# Research on Three-Dimensional Integrated Guidance and Control Design with Multiple Constraints

Authors: Zhenhua Fu; Kuanqiao Zhang; Suochang Yang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

In this paper, a fixed-time convergence-integrated guidance and control (IGC) method is proposed, which considers terminal line-of-sight (LOS) angle constraint, full-state constraints, and input saturation. Firstly, an IGC design model considering the full coupling of three channels is constructed, and a fixed-time convergence disturbance observer is used to estimate and compensate the unknown disturbances in the model. Secondly, based on the fixed-time stability theory, sliding mode control, and backstepping control, a novel IGC scheme is carried out, and the second-order instruction filter is used to restrict the system states and control instructions. Furthermore, the fixed-time stability of the close-loop system is proved and the expression of convergence time upper bound is given. Finally, comparison simulation and Monte Carlo simulation verify the effectiveness and superiority of the proposed IGC algorithm.


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## Body

## 1. Introduction

The guidance and control system (GCS) is the core system of missile. The classical GCS is designed individually, which includes guidance loop and control loop. In contrast, the IGC design scheme takes guidance loop and control loop as a whole. Since the IGC design method can take advantage of the coupling relationship between guidance system and control system, as well as the comprehensive information, such as the information on the relative motion of the missile and the target, the attitude angle, and overload of the missile, it can significantly enhance the missile performance compared with the classical scheme [1, 2].There were many theories and methods for IGC scheme, including small gain theory [3], predictive control [4], variable structure control [5], adaptive control [6], and backstepping control [7]. Among them, backstepping control method was used widely. However, backstepping control had the problem that was called “exponential expansion,” which was due to the need to differentiate the virtual control quantity multiple times. To overcome this problem, dynamic surface control was proposed and applied to IGC design [8]. The derivatives of virtual quantities were obtained by introducing a first-order filter. In [9], a 3D IGC scheme is proposed by using sliding mode control and backstepping control. The scheme used the Nussbaum function and an auxiliary system to solve the problem of input saturation. However, it was a three-channel independent design method. In [10], the dynamic surface control and barrier Lyapunov function were used in IGC design, and a neural network disturbance observer was applied to estimate the system interferences.Many missiles, such as antitank missiles and antiaircraft missiles, can increase their warhead damage by controlling the terminal impact angles when attacking the target [11]. Therefore, terminal angle constraint such as LOS angle constraint can be considered in IGC design to improve the combat effectiveness of missile. In [12], a robust IGC design method with terminal LOS angle constraint was obtained based on dynamic inverse control, and the system uncertainties were restrained by introducing sliding mode control. The sign function was replaced by continuous high gain saturation function to solve the chattering problem inherent in sliding mode control. But it also changed the original structure of sliding mode control and reduced the robustness. In [13], an IGC scheme with terminal LOS angle constraint was designed based on dynamic surface control, and adaptive control was adopted to compensate system disturbances. However, dynamic surface control-based IGC design could only ensure that the system states converge asymptotically, but the missile attack time is limited.Because of the aerodynamic structure limitation, the missile system state variables, such as attack angle and Euler angular rates, should change within certain ranges during the flight. In the meanwhile, because of the actuator physical limitation, the control ability generated by the actuator is limited. Taking pneumatic rudder as an example, both positive and negative rudder deflection angles have upper bounds. If rudder deflection angles and system state variables exceed the allowed ranges, the missile guidance and control performance will be negatively affected. Even the missile will be uncontrollable and unstable, leading to off-target [14]. Therefore, state constraints and control constraint should be considered in the IGC design. In [15], dynamic surface control was used to carry out an IGC system, and the model uncertainty and disturbance were estimated by a proposed finite-time convergence disturbance observer. In addition, the Nussbaum function was used to deal with the control constraint, which was also known as input saturation. In [16], aiming at state constraints, an IGC system was designed based on dynamic surface control and barrier Lyapunov function. In order to solve the state constraints, a saturation function was introduced into the design to ensure that the state variables did not change beyond the constraint ranges.The above analysis shows that most of the literatures on IGC design research aim to solve unitary constraint problem, such as terminal LOS angle constraint, input saturation, state constraints, and finite-time convergence. However, the missile GCS was faced with a variety of constraints. As far as we know, there are few literatures dealing with the problems of input saturation, full-state constraints, terminal LOS angle constraint, and global finite-time convergence in 3D IGC scheme. For this issue, a fixed-time IGC algorithm for STT missile is proposed in this paper, which considers terminal LOS angle constraint, input saturation, and full-state constraints. The main contributions are as follows:(a)
A 3D IGC design model with multiple constraints is established rather than the single plane model established in [5](b)
A sliding mode surface which can strictly converge in fixed-time is designed to solve terminal angle constraint rather than common asymptotic stability(c)
The second-order instruction filter is constructed to estimate the derivative of the virtual commands, which effectively relieve the computation burden in backstepping design, and guarantee virtual commands in prescribed ranges. Furthermore, an auxiliary system is designed to eliminate the tracking error of the filter(d)
Different from the general asymptotic or finite-time convergence IGC algorithms proposed in [15–17], a 3D IGC algorithm is proposed to achieve global fixed-time convergence in the presence of extern disturbances, terminal angle constraint, full-state constraints, and input saturation


## 2. Problem Description and Preliminaries

### 2.1. Problem Description

Figure1 establishes the relative motion of missile and target in 3D inertial coordinate system Oxyz, where Ox4y4z4 is the LOS coordinate system. M represent the missile and T represent target. ε is LOS elevation angle and η is the azimuth angle. R is relative distance of missile and target. Define am4=amx4amy4amz4T and at4=atx4aty4atz4T which are acceleration vectors of missile and target in Ox4y4z4, respectively.Figure 1
Relative motion of missile and target.The missile and target relative motion equations [18] are
(1)2Ṙε̇+Rε¨+Rη̇2sinεcosε=aty4−amy4,−2Ṙη̇cosε−Rη¨cosε+2Rε̇η̇sinε=atz4−amz4.Defineam2=amx2amy2amz2T, which is missile acceleration vector in ballistic coordinate system. It can be obtained from the missile dynamic equations as
(2)amy2=PsinαcosγV+cosαsinβsinγV+YcosγV−ZsinγV−mgcosθmm,amz2=PsinαsinγV+cosαsinβcosγV+YsinγV+ZcosγVm,where m is missile mass, θm is fight path angle, α is attack angle, β is sideslip angle, γV is bank angle, Y is aerodynamic lift force, Z is lateral force, P is thrust force, and g is gravitational acceleration.From the coordinate transformation, we can get(3)am4=Lη,εL−1ψVm,θmam2,Lη,ε=cosεcosηsinε−cosεsinη−sinεcosηcosεsinεsinηsinη0cosη,LψVm,θm=cosθmcosψVmsinθm−cosθmsinψVm−sinθmcosψVmcosθmsinθmsinψVmsinψVm0cosψVm,where ψVm is the heading angle of missile.The aerodynamic forcesY and Z are modeled as
(4)Y=CyααQS+Yδz+dY,Z=CzββQS+Zδy+dZ,where Yδz and Zδy are lift force and lateral force produced by the rudder angles, Cyα and Czβ are aerodynamic coefficient with respect to α and β, Q=ρVm2/2 is dynamic pressure, S stands for the reference area, ρ is atmospheric density, and dY and dZ are the unknown bounded uncertainties including modeling error and model uncertainties, here treated as disturbance variables.Since the rudder deflection angles contribute little to the aerodynamic forces,Yδz and Zδy can be regarded as small quantities [19].In this paper, the IGC design applies only to the terminal guidance stage. Therefore, assuming that the missile flies without power and the flight speed is approximately invariant, the velocity change rate is regarded as interference. For STT missile, its required bank angle control is zero. So it can be assumed to be a small quantity, andsinγV≈0 and cosγV≈1. Define new state variables x1=ε̇η̇T and x2#=αβT. Combined with (1) to (4), the relative motion equation in 3D space can be described as
(5)ẋ1=f1x1+g1x1x2#+d1t,with
(6)f1x1=−2ṘRε̇−η̇2sinεcosε+fθmRgcosθm−2ṘRη̇+2ε̇η̇tanε+sinθmsinη−ψVmRcosεgcosθm,g1x1=−fθmmRCyαQS−sinεsinη−ψVmmRCzβQS−sinθmsinη−ψVmmRcosεCyαQScosη−ψVmmRcosεCzβQS,d1=aty4R−dYfθm+dZsinεsinη−ψVmmR−atz4Rcosε−dYsinθmsinη−ψVm−dZcosη−ψVmmRcosε,fθm=cosθmcosε+sinθmsinεcosη−ψVm.As is well known thatR is bounded, and target acceleration and aerodynamic parameters (Yδz, Zδy, dY, and dZ) are bounded, so d1t is bounded.According to missile dynamic equations [20], the missile control model can be described as
(7)α̇=−ωxtanβcosα+ωytanβsinα+ωz−CyααQS−mgcosθmmVmcosβ+Δα,β̇=ωxsinα+ωycosα+CzββQSmVm+Δβ,γ̇=ωx−ωytanϑcosγ+ωztanϑsinγ+Δγ,ω̇x=mxαα+mxββ+mxδxδxQSLJx+Jy−JzJxωyωz+Δωx,ω̇y=myββ+myδyδyQSLJy+Jz−JxJyωzωx+Δωy,ω̇z=mzαα+mzδzδzQSLJz+Jx−JyJzωxωy+Δωz,where γ is roll angle, Jx, Jy, and Jz are the inertia components, ωx, ωy, and ωz are Euler angular rates, mxα, mzα, mxβ, myβ, mxδx, myδy, and mzδz are moment aerodynamic coefficients with respect to α, β, δx, δy, and δz, which δx, δy, and δz are rudder angles, Δα, Δβ, Δγ, Δωx, Δωy, and Δωz are external disturbances and modeling errors, and ϑ is missile pitch angle.The missile terminal LOS angle constraint [21] can be described as
(8)εtf=εd,ηtf=ηd,where tf is the guidance final moment and εd and ηd are expected terminal LOS angles.Define new state variablesx0=ε−εdη−ηdT, x2=αβγT, x3=ωxωyωzT, and u=δxδyδzT. Combining with (5) and (7), the IGC design model with terminal LOS angle constraint is established as
(9)ẋ0=x1,ẋ1=f1x1+g1x1x2#+d1t,ẋ2=f2x2+g2ϑ,x2x3+d2t,ẋ3=f3x2,x3+g3u+d3t,with
(10)f2x2=−CyααQS−mgcosθmmVmcosβCzββQSmVm0,g2ϑ,x2=−tanβcosαtanβsinα1sinαcosα01−tanϑcosγtanϑsinγ,f3x2,x3=mxαα+mxββQSLJx+Jy−JzJxωyωzmyββQSLJy+Jz−JxJyωzωxmzααQSLJz+Jx−JyJzωxωy,g3=mxδxQSLJx000myδyQSLJy000mzδzQSLJz,d2t=ΔαΔβΔγ,d3t=ΔωxΔωyΔωz.Due to the existence of factors such as large target maneuvers, unsatisfactory initial states in the shift phase of middle and final guidance, external interference, and system uncertainty, the state variables such asα, β, γ, ωx, ωy, and ωz may exceed the allowable ranges, which will worsen the dynamic quality of the system, affect the control performance, and even lead to control divergence.In this paper, the research goal is to design an IGC algorithm that makes the missile system state variables and rudder angles not exceed the allowed ranges and the LOS angles and angular rates converge in fixed-time.


### 2.2. Preliminaries

The following definitions and lemmas are given for analysis convenience.Definition 1.
Definey=y1,y2,⋯,ynT, sgn⋅ is sign function and sgn0=0, sgnay=yasgny, and ya=diagy1a,⋯,yna.Definition 2.
For a nonlinear systemẋt=fxt, x0=x0, if there is a moment Tmax>0, for any x0∈R, t>Tmax, satisfying xt=0, the system is fixed-time stable.Lemma 3 (see [22, 23]).
For the systemẏ=−a1sgnb1y−a2sgnb2y, and a1>0, a2>0, 0<b1<1, and b2>1, then the system is stable in fixed-time and the convergence time satisfies
(11)T1<1a11−b1+1a2b2−1.
In addition, if the system has a small disturbance, that isẏ=−a1sgnb1y−a2sgnb2y+ς, ς is a small positive number, then the system can converge to the neighborhood of the origin Ω=y≤2ϑa1ϑb1+a2b1ϑb2=ς in fixed-time, and the convergence time satisfies
(12)T2<1a12b1−11−b1+1a2b2−1.Lemma 4 (see [24]).
Assuming that the Lyapunov functionVx satisfies V̇x≤−a1Vb1x−a2Vb2x, and a1>0, a2>0, 0<b1<1, and b2>1, then the system is stable in fixed-time and the convergence time satisfies
(13)T3<1a11−b1+1a2b2−1.Lemma 5 (see [25]).
For any real numberxi, i=1,2,⋯,n, there are real numbers 0<a<1 and b>1, such that the following equations are true
(14)∑i=1nxia≤∑i=1nxia,∑i=1nxib≤nb−1∑i=1nxib.
According to Lemma5, the following equations hold
(15)xTsgnax=∑i=1nxi2a+1/2≥∑i=1nxi2a+1/2=xa+1,xTsgnbx≥n1−b+1/2∑i=1nxi2b+1/2=n1−b/2xb+1.


## 2.1. Problem Description

Figure1 establishes the relative motion of missile and target in 3D inertial coordinate system Oxyz, where Ox4y4z4 is the LOS coordinate system. M represent the missile and T represent target. ε is LOS elevation angle and η is the azimuth angle. R is relative distance of missile and target. Define am4=amx4amy4amz4T and at4=atx4aty4atz4T which are acceleration vectors of missile and target in Ox4y4z4, respectively.Figure 1
Relative motion of missile and target.The missile and target relative motion equations [18] are
(1)2Ṙε̇+Rε¨+Rη̇2sinεcosε=aty4−amy4,−2Ṙη̇cosε−Rη¨cosε+2Rε̇η̇sinε=atz4−amz4.Defineam2=amx2amy2amz2T, which is missile acceleration vector in ballistic coordinate system. It can be obtained from the missile dynamic equations as
(2)amy2=PsinαcosγV+cosαsinβsinγV+YcosγV−ZsinγV−mgcosθmm,amz2=PsinαsinγV+cosαsinβcosγV+YsinγV+ZcosγVm,where m is missile mass, θm is fight path angle, α is attack angle, β is sideslip angle, γV is bank angle, Y is aerodynamic lift force, Z is lateral force, P is thrust force, and g is gravitational acceleration.From the coordinate transformation, we can get(3)am4=Lη,εL−1ψVm,θmam2,Lη,ε=cosεcosηsinε−cosεsinη−sinεcosηcosεsinεsinηsinη0cosη,LψVm,θm=cosθmcosψVmsinθm−cosθmsinψVm−sinθmcosψVmcosθmsinθmsinψVmsinψVm0cosψVm,where ψVm is the heading angle of missile.The aerodynamic forcesY and Z are modeled as
(4)Y=CyααQS+Yδz+dY,Z=CzββQS+Zδy+dZ,where Yδz and Zδy are lift force and lateral force produced by the rudder angles, Cyα and Czβ are aerodynamic coefficient with respect to α and β, Q=ρVm2/2 is dynamic pressure, S stands for the reference area, ρ is atmospheric density, and dY and dZ are the unknown bounded uncertainties including modeling error and model uncertainties, here treated as disturbance variables.Since the rudder deflection angles contribute little to the aerodynamic forces,Yδz and Zδy can be regarded as small quantities [19].In this paper, the IGC design applies only to the terminal guidance stage. Therefore, assuming that the missile flies without power and the flight speed is approximately invariant, the velocity change rate is regarded as interference. For STT missile, its required bank angle control is zero. So it can be assumed to be a small quantity, andsinγV≈0 and cosγV≈1. Define new state variables x1=ε̇η̇T and x2#=αβT. Combined with (1) to (4), the relative motion equation in 3D space can be described as
(5)ẋ1=f1x1+g1x1x2#+d1t,with
(6)f1x1=−2ṘRε̇−η̇2sinεcosε+fθmRgcosθm−2ṘRη̇+2ε̇η̇tanε+sinθmsinη−ψVmRcosεgcosθm,g1x1=−fθmmRCyαQS−sinεsinη−ψVmmRCzβQS−sinθmsinη−ψVmmRcosεCyαQScosη−ψVmmRcosεCzβQS,d1=aty4R−dYfθm+dZsinεsinη−ψVmmR−atz4Rcosε−dYsinθmsinη−ψVm−dZcosη−ψVmmRcosε,fθm=cosθmcosε+sinθmsinεcosη−ψVm.As is well known thatR is bounded, and target acceleration and aerodynamic parameters (Yδz, Zδy, dY, and dZ) are bounded, so d1t is bounded.According to missile dynamic equations [20], the missile control model can be described as
(7)α̇=−ωxtanβcosα+ωytanβsinα+ωz−CyααQS−mgcosθmmVmcosβ+Δα,β̇=ωxsinα+ωycosα+CzββQSmVm+Δβ,γ̇=ωx−ωytanϑcosγ+ωztanϑsinγ+Δγ,ω̇x=mxαα+mxββ+mxδxδxQSLJx+Jy−JzJxωyωz+Δωx,ω̇y=myββ+myδyδyQSLJy+Jz−JxJyωzωx+Δωy,ω̇z=mzαα+mzδzδzQSLJz+Jx−JyJzωxωy+Δωz,where γ is roll angle, Jx, Jy, and Jz are the inertia components, ωx, ωy, and ωz are Euler angular rates, mxα, mzα, mxβ, myβ, mxδx, myδy, and mzδz are moment aerodynamic coefficients with respect to α, β, δx, δy, and δz, which δx, δy, and δz are rudder angles, Δα, Δβ, Δγ, Δωx, Δωy, and Δωz are external disturbances and modeling errors, and ϑ is missile pitch angle.The missile terminal LOS angle constraint [21] can be described as
(8)εtf=εd,ηtf=ηd,where tf is the guidance final moment and εd and ηd are expected terminal LOS angles.Define new state variablesx0=ε−εdη−ηdT, x2=αβγT, x3=ωxωyωzT, and u=δxδyδzT. Combining with (5) and (7), the IGC design model with terminal LOS angle constraint is established as
(9)ẋ0=x1,ẋ1=f1x1+g1x1x2#+d1t,ẋ2=f2x2+g2ϑ,x2x3+d2t,ẋ3=f3x2,x3+g3u+d3t,with
(10)f2x2=−CyααQS−mgcosθmmVmcosβCzββQSmVm0,g2ϑ,x2=−tanβcosαtanβsinα1sinαcosα01−tanϑcosγtanϑsinγ,f3x2,x3=mxαα+mxββQSLJx+Jy−JzJxωyωzmyββQSLJy+Jz−JxJyωzωxmzααQSLJz+Jx−JyJzωxωy,g3=mxδxQSLJx000myδyQSLJy000mzδzQSLJz,d2t=ΔαΔβΔγ,d3t=ΔωxΔωyΔωz.Due to the existence of factors such as large target maneuvers, unsatisfactory initial states in the shift phase of middle and final guidance, external interference, and system uncertainty, the state variables such asα, β, γ, ωx, ωy, and ωz may exceed the allowable ranges, which will worsen the dynamic quality of the system, affect the control performance, and even lead to control divergence.In this paper, the research goal is to design an IGC algorithm that makes the missile system state variables and rudder angles not exceed the allowed ranges and the LOS angles and angular rates converge in fixed-time.


## 2.2. Preliminaries

The following definitions and lemmas are given for analysis convenience.Definition 1.
Definey=y1,y2,⋯,ynT, sgn⋅ is sign function and sgn0=0, sgnay=yasgny, and ya=diagy1a,⋯,yna.Definition 2.
For a nonlinear systemẋt=fxt, x0=x0, if there is a moment Tmax>0, for any x0∈R, t>Tmax, satisfying xt=0, the system is fixed-time stable.Lemma 3 (see [22, 23]).
For the systemẏ=−a1sgnb1y−a2sgnb2y, and a1>0, a2>0, 0<b1<1, and b2>1, then the system is stable in fixed-time and the convergence time satisfies
(11)T1<1a11−b1+1a2b2−1.
In addition, if the system has a small disturbance, that isẏ=−a1sgnb1y−a2sgnb2y+ς, ς is a small positive number, then the system can converge to the neighborhood of the origin Ω=y≤2ϑa1ϑb1+a2b1ϑb2=ς in fixed-time, and the convergence time satisfies
(12)T2<1a12b1−11−b1+1a2b2−1.Lemma 4 (see [24]).
Assuming that the Lyapunov functionVx satisfies V̇x≤−a1Vb1x−a2Vb2x, and a1>0, a2>0, 0<b1<1, and b2>1, then the system is stable in fixed-time and the convergence time satisfies
(13)T3<1a11−b1+1a2b2−1.Lemma 5 (see [25]).
For any real numberxi, i=1,2,⋯,n, there are real numbers 0<a<1 and b>1, such that the following equations are true
(14)∑i=1nxia≤∑i=1nxia,∑i=1nxib≤nb−1∑i=1nxib.
According to Lemma5, the following equations hold
(15)xTsgnax=∑i=1nxi2a+1/2≥∑i=1nxi2a+1/2=xa+1,xTsgnbx≥n1−b+1/2∑i=1nxi2b+1/2=n1−b/2xb+1.


## 3. IGC Design and Stability Analysis

The IGC model (9) established in this paper is a strict-feedback nonlinear system with uncertainty, so the backstepping control design method is used to accomplish the IGC scheme. The backstepping control is improved based on fixed-time stability theory to let the system states fixed-time convergence. A second-order instruction filter is introduced into backstepping control to overcome “differential expansion.” On the one hand, the derivatives of virtual control quantities are realized through integration, and the noise influence on the system is effectively reduced. On the other hand, the system states and actual control instructions are constrained to ensure that they do not exceed the allowable ranges.


### 3.1. Novel IGC Law Design

For the missile attacking the target accurately with desired terminal LOS angle, that is, the system statesx0 and x1 fixed-time converge to zero, based on Lemma 5 and piecewise sliding mode surface idea, a fixed-time convergence sliding mode is designed as
(16)s1=x1+k1sgna1x0+k2φx0,where φx0=φx01,φx02T,
(17)φx0i=x0ia2,x0i≥δ,λ1x0i+λ2x1i3,x1i<δ,where a1>1, 0<a2=p1/p2<1, ki=diagki1,ki2 is the positive definite matrix to be designed, i=1,2, δ>0, p1, and p2 are odd numbers, λ1=3−a2δα2−1/2, and λ2=a2−1δα2−3/2.The time derivative of (16) is given by
(18)ṡ1=ẋ1+k1a1x0a1−1x1+k2φ′x0x1,with
(19)φ′x0i=a2x0ia2−1,x0i≥δ,λ1+3λ2x0i2,x0i<δ,i=1,2.Based on fixed-time stability theory, backstepping control, and the above sliding mode surface, the design process of the novel IGC scheme with multiple constraints are as follows:Step 1.
Define the tracking error as(20)s1=x1+k1sgna1x0+k2φx0.
The time derivative ofs1 is given by
(21)ṡ1=ẋ1+k1a1x0a1−1x1+k2φ′x0x1=f1+g1x2#+d1+k1a1x0a1−1x1+k2φ′x0x1.
Design the virtual control law ofx2# as
(22)x2c#=−g1−1f1+d^1+k1a1x0a1−1x1+k2φ′x0x1+k3sgnb1s1+k4sgnb2s1+v1,where b1>1, 0<b2=p3/p4<1, p3 and p4 are odd numbers, k3 and k4 are positive definite diagonal matrixes to be designed, and v1 is filter tracking error compensation, which is used to compensate the influence of instruction filter on virtual control law, and its definition will be given later. d^1 is the estimation of d1. Here, the observer proposed in [26] is used to estimate the disturbances, and the specific expression is
(23)x^̇1=f1x1+g1x1x2#+d^1,d^1=λ11sgnγ1x~1+λ12sgnγ2x~1+λ13∫0tsgnx~1τdτ,where λ1i>0i=1,2,3, x~1=x1−x^1, and x^1 is the estimation of x1.
In order to avoid “differential expansion” and considering the limitation of virtual control variables, a new virtual control law was introduced by referring to the dynamic surface design method.x2d# and its derivative ẋ2d# are obtained by passing through a second-order instruction filter. The state space expression of the filter is
(24)q̇1q̇2=q22ζωnωn22ζωnSMxc−q1−q2,(25)xdẋd=q1q2,where SM⋅ is the saturation function, which is defined as
(26)SMx=Msgnx,x≥M,x,x<M.
It can be seen from (24) and (25) that the calculation of ẋd does not need a differentiator, thus avoiding the problem of “differential expansion.” If xc is bounded, then both xd and ẋd are continuously bounded. By selecting a larger natural frequency ωn, xd−xc can be made small. However, if it is too large, it will bring high frequency noise to the system, so compromise should be considered when selecting parameters.
The expression ofv1 is
(27)ν̇1=−l1sgnb1v1+sgnb2v1+v1s1Tg1x2d#−x2c#v12,v1>μ1,0,v1≤μ1,where l1 is the positive definite diagonal matrix to be designed and μ1 is a smaller positive number.Step 2.
Define the tracking error ass2=x2−x2d and x2d=x2d#;0. The time derivative of s2 is given by
(28)ṡ2=ẋ2−ẋ2d=f2+g2x3+d2−ẋ2d.
The virtual control quantity ofx3 is designed as
(29)x3c=−g2−1f2+ẋ2d+d^2+k5sgnb1s2+k6sgnb2s2+v2+χ,where k5 and k6 are positive definite diagonal matrixes to be designed, χ=g1Ts10T.
Letx3c pass through the second-order filter to get x3d and ẋ3d. The expression of v2 is
(30)ν̇2=−l2sgnb1v2+sgnb2v2+v2s2Tg2x3d−x3cv22,v2>μ2,0,v2≤μ2.Step 3.
The tracking error is defined ass3=x3−x3d. The time derivative of s3 is given by
(31)ṡ3=ẋ3−ẋ3d=f3+g3u+d3−ẋ3d.
The virtual control law ofu is designed as
(32)uc=−g3−1f3+ẋ3d+d^3+k7sgnb1s2+k8sgnb2s2+v3+g2Ts2.
Letuc pass through the second-order filter to get the actual IGC law u. The expression of v3 is
(33)ν̇3=−l3sgnb1v3+sgnb2v3−v3s3Tg3u−ucv32,v3>μ3,0,v3≤μ3.
The IGC algorithm structure is illustrated in Figure2.Figure 2
Multiple constraint fixed-time IGC algorithm structure.


### 3.2. Stability Analysis

Theorem 6.
For the system (9), the proposed novel IGC law from (20) to (33) can guarantee that the system states x0 and x1 converge to the origin neighborhood in fixed-time, and the closed-loop IGC system is stable in spite of disturbances.Proof.
From (21) to (32), we can obtain
(34)ṡ1=f1+g1s2#+x2d#−x2c#+x2c#+d1+k1a1x0a1−1x1+k2φ′x0x1=g1s2#+x2d#−x2c#−k3sgnb1s1−k4sgnb2s1−v1+d~1,ṡ2=f2+g2s3+x3d−x3c+x3c+d2−ẋ2d=g2s3+x3d−x3c−k5sgnb1s2−k6sgnb2s2−v2−χ+d~2,ṡ3=f3+g3u−uc+uc+d3−ẋ3d=g3u−uc−g2Ts2−k7sgnb1s3−k8sgnb2s3−v3+d~3,where d~i=di−d^ii=1,2,3 and s2#=s21s22T.
Construct the Lyapunov function(35)V=12∑i=13siTsi+viTvi.
The time derivative ofV is obtained as
(36)V̇=∑i=13siTṡi+viTvi=s1Tg1s2#+x2d#−x2c#+d~1−k3sgnb1s1−k4sgnb2s1−v1+s2Tg2s3+x3d−x3c+d~2−k5sgnb1s2−k6sgnb2s2−v2−χ+s3Tg3u−uc+d~3−k7sgnb1s3−k8sgnb2s3−v3−g2Ts2−v1Tl1sgnb1v1+sgnb2v1−s1Tg1x2d#−x2c#−v2Tl2sgnb1v2+sgnb2v2−s2Tg2x3d−x3c−v3Tl3sgnb1v3+sgnb2v3−s3Tg3u−uc=s1Td~1−s1Tk3sgnb1s1−s1Tk4sgnb2s1−s1Tv1+s2Td~2−s2Tk5sgnb1s2−s2Tk6sgnb2s2−s2Tv2+s3Td~3−s3Tk7sgnb1s3−s3Tk8sgnb2s3−s3Tv3−v1Tl1sgnb1v1+sgnb2v1−v2Tl2sgnb1v2+sgnb2v2−v3Tl3sgnb1v3+sgnb2v3.
According to (15), it can be obtained as
(37)V̇≤−31−b1/2∑i=13λmink2i+1sib1+1+λminlivib1+1−∑i=13λmink2i+2sib2+1+λminlivib2+1+∑i=13siTd~i−siTvi.
Based on the Young inequality, we can get(38)siTd~i≤12siTsi+12d~iTd~i,(39)−siTvi≤12siTsi+12viTvi.
Consider the following inequality:(40)p≤pm1+pm2,where p≥0, m1>1, and 0<m2<1. Then, the following inequality holds
(41)−n1pm1−n2pm2+n3p≤−n1−n3pm1−n2−n3pm2,where n1>0, n2>0, and n3>0.
Combining with (38), (39), and (41), equation (37) can be written as
(42)V̇≤−31−b1/2∑i=13λmink2i+1−1sib1+1+λminli−12vib1+1−∑i=13λmink2i+2−1sib2+1+λminli−12vib2+1+12∑i=13d~iTd~i.
Select appropriate parameters satisfying(43)λmink2i+1−1≥18b1−1/2k,λminli−12≥18b1−1/2k,λmink2i+2−1≥k,σ=12d~1Td~1+d~2Td~2+d~3Td~3,i=1,2,3,where k>0.
Then, equation (42) can be written as
(44)V̇≤−6b1−1/2k∑i=13si2b1+1/2+vi2b1+1/2−k∑i=13si2b2+1/2+vi2b2+1/2+σ.
By Lemma5 and (44), one can obtain
(45)V̇≤−k∑i=13si2+vi2b1+1/2+σ−k∑i=13si2+vi2b2+1/2=−kVb1+1/2−kVb2+1/2+σ.
Sinceb1+1/2>1 and 1/2<b2+1/2<1, according to Lemma 4, V can fixed-time converge to a small neighborhood as
(46)Ω1=V≤2ϑ1kϑ1b1+1/2+kϑ1b2+1/2=σ.
And the convergence time satisfies(47)t1<1kb1−1+1k2b2+1/2−11−b2.
Combining with (46) and (35), s1 can converge in fixed-time t1 to the region s1≤2ϑ1. Define Δ=2ϑ1, then s1i≤Δ, i=1,2. According to the region where the value of x0i is located, the convergence region of x0i and x1i can be obtained as follows.
(a)
Whenx0i>δ, it can be obtained from (16) that(48)ẋ0i=−k1isgna1x0i−k2isgna2x0i+ϕi,ϕi≤Δ
According to Lemma4, x0i can converge to the region Ω2i=x0i≤2ϑ2ik1iϑ2ia1+k2iϑ2ia2≤Δ, x1i can converge to the region
(49)x1i≤k1ix0ia1+k2ix0ia2+ϕi≤k1i2ϑ2ia1+k2i2ϑ2ia2+Δ,and convergence time satisfies
(50)t2<1k1ia1−1+1k2i2a2−11−a2.(b)
Whenx0i<δ, it can be obtained from (16) that(51)x1i≤k1ix0ia1+k2iλ1x0i+λ2x1i3+ϕi<k1iδa1+k2iδa2+Δ
Based on the above analyses,x0i and x1i can converge to the neighborhood
(52)x0i≤κi=max2ϑ2i,δ,k1iϑ2ia1+k2iϑ2ia2≤Δ,x1i≤k1κia1+k2κia2+Δ,with fixed-time. In addition, we can get the convergence time which satisfies
(53)t≤t1+t2<tm=1kb1−1+1k2b2+1/2−11−b2+1k1ia1−1+1k2i2a2−11−a2.
It can be seen from (53) that the upper bound of convergence time tm increases with the increase of k, k1i, k2i, a1, and b1 and increases with the decrease of a2 and b2. According to (43), k is related to the control parameter matrices kii=3,4,⋯,8 and ljj=1,2,3. Therefore, tm only depends on the control parameters, independent of the system states.
Although the convergence speed of the IGC system states will increase as the parametersa1, b1, kii=1,2,⋯,8, and ljj=1,2,3 increase and decrease as the parameters a2 and b2 increase, excessive pursuit of convergence speed will produce large control instructions, requiring the missile to provide a large overload, and the missile available overload is limited, so compromise should be considered in parameter selection.
This completes the proof.


## 3.1. Novel IGC Law Design

For the missile attacking the target accurately with desired terminal LOS angle, that is, the system statesx0 and x1 fixed-time converge to zero, based on Lemma 5 and piecewise sliding mode surface idea, a fixed-time convergence sliding mode is designed as
(16)s1=x1+k1sgna1x0+k2φx0,where φx0=φx01,φx02T,
(17)φx0i=x0ia2,x0i≥δ,λ1x0i+λ2x1i3,x1i<δ,where a1>1, 0<a2=p1/p2<1, ki=diagki1,ki2 is the positive definite matrix to be designed, i=1,2, δ>0, p1, and p2 are odd numbers, λ1=3−a2δα2−1/2, and λ2=a2−1δα2−3/2.The time derivative of (16) is given by
(18)ṡ1=ẋ1+k1a1x0a1−1x1+k2φ′x0x1,with
(19)φ′x0i=a2x0ia2−1,x0i≥δ,λ1+3λ2x0i2,x0i<δ,i=1,2.Based on fixed-time stability theory, backstepping control, and the above sliding mode surface, the design process of the novel IGC scheme with multiple constraints are as follows:Step 1.
Define the tracking error as(20)s1=x1+k1sgna1x0+k2φx0.
The time derivative ofs1 is given by
(21)ṡ1=ẋ1+k1a1x0a1−1x1+k2φ′x0x1=f1+g1x2#+d1+k1a1x0a1−1x1+k2φ′x0x1.
Design the virtual control law ofx2# as
(22)x2c#=−g1−1f1+d^1+k1a1x0a1−1x1+k2φ′x0x1+k3sgnb1s1+k4sgnb2s1+v1,where b1>1, 0<b2=p3/p4<1, p3 and p4 are odd numbers, k3 and k4 are positive definite diagonal matrixes to be designed, and v1 is filter tracking error compensation, which is used to compensate the influence of instruction filter on virtual control law, and its definition will be given later. d^1 is the estimation of d1. Here, the observer proposed in [26] is used to estimate the disturbances, and the specific expression is
(23)x^̇1=f1x1+g1x1x2#+d^1,d^1=λ11sgnγ1x~1+λ12sgnγ2x~1+λ13∫0tsgnx~1τdτ,where λ1i>0i=1,2,3, x~1=x1−x^1, and x^1 is the estimation of x1.
In order to avoid “differential expansion” and considering the limitation of virtual control variables, a new virtual control law was introduced by referring to the dynamic surface design method.x2d# and its derivative ẋ2d# are obtained by passing through a second-order instruction filter. The state space expression of the filter is
(24)q̇1q̇2=q22ζωnωn22ζωnSMxc−q1−q2,(25)xdẋd=q1q2,where SM⋅ is the saturation function, which is defined as
(26)SMx=Msgnx,x≥M,x,x<M.
It can be seen from (24) and (25) that the calculation of ẋd does not need a differentiator, thus avoiding the problem of “differential expansion.” If xc is bounded, then both xd and ẋd are continuously bounded. By selecting a larger natural frequency ωn, xd−xc can be made small. However, if it is too large, it will bring high frequency noise to the system, so compromise should be considered when selecting parameters.
The expression ofv1 is
(27)ν̇1=−l1sgnb1v1+sgnb2v1+v1s1Tg1x2d#−x2c#v12,v1>μ1,0,v1≤μ1,where l1 is the positive definite diagonal matrix to be designed and μ1 is a smaller positive number.Step 2.
Define the tracking error ass2=x2−x2d and x2d=x2d#;0. The time derivative of s2 is given by
(28)ṡ2=ẋ2−ẋ2d=f2+g2x3+d2−ẋ2d.
The virtual control quantity ofx3 is designed as
(29)x3c=−g2−1f2+ẋ2d+d^2+k5sgnb1s2+k6sgnb2s2+v2+χ,where k5 and k6 are positive definite diagonal matrixes to be designed, χ=g1Ts10T.
Letx3c pass through the second-order filter to get x3d and ẋ3d. The expression of v2 is
(30)ν̇2=−l2sgnb1v2+sgnb2v2+v2s2Tg2x3d−x3cv22,v2>μ2,0,v2≤μ2.Step 3.
The tracking error is defined ass3=x3−x3d. The time derivative of s3 is given by
(31)ṡ3=ẋ3−ẋ3d=f3+g3u+d3−ẋ3d.
The virtual control law ofu is designed as
(32)uc=−g3−1f3+ẋ3d+d^3+k7sgnb1s2+k8sgnb2s2+v3+g2Ts2.
Letuc pass through the second-order filter to get the actual IGC law u. The expression of v3 is
(33)ν̇3=−l3sgnb1v3+sgnb2v3−v3s3Tg3u−ucv32,v3>μ3,0,v3≤μ3.
The IGC algorithm structure is illustrated in Figure2.Figure 2
Multiple constraint fixed-time IGC algorithm structure.


## 3.2. Stability Analysis

Theorem 6.
For the system (9), the proposed novel IGC law from (20) to (33) can guarantee that the system states x0 and x1 converge to the origin neighborhood in fixed-time, and the closed-loop IGC system is stable in spite of disturbances.Proof.
From (21) to (32), we can obtain
(34)ṡ1=f1+g1s2#+x2d#−x2c#+x2c#+d1+k1a1x0a1−1x1+k2φ′x0x1=g1s2#+x2d#−x2c#−k3sgnb1s1−k4sgnb2s1−v1+d~1,ṡ2=f2+g2s3+x3d−x3c+x3c+d2−ẋ2d=g2s3+x3d−x3c−k5sgnb1s2−k6sgnb2s2−v2−χ+d~2,ṡ3=f3+g3u−uc+uc+d3−ẋ3d=g3u−uc−g2Ts2−k7sgnb1s3−k8sgnb2s3−v3+d~3,where d~i=di−d^ii=1,2,3 and s2#=s21s22T.
Construct the Lyapunov function(35)V=12∑i=13siTsi+viTvi.
The time derivative ofV is obtained as
(36)V̇=∑i=13siTṡi+viTvi=s1Tg1s2#+x2d#−x2c#+d~1−k3sgnb1s1−k4sgnb2s1−v1+s2Tg2s3+x3d−x3c+d~2−k5sgnb1s2−k6sgnb2s2−v2−χ+s3Tg3u−uc+d~3−k7sgnb1s3−k8sgnb2s3−v3−g2Ts2−v1Tl1sgnb1v1+sgnb2v1−s1Tg1x2d#−x2c#−v2Tl2sgnb1v2+sgnb2v2−s2Tg2x3d−x3c−v3Tl3sgnb1v3+sgnb2v3−s3Tg3u−uc=s1Td~1−s1Tk3sgnb1s1−s1Tk4sgnb2s1−s1Tv1+s2Td~2−s2Tk5sgnb1s2−s2Tk6sgnb2s2−s2Tv2+s3Td~3−s3Tk7sgnb1s3−s3Tk8sgnb2s3−s3Tv3−v1Tl1sgnb1v1+sgnb2v1−v2Tl2sgnb1v2+sgnb2v2−v3Tl3sgnb1v3+sgnb2v3.
According to (15), it can be obtained as
(37)V̇≤−31−b1/2∑i=13λmink2i+1sib1+1+λminlivib1+1−∑i=13λmink2i+2sib2+1+λminlivib2+1+∑i=13siTd~i−siTvi.
Based on the Young inequality, we can get(38)siTd~i≤12siTsi+12d~iTd~i,(39)−siTvi≤12siTsi+12viTvi.
Consider the following inequality:(40)p≤pm1+pm2,where p≥0, m1>1, and 0<m2<1. Then, the following inequality holds
(41)−n1pm1−n2pm2+n3p≤−n1−n3pm1−n2−n3pm2,where n1>0, n2>0, and n3>0.
Combining with (38), (39), and (41), equation (37) can be written as
(42)V̇≤−31−b1/2∑i=13λmink2i+1−1sib1+1+λminli−12vib1+1−∑i=13λmink2i+2−1sib2+1+λminli−12vib2+1+12∑i=13d~iTd~i.
Select appropriate parameters satisfying(43)λmink2i+1−1≥18b1−1/2k,λminli−12≥18b1−1/2k,λmink2i+2−1≥k,σ=12d~1Td~1+d~2Td~2+d~3Td~3,i=1,2,3,where k>0.
Then, equation (42) can be written as
(44)V̇≤−6b1−1/2k∑i=13si2b1+1/2+vi2b1+1/2−k∑i=13si2b2+1/2+vi2b2+1/2+σ.
By Lemma5 and (44), one can obtain
(45)V̇≤−k∑i=13si2+vi2b1+1/2+σ−k∑i=13si2+vi2b2+1/2=−kVb1+1/2−kVb2+1/2+σ.
Sinceb1+1/2>1 and 1/2<b2+1/2<1, according to Lemma 4, V can fixed-time converge to a small neighborhood as
(46)Ω1=V≤2ϑ1kϑ1b1+1/2+kϑ1b2+1/2=σ.
And the convergence time satisfies(47)t1<1kb1−1+1k2b2+1/2−11−b2.
Combining with (46) and (35), s1 can converge in fixed-time t1 to the region s1≤2ϑ1. Define Δ=2ϑ1, then s1i≤Δ, i=1,2. According to the region where the value of x0i is located, the convergence region of x0i and x1i can be obtained as follows.
(a)
Whenx0i>δ, it can be obtained from (16) that(48)ẋ0i=−k1isgna1x0i−k2isgna2x0i+ϕi,ϕi≤Δ
According to Lemma4, x0i can converge to the region Ω2i=x0i≤2ϑ2ik1iϑ2ia1+k2iϑ2ia2≤Δ, x1i can converge to the region
(49)x1i≤k1ix0ia1+k2ix0ia2+ϕi≤k1i2ϑ2ia1+k2i2ϑ2ia2+Δ,and convergence time satisfies
(50)t2<1k1ia1−1+1k2i2a2−11−a2.(b)
Whenx0i<δ, it can be obtained from (16) that(51)x1i≤k1ix0ia1+k2iλ1x0i+λ2x1i3+ϕi<k1iδa1+k2iδa2+Δ
Based on the above analyses,x0i and x1i can converge to the neighborhood
(52)x0i≤κi=max2ϑ2i,δ,k1iϑ2ia1+k2iϑ2ia2≤Δ,x1i≤k1κia1+k2κia2+Δ,with fixed-time. In addition, we can get the convergence time which satisfies
(53)t≤t1+t2<tm=1kb1−1+1k2b2+1/2−11−b2+1k1ia1−1+1k2i2a2−11−a2.
It can be seen from (53) that the upper bound of convergence time tm increases with the increase of k, k1i, k2i, a1, and b1 and increases with the decrease of a2 and b2. According to (43), k is related to the control parameter matrices kii=3,4,⋯,8 and ljj=1,2,3. Therefore, tm only depends on the control parameters, independent of the system states.
Although the convergence speed of the IGC system states will increase as the parametersa1, b1, kii=1,2,⋯,8, and ljj=1,2,3 increase and decrease as the parameters a2 and b2 increase, excessive pursuit of convergence speed will produce large control instructions, requiring the missile to provide a large overload, and the missile available overload is limited, so compromise should be considered in parameter selection.
This completes the proof.


## 4. Numerical Simulation Results and Analysis

The missile 6-DOF simulation is implemented to illustrate the performance of the proposed IGC (PIGC) design method. Table1 shows the initial conditions of missile and target. Table 2 shows the missile dynamic parameters. The states and actuator constraints are described in Table 3. The lateral acceleration of the target is set as atz2=10sinπt/2 m/s2. Set the desired terminal LOS angles εd=−70° and ηd=0°.Table 1
The initial condition for missile and target.
Symbol
Value
Symbol
Value
Symbol
Value
x
m
0 m
x
t
5000 m
α
8 deg
y
m
5000 m
y
t
1000 m
β
4 deg
z
m
0 m
z
t
2000 m
γ
8 deg
V
m
600 m/s
V
t
50 m/s
ω
x
20 deg/s
θ
m
15 deg
θ
t
0 deg
ω
y
-20 deg/s
ψ
m
-5 deg
ψ
t
0 deg
ω
z
-30 deg/sTable 2
Missile dynamic parameters.
Symbol
Value
Symbol
Value
Symbol
Value
m
1200 kg
m
x
α
0.45
C
y
α
57.15
S
0.42 m2
m
x
β
-0.38
C
y
β
-0.081
L
0.69 m
m
x
δ
x
2.13
C
y
δ
z
5.75
J
x
100 kg·m2
m
y
β
-27.30
C
z
α
0.09
J
y
5800 kg·m2
m
y
δ
y
-26.60
C
z
β
-56.32
J
z
5700 kg·m2
m
z
α
-28.15
C
z
δ
y
-5.6
g
9.8 m/s2
m
z
δ
z
-27.90Table 3
Constraints of states and actuators.
Symbol
Range
Symbol
Range
α
[-10, 10] deg
ω
x, ωz
[-40, 40] deg/s
β
[-5, 5] deg
ω
y
[-30, 30] deg/s
γ
[-10, 10] deg
δ
x, δy, δz
[-12, 12] degThere is an uncertainty 0.1sin (0.2πt) for the missile inertia components and 0.2sin (0.2πt) for aerodynamic coefficients. Set Δα=Δβ=Δγ=0.2sintrad/s and Δωx=Δωy=Δωz=sintrad/s2.The design parameters for PIGC are set ask1=diag0.6,1, k2=diag1,1, a1=3, a2=3/5, b1=2, b2=0.5,μ1=μ2=μ3=0.01, δ=0.01, k3=k4=diag3,4, k5=k6=diag8,12,12, k7=k8=diag20,10,10, ζ=diag0.6,0.7,0.6, ωn=diag10,20,10, l1=diag3,2,5, l2=diag10,8,3, and l3=diag10,10,5. According to (53), it can be calculated that tm≈13.44s.


### 4.1. Case 1: Comparison Simulation with Existing Methods

The robust IGC design method (RIGC) proposed in [12] and the new dynamic surface control based IGC design method (NDIGC) proposed in [15] are introduced for comparative simulation analysis.Missile and target trajectories, LOS angular rates, and relative distances are shown in Figures3 and 4. The missile trajectories are relatively smooth under the three IGC laws. PIGC can make the LOS angular rates converge stably to near zero within fixed-time tm and has a significant advantage in zeroing angular rates, thus leading to a small miss distance (see Table 4).Figure 3
Trajectories of missile and target in case 1.Figure 4
LOS angular rates and relative distances in case 1.Table 4
Simulation results of three IGC laws.
IGC law
NDIGC
RIGC
PIGC
Intercept time (s)
14.87
14.91
14.77
Miss distance (m)
1.33
2.31
0.71
LOS elevation angle error (deg)
0.28
0.34
0.05
LOS azimuth angle error (deg)
0.12
0.21
0.04LOS angles are presented in Figure5. It can be seen that the LOS angles can gradually converge to the expected value under the three IGC laws. The LOS angles of PIGC converge fastest. RIGC adopts sliding mode control to suppress disturbances such as model uncertainty and target maneuver. When there are large model uncertainties and disturbances, the convergence performance of RIGC is greatly affected. NDIGC selects linear sliding mode surface, which can only ensure the states convergent asymptotically. Therefore, PIGC is superior in terms of angle constraints and convergence.Figure 5
LOS elevation and azimuth angles in case 1.The attack angles, sideslip angles, and roll angles curves are given in Figure6. The three Euler angular rates are presented in Figure 7. The angles and angular rates can converge to near zero under three IGC algorithms. Compared with PIGC scheme, larger attack angle, sideslip angle, and roll angle result in larger pitch, yaw, and roll angular rates under RIGC and NDIGC. Moreover, the PCIGC can ensure that the system states do not exceed the constraint ranges.Figure 6
Attack, sideslip, and roll angles in case 1.Figure 7
Roll, yaw, and pitch angular rates in case 1.The rudder angle curves are shown in Figure8. Compared with RIGC, NDIGC and PIGC can restrict the rudder angles within the constraints. PIGC, in particular, can ensure that the rudder angles converge steadily to near zero in finite-time.Figure 8
Roll, yaw, and pitch channel rudder angles in case 1.The simulation results in Figures3, 5, and 6 seem divergent at the intercept time. This is because the relative motions of the missile and target change dramatically at the end, so the LOS angular rate changes dramatically and diverges at the last moment of the simulation. At the same time, as the system state constraints such as attack angle are not considered and corresponding virtual control instructions are not restricted, system states of RIGC and NDIGC cannot converge stably and diverge at the end of simulation.Table4 shows the simulation results of three IGC laws in terms of intercept times, miss distances, and LOS angle errors. It can be seen that PIGC has the smallest intercept time, miss distance, and angle errors. PIGC can impact the target accurately with shorter time, smaller miss distance, and higher angle constraint accuracy.From the simulation results, it can be concluded that PIGC can effectively solve terminal angle constraints, state constraints, and control constraints and has fixed-time convergence, which is superior to NDIGC and RIGC.


### 4.2. Case 2: Monte Carlo Simulation

In this case, a missile intercepts a ground target. In case 1, it is assumed that the missile velocity is constant. But in a real scenario, the missile’s speed would certainly change because of aerodynamic drag and gravity. Therefore, in order to be closer to engineering practice, the simulation under time-varying missile velocity will be considered in this case. At the same time, In order to further verify the robustness of PIGC algorithm, Monte Carlo simulation was carried out with 100 simulation times. Simulation conditions are as follows:(1)
The aerodynamic coefficients vary from -10% to +10% of the nominal value and follow normal distribution(2)
The mass and moment of inertia vary from -10% to +10% of the nominal value and follow normal distribution(3)
The initial position coordinatesym and zm of the missile vary randomly between 4500 m~5500 m and 0 m~500 m, respectively, subjected to normal distribution. The target position is (5000 m, 0 m, 1000 m)(4)
The missile initial fight path angleθm and heading angle ψVm vary randomly between 0 deg~10 deg and -5 deg~0 deg, respectively, subjected to normal distribution(5)
The mathematical expression of aerodynamic resistance is(54)X=Cx0+Cxαα+Cxββ+Cxδxδx+Cxδyδy+CxδzδzQsThe relevant aerodynamic coefficients are set asCx0=0.32, Cxα=0.21, Cxβ=0.19, Cxδx=0.05, Cxδy=0.07, and Cxδz=0.06.The Monte Carlo simulation results are exhibited in Figures9–22.Figure 9
Trajectories of missile in case 2.Figure 10
Relative distances of missile and target in case 2.Figure 11
Missile velocities in case 2.Figure 12
LOS elevation angles in case 2.Figure 13
LOS azimuth angles in case 2.Figure 14
Attack angles in case 2.Figure 15
Sideslip angles in case 2.Figure 16
Roll angles in case 2.Figure 17
Roll angular rates in case 2.Figure 18
Yaw angular rates in case 2.Figure 19
Pitch angular rates in case 2.Figure 20
Roll channel rudder angles in case 2.Figure 21
Yaw channel rudder angles in case 2.Figure 22
Pitch channel rudder angles in case 2.As shown in Figures9 and 10, perfect attacking target with small miss distance is achieved in each scenario. The mean and standard deviation of miss distance are found as 1.2639 m and 0.5019 m, respectively. It can be seen in Figure 11 that the missile’s speed is reduced by air resistance.In Figures12 and 13, the LOS angles finite-time converge to the desired angles, and the convergence time is all less than tm that embodies the fixed-time convergence characteristic of PIGC. In Figures 14–22, the system states and rudder angles are all finite-time converge to near zero and never violate the constraints. In addition, there are problems of nonsmoothness at about 10 to 11 seconds in Figures 14, 19, and 22 and at about 7 to 8 seconds in Figures 15, 16, 17, 18, 20, and 21. These were because the nonsingular terminal surface (20) proposed in this paper is a piecewise function with switching term, so when the sliding mode surface is in the switching position, the control instructions have small fluctuations; thus, the curves of the system states will appear not smooth.It can be seen from these figures that PIGC achieves satisfactory performance under different initial fight conditions and uncertainties of the missile-related parameters. Monte Carlo simulation shows that the proposed IGC algorithm is robust.


## 4.1. Case 1: Comparison Simulation with Existing Methods

The robust IGC design method (RIGC) proposed in [12] and the new dynamic surface control based IGC design method (NDIGC) proposed in [15] are introduced for comparative simulation analysis.Missile and target trajectories, LOS angular rates, and relative distances are shown in Figures3 and 4. The missile trajectories are relatively smooth under the three IGC laws. PIGC can make the LOS angular rates converge stably to near zero within fixed-time tm and has a significant advantage in zeroing angular rates, thus leading to a small miss distance (see Table 4).Figure 3
Trajectories of missile and target in case 1.Figure 4
LOS angular rates and relative distances in case 1.Table 4
Simulation results of three IGC laws.
IGC law
NDIGC
RIGC
PIGC
Intercept time (s)
14.87
14.91
14.77
Miss distance (m)
1.33
2.31
0.71
LOS elevation angle error (deg)
0.28
0.34
0.05
LOS azimuth angle error (deg)
0.12
0.21
0.04LOS angles are presented in Figure5. It can be seen that the LOS angles can gradually converge to the expected value under the three IGC laws. The LOS angles of PIGC converge fastest. RIGC adopts sliding mode control to suppress disturbances such as model uncertainty and target maneuver. When there are large model uncertainties and disturbances, the convergence performance of RIGC is greatly affected. NDIGC selects linear sliding mode surface, which can only ensure the states convergent asymptotically. Therefore, PIGC is superior in terms of angle constraints and convergence.Figure 5
LOS elevation and azimuth angles in case 1.The attack angles, sideslip angles, and roll angles curves are given in Figure6. The three Euler angular rates are presented in Figure 7. The angles and angular rates can converge to near zero under three IGC algorithms. Compared with PIGC scheme, larger attack angle, sideslip angle, and roll angle result in larger pitch, yaw, and roll angular rates under RIGC and NDIGC. Moreover, the PCIGC can ensure that the system states do not exceed the constraint ranges.Figure 6
Attack, sideslip, and roll angles in case 1.Figure 7
Roll, yaw, and pitch angular rates in case 1.The rudder angle curves are shown in Figure8. Compared with RIGC, NDIGC and PIGC can restrict the rudder angles within the constraints. PIGC, in particular, can ensure that the rudder angles converge steadily to near zero in finite-time.Figure 8
Roll, yaw, and pitch channel rudder angles in case 1.The simulation results in Figures3, 5, and 6 seem divergent at the intercept time. This is because the relative motions of the missile and target change dramatically at the end, so the LOS angular rate changes dramatically and diverges at the last moment of the simulation. At the same time, as the system state constraints such as attack angle are not considered and corresponding virtual control instructions are not restricted, system states of RIGC and NDIGC cannot converge stably and diverge at the end of simulation.Table4 shows the simulation results of three IGC laws in terms of intercept times, miss distances, and LOS angle errors. It can be seen that PIGC has the smallest intercept time, miss distance, and angle errors. PIGC can impact the target accurately with shorter time, smaller miss distance, and higher angle constraint accuracy.From the simulation results, it can be concluded that PIGC can effectively solve terminal angle constraints, state constraints, and control constraints and has fixed-time convergence, which is superior to NDIGC and RIGC.


## 4.2. Case 2: Monte Carlo Simulation

In this case, a missile intercepts a ground target. In case 1, it is assumed that the missile velocity is constant. But in a real scenario, the missile’s speed would certainly change because of aerodynamic drag and gravity. Therefore, in order to be closer to engineering practice, the simulation under time-varying missile velocity will be considered in this case. At the same time, In order to further verify the robustness of PIGC algorithm, Monte Carlo simulation was carried out with 100 simulation times. Simulation conditions are as follows:(1)
The aerodynamic coefficients vary from -10% to +10% of the nominal value and follow normal distribution(2)
The mass and moment of inertia vary from -10% to +10% of the nominal value and follow normal distribution(3)
The initial position coordinatesym and zm of the missile vary randomly between 4500 m~5500 m and 0 m~500 m, respectively, subjected to normal distribution. The target position is (5000 m, 0 m, 1000 m)(4)
The missile initial fight path angleθm and heading angle ψVm vary randomly between 0 deg~10 deg and -5 deg~0 deg, respectively, subjected to normal distribution(5)
The mathematical expression of aerodynamic resistance is(54)X=Cx0+Cxαα+Cxββ+Cxδxδx+Cxδyδy+CxδzδzQsThe relevant aerodynamic coefficients are set asCx0=0.32, Cxα=0.21, Cxβ=0.19, Cxδx=0.05, Cxδy=0.07, and Cxδz=0.06.The Monte Carlo simulation results are exhibited in Figures9–22.Figure 9
Trajectories of missile in case 2.Figure 10
Relative distances of missile and target in case 2.Figure 11
Missile velocities in case 2.Figure 12
LOS elevation angles in case 2.Figure 13
LOS azimuth angles in case 2.Figure 14
Attack angles in case 2.Figure 15
Sideslip angles in case 2.Figure 16
Roll angles in case 2.Figure 17
Roll angular rates in case 2.Figure 18
Yaw angular rates in case 2.Figure 19
Pitch angular rates in case 2.Figure 20
Roll channel rudder angles in case 2.Figure 21
Yaw channel rudder angles in case 2.Figure 22
Pitch channel rudder angles in case 2.As shown in Figures9 and 10, perfect attacking target with small miss distance is achieved in each scenario. The mean and standard deviation of miss distance are found as 1.2639 m and 0.5019 m, respectively. It can be seen in Figure 11 that the missile’s speed is reduced by air resistance.In Figures12 and 13, the LOS angles finite-time converge to the desired angles, and the convergence time is all less than tm that embodies the fixed-time convergence characteristic of PIGC. In Figures 14–22, the system states and rudder angles are all finite-time converge to near zero and never violate the constraints. In addition, there are problems of nonsmoothness at about 10 to 11 seconds in Figures 14, 19, and 22 and at about 7 to 8 seconds in Figures 15, 16, 17, 18, 20, and 21. These were because the nonsingular terminal surface (20) proposed in this paper is a piecewise function with switching term, so when the sliding mode surface is in the switching position, the control instructions have small fluctuations; thus, the curves of the system states will appear not smooth.It can be seen from these figures that PIGC achieves satisfactory performance under different initial fight conditions and uncertainties of the missile-related parameters. Monte Carlo simulation shows that the proposed IGC algorithm is robust.


## 5. Conclusion

In this paper, a novel fixed-time IGC algorithm has been proposed for terminal angle constraint, full-state constraints, and control constraint. The IGC design model considering multiple constraints is established, and the disturbance observer is used to estimate and compensate the model disturbance. The second-order instruction filter is introduced to overcome the “differential expansion,” system state constraints, and control constraint. The fixed-time convergence and the stability of the IGC scheme are strictly proved. Simulation results verify the effectiveness and superiority of the proposed IGC algorithm.The IGC algorithm proposed in this paper contains many design parameters, which can be further studied on parameter optimization.



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*Source: 6296770-2022-07-18.xml*