CN105865272A

CN105865272A - Integrated control method used for semi-strapdown guided missile

Info

Publication number: CN105865272A
Application number: CN201610364808.7A
Authority: CN
Inventors: 易科; 陈建; 任章; 李清东; 晏涛; 贾晓洪; 吴军彪; 周卫文
Original assignee: Beihang University; Aviation Industry Corp of China AVIC; China Airborne Missile Academy
Current assignee: Beihang University; Aviation Industry Corp of China AVIC; China Airborne Missile Academy
Priority date: 2016-05-27
Filing date: 2016-05-27
Publication date: 2016-08-17
Anticipated expiration: 2036-05-27
Also published as: CN105865272B

Abstract

The invention discloses an integrated control method for a semi-strapdown guided missile, which is divided into the following steps: Step 1: Establish an integrated mathematical model for the stability tracking of the seeker and the attitude control of the missile body, including the kinematic model of the attitude of the missile body, Frame kinematics model, angle tracking system model, integrated mathematical model; Step 2: Design integrated controller configuration; Step 3: Design nonlinear dynamic inverse controller; Step 3: Controller design based on nonlinear dynamic inverse . According to the integrated model, the present invention designs an integrated control configuration based on inner and outer loops, solves the coupling problem between control loops, reduces the conservatism of control system design, and improves the comprehensive performance of the system.

Description

An integrated control method for semi-strapdown guided missile

技术领域technical field

本发明属于武器技术、导引头技术、控制方法领域，涉及空空导弹控制系统设计研究，具体涉及一种半捷联制导导弹的一体化控制方法。The invention belongs to the fields of weapon technology, seeker technology and control methods, and relates to the design and research of air-to-air missile control systems, in particular to an integrated control method for semi-strapdown guided missiles.

背景技术Background technique

导引头稳定跟踪控制是半捷联导引头控制的核心技术，为了研究半捷联稳定跟踪算法，需要了解弹体与伺服框架的动力学、运动学关系，并对半捷联姿态算法进行深入研究。传统的半捷联导引头稳定控制系统和弹体姿态控制系统设计，为便于分析考察各个分系统的性能，通常将导引头稳定跟踪控制系统、弹体姿态控制系统作为两个独立的部分，割裂开来分别进行研究，采用的是两回路独立设计思想。在这种分离设计思想下，通常都假设半捷联稳定控制系统与弹体姿态控制系统是可解耦的，这样就可以把问题分解为对两个低阶子系统的设计，大大降低了设计难度。Seeker stable tracking control is the core technology of semi-strapdown seeker control. In order to study the semi-strapdown stable tracking algorithm, it is necessary to understand the dynamics and kinematics relationship between the missile body and the servo frame, and carry out the semi-strapdown attitude algorithm study in depth. The traditional semi-strapdown seeker stability control system and projectile attitude control system are designed. In order to facilitate the analysis and investigation of the performance of each subsystem, the seeker stability tracking control system and projectile attitude control system are usually regarded as two independent parts. , separated to study separately, using the independent design idea of two loops. Under this separation design idea, it is usually assumed that the semi-strapdown stability control system and the missile attitude control system can be decoupled, so that the problem can be decomposed into the design of two low-order subsystems, which greatly reduces the design difficulty.

实际上，半捷联导引头的特殊结构使得弹体与导引头稳定平台框架之间耦合严重。当目标进行大机动时，对于依靠气动舵来实现法向过载控制的半捷联成像导弹而言，导弹需要大的姿态调整以获得大的过载来应对目标的大机动。如果半捷联导引头稳定跟踪控制系统的响应不够及时，导弹姿态调整在半捷联体制下可能会导致目标脱离导引头视场，尤其在制导末段随着弹目相对距离的接近，这种现象更为严重。这主要是由于寄生回路的存在，使得导引头稳定跟踪控制回路和弹体姿态控制回路产生严重耦合。因此，半捷联红外成像制导系统要实现对目标的稳定跟踪，需要通过半捷联导引头稳定跟踪控制系统和弹体姿态控制系统共同协调控制来实现，才能从根本上解决半捷联导引头的工程应用问题。In fact, the special structure of the semi-strapdown seeker makes the coupling between the missile body and the stabilizer platform frame of the seeker serious. When the target performs a large maneuver, for the semi-strapdown imaging missile that relies on the aerodynamic rudder to achieve normal overload control, the missile needs a large attitude adjustment to obtain a large overload to cope with the large maneuver of the target. If the response of the semi-strapdown seeker stabilization tracking control system is not timely enough, the missile attitude adjustment may cause the target to leave the seeker's field of view under the semi-strapdown system, especially at the end of the guidance as the relative distance of the missile and target approaches, This phenomenon is more serious. This is mainly due to the existence of the parasitic loop, which causes serious coupling between the seeker stability tracking control loop and the projectile attitude control loop. Therefore, in order to realize the stable tracking of the target by the semi-strapdown infrared imaging guidance system, it needs to be realized through the coordinated control of the semi-strapdown seeker stable tracking control system and the projectile attitude control system, so as to fundamentally solve the problem of semi-strapdown guidance. Leading engineering application problems.

发明内容Contents of the invention

本发明的目的是为了解决上述问题，采用了一体化设计方案，通过建立半捷联导引头与弹体姿态控制一体化模型，使用非线性动态逆控制技术，达到了两个分系统之间的协调控制效果，具有弹体姿态响应速度快、导引头跟踪误差小等优势，对半捷联导弹控制系统的工程设计具有指导性意义。The purpose of the present invention is to solve the above-mentioned problems, adopting an integrated design scheme, by establishing an integrated model of semi-strapdown seeker and projectile attitude control, and using nonlinear dynamic inverse control technology to achieve a balance between the two subsystems. It has the advantages of fast missile attitude response and small seeker tracking error, which has guiding significance for the engineering design of the semi-strapdown missile control system.

本发明的一种用于半捷联制导导弹的一体化控制方法，分以下步骤：A kind of integrated control method that is used for semi-strapdown guided missile of the present invention, divides following steps:

步骤1：建立导引头稳定跟踪与弹体姿态控制一体化数学模型，包括弹体姿态运动学模型、框架运动学模型、角跟踪系统模型、一体化数学模型；Step 1: Establish the integrated mathematical model of seeker stability tracking and projectile attitude control, including projectile attitude kinematics model, frame kinematics model, angle tracking system model, and integrated mathematical model;

步骤2：设计一体化控制器构型；Step 2: Design the configuration of the integrated controller;

步骤3：基于非线性动态逆的控制器设计。Step 3: Controller design based on nonlinear dynamic inverse.

本发明通过以上三个步骤，即首先建立一体化模型，然后在此基础上设计基于内外环的一体化控制器构型，最后分别设计内外环控制回路的非线性动态逆控制律。最终解决导引头稳定跟踪控制回路与弹体姿态控制回路之间的耦合问题，提高了半捷联制导导弹的控制效果。Through the above three steps, the present invention first establishes an integrated model, then designs an integrated controller configuration based on inner and outer loops on this basis, and finally designs nonlinear dynamic inverse control laws of inner and outer loop control loops respectively. Finally, the coupling problem between the seeker stability tracking control loop and the body attitude control loop is solved, and the control effect of the semi-strapdown guided missile is improved.

本发明的优点在于：The advantages of the present invention are:

(1)通过分析了导引头稳定跟踪控制回路与弹体姿态控制回路之间的耦合关系，在这基础上建立了导引头稳定跟踪与弹体姿态控制一体化数学模型，为后续一体化控制器设计提供了数学模型与理论基础；(1) By analyzing the coupling relationship between the seeker stable tracking control loop and the missile attitude control loop, on this basis, the integrated mathematical model of the seeker stable tracking and missile attitude control is established, which will be used for the subsequent integration The controller design provides a mathematical model and a theoretical basis;

(2)根据一体化模型，设计的基于内外环的一体化控制构型，解决了控制回路之间的耦合问题，降低了控制系统设计的保守性，提高了系统的综合性能；(2) According to the integrated model, the integrated control configuration based on the inner and outer loops is designed, which solves the coupling problem between the control loops, reduces the conservatism of the control system design, and improves the comprehensive performance of the system;

(3)采用动态逆控制理论设计的控制律，该控制律能够保证导引头稳定跟踪与弹体姿态一体化模型的快速性、收敛性与稳定性。(3) The control law designed by dynamic inverse control theory can ensure the rapidity, convergence and stability of the integrated model of seeker stable tracking and projectile attitude.

附图说明Description of drawings

图1为本发明的方法流程图；Fig. 1 is method flowchart of the present invention;

图2为弹体坐标系与探测坐标系之间的关系图；Fig. 2 is a diagram of the relationship between the projectile coordinate system and the detection coordinate system;

图3为半捷联导引头光轴指向空间几何关系图；Fig. 3 is a diagram of the spatial geometric relationship of the optical axis of the semi-strapdown seeker;

图4为双回路非线性动态逆一体化控制器框图；Fig. 4 is a block diagram of a dual-loop nonlinear dynamic inverse integrated controller;

图5为非线性动态逆控制流程示意图；Fig. 5 is a schematic diagram of a nonlinear dynamic inverse control flow;

图6为α、β、γ_v的跟踪误差图；Figure 6 is the tracking error diagram of α, β, γ _v ;

图7为实际舵偏角图；Figure 7 is the actual rudder deflection angle diagram;

图8为实际失调角图；Figure 8 is the actual misalignment angle diagram;

图9为实际框架角速度意图；Figure 9 is the actual frame angular velocity diagram;

图10为实际框架角图。Figure 10 is a corner view of the actual frame.

具体实施方式detailed description

下面将结合附图和实施例对本发明作进一步的详细说明。The present invention will be further described in detail with reference to the accompanying drawings and embodiments.

本发明采用了一体化设计方法，通过建立半捷联导引头与弹体姿态控制一体化模型，使用非线性动态逆控制技术，达到了两个分系统之间的协调控制效果，具有弹体姿态响应速度快、导引头跟踪误差小等优势，对半捷联导弹控制系统的工程设计具有指导性意义。The present invention adopts an integrated design method, establishes an integrated model of semi-strapdown seeker and projectile attitude control, and uses nonlinear dynamic inverse control technology to achieve a coordinated control effect between the two sub-systems. The advantages of fast attitude response and small seeker tracking error have guiding significance for the engineering design of the semi-strapdown missile control system.

本发明是一种用于半捷联制导导弹的一体化控制方法，流程如图1所示，包括以下几个步骤：The present invention is an integrated control method for semi-strapdown guided missiles, the process flow shown in Figure 1, including the following steps:

步骤1：建立导引头稳定跟踪与弹体姿态控制一体化数学模型，包括弹体姿态运动学模型、框架运动学模型、角跟踪系统模型、一体化数学模型。Step 1: Establish the integrated mathematical model of seeker stability tracking and projectile attitude control, including projectile attitude kinematics model, frame kinematics model, angle tracking system model, and integrated mathematical model.

(1)弹体姿态运动学模型(1) Body posture kinematics model

在弹目相对运动中，导弹根据制导指令给出需用过载指令，计算需用过载所需的α、β、γ_v可以直接由动力学关系求得。省略中间推导过程，直接给出导弹气流角微分方程(α、β、γ_v的微分方程)：In the relative motion of the projectile and the target, the missile gives the required overload command according to the guidance command, and the α, β, and γ _v required to calculate the required overload can be obtained directly from the dynamic relationship. The intermediate derivation process is omitted, and the missile airflow angle differential equation (differential equation of α, β, γ _v ) is directly given:

$[\begin{matrix} \overset{\cdot &Center Dot;}{α α} \\ \overset{\cdot &Center Dot;}{β β} \\ {\overset{\cdot \cdot}{γ γ}}_{v v} \end{matrix}] = = [\begin{matrix} \frac{{mgcosγ mgcosγ}_{v v} cos cos θ θ - - L L}{m m V V cos cos β β} \\ \frac{{mgsinγ mgsinγ}_{v v} cos cos θ θ + + Y Y}{m m V V} \\ - - {\overset{\cdot &Center Dot;}{ψ ψ}}_{v v} sin sin θ θ - - \frac{(({mgcosγ mgcosγ}_{v v} cos cos θ θ - - L L)) tan the tan β β}{m m V V} \end{matrix}] + + [\begin{matrix} - - cos cos α α tan the tan β β & sin sin α α tan the tan β β & 11 \\ sin sin α α & cos cos α α & 00 \\ cos cos α α sec sec β β & - - sin sin α α sec sec β β & 00 \end{matrix}] [\begin{matrix} {ω ω}_{m m x x} \\ {ω ω}_{m m y the y} \\ {ω ω}_{m m z z} \end{matrix}] - - - - - - ((11))$

式中，α,β,γ_v分别为攻角、侧滑角、倾侧角；θ,ψ_v分别为弹道倾角、弹道偏角；L,Y分别为升力和侧向力；ω_mx,ω_my,ω_mz为弹体体轴姿态角速度；m为导弹质量，g为重力加速度，V为导弹飞行速度。In the formula, α, β, γ _v are attack angle, sideslip angle, roll angle respectively; θ, ψ _v are ballistic inclination angle, ballistic deflection angle respectively; L, Y are lift force and lateral force respectively; ω _mx , ω _my , ω _mz is the attitude angular velocity of the projectile body axis; m is the mass of the missile, g is the acceleration of gravity, and V is the flight speed of the missile.

战术导弹的外形一般都是轴对称的，导弹对弹体坐标系各轴的惯量积为零。可列出姿态角速度微分方程如下：The shape of tactical missiles is generally axisymmetric, and the product of inertia of the missile to each axis of the missile body coordinate system is zero. The attitude angular velocity differential equation can be listed as follows:

$[\begin{matrix} {\overset{\cdot &Center Dot;}{ω ω}}_{m m x x} \\ {\overset{\cdot &Center Dot;}{ω ω}}_{m m y the y} \\ {\overset{\cdot &Center Dot;}{ω ω}}_{m m z z} \end{matrix}] = = [\begin{matrix} \frac{{M m}_{x x 00}}{{I I}_{x x}} - - \frac{(({I I}_{z z} {I I}_{y the y})) {ω ω}_{m m z z} {ω ω}_{m m y the y}}{{I I}_{x x}} \\ \frac{{M m}_{y the y 00}}{{I I}_{y the y}} - - \frac{(({I I}_{x x} - - {I I}_{z z})) {ω ω}_{m m x x} {ω ω}_{m m z z}}{{I I}_{y the y}} \\ \frac{{M m}_{z z 00}}{{I I}_{z z}} - - \frac{(({I I}_{y the y} - - {I I}_{x x})) {ω ω}_{m m y the y} {ω ω}_{m m x x}}{{I I}_{z z}} \end{matrix}] + + [\begin{matrix} \frac{{QSL QSL}_{r r} {m m}_{x x}^{{δ δ}_{x x}}}{{I I}_{x x}} & 00 & 00 \\ 00 & \frac{{QSL QSL}_{r r} {m m}_{y the y}^{{δ δ}_{y the y}}}{{I I}_{y the y}} & 00 \\ 00 & 00 & \frac{{QSL QSL}_{r r} {m m}_{z z}^{{δ δ}_{z z}}}{{I I}_{z z}} \end{matrix}] [\begin{matrix} {δ δ}_{x x} \\ {δ δ}_{y the y} \\ {δ δ}_{z z} \end{matrix}] - - - - - - ((22))$

式中，M_x0,M_y0,M_z0为零舵偏状态下的气动力矩；为滚转力矩系数对δ_x的导数，为滚转力矩系数对δ_y的导数，为滚转力矩系数对δ_z的导数；动压Q＝0.5ρV²，ρ为空气密度；S为导弹参考面积；L_r为导弹参考长度；δ_x,δ_y,δ_z为滚转、偏航、俯仰三通道的舵偏角；I_x,I_y,I_z分别为导弹三个方向的转动惯量。In the formula, M _x0 , M _y0 , M _z0 are the aerodynamic moments in the state of zero rudder deviation; is the derivative of the rolling moment coefficient to δ _x , is the derivative of the rolling moment coefficient to δ _y , is the derivative of rolling moment coefficient to δ _z ; dynamic pressure Q=0.5ρV ² , ρ is air density; S is missile reference area; L _r is missile reference length; δ _x , δ _y , δ _z are roll, deflection are the rudder deflection angles of the three channels of navigation and pitch; I _x , I _y , and I _z are the moments of inertia of the missile in three directions, respectively.

(2)框架运动学模型(2) Frame kinematics model

导引头光轴内框安装在外框上，外框架基座与弹体固联。根据刚体运动学原理，导引头光轴的空间运动是基座的运动与框架转动的复合运动，外框的运动是基座运动与外框自身转动的合成，内框的运动是外框耦合运动与内框自身转动共同引起。弹体的姿态运动通过几何约束和摩擦耦合到导引头运动中，中间存在复杂的几何运动关系传递，如图2所示。The inner frame of the optical axis of the seeker is installed on the outer frame, and the base of the outer frame is firmly connected with the missile body. According to the principle of rigid body kinematics, the spatial motion of the optical axis of the seeker is the compound motion of the base motion and the frame rotation, the motion of the outer frame is the synthesis of the base motion and the rotation of the outer frame itself, and the motion of the inner frame is the coupling of the outer frame The movement is caused jointly with the rotation of the inner frame itself. The attitude motion of the projectile is coupled to the seeker motion through geometric constraints and friction, and there is a complex geometric motion relationship in the middle, as shown in Figure 2.

导引头光轴中心在空间中的运动为：The movement of the optical axis center of the seeker in space is:

ω_d＝ω_dm+ω_ds (3) _ωd = _ωdm + _ωds (3)

式中，In the formula,

${ω ω}_{d d m m} = = [\begin{matrix} {ω ω}_{m m x x} {cosλ cosλ}_{z z} {cosλ cosλ}_{y the y} + + {ω ω}_{m m y the y} {sinλ sinλ}_{z z} - - {ω ω}_{m m z z} {cosλ cosλ}_{z z} {sinλ sinλ}_{y the y} \\ - - {ω ω}_{m m x x} {sinλ sinλ}_{z z} {cosλ cosλ}_{y the y} + + {ω ω}_{m m y the y} {cosλ cosλ}_{z z} + + {ω ω}_{m m z z} {sinλ sinλ}_{z z} {sinλ sinλ}_{y the y} \\ {ω ω}_{m m x x} {sinλ sinλ}_{y the y} + + {ω ω}_{m m z z} {cosλ cosλ}_{y the y} \end{matrix}] {ω ω}_{d d s the s} = = [\begin{matrix} {\overset{\cdot &Center Dot;}{λ λ}}_{y the y} {sinλ sinλ}_{z z} \\ {\overset{\cdot &Center Dot;}{λ λ}}_{y the y} {cosλ cosλ}_{z z} \\ {\overset{\cdot \cdot}{λ λ}}_{z z} \end{matrix}]$

其中，ω_d为光轴角速度在探测坐标系中的投影，ω_dm为弹体角速度在探测坐标系中的投影，ω_ds为导引头伺服框架角速度在探测坐标系中的投影，ω_mx,ω_my,ω_mz为弹体体轴角速度，λ_y,λ_z为半捷联稳定平台的内外框架角。Among them, ω _d is the projection of the angular velocity of the optical axis in the detection coordinate system, ω _dm is the projection of the projectile angular velocity in the detection coordinate system, ω _ds is the projection of the seeker servo frame angular velocity in the detection coordinate system, ω _mx , ω _my , ω _mz are the angular velocities of the projectile body axis, λ _y , λ _z are the inner and outer frame angles of the semi-strapdown stabilized platform.

(3)角跟踪系统模型(3) Angle tracking system model

为了简单描述，以二维空间为例，导引头空间角关系如图3示，q为弹目视线角，λ为框架角，θ为弹体姿态角，为光轴与基准线夹角，ε为光轴与目标视线的夹角，即失调角。For a simple description, taking two-dimensional space as an example, the spatial angle relationship of the seeker head is shown in Figure 3, q is the line of sight angle of the projectile, λ is the frame angle, and θ is the attitude angle of the projectile. is the angle between the optical axis and the reference line, ε is the angle between the optical axis and the target line of sight, that is, the misalignment angle.

所谓角跟踪系统，通过控制导引头上的滚转和俯仰两个方向的框架转动，使得导引头光轴指向始终跟踪弹目视线方向，亦即最终目的是使导引头光轴与弹目视线重合。但由于目标一直处于机动状态，因此导引头光轴与弹目视线角之间总存在偏差，这个偏差也就是失调角。光轴沿线与弹目视线连线之间的误差角可以通过红外成像导引头测量获得。根据跟踪原理，采用小角度近似并忽略三阶小项，可得到三维跟踪角误差微分方程：The so-called angle tracking system, by controlling the frame rotation on the seeker head in the two directions of roll and pitch, makes the optical axis of the seeker head always track the line of sight of the projectile, that is, the ultimate goal is to make the optical axis of the seeker The lines of sight coincide. However, since the target is always in a maneuvering state, there is always a deviation between the optical axis of the seeker and the line of sight angle of the projectile, and this deviation is also the misalignment angle. The error angle between the line along the optical axis and the line of sight of the projectile can be obtained by measuring the infrared imaging seeker. According to the tracking principle, the three-dimensional tracking angle error differential equation can be obtained by using small angle approximation and ignoring the third-order small term:

$\{\begin{matrix} {\overset{\cdot \cdot}{ϵ ϵ}}_{y the y} = = {ω ω}_{y the y} - - {ω ω}_{d d y the y} + + {ω ω}_{d d x x} {ϵ ϵ}_{z z} \\ {\overset{\cdot &Center Dot;}{ϵ ϵ}}_{z z} = = {ω ω}_{z z} - - {ω ω}_{d d z z} - - {ω ω}_{d d x x} {ϵ ϵ}_{y the y} \end{matrix} - - - - - - ((44))$

式中，ε_y,ε_z为失调角，ω_y,ω_z为视线角速度，ω_dx,ω_dy,ω_dz为光轴角速度。In the formula, ε _y , ε _z are misalignment angles, ω _y , ω _z are line-of-sight angular velocities, ω _dx , ω _dy , ω _dz are optical axis angular velocities.

(4)导引头稳定跟踪与弹体姿态控制一体化数学模型(4) Integrated mathematical model of seeker stability tracking and projectile attitude control

本发明提出姿态控制与导引头稳定跟踪控制进行一体化设计方法。根据制导指令、导引头测量的弹目相对运动信息以及惯导元件测量的弹体姿态信息和导引头框架角位置测量信息，通过控制律，直接输出舵偏角改变弹体姿态产生机动过载来打击目标，同时驱动导引头框架运动，控制光轴实现对目标的准确跟踪。无论这种映射关系是线性的，还是非线性的，都可通过对制导指令、相对运动测量信息进行跟踪来达到同时协调控制弹体姿态产生可用过载、以及令光轴稳定指向目标的目的。The invention proposes an integrated design method for attitude control and seeker head stability tracking control. According to the guidance command, the relative motion information of the projectile measured by the seeker, the attitude information of the missile body measured by the inertial navigation element and the angular position measurement information of the seeker frame, through the control law, the rudder deflection angle is directly output to change the attitude of the missile body to generate maneuvering overload To hit the target, and at the same time drive the seeker frame to move, control the optical axis to achieve accurate tracking of the target. Regardless of whether the mapping relationship is linear or nonlinear, the purpose of coordinating and controlling the attitude of the missile body to generate available overload and stably pointing the optical axis to the target can be achieved by tracking the guidance command and relative motion measurement information.

在前三步的基础上建立半捷联导引头控制与姿态控制一体化数学模型，将一体化模型写成如下级联仿射非线性系统：On the basis of the first three steps, the integrated mathematical model of semi-strapdown seeker control and attitude control is established, and the integrated model is written as the following cascaded affine nonlinear system:

$\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {f f}_{11} (({x x}_{11})) + + {g g}_{11} (({x x}_{11})) [\begin{matrix} {x x}_{22} \\ {u u}_{11} \end{matrix}] \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = {f f}_{22} (({x x}_{22})) + + {g g}_{22} (({x x}_{22})) {u u}_{22} \end{matrix} - - - - - - ((55))$

式中，In the formula,

x₁＝[α β γ_v ε_y ε_z]^T， x ₁ =[α β γ _v ε _y ε _z ] ^T ,

x₂＝[ω_mx ω_my ω_mz]^T，u₂＝[δ_x δ_y δ_z]^T，x ₂ =[ω _mx ω _my ω _mz ] ^T , u ₂ =[δ _x δ _y δ _z ] ^T ,

${f f}_{11} (({x x}_{11})) = = [\begin{matrix} \frac{{mgcosγ mgcosγ}_{v v} cos cos θ θ - - L L}{m m V V cos cos β β} \\ \frac{{mgsinγ mgsinγ}_{v v} cos cos θ θ + + Y Y}{m m V V} \\ - - {\overset{\cdot &Center Dot;}{ψ ψ}}_{v v} sin sin θ θ - - \frac{(({mgcosγ mgcosγ}_{v v} cos cos θ θ - - L L)) tan the tan β β}{m m V V} \\ {ω ω}_{y the y} \\ {ω ω}_{z z} \end{matrix}],, \begin{matrix} {g g}_{11} ((x x)) = = [\begin{matrix} - - cos cos α α tan the tan β β & sin sin α α tan the tan β β \\ sin sin α α & cos cos α α \\ cos cos α α sec sec β β & - - sin sin α α sec sec β β \\ {sinλ sinλ}_{z z} {cosλ cosλ}_{y the y} + + {ϵ ϵ}_{z z} {cosλ cosλ}_{z z} {cosλ cosλ}_{y the y} & - - {cosλ cosλ}_{z z} + + {ϵ ϵ}_{z z} {sinλ sinλ}_{z z} \\ - - (({sinλ sinλ}_{y the y} + + {ϵ ϵ}_{y the y} {cosλ cosλ}_{z z} {cosλ cosλ}_{y the y})) & - - {ϵ ϵ}_{y the y} {sinλ sinλ}_{z z} \end{matrix} \\ \begin{matrix} 11 & 00 & 00 \\ 00 & 00 & 00 \\ 00 & 00 & 00 \\ - - (({sinλ sinλ}_{z z} {sinλ sinλ}_{y the y} + + {ϵ ϵ}_{z z} {cosλ cosλ}_{z z} {sinλ sinλ}_{y the y})) & - - {cosλ cosλ}_{z z} + + {ϵ ϵ}_{z z} {sinλ sinλ}_{z z} & 00 \\ - - {cosλ cosλ}_{y the y} + + {ϵ ϵ}_{y the y} {cosλ cosλ}_{z z} {sinλ sinλ}_{y the y} & - - {ϵ ϵ}_{y the y} {sinλ sinλ}_{z z} & - - 11 \end{matrix}] \end{matrix},,$

${f f}_{22} ((x x)) = = [\begin{matrix} \frac{{M m}_{x x 00}}{{I I}_{x x}} - - \frac{(({I I}_{z z} - - {I I}_{y the y})) {ω ω}_{m m z z} {ω ω}_{m m y the y}}{{I I}_{x x}} \\ \frac{{M m}_{y the y 00}}{{I I}_{y the y}} - - \frac{(({I I}_{x x} - - {I I}_{z z})) {ω ω}_{m m x x} {ω ω}_{m m z z}}{{I I}_{y the y}} \\ \frac{{M m}_{z z 00}}{{I I}_{z z}} - - \frac{(({I I}_{y the y} - - {I I}_{x x})) {ω ω}_{m m y the y} {ω ω}_{m m x x}}{{I I}_{z z}} \end{matrix}],, {g g}_{22} ((x x)) = = [\begin{matrix} \frac{{QSL QSL}_{r r} {m m}_{x x}^{{δ δ}_{x x}}}{{I I}_{x x}} & 00 & 00 \\ 00 & \frac{{QSL QSL}_{r r} {m m}_{y the y}^{{δ δ}_{y the y}}}{{I I}_{y the y}} & 00 \\ 00 & 00 & \frac{{QSL QSL}_{r r} {m m}_{z z}^{{δ δ}_{z z}}}{{I I}_{z z}} \end{matrix}] . .$

步骤2：一体化控制器构型设计Step 2: Integrated controller configuration design

基于奇异摄动理论设计导引头稳定跟踪与弹体姿态一体化控制构型如图4所示。制导系统给出的气流角指令α_c,β_c,γ_vc，框架角位置测量传感器测得的框架角位置λ_y,λ_z，弹目视线角q_y,q_z经过滤波器得到的弹目视线转率ω_y,ω_z，以及导引头接收机得到的跟踪角误差ε_y,ε_z，通过外环控制器输出框架角速度控制信号ω_λyc,ω_λzc，控制光轴指向实时跟踪弹目视线，同时给出姿态角速度的伪控制量ω_mxc,ω_myc,ω_mzc，再送给内环姿态角速度控制系统，控制气动舵实现对ω_mxc,ω_myc,ω_mzc的快速跟踪。Based on the singular perturbation theory, the integrated control configuration of seeker stability tracking and projectile attitude is designed as shown in Fig. 4. The airflow angle command α _c , β _c , γ _vc given by the guidance system, the frame angular position λ _y , λ _z measured by the frame angular position measurement sensor, and the projectile sight angle q _y , q _z obtained through the filter The line-of-sight rotation rate ω _y , ω _z , and the tracking angle error ε _y , ε _z obtained by the seeker receiver, output the frame angular velocity control signal ω _λyc , ω _λzc through the outer loop controller, and control the optical axis to point to the real-time tracking target At the same time, the pseudo control variables of attitude angular velocity ω _mxc , ω _myc , ω _{mzc are given} , and then sent to the inner ring attitude angular velocity control system to control the aerodynamic rudder to realize fast tracking of ω _mxc , ω _myc , ω _mzc .

步骤3：基于非线性动态逆的控制器设计Step 3: Controller design based on nonlinear dynamic inverse

设计动态逆控制律：Design a dynamic inverse control law:

对如下仿射非线性系统：For the following affine nonlinear system:

$\{\begin{matrix} \overset{\cdot &Center Dot;}{x x} = = f f ((x x)) + + g g ((x x)) u u \\ y the y = = h h ((x x)) \end{matrix} - - - - - - ((66))$

设计非线性动态逆控制器：Design a nonlinear dynamic inverse controller:

$u u = = {g g}^{- - 11} ((x x)) [[{\overset{\cdot \cdot}{x x}}_{d d e e s the s} - - f f ((x x))]] - - - - - - ((77))$

式中，期望的动态这里又称为伪控制量，记为关于伪控制量的选取，本发明采用如下方式选取：where the desired dynamic Here is also called the pseudo-control quantity, denoted as Regarding the selection of the pseudo control quantity, the present invention adopts the following method to select:

$v v = = {\overset{\cdot &Center Dot;}{x x}}_{d d e e s the s} = = {K K}_{c c} (({x x}_{c c} - - x x)) - - - - - - ((88))$

图5简要说明了动态逆的实现过程。Figure 5 briefly illustrates the implementation process of the dynamic inverse.

下面设计一体化模型的内外环回路进行非线性动态控制器。Next, the inner and outer loops of the integrated model are designed for nonlinear dynamic controller.

(1)设计外环动态逆控制器(1) Design the outer loop dynamic inverse controller

定义外回路的状态变量为[α β γ_v ε_y ε_z]^T，输入控制变量为，[ω_mx ω_my ω_mzω_λy ω_λz]^T。设计外环动态逆控制律如下：Define the state variable of the outer loop as [α β γ _v ε _y ε _z ] ^T , and the input control variable as [ω _mx ω _my ω _mz ω _λy ω _λz ] ^T . The dynamic inverse control law of the outer loop is designed as follows:

$[\begin{matrix} {ω ω}_{m m x x c c} \\ {ω ω}_{m m y the y c c} \\ {ω ω}_{m m z z c c} \\ {ω ω}_{λ λ y the y} \\ {ω ω}_{λ λ z z} \end{matrix}] = = {g g}_{11}^{- - 11} (({x x}_{11})) (([\begin{matrix} {v v}_{α α} \\ {v v}_{β β} \\ {v v}_{{γ γ}_{v v}} \\ {v v}_{{ϵ ϵ}_{y the y}} \\ {v v}_{{ϵ ϵ}_{z z}} \end{matrix}] - - f f (({x x}_{11})))) - - - - - - ((99))$

式中：v_α,v_β,分别为状态变量α,β,γ_v,ε_y,ε_z的期望动态形式如式(8)，ω_mxc,ω_myc,ω_mzc为内环期望的弹体姿态角速度指令，作为内环的伪控制量。这样就得到了外回路的动态逆控制器形式。Where: v _α , v _β , are the expected dynamic forms of the state variables α, β, γ _v , ε _y , ε _z respectively, as shown in Equation (8), ω _mxc , ω _myc , ω _mzc are the desired attitude angular velocity commands of the inner loop, as the pseudo Control amount. In this way, the dynamic inverse controller form of the outer loop is obtained.

(2)设计内环动态逆控制器(2) Design the inner loop dynamic inverse controller

内回路控制指令为外回路伪控制量ω_mxc,ω_myc,ω_mzc，输入控制变量为δ_x,δ_y,δ_z。设计内环动态控制律如下：The inner loop control command is the outer loop pseudo control quantity ω _mxc , ω _myc , ω _mzc , and the input control variable is δ _x , δ _y , δ _z . The dynamic control law of the inner loop is designed as follows:

$[\begin{matrix} {δ δ}_{x x} \\ {δ δ}_{y the y} \\ {δ δ}_{z z} \end{matrix}] = = {g g}_{22}^{- - 11} (({x x}_{22})) (([\begin{matrix} {v v}_{{ω ω}_{m m x x}} \\ {v v}_{{ω ω}_{m m y the y}} \\ {v v}_{{ω ω}_{m m z z}} \end{matrix}] - - f f (({x x}_{22})))) - - - - - - ((1010))$

式中：分别为状态变量ω_mxc,ω_myc,ω_mzc的期望动态形式如式(8)。In the formula: are respectively the expected dynamic forms of the state variables ω _mxc , ω _myc , ω _mzc as in formula (8).

以上两步得到了基于内外回路非线性动态逆设计方法的导引头稳定跟踪与弹体姿态一体化控制器。In the above two steps, an integrated controller based on the nonlinear dynamic inverse design method of the inner and outer loops is obtained for the seeker's stable tracking and projectile attitude.

仿真案例Simulation case

设弹目视线角偏角转率为频率0.1Hz，幅值5°/s的余弦曲线，弹目视线倾角转率为频率0.1Hz，幅值10°/s的余弦曲线。攻角、侧滑角、滚转角的指令在仿真初始时刻给15°,10°和5°的阶跃信号。仿真初始参数如表1所示。It is assumed that the deflection angle rotation rate of the bullet-eye sight angle is a cosine curve with a frequency of 0.1Hz and an amplitude of 5°/s, and the rotation rate of the bullet-eye sight angle is a cosine curve with a frequency of 0.1Hz and an amplitude of 10°/s. The commands of angle of attack, angle of sideslip and roll angle give 15°, 10° and 5° step signals at the initial moment of simulation. The initial parameters of the simulation are shown in Table 1.

表1仿真初始条件Table 1 Simulation initial conditions

仿真算法采用定步长四阶龙格库塔法，仿真步长为1ms,仿真时间为5s。仿真结果如图6、图7、图8、图9、图10所示。The simulation algorithm adopts the fourth-order Runge-Kutta method with fixed step size, the simulation step size is 1ms, and the simulation time is 5s. The simulation results are shown in Figure 6, Figure 7, Figure 8, Figure 9, and Figure 10.

从图6可以看出，攻角、侧滑角、速度滚转角α、β、γ_v的跟踪误差在0.7s内收敛，偏差接近于0；从图7可以看出，由于初始误差比较大，导致初始时刻舵面的偏角较大，但随着跟踪误差的减小，舵偏角逐渐减小到正常工作区间。这表明设计的一体化控制器对弹体姿态的控制具有满意的动态效果和较小的跟踪误差。It can be seen from Fig. 6 that the tracking errors of angle of attack, sideslip angle, and speed roll angle α, β, and γ _v converge within 0.7s, and the deviation is close to 0; it can be seen from Fig. 7 that due to the relatively large initial error, As a result, the deflection angle of the rudder surface is relatively large at the initial moment, but as the tracking error decreases, the deflection angle of the rudder gradually decreases to the normal working range. This shows that the designed integrated controller has satisfactory dynamic effect and small tracking error on the attitude control of the projectile.

从图8、图9、图10可以看出，导引头失调角在0.7s内收敛趋近于0，位标器能快速准确的跟踪目标，目标位于导引头光轴的中心。整个稳定跟踪过程中，框架角瞬时速率最大为150deg/s，框架角最大为40deg,这都是符合导引头伺服框架设计要求的。考虑了弹体姿态运动与位标器运动之间耦合的一体化控制能够保证导引头光轴对目标的快速跟踪，具有较小的失调角。It can be seen from Fig. 8, Fig. 9, and Fig. 10 that the misalignment angle of the seeker converges to 0 within 0.7s, and the position marker can track the target quickly and accurately, and the target is located in the center of the optical axis of the seeker. During the whole stable tracking process, the maximum instantaneous rate of the frame angle is 150deg/s, and the maximum frame angle is 40deg, which is in line with the design requirements of the seeker servo frame. The integrated control considering the coupling between the projectile attitude motion and the marker motion can ensure the optical axis of the seeker to track the target quickly, with a small misalignment angle.

Claims

1. An integrated control method for semi-strapdown guided missiles, comprising the following steps:

Step 1: Establish the integrated mathematical model of seeker stability tracking and projectile attitude control, including projectile attitude kinematics model, frame kinematics model, angle tracking system model, and integrated mathematical model;

(1) Kinematics model of projectile attitude;

The differential equation of missile airflow angle is:

[\begin{matrix} \overset{\cdot &Center Dot;}{α α} \\ \overset{\cdot &Center Dot;}{β β} \\ {\overset{\cdot &Center Dot;}{γ γ}}_{v v} \end{matrix}] = = [\begin{matrix} \frac{m m g g {cosγ cosγ}_{v v} c c o o s the s θ θ - - L L}{m m V V cos cos β β} \\ \frac{m m g g {sinγ sinγ}_{v v} c c o o s the s θ θ + + Y Y}{m m V V} \\ - - {\overset{\cdot &Center Dot;}{ψ ψ}}_{v v} s the s i i n no θ θ - - \frac{((m m g g {cosγ cosγ}_{v v} c c o o s the s θ θ - - L L)) t t a a n no β β}{m m V V} \end{matrix}] + + [\begin{matrix} - - c c o o s the s α α t t a a n no β β & sin sin α α t t a a n no β β & 11 \\ sin sin α α & c c o o s the s α α & 00 \\ cos cos α α sec sec β β & - - sin sin α α sec sec β β & 00 \end{matrix}] [\begin{matrix} {ω ω}_{m m x x} \\ {ω ω}_{m m y the y} \\ {ω ω}_{m m z z} \end{matrix}] - - - - - - ((11))

In the formula, α, β, γ _v are attack angle, sideslip angle, roll angle respectively; θ, ψ _v are ballistic inclination angle, ballistic deflection angle respectively; L, Y are lift force and lateral force respectively; ω _mx , ω _my , ω _mz is the attitude angular velocity of the projectile body axis; m is the mass of the missile, g is the acceleration of gravity, and V is the flight speed of the missile;

The attitude angular velocity differential equation is:

[\begin{matrix} {\overset{\cdot \cdot}{ω ω}}_{m m x x} \\ {\overset{\cdot \cdot}{ω ω}}_{m m y the y} \\ {\overset{\cdot &Center Dot;}{ω ω}}_{m m z z} \end{matrix}] = = [\begin{matrix} \frac{{M m}_{x x 00}}{{I I}_{x x}} - - \frac{(({I I}_{z z} - - {I I}_{y the y})) {ω ω}_{m m z z} {ω ω}_{m m y the y}}{{I I}_{x x}} \\ \frac{{M m}_{y the y 00}}{{I I}_{y the y}} - - \frac{(({I I}_{x x} - - {I I}_{z z})) {ω ω}_{m m x x} {ω ω}_{m m z z}}{{I I}_{y the y}} \\ \frac{{M m}_{z z 00}}{{I I}_{z z}} - - \frac{(({I I}_{y the y} - - {I I}_{x x})) {ω ω}_{m m y the y} {ω ω}_{m m x x}}{{I I}_{z z}} \end{matrix}] + + [\begin{matrix} \frac{{QSL QSL}_{r r} {m m}_{x x}^{{δ δ}_{x x}}}{{I I}_{x x}} & 00 & 00 \\ 00 & \frac{{QSL QSL}_{r r} {m m}_{y the y}^{{δ δ}_{y the y}}}{{I I}_{y the y}} & 00 \\ 00 & 00 & \frac{{QSL QSL}_{r r} {m m}_{z z}^{{δ δ}_{z z}}}{{I I}_{z z}} \end{matrix}] [\begin{matrix} {δ δ}_{x x} \\ {δ δ}_{y the y} \\ {δ δ}_{z z} \end{matrix}] - - - - - - ((22))

In the formula, M _x0 , M _y0 , M _z0 are the aerodynamic moments in the state of zero rudder deviation; is the derivative of the rolling moment coefficient to δ _x , is the derivative of the rolling moment coefficient to δ _y , is the derivative of rolling moment coefficient to δ _z ; dynamic pressure Q=0.5ρV ² , ρ is air density; S is missile reference area; L _r is missile reference length; δ _x , δ _y , δ _z are roll, deflection The rudder deflection angle of the three channels of navigation and pitch; I _x , I _y , and I _z are the moments of inertia of the missile in three directions respectively;

(2) Frame kinematics model;

The movement of the optical axis center of the seeker in space is:

_ωd = _ωdm + _ωds (3)

In the formula,

{ω ω}_{d d m m} = = [\begin{matrix} {ω ω}_{m m x x} {cosλ cosλ}_{z z} {cosλ cosλ}_{y the y} + + {ω ω}_{m m y the y} {sinλ sinλ}_{z z} - - {ω ω}_{m m z z} {cosλ cosλ}_{z z} {sinλ sinλ}_{y the y} \\ - - {ω ω}_{m m x x} {sinλ sinλ}_{z z} {cosλ cosλ}_{y the y} + + {ω ω}_{m m y the y} {cosλ cosλ}_{z z} + + {ω ω}_{m m z z} {sinλ sinλ}_{z z} {sinλ sinλ}_{y the y} \\ {ω ω}_{m m x x} {sinλ sinλ}_{y the y} + + {ω ω}_{m m z z} {cosλ cosλ}_{y the y} \end{matrix}] {ω ω}_{d d x x} = = [\begin{matrix} {\overset{\cdot \cdot}{λ λ}}_{y the y} {sinλ sinλ}_{z z} \\ {\overset{\cdot \cdot}{λ λ}}_{y the y} {cosλ cosλ}_{z z} \\ {\overset{\cdot \cdot}{λ λ}}_{z z} \end{matrix}]

Among them, ω _d is the projection of the angular velocity of the optical axis in the detection coordinate system, ω _dm is the projection of the projectile angular velocity in the detection coordinate system, ω _ds is the projection of the seeker servo frame angular velocity in the detection coordinate system, ω _mx , ω _my , ω _mz are the angular velocity of projectile body axis, λ _y , λ _z are the inner and outer frame angles of the semi-strapdown stable platform;

(3) Angle tracking system model;

The differential equation of 3D tracking angle error is:

\{\begin{matrix} {\overset{\cdot \cdot}{ϵ ϵ}}_{y the y} = = {ω ω}_{y the y} - - {ω ω}_{d d y the y} + + {ω ω}_{d d x x} {ϵ ϵ}_{z z} \\ {\overset{\cdot \cdot}{ϵ ϵ}}_{z z} = = {ω ω}_{z z} - - {ω ω}_{d d z z} - - {ω ω}_{d d x x} {ϵ ϵ}_{y the y} \end{matrix} - - - - - - ((44))

In the formula, ε _y , ε _z are misalignment angles, ω _y , ω _z are line-of-sight angular velocities, ω _dx , ω _dy , ω _dz are optical axis angular velocities;

(4) The integrated mathematical model of seeker stability tracking and projectile attitude control;

Establish the integrated mathematical model of semi-strapdown seeker control and attitude control:

\{\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = {f f}_{11} (({x x}_{11})) + + {g g}_{11} (({x x}_{11})) [\begin{matrix} {x x}_{22} \\ {u u}_{11} \end{matrix}] \\ {\overset{\cdot \cdot}{x x}}_{22} = = {f f}_{22} (({x x}_{22})) + + {g g}_{22} (({x x}_{22})) {u u}_{22} \end{matrix} - - - - - - ((55))

In the formula,

x ₁ =[α β γ _v ε _y ε _z ] ^T ,

x ₂ =[ω _mx ω _my ω _mz ] ^T , u ₂ =[δ _x δ _y δ _z ] ^T ,

{f f}_{11} (({x x}_{11})) = = [\begin{matrix} \frac{m m g g {cosγ cosγ}_{v v} cos cos θ θ - - L L}{m m V V cos cos β β} \\ \frac{m m g g {sinγ sinγ}_{v v} cos cos θ θ + + Y Y}{m m V V} \\ - - {\overset{\cdot \cdot}{ψ ψ}}_{v v} sin sin θ θ - - \frac{((m m g g {cosγ cosγ}_{v v} cos cos θ θ - - L L)) tan the tan β β}{m m V V} \\ {ω ω}_{y the y} \\ {ω ω}_{z z} \end{matrix}],, \begin{matrix} {g g}_{11} ((x x)) = = [\begin{matrix} - - cos cos α α tan the tan β β & sin sin α α tan the tan β β \\ sin sin α α & cos cos α α \\ cos cos α α sec sec β β & - - sin sin α α sec sec β β \\ {sinλ sinλ}_{z z} {cosλ cosλ}_{y the y} + + {ϵ ϵ}_{z z} {cosλ cosλ}_{z z} {cosλ cosλ}_{y the y} & - - {cosλ cosλ}_{z z} + + {ϵ ϵ}_{z z} {sinλ sinλ}_{z z} \\ - - (({sinλ sinλ}_{y the y} + + {ϵ ϵ}_{y the y} {cosλ cosλ}_{z z} {cosλ cosλ}_{y the y})) & - - {ϵ ϵ}_{y the y} {sinλ sinλ}_{z z} \end{matrix} \\ \begin{matrix} 11 & 00 & 00 \\ 00 & 00 & 00 \\ 00 & 00 & 00 \\ - - (({sinλ sinλ}_{z z} {cosλ cosλ}_{y the y} + + {ϵ ϵ}_{z z} {cosλ cosλ}_{z z} {sinλ sinλ}_{y the y})) & - - {cosλ cosλ}_{z z} + + {ϵ ϵ}_{z z} {sinλ sinλ}_{z z} & 00 \\ - - {cosλ cosλ}_{y the y} + + {ϵ ϵ}_{y the y} {cosλ cosλ}_{z z} {sinλ sinλ}_{y the y} & - - {ϵ ϵ}_{y the y} {sinλ sinλ}_{z z} & - - 11 \end{matrix}],, \end{matrix}

{f f}_{22} ((x x)) = = [\begin{matrix} \frac{{M m}_{x x 00}}{{I I}_{x x}} - - \frac{(({I I}_{z z} - - {I I}_{y the y})) {ω ω}_{m m z z} {ω ω}_{m m y the y}}{{I I}_{x x}} \\ \frac{{M m}_{y the y 00}}{{I I}_{y the y}} - - \frac{(({I I}_{x x} - - {I I}_{z z})) {ω ω}_{m m x x} {ω ω}_{m m z z}}{{I I}_{y the y}} \\ \frac{{M m}_{z z 00}}{{I I}_{z z}} - - \frac{(({I I}_{y the y} - - {I I}_{x x})) {ω ω}_{m m y the y} {ω ω}_{m m x x}}{{I I}_{z z}} \end{matrix}],, {g g}_{22} ((x x)) = = [\begin{matrix} \frac{{QSL QSL}_{r r} {m m}_{x x}^{{δ δ}_{x x}}}{{I I}_{x x}} & 00 & 00 \\ 00 & \frac{{QSL QSL}_{r r} {m m}_{y the y}^{{δ δ}_{y the y}}}{{I I}_{y the y}} & 00 \\ 00 & 00 & \frac{{QSL QSL}_{r r} {m m}_{z z}^{{δ δ}_{z z}}}{{I I}_{z z}} \end{matrix}];;

Step 2: Design the configuration of the integrated controller;

Assume that the airflow angle command α _c , β _c , γ _vc given by the guidance system, the frame angular position λ _y , λ _z measured by the frame angular position measurement sensor, and the bombardment line-of-sight angle q _y , q _z obtained through the filter The visual line-of-sight rate ω _y , ω _z , and the tracking angle error ε _y , ε _z obtained by the seeker receiver, output frame angular velocity control signals ω _λyc , ω _λzc through the outer loop controller, and control the optical axis to point to the real-time tracking projectile. At the same time, the pseudo-control variables ω _mxc , ω _myc , ω _mzc of the attitude angular velocity are given, and then sent to the inner ring attitude angular velocity control system to control the aerodynamic rudder to achieve fast tracking of ω _mxc , ω _myc , ω _mzc ;

Step 3: Controller design based on nonlinear dynamic inverse;

(1) Design the outer loop dynamic inverse controller;

Suppose the state variable of the outer loop is [α β γ _v ε _y ε _z ] ^T , and the input control variable is [ω _mx ω _my ω _mz ω _λy ω _λz ] ^T ; then the outer loop dynamic inverse controller is:

[\begin{matrix} {ω ω}_{m m x x c c} \\ {ω ω}_{m m y the y c c} \\ {ω ω}_{m m z z c c} \\ {ω ω}_{λ λ y the y} \\ {ω ω}_{λ λ z z} \end{matrix}] = = {g g}_{11}^{- - 11} (({x x}_{11})) (([\begin{matrix} {v v}_{α α} \\ {v v}_{β β} \\ {v v}_{{γ γ}_{v v}} \\ {v v}_{{ϵ ϵ}_{y the y}} \\ {v v}_{{ϵ ϵ}_{z z}} \end{matrix}] - - f f (({x x}_{11})))) - - - - - - ((99))

In the formula: are the expected dynamic forms of the state variables α, β, γ _v , ε _y , ε _z respectively, and ω _mxc , ω _myc , ω _mzc are the expected angular velocity commands of the projectile body attitude of the inner loop, which are used as pseudo-control quantities of the inner loop;

(2) Design the inner loop dynamic inverse controller;

Let the inner-loop control command be the pseudo-control quantity ω _mxc , ω _myc , ω _mzc of the outer loop, and the input control variables be δ _x , δ _y , δ _z ;

Then the inner loop dynamic inverse controller:

[\begin{matrix} {δ δ}_{x x} \\ {δ δ}_{y the y} \\ {δ δ}_{z z} \end{matrix}] = = {g g}_{22}^{- - 11} (({x x}_{22})) (([\begin{matrix} {v v}_{{ω ω}_{m m x x}} \\ {v v}_{{ω ω}_{m m y the y}} \\ {v v}_{{ω ω}_{m m z z}} \end{matrix}] - - f f (({x x}_{22})))) - - - - - - ((1010))

In the formula: are respectively the expected dynamic forms of the state variables ω _mxc , ω _myc , ω _mzc as in formula (8);

The outer-loop dynamic inverse controller and the inner-loop dynamic inverse controller are respectively obtained through formulas (9) and (10), and the integrated control of the missile is completed.