# Integrated Guidance and Control of Interceptor Missile Based on Asymmetric Barrier Lyapunov Function

Authors: Xiang Liu; Xiaogeng Liang
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

In this study, a novel integrated guidance and control (IGC) algorithm based on an IGC method and the asymmetric barrier Lyapunov function is designed; this algorithm is designed for the interceptor missile which uses a direct-force/aerodynamic-force control scheme. First, by considering the coupling between the pitch and the yaw channels of the interceptor missile, an IGC model of these channels is established, and a time-varying gain extended state observer (TVGESO) is designed to estimate unknown interferences in the model. Second, by considering the system output constraint problem, an asymmetric barrier Lyapunov function and a dynamic surface sliding-mode control method are employed to design the control law of the pitch and yaw channels to obtain the desired control moments. Finally, in light of redundancy in such actuators as aerodynamic rudders and jet devices, a dynamic control allocation algorithm is designed to assign the desired control moments to the actuators. Moreover, the results of simulations show that the IGC algorithm based on the asymmetric barrier Lyapunov function for the interceptor missile allows the outputs to meet the constraints and improves the stability of the control system of the interceptor missile.


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## Body

## 1. Introduction

The interceptor missile plays an important role in modern anti-missile systems. Given the rapid development of hypersonic aircraft, demands on accurate guidance and control of the interceptor missile has become increasingly stringent. To cope with high-speed aircraft with a strong penetration capability, the interceptor missile generally adopts a direct-force/aerodynamic-force control scheme that can accelerate the speed of command response. The guidance and control system of the interceptor missile is a highly dynamic and multivariate system of strong coupling, rapid temporal change, and uncertainty, thereby making it critical to the examination of the guidance and control of interceptor missile at high speed with strict control requirements.Integrated guidance and control (IGC) refers to using the relative interceptor–target motion information and the dynamic information of the interceptor missile to generate a control force that drives it to strike the target. IGC allows for a rational allocation of interceptor missile control capability to not only maintain the interceptor missile’s flight attitude but also improve the precision of its guidance [1, 2]. The authors in [3, 4] explored the IGC control law by using high-order sliding-mode control methods and backstepping control algorithms. On the basis of backstepping control algorithms, the authors in [5, 6] considered saturation factors and introduced dynamic models of actuators models to design algorithms for longitudinal and anti-saturation IGC for aircraft. An adaptive sliding-mode control method was used to implement the design of an IGC system on the pitch plane to configure missiles to attack ground targets [7], but the problem of target mobility was not considered. Based on a backstepping method, the missile control loop was regarded as a second-order element in the study [8], which introduced a first-order integral filter to estimate the derivative of the input to the virtual control and design the control law. Based on a three-dimensional (3D) ICG model, a robust adaptive backstepping method was implemented to design an IGC algorithm for a missile [9]. Methods of IGC design have been widely used in guidance and control systems of aircraft [10], missiles [11–13], and unmanned aerial vehicles [14, 15]. In recent years, researchers have combined this method with modern control theories, such as dynamic surface control [16], optimal control [17], and predictive control [18], to generate methods of IGC design for aircrafts. However, most prevalent IGC design methods do not consider the coupling relationship between the pitch and the yaw channels. Moreover, to improve the stability of the guidance process, constraints concerning the angular velocities of the line of sight during the guidance of the interceptor missile should be considered.Nonlinearity commonly exists in the IGC system of the interceptor missile. In recent years, with practical engineering problems’ increasing demand on the control performance, there has been considerable progress in the development of nonlinear control theory, especially in adaptive control [19], neural network control [20], fuzzy control [21], etc. All of these have laid a solid foundation for the in-depth research of nonlinear control theory. However, in actual engineering applications, the use of nonlinear systems is always subject to input, output, and state constraints, among others, and violation of these constraints can result in the control system’s downgraded performance. Therefore, it has now become an important research direction, when constructing the control system to consider the effect of these constraints and to properly handle them in the controller design process. The authors in [22] proposed an adaptive neural network constrained control algorithm for single-input/single-output nonlinear stochastic switching systems; this algorithm constructed traditional Lyapunov function to handle constraint control, which achieved good results. As barrier Lyapunov function does not require an exact solution of the system, the constrained control method based on barrier Lyapunov function has been widely used in state constraint and output constraint problems in recent years. Barrier Lyapunov function is a special type of continuous function, unlike traditional Lyapunov function which is radially unbounded, in barrier Lyapunov function, when the parameters approach the limit value, the function value will tend to infinity to ensure that the control system satisfies the constraints [23]. Barrier Lyapunov function can be utilized to satisfy the constraints of both symmetric and asymmetric constraint controls, even when the constraint is a time-varying asymmetric one. According to the authors in [24], barrier Lyapunov function has been used to solve constraint problems in a hybrid PDE-ODE system that describes a nonuniform gantry crane system. The authors in [25] proposed an adaptive fuzzy neural network control method using impedance learning for a constrained robot system based on barrier Lyapunov function. According to the authors in [26, 27], barrier Lyapunov functions have been used to solve constraint problems in nonlinear and uncertain systems and to expand the definitions of the constraints. Methods based on these functions can effectively solve problems of symmetrically and asymmetrically constrained control. The authors in [28, 29] have expanded output control constraints to include time-varying outputs while relaxing the limitations on the initial values of control systems. By combining barrier Lyapunov functions with dynamic surface control technologies, some studies [30, 31] have proposed barrier Lyapunov function-based methods suitable for constrained dynamic surface control to solve the computational inflation problem caused by backstepping control. Researchers subsequently applied this method to brushless DC motors [32], plane braking systems [33], and hypersonic aircraft [34, 35] to achieve satisfactory results in terms of constrained control. However, few studies have investigated the application of this method to interceptor missile control. Design methods based on barrier Lyapunov functions are advantageous because they can solve the output constraint problem of interceptor missile guidance control systems and improve their stability.In view of the above analysis, to aim at the interceptor missile which uses a direct-force/aerodynamic-force control scheme, and to consider the coupling relationship between the pitch and the yaw channels as well as the constraints on the system’s output, this study proposes an IGC algorithm based on the asymmetric barrier Lyapunov function. First, a time-varying gain extended state observer (TVGESO) is designed to estimate interferences in the system. Second, an asymmetric barrier Lyapunov function and a dynamic surface sliding-mode control method, respectively, are used to design control laws for the interceptor missile to obtain the desired moments. Finally, a dynamic control allocation algorithm is designed to allocate the desired control moments. The results of simulations show that the proposed algorithm enables the outputs to meet the constraints and improves the stability of the interceptor missile control systems.


## 2. IGC Model of Interceptor Missile

The relationship of relative motion between the interceptor missile and its target in 3D space is shown in Figure1.Figure 1
The relative relationship between the interceptor missile and its target in 3D space.In the figure,Oxyz refers to an inertial coordinate system and Ox4y4z4 refers to a line-of-sight coordinate system, respectively; M and T refer to the interceptor missile and the target, respectively; qε and qβ are the vertical and horizontal angles of the line of sight with the interceptor missile and the target, respectively, and r denotes the relative distance between the interceptor missile and the target. The model of the relative motion of the interceptor missile and target is as follows:
(1)rq¨ε=−2ṙq̇ε−rq̇β2sinqεcosqε−atε−am4ε,r¨=rq̇ε2+rq̇β2cos2qε+atr−am4r,−rq¨βcosqε=2ṙq̇βcosqε−2rq̇εq̇βsinqε+atβ−am4β,where q̇ε and q̇β denote the vertical and horizontal angular velocities of the line of sight with the interceptor missile and the target, respectively; am4ε and am4β denote the longitudinal and lateral accelerations of the interceptor missile, respectively; and atε and atβ denote the longitudinal and lateral accelerations of the target, respectively.The interceptor missile uses a direct-force/aerodynamic-force control scheme. Assuming that the direct force is adjustable and continuous, the angles of deflection of the rudder equivalent to the direct force in the pitch and the yaw channels are, respectively, defined as(2)δsy=FyFsmax,δsz=FzFsmax,where Fz and Fy denote the thrust generated by the jet device, and Fsmax is the maximum steady-state thrust of the jet devices.In light of the coupling relationship between the pitch and yaw channels of the interceptor missile, its dynamic model is expressed as follows:(3)α̇=ωz+ωytanβsinα−qSCyααcosαmVmcosβ+dα,β̇=ωycosα+qSCyααsinαsinβmVm+qSCzββcosβmVm+dβ,ω̇z=qSL¯mzααJz+qSL¯mzωzωzJz+qSL¯mzδzδzJz+1+KzFyLmJz+dωz,ω̇y=qSL¯myββJy+qSL¯myωyωyJy+qSL¯myδyδyJy+1+KyFzLmJy+dωy,(4)am3ε=qSCyααm,am3β=qSCzββm,where S is the reference area of the interceptor missile; q is the dynamic pressure; Vm is the speed of the interceptor missile; L¯ is its reference length; Lm is the average distance between the jet device and its center of mass; m is the mass of the interceptor missile; α and β denote the attack angle and the sideslip angle, respectively; ωz and ωy denote the angular velocities of the pitch and the yaw, respectively; δz and δy denote the angles of deflection of the aerodynamic rudder; Kz and Ky denote the amplification factors of moment (used to describe the effect of the mutual interference between lateral jets and incoming flow on the aerodynamic moment of the interceptor missile); dα, dωz, dβ, and dωy refer to the disturbances and uncertain interferences at each link of the system; Jz and Jy refer to the moments of inertia; Cyα, Czβ, mzα, mzωz, mzδz, myβ, myωy, and myδy refer to the relevant aerodynamic forces and coefficients of moment; and am3ε and am3β refer to the vertical and lateral overloads of the interceptor missile, respectively.Assuming that the angle of the line of sight of the interceptor missile in the terminal guidance stage changes slightly and that the angle of line of sight and direction of velocity of the interceptor missile are relatively small, letam3ε=am4ε and am3β=am4β. According to Equations (1)–(4), by defining xz1=qε, xz2=q̇ε, xz3=α, xz4=ωz, xy1=qβ, xy2=q̇β, xy3=β, and xy4=ωy, one can have the nonlinear IGC model in the pitch channel for the interceptor missile:
(5)ẋz2=fz2xz2+gz2xz3−xy22sinxz1cosxz1+atεr,ẋz3=fz3xz3+xz4+dz3,ẋz4=fz4xz3+fz5xz4+gz3uz+dz4,yz=xz2,where fz2=−2ṙ/r, fz3=−qSCyα/mVm, fz4=qSL¯mzα/Jz, fz5=qSL¯mzωz/Jz, gz2=−qSCyα/mr, gz3=1/Jz, dz3=ωytanβsinα+dα, dz4=dωz, and uz=qSL¯mzδzδz+1+KzFyLm, with uz representing the moment generated jointly by both the aerodynamic rudders and the jet devices in the pitch channels.Similarly, the nonlinear IGC model for the interceptor missile in its yaw channel is as follows:(6)ẋy2=fy2xy2+gy2xy3+2xz2xy2tanxz1−atβrcosxz1,ẋy3=fy3xy3+xy4+dy3,ẋy4=fy4xy3+fy5xy4+gy3uy+dy4,yy=xy2,where fy2=−2ṙ/r, fy3=qSCyβ/mVm, fy4=qSL¯myβ/Jy, fy5=qSL¯myωy/Jy, gy2=qSCyβ/mrcosxz1, gy3=1/Jy, dy3=qSCyααsinαsinβ/mVm+dβ, dy4=dωy, and uy=qSL¯myδyδy+1+KyFzLm, with uy representing the moment generated jointly by both the aerodynamic rudders and the jet devices in the yaw channel.Assumption 1.
The unknown interferencesdz3, dz4, dy3, and dy4 in the IGC models in Equations (5) and (6) of the interceptor missile are continuously differentiable, and the derivatives are bounded.


## 3. Design of TVGESO

A TVGESO can estimate nonlinear uncertainties in the model of the system and feed the estimates back into the control system for compensation. To eliminate the effects of unknown uncertain interferencesatε, atβ, dz3, dz4, dy3, and dy4 in the system models in Equations (5) and (6) on the control system of the interceptor missile, a TVGESO is designed to estimate these interferences.By definingvr=ṙ, vε=rq̇ε, and vβ=rq̇βcosqε, one obtains
(7)v̇ε=ṙq̇ε−2ṙq̇ε−rq̇β2sinqεcosqε+atε−am4ε=−vrvεr−vβ2tanqεr−am4ε+atε,(8)v̇β=ṙq̇βcosqε+rq¨βcosqε−rq̇εq̇βsinqε=−vrvβr+vεvβtanqεr+am4β−atβ,Considering the system in Equation (5) and Equation (7), one can design the following TVGESO to estimate acceleration atε of the target:
(9)ez21=vz21−vε,ez22=vz22−atε,v̇z21=vz22−λz21ez21+g¯z2,v̇z22=−λz22ez21,where g¯z2=−vrvε/r−vβ2tanqε/r−qSCyα/mα; vz21 and vz22, respectively, are the estimated values of vε and atε; ez21 and ez22 are the estimated errors; and λz21 and λz22 are the time-varying gain coefficients designed for the state observer. They are defined as λz21=2Lt and λz22=L2t, respectively. Function Lt is defined as
(10)L̇t=γez21,ez21≠0,0,ez21=0,L0=λz0,where γ is the adaptive coefficient and is greater than zero. As indicated in the literature [36], appropriate values of the coefficient can ensure that the error system of the TVGESO is stable for a limited time.Similarly, by estimating interferencesdz3 and dz4 of the angle-of-attack loop and the pitch angular velocity loop, respectively, in Equation (5), one can obtain the following:
(11)ez31=vz31−xz3,ez32=vz32−dz3,v̇z31=vz32−λz31ez31+g¯z3,v̇z32=−λz32ez31,(12)ez41=vz41−xz4,ez42=vz42−dz4,v̇z41=vz42−λz41ez41+g¯z4,v̇z42=−λz42ez41,where g¯z3=fz3xz3+xz4 and g¯z4=fz4xz3+fz5xz4+gz3uz, the interferences dz3 and dz4 are estimated as vz32 and vz42, respectively, and the estimation errors for them are denoted by ez32 and ez42, respectively.According to Equations (9)–(12), the interferences atβ, dy3, and dy4 in the system in Equation (6) have estimated values of vy22, vy32, and vy42, respectively, with the estimation errors of ey22, ey32, and ey42, respectively.


## 4. Design of the Dynamic Surface Sliding-Mode Control Law Based on Asymmetric Barrier Lyapunov Function

Let us defineζ as an open region containing the origin and the barrier Lyapunov function Vx as a scalar function defined in ζ for the system ẋ=fx,t. It also has the following characteristics: (1) smooth and positive definite, (2) has a first-order continuous partial derivative at each point in ζ, (3) tends to infinity when x approaches the edge of ζ, and (4) satisfies the expression Vxt≤b for ∀t≥0 if x0∈ζ, where b>0.Assumption 2.
For anyt>0, there exist constants K¯ci and K¯cii=0,1,2 that satisfy k¯c1≥K¯c0 and k¯c1≤K¯c0, with their derivatives satisfying k¯ciit≥K¯ci, k¯ciit≤K¯ci, i=1,2, and ∀t≥0.Assumption 3.
For anyk¯c1 和 k¯c1, there exist functions Y¯0t and Y¯0t as well as positive constants Y1 and Y2 satisfying Y¯0t>k¯c1t and k¯c1t<Y¯0t, such that they make the system track command x2d=diagxz2d,xy2d, and its time derivative satisfies k¯c1t≤x2dt≤k¯c1t and ẋ2dt≤Y1 as well as x¨2dt≤Y2 for ∀t≥0. There exists a continuous set satisfying Mx2d=x2d∈ℝ:xd2+ẋd2+x¨d2≤ϕx2d⊂ℝ3.Given that the IGC model of interceptor missile is a mismatching and uncertain system, to enable the guidance and control system to accurately pursue the target, and not violating the constraints on the control system, the control law allows using a dynamic surface sliding mode algorithm based on the asymmetric barrier Lyapunov function. It can enable the control system to pursue the target highly precisely, meanwhile ensuring that the closed-loop system is consistent and ultimately bounded, and the tracking error converges to a small set.


### 4.1. Design of Control Law for Pitch Channel

For the system in Equation (5), define xz2d as the system’s track command signal.
(1)
Define the first dynamic error surface:
(13)sz2=xz2−xz2d.Taking the derivative ofsz2, one obtains the error dynamic equation:
(14)ṡz2=fz2xz2−xy22sinxz1cosxz1+atεr−ẋz2d+gz2xz3.Because the backstepping method does not have a perfect solution to the expansion of items and the problems caused by the expansion of items in the derivation process of the virtual control, this shortcoming is particularly prominent in the higher-order system. By using the dynamic surface control method and using the first-order filter to calculate the derivative of the virtual control, the expansion of the differential items can be eliminated and the controller and parameters can be designed simply [37]. Introduce virtual control variables xz3∗ and x¯z3∗. To avoid complicated calculations of the expansion of the number of items during the derivation of the virtual control variables, the virtual control variable xz3∗ before filtering is passed through a first-order low-pass filter to become the virtual control variable x¯z3∗:
(15)τz3x¯̇z3∗+x¯z3∗=xz3∗,x¯z3∗0=xz3∗0.In the above expression,τz3 is the filter’s time constant, τz3>0, and the filtering error is defined as yz3=x¯z3∗−xz3∗.Considering boundary layer errors of the dynamic surface, one can construct the following asymmetric barrier Lyapunov function:(16)Vz1=1−p1sz22logka112ka112−sz22+p1sz22logkb112kb112−sz22+12yz32,where
(17)p1sz2=10<sz2,0sz2≤0.ka11=xz2d−k¯c1, kb11=k¯c1−xz2d, log• represents a natural logarithm, and k¯c1 and k¯c1 represent output constraints.Given the independence characteristic of the output constraintsk¯c1 and k¯c1, the tracking error constraints ka11 and kb11 can be designed independently. When constraints ka11 and kb11 are constant, and ka11≠kb11, the output constrained control can be extended to a static asymmetric constraint, whereas the output constraint becomes a symmetric constraint when ka11=kb11. This means that the initial output can be changed depending on the setting of the constraint. It is evident that the asymmetric barrier Lyapunov function relaxes the constraint on the initial condition of the output.As shown by Equation (16), the expression Vz1≥0 simplifies to Vz1=0 if and only if sz2=0 and x¯z3∗0=xz3∗0 simultaneously. Therefore, Vz1 is a positive-definite function in the range −ka11<sz2<kb11. Moreover, given limsz2→0+dVz1/dsz2=limsz2→0−dVz1/dsz2=0, Vz1 is a piecewise continuously differentiable function in the ranges sz2∈−ka11,0 and sz2∈0,kb11. Therefore, Vz1 is a valid Lyapunov function that can ensure that the system’s output error is constrained in the ranges sz2∈−ka11,0 and sz2∈0,kb11.Substituting the estimated values of TVGESO in Equation (9), one can design the virtual control variable for the first dynamic surface as shown in Equation (18):
(18)xz3∗=−gz2−1fz2xz2−xy22sinxz1cosxz1+vz22r−ẋz2d−kz2+k¯z2sz2,where kz2>0, gz2>0, and r>0, and the control coefficient k¯z2 is defined as
(19)k¯z2=1−p1ka112+p1kb112.According to Equation (15), the derivative of the virtual control variable for the error surface after filtering is
(20)x¯̇z3∗=−τz3−1x¯z3∗−xz3∗.
(2)
Define the second dynamic error surface as
(21)sz3=xz3−x¯z3∗.By taking the derivation ofsz3, one can obtain the equation for the dynamic error as follows:
(22)ṡz3=ẋz3−x¯̇z3∗=fz3xz3+xz4+dz3−x¯̇z3∗.Introduce two virtual control variablesxz4∗ and x¯z4∗. By passing the virtual control variable xz4∗ through a first-order low-pass filter, one obtains the virtual control variable x¯z4∗:
(23)τz4x¯̇z4∗+x¯z4∗=xz4∗,x¯z4∗0=xz4∗0.In Equation (23), τz4 denotes the filter time constant, τz4>0, and the filtering error is defined as yz4=x¯z4∗−xz4∗.Construct the following Lyapunov function:(24)Vz2=12sz32+12yz42.By substituting the estimated values of TVGESO into Equation (11), one can design the virtual control variable for the second dynamic surface as shown in Equation (25):
(25)xz4∗=−fz3xz3−vz32+x¯̇z3∗−kz3sz3−μgz2sz2,where kz3>0 and μ=1−p1/ka112−sz22+p1/kb112−sz22.According to Equation (23), the derivative of the virtual control variable for the error surface after filtering is
(26)x¯̇z4∗=−τz4−1x¯z4∗−xz4∗.
(3)
Define the third dynamic error surface:
(27)sz4=xz4−x¯z4∗.By taking the derivative ofsz4, one can obtain the equation of error dynamic as follows:
(28)ṡz4=ẋz4−x¯̇z4∗=fz4xz3+fz5x4+gz3uz+dz4−x¯̇z4∗.Construct the following Lyapunov function:(29)Vz3=12sz42.By substitutingvz42 with the estimated values of TVGESO in Equation (12), one can design the dynamic sliding-mode control law of the interceptor missile as shown in Equation (30):
(30)uz=gz3−1−fz4xz3−fz5xz4−vz42+x¯̇z4∗−kz4sz4−kz5sz4∂zsgnsz4,where kz4>0, kz5>0, 0<∂z<1, and gz3>0.


### 4.2. Stability Analysis of Control Law for Pitch Channel

(31)V1S1⟶∞,γ1ϖ≤Uϖ≤γ2ϖ,where γ1 and γ2 are K∞ functions. Suppose Vη≔V1S1+Uϖ and S10∈S1. If the inequality,
(32)V̇=∂V∂ηh≤−cV+υ,is valid in the domain θ¯∈N, and c and υ are positive constants, then S1t is still in the set for ∀t∈0,∞.Lemma 1 (see [20]).
For any two functionskat and kbt, suppose S1≔S1∈ℝ:−kat<S1<kbt⊂ℝ and N≔ℝl×S1⊂ℝl+1 to be open sets. For system η̇=ht,η with η≔ϖ,S1T∈N as state, function h:ℝ+×N→ℝl+1 is piecewise continuous with respect to t, and S1 satisfies the Lipschitz condition in ℝ+×N. If there exist two functions U:ℝl→ℝ+ and V1:S1→ℝ+ that are continuous and positively definite in their domains, respectively, the following two expressions are valid when S1→−kat or S1→kbt:Theorem 1.
For the closed-loop system Equation (5), if the virtual control variables and the control law satisfy Equations (18), (25), and (30), then for any set Mz0 to which the initial condition yz0 belongs, there always exists a sufficiently large set Mz≔k¯c1t<−ka11+xz2dt≤yzt≤kb11+xz2dt<k¯c1t that can prevent the system’s output constraints from being violated. Moreover, by selecting appropriate design parameters, it is possible to bound all closed-loop signals of the closed-loop system Equation (5) and make the output error converge in the neighborhood of the origin.Proof 1.
According to Equations (16), (24), and (29), one can construct a Lyapunov function that is a combination of asymmetric barrier Lyapunov functions and traditional Lyapunov functions as follows:
(33)Vz=Vz1+Vz2+Vz3=1−p1sz22logka112ka112−sz22+p1sz22logkb112kb112−sz22+12yz32+12sz32+12yz42+12sz42.By taking derivatives on both sides of the Lyapunov function in Equation (33), one obtains Equation (34):
(34)V̇z=μsz2ṡz2+yz3ẏz3+sz3ṡz3+yz4ẏz4+sz4ṡz4,where μ=1−p1/ka112−sz22+p1/kb112−sz22.Define the estimated error of the TVGESO system to satisfy Equation (35):
(35)∣ez22∣<Nz2,∣ez32∣<Nz3,∣ez42∣<Nz4,where Nz2, Nz3, and Nz4 are positive constants.It can be seen that the filtering errors are(36)yz3=x¯z3∗−xz3∗,yz4=x¯z4∗−xz4∗.By taking derivatives ofyz3 and yz4, one obtains the dynamic filtering errors:
(37)ẏz3=−τz3−1yz3−ẋz3∗,ẏz4=−τz4−1yz4−ẋz4∗.From Equations (13)–(30) and (36), one has
(38)xz2=sz2+xz2d,xz3=sz3+x¯z3∗=sz3+yz3+xz3∗,xz4=sz4+x¯z4∗=sz4+yz4+xz4∗.From Equations (13)–(28) and (36)–(38), one has
(39)ṡz2=fz2xz2−xy22sinxz1cosxz1+atεr−ẋz2d+gz2sz3+yz3+xz3∗=gz2sz3+gz2yz3−gz2kz2+k¯z2sz2−e~z22,where e~z22=vz22−atε/r, and it is assumed that e~z22<N~z2, N~z2 is a positive constant, and μN~z2>0.
(40)ṡz3=fz3xz3+sz4+yz4+xz4∗+dz3−ẋz3∗=sz4+yz4−kz3sz3−ez31−μgz2sz2,(41)ṡz4=fz4xz3+fz5xz4+gz3uz+dz4−xz4∗=−kz4sz4−kz5sz4∂zsgnsz4−ez41.According to Young’s inequality and Equations (39)–(41), one has
(42)sz2ṡz2=sz2gz2sz3+gz2yz3−gz2kz2+k¯z2sz2−e~z22≤12+12gz2−gz2kz2−gz2k¯z2sz22+12gz2yz32+12N~z22+gz2sz2sz3,(43)sz3ṡz3=sz3sz4+yz4−kz3sz3−ez31−μgz2sz2≤32−kz3sz32+12sz42+12yz42+12Nz32−μgz2sz2sz3,(44)sz4ṡz4=sz4−kz4sz4−kz5sz4∂zsgnsz4−ez41≤12−kz4+32kz5sz42+12Nz42+12kz5.It is clear that the variables in the system model and their derivatives are all bounded. If there exist continuous functionsz~z3 and z~z4 that satisfy z~z3>0 and z~z4>0, the variables ẋz3∗ and ẋz4∗ satisfy
(45)ẋz3∗≤z~z3,ẋz4∗≤z~z4.According to Young’s inequality and Equations (36)–(37) and (45), one has
(46)yz3ẏz3=yz3−τz3−1yz3−ẋz3∗≤−τz3−1yz32+12yz32+z~z32,(47)yz4ẏz4=yz4−τz4−1yz4−ẋz4∗≤−τz4−1yz42+12yz42+z~z42.By substituting Equations (42)–(47) into Equation (34) and arranging the terms, one has
(48)V̇z=μsz2ṡz2+yz3ẏz3+sz3ṡz3+yz4ẏz4+sz4ṡz4≤−μgz2kz2+gz2k¯z2−12gz2−12sz22−kz3−32−12μsz32−kz4−1−32kz5sz42−τz3−1−12−12μgz2yz32−τz4−1−1yz42+12μN~z22+12Nz32+12Nz42+12z~z32+12z~z42+12kz5.For convenience of description, define the following parameters:(49)Kz2=gz2kz2+gz2k¯z2−12gz2−12,Kz3=kz3−32−12μ,Kz4=kz4−1−32kz5,σz3=τz3−1−12−12μgz2,σz4=τz4−1−1,A~=12μN~z22+12Nz32+12Nz42+12z~z32+12z~z42+12kz5.Then, Equation (48) can be rewritten as
(50)V̇z≤−μKz2sz22−Kz3sz32−Kz4s42−σz3yz32−σz4yz42+A~.In the set−ka11t<sz2<kb11t, one has
(51)−1−p1ka112−sz22+p1kb112−sz22sz22≤−1−p1logka112ka112−sz22+p1logkb112kb112−sz22.Then, Equation (50) can be further rewritten as
(52)V̇z≤−1−p1logka112ka112−sz22+p1logkb112kb112−sz22Kz2−Kz3sz32−Kz4sz42−σz3yz32−σz4yz42+A~.Define positive-definite matrixQ as
(53)Q=Kz2000Kz3000Kz4.By selectingκ=min2λminQ,2σz3,2σz4, where κ is a positive constant and λminQ denotes the minimum eigenvalue of matrix Q, one has
(54)V̇z≤−κVz+A~.Define two setsMz0≔yz∈ℝ:k¯c0≤yz0≤k¯c0⊂ℝ and Mz≔yz∈ℝ:k¯c1≤yzt≤k¯c1⊂ℝ that satisfy k¯c1t>k¯c0+Y¯0+xz2d0⊂ℝ and k¯c1t<k¯c0+Y¯0+xz2d0⊂ℝ. From sz20=yz0−xz2d0, one has
(55)−ka11≤sz20≤kb11.Therefore, setMz is an invariant set. According to Lemma 1, it can be seen that given V̇z≤−κVz+A~, the following expression is valid for ∀t≥0:
(56)−ka11≤sz2t≤kb11.According to Equations (13) and (56), one has the following expression for ∀t≥0:
(57)k¯c1t<−ka11+xz2dt≤yzt≤kb11+xz2dt<k¯c1t.Therefore, it can be concluded that for any initial compact setMz0 defined by yz0∈Mz0, there is always a sufficiently large compact set Mz to make yz∈Mz for ∀t≥0.Define parameters according to Equation (48) to satisfy the following rules:
(58)kz2≥12gz2−1+12−k¯z2+12κ,kz3≥32+12μ+12κ,kz4≥32kz5+1+12κ,τz3−1≥12+12μgz2+12κ,τz4−1≥1+12κ.Multiply both sides of Equation (54) by eκt and arrange the terms to obtain the following expression:
(59)Vzt≤κVz0−A~e−κt+A~κ.Therefore, it is clear that by designing parameterskz2, kz3, kz4, and kz5 and parameters τz3 and τz4, it is possible to ensure that all closed-loop signals of the system are bounded. Opting to increase kz2, kz3, kz4, and kz5 and decrease τz3 and τz4 can ensure that κ is sufficiently large to make the filtering errors and the error surface small enough to control accuracy. Q.E.D.Remark 1.
Theoretically, larger values of the designed parameterskz2, kz3, kz4, and kz5; smaller ones of τz3 and τz4; the final boundary of the resulting error surfaces sz2, sz3, and sz4; and filtering errors yz3 and yz4 indicate an increase in the precision of control. However, in practice, very large values of kz2, kz3, kz4, and kz5, and very small ones of τz3 and τz4 can easily lead to input saturation in interceptor missile control systems, which triggers the saturation nonlinearity of the system such that the required overloads of the interceptor missile are beyond the available capacity, resulting in a reduction in the system’s control performance. Moreover, given the physical limitations of the low-pass filter, τz3 and τz4 should not be chosen to be arbitrarily small. Therefore, the parameters of the control algorithm should be properly selected in light of practical considerations.


### 4.3. Design of Control Law for Yaw Channel

Based on the design of the control law for the pitch channel, one can substitutevy22, vy32, and vy42—the estimated interferences of the TVGESO—into the system of Equation (6) and define xy2d as the system’s track command sign for the following form of control law for the yaw channel:
(60)sy2=xy2−xy2d,τy3x¯̇y3∗+x¯y3∗=xy3∗,x¯y3∗0=xy3∗0,xy3∗=−gy2−1fy2xy2+2xz2xy2tanxz1−vy22rcosxz1−ẋy2d−ky2+k¯y2sy2,sy3=xy3−x¯y3∗,τy4x¯̇y4∗+x¯y4∗=xy4∗,x¯y4∗0=xy4∗0,xy4∗=−fy3xy3−vy32+x¯̇y3∗−ky3sy3−μ~gy2sy2,sy4=xy4−x¯y4∗,uy=gy3−1−fy4xy3−fy5xy4−vy42+x¯̇y4∗−ky4sy4−ky5sy4∂ysgnsy4,where k¯y2=1−p2ka122+p2kb122,
(61)p2sy2=10<sy2,0sy2≤0,ka12=xy2d−k¯c1, kb12=k¯c1−xy2d, μ~=1−p2/ka122−sy22+p2/kb122−sy22, ky2>0, ky3>0, ky4>0, ky5>0, 0<∂y<1, gy2>0, gy3>0, and r>0. The parameters sy2, sy3, and sy4 are error surfaces; τy3 and τy4 are filter time constants, τy3>0 and τy4>0; xy3∗ and xy4∗ are virtual control variables of the system Equation (6); and x¯y3∗ and x¯y4∗, respectively, are virtual control variables after filtering.According to the stability analysis method in Equations (33)–(59), it can be proven that the closed-loop system of Equation (6) is stable and the system’s control precision can be achieved by designing appropriate parameters.


## 4.1. Design of Control Law for Pitch Channel

For the system in Equation (5), define xz2d as the system’s track command signal.
(1)
Define the first dynamic error surface:
(13)sz2=xz2−xz2d.Taking the derivative ofsz2, one obtains the error dynamic equation:
(14)ṡz2=fz2xz2−xy22sinxz1cosxz1+atεr−ẋz2d+gz2xz3.Because the backstepping method does not have a perfect solution to the expansion of items and the problems caused by the expansion of items in the derivation process of the virtual control, this shortcoming is particularly prominent in the higher-order system. By using the dynamic surface control method and using the first-order filter to calculate the derivative of the virtual control, the expansion of the differential items can be eliminated and the controller and parameters can be designed simply [37]. Introduce virtual control variables xz3∗ and x¯z3∗. To avoid complicated calculations of the expansion of the number of items during the derivation of the virtual control variables, the virtual control variable xz3∗ before filtering is passed through a first-order low-pass filter to become the virtual control variable x¯z3∗:
(15)τz3x¯̇z3∗+x¯z3∗=xz3∗,x¯z3∗0=xz3∗0.In the above expression,τz3 is the filter’s time constant, τz3>0, and the filtering error is defined as yz3=x¯z3∗−xz3∗.Considering boundary layer errors of the dynamic surface, one can construct the following asymmetric barrier Lyapunov function:(16)Vz1=1−p1sz22logka112ka112−sz22+p1sz22logkb112kb112−sz22+12yz32,where
(17)p1sz2=10<sz2,0sz2≤0.ka11=xz2d−k¯c1, kb11=k¯c1−xz2d, log• represents a natural logarithm, and k¯c1 and k¯c1 represent output constraints.Given the independence characteristic of the output constraintsk¯c1 and k¯c1, the tracking error constraints ka11 and kb11 can be designed independently. When constraints ka11 and kb11 are constant, and ka11≠kb11, the output constrained control can be extended to a static asymmetric constraint, whereas the output constraint becomes a symmetric constraint when ka11=kb11. This means that the initial output can be changed depending on the setting of the constraint. It is evident that the asymmetric barrier Lyapunov function relaxes the constraint on the initial condition of the output.As shown by Equation (16), the expression Vz1≥0 simplifies to Vz1=0 if and only if sz2=0 and x¯z3∗0=xz3∗0 simultaneously. Therefore, Vz1 is a positive-definite function in the range −ka11<sz2<kb11. Moreover, given limsz2→0+dVz1/dsz2=limsz2→0−dVz1/dsz2=0, Vz1 is a piecewise continuously differentiable function in the ranges sz2∈−ka11,0 and sz2∈0,kb11. Therefore, Vz1 is a valid Lyapunov function that can ensure that the system’s output error is constrained in the ranges sz2∈−ka11,0 and sz2∈0,kb11.Substituting the estimated values of TVGESO in Equation (9), one can design the virtual control variable for the first dynamic surface as shown in Equation (18):
(18)xz3∗=−gz2−1fz2xz2−xy22sinxz1cosxz1+vz22r−ẋz2d−kz2+k¯z2sz2,where kz2>0, gz2>0, and r>0, and the control coefficient k¯z2 is defined as
(19)k¯z2=1−p1ka112+p1kb112.According to Equation (15), the derivative of the virtual control variable for the error surface after filtering is
(20)x¯̇z3∗=−τz3−1x¯z3∗−xz3∗.
(2)
Define the second dynamic error surface as
(21)sz3=xz3−x¯z3∗.By taking the derivation ofsz3, one can obtain the equation for the dynamic error as follows:
(22)ṡz3=ẋz3−x¯̇z3∗=fz3xz3+xz4+dz3−x¯̇z3∗.Introduce two virtual control variablesxz4∗ and x¯z4∗. By passing the virtual control variable xz4∗ through a first-order low-pass filter, one obtains the virtual control variable x¯z4∗:
(23)τz4x¯̇z4∗+x¯z4∗=xz4∗,x¯z4∗0=xz4∗0.In Equation (23), τz4 denotes the filter time constant, τz4>0, and the filtering error is defined as yz4=x¯z4∗−xz4∗.Construct the following Lyapunov function:(24)Vz2=12sz32+12yz42.By substituting the estimated values of TVGESO into Equation (11), one can design the virtual control variable for the second dynamic surface as shown in Equation (25):
(25)xz4∗=−fz3xz3−vz32+x¯̇z3∗−kz3sz3−μgz2sz2,where kz3>0 and μ=1−p1/ka112−sz22+p1/kb112−sz22.According to Equation (23), the derivative of the virtual control variable for the error surface after filtering is
(26)x¯̇z4∗=−τz4−1x¯z4∗−xz4∗.
(3)
Define the third dynamic error surface:
(27)sz4=xz4−x¯z4∗.By taking the derivative ofsz4, one can obtain the equation of error dynamic as follows:
(28)ṡz4=ẋz4−x¯̇z4∗=fz4xz3+fz5x4+gz3uz+dz4−x¯̇z4∗.Construct the following Lyapunov function:(29)Vz3=12sz42.By substitutingvz42 with the estimated values of TVGESO in Equation (12), one can design the dynamic sliding-mode control law of the interceptor missile as shown in Equation (30):
(30)uz=gz3−1−fz4xz3−fz5xz4−vz42+x¯̇z4∗−kz4sz4−kz5sz4∂zsgnsz4,where kz4>0, kz5>0, 0<∂z<1, and gz3>0.


## 4.2. Stability Analysis of Control Law for Pitch Channel

(31)V1S1⟶∞,γ1ϖ≤Uϖ≤γ2ϖ,where γ1 and γ2 are K∞ functions. Suppose Vη≔V1S1+Uϖ and S10∈S1. If the inequality,
(32)V̇=∂V∂ηh≤−cV+υ,is valid in the domain θ¯∈N, and c and υ are positive constants, then S1t is still in the set for ∀t∈0,∞.Lemma 1 (see [20]).
For any two functionskat and kbt, suppose S1≔S1∈ℝ:−kat<S1<kbt⊂ℝ and N≔ℝl×S1⊂ℝl+1 to be open sets. For system η̇=ht,η with η≔ϖ,S1T∈N as state, function h:ℝ+×N→ℝl+1 is piecewise continuous with respect to t, and S1 satisfies the Lipschitz condition in ℝ+×N. If there exist two functions U:ℝl→ℝ+ and V1:S1→ℝ+ that are continuous and positively definite in their domains, respectively, the following two expressions are valid when S1→−kat or S1→kbt:Theorem 1.
For the closed-loop system Equation (5), if the virtual control variables and the control law satisfy Equations (18), (25), and (30), then for any set Mz0 to which the initial condition yz0 belongs, there always exists a sufficiently large set Mz≔k¯c1t<−ka11+xz2dt≤yzt≤kb11+xz2dt<k¯c1t that can prevent the system’s output constraints from being violated. Moreover, by selecting appropriate design parameters, it is possible to bound all closed-loop signals of the closed-loop system Equation (5) and make the output error converge in the neighborhood of the origin.Proof 1.
According to Equations (16), (24), and (29), one can construct a Lyapunov function that is a combination of asymmetric barrier Lyapunov functions and traditional Lyapunov functions as follows:
(33)Vz=Vz1+Vz2+Vz3=1−p1sz22logka112ka112−sz22+p1sz22logkb112kb112−sz22+12yz32+12sz32+12yz42+12sz42.By taking derivatives on both sides of the Lyapunov function in Equation (33), one obtains Equation (34):
(34)V̇z=μsz2ṡz2+yz3ẏz3+sz3ṡz3+yz4ẏz4+sz4ṡz4,where μ=1−p1/ka112−sz22+p1/kb112−sz22.Define the estimated error of the TVGESO system to satisfy Equation (35):
(35)∣ez22∣<Nz2,∣ez32∣<Nz3,∣ez42∣<Nz4,where Nz2, Nz3, and Nz4 are positive constants.It can be seen that the filtering errors are(36)yz3=x¯z3∗−xz3∗,yz4=x¯z4∗−xz4∗.By taking derivatives ofyz3 and yz4, one obtains the dynamic filtering errors:
(37)ẏz3=−τz3−1yz3−ẋz3∗,ẏz4=−τz4−1yz4−ẋz4∗.From Equations (13)–(30) and (36), one has
(38)xz2=sz2+xz2d,xz3=sz3+x¯z3∗=sz3+yz3+xz3∗,xz4=sz4+x¯z4∗=sz4+yz4+xz4∗.From Equations (13)–(28) and (36)–(38), one has
(39)ṡz2=fz2xz2−xy22sinxz1cosxz1+atεr−ẋz2d+gz2sz3+yz3+xz3∗=gz2sz3+gz2yz3−gz2kz2+k¯z2sz2−e~z22,where e~z22=vz22−atε/r, and it is assumed that e~z22<N~z2, N~z2 is a positive constant, and μN~z2>0.
(40)ṡz3=fz3xz3+sz4+yz4+xz4∗+dz3−ẋz3∗=sz4+yz4−kz3sz3−ez31−μgz2sz2,(41)ṡz4=fz4xz3+fz5xz4+gz3uz+dz4−xz4∗=−kz4sz4−kz5sz4∂zsgnsz4−ez41.According to Young’s inequality and Equations (39)–(41), one has
(42)sz2ṡz2=sz2gz2sz3+gz2yz3−gz2kz2+k¯z2sz2−e~z22≤12+12gz2−gz2kz2−gz2k¯z2sz22+12gz2yz32+12N~z22+gz2sz2sz3,(43)sz3ṡz3=sz3sz4+yz4−kz3sz3−ez31−μgz2sz2≤32−kz3sz32+12sz42+12yz42+12Nz32−μgz2sz2sz3,(44)sz4ṡz4=sz4−kz4sz4−kz5sz4∂zsgnsz4−ez41≤12−kz4+32kz5sz42+12Nz42+12kz5.It is clear that the variables in the system model and their derivatives are all bounded. If there exist continuous functionsz~z3 and z~z4 that satisfy z~z3>0 and z~z4>0, the variables ẋz3∗ and ẋz4∗ satisfy
(45)ẋz3∗≤z~z3,ẋz4∗≤z~z4.According to Young’s inequality and Equations (36)–(37) and (45), one has
(46)yz3ẏz3=yz3−τz3−1yz3−ẋz3∗≤−τz3−1yz32+12yz32+z~z32,(47)yz4ẏz4=yz4−τz4−1yz4−ẋz4∗≤−τz4−1yz42+12yz42+z~z42.By substituting Equations (42)–(47) into Equation (34) and arranging the terms, one has
(48)V̇z=μsz2ṡz2+yz3ẏz3+sz3ṡz3+yz4ẏz4+sz4ṡz4≤−μgz2kz2+gz2k¯z2−12gz2−12sz22−kz3−32−12μsz32−kz4−1−32kz5sz42−τz3−1−12−12μgz2yz32−τz4−1−1yz42+12μN~z22+12Nz32+12Nz42+12z~z32+12z~z42+12kz5.For convenience of description, define the following parameters:(49)Kz2=gz2kz2+gz2k¯z2−12gz2−12,Kz3=kz3−32−12μ,Kz4=kz4−1−32kz5,σz3=τz3−1−12−12μgz2,σz4=τz4−1−1,A~=12μN~z22+12Nz32+12Nz42+12z~z32+12z~z42+12kz5.Then, Equation (48) can be rewritten as
(50)V̇z≤−μKz2sz22−Kz3sz32−Kz4s42−σz3yz32−σz4yz42+A~.In the set−ka11t<sz2<kb11t, one has
(51)−1−p1ka112−sz22+p1kb112−sz22sz22≤−1−p1logka112ka112−sz22+p1logkb112kb112−sz22.Then, Equation (50) can be further rewritten as
(52)V̇z≤−1−p1logka112ka112−sz22+p1logkb112kb112−sz22Kz2−Kz3sz32−Kz4sz42−σz3yz32−σz4yz42+A~.Define positive-definite matrixQ as
(53)Q=Kz2000Kz3000Kz4.By selectingκ=min2λminQ,2σz3,2σz4, where κ is a positive constant and λminQ denotes the minimum eigenvalue of matrix Q, one has
(54)V̇z≤−κVz+A~.Define two setsMz0≔yz∈ℝ:k¯c0≤yz0≤k¯c0⊂ℝ and Mz≔yz∈ℝ:k¯c1≤yzt≤k¯c1⊂ℝ that satisfy k¯c1t>k¯c0+Y¯0+xz2d0⊂ℝ and k¯c1t<k¯c0+Y¯0+xz2d0⊂ℝ. From sz20=yz0−xz2d0, one has
(55)−ka11≤sz20≤kb11.Therefore, setMz is an invariant set. According to Lemma 1, it can be seen that given V̇z≤−κVz+A~, the following expression is valid for ∀t≥0:
(56)−ka11≤sz2t≤kb11.According to Equations (13) and (56), one has the following expression for ∀t≥0:
(57)k¯c1t<−ka11+xz2dt≤yzt≤kb11+xz2dt<k¯c1t.Therefore, it can be concluded that for any initial compact setMz0 defined by yz0∈Mz0, there is always a sufficiently large compact set Mz to make yz∈Mz for ∀t≥0.Define parameters according to Equation (48) to satisfy the following rules:
(58)kz2≥12gz2−1+12−k¯z2+12κ,kz3≥32+12μ+12κ,kz4≥32kz5+1+12κ,τz3−1≥12+12μgz2+12κ,τz4−1≥1+12κ.Multiply both sides of Equation (54) by eκt and arrange the terms to obtain the following expression:
(59)Vzt≤κVz0−A~e−κt+A~κ.Therefore, it is clear that by designing parameterskz2, kz3, kz4, and kz5 and parameters τz3 and τz4, it is possible to ensure that all closed-loop signals of the system are bounded. Opting to increase kz2, kz3, kz4, and kz5 and decrease τz3 and τz4 can ensure that κ is sufficiently large to make the filtering errors and the error surface small enough to control accuracy. Q.E.D.Remark 1.
Theoretically, larger values of the designed parameterskz2, kz3, kz4, and kz5; smaller ones of τz3 and τz4; the final boundary of the resulting error surfaces sz2, sz3, and sz4; and filtering errors yz3 and yz4 indicate an increase in the precision of control. However, in practice, very large values of kz2, kz3, kz4, and kz5, and very small ones of τz3 and τz4 can easily lead to input saturation in interceptor missile control systems, which triggers the saturation nonlinearity of the system such that the required overloads of the interceptor missile are beyond the available capacity, resulting in a reduction in the system’s control performance. Moreover, given the physical limitations of the low-pass filter, τz3 and τz4 should not be chosen to be arbitrarily small. Therefore, the parameters of the control algorithm should be properly selected in light of practical considerations.


## 4.3. Design of Control Law for Yaw Channel

Based on the design of the control law for the pitch channel, one can substitutevy22, vy32, and vy42—the estimated interferences of the TVGESO—into the system of Equation (6) and define xy2d as the system’s track command sign for the following form of control law for the yaw channel:
(60)sy2=xy2−xy2d,τy3x¯̇y3∗+x¯y3∗=xy3∗,x¯y3∗0=xy3∗0,xy3∗=−gy2−1fy2xy2+2xz2xy2tanxz1−vy22rcosxz1−ẋy2d−ky2+k¯y2sy2,sy3=xy3−x¯y3∗,τy4x¯̇y4∗+x¯y4∗=xy4∗,x¯y4∗0=xy4∗0,xy4∗=−fy3xy3−vy32+x¯̇y3∗−ky3sy3−μ~gy2sy2,sy4=xy4−x¯y4∗,uy=gy3−1−fy4xy3−fy5xy4−vy42+x¯̇y4∗−ky4sy4−ky5sy4∂ysgnsy4,where k¯y2=1−p2ka122+p2kb122,
(61)p2sy2=10<sy2,0sy2≤0,ka12=xy2d−k¯c1, kb12=k¯c1−xy2d, μ~=1−p2/ka122−sy22+p2/kb122−sy22, ky2>0, ky3>0, ky4>0, ky5>0, 0<∂y<1, gy2>0, gy3>0, and r>0. The parameters sy2, sy3, and sy4 are error surfaces; τy3 and τy4 are filter time constants, τy3>0 and τy4>0; xy3∗ and xy4∗ are virtual control variables of the system Equation (6); and x¯y3∗ and x¯y4∗, respectively, are virtual control variables after filtering.According to the stability analysis method in Equations (33)–(59), it can be proven that the closed-loop system of Equation (6) is stable and the system’s control precision can be achieved by designing appropriate parameters.


## 5. Dynamic Control Allocation Algorithm

Based on Equations (30) and (60), one can obtain the control moments jointly generated by the aerodynamic rudders and the jet devices using control inputs uz and uy as the desired control moments. Therefore, it is necessary to use a dynamic control allocation technique to distribute the desired control moments to the actuators.Define virtual control momentu as
(62)u=Bδ,where u=uz,uyT, δ=δz,δsy,δy,δszT,
(63)B=Bz1Bz20000By1By2T,Bz1=qSL¯mzδz, Bz2=1+KzFsmaxLm, By1=qSL¯myδy, and By2=1+KyFsmaxLm.Given that actuators are subject to physical constraints, such as structural and load-related constraints, the range of deflection and speed are limited. Define the control moment as in the feasible rangeδmint≤δt≤δmaxt. To achieve stable control performance, define the rate of change of the control moment to satisfy δ̇mint≤δ̇t≤δ̇maxt, where δmin and δmax are the minimum and maximum positional constraints of the actuators, respectively, with δ̇min and δ̇max being, respectively, the minimum and maximum speed constraints. Using T to denote sampling time, one can rewrite the feasible range of the actuators as
(64)δ¯t≤δt≤δ¯t,where δ¯t=maxδmin,δt−T+δ̇minT and δ¯t=minδmax,δt−T+δ̇maxT.To achieve stable and smooth actuator trajectories to suppress the impacts of noise and interference on the controller, a dynamic control allocation algorithm is designed to solve the allocation problem between the aerodynamic rudders and the jet devices.The dynamic control allocation problem can be expressed as the following hybrid optimization problem:(65)minδtJ0+J1+J2,subjecttou=Bδ,δ¯t≤δt≤δ¯t,where J0=W0δt2, J2=W2δt−δt−T2+W2δt−δt−2T2, J1=W1δt−δst2, W0=W0T>0, W1=W1T>0, and W2=W2T>0.· represents a norm defined as x=xTx, δst denotes the desired control command of the actuators, J0 denotes the minimum energy principle, J1 denotes the minimum error principle for the desired control commands of the actuators, and J2 denotes the minimum transition rate principle for the relative sampling time of the expected control commands.Define diagonal matricesR0, R1, and R2 with corresponding dimensions
(66)R0=diagR01,R02,R03,R04,R1=diagR11,R12,R13,R14,R2=diagR21,R22,R23,R24,where
(67)R0i=δ¯itδit,δit>δ¯it,1,δ¯it≤δit≤δ¯itδ¯itδit,δit<δ¯it,,R1i=δ¯sitδsit,δsit>δ¯sit,1,δ¯sit≤δsit≤δ¯sitδ¯sitδsit,δsit<δ¯sit,,and(68)R2i=δ¯it−Tδit−T,δit−T>δ¯it−T,1,δ¯it−T≤δit−T≤δ¯it−T,δ¯it−Tδit−T,δit−T<δ¯it−T,with δit, δ¯it, δ¯it, δsit, δ¯sit, δ¯sit, δit−T, δ¯it−T, and δ¯it−T the elements of the corresponding vectors i∈1,2,3,4.Theorem 2.
The hybrid optimization problem in Equation (65) is equivalent to
(69)minδtJ¯0+J¯1+J¯2,subjecttou=Bδ,δ¯t≤δt≤δ¯t,where J¯0=R0W0δt2, J¯2=R2W2δt−δt−T2+W2δt−δt−2T2, and J¯1=R1W1δt−δst2, and the general solution is given in Equation (70):
(70)δt=Eδst+F1δt−T+F2δt−2T+Gut,where E=I−GBW−2R12W12, F1=I−GBW−2R22W22, F2=I−GBW−2W22, G=W−2BTBW−2BT−1, and W=R02W02+R12W12+R22W22+W221/2.Proof 2.
From Equation (69), one has
(71)J¯=J¯0+J¯1+J¯2=δTtR02W02δ+δt−δstTR12W12δt−δst+δt−δt−TTR22W22δt−δt−T+δt−δt−2TTW22δt−δt−2T=Wδt−δ0∗t2,where δ0∗t=W−2R12W12δst+R22W22δt−T+W22δt−2T. Therefore, the optimization problem in Equation (69) is equivalent to
(72)minδt=Wδt−δ0∗t2,subjecttou=Bδ,δ¯t≤δt≤δ¯t.According to the weighted pseudo-inverse theorem [38], the general solution of the optimization problem in Equation (69) is
(73)δt=I−GBδ0∗t+Gut.By substituting Equation (71) into Equation (73), one can find that Theorem 2 is valid. Q.E.D.Remark 2.
By including the minimum error term related to the error between the command of the given step and those one or two steps prior, the purpose is to reduce chattering in the system and smoothen the control commands. By increasing the coefficient matricesR0, R1, and R2, one can solve the constraint problem of control moment to solve the optimization problem.


## 6. Simulation Analysis

To verify the effectiveness of the asymmetric barrier Lyapunov function-based IGC algorithm designed in this study for the interceptor missile, it was assumed that given an interceptor missile, its initial position wasXm,Ym,Zm=0m,0m,0m, its initial velocity was Vm0=1600m/s, and the initial position of the target was Xt,Yt,Zt=5000m,6000m,4000m, and the initial velocity was Vt0=1200m/s. The rudder deflection angle of the interceptor missile was limited in the range -30° to +30°, the angular velocity of the rudder’s deflection was limited in the range -300°/s to +300°/s, the maximum steady-state thrust of the jet devices was Fsmax=4800N, and the their moment amplification factor was Kz=Ky=0.3. The asymmetric output constraints were k¯c1=−4 and k¯c1=2, and the time-varying asymmetric output constraints were k¯c1=−4+0.2cost and k¯c1=2+0.1sint. The aerodynamic parameters of the interceptor missile are shown in Table 1.Table 1
Aerodynamic parameters of the interceptor missile.
Parameter
Value
Parameter
Value
Jz
247.43 kg·m2
Czβ
19.79
Jy
247.43 kg·m2
mzα
-15.26
m
204 kg
myβ
-15.26
L¯
0.22 m
mzωz
-0.16
Lm
2 m
myωy
-0.16
S
0.0409 m2
mzδz
-11.81
Cyα
-19.79
myδy
-11.81The method designed in this study (i.e., ABLFIGC+DCA) was compared with the combined use of IGC and the traditional dynamic surface sliding-mode control law (i.e., IGC+DCA). Assuming that the interferences in the system weredz3=dz4=dy3=dy4=0.02sint, target interception was maneuvered in the following two scenarios:Scenario 1.
The target was in uniform motion:atε=atβ=0m/s2.Scenario 2.
The target was in accelerated motion:atε=atβ=10m/s2.When the asymmetric output constraints werek¯c1=−4 and k¯c1=2, the simulation comparison plots of Scenario 1 are shown in Figures 2–7.Figure 2
Curves of angular velocities of vertical line of sight of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 3
Curves of angular velocities of horizontal line of sight of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 4
Attack angle curves of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 5
Sideslip angle curves of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 6
Pitch angular velocity curves of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 7
Yaw angular velocity curves of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figures2 and 3, respectively, show the curves of angular velocities of the vertical and horizontal lines of sight of the interceptor missile in Scenario 1. It is clear that a certain degree of chattering occurred during convergence when the traditional IGC algorithm was used, and the two angular velocities exhibited large jumps in the late stage of guidance. By contrast, the ABLFIGC+DCA algorithm designed in this study caused the two angular velocities to converge more quickly to steady-state values in a smoother convergence process that met the asymmetric output constraints k¯c1 and k¯c1, thus improving the stability of the guidance and control system and showing good robustness against external interferences.Figures4–7 show the curves of the attack angle, sideslip angle, and angular velocities of the pitch and yaw of the interceptor missile in Scenario 1, respectively. A certain degree of chattering appeared due to interference when the traditional IGC algorithm was used, and the above four parameters all exhibited large jumps in the late stage of guidance. By contrast, the ABLFIGC+DCA algorithm designed in this study renders the entire convergence process smoother, with the above four parameters stable and not recording large jumps in the late stage of guidance. Moreover, this algorithm showed good robustness against external interference.The simulation comparison plots of Scenario2 are shown in Figures 8–13.Figure 8
Curves of angular velocities of vertical line of sight of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 9
Curves of angular velocities of the horizontal line of sight of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 10
Attack angle curves of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 11
Sideslip angle curves of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 12
Pitch angular velocity curves of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 13
Yaw angular velocity curves of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figures8–13 show the simulation comparison plots of the target motion Scenario 2. From Figures 8 and 9, it is clear that the ABLFIGC+DCA algorithm smoothened convergence and was free of large jumps in the guidance process, met the asymmetric output constraints k¯c1 and k¯c1, and showed good robustness against external interference. As shown in Figures 10–13, the ABLFIGC+DCA algorithm ensured that the flight attitude of the interceptor missile was stable and changed smoothly.The results of comparison of the two scenarios in terms of the target miss distance and interception time of the interceptor missile are shown in Table2.Table 2
Comparison of simulation results between the two scenarios.
Scenario
Control mode
Miss distance (m)
Interception time (s)
1
ABLFIGC+DCA
0.608
15.21
IGC + DCA
0.792
15.26
2
ABLFIGC+DCA
0.854
21.42
IGC + DCA
0.992
21.48As shown in Table2, the algorithm designed in this study had a smaller miss distance and interception time than the IGC algorithm subject to the traditional dynamic sliding-mode control law.For Scenario1, the simulation comparison plots of δz, δsy, δy, and δsz of the actuators obtained by the dynamic control allocation algorithm are shown in Figures 14–17.Figure 14
Curves of elevator deflection angleδz: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 15
Curves of rudder deflection angleδy: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 16
Curves of equivalent rudder deflection angleδsz: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 17
Curves of equivalent rudder deflection angleδsy: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figures14–17 show the curves of the elevator deflection angle δz, the rudder deflection angle rudder angle δy, and the equivalent rudder deflection angles δsz and δsy, respectively, all of which were obtained via the dynamic control allocation algorithm. As shown in these figures, the ABLFIGC+DCA algorithm rendered the actuator outputs bounded and caused them to alter smoothly, had good robustness against external interferences, and did not incur a large jump in the late stage of guidance, indicating a good control allocation performance.When the time-varying asymmetric output constraints werek¯c1=−4+0.2cost and k¯c1=2+0.1sint, due to the attack angle, sideslip angle, and angular velocities of the pitch and yaw of the interceptor missile which are similar to the above, therefore, only the angular velocities of the vertical and horizontal lines of sight of the interceptor missile are compared. The simulation comparison plots of Scenario 1 are shown in Figures 18 and 19.Figure 18
Curves of angular velocities of vertical line of sight of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 19
Curves of angular velocities of horizontal line of sight of the interceptor missile in Scenario1: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.The simulation comparison plots of Scenario2 are shown in Figures 20 and 21.Figure 20
Curves of angular velocities of vertical line of sight of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figure 21
Curves of angular velocities of horizontal line of sight of the interceptor missile in Scenario2: ABLFIGC+DCA refers to a combination of the proposed asymmetric barrier Lyapunov function-based IGC algorithm and the dynamic control allocation algorithm; IGC+DCA refers to a combination of the IGC algorithm subject to the traditional dynamic surface sliding-mode control law and the dynamic control allocation algorithm.Figures18–21, respectively, show the curves of angular velocities of the vertical and horizontal lines of sight of the interceptor missile in Scenario 1 and Scenario 2. From Figures 18 and 21, it is clear that the ABLFIGC+DCA algorithm smoothened convergence and was free of large jumps in the guidance process, met the asymmetric output constraints k¯c1=−4+0.2cost and k¯c1=2+0.1sint, and showed good robustness against external interference.As shown by the above simulation results, the asymmetric barrier Lyapunov function-based IGC algorithm proposed in this study for the interceptor missile, when compared with the conventional IGC algorithm, can enable the outputs to meet constraints and the system to have better control performance and anti-interference ability, thus improving the stability of the guidance and control systems for the interceptor missile.


## 7. Conclusion

In the context of the interceptor missile which uses a direct-force/aerodynamic-force control scheme, by considering the coupling relationship between the pitch and yaw channels of the interceptor missile as well as the output constraint problems of the system. This study combined the IGC method with an asymmetric barrier Lyapunov function to design an asymmetric barrier Lyapunov function-based IGC algorithm for the interceptor missile. Compared with traditional algorithms, by adopting the asymmetric barrier Lyapunov function, the proposed algorithm relaxed the constraints on the initial conditions of the system. In addition, the study utilized the dynamic surface sliding-mode control law and a TVGESO to design an IGC algorithm for interceptor missile, which not only reduced the system’s requirements for high-order differentiability of the stability function but also enhanced the control system’s resistance to unknown interferences. The study also considered the existing redundancy of the aerodynamic rudder and the reaction jet device in the actuator and consequently designed a dynamic control allocation algorithm to allocate the desired control moments to the actuator, thereby improving its efficiency. Finally, the simulation results showed that the proposed algorithm demonstrated relatively good dynamic characteristics and was able to satisfy the interceptor missile IGC system’s requirements for accuracy and stability without breaking the constraints.



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*Source: 8531584-2019-04-28.xml*