) - - - - - - ((22 twenty two))$

由于V(0)和V(t)都是有界的，容易得到：Since both V(0) and V(t) are bounded, it is easy to get:

${&Integral; &Integral;}_{00}^{t t} Ξ ξ ((t t)) dτ dτ \leq \leq V V ((00)) \leq \leq \infty \infty - - - - - - ((23 twenty three))$

因此，根据Barbalat引理可知，

这表明：当t→∞时，S(t)→0。直观上看，由

不能推导出然而，再入飞行器姿态控制闭环系统的渐进稳定性可解释如下：Therefore, according to Barbalat's lemma,

This shows that: when t→∞, S(t)→0. Intuitively, by

cannot be deduced However, the asymptotic stability of the RV attitude control closed-loop system can be explained as follows:

根据自适应律表达式（16）可知，如果S(t)≠0，切换增益η会一直增大。因此，必然存在一个时刻t_F>0，η增大到一个足够大的值，满足滑动模态的到达条件（比如η_j>Δv_imax,j=α,β,μ），使得滑动模态建立，即存在

从而根据S(t)=0可得：According to the adaptive law expression (16), if S(t)≠0, the switching gain η will always increase. Therefore, there must be a time t _F >0, and η increases to a sufficiently large value to meet the arrival condition of the sliding mode (such as η _j >Δv _imax ,j=α,β,μ), so that the sliding mode is established , that exists

Therefore, according to S(t)=0, it can be obtained:

$\overset{~ ~}{Ω Ω} ((t t)) = = {e e}^{- - λt λt} ((Λt Λt + + {I I}_{33 \times \times 33})) \overset{~ ~}{Ω Ω} ((00)) - - - - - - ((24 twenty four))$

由式（24）可知，

即闭环姿态控制系统是渐进稳定的。From formula (24), we can see that

That is, the closed-loop attitude control system is asymptotically stable.

对于采用修正的连续化自适应指数时变滑模控制律（17）以及自适应增益算法（19）的情况，此时闭环系统是一致有界稳定的。For the case of using the modified continuous adaptive exponential time-varying sliding mode control law (17) and the adaptive gain algorithm (19), the closed-loop system is uniformly bounded and stable.

实施例Example

1）建立再入飞行器六自由度十二状态方程作为被控对象模型，将方程中涉及到的三个气流姿态角（迎角α，侧滑角β，倾侧角μ）运动学方程和三个绕机体轴旋转的角速度（滚转角速度ω_x，偏航角速度ω_y，俯仰角速度ω_z动力学方程）写成仿射非线性形式（3）；1) Establish the six-degree-of-freedom twelve-state equation of the re-entry vehicle as the model of the controlled object, and combine the three airflow attitude angles (angle of attack α, sideslip angle β, and roll angle μ) kinematic equations involved in the equation with the three The angular velocity of rotation around the body axis (roll angular velocity ω _x , yaw angular velocity ω _y , pitch angular velocity ω _z dynamic equation) is written in the affine nonlinear form (3);

2）对上述仿射非线性系统进行反馈线性化处理，得到解耦的再入飞行器三通道数学模型；2) Perform feedback linearization processing on the above-mentioned affine nonlinear system to obtain a decoupled three-channel mathematical model of the re-entry vehicle;

3）构造指数时变滑模函数式（9）和相应的控制律（11），控制律中的切换增益通过式（13）在线计算。3) Construct the exponential time-varying sliding mode function formula (9) and the corresponding control law (11), and the switching gain in the control law is calculated online through formula (13).

4）根据式（14）计算得到控制力矩指令。由于控制力矩不能直接施加到再入飞行模型中，需要根据气动力矩的拟合表达式（2）进行相应的逆运算，获得真实舵面偏转角指令[δ_x δ_y δ_z]^T。4) Calculate the control torque command according to formula (14). Since the control torque cannot be directly applied to the re-entry flight model, it is necessary to perform the corresponding inverse calculation according to the fitting expression (2) of the aerodynamic torque to obtain the real steering surface deflection angle command [δ _x δ _y δ _z ] ^T .

5）将上一步中得到舵面偏转角指令输入到再入飞行器进行姿态控制。5) Input the rudder surface deflection angle command obtained in the previous step to the reentry vehicle for attitude control.

为了验证本发明在再入飞行器姿态控制中应用时的优越性，与自适应普通滑模控制器的控制效果进行对比。In order to verify the superiority of the present invention when it is applied in the attitude control of the re-entry vehicle, it is compared with the control effect of the adaptive ordinary sliding mode controller.

普通滑模面的定义为：The general sliding mode surface is defined as:

$S S ((t t)) = = \overset{\cdot &Center Dot;}{\overset{~ ~}{Ω Ω}} + + Λ Λ \overset{~ ~}{Ω Ω} - - - - - - ((2525))$

相应的，滑模姿态控制律为：Correspondingly, the sliding mode attitude control law is:

$v v = = {\overset{\cdot &Center Dot; \cdot \cdot}{Ω Ω}}_{c c} - - Λ Λ \overset{\cdot \cdot}{\overset{~ ~}{Ω Ω}} - - ηsat ηsat ((S S ((t t)))) - - - - - - ((2626))$

其中，切换增益的自适应算法与自适应指数时变滑模时一致，采用式（13）。Among them, the adaptive algorithm of switching gain is consistent with the adaptive exponential time-varying sliding mode, using formula (13).

本发明在Matlab2009a环境下进行仿真验证。飞行初始状态如下：初始高度为28km，速度2000m/s，姿态角初始值为[1°,1°,1°]^T，舵面偏转角限制为±20°。姿态角给定指令为：[α_c,β_c,μ_c]^T=[4°,0°,20°]^T，进一步，为了验证所设计控制律的鲁棒性，考虑-30%的大气密度拉偏以及如下所示的高频外部扰动（直接施加于三轴的控制力矩上）：The present invention carries out simulation verification under the environment of Matlab2009a. The initial state of the flight is as follows: the initial altitude is 28km, the speed is 2000m/s, the initial attitude angle is [1°, 1°, 1°] ^T , and the rudder deflection angle is limited to ±20°. Attitude angle given command is: [α _c , β _c , μ _c ] ^T = [4°,0°,20°] ^T , further, in order to verify the robustness of the designed control law, consider -30% atmospheric Density pull and high frequency external perturbations (applied directly to the control torques of the three axes) as follows:

d=[100sin(t) 100sin(t) 100sin(t)]^TN·m。d=[100sin(t) 100sin(t) 100sin(t)] ^T N m.

控制器参数选择：普通滑模面参数与指数时变滑模面参数选择相同，滑模面斜率λ=4，边界层厚度

自适应参数k=0.0005。Selection of controller parameters: The parameters of the ordinary sliding mode surface are the same as those of the exponential time-varying sliding mode surface, the slope of the sliding mode surface is λ=4, and the thickness of the boundary layer is

The adaptive parameter k=0.0005.

运用自适应指数时变滑模控制器和自适应普通滑模控制器时再入飞行器姿态角响应曲线如图3所示。从图中可以看出，二者均能实现姿态角的稳定跟踪控制。另外，采用普通滑模控制时，姿态角响应速度较快，这是由于普通滑模控制有明显的到达阶段，一开始时控制量较大。Figure 3 shows the attitude angle response curve of the reentry vehicle when using the adaptive exponential time-varying sliding mode controller and the adaptive ordinary sliding mode controller. It can be seen from the figure that both of them can realize the stable tracking control of the attitude angle. In addition, when the ordinary sliding mode control is adopted, the response speed of the attitude angle is faster. This is because the ordinary sliding mode control has an obvious arrival stage, and the control amount is relatively large at the beginning.

图4给出了分别采用自适应指数时变滑模控制器和自适应普通滑模控制器时姿态角速度响应对比曲线。从图中可以看出，开始阶段采用自适应普通滑模控制时，姿态角速度较大，容易引起控制量饱和。Figure 4 shows the comparison curves of the attitude angular velocity response when the adaptive exponential time-varying sliding mode controller and the adaptive ordinary sliding mode controller are respectively used. It can be seen from the figure that when the adaptive ordinary sliding mode control is adopted in the initial stage, the angular velocity of the attitude is relatively large, which easily causes the saturation of the control amount.

图5给出了分别采用自适应指数时变滑模控制器和自适应普通滑模控制器时舵面偏转角响应对比曲线。从图中可以看出，采用普通滑模控制时舵面偏转角在开始阶段会出现饱和现象，这是由于其自适应切换增益较大而引起的；而采用指数时变滑模控制时不但舵面偏转角没有出现饱和现象，且舵面偏转角指令较平滑。这是将指数时变滑模控制与自适应算法结合带来的优点。Figure 5 shows the comparison curves of the deflection angle response of the rudder surface when the adaptive exponential time-varying sliding mode controller and the adaptive ordinary sliding mode controller are respectively used. It can be seen from the figure that when using ordinary sliding mode control, the rudder deflection angle will be saturated at the beginning stage, which is caused by the large adaptive switching gain; while using exponential time-varying sliding mode control not only There is no saturation phenomenon in the surface deflection angle, and the rudder surface deflection angle command is relatively smooth. This is the advantage brought by the combination of exponential time-varying sliding mode control and adaptive algorithm.

图6给出了分别采用自适应指数时变滑模控制器和自适应普通滑模控制器时自适应切换增益对比曲线。从图中可以看出，采用自适应普通滑模滑模控制的切换增益值明显要大于采用自适应指数时变滑模控制时的值。这主要是由于采用普通滑模控制时开始阶段偏离滑模面较远，根据式（13）可知，切换增益会迅速增大到一个较大的值（远远大于系统存在的实际扰动）；而指数时变滑模控制时，由于初始时刻就处于滑模面上，不会引起自适应切换增益的迅速增大。Figure 6 shows the comparison curves of the adaptive switching gain when the adaptive exponential time-varying sliding mode controller and the adaptive ordinary sliding mode controller are used respectively. It can be seen from the figure that the switching gain value of the adaptive ordinary sliding mode sliding mode control is obviously greater than that of the adaptive exponential time-varying sliding mode control. This is mainly due to the fact that the initial stage deviates far from the sliding mode surface when using ordinary sliding mode control. According to formula (13), it can be seen that the switching gain will rapidly increase to a larger value (far greater than the actual disturbance in the system); and In exponential time-varying sliding mode control, since the initial moment is on the sliding mode surface, the adaptive switching gain will not increase rapidly.

图7给出了分别采用自适应指数时变滑模控制器和自适应普通滑模控制器时滑模面响应对比曲线，可以看出，采用普通滑模控制存在明显的到达阶段。Figure 7 shows the comparison curves of the sliding mode surface response when using the adaptive exponential time-varying sliding mode controller and the adaptive ordinary sliding mode controller respectively. It can be seen that there is an obvious arrival stage when using the ordinary sliding mode control.

Claims

1. The attitude control method of the reentry vehicle self-adaptive index time-varying sliding mode is characterized by comprising the following steps: the method comprises the following steps:

step 1, taking an unpowered reentry aircraft model which is symmetrical about an x-O-y plane of a body coordinate system as an object, establishing an attitude motion equation:

\overset{\cdot}{α} = ω_{z}

\overset{\cdot}{β} = ω_{x} \sin α + ω_{y} \cos α

\overset{\cdot}{μ} = ω_{x} \cos α - ω_{y} \sin α

{\overset{\cdot}{ω}}_{x} = \frac{I_{yy}}{I^{*}} M_{x} + \frac{I_{xy}}{I^{*}} M_{y} - \frac{I_{yy} (I_{zz} - I_{yy}) - I_{xy}^{2}}{I^{*}} ω_{y} ω_{z} - \frac{I_{xy} (I_{yy} + I_{xx} - I_{zz})}{I^{*}} ω_{x} ω_{z} - - - (1)

{\overset{\cdot}{ω}}_{y} = \frac{I_{xy}}{I^{*}} M_{x} + \frac{I_{xx}}{I^{*}} M_{y} - \frac{I_{xx} (I_{xx} - I_{zz}) + I_{xy}^{2}}{I^{*}} ω_{x} ω_{z} + \frac{I_{xy} (I_{xx} + I_{yy} - I_{zz})}{I^{*}} ω_{y} ω_{z}

{\overset{\cdot}{ω}}_{z} = \frac{1}{I_{zz}} M_{z} - \frac{I_{yy} - I_{xx}}{I_{zz}} ω_{x} ω_{y} - \frac{I_{xy}}{I_{zz}} (ω_{y}^{2} - ω_{x}^{2})

in the formula, alpha, beta and mu are respectively an attack angle, a sideslip angle and a roll angle; omega_x,ω_y,ω_zRoll, yaw and pitch velocities, respectively; i is_xx,I_yy,I_zz,I_xyRespectively, the moment of inertia and the inertia product I about the x, y and z axes under the body coordinate system_xz=I_yz=0，

M_x,M_y,M_zRespectively are aerodynamic moment under a coordinate system of the machine body; wherein, the aerodynamic moment is:

M_{i} = \hat{q} Sl C_{mi} (α, β, Ma, δ_{x}, δ_{y}, δ_{z}), i = x, y, z - - - (2)

in the formula:

the pressure is dynamic pressure, rho is atmospheric density, and V is the flying speed of the aircraft; s, l is the reference area and the reference length of the aircraft respectively; delta_x,δ_y，δ_zRespectively an aileron, a rudder and an elevator; c_mx,C_my,C_mzRoll, yaw and pitch moment coefficients, respectively, with respect to α, β, δ_x,δ_y,δ_zAnd mach number Ma;

in the reentry process, BTT control is adopted, the sideslip angle is maintained near a zero value, sin beta is approximately equal to 0, tan beta is approximately equal to 0, and cos beta is approximately equal to 1;

step 2, rewriting the reentry aircraft model established in the step 1 into a MIMO affine nonlinear system form:

\overset{\cdot}{x} = f (x) + g (x) u - - - (3)

Ω=h(x)

wherein x = [ α β μ ω =_x ω_y ω_z]^TIs the state vector, Ω = [ α β μ =]^TIs a system output variable, u = [ M = [)_x M_y M_z]^TIs the calculated aerodynamic moment and control surface deflection angle instruction [ delta ]_x,δ_y,δ_z]^TCalculated by inverting the aerodynamic momentTo; f (x) = [ f₁(x)…f₆(x)]^TIs a 6 × 1 dimensional matrix, g (x) = [ g [)₁(x) g₂(x) g₃(x)]^TIs a 6 × 3 dimensional matrix, h (x) = [ h%₁(x) h₂(x) h₃(x)]^TIs a 3 x 1 dimensional matrix; wherein,

\{\begin{matrix} f_{1} (x) = ω_{z} \\ f_{2} (x) = ω_{x} \sin α + ω_{y} \cos α \\ f_{3} (x) = ω_{x} \cos α - ω_{y} \sin α \\ f_{4} (x) = - \frac{I_{yy} (I_{zz} - I_{yy}) - I_{xy}^{2}}{I^{*}} ω_{y} ω_{z} - \frac{I_{xy} (I_{yy} + I_{xx} - I_{zz})}{I^{*}} ω_{x} ω_{z} \\ f_{5} (x) = - \frac{I_{xx} (I_{xx} - I_{zz}) + I_{xy}^{2}}{I^{*}} ω_{x} ω_{z} + \frac{I_{xy} (I_{xx} + I_{yy} - I_{zz})}{I^{*}} ω_{y} ω_{z} \\ f_{6} (x) = - \frac{I_{yy} - I_{xx}}{I_{zz}} ω_{x} ω_{y} - \frac{I_{xy}}{I_{zz}} (ω_{y}^{2} - ω_{x}^{2}) \end{matrix},

\{\begin{matrix} g_{1} (x) = {[\begin{matrix} 0 & 0 & 0 & \frac{I_{yy}}{I^{*}} & \frac{I_{xy}}{I^{*}} & 0 \end{matrix}]}^{T} \\ g_{2} (x) = {[\begin{matrix} 0 & 0 & 0 & \frac{I_{xy}}{I^{*}} & \frac{I_{xx}}{I^{*}} & 0 \end{matrix}]}^{T} \\ g_{3} (x) = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & \frac{1}{I_{zz}} \end{matrix}]}^{T} \end{matrix};

and 3, aiming at the affine nonlinear system obtained in the step 2, applying a feedback linearization theory to carry out linearization treatment to obtain a three-channel linearization model of pitching, rolling and yawing:

\overset{\cdot \cdot}{Ω} = F (x) + E (x) U - - - (4)

in the formula,

F (x) = [\begin{matrix} f_{3} (x) \\ \sin α \cdot f_{1} (x) + \cos α \cdot f_{2} (x) + (ω_{x} \cos α - ω_{y} \sin α) \cdot ω_{z} \\ \cos α \cdot f_{1} (x) - \sin α \cdot f_{2} (x) - (ω_{x} \sin α + ω_{y} \cos α) \cdot ω_{z} \end{matrix}],

E (x) = [\begin{matrix} 0 & 0 & \frac{1}{I_{zz}} \\ \frac{I_{yy} \sin α}{I^{*}} + \frac{I_{xy} \cos α}{I^{*}} & \frac{I_{xy} \sin α}{I^{*}} + \frac{I_{xx} \cos α}{I^{*}} & 0 \\ \frac{I_{yy} \cos α}{I^{*}} - \frac{I_{xy} \sin α}{I^{*}} & \frac{I_{xy} \cos α}{I^{*}} - \frac{I_{xx} \sin α}{I^{*}} & 0 \end{matrix}],

U=[u₁ u₂ u₃]^T=[M_x M_y M_z]^T；

the selected control law form is:

U=E^-1(v-F)（5）

wherein v = [ v =₁ v₂ v₃]^TIs an auxiliary control amount;

substituting the control law into the linearization model to obtain a decoupled integrator in the form of:

\overset{\cdot \cdot}{Ω} = v - - - (6)

when there is parameter uncertainty and external disturbances in the affine nonlinear system of the reentry aircraft model, the feedback linearized system model is represented as:

\overset{\cdot \cdot}{Ω} = F + ΔF + (E + ΔE) U - - - (7)

Δv=[Δv₁ Δv₂ Δv₃]^Trepresents the polymerization disturbance: Δ v = Δ F + Δ EU; and Δ v is present_1max,Δv_2max,Δv_3maxSo that | Δ v₁|≤Δv_1max，|Δv₂|≤Δv_2max，|Δv₃|≤Δv_3max；

Substituting the selected control law form and the aggregate disturbance delta v into the system model subjected to feedback linearization to obtain a reentry aircraft feedback linearization system considering parameter uncertainty and disturbance, wherein the reentry aircraft feedback linearization system comprises the following steps:

\overset{\cdot \cdot}{Ω} = v + Δv - - - (8)

step 4, designing a self-adaptive index time-varying sliding mode controller aiming at the linearization system obtained in the step 3;

firstly, selecting an exponential time-varying sliding mode surface:

S (t) = \overset{\cdot}{\tilde{Ω}} + Λ \tilde{Ω} + A e^{- at} - - - (9)

in the formula,

for systematic tracking error, Ω_c=[α_c β_c μ_c]^TAttitude command given for guidance loop, s (t) = [ s ]_α(t)s_β(t)s_μ(t)]^T∈R³Is a sliding mode surface function vector, A belongs to R³For the parameter matrix related to the initial value of the system state, Λ = diag { λ }₁,λ₂,λ₃}∈R^3×3Representing the slope of the slip form surface, a ∈ R⁺Represents the approaching speed, lambda, of the time-varying slip form surface to the time-invariant slip form surface₁=λ₂=λ₃=a=λ；S(0)=0_3×1The value of A is:

A = - \overset{\cdot}{\tilde{Ω}} (0) - Λ \tilde{Ω} (0) = - Λ \tilde{Ω} (0) - - - (10)

then, designing a modified adaptive exponential time-varying sliding mode controller in the form of:

v = v_{eq} + v_{sw} = {\overset{\cdot \cdot}{Ω}}_{c} - Λ \overset{\cdot}{\tilde{Ω}} + Aλ e^{- λt} - ηsat (S (t)) - - - (11)

in the formula,

denotes equivalent control, v_sw= η sat (s (t))) indicates switching control, η = diag { η =_α,η_β,η_μThe gain is the switching gain of sliding mode control; the saturation function sat (-) and the handover gain adaptation algorithm are respectively expressed as:

{\overset{\cdot}{η}}_{j} = \frac{1}{k_{j}} (- σ_{j} η_{j} + | s_{j} (t) |), j = α, β, μ - - - (13)

wherein,

denotes the boundary layer thickness, σ_jIs a small positive constant, k_j>0 is the adaptive rate; eta_jAdaptive speed of (k)_jControl of (2);

step 5, obtaining a control moment instruction of attitude control of the reentry vehicle according to the step 4:

U＝E^-1(v-F)（14）

then, the control moment is distributed to the pneumatic control surface by combining the pneumatic moment to obtain a control surface deflection angle instruction [ delta ] required by attitude control_x δ_y δ_z]^T；

Step 6, the control surface deflection angle instruction [ delta ] obtained in the step 5 is processed_x δ_y δ_z]^TInputting the attitude data into a reentry vehicle to control the attitude; at the same time, the aircraft control system outputs real-time flight states (α, β, μ, ω)_x,ω_y，ω_z) And input to the attitude control system as a feedback state; repeating the steps 2-6, thereby realizing the existence of the systemUnder the condition of parameter uncertainty and external disturbance, the deflection angle [ delta ] of the control surface is controlled_x δ_y δ_z]^TAttitude command Ω given to guidance ring_c=[α_c β_c μ_c]^TAnd (6) tracking.

2. The reentry vehicle adaptive index time-varying sliding mode attitude control method according to claim 1, characterized in that: the origin O of the aircraft body coordinate system is taken at the mass center of the aircraft, the axis Ox is coincident with the longitudinal axis of the aircraft body, and the pointing head is positive; the Oy axis is positioned in the longitudinal symmetry plane of the machine body and is vertical to the Ox axis, and the pointing direction is positive; the Oz axis is perpendicular to the Oxy plane, and the direction is determined according to a right-hand rectangular coordinate system.

3. The reentry vehicle adaptive index time-varying sliding mode attitude control method according to claim 1, characterized in that: k in step 4_jIs constant, and k_α=k_β=k_μ=k。