# Integrated Guidance and Control with Input Saturation and Impact Angle Constraint

Authors: Zhenhua Fu; Kuanqiao Zhang; Qintao Gan; Suochang Yang
Journal: Discrete Dynamics in Nature and Society
 (2020)

---

## Abstract

Aiming at the problem of impact angle constraint and input saturation, an integrated guidance and control (IGC) algorithm with impact angle constraint and input saturation is proposed. A three-channel independent design model of missile IGC with impact angle constraint is established, and an extended state observer with fast finite-time convergence is designed to estimate and compensate model errors and coupling relationship between channels. Based on the nonsingular terminal sliding mode control and backstepping control, the IGC three-channel independent design is completed. Nussbaum function and an auxiliary system are introduced to deal with the input saturation. The Lyapunov function is constructed to prove the finite-time convergence of the IGC algorithm. The missile six-degree-of-freedom simulation results show the effectiveness and superiority of the IGC algorithm.


---

## Body

## 1. Introduction

The traditional design of missile guidance and control system is based on the spectrum separation, the guidance system and control system are designed separately, and the coupling relationship between them is not considered. At the end of the guidance phase, due to the drastic change of the relative motion between the missile and the target, it is difficult for the control system to meet the requirement of fast response of the guidance system. If the velocity and acceleration of the target are large, this design method will lead to system instability and large miss distance [1]. In order to solve this problem, the concept of integrated guidance and control (IGC) design is proposed. The main idea is to design the guidance system and the control system together and consider the coupling relationship between them. The control instruction is generated directly from the relative motion information of the missile and the target [2]. This algorithm makes full use of the comprehensive information such as line-of-sight (LOS) angle, altitude, overload, and so on and can effectively improve the guidance and control performance.In recent years, many literatures have carried out IGC design, using various control technologies for integrated design, such as small gain theory [3], θ-D method [4], sliding mode control [5], adaptive control [6], and so on. In [7], an IGC model is established. The target maneuvering, model nonlinearity, and system disturbance are regarded as the total disturbances. The second-order disturbance observer is used to estimate the disturbances, and the sliding mode control is used for IGC design. The stability of the system is proved by the Lyapunov stability theory.Because the IGC model has strict feedback form, the backstepping control is widely used in the IGC design. In [8], the nonlinear model of guidance and control system in longitudinal plane is established. The IGC design is carried out based on backstepping control technology, and the adaptive law is used to estimate the uncertainty of the model. However, the backstepping control needs to derive the virtual control quantity, and there exists the problem of exponential expansion. In order to solve this problem, dynamic surface control is used in the IGC design [9, 10], and the problem of multiple derivation of virtual control quantity is solved by introducing first-order filter. In [11], the IGC design is completed by combining dynamic surface control, neural network disturbance observer, and barrier Lyapunov function.In order to increase the damage efficiency of warhead, many missiles (such as some anti–ship missiles, anti–tank missiles, air defense missiles, etc.) need to hit the target with impact angle constraints [12]. Therefore, the impact angle constraint needs to be considered in IGC design. Based on sliding mode control and dynamic inverse control, a robust dynamic inverse design method is proposed to compensate the system uncertainty in [13]. Then, this method is combined with dynamic surface control to complete the three-dimensional IGC design with impact angle constraint. In order to avoid the high-frequency buffeting, the continuous approximate function is used to replace the symbolic function, but the inherent structure of sliding mode control is changed so that the robustness of the system is reduced. In [14], adaptive control is used to compensate the system interference, and combined with dynamic surface control, the IGC design with impact angle constraint is completed. The traditional IGC design based on dynamic surface control can only guarantee the gradual convergence of the system states to the expected value, and the rapidity of IGC algorithm is reduced due to the introduction of first-order filter.Due to the physical constraints of the missile actuator, the large maneuvering of the target, the internal disturbance of the system, and the external disturbance, it is easy to increase the amplitude of the control quantity, which may reach the upper bound of the actuator constraint and lead to the control input saturation. The input saturation problem may lead to the degradation of control performance and even some unpredictable results. Therefore, it is necessary to consider the problem of input saturation in IGC design. In order to solve the input saturation of nonlinear systems, Chen et al. [15] regard the part beyond the control quantity as external disturbance and use the finite-time convergent disturbance observer to estimate and compensate the disturbance. Then, a robust tracking controller is designed based on terminal sliding mode control. Liang et al. [16] introduce an auxiliary system into the IGC model in longitudinal channel and employ the Nussbaum function to deal with the input saturation problem. Based on the dynamic surface control, they complete IGC design in longitudinal channel. For the input saturation problem, an improved saturation function and an auxiliary system are used in [17], and the auxiliary system states are applied to the design of integrated control law and stability analysis. At the same time, aiming at the target maneuvering, external disturbance, and model uncertainty, a finite-time convergent disturbance observer is introduced to estimate and compensate.On the basis of the above analysis and considering the impact angle constraint, input saturation, and model uncertainty, a finite-time convergent IGC algorithm against maneuvering target is presented for STT missile in this paper. The main contributions of this paper are as follows:(1)
An IGC design model considering impact angle constraint, model uncertainty, and input saturation is established by combining the missile dynamics model and motion kinematics between the missile and target.(2)
An extended state observer (ESO) with finite-time convergence is improved to estimate and compensate the disturbances caused by target maneuvering, system disturbance, and model uncertainty.(3)
Based on the improved ESO and backstepping control, a novel IGC algorithm with impact angle constraint is proposed, and the Nussbaum function is used to deal with the input saturation.(4)
Based on Lyapunov stability theory and finite-time stability theory, the stability and global finite-time convergence of the proposed IGC algorithm are proved.


## 2. Problem Formulation and Preliminaries

In this section, the IGC model is presented and the related definition and lemmas are given. The IGC model is established based on the missile dynamics model and motion kinematics between the missile and target. The related definition and lemmas are presented to facilitate the IGC law design and stability analysis.


### 2.1. Problem Formulation

The relative motion model of the missile and target is established in the three-dimensional inertial coordinate system, as shown in Figure1. Oxyz is the inertial coordinate system; Ox4y4z4 is the LOS coordinate system. M and T represent the missile and target, respectively. ε and η are the LOS elevation angle and azimuth angle, respectively, and R is the relative distance between the missile and target. r is the projection of R on the horizontal plane of the inertial coordinate system; that is, r=Rcosε.Figure 1
3D missile and target relative motion.The relative motion equation of missile and target in three dimension can be described as [18](1)2R˙ε˙+Rε¨+Rη˙2sinεcosε=aty4−amy4,−2R˙η˙cosε−Rη¨cosε+2Rε˙η˙sinε=atz4−amz4,where at4=atx4aty4atz4T and am4=amx4amy4amz4T are the acceleration vectors in LOS coordinate system of missile and target, respectively.Letam2=amx2amy2amz2T be the acceleration vector of missile in ballistic coordinate system. According to the missile dynamic equation, we can obtain amy2 and amz2 as(2)amy2=Ym−gcosθm,amz2=Zm,where m is the missile mass, g is the gravity acceleration, and θm is the missile trajectory inclination angle. Y and Z are lift and lateral forces, which are given as(3)Y=CyααQS+Yδz+dY,Z=CzββQS+Zδy+dZ,where α and β are attack angle and sideslip angle of missile, respectively, Yδz and Zδy are lift and lateral force produced by rudder deflection angle, respectively, Cyα is the derivative of the lift force coefficient with respect to α, Czβ is the derivative of the lateral force coefficient with respect to β, S is the aerodynamics reference area, Q=ρVm2/2 is the dynamic pressure, and dY and dZ are unknown and bounded uncertain terms.Invoking (1) and (2), the relative motion equation is established as(4)ε¨=−2R˙Rε˙−CyαQSmRα+dε,η¨=−2r˙rη˙+CzβQSmrβ+dη,with(5)dε=aty4R−Yδz+dYmR+gcosθmR−η˙2sinεcosε,dη=−atz4r+Zδy+dZmr,where dε and dη are unknown and bounded uncertainties.Remark 1.
R, r, aty4, atz4, Yδz, Zδy, dY, and dZ are all bounded. Therefore, it can be known from (5) that dε and dη are bounded.
The missile dynamics model can be established as [19](6)α˙=ωz1−ωx1tanβcosα−CyααQS−mgcosθmmVmcosβ+ωy1tanβsinα+Δα,β˙=ωx1sinα+ωy1cosα+CyββQSmVm+Δβ,γ˙=ωx1−tanϑωy1cosγ−ωz1sinγ+Δγ,ω˙x1=mx1αα+mx1ββ+mx1δxδxQSLJx1+Jy1−Jz1Jx1ωy1ωz1+Δωx1,ω˙y1=my1ββ+my1δyδyQSLJy1+Jz1−Jx1Jy1ωz1ωx1+Δωy1,ω˙z1=mz1αα+mz1δzδzQSLJz1+Jx1−Jy1Jz1ωx1ωy1+Δωz1,where ϑ and γ are pitch and roll angle, respectively; Jx1, Jy1, and Jz1 are roll, yaw, and pitch moments of inertia, respectively; ωx1, ωy1, and ωz1 are roll, yaw, and pitch angular rates, respectively; mx1α and mz1α are the derivatives of roll and pitch moment coefficients with respect to α, respectively; mx1β and my1β are the derivatives of roll and pitch moment coefficients with respect to β, respectively; mx1δx, my1δy, and mz1δz are the derivatives of roll, yaw, and pitch moment coefficients with respect to δx, δy, and δz, respectively; and Δα, Δβ, Δγ, Δωx1, Δωy1, and Δωz1 are external disturbance and modeling errors.
According to the principle of considering the main factors and treating the secondary factors as uncertainty, the missile dynamics model (6) is simplified to(7)α˙=ωz1−CyαQSmVmα+dα,β˙=ωy1+CzβQSmVmβ+dβ,γ˙=ωx1+dγ,ω˙x1=mx1δxQSLJx1δx+d¯ωx1,ω˙y1=my1βQSLJy1β+my1δyQSLJy1δy+d¯ωy1,ω˙z1=mz1αQSLJz1α+mz1δzQSLJz1δz+d¯ωz1,with(8)dα=Δα−ωx1tanβcosα+ωy1tanβsinα+CyαQSmVmα−CyααQS−mgcosθmmVmcosβ,dβ=Δβ+ωx1sinα+ωy1cosα−1,dγ=Δγ−tanϑωy1cosγ−ωz1sinγ,d¯ωx1=mx1αα+mx1ββQSLJx1+Jy1−Jz1Jx1ωy1ωz1+Δωx1,d¯ωy1=Δωy1+Jz1−Jx1Jy1ωz1ωx1,d¯ωz1=Δωz1+Jx1−Jy1Jz1ωx1ωy1.
Taking the longitudinal plane as an example, the missile impact angleθd is defined as the included angle between the missile and the target velocity vectors at the engagement time, and the impact angle corresponds to the terminal LOS angle as follows [20, 21]:(9)εtf=θttf−arctansinθdcosθd−vt/vm.
So, the impact angle constraint can be transformed into the terminal LOS angle constraint.
Letx11=ε−εd, x12=ε˙, x13=α, and x14=ωz1, where εd is the expected impact LOS elevation angle at the engagement time. The control model of the IGC system in the pitch channel with impact angle constraint can be obtained by combining (4) and (7) as follows:(10)x˙11=x12,x˙12=a22x12+a23x13+dε,x˙13=x14+a33x13+dα,x˙14=a43x13+bδz+d¯ωz1,with(11)a22=−2R˙R,a23=−CyαQSmR,a33=−CyαQSmVm,a43=mz1αQSLJz1,b1=mz1δzQSLJz1.
Due to the large maneuver of the target, the uncertainty of the missile itself, and external interference, the amplitude of the missile control command may become larger, reaching the upper limit of the actuator constraint, and the control quantity may be saturated. The input saturation problem will make the dynamic quality of the guidance system worse, lead to the control performance decline and even destroy the system stability, and lead to system crash. Therefore, it is necessary to consider the saturation of actuators in the IGC design. Considering the input saturation, the control model of the IGC system in the pitch channel (10) can be rewritten as(12)x˙11=x12,x˙12=a22x12+a23x13+dε˙,x˙13=x14+a33x13+dα,x˙14=a43x13+bsatδz+d¯ωz1,where satδz is the actual pitch rudder deflection angle. satδz is defined as(13)satδz=δz,δz<δzmax,δzmaxsgnδz,δz≥δzmax,where δzmax is the known upper bound of δz, that is, the maximum rudder deflection angle.
Considering the actuator saturation,satδz is not differentiable at δz=δzmax. In order to enable the backstepping design method to be applied to IGC design, the following hyperbolic tangent function is used to smooth the saturation function [22] as follows:(14)gδz=δzmaxtanhδzδzmax=δimaxeδz/δzmax−eδz/δzmaxeδz/δzmax+eδz/δzmax.
Letdz=satδz−gδz; then(15)satδz=dz+gδz.(16)dz=satδz−gδz≤δzmax1−tanhδzδzmax=0.238δzmax,so dz is bounded. Combined with (12), (14), and (15), the control model of the IGC system in the pitch channel considering input saturation is(17)x˙11=x12,x˙12=a22x12+a23x13+dε,x˙13=x14+a33x13+dα,x˙14=a43x13+b1gδz+dωz1,where dωz1=d¯ωz1+dz.
Similar to the control model of the IGC system in the pitch channel, the control model in yaw channel can be established by combining (4) and (7) as follows:(18)x˙21=x22,x˙22=a¯22x22+a¯23x23+dη,x˙23=x24+a¯33x23+dβ,x˙24=a¯43x23+b2gδy+dωy1,with(19)x21=η−ηd,x22=η˙,x23=β,x24=ωy1,a¯22=−2r˙r,a¯23=CzβQSmr,a¯33=CzβQSmVm,a¯43=my1βQSLJy1,b2=my1δyQSLJy1,where ηd is the expected LOS azimuth angle.
According to (7), the control model in the roll channel can be established as(20)x˙31=x32+dγ,x˙32=b3gδx+dωx1,with(21)x31=γ,x32=ωx1,b3=mx1δxQSLJx1.


### 2.2. Preliminaries

For the convenience of the following analysis, the relative definitions and lemmas are introduced in this section.Definition 1.
For easy writing, definexa=xasgnx, where sgn⋅ is symbolic function, and sgn0=0.Lemma 1 (see [23]).
Assume that there is a Lyapunov functionVx satisfying V˙x≤−β1Vα1x−β2Vx for any 0<α1<1, β1>0, β2>0; then the origin of the system is finite-time convergent, and the convergence time satisfies(22)T1≤1β21−α1ln1+β2V1−α1x0β1.Lemma 2 (see [24]).
Assume that a Lyapunov functionVx satisfying V˙x≤−β1Vα1x−β2Vx+β3Vα2x for any 0<α2<1, α1>α2, β3>0, and then the system states are stable in finite-time, and the convergence time satisfies(23)T2≤1β2−θ21−α1ln1+β2−θ2V1−α1x0β1−θ1,where θ1∈0,β1, θ2∈0,β2. Convergence domain satisfies(24)Ω=xθ1Vα1−α2x+θ2V1−α1x<β3.Lemma 3 (see [25]).
For any real numberxi, i=1,2,…,n, there exists a real number 0<α<1 such that the following inequality holds:(25)∑i=1nxiα≤∑i=1nxiα.Lemma 4 (see [26]).
Suppose thatV⋅ and χ⋅ are smooth functions defined in 0,tf, and Vt≥0, Nχ is a Nussbaum gain function, if the inequality is satisfied(26)V≤V0e−Kt+LK1−e−Kt+e−KtM∫0tψtNχ−1χ˙e−Kτdτ,where K>0, L>0, M>0, and ψt>0; then V⋅ and χ⋅ are bounded in 0,tf.


## 2.1. Problem Formulation

The relative motion model of the missile and target is established in the three-dimensional inertial coordinate system, as shown in Figure1. Oxyz is the inertial coordinate system; Ox4y4z4 is the LOS coordinate system. M and T represent the missile and target, respectively. ε and η are the LOS elevation angle and azimuth angle, respectively, and R is the relative distance between the missile and target. r is the projection of R on the horizontal plane of the inertial coordinate system; that is, r=Rcosε.Figure 1
3D missile and target relative motion.The relative motion equation of missile and target in three dimension can be described as [18](1)2R˙ε˙+Rε¨+Rη˙2sinεcosε=aty4−amy4,−2R˙η˙cosε−Rη¨cosε+2Rε˙η˙sinε=atz4−amz4,where at4=atx4aty4atz4T and am4=amx4amy4amz4T are the acceleration vectors in LOS coordinate system of missile and target, respectively.Letam2=amx2amy2amz2T be the acceleration vector of missile in ballistic coordinate system. According to the missile dynamic equation, we can obtain amy2 and amz2 as(2)amy2=Ym−gcosθm,amz2=Zm,where m is the missile mass, g is the gravity acceleration, and θm is the missile trajectory inclination angle. Y and Z are lift and lateral forces, which are given as(3)Y=CyααQS+Yδz+dY,Z=CzββQS+Zδy+dZ,where α and β are attack angle and sideslip angle of missile, respectively, Yδz and Zδy are lift and lateral force produced by rudder deflection angle, respectively, Cyα is the derivative of the lift force coefficient with respect to α, Czβ is the derivative of the lateral force coefficient with respect to β, S is the aerodynamics reference area, Q=ρVm2/2 is the dynamic pressure, and dY and dZ are unknown and bounded uncertain terms.Invoking (1) and (2), the relative motion equation is established as(4)ε¨=−2R˙Rε˙−CyαQSmRα+dε,η¨=−2r˙rη˙+CzβQSmrβ+dη,with(5)dε=aty4R−Yδz+dYmR+gcosθmR−η˙2sinεcosε,dη=−atz4r+Zδy+dZmr,where dε and dη are unknown and bounded uncertainties.Remark 1.
R, r, aty4, atz4, Yδz, Zδy, dY, and dZ are all bounded. Therefore, it can be known from (5) that dε and dη are bounded.
The missile dynamics model can be established as [19](6)α˙=ωz1−ωx1tanβcosα−CyααQS−mgcosθmmVmcosβ+ωy1tanβsinα+Δα,β˙=ωx1sinα+ωy1cosα+CyββQSmVm+Δβ,γ˙=ωx1−tanϑωy1cosγ−ωz1sinγ+Δγ,ω˙x1=mx1αα+mx1ββ+mx1δxδxQSLJx1+Jy1−Jz1Jx1ωy1ωz1+Δωx1,ω˙y1=my1ββ+my1δyδyQSLJy1+Jz1−Jx1Jy1ωz1ωx1+Δωy1,ω˙z1=mz1αα+mz1δzδzQSLJz1+Jx1−Jy1Jz1ωx1ωy1+Δωz1,where ϑ and γ are pitch and roll angle, respectively; Jx1, Jy1, and Jz1 are roll, yaw, and pitch moments of inertia, respectively; ωx1, ωy1, and ωz1 are roll, yaw, and pitch angular rates, respectively; mx1α and mz1α are the derivatives of roll and pitch moment coefficients with respect to α, respectively; mx1β and my1β are the derivatives of roll and pitch moment coefficients with respect to β, respectively; mx1δx, my1δy, and mz1δz are the derivatives of roll, yaw, and pitch moment coefficients with respect to δx, δy, and δz, respectively; and Δα, Δβ, Δγ, Δωx1, Δωy1, and Δωz1 are external disturbance and modeling errors.
According to the principle of considering the main factors and treating the secondary factors as uncertainty, the missile dynamics model (6) is simplified to(7)α˙=ωz1−CyαQSmVmα+dα,β˙=ωy1+CzβQSmVmβ+dβ,γ˙=ωx1+dγ,ω˙x1=mx1δxQSLJx1δx+d¯ωx1,ω˙y1=my1βQSLJy1β+my1δyQSLJy1δy+d¯ωy1,ω˙z1=mz1αQSLJz1α+mz1δzQSLJz1δz+d¯ωz1,with(8)dα=Δα−ωx1tanβcosα+ωy1tanβsinα+CyαQSmVmα−CyααQS−mgcosθmmVmcosβ,dβ=Δβ+ωx1sinα+ωy1cosα−1,dγ=Δγ−tanϑωy1cosγ−ωz1sinγ,d¯ωx1=mx1αα+mx1ββQSLJx1+Jy1−Jz1Jx1ωy1ωz1+Δωx1,d¯ωy1=Δωy1+Jz1−Jx1Jy1ωz1ωx1,d¯ωz1=Δωz1+Jx1−Jy1Jz1ωx1ωy1.
Taking the longitudinal plane as an example, the missile impact angleθd is defined as the included angle between the missile and the target velocity vectors at the engagement time, and the impact angle corresponds to the terminal LOS angle as follows [20, 21]:(9)εtf=θttf−arctansinθdcosθd−vt/vm.
So, the impact angle constraint can be transformed into the terminal LOS angle constraint.
Letx11=ε−εd, x12=ε˙, x13=α, and x14=ωz1, where εd is the expected impact LOS elevation angle at the engagement time. The control model of the IGC system in the pitch channel with impact angle constraint can be obtained by combining (4) and (7) as follows:(10)x˙11=x12,x˙12=a22x12+a23x13+dε,x˙13=x14+a33x13+dα,x˙14=a43x13+bδz+d¯ωz1,with(11)a22=−2R˙R,a23=−CyαQSmR,a33=−CyαQSmVm,a43=mz1αQSLJz1,b1=mz1δzQSLJz1.
Due to the large maneuver of the target, the uncertainty of the missile itself, and external interference, the amplitude of the missile control command may become larger, reaching the upper limit of the actuator constraint, and the control quantity may be saturated. The input saturation problem will make the dynamic quality of the guidance system worse, lead to the control performance decline and even destroy the system stability, and lead to system crash. Therefore, it is necessary to consider the saturation of actuators in the IGC design. Considering the input saturation, the control model of the IGC system in the pitch channel (10) can be rewritten as(12)x˙11=x12,x˙12=a22x12+a23x13+dε˙,x˙13=x14+a33x13+dα,x˙14=a43x13+bsatδz+d¯ωz1,where satδz is the actual pitch rudder deflection angle. satδz is defined as(13)satδz=δz,δz<δzmax,δzmaxsgnδz,δz≥δzmax,where δzmax is the known upper bound of δz, that is, the maximum rudder deflection angle.
Considering the actuator saturation,satδz is not differentiable at δz=δzmax. In order to enable the backstepping design method to be applied to IGC design, the following hyperbolic tangent function is used to smooth the saturation function [22] as follows:(14)gδz=δzmaxtanhδzδzmax=δimaxeδz/δzmax−eδz/δzmaxeδz/δzmax+eδz/δzmax.
Letdz=satδz−gδz; then(15)satδz=dz+gδz.(16)dz=satδz−gδz≤δzmax1−tanhδzδzmax=0.238δzmax,so dz is bounded. Combined with (12), (14), and (15), the control model of the IGC system in the pitch channel considering input saturation is(17)x˙11=x12,x˙12=a22x12+a23x13+dε,x˙13=x14+a33x13+dα,x˙14=a43x13+b1gδz+dωz1,where dωz1=d¯ωz1+dz.
Similar to the control model of the IGC system in the pitch channel, the control model in yaw channel can be established by combining (4) and (7) as follows:(18)x˙21=x22,x˙22=a¯22x22+a¯23x23+dη,x˙23=x24+a¯33x23+dβ,x˙24=a¯43x23+b2gδy+dωy1,with(19)x21=η−ηd,x22=η˙,x23=β,x24=ωy1,a¯22=−2r˙r,a¯23=CzβQSmr,a¯33=CzβQSmVm,a¯43=my1βQSLJy1,b2=my1δyQSLJy1,where ηd is the expected LOS azimuth angle.
According to (7), the control model in the roll channel can be established as(20)x˙31=x32+dγ,x˙32=b3gδx+dωx1,with(21)x31=γ,x32=ωx1,b3=mx1δxQSLJx1.


## 2.2. Preliminaries

For the convenience of the following analysis, the relative definitions and lemmas are introduced in this section.Definition 1.
For easy writing, definexa=xasgnx, where sgn⋅ is symbolic function, and sgn0=0.Lemma 1 (see [23]).
Assume that there is a Lyapunov functionVx satisfying V˙x≤−β1Vα1x−β2Vx for any 0<α1<1, β1>0, β2>0; then the origin of the system is finite-time convergent, and the convergence time satisfies(22)T1≤1β21−α1ln1+β2V1−α1x0β1.Lemma 2 (see [24]).
Assume that a Lyapunov functionVx satisfying V˙x≤−β1Vα1x−β2Vx+β3Vα2x for any 0<α2<1, α1>α2, β3>0, and then the system states are stable in finite-time, and the convergence time satisfies(23)T2≤1β2−θ21−α1ln1+β2−θ2V1−α1x0β1−θ1,where θ1∈0,β1, θ2∈0,β2. Convergence domain satisfies(24)Ω=xθ1Vα1−α2x+θ2V1−α1x<β3.Lemma 3 (see [25]).
For any real numberxi, i=1,2,…,n, there exists a real number 0<α<1 such that the following inequality holds:(25)∑i=1nxiα≤∑i=1nxiα.Lemma 4 (see [26]).
Suppose thatV⋅ and χ⋅ are smooth functions defined in 0,tf, and Vt≥0, Nχ is a Nussbaum gain function, if the inequality is satisfied(26)V≤V0e−Kt+LK1−e−Kt+e−KtM∫0tψtNχ−1χ˙e−Kτdτ,where K>0, L>0, M>0, and ψt>0; then V⋅ and χ⋅ are bounded in 0,tf.


## 3. Fast Finite-Time Disturbance Observer

Extended state observer is an effective method to estimate system uncertainties and external disturbances. It regards system uncertainties and external disturbances as total disturbance and extends them to a new variable. Then an observer is designed to estimate the disturbance. ESO has little dependence on the model and can be calculated in real time, so it is widely used. However, the traditional ESO regards the total disturbance as a constant or a slowly varying quantity, so the derivative of the total disturbance is zero. This method hinders the further improvement of ESO. For time-varying disturbance, by increasing the ESO gain, the satisfactory disturbance estimation effect can be obtained. However, when there is measurement noise in the output of the system, high gain will lead to the amplification of measurement noise. Based on the design idea of ESO, a fast finite-time convergent ESO is improved by using the supertwisting algorithm [27], which has the advantages of ESO and sliding mode control. In addition, the supertwisting algorithm can reduce the chattering phenomenon of sliding mode control. The improved ESO can not only estimate the total disturbance, but also has robustness and finite-time convergence.Consider the following first-order nonlinear system:(27)x˙=u+d,where u is the system input, and d is the total disturbance of the system.Assumption 1.
d is continuously differentiable, and its differential satisfies d˙≤ς¯.
Based on the supertwisting algorithm, a fast finite-time convergent ESO is improved for system (27) as follows:(28)z˙1=u+k1e1a+e1,z˙2=k2e12a−1+a+1e1a+e1,where e1=x−z1, k1>0, k2>0, 1/2<a<1.
Invoking (27) and (28), we have(29)e˙1=e2−k1e1a+e1,e˙2=d˙−k2e12a−1+a+1e1a+e1,where e2=d−z2.Remark 2.
Supertwisting algorithm can greatly weaken chattering and has strong robustness and high-precision control performance. However, the algorithm has shortcomings as follows: (1) the control law is continuous function, but not smooth function, affecting control performance; (2) when system states are far from the equilibrium point, the convergence rate is slow. Aiming at these disadvantages, a linear term is introduced to accelerate convergence speed and ensure smoothness of function. From (29), we can see that the linear term e1 in (28) will accelerate convergence speed when system states are far away from equilibrium point. When system approaches equilibrium point, the nonlinear term e1a plays a major role and accelerates convergence speed of system. Therefore, compared with traditional supertwisting ESO, the fast finite-time convergent ESO (28) has faster convergence speed.Proposition 1.
For system (27) with Assumption 1, the ESO (28) is employed to estimate the disturbance of the system. Its estimation error can finite-time converge to the region as follows:(30)Ω1=e1,e2θ1V11−1/2a+θ2V11/2a−1/2<λ3.Proof.
Define a new vector as(31)e¯=e1a+e1,e2T.
Differentiatinge¯ with respect to time gives(32)e¯˙=ae1a−1+1e2−k1e1a+e1−k2ae1a−1+1e1a+e1+0d˙=−ae1a−1+1Ae¯+ς,with(33)A=k1−1k20,ς=0d˙.
It is easy to prove thatA is a Hurwitz matrix, so there exists a positive definite symmetric matrix P satisfying ATP+PA=Q, where Q is a symmetric positive definite matrix.
Construct the Lyapunov function as(34)V1=e¯TPe¯.
SinceP is a positive definite matrix, we have(35)λminPe¯2≤V1≤λmaxPe¯2.
DifferentiatingV1 with respect to time and substituting (32) into it gives(36)V˙1=−e¯Tae1a−1+1ATP+PAe¯+ςTPe¯+e¯TPς≤−ae1a−1e¯TQe¯−e¯TQe¯+2e¯Pς.
Noting thate1≤e¯1/a and ς=d˙≤ς¯, invoking (35) and (36) yields(37)V˙1≤−aλminQe¯3−1/a−λminQe¯2+2ς¯e¯P≤−aλminQλmax1/2a−3/2PV13/2−1/2a−λminQλmax−1PV1+2ς¯λmin−1/2PPV11/2=−λ1V13/2−1/2a−λ2V1+λ3V11/2,where λ1=aλminQλmax1/2a−3/2P, λ2=λminQλmax−1P, λ3=2ς¯λmin−1/2PP.
It can be obtained from Lemma1 that the proposed ESO (28) is stable, and the observer errors e1 and e2 will converge to a small neighborhood of zero in a finite time, and the convergence time satisfies(38)t1≤2aλ2−θ21−alnλ2−θ2V1/2a−1/20λ1−θ1,where θ1∈0,λ1, θ2∈0,λ2. The convergence domain satisfies(39)Ω1=e1,e2θ1V11−1/2a+θ2V11/2a−1/2<λ3.The proof is completed.
Taking the pitch channel as an example, the proposed ESO (28) is used to estimate the uncertainties in the model (12) as follows:(40)x^˙12=a22x12+a23x13+k11x˜12a+x˜12,d^˙ε=k12x˜122a−1+a+1x˜12a+x˜12,x^˙13=x14+a33x13+k21x˜13a+x˜13,d^˙α=k22x˜132a−1+a+1x˜13a+x˜13,x^˙14=a43x13+bsatδz+k31x˜14a+x˜14,d^˙ωz1=k32x˜142a−1+a+1x˜14a+x˜14,where x˜12=x12−x^12, x˜13=x13−x^13, x˜13=x14−x^14, x^1i is the estimation of x1i, i=2,3,4, and d^j is the estimation of dj, j=ε,α,ωz1.


## 4. IGC Design with Input Saturation

### 4.1. Design of Pitch Channel Controller and Finite-Time Convergence Analysis

In order to solve the nonlinear saturation problem, the following auxiliary systems are introduced as(41)δ˙z=−μ1δz+δzc.Then, the control model of the IGC system in the pitch channel with input saturation and impact angle constraint is obtained as(42)x˙11=x12,x˙12=a22x12+a23x13+dε,x˙13=x14+a33x13+dα,x˙14=a43x13+b1gδz+dωz1,δ˙z=−μ1δz+δzc.The model (42) has a strict feedback structure, so the backstepping method combined with terminal sliding mode control can be used to design the controller. The specific design steps of the IGC algorithm are given below:Step 1. Select the following nonsingular terminal sliding surface with impact angle constraint [28] as(43)s11=x12+ρ1x11+ρ2φx11,with(44)φx11=x11b,x11≥σ,λ1x11+λ2x113,x11<σ,whereλ1=3−bσb−1/2, λ2=b−1σb−3/2, σ>0, and 0<b=p1/p2<1; p1 and p2 are positive odd numbers.Differentiatings11 with respect to time gives(45)s˙11=x˙12+ρ1x12+ρ2φ′x11x12,with(46)φ′x11=bx11b−1,x11≥σ,λ1+3λ2x112,x11<σ.Substituting (42) into (45), we have(47)s˙11=a22x12+a23x13+dε+ρ1+ρ2φ′x11x12.The virtual control lawx13c is designed as(48)x13c=−1a23a22+ρ1b1x11b1−1+ρ2φ′x11x12+d^ε+l11s11+l12s11c.A new virtual control lawx13d is introduced to avoid the differential expansion problem caused by the derivation of the control law and to ensure the fast convergence of the system in finite time. x13d is generated by x13c. A low-pass filter is designed to generate x13d and its derivatives x˙13d as(49)τ1x˙13d=−y11−y11c,wherey11=x13d−x13c, x13d0=x13c0.Remark 3.
The dynamic surface control used a first-order linear filterτ1x˙13d=y1 to solve the differential expansion problem of backstepping control. The performance of the filter is related to τ1. Reducing τ1 can improve the convergence speed and filtering accuracy, but too large will amplify the measurement noise. The filter (49) can improve the convergence speed and filtering accuracy by reducing τ1 and c, which reduces the selection requirement of τ1, and can meet the requirements of finite-time convergence.Step 2. Define the sliding mode surface as(50)s12=x13−x13d.Differentiatings12 with respect to time gives(51)s˙12=x˙13−x˙13d=x14+a33x13+dα−x˙13d.The virtual control lawx14c is designed as(52)x14c=−a33x13−d^α+x˙13d−l21s12−l22s12c−a23s11.Design the first-order nonlinear filter as(53)τ2x˙14d=−y12−y12c,wherey12=x14d−x14c, x14d0=x14c0.Step 3. Define the sliding mode surface as(54)s13=x14−x14d.Differentiatings13 with respect to time gives(55)s˙13=x˙14−x˙14d=a43x13+b1gδz+dωz1−x˙14d.The virtual control lawx15c is designed as(56)x15c=−1b1a43x13+d^ωz1−x˙14d+l31s13+l32s13c+s12.Design the first-order nonlinear filter as(57)τ3x˙15d=−y13−y13c,wherey13=x15d−x15c, x15d0=x15c0.Step 4. Define the sliding mode surface as(58)s14=gδz−x15d.
Differentiatings14 with respect to time gives(59)s˙14=g′δz−μ1δz+δzc−x˙15d,where g′δz=∂gδz/∂δz=4/eδz/δzmax+eδz/δzmax2.
The actual control lawδzc is designed as(60)δzc=Nχ1δ¯zc,δ¯zc=−l41s14−l42s14c+μ1g′δzδz+x˙15d−b1s13,with(61)Nχ1=eχ2cosχ,χ˙1=γχs14δ¯zc,where γχ>0 is the design parameter.Proposition 2.
For system (17), the designed IGC law (60) can make the states of the pitch channel closed-loop system converge to a smaller neighborhood of the origin in finite time.Proof.
From (43) to (60), we have(62)s˙11=a22x12+a23s12+y11+x13c+dε+ρ1+ρ2φ′x11x12=−l11s11−l12s11c+d˜ε+a23s12+y11,(63)s˙12=s13+y12+x14c+a33x13+dα−x˙13d=−l21s12−l22s12c+d˜α+s12+y12−a23s11,(64)s˙13=a43x13+bs14+y13+x15c+dωz1−x˙14d=−l31s13−l32s13c+d˜ωz1+bs14+y13−s12,(65)s˙14=g′δz−μ1δz+δzc+δ¯zc−δ¯zc−x˙15d=−l41s14−l42s14c−b1s13+g′δzNχ1−1δ¯zc,(66)y˙1i=x˙1id−x˙1ic=−1τiy1i+y1ic−x˙1ic,i=1,2,3,(67)y1iy˙1i=−1τiy1i2+y1i1+c−y1ix˙1ic≤−1τiy1i2+y1i1+c+y1ix˙1ic.
There exists a positive real numberξ1i such that x˙1ic≤ξ1i [29], so we can rewrite (67) as(68)y1iy˙1i≤−1τiy1i2+y1i1+c+y1iξ1i.
Construct the Lyapunov function as(69)V2=12∑i=14s1i2+12∑i=13y1i2.
DifferentiatingV2 with respect to time gives(70)V˙2=∑i=14s1is˙1i+∑i=13y1iy˙1i≤∑i=13−1τiy1i2+y1i1+c+y1iξ1i+s11−l11s11−l12s11c+d˜ε+a23s12+y11+s12−l21s12−l22s12c+d˜α+s13+y12−a23s11+s13−l31s13−l32s13c+d˜ωz1+bs14+y13−s12+s14−l41s14−l42s14c−bs13+g′δzNχ1−1δ¯zc=∑i=14−li1s1i2−li2s1i1+c+a23s11y11+s12y21+bs13y31+∑i=13−1τiy1i2+y1i1+c+y1iξ1i+s11d˜ε+s12d˜α+s13d˜ωz1+χ˙1γχg′δzNχ1−1.
Based on Young’s inequality, the inequality (70) can be rewritten as(71)V˙2≤∑i=14−li1s1i2−li2s1i1+c+a23s112+y1122+s122+y1222+bs132+y1322+s112+d˜ε2+s122+d˜α2+s132+d˜ωz122+∑i=13−1τiy1i2+y1i1+c+ξi12y1i2+14+χ˙1γχg′δzNχ1−1=−l11−a23+12s112−l21−1s122−l31−b+12s132−l41s142−∑i=14li2s1i1+c−1τ1−a232−ξ112y112−1τ2−12−ξ22y122−1τ3−b2−ξ32y132−∑i=13y1i1+cτi+ζ1+ζ2,with(72)ζ1=d˜ε2+d˜α2+d˜ωz12+32,ζ2=χ˙1γχg′δzNχ1−1.
Let(73)l11≥a23+12+12k,l21≥1+12k,l31≥b+12+12k,l41≥12k,li2≥12k,i=1,2,3,4,1τ1≥a23+ξ122+12k,1τ2≥1+ξ222+12k,1τ3≥b+ξ322+12k,where k>0; we can obtain(74)V˙2≤−kV2−k2∑i=14s1i21+c/2+∑i=13y1i21+c/2+ζ1+ζ2.
According to Lemma3, we have(75)V˙2≤−kV2−k2∑i=14s1i2+∑i=13y1i21+c/2+ζ=−kV2−kV21+c/2+ζ,where ζ=ζ1+ζ2.
Due toV2≥0, the following inequality holds:(76)V˙2≤−kV2+ζ.
Solving the inequality (76) results in(77)V2≤V20e−kt+ζ1k1−e−kt+e−Ktγχ∫0tg′δzNχ1−1χ˙e−kτdτ.
According to Lemma4, it can be seen that V2 and χ1 are bounded, and further, s1, s2, s3, s4, y1, y2, y3, and ζ are bounded. The following two forms can be obtained from (75) as(78)V˙2≤−k−ζV2V2−kV21+c/2.(79)V˙2≤−kV2−k−ζV21+c/2V21+c/2.
For (78), if k−ζ/V2>0, according to Lemma 1, V2 is finite-time convergent, and the convergence region is(80)V2≤Δ1=ζk.
For (79), similar to the analysis of (78), V2 can finite-time converge to the region(81)V2≤Δ2=ζk2/1+c.
Invoking (80) and (81), it is obtained that V2 can finite-time converge to the region Δ=minΔ1,Δ2. Therefore, s11 can finite-time converge to the region s11≤Δ=ϕ, and the convergence regions of x11 and x12 are discussed as follows:(1)
Ifx1≥σ, we can rewrite (43) as(82)x12=x˙11=−ρ1x11−ρ2x11b+ϕ1,ϕ1≤ϕ.Construct the Lyapunov function as(83)V2=x112.DifferentiatingV2 with respect to time and substituting (82) into it gives(84)V˙2=2x11x˙11=−2ρ1x12−2ρ2xib+1+2ϕ1x11≤−2ρ1x12−2ρ2xib+1+2ϕ1x11+2ϕx11=−2ρ1V2−2k2V2b+1/2+2ϕV21/2.From Lemma2, it can be seen that x1 can converge to region Ω1=x11=ϑ1θ1ϑ1b+θ2ϑ11−b<2ϕ in finite time, where θ1∈0,2ρ2, θ2∈0,2ρ1.By choosing parametersk1 and k2 reasonably, ϑ1≤δ can be achieved. At this time, the convergence domain of x1 is(85)x11≤maxϑ1,σ=δ.Combining (82), the convergence domain of x12 is(86)x12≤ρ1x11+ρ2x11b+ϕ1≤k1δ+k2δb+ϕ.(2)
Ifx1≤σ, we can rewrite (43) as(87)x12=x˙1=−ρ1x11−ρ2λ1x1+λ2x13+ϕ1≤ρ1x11−ρ2λ1x1+λ2x13+ϕ1≤ρ1δ+k2δb+ϕ.
In summary,x1 and x12 can converge to a small neighborhood of the origin in finite time, and the neighborhood is(88)x11≤δ,x12≤ρ1δ+k2δb+ϕ.The proof is complete.


### 4.2. Design of Yaw and Roll Channel Controller

Referring to the design process of the pitch channel, the integrated guidance control law of the yaw channel is designed as(89)s21=x22+ρ1sgnbx21+ρ2φx21,x23c=−1a¯23a¯22+ρ1+ρ2φ′x21x22+d^η+m11s21+m12sgncs21,τ1x˙23d=−x23d−x23c−sgncx23d−x23c,s22=x23−x23d,x24c=−a¯33x23−d^β+x˙23d−m21s22−m22sgncs22−a¯23s21,τ2x˙24d=−x24d−x24c−sgncx24d−x24c,s23=x24−x24d,x25c=−1bya¯43x23+d^ωy1−x˙24d+m31s23+m32sgncs23+s22,τ3x˙25d=−x25d−x25c−sgncx25d−x25c,s24=gδy−x25d,δyc=Nχ2δ¯yc,δ¯yc=−m41s24−m42sgncs24+μ2g′δyδy+x˙25d−bys23,where Nχ2=eχ22cosχ2, χ˙2=γχ2s24δ¯yc, μ2>0, γχ2>0, and mij>0; i=1,2,3,4, j=1,2.The control purpose of the roll channel is to prevent the missile body from rolling. The IGC algorithm for the roll channel is(90)s31=x31,x32c=−d^γ−n21s31−n22sgncs31,τ1x˙32d=−x32d−x32c−sgncx32d−x32,s32=x32−x32d,x33c=−1bxd^ωx1−x˙32d+n31s23+n32sgncs23s23+s31,τ2x˙33d=−x33d−x33c−sgncx33d−x33c,s33=gδx−x33d,δxc=Nχ3δ¯xc,δ¯xc=−n41s33−n42sgncs33+μ3g′δxδy+x˙33d−bxs32,where Nχ3=eχ32cosχ3, χ˙3=γχ3s33δ¯xc, μ3>0, γχ3>0, and nij>0, i=1,2,3,4, j=1,2.


## 4.1. Design of Pitch Channel Controller and Finite-Time Convergence Analysis

In order to solve the nonlinear saturation problem, the following auxiliary systems are introduced as(41)δ˙z=−μ1δz+δzc.Then, the control model of the IGC system in the pitch channel with input saturation and impact angle constraint is obtained as(42)x˙11=x12,x˙12=a22x12+a23x13+dε,x˙13=x14+a33x13+dα,x˙14=a43x13+b1gδz+dωz1,δ˙z=−μ1δz+δzc.The model (42) has a strict feedback structure, so the backstepping method combined with terminal sliding mode control can be used to design the controller. The specific design steps of the IGC algorithm are given below:Step 1. Select the following nonsingular terminal sliding surface with impact angle constraint [28] as(43)s11=x12+ρ1x11+ρ2φx11,with(44)φx11=x11b,x11≥σ,λ1x11+λ2x113,x11<σ,whereλ1=3−bσb−1/2, λ2=b−1σb−3/2, σ>0, and 0<b=p1/p2<1; p1 and p2 are positive odd numbers.Differentiatings11 with respect to time gives(45)s˙11=x˙12+ρ1x12+ρ2φ′x11x12,with(46)φ′x11=bx11b−1,x11≥σ,λ1+3λ2x112,x11<σ.Substituting (42) into (45), we have(47)s˙11=a22x12+a23x13+dε+ρ1+ρ2φ′x11x12.The virtual control lawx13c is designed as(48)x13c=−1a23a22+ρ1b1x11b1−1+ρ2φ′x11x12+d^ε+l11s11+l12s11c.A new virtual control lawx13d is introduced to avoid the differential expansion problem caused by the derivation of the control law and to ensure the fast convergence of the system in finite time. x13d is generated by x13c. A low-pass filter is designed to generate x13d and its derivatives x˙13d as(49)τ1x˙13d=−y11−y11c,wherey11=x13d−x13c, x13d0=x13c0.Remark 3.
The dynamic surface control used a first-order linear filterτ1x˙13d=y1 to solve the differential expansion problem of backstepping control. The performance of the filter is related to τ1. Reducing τ1 can improve the convergence speed and filtering accuracy, but too large will amplify the measurement noise. The filter (49) can improve the convergence speed and filtering accuracy by reducing τ1 and c, which reduces the selection requirement of τ1, and can meet the requirements of finite-time convergence.Step 2. Define the sliding mode surface as(50)s12=x13−x13d.Differentiatings12 with respect to time gives(51)s˙12=x˙13−x˙13d=x14+a33x13+dα−x˙13d.The virtual control lawx14c is designed as(52)x14c=−a33x13−d^α+x˙13d−l21s12−l22s12c−a23s11.Design the first-order nonlinear filter as(53)τ2x˙14d=−y12−y12c,wherey12=x14d−x14c, x14d0=x14c0.Step 3. Define the sliding mode surface as(54)s13=x14−x14d.Differentiatings13 with respect to time gives(55)s˙13=x˙14−x˙14d=a43x13+b1gδz+dωz1−x˙14d.The virtual control lawx15c is designed as(56)x15c=−1b1a43x13+d^ωz1−x˙14d+l31s13+l32s13c+s12.Design the first-order nonlinear filter as(57)τ3x˙15d=−y13−y13c,wherey13=x15d−x15c, x15d0=x15c0.Step 4. Define the sliding mode surface as(58)s14=gδz−x15d.
Differentiatings14 with respect to time gives(59)s˙14=g′δz−μ1δz+δzc−x˙15d,where g′δz=∂gδz/∂δz=4/eδz/δzmax+eδz/δzmax2.
The actual control lawδzc is designed as(60)δzc=Nχ1δ¯zc,δ¯zc=−l41s14−l42s14c+μ1g′δzδz+x˙15d−b1s13,with(61)Nχ1=eχ2cosχ,χ˙1=γχs14δ¯zc,where γχ>0 is the design parameter.Proposition 2.
For system (17), the designed IGC law (60) can make the states of the pitch channel closed-loop system converge to a smaller neighborhood of the origin in finite time.Proof.
From (43) to (60), we have(62)s˙11=a22x12+a23s12+y11+x13c+dε+ρ1+ρ2φ′x11x12=−l11s11−l12s11c+d˜ε+a23s12+y11,(63)s˙12=s13+y12+x14c+a33x13+dα−x˙13d=−l21s12−l22s12c+d˜α+s12+y12−a23s11,(64)s˙13=a43x13+bs14+y13+x15c+dωz1−x˙14d=−l31s13−l32s13c+d˜ωz1+bs14+y13−s12,(65)s˙14=g′δz−μ1δz+δzc+δ¯zc−δ¯zc−x˙15d=−l41s14−l42s14c−b1s13+g′δzNχ1−1δ¯zc,(66)y˙1i=x˙1id−x˙1ic=−1τiy1i+y1ic−x˙1ic,i=1,2,3,(67)y1iy˙1i=−1τiy1i2+y1i1+c−y1ix˙1ic≤−1τiy1i2+y1i1+c+y1ix˙1ic.
There exists a positive real numberξ1i such that x˙1ic≤ξ1i [29], so we can rewrite (67) as(68)y1iy˙1i≤−1τiy1i2+y1i1+c+y1iξ1i.
Construct the Lyapunov function as(69)V2=12∑i=14s1i2+12∑i=13y1i2.
DifferentiatingV2 with respect to time gives(70)V˙2=∑i=14s1is˙1i+∑i=13y1iy˙1i≤∑i=13−1τiy1i2+y1i1+c+y1iξ1i+s11−l11s11−l12s11c+d˜ε+a23s12+y11+s12−l21s12−l22s12c+d˜α+s13+y12−a23s11+s13−l31s13−l32s13c+d˜ωz1+bs14+y13−s12+s14−l41s14−l42s14c−bs13+g′δzNχ1−1δ¯zc=∑i=14−li1s1i2−li2s1i1+c+a23s11y11+s12y21+bs13y31+∑i=13−1τiy1i2+y1i1+c+y1iξ1i+s11d˜ε+s12d˜α+s13d˜ωz1+χ˙1γχg′δzNχ1−1.
Based on Young’s inequality, the inequality (70) can be rewritten as(71)V˙2≤∑i=14−li1s1i2−li2s1i1+c+a23s112+y1122+s122+y1222+bs132+y1322+s112+d˜ε2+s122+d˜α2+s132+d˜ωz122+∑i=13−1τiy1i2+y1i1+c+ξi12y1i2+14+χ˙1γχg′δzNχ1−1=−l11−a23+12s112−l21−1s122−l31−b+12s132−l41s142−∑i=14li2s1i1+c−1τ1−a232−ξ112y112−1τ2−12−ξ22y122−1τ3−b2−ξ32y132−∑i=13y1i1+cτi+ζ1+ζ2,with(72)ζ1=d˜ε2+d˜α2+d˜ωz12+32,ζ2=χ˙1γχg′δzNχ1−1.
Let(73)l11≥a23+12+12k,l21≥1+12k,l31≥b+12+12k,l41≥12k,li2≥12k,i=1,2,3,4,1τ1≥a23+ξ122+12k,1τ2≥1+ξ222+12k,1τ3≥b+ξ322+12k,where k>0; we can obtain(74)V˙2≤−kV2−k2∑i=14s1i21+c/2+∑i=13y1i21+c/2+ζ1+ζ2.
According to Lemma3, we have(75)V˙2≤−kV2−k2∑i=14s1i2+∑i=13y1i21+c/2+ζ=−kV2−kV21+c/2+ζ,where ζ=ζ1+ζ2.
Due toV2≥0, the following inequality holds:(76)V˙2≤−kV2+ζ.
Solving the inequality (76) results in(77)V2≤V20e−kt+ζ1k1−e−kt+e−Ktγχ∫0tg′δzNχ1−1χ˙e−kτdτ.
According to Lemma4, it can be seen that V2 and χ1 are bounded, and further, s1, s2, s3, s4, y1, y2, y3, and ζ are bounded. The following two forms can be obtained from (75) as(78)V˙2≤−k−ζV2V2−kV21+c/2.(79)V˙2≤−kV2−k−ζV21+c/2V21+c/2.
For (78), if k−ζ/V2>0, according to Lemma 1, V2 is finite-time convergent, and the convergence region is(80)V2≤Δ1=ζk.
For (79), similar to the analysis of (78), V2 can finite-time converge to the region(81)V2≤Δ2=ζk2/1+c.
Invoking (80) and (81), it is obtained that V2 can finite-time converge to the region Δ=minΔ1,Δ2. Therefore, s11 can finite-time converge to the region s11≤Δ=ϕ, and the convergence regions of x11 and x12 are discussed as follows:(1)
Ifx1≥σ, we can rewrite (43) as(82)x12=x˙11=−ρ1x11−ρ2x11b+ϕ1,ϕ1≤ϕ.Construct the Lyapunov function as(83)V2=x112.DifferentiatingV2 with respect to time and substituting (82) into it gives(84)V˙2=2x11x˙11=−2ρ1x12−2ρ2xib+1+2ϕ1x11≤−2ρ1x12−2ρ2xib+1+2ϕ1x11+2ϕx11=−2ρ1V2−2k2V2b+1/2+2ϕV21/2.From Lemma2, it can be seen that x1 can converge to region Ω1=x11=ϑ1θ1ϑ1b+θ2ϑ11−b<2ϕ in finite time, where θ1∈0,2ρ2, θ2∈0,2ρ1.By choosing parametersk1 and k2 reasonably, ϑ1≤δ can be achieved. At this time, the convergence domain of x1 is(85)x11≤maxϑ1,σ=δ.Combining (82), the convergence domain of x12 is(86)x12≤ρ1x11+ρ2x11b+ϕ1≤k1δ+k2δb+ϕ.(2)
Ifx1≤σ, we can rewrite (43) as(87)x12=x˙1=−ρ1x11−ρ2λ1x1+λ2x13+ϕ1≤ρ1x11−ρ2λ1x1+λ2x13+ϕ1≤ρ1δ+k2δb+ϕ.
In summary,x1 and x12 can converge to a small neighborhood of the origin in finite time, and the neighborhood is(88)x11≤δ,x12≤ρ1δ+k2δb+ϕ.The proof is complete.


## 4.2. Design of Yaw and Roll Channel Controller

Referring to the design process of the pitch channel, the integrated guidance control law of the yaw channel is designed as(89)s21=x22+ρ1sgnbx21+ρ2φx21,x23c=−1a¯23a¯22+ρ1+ρ2φ′x21x22+d^η+m11s21+m12sgncs21,τ1x˙23d=−x23d−x23c−sgncx23d−x23c,s22=x23−x23d,x24c=−a¯33x23−d^β+x˙23d−m21s22−m22sgncs22−a¯23s21,τ2x˙24d=−x24d−x24c−sgncx24d−x24c,s23=x24−x24d,x25c=−1bya¯43x23+d^ωy1−x˙24d+m31s23+m32sgncs23+s22,τ3x˙25d=−x25d−x25c−sgncx25d−x25c,s24=gδy−x25d,δyc=Nχ2δ¯yc,δ¯yc=−m41s24−m42sgncs24+μ2g′δyδy+x˙25d−bys23,where Nχ2=eχ22cosχ2, χ˙2=γχ2s24δ¯yc, μ2>0, γχ2>0, and mij>0; i=1,2,3,4, j=1,2.The control purpose of the roll channel is to prevent the missile body from rolling. The IGC algorithm for the roll channel is(90)s31=x31,x32c=−d^γ−n21s31−n22sgncs31,τ1x˙32d=−x32d−x32c−sgncx32d−x32,s32=x32−x32d,x33c=−1bxd^ωx1−x˙32d+n31s23+n32sgncs23s23+s31,τ2x˙33d=−x33d−x33c−sgncx33d−x33c,s33=gδx−x33d,δxc=Nχ3δ¯xc,δ¯xc=−n41s33−n42sgncs33+μ3g′δxδy+x˙33d−bxs32,where Nχ3=eχ32cosχ3, χ˙3=γχ3s33δ¯xc, μ3>0, γχ3>0, and nij>0, i=1,2,3,4, j=1,2.


## 5. Simulation Analysis

In this section, the performance of the proposed integrated guidance and control design method (PIGC) is verified based on the missile six-degree-of-freedom simulation. In the inertial coordinate system, the initial missile and target positions are set to (0 m, 5000 m, 0 m) and (8000 m, 1000 m, 500 m), and the speed of the missile isvm = 800 m/s. The missile’s initial ballistic inclination angle, ballistic deflection angle, angle of attack, sideslip angle, roll angle, pitch angle rate, yaw angle rate, and roll angle rate are θm0 = −10°, ψvm0 = −5°, α0 = 10°, β0 = 5°, ωx10 = 20°/s, ωy10 = −20°/s, and ωz10 = 30°/s. The termination condition of the simulation is that y of the missile in the inertial coordinate system is less than or equal to 0. The simulation step size is 0.01 s, and the fourth-order Runge–Kutta method is used to solve the model.The controller parameters are selected asρ1=0.6, ρ2=1, b=3/5, c=0.5, l11=l12=3, l21=l22=8, l31=l32=20, l41=l42=15, m11=m12=3, m21=m22=12, m31=m32=10, m41=m42=10, μ1=μ2=μ3=5, n21=n22=12, n31=n32=10, n41=n42=10, γχ1=γχ2=γχ3=0.001, δ=0.01, τ1=τ2=τ3=0.1.Case 1.
Attack target under different impact angle constraints.
Set the change range of the rudder deflection angle to [−15°, 15°], the expected LOS inclination angles are −50°, −60°, −70°, and −80°, and the expected LOS deflection angles are 10°, 5°, 0°, and −10°. The simulation results are shown in Figures2 to 6.
It can be seen from Figure2 that, for different expected LOS angles, the missile can effectively hit the target, and as the initial LOS angle deviation increases, the ballistic curvature also increases accordingly. It can be seen from Figure 3 that the LOS angles ε and η can effectively converge to the vicinity of the expected values εd and ηd in finite time, reflecting the finite-time convergence characteristics of PIGC. In these four cases, the miss distances were 0.55 m, 0.54 m, 0.77 m, and 0.85 m, indicating that PIGC has high hit precision.
As can be seen from Figures4 and 5, the missile’s attack angle, sideslip angle, roll angle, and attitude angular velocity can converge to near zero in finite time. However, there is a certain oscillation phenomenon in the early stage, which is mainly due to the finite-time convergence of the LOS angle and LOS angular rate. So, the missile needs a larger overload, that is, a larger rudder deflection angle. However, the initial rudder deflection angle is zero, so the rudder deflection angle in the early stage will change greatly, which will cause a large change for system states such as the attack angle. With the increase of the initial LOS angle error, the convergence time of the states also increases, which is mainly because the convergence time is related to the initial value of the LOS angle error. The larger the initial error, the longer the convergence time.
As can be seen from Figure6, the rudder deflection angle does not exceed the maximum limit and gradually converged to near zero in the later stages. From the simulation results, it can be seen that under different impact angle constraints, PIGC can accurately hit the target with the expected LOS angle, which shows the effectiveness of PIGC in dealing with the problem of impact angle constraints and input saturation.Figure 2
3D missile and target trajectory curves of Case1.Figure 3
Curves representing LOS elevation and azimuth angles, distance between missile and target of Case1.
(a)(b)(c)Figure 4
Curves representing attack, sideslip, and roll angles of Case1.
(a)(b)(c)Figure 5
Curves representing pitch, yaw, and roll angular rates of Case1.
(a)(b)(c)Figure 6
Curves representing rudder deflection angles of Case1.
(a)(b)(c)Case 2.
Comparative simulation with existing IGC algorithm.
The IGC algorithm based on traditional dynamic surface control (DIGC) and the robust IGC algorithm (RIGC) proposed in [13] are introduced in the simulation. Set the change range of the rudder deflection angle to [−10°, 10°], and the expected LOS inclination and deflection angles are −60° and 0°, respectively. It is assumed that there is uncertainty of 0.1 sin (0.2 πt) in the moment of inertia of the missile, and there is uncertainty of 0.2 sin (0.2 πt) in the aerodynamic moment and aerodynamic moment coefficient. The simulation results are shown in Figures 7 to 11 and Table 1.
As can be seen from Figure7, under the three IGC algorithms, the missile can smoothly fly towards the target. It can be seen from Figure 8 that the three IGC algorithms can gradually make the LOS angles ε and η gradually converge near the expected values εd and ηd. Among them, PIGC has the fastest convergence speed, reflecting its fast finite-time convergence characteristics. The LOS angles of the DIGC and RIGC have certain divergence trends at the end of the guidance phase, especially the DIGC divergence is more obvious. This is mainly because PIGC uses ESO to compensate the system disturbance and uncertainty, which enhances the robustness of the system. However, the robustness of DIGC and RIGC at the end of guidance phase cannot effectively counteract interference, and therefore divergences occur. The PIGC has the smallest miss distances of the three IGC algorithms, followed by RIGC, indicating that PIGC has higher guidance precision.
It can be seen from Figures9 and 10 that under PIGC and RIGC, the attack angle, sideslip angle, roll angle, and attitude angular velocity of the missile can converge to near zero in finite time. PIGC converges faster and its convergence accuracy is higher. As can be seen from Figure 11, the rudder deflection angles of the three channels of PIGC do not exceed the maximum limit, and they gradually converge to near zero in the later stages. This is because PIGC has considered and dealt with the problem of input constraint. DIGC and RIGC’s pitch and yaw channel rudder deflection angles have reached the upper limit of amplitude during the initial stage of guidance phase.
Table1 shows the simulation results of attack times, LOS angle errors, and miss distances under three IGC algorithms. It can be seen that compared to DIGC and RIGC, PIGC’s attack time, LOS angle errors, and miss distance are smallest, and it has high guidance precision and angular constraint precision.
Based on the analysis of the simulation results of the two simulation cases, it can be concluded that, under different impact angle constraints, PIGC can accurately hit the target with expected impact angle, and the miss distance is within 1 m. In addition, the state variables such as the LOS angle and the attack angle can be quickly converged in finite-time, and the rudder deflection angle is within the set maximum amplitude range. Compared with DIGC and RIGC, PIGC can hit the target with shorter time, smaller miss distance, and higher angle constraint precision, and its state variables converge faster. Simulation results show the effectiveness and superiority of PIGC.Figure 7
3D missile and target trajectory curves of Case2.Figure 8
Curves representing LOS elevation and azimuth angles, distance between missile and target of Case2.
(a)(b)(c)Figure 9
Curves representing attack, sideslip, and roll angles of Case2.
(a)(b)(c)Figure 10
Curves representing pitch, yaw, and roll angular rates of Case2.
(a)(b)(c)Figure 11
Curves representing rudder deflection angles of Case2.
(a)(b)(c)Table 1
Simulation results of Case2.
Control lawAttack time (s)Miss distance (m)LOS elevation angle deviation (°)LOS azimuth angle deviation (°)DIGC14.692.220.260.15RIGC14.991.187.944.71PIGC14.510.760.040.05


## 6. Conclusion

In this paper, a three-channel IGC algorithm with finite-time convergence is proposed. Through theoretical analysis and simulation verification, the following conclusions are reached:(1)
The proposed IGC algorithm can achieve finite-time convergence of the system states. Under different impact angle constraints, the missile can be guaranteed to hit the target accurately with expected impact angle. Compared with the existing IGC algorithms, it has higher guidance precision and better convergence performance.(2)
The Nussbaum gain function and an auxiliary system are used to effectively handle the problem of input saturation, which can ensure that the rudder deflection angle does not exceed the set range.



---

*Source: 5917983-2020-09-15.xml*