# Fixed-Time Disturbance Observer-Based Adaptive Finite-Time Guidance Law Design considering Impact Angle Constraint and Autopilot Dynamics

Authors: Yunjie Wu; Fei Ma; Xiaofei Yang; Siqi Wang; Xiaodong Liu
Journal: Complexity
 (2021)

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## Abstract

This paper aims to establish an effective guidance law to accomplish the interception guidance mission for a missile intercepting a target with impact angle constraint and autopilot dynamics. To achieve this purpose, a fixed-time disturbance observer-based adaptive finite-time guidance law is presented. First, a fixed-time disturbance observer (FTDO) is designed to guarantee the fast estimation of the lumped disturbance caused by the target maneuver. Then, the FTDO-based adaptive integral sliding mode backstepping (AISMB) guidance law is constructed for the interception guidance problem. Besides, several adaptive laws are established to estimate the derivative of virtual control inputs, making the “differential explosion problem” of conventional backstepping get avoided. The finite-time convergence characteristic of the closed-loop system is analyzed by utilizing the Lyapunov stability theory. Finally, the simulation examples are conducted to demonstrate the effectiveness of the proposed composite guidance law.


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## Body

## 1. Introduction

The high maneuverability of the targets can bring some great difficulties in the interception and defense in modern war [1]. Therefore, it is necessary to develop some innovative guidance methods [2] for the missiles to improve the interception efficiency against such targets. To achieve this goal, the missiles are usually required to complete the target interception at an expected impact angle while ensuring the small miss distance. By applying this guidance strategy, the efficiency of warheads and striking efficiency can be improved [3].For a long time, the proportional navigation (PN) guidance law is one of the widely attractive methods which possesses simple calculation and easy implementation for the practical engineering [4]. On this basis, scholars have designed varieties of modified PN-based guidance laws, such as pure proportional navigation (PPN) guidance law [5], biased proportional navigation (BPN) guidance law [6], and generalized proportional navigation (GPN) guidance law [7]. By introducing an argument term to PN, Kim et al. [6] can intercept the target at the desired angle. In [7], the GPN is designed to achieve the guidance mission in the presence of small maneuvering and nonmaneuvering target. However, when facing a highly maneuvering target, few of these mentioned methods can simultaneously guarantee the small miss distance and desired impact angle. This disadvantage encourages the researchers to further develop some advanced methods for the guidance law, i.e., optimal control [8], sliding mode control [9], and disturbance observer-based control [10, 11].The optimal control-based guidance law can achieve accurate attack at an expected impact angle under the premise of minor control energy. However, the design process of this guidance law depends on the estimation of time-to-go [12]. Though some scholars have proposed an excellent estimation of time-to-go and applied it to achieve the guidance mission with specified attack time [13], this guidance scheme cannot be suitable for other nonlinear schemes. As one of the most promising methods, the sliding mode control- (SMC-) based guidance law can provide the demanding of miss distance and terminal impact angle constraint without the prior knowledge of time-to-go. In addition, this method has great robustness to external disturbances and parameter uncertainties [14]. In [15], a guidance law for intercepting the target with evasive maneuver is proposed by applying the second-order sliding mode control method. Moreover, Ebrahimi et al. [16] combined optimal control and sliding mode control to construct a composite guidance law. Although these traditional SMC guidance laws can achieve robust guidance with simple forms, the state of the closed-loop system can only satisfy asymptotical stability. However, in some practical scenarios, the interception only lasts for several seconds. In this situation, the guidance law should have faster convergence speed, while the previous asymptotical stability cannot guarantee this property.Subsequently, the finite-time stability theory [17] has been extended to the guidance and control of the spacecrafts for the purpose of addressing the aforesaid control issue. As known, the finite-time convergence theory-based controller can acquire the finite-time convergence property so as to drive the system state into the origin within a predefined convergence time [18]. Notably, the terminal sliding mode control is an excellent method to ensure this finite-time property. In [19], a finite-time convergent guidance law is presented to ensure that the line-of-sight (LOS) angular rate and LOS error converge to zero in finite time. Thus, the accurate interception is achieved, and the expected impact angle constraint is also satisfied. Based on the terminal sliding mode control theory, a guidance law [20] satisfying different initial conditions is designed to intercept the constant speed targets. Although abovementioned papers can solve the interception guidance problem with impact angle constraints, the target maneuvers are only considered as nonmoving or constant speed moving, which exhibits certain limitations in the practical engagements. Besides, traditional terminal sliding mode control may show the singularity when the sliding modes converge to zero [21]. Therefore, an adaptive nonsingular fast terminal sliding mode guidance law is developed by [22] to intercept the maneuvering targets with impact angle constraints and improve the singular problem of conventional terminal sliding mode control. Furthermore, one novel nonsingular terminal sliding mode control is designed to ensure that the missile strikes the target with the terminal impact angle constraint [23].Strictly speaking, the autopilot dynamics of interceptor missiles can usually bring substantial influences to the actual guidance system [24], while most of the aforementioned studies do not focus on this issue. Thus, a finite-time guidance law considering the dynamics of the missile’s autopilot as a first-order lag is designed in [25]. However, the autopilot lag would be more complicated in the actual situation and can usually be modeled as a second-order dynamics [26]. In [27], a robust continuous guidance law with the terminal angle constraint is proposed in the presence of second-order autopilot dynamics. In addition, Zhao et al. [28] presented one output feedback sliding mode guidance law by applying the sliding mode theory and dynamic surface control, in which the adaptive technique is utilized to acquire the estimation of target’s maneuver.To further improve the disturbance rejection capability, guidance laws based on the disturbance observer, i.e., nonlinear disturbance observer (NDO) [29], extended disturbance observer (ESO) [30], and sliding mode disturbance observer (SMO) [31], have been broadly applied to handle this issue. However, the convergence speed of the disturbance observers in aforesaid research studies is mainly depended on the observers’ gains, and the transient process will be significantly deteriorated [32].In recent years, fixed-time stability gets attractive and provides an effective strategy to construct the disturbance observer [33]. Theoretically speaking, the disturbance observers with fixed-time stability can guarantee that the estimation errors converge to zero in a prescribed fixed-time regardless of the initial conditions, indicating that these observers would obtain faster estimation [34]. In [35], a fixed-time differentiator-based disturbance observer is designed, which can ensure the exact estimation for the missile’s control system. Besides, Zuo [36] extended the fixed-time stability technique to high-order systems and presented the general formulations of the fixed-time disturbance observer. Furthermore, the fixed-time disturbance observer (FTDO) is also developed in [37] so as to acquire the excellent observation of lumped disturbances. It is obvious that the fixed-time disturbance observer has shown quite impressive estimation performance [38].Inspired by above discussions, this study attempts to employ the fixed-time disturbance observer to construct the composite adaptive finite time guidance law for intercepting highly maneuvering targets considering impact angle constraint and autopilot dynamics. Main contributions of this paper are summarized as follows:(1)
The fixed-time disturbance observer (FTDO) is utilized to estimate the lumped disturbance caused by target maneuver. Different from the most existing ESO [10] and SMO [31], the proposed FTDO can achieve fixed-time stability, and the convergence time is independent of the initial conditions.(2)
Based on the FTDO, an adaptive integral sliding mode backstepping (AISMB) guidance law is proposed to accomplish the interception mission, and the closed-loop guidance system can converge a small neighborhood around the origin in a finite time. The switching functions are introduced into the sliding mode reaching laws, which can guarantee the reaching property, meanwhile avoid the possible singularity problem. Moreover, the adopted adaptive laws can modify the control gains so as to reduce the control energy.(3)
In contrast to the conventional backstepping methods [28], the proposed adaptive laws in AISMB are employed to estimate the derivative of virtual control inputs in the recursive design process, and the “explosion of terms” can be conquered accordingly.The remaining part of this paper is organized as follows. The missile-target engagement motion equations, autopilot dynamics, and control-oriented model are described in Section2. Section 3 presents the composite guidance scheme and the corresponding stability analysis of the closed-loop system. Numerical simulations are implemented in Section 4 so as to demonstrate the effectiveness of the proposed guidance law. Finally, some conclusion remarks are presented in Section 5.


## 2. Problem Formulation

### 2.1. Motion Kinematics and Relative Motion Dynamics

In this study, we consider the planar interception guidance problem with impact angle constraint and autopilot dynamics. The 2-dimensional (2D) homing engagement geometry for a missile intercepts a target, as depicted in Figure1.Figure 1
Two-dimensional engagement geometry.The velocity of the missile and target are represented byVm and Vt, respectively. am and at are used to represent the corresponding normal acceleration of the missile and target, respectively. γm and γt denote the flight path angle (FPA) of the missile and target, respectively. Besides, the LOS angle and relative disturbance between the missile and target are denoted by θ and r, respectively. Then, the kinematic engagement equations can be described by Kumar et al. [39]:(1)r˙=Vtcosθt−Vmcosθm,(2)rθ˙=Vtsinθt−Vmsinθm,(3)γ˙m=amVm,(4)γ˙t=atVt,where θm=γm−θ, θt=γt−θ.Calculating the time derivatives of (1) and (2) yields(5)r¨=rθ˙2+atsinθt−amsinθm,(6)θ¨=−2r˙rθ˙+cosθtrat−cosθmram.Remark 1.
Generally, the acceleration along the missile’s velocity cannot be controlled in the terminal guidance phase, and only equation (6) is considered to design the guidance law. Besides, from (6), cosθm=0 will be satisfied when θm=±π/2, which can bring about control singularity and cause the failure of missile interception finally. However, if θm=±π/2, γ˙m−θ˙≠0 will hold. Thus, θm=±π/2 is not the stable equilibrium point, and the system trajectories will just cross this unstable point [20]. On this basis, am can be applied to control the LOS angle θ.
During the design process of the guidance law, we assume that the missile and the target are point masses moving in constant velocitiesVm and Vt, and the speed ratio of target-to-missile satisfies υ=Vt/Vm<1. Besides, to avoid the possible singularity analyzed in Remark 1, equation (6) is rewritten as(7)θ¨=−2r˙rθ˙−1ram+φ,where φ=cosθt/rat−cosθm/ram+1/ram.
When the missile hit the target, following relation exists [39]:(8)Vmsinγmf−θf=Vtsinγtf−θf,where γmf, γtf, and θf represent the final FPA of the missile, final FPA of the target, and final expected LOS angle, respectively.
Furthermore, with equation (2), rθ˙=0 will hold. Thus, the relation between the final LOS angle θf and designed impact angle θimp can be regarded as a one-to-one correspondence formulated by(9)θf=γtf−arctansinθimpcosθimp−υ,where θimp=γtf−γmf.
More specifically, whenθimp=nπ,n=±1,±2,…, the head-on and tail-chase intercept scenarios can be considered.Remark 2.
Basically, the missile can successfully achieve the interception by hit-to-kill impact, and the relative distance at the impact time is not equal to zero. Therefore, whenr≤r0∈rmin,rmax is satisfied, it can be considered that the missile has completed the interception mission.
Typically, the dynamics of autopilot can be modeled as the following second-order system [24]:(10)a¨m=−2ξωn−ωn2a˙m+ωn2amc,where ξ denotes the damping ratio, ωn denotes the natural frequency, and amc denotes the actual guidance command.


### 2.2. Control-Oriented Model

Considering autopilot dynamics (10), the acceleration command amc can be designed to achieve the target interception at the expected LOS angle θf. Therefore, let x1=θ−θf, x2=θ˙, x3=am,x4=a˙m,andu=amc, and the control-oriented model can be described by(11)x˙1=x2,x˙2=f2+g2x3+d2,x˙3=x4,x˙4=f4+g4u,where f2=−2r˙/rx2,g2=−1/r,d2=φ,f4=−2ξωnx3−ωn2x32,andg4=ωn2.Assumption 1.
The target acceleration and its derivative are bounded, that is,at≤atmax and a˙t≤a˙tmax.Assumption 2.
The derivative of the disturbanced˙2 is bounded and satisfies d˙2≤d2max.
Finally, the purpose of this guidance law design problem is to ensure the small miss distance and small LOS error by driving the statex1 and x2 to the small neighborhood of zero in finite time.


## 2.1. Motion Kinematics and Relative Motion Dynamics

In this study, we consider the planar interception guidance problem with impact angle constraint and autopilot dynamics. The 2-dimensional (2D) homing engagement geometry for a missile intercepts a target, as depicted in Figure1.Figure 1
Two-dimensional engagement geometry.The velocity of the missile and target are represented byVm and Vt, respectively. am and at are used to represent the corresponding normal acceleration of the missile and target, respectively. γm and γt denote the flight path angle (FPA) of the missile and target, respectively. Besides, the LOS angle and relative disturbance between the missile and target are denoted by θ and r, respectively. Then, the kinematic engagement equations can be described by Kumar et al. [39]:(1)r˙=Vtcosθt−Vmcosθm,(2)rθ˙=Vtsinθt−Vmsinθm,(3)γ˙m=amVm,(4)γ˙t=atVt,where θm=γm−θ, θt=γt−θ.Calculating the time derivatives of (1) and (2) yields(5)r¨=rθ˙2+atsinθt−amsinθm,(6)θ¨=−2r˙rθ˙+cosθtrat−cosθmram.Remark 1.
Generally, the acceleration along the missile’s velocity cannot be controlled in the terminal guidance phase, and only equation (6) is considered to design the guidance law. Besides, from (6), cosθm=0 will be satisfied when θm=±π/2, which can bring about control singularity and cause the failure of missile interception finally. However, if θm=±π/2, γ˙m−θ˙≠0 will hold. Thus, θm=±π/2 is not the stable equilibrium point, and the system trajectories will just cross this unstable point [20]. On this basis, am can be applied to control the LOS angle θ.
During the design process of the guidance law, we assume that the missile and the target are point masses moving in constant velocitiesVm and Vt, and the speed ratio of target-to-missile satisfies υ=Vt/Vm<1. Besides, to avoid the possible singularity analyzed in Remark 1, equation (6) is rewritten as(7)θ¨=−2r˙rθ˙−1ram+φ,where φ=cosθt/rat−cosθm/ram+1/ram.
When the missile hit the target, following relation exists [39]:(8)Vmsinγmf−θf=Vtsinγtf−θf,where γmf, γtf, and θf represent the final FPA of the missile, final FPA of the target, and final expected LOS angle, respectively.
Furthermore, with equation (2), rθ˙=0 will hold. Thus, the relation between the final LOS angle θf and designed impact angle θimp can be regarded as a one-to-one correspondence formulated by(9)θf=γtf−arctansinθimpcosθimp−υ,where θimp=γtf−γmf.
More specifically, whenθimp=nπ,n=±1,±2,…, the head-on and tail-chase intercept scenarios can be considered.Remark 2.
Basically, the missile can successfully achieve the interception by hit-to-kill impact, and the relative distance at the impact time is not equal to zero. Therefore, whenr≤r0∈rmin,rmax is satisfied, it can be considered that the missile has completed the interception mission.
Typically, the dynamics of autopilot can be modeled as the following second-order system [24]:(10)a¨m=−2ξωn−ωn2a˙m+ωn2amc,where ξ denotes the damping ratio, ωn denotes the natural frequency, and amc denotes the actual guidance command.


## 2.2. Control-Oriented Model

Considering autopilot dynamics (10), the acceleration command amc can be designed to achieve the target interception at the expected LOS angle θf. Therefore, let x1=θ−θf, x2=θ˙, x3=am,x4=a˙m,andu=amc, and the control-oriented model can be described by(11)x˙1=x2,x˙2=f2+g2x3+d2,x˙3=x4,x˙4=f4+g4u,where f2=−2r˙/rx2,g2=−1/r,d2=φ,f4=−2ξωnx3−ωn2x32,andg4=ωn2.Assumption 1.
The target acceleration and its derivative are bounded, that is,at≤atmax and a˙t≤a˙tmax.Assumption 2.
The derivative of the disturbanced˙2 is bounded and satisfies d˙2≤d2max.
Finally, the purpose of this guidance law design problem is to ensure the small miss distance and small LOS error by driving the statex1 and x2 to the small neighborhood of zero in finite time.


## 3. Guidance Law Design and Analysis

In this section, the design process of the guidance scheme is described in detail. Specifically, the fixed-time disturbance observer is employed to estimate the lumped disturbance. Then, the FTDO-based guidance law is developed by combing the adaptive theory, integral sliding mode control, and backstepping control. The control structure of the guidance scheme is shown in Figure2.Figure 2
Control diagram of the composite guidance strategyNotation 1.
In this paper,sigxp=signxxp, where signx is the sign function.
Before the conduction of the guidance law, some necessary definitions and lemmas are introduced as follows.Definition 1 (see [40]).
Consider the system(12)x˙=fx,t,f0,t=0,x∈ℝn,where f:N0⟶ℝn is continuous on the open neighborhood N0 of the origin x=0. For any t0, given the initial state x0, which relies on the settling time Tx0, then system (12) exists in a solution x=φt,t0,x0. If t∈t0,Tx0, φt,t0,x0∈N/0 with limt⟶Tx0φt,t0,x0=0 holds, and if t>Tx0, φt,t0,x0=0 will hold. The origin of (12) possesses locally finite-time stability in the neighborhood N⊆N0 if it is also Lyapunov stable. This solution is said to be globally finite-time stable if N=N0=ℝn.Remark 3.
From above analysis, the finite-time stability will imply the asymptotic stability of the origin, and the convergence time can be guaranteed by a prescribed timeT=Tx0. If the system initial state x0 can be obtained, the settling time can be evaluated by Tx0, and the system can achieve finite-time stability. However, if the system state x0 is not available, the settling time cannot be estimated by Tx0 in advance. In this situation, it is possible that Tx0 will become large if the initial state is very large. Thus, to remedy this shortcoming, Polyakov [41] proposes the fixed-time stability which can guarantee that the settling time T=Tx0 will be independent of the initial condition x0.Definition 2 (see [41]).
Based on Definition1, the origin of (12) is said to be fixed-time stable if it satisfies globally finite-time stable and the settling time Tx0 is bounded, that is, for any initial state x0, there exists Tmax>0 such that Tx0<Tmax.Remark 4.
The settling time of fixed-time stability is regardless of the initial state, while finite time is the opposite. This property implies that the convergence time can be determined by a prescribed manner.Lemma 1 (see [21]).
For system (12) with continuous positive definite function Vx, if the following equation(13)V˙x≤−τVxη,where τ>0and0<η<1, holds, then the system will converge to the origin in finite time bounded by(14)t≤V1−ηx0τ1−η.Lemma 2 (see [1]).
Similarly, ifVx defined in Lemma 1 satisfies(15)V˙x≤−aVx−bVxγ,where a>0,b>0,and0<γ<1, the system converges to the origin in finite time bounded by(16)t≤1a1−γlnaVx01−γ+bb.Lemma 3 (see [28]).
For positive real numbersy1,y2,…,yn and 0<p<1, the following inequality is satisfied:(17)y12+y22+⋯+yn2p≤y1p+y2p+⋯+ynp2.Lemma 4 (see [42, 43]).
For the following system,(18)s=e˙n+knsigen−1αn+⋯+k1sige1α1,where ki and αii=1,2,…,n are constants, e˙i=ei+1,i=1,2,…,n−1. If ki is selected such that the polynomial pn+knpn−1+⋯+k2p+k1 is Hurwitz and αi are positive constants satisfying(19)α1=α,n=1,αi−1=αiαi+12αi+1−αi,i=2,…,n,where αn+1=1andαn=α∈0,1, then the system will converge to its equilibrium point in finite time. When n=2, the settling time can be estimated by(20)t≤3+α1V01−α1/3+α1K1−α1,where V0 denotes the initial value of the selected Lyapunov function V and K is a positive constant.


### 3.1. The Design of Fixed-Time Disturbance Observer

As is known to all, the lumped disturbanced2 will degrade the performance of the missile guidance and control system. Inspired by [44], a disturbance observer with fixed-time convergence characteristics is proposed to estimate the disturbances. This fixed-time disturbance observer can drive the estimation error to a small neighborhood of the origin within a bounded time regardless of the initial state. Therefore, we can obtain following theorem.Theorem 1.
For guidance system (11), the unknown term d2 satisfies d˙2<L2, where L2 is a positive constant. The fixed-time disturbance observer can be described as(21)x^˙2=f2+g2x3+d^2,d^2=k21sige21p+k22sige21q+∫0tk23signe21τdτ,where 0<p<1, q>1, x^2 is the estimation of x2, e21=x2−x^2 represents the estimate error, and k2jj=1,2,and3 are all positive designed parameters that satisfy the follow conditions:(22)M2=k23+L2,m2=ki3−L2,h2=1k21+2em2k211/3,k23>L2,k21h>M2,with e representing the nature constant.Proof.
Taking the derivative ofe21, one can obtain(23)e˙21=x˙2−x^˙2=d2−d^2=d2−k21sige21p−k22sige21q−∫0tk23signe21τdτ.
Definee22=d2−∫0tk23signe21τdτ, and equation (23) can be rewritten as(24)e˙21=−k21sige21p−k22sige21q+e22.
Then, calculating the derivative ofe22, one has(25)e˙22=−k33signe21+d˙2.
According to equations (24) and (25), the following error dynamics can be obtained:(26)e˙21=−k21sige21p−k22sige21q+e22,e˙22=−k33signe21+d˙2.
On the basis of the result in [44], if the observer gains satisfy condition (22), e21 and e22 can uniformly converge to the origin within a fixed-time bounded by(27)to≤1k221q−1−1k2111−p1+M21−M2h2/k21m2.
This result means thate21=0 and e22=0 for t>to. Recalling (23), it can be obtained that e˙21=d2−d^2=0. Therefore, the equivalent disturbance can be exactly estimated after t>to. This completes the proof of Theorem 1.Remark 5.
Compared with SMO and ESO, the applied FTDO can achieve the fixed-time stability, and the settling time is uniformly bounded and independent of the initial state. Besides, as shown in equation (21), the sign function is integrated in the integral term, which can guarantee the continuity of the estimation value and eliminate the chattering problem. This characteristic will be confirmed by the subsequent simulation results.


### 3.2. FTDO-Based Adaptive Integral Sliding Mode Backstepping Guidance Law Design

As shown in the previous section, the estimation of unknown disturbance can be obtained by the proposed FTDO in a short time regardless of the initial conditions. Based on this merit, the following composite guidance law is developed in detail.


#### 3.2.1. Finite Time Integral Siding Mode Control

Generally speaking, the small miss distance and the desired LOS angle can be satisfied by drivingx1 and x2 to zero [17]. Therefore, the sliding surface should contain x1 and x2. In this situation, we introduce a novel integral sliding surface s2 defined as(28)s2=x2−x20+∫λ11sigx1α1+λ12sigx2α2dτ,where λ11>0, λ12>0, 0<α1=α<1,andα2=2α1/α1+1.Combing the second equation of (11) and (28), it is clear that(29)s˙2=x˙2+λ11sigx1α1+λ12sigx2α2=f2+g2x3+d2+λ11sigx1α1+λ12sigx2α2.To stabilize the subsystem ofx2, the virtual control law can be designed as(30)x3d=g2−1−f2−λ11sigx1α1−λ12sigx2α2−k2F2s2−λ^2s2−d^2,where k2>0 is the designed parameter, d^2 is estimated by the designed FTDO, and F2s2 is constructed as(31)F2s2=sigs2β,s2≥ϖ2,τ21s2+τ22s22signs2,s2<ϖ2,with 0<β<1, where ϖ2 is a small enough positive constant, τ21=2−βϖ2β−1, τ22=1−βϖ2β−2, and λ^2 is an adaptive gain designed as(32)λ^˙2=κ2s22−σ2λ^2,s2≥δ20,else,with κ2>0, σ2>0, and δ2>0.Remark 6.
By introducing switching function (31) [45], the potential singularity that appeared in the derivative of term sigs2β will get avoided. For the case s2≥ϖ2, F˙2s2=βs2β−1s˙2 is nonsingular, while for the case s2<ϖ2, F˙2s2=τ21s˙2+2τ22s2s˙2 will also not to be singular when s2=0 and s˙2≠0. However, the subsystem will approach zero with worse convergence property when s2<ϖ2. Thus, we should select small enough to guarantee the power rate reaching law property, meanwhile avoid the singularity problem.
Define the state error ofx3 as(33)s3=x3−x3d.
Then, substituting (33) and (30) into (29), one can obtain that(34)s˙2=f2+g2x3d+s3+d2+λ11sigx1α1+λ12sigx2α2=f2+g2s3+g2x3d+d2+λ11sigx1α1+λ12sigx2α2=−k2F2s2−λ^2s2+d2−d^2+g2s3.


#### 3.2.2. Backstepping Strategy

The backstepping scheme contains a step-by-step construction of the new subsystem by the state errorsi=xi−xidi=3,4, where xid is the desired control command for state xi [29]. Based on this strategy, calculating the derivative of s3 yields(35)s˙3=x˙3−x˙3d=x4−x˙3d.It is obvious that the derivative ofx3d should be compensated in the subsequent design step. However, it is difficult to calculate the exact value of x˙3d, and the complexity will increase as the order of the system increases. In order to solve this “differential explosion” problem, x˙id will be considered as an uncertain term with the following reasonable assumption and finally resolved by the predesigned adaptive laws.Assumption 3.
(see [29]). x3d and x4d are bounded, satisfying xid≤χii=3and4 where χ3 and χ4 are all positive constants.
Thus, the virtual controlx4d can be designed as(36)x4d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2,where k3>0 is the designed parameter and F3s3 is constructed as(37)F3s3=sigs3β,s3≥ϖ3,τ31s3+τ32s32signs3,s3<ϖ3,with ϖ3 is a small enough positive constant, where τ31=2−βϖ3β−1, τ32=1−βϖ3β−2, and λ^3 is an adaptive gain designed as(38)λ^˙3=κ3s32−σ3λ^3,s3≥δ3,0,else,with κ2>0, σ2>0, and δ2>0. ρ^3 is an adaptive parameter designed as(39)ρ^˙3=γ3s322ε32−ξ3ρ^3,with ε3>0,γ3>0,andξ3>0.
Substituting (36) into (35), one obtains(40)s˙3=x4−x˙3d=x4d+s4−x˙3d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2−x˙3d+s4.
Similarly, the state error forx4 is defined as(41)s4=x4−x4d.
Calculating the time derivative ofs4 yields(42)s˙4=x˙4−x˙4d=f4+g4u+d4−x˙4d.
Then, the actual control law is designed as(43)u=g4−1−f4−s3−ρ^4s42ε42−k4F4s4−λ^4s4,where k4>0 is the designed parameter and F4s4 is constructed as(44)F4s4=sigs4β,s4≥ϖ4,τ41s4+τ42s42signs4,s4<ϖ4,with ϖ4 is a small enough positive constant, where τ41=2−βϖ4β−1 and τ42=1−βϖ4β−2. λ^4 is an adaptive gain designed as(45)λ^˙4=κ4s42−σ4λ^4,s4≥δ4,0,else,with κ4>0, σ4>0, and δ4>0. ρ^4 is an adaptive parameter designed as(46)ρ^˙4=γ4s422ε42−ξ4ρ^4,with ε3>0,γ3>0,andξ3>0.
Substituting (43) into (42), one has(47)s˙4=f4+g4u+d4−x˙4d=−s3−ρ^4s42ε42−k4F4s4−λ^4s4−x˙4d.
Based on the above design steps, the proposed FTDO-based guidance law can be summarized as(48)s2=x2−x20+∫λ11sigx1α1+λ12sigx2α2dτ,x3d=g2−1−f2−λ11sigx1α1−λ12sigx2α2−k2F2s2−λ^2s2−d^2,s3=x3−x3d,x4d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2,s4=x4−x4d,u=g4−1−f4−s3−ρ^4s42ε42−k4F4s4−λ^4s4,with functions Fisii=2,3,and4 defined as(49)Fisi=sigsiβ,si≥ϖi,τi1s2+τi2si2signsi,si<ϖi,and adaptive laws:(50)λ^˙i=κisi2−σiλ^i,si≥δi,0,else,i=2,3,and4,(51)ρ^˙j=γjsj22εj2−ξjρ^j,j=3and4.Theorem 2.
For guidance system (11) with proposed AISMB guidance law (48), FTDO (21), as well as adaptive laws (50) and (51), the closed-loop system can achieve finite time stability and the sliding modes sii=2,3,and4 and states x1 and x2 are guaranteed to converge to the small neighborhood around the origin in finite time.Proof.
Selecting the Lyapunov candidate as(52)V1=12∑i=24si2+12∑i=24λ˜i2κi+12∑j=3,4ρ˜j2γj,where λ˜i=λ¯i−λ^, ρ˜j=ρ¯j−ρ^j, λ¯i>0,andρ¯i>0.
Its time derivative is(53)V˙1=∑i=24sis˙i−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj.
Substituting equations (34), (40), and (47) into (53), one obtains(54)V˙1=s2−k2F2s2−λ^2s2+d2−d^2+g2s3+s3−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2−x˙3d+s4+s4−s3−ρ^4s42ε42−k4F4s4−λ^4s4−x˙4d−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj,where e2=d2−d^2.
Sinceϖi are sufficient small positive constants, (54) can be rewritten as(55)V˙1=−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2−ρ^3s322ε32−x˙3ds3−ρ^4s422ε42−s4x˙4d−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj.
With adaptive laws (50) and (51), (55) becomes(56)V˙1=−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2−ρ^3s322ε32−x˙3ds3−ρ^4s422ε42−s4x˙4d−∑i=24λ˜iκiκisi2−σiλ^i−∑j=3,4ρ˜jγjγjsj22εj2−ξjρ^j=−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4−xjdsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
According to Assumption3, one has(57)V˙1≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4−xjdsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4χjsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
Notingsjχj≤sj2/2εj2χj2+εj2/2 and s2e2≤1/2ν2s22+ν2/2e22 (ν2 is a positive constant which is only used for analysis), one can get(58)V˙1≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+12ν2s22+ν22e22+∑j=3,4sj22εj2χj2+εj22−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
By the definition ofλ˜i and ρ˜j, it can be obtained that(59)λ˜iλ^i≤−12λ˜i2+12λ¯i2,ρ˜jρ^j≤−12ρ˜j2+12ρ¯j2.
Substituting (59) into (58) gives(60)V˙1≤−∑i=24kisiβ+1−λ¯2s22−λ¯3+ρ¯32ε32s32−λ¯4+ρ¯42ε42s42−∑i=24σi2κiλ˜i2−∑j=3,4ξj2γjρ˜j2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422.
Applyingxμyθ≤μ/μ+θxμ+θ+θ/μ+θι−μ/θyμ+θ and taking x=si2, y=1, μ=β+1/2,θ=1−β/2,andι=2/β+1 into account, one has(61)si2β+1/2≤β+12si2+1−β2β+121+β/1−β.
Similarly, one can obtain that(62)∑i=24σi2κiλ˜i2β+1/2≤β+12∑i=24σi2κiλ˜i2+1−β2β+121+β/1−β,(63)∑j=3,4ξj2γjρ˜j2β+1/2≤β+12∑j=3,4ξj2γjρ˜j2+1−β2β+121+β/1−β.
Substituting equations (61)–(63) into (60) yields(64)V˙1≤−∑i=24kisi2β+1/2−λ¯22β+1s22β+1/2−ι−λ¯3+ρ¯32ε322β+1s32β+1/2−ι−λ¯4+ρ¯42ε422β+1s42β+1/2−ι−2β+1∑i=24σi2κiλ˜i2β+1/2−ι−2β+1∑j=3,4ξj2γjρ˜j2β+1/2−ι+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422,where ι=1−β/2β+1/22β/1−β>0.
With Lemma3, (64) becomes(65)V˙1≤−k2+2λ¯2β+12β+1/212s22β+1/2−k3+2β+1λ¯3+ρ¯32ε322β+1/212s32β+1/2−k4+2β+1λ¯4+ρ¯42ε422β+1/212s42β+1/2−2β+1σ0β+1/212∑i=241κiλ˜i2β+1/2−2β+1ξ0β+1/212∑j=3,41γjρ˜j2β+1/2+ιλ¯2+λ¯3+ρ¯32ε32+λ¯4+ρ¯42ε42+2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422,where σ0=minσi>0 and ξ0=minξj>0.
By choosing appropriate parameters, following conditions will hold:(66)Ψ1=mink2+2λ¯2β+12β+1/2,k3+2β+1λ¯3+ρ¯32ε322β+1/2,k4+2β+1λ¯4+ρ¯42ε422β+1/2>0,Ψ2=2β+1σ0β+1/2>0,Ψ3=2β+1ξ0β+1/2>0,Ψ0=ιλ¯2+λ¯3+ρ¯32ε32+λ¯4+ρ¯42ε42+2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422>0.
Then, (65) can be rewritten as(67)V˙1≤−Ψ112∑i=24si2β+1/2−Ψ212∑i=241κiλ˜i2β+1/2−Ψ312∑i=341γiρ˜i2β+1/2+Ψ0≤−ΨV1β+1/2,where Ψ=minΨ1,Ψ2,Ψ3>0.
According to Lemma1, the system will converge to the equilibrium point in finite time and the settling time is given by(68)tr≤V11−βx0Ψ1−β.
Furthermore, all the state in the closed-loop guidance system are uniformly ultimately bounded.
Then, consider another Lyapunov function:(69)Vs=∑i=2412si2.
With (34), (40), and (47), the time derivative of Vs becomes(70)V˙s=∑i=24sis˙i≤−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2.
Sinces2e2≤1/2ν2s22+ν2/2e22 with ν2 is a positive constant, we have(71)V˙s≤−∑i=24kisiβ+1−∑i=24λ^isi2+12ν2s22+ν22e22≤−∑i=24kisiβ+1−λ^2−12ν2s22−λ^3s32−λ^4s42+ν22e22≤−2β+1/2mink2,k3,k4∑i=24si22β+1/2−2minλ^2−12ν2,λ^3,λ^4∑i=24si22+ν22e22.
From (71), if we choose appropriate adaptive laws for λ^ii=2,3,and4 and large enough ν2 to guarantee λ^2−1/2ν2>0, λ^3>0, and λ^4>0, one can obtain that(72)V˙s≤−Γ1Vsβ+1/2−Γ2Vs+Γ0≤−Γ1Vsβ+1/2−Γ2V,where Γ1=21+β/2minkii=2,3,4>0 and Γ2=2minλ^2−1/2ν2,λ^3,λ^4>0.
Thus, according to Lemma2, si will converge to zero in finite time bounded by(73)ts≤2Γ21−βlnΓ2Vs01−β/2+Γ1Γ1.
Whens2=0 and s˙2=0 are achieved, the following equation can be obtained:(74)s˙2=x˙2+λ11sigx1α1+λ12sigx2α2=0.
Namely,(75)x˙2=−λ11sigx1α1−λ12sigx2α2.
Sincex˙1=x2, (75) can be rewritten as(76)x¨1+λ12sigx˙1α2+λ11sigx1α1=0.
According to Lemma4, once parameters λ12 and λ11 are selected to make polynomial p2+λ12pα2+λ1pα1 Hurwitz, x1 and x2 can converge to the equilibrium point in finite time which can be calculated as(77)tx≤3+α1V01−α1/3+α1x10,x20K1−α1,where K is a positive constant.
Therefore, the designed composite guidance law can guarantee the finite-time stability of the closed-loop system and drive the sliding modessi and states x1 and x2 to converge to zero in finite time with the total settling time tc=to+tr+ts+tx.
The proof of Theorem2 has been completed.Remark 7.
In most existing literature, the “differential explosion problem” of the traditional backstepping method is solved by introducing the first-order filter or tracking differentiator. In this study, the proposed adaptive laws are applied to estimate the derivative of virtual control inputs.Remark 8.
For the global guidance system,λ11,λ12,α1,β1,κi,σi,εi,γj,εj,andξji=2,3,and4;j=3and4 are all positive parameters to be regulated. Specifically, ki can be selected large enough to stabilize each subsystem. For subsystem x2, κ2 is selected large enough and σ2 is selected small enough, while for the subsystems x3 and x4, regulation laws of κ3,κ4 and σ3,σ4 are the opposite. γj and ξj can be selected small enough, while εi can be selected large enough. With the satisfactory parameters, the convergence characteristic of the closed-loop system can be guaranteed. In other words, the state errors can converge to any small interval of the origin. Besides, when the system gets convergent, λ11,λ12, α1,andα2 will further balance the LOS error and miss distance. As λ11 and λ12 increase, the miss distance will increase and the LOS error decreases. Besides, oversized λ11 and λ12 will result in the failure of the mission. Thus, appropriate small λ11 and λ12 should be determined. The parameters α1 and β mainly affect the convergence rate. Large α1 and β will lead to the oscillation of the transient process, while small α1 and β will degrade the convergence speed and bring about the interception failure. Therefore, when tuning these parameters, selecting proper λ11,λ12, α1, and β first so as to guarantee the relatively fast convergence rate with acceptable miss distance and LOS error. Then, choosing appropriate ki, κi, σi, εi, γj,εj,andξj to stabilize each subsystem. Finally, these parameters can be fine-tuned to improve comprehensive performance.
To further explain the tuning process, the corresponding main tuning steps can be described as Table1.Table 1
The main parameter tuning steps.
(i) Selectλ11 and λ12 to make the polynomial p2+λ12pα2+λ1pα1 Hurwitz and set α1 and β as 0.5(ii) Fixλ11, λ12, α1, and β, set κ2, σ2, and ε2as zero, and then tune the parameter k2 to stabilize the subsystem x2; if this subsystem looks stable or has a stable trend, go to step iii; otherwise, continue this step(iii) Adjust the parameterκ2, σ2, and ε2 to further stabilize the subsystem x2; if this subsystem looks stable or has a stable trend, go to step iv; otherwise, continue this step(iv) Fixλ11, λ12, α1, β, k2, κ2, σ2, and ε2, set κ3, σ3, and ε3 as zero, and then tune the parameters k3,γ3,ε3,andξ3to stabilize the subsystem x3; if this subsystem looks stable or has a stable trend, go to step v; otherwise, continue this step(v) Adjust the parameterκ3, σ3, and ε3 to further stabilize the subsystem x3; if this subsystem looks stable or has a stable trend, go to step vi; otherwise, continue this step(vi) Fixλ11, λ12, α1, β, k2, κ2, σ2, ε2, k3, κ3, σ3, ε3, γ3,ε3,andξ3, set κ4, σ4, and ε4 as zero, and then tune the parameters k4, γ4,ε4,andξ4 to stabilize the subsystem x4; if this subsystem looks stable or has a stable trend, go to step vii; otherwise, continue this step(vii) Adjust the parameterκ4, σ4, and ε4 to further stabilize the subsystem x4; if this subsystem looks stable or has a stable trend, go to step viii; otherwise, continue this step(viii) In this step, the stability of closed-loop is guaranteed; then, re-adjust the parametersλ11, λ12, α1, and β to balance the miss distance, LOS error, and convergence speed; these parameters will be further fine-tuned so as to achieve better overall performance


## 3.1. The Design of Fixed-Time Disturbance Observer

As is known to all, the lumped disturbanced2 will degrade the performance of the missile guidance and control system. Inspired by [44], a disturbance observer with fixed-time convergence characteristics is proposed to estimate the disturbances. This fixed-time disturbance observer can drive the estimation error to a small neighborhood of the origin within a bounded time regardless of the initial state. Therefore, we can obtain following theorem.Theorem 1.
For guidance system (11), the unknown term d2 satisfies d˙2<L2, where L2 is a positive constant. The fixed-time disturbance observer can be described as(21)x^˙2=f2+g2x3+d^2,d^2=k21sige21p+k22sige21q+∫0tk23signe21τdτ,where 0<p<1, q>1, x^2 is the estimation of x2, e21=x2−x^2 represents the estimate error, and k2jj=1,2,and3 are all positive designed parameters that satisfy the follow conditions:(22)M2=k23+L2,m2=ki3−L2,h2=1k21+2em2k211/3,k23>L2,k21h>M2,with e representing the nature constant.Proof.
Taking the derivative ofe21, one can obtain(23)e˙21=x˙2−x^˙2=d2−d^2=d2−k21sige21p−k22sige21q−∫0tk23signe21τdτ.
Definee22=d2−∫0tk23signe21τdτ, and equation (23) can be rewritten as(24)e˙21=−k21sige21p−k22sige21q+e22.
Then, calculating the derivative ofe22, one has(25)e˙22=−k33signe21+d˙2.
According to equations (24) and (25), the following error dynamics can be obtained:(26)e˙21=−k21sige21p−k22sige21q+e22,e˙22=−k33signe21+d˙2.
On the basis of the result in [44], if the observer gains satisfy condition (22), e21 and e22 can uniformly converge to the origin within a fixed-time bounded by(27)to≤1k221q−1−1k2111−p1+M21−M2h2/k21m2.
This result means thate21=0 and e22=0 for t>to. Recalling (23), it can be obtained that e˙21=d2−d^2=0. Therefore, the equivalent disturbance can be exactly estimated after t>to. This completes the proof of Theorem 1.Remark 5.
Compared with SMO and ESO, the applied FTDO can achieve the fixed-time stability, and the settling time is uniformly bounded and independent of the initial state. Besides, as shown in equation (21), the sign function is integrated in the integral term, which can guarantee the continuity of the estimation value and eliminate the chattering problem. This characteristic will be confirmed by the subsequent simulation results.


## 3.2. FTDO-Based Adaptive Integral Sliding Mode Backstepping Guidance Law Design

As shown in the previous section, the estimation of unknown disturbance can be obtained by the proposed FTDO in a short time regardless of the initial conditions. Based on this merit, the following composite guidance law is developed in detail.


### 3.2.1. Finite Time Integral Siding Mode Control

Generally speaking, the small miss distance and the desired LOS angle can be satisfied by drivingx1 and x2 to zero [17]. Therefore, the sliding surface should contain x1 and x2. In this situation, we introduce a novel integral sliding surface s2 defined as(28)s2=x2−x20+∫λ11sigx1α1+λ12sigx2α2dτ,where λ11>0, λ12>0, 0<α1=α<1,andα2=2α1/α1+1.Combing the second equation of (11) and (28), it is clear that(29)s˙2=x˙2+λ11sigx1α1+λ12sigx2α2=f2+g2x3+d2+λ11sigx1α1+λ12sigx2α2.To stabilize the subsystem ofx2, the virtual control law can be designed as(30)x3d=g2−1−f2−λ11sigx1α1−λ12sigx2α2−k2F2s2−λ^2s2−d^2,where k2>0 is the designed parameter, d^2 is estimated by the designed FTDO, and F2s2 is constructed as(31)F2s2=sigs2β,s2≥ϖ2,τ21s2+τ22s22signs2,s2<ϖ2,with 0<β<1, where ϖ2 is a small enough positive constant, τ21=2−βϖ2β−1, τ22=1−βϖ2β−2, and λ^2 is an adaptive gain designed as(32)λ^˙2=κ2s22−σ2λ^2,s2≥δ20,else,with κ2>0, σ2>0, and δ2>0.Remark 6.
By introducing switching function (31) [45], the potential singularity that appeared in the derivative of term sigs2β will get avoided. For the case s2≥ϖ2, F˙2s2=βs2β−1s˙2 is nonsingular, while for the case s2<ϖ2, F˙2s2=τ21s˙2+2τ22s2s˙2 will also not to be singular when s2=0 and s˙2≠0. However, the subsystem will approach zero with worse convergence property when s2<ϖ2. Thus, we should select small enough to guarantee the power rate reaching law property, meanwhile avoid the singularity problem.
Define the state error ofx3 as(33)s3=x3−x3d.
Then, substituting (33) and (30) into (29), one can obtain that(34)s˙2=f2+g2x3d+s3+d2+λ11sigx1α1+λ12sigx2α2=f2+g2s3+g2x3d+d2+λ11sigx1α1+λ12sigx2α2=−k2F2s2−λ^2s2+d2−d^2+g2s3.


### 3.2.2. Backstepping Strategy

The backstepping scheme contains a step-by-step construction of the new subsystem by the state errorsi=xi−xidi=3,4, where xid is the desired control command for state xi [29]. Based on this strategy, calculating the derivative of s3 yields(35)s˙3=x˙3−x˙3d=x4−x˙3d.It is obvious that the derivative ofx3d should be compensated in the subsequent design step. However, it is difficult to calculate the exact value of x˙3d, and the complexity will increase as the order of the system increases. In order to solve this “differential explosion” problem, x˙id will be considered as an uncertain term with the following reasonable assumption and finally resolved by the predesigned adaptive laws.Assumption 3.
(see [29]). x3d and x4d are bounded, satisfying xid≤χii=3and4 where χ3 and χ4 are all positive constants.
Thus, the virtual controlx4d can be designed as(36)x4d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2,where k3>0 is the designed parameter and F3s3 is constructed as(37)F3s3=sigs3β,s3≥ϖ3,τ31s3+τ32s32signs3,s3<ϖ3,with ϖ3 is a small enough positive constant, where τ31=2−βϖ3β−1, τ32=1−βϖ3β−2, and λ^3 is an adaptive gain designed as(38)λ^˙3=κ3s32−σ3λ^3,s3≥δ3,0,else,with κ2>0, σ2>0, and δ2>0. ρ^3 is an adaptive parameter designed as(39)ρ^˙3=γ3s322ε32−ξ3ρ^3,with ε3>0,γ3>0,andξ3>0.
Substituting (36) into (35), one obtains(40)s˙3=x4−x˙3d=x4d+s4−x˙3d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2−x˙3d+s4.
Similarly, the state error forx4 is defined as(41)s4=x4−x4d.
Calculating the time derivative ofs4 yields(42)s˙4=x˙4−x˙4d=f4+g4u+d4−x˙4d.
Then, the actual control law is designed as(43)u=g4−1−f4−s3−ρ^4s42ε42−k4F4s4−λ^4s4,where k4>0 is the designed parameter and F4s4 is constructed as(44)F4s4=sigs4β,s4≥ϖ4,τ41s4+τ42s42signs4,s4<ϖ4,with ϖ4 is a small enough positive constant, where τ41=2−βϖ4β−1 and τ42=1−βϖ4β−2. λ^4 is an adaptive gain designed as(45)λ^˙4=κ4s42−σ4λ^4,s4≥δ4,0,else,with κ4>0, σ4>0, and δ4>0. ρ^4 is an adaptive parameter designed as(46)ρ^˙4=γ4s422ε42−ξ4ρ^4,with ε3>0,γ3>0,andξ3>0.
Substituting (43) into (42), one has(47)s˙4=f4+g4u+d4−x˙4d=−s3−ρ^4s42ε42−k4F4s4−λ^4s4−x˙4d.
Based on the above design steps, the proposed FTDO-based guidance law can be summarized as(48)s2=x2−x20+∫λ11sigx1α1+λ12sigx2α2dτ,x3d=g2−1−f2−λ11sigx1α1−λ12sigx2α2−k2F2s2−λ^2s2−d^2,s3=x3−x3d,x4d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2,s4=x4−x4d,u=g4−1−f4−s3−ρ^4s42ε42−k4F4s4−λ^4s4,with functions Fisii=2,3,and4 defined as(49)Fisi=sigsiβ,si≥ϖi,τi1s2+τi2si2signsi,si<ϖi,and adaptive laws:(50)λ^˙i=κisi2−σiλ^i,si≥δi,0,else,i=2,3,and4,(51)ρ^˙j=γjsj22εj2−ξjρ^j,j=3and4.Theorem 2.
For guidance system (11) with proposed AISMB guidance law (48), FTDO (21), as well as adaptive laws (50) and (51), the closed-loop system can achieve finite time stability and the sliding modes sii=2,3,and4 and states x1 and x2 are guaranteed to converge to the small neighborhood around the origin in finite time.Proof.
Selecting the Lyapunov candidate as(52)V1=12∑i=24si2+12∑i=24λ˜i2κi+12∑j=3,4ρ˜j2γj,where λ˜i=λ¯i−λ^, ρ˜j=ρ¯j−ρ^j, λ¯i>0,andρ¯i>0.
Its time derivative is(53)V˙1=∑i=24sis˙i−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj.
Substituting equations (34), (40), and (47) into (53), one obtains(54)V˙1=s2−k2F2s2−λ^2s2+d2−d^2+g2s3+s3−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2−x˙3d+s4+s4−s3−ρ^4s42ε42−k4F4s4−λ^4s4−x˙4d−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj,where e2=d2−d^2.
Sinceϖi are sufficient small positive constants, (54) can be rewritten as(55)V˙1=−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2−ρ^3s322ε32−x˙3ds3−ρ^4s422ε42−s4x˙4d−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj.
With adaptive laws (50) and (51), (55) becomes(56)V˙1=−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2−ρ^3s322ε32−x˙3ds3−ρ^4s422ε42−s4x˙4d−∑i=24λ˜iκiκisi2−σiλ^i−∑j=3,4ρ˜jγjγjsj22εj2−ξjρ^j=−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4−xjdsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
According to Assumption3, one has(57)V˙1≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4−xjdsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4χjsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
Notingsjχj≤sj2/2εj2χj2+εj2/2 and s2e2≤1/2ν2s22+ν2/2e22 (ν2 is a positive constant which is only used for analysis), one can get(58)V˙1≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+12ν2s22+ν22e22+∑j=3,4sj22εj2χj2+εj22−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
By the definition ofλ˜i and ρ˜j, it can be obtained that(59)λ˜iλ^i≤−12λ˜i2+12λ¯i2,ρ˜jρ^j≤−12ρ˜j2+12ρ¯j2.
Substituting (59) into (58) gives(60)V˙1≤−∑i=24kisiβ+1−λ¯2s22−λ¯3+ρ¯32ε32s32−λ¯4+ρ¯42ε42s42−∑i=24σi2κiλ˜i2−∑j=3,4ξj2γjρ˜j2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422.
Applyingxμyθ≤μ/μ+θxμ+θ+θ/μ+θι−μ/θyμ+θ and taking x=si2, y=1, μ=β+1/2,θ=1−β/2,andι=2/β+1 into account, one has(61)si2β+1/2≤β+12si2+1−β2β+121+β/1−β.
Similarly, one can obtain that(62)∑i=24σi2κiλ˜i2β+1/2≤β+12∑i=24σi2κiλ˜i2+1−β2β+121+β/1−β,(63)∑j=3,4ξj2γjρ˜j2β+1/2≤β+12∑j=3,4ξj2γjρ˜j2+1−β2β+121+β/1−β.
Substituting equations (61)–(63) into (60) yields(64)V˙1≤−∑i=24kisi2β+1/2−λ¯22β+1s22β+1/2−ι−λ¯3+ρ¯32ε322β+1s32β+1/2−ι−λ¯4+ρ¯42ε422β+1s42β+1/2−ι−2β+1∑i=24σi2κiλ˜i2β+1/2−ι−2β+1∑j=3,4ξj2γjρ˜j2β+1/2−ι+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422,where ι=1−β/2β+1/22β/1−β>0.
With Lemma3, (64) becomes(65)V˙1≤−k2+2λ¯2β+12β+1/212s22β+1/2−k3+2β+1λ¯3+ρ¯32ε322β+1/212s32β+1/2−k4+2β+1λ¯4+ρ¯42ε422β+1/212s42β+1/2−2β+1σ0β+1/212∑i=241κiλ˜i2β+1/2−2β+1ξ0β+1/212∑j=3,41γjρ˜j2β+1/2+ιλ¯2+λ¯3+ρ¯32ε32+λ¯4+ρ¯42ε42+2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422,where σ0=minσi>0 and ξ0=minξj>0.
By choosing appropriate parameters, following conditions will hold:(66)Ψ1=mink2+2λ¯2β+12β+1/2,k3+2β+1λ¯3+ρ¯32ε322β+1/2,k4+2β+1λ¯4+ρ¯42ε422β+1/2>0,Ψ2=2β+1σ0β+1/2>0,Ψ3=2β+1ξ0β+1/2>0,Ψ0=ιλ¯2+λ¯3+ρ¯32ε32+λ¯4+ρ¯42ε42+2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422>0.
Then, (65) can be rewritten as(67)V˙1≤−Ψ112∑i=24si2β+1/2−Ψ212∑i=241κiλ˜i2β+1/2−Ψ312∑i=341γiρ˜i2β+1/2+Ψ0≤−ΨV1β+1/2,where Ψ=minΨ1,Ψ2,Ψ3>0.
According to Lemma1, the system will converge to the equilibrium point in finite time and the settling time is given by(68)tr≤V11−βx0Ψ1−β.
Furthermore, all the state in the closed-loop guidance system are uniformly ultimately bounded.
Then, consider another Lyapunov function:(69)Vs=∑i=2412si2.
With (34), (40), and (47), the time derivative of Vs becomes(70)V˙s=∑i=24sis˙i≤−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2.
Sinces2e2≤1/2ν2s22+ν2/2e22 with ν2 is a positive constant, we have(71)V˙s≤−∑i=24kisiβ+1−∑i=24λ^isi2+12ν2s22+ν22e22≤−∑i=24kisiβ+1−λ^2−12ν2s22−λ^3s32−λ^4s42+ν22e22≤−2β+1/2mink2,k3,k4∑i=24si22β+1/2−2minλ^2−12ν2,λ^3,λ^4∑i=24si22+ν22e22.
From (71), if we choose appropriate adaptive laws for λ^ii=2,3,and4 and large enough ν2 to guarantee λ^2−1/2ν2>0, λ^3>0, and λ^4>0, one can obtain that(72)V˙s≤−Γ1Vsβ+1/2−Γ2Vs+Γ0≤−Γ1Vsβ+1/2−Γ2V,where Γ1=21+β/2minkii=2,3,4>0 and Γ2=2minλ^2−1/2ν2,λ^3,λ^4>0.
Thus, according to Lemma2, si will converge to zero in finite time bounded by(73)ts≤2Γ21−βlnΓ2Vs01−β/2+Γ1Γ1.
Whens2=0 and s˙2=0 are achieved, the following equation can be obtained:(74)s˙2=x˙2+λ11sigx1α1+λ12sigx2α2=0.
Namely,(75)x˙2=−λ11sigx1α1−λ12sigx2α2.
Sincex˙1=x2, (75) can be rewritten as(76)x¨1+λ12sigx˙1α2+λ11sigx1α1=0.
According to Lemma4, once parameters λ12 and λ11 are selected to make polynomial p2+λ12pα2+λ1pα1 Hurwitz, x1 and x2 can converge to the equilibrium point in finite time which can be calculated as(77)tx≤3+α1V01−α1/3+α1x10,x20K1−α1,where K is a positive constant.
Therefore, the designed composite guidance law can guarantee the finite-time stability of the closed-loop system and drive the sliding modessi and states x1 and x2 to converge to zero in finite time with the total settling time tc=to+tr+ts+tx.
The proof of Theorem2 has been completed.Remark 7.
In most existing literature, the “differential explosion problem” of the traditional backstepping method is solved by introducing the first-order filter or tracking differentiator. In this study, the proposed adaptive laws are applied to estimate the derivative of virtual control inputs.Remark 8.
For the global guidance system,λ11,λ12,α1,β1,κi,σi,εi,γj,εj,andξji=2,3,and4;j=3and4 are all positive parameters to be regulated. Specifically, ki can be selected large enough to stabilize each subsystem. For subsystem x2, κ2 is selected large enough and σ2 is selected small enough, while for the subsystems x3 and x4, regulation laws of κ3,κ4 and σ3,σ4 are the opposite. γj and ξj can be selected small enough, while εi can be selected large enough. With the satisfactory parameters, the convergence characteristic of the closed-loop system can be guaranteed. In other words, the state errors can converge to any small interval of the origin. Besides, when the system gets convergent, λ11,λ12, α1,andα2 will further balance the LOS error and miss distance. As λ11 and λ12 increase, the miss distance will increase and the LOS error decreases. Besides, oversized λ11 and λ12 will result in the failure of the mission. Thus, appropriate small λ11 and λ12 should be determined. The parameters α1 and β mainly affect the convergence rate. Large α1 and β will lead to the oscillation of the transient process, while small α1 and β will degrade the convergence speed and bring about the interception failure. Therefore, when tuning these parameters, selecting proper λ11,λ12, α1, and β first so as to guarantee the relatively fast convergence rate with acceptable miss distance and LOS error. Then, choosing appropriate ki, κi, σi, εi, γj,εj,andξj to stabilize each subsystem. Finally, these parameters can be fine-tuned to improve comprehensive performance.
To further explain the tuning process, the corresponding main tuning steps can be described as Table1.Table 1
The main parameter tuning steps.
(i) Selectλ11 and λ12 to make the polynomial p2+λ12pα2+λ1pα1 Hurwitz and set α1 and β as 0.5(ii) Fixλ11, λ12, α1, and β, set κ2, σ2, and ε2as zero, and then tune the parameter k2 to stabilize the subsystem x2; if this subsystem looks stable or has a stable trend, go to step iii; otherwise, continue this step(iii) Adjust the parameterκ2, σ2, and ε2 to further stabilize the subsystem x2; if this subsystem looks stable or has a stable trend, go to step iv; otherwise, continue this step(iv) Fixλ11, λ12, α1, β, k2, κ2, σ2, and ε2, set κ3, σ3, and ε3 as zero, and then tune the parameters k3,γ3,ε3,andξ3to stabilize the subsystem x3; if this subsystem looks stable or has a stable trend, go to step v; otherwise, continue this step(v) Adjust the parameterκ3, σ3, and ε3 to further stabilize the subsystem x3; if this subsystem looks stable or has a stable trend, go to step vi; otherwise, continue this step(vi) Fixλ11, λ12, α1, β, k2, κ2, σ2, ε2, k3, κ3, σ3, ε3, γ3,ε3,andξ3, set κ4, σ4, and ε4 as zero, and then tune the parameters k4, γ4,ε4,andξ4 to stabilize the subsystem x4; if this subsystem looks stable or has a stable trend, go to step vii; otherwise, continue this step(vii) Adjust the parameterκ4, σ4, and ε4 to further stabilize the subsystem x4; if this subsystem looks stable or has a stable trend, go to step viii; otherwise, continue this step(viii) In this step, the stability of closed-loop is guaranteed; then, re-adjust the parametersλ11, λ12, α1, and β to balance the miss distance, LOS error, and convergence speed; these parameters will be further fine-tuned so as to achieve better overall performance


## 3.2.1. Finite Time Integral Siding Mode Control

Generally speaking, the small miss distance and the desired LOS angle can be satisfied by drivingx1 and x2 to zero [17]. Therefore, the sliding surface should contain x1 and x2. In this situation, we introduce a novel integral sliding surface s2 defined as(28)s2=x2−x20+∫λ11sigx1α1+λ12sigx2α2dτ,where λ11>0, λ12>0, 0<α1=α<1,andα2=2α1/α1+1.Combing the second equation of (11) and (28), it is clear that(29)s˙2=x˙2+λ11sigx1α1+λ12sigx2α2=f2+g2x3+d2+λ11sigx1α1+λ12sigx2α2.To stabilize the subsystem ofx2, the virtual control law can be designed as(30)x3d=g2−1−f2−λ11sigx1α1−λ12sigx2α2−k2F2s2−λ^2s2−d^2,where k2>0 is the designed parameter, d^2 is estimated by the designed FTDO, and F2s2 is constructed as(31)F2s2=sigs2β,s2≥ϖ2,τ21s2+τ22s22signs2,s2<ϖ2,with 0<β<1, where ϖ2 is a small enough positive constant, τ21=2−βϖ2β−1, τ22=1−βϖ2β−2, and λ^2 is an adaptive gain designed as(32)λ^˙2=κ2s22−σ2λ^2,s2≥δ20,else,with κ2>0, σ2>0, and δ2>0.Remark 6.
By introducing switching function (31) [45], the potential singularity that appeared in the derivative of term sigs2β will get avoided. For the case s2≥ϖ2, F˙2s2=βs2β−1s˙2 is nonsingular, while for the case s2<ϖ2, F˙2s2=τ21s˙2+2τ22s2s˙2 will also not to be singular when s2=0 and s˙2≠0. However, the subsystem will approach zero with worse convergence property when s2<ϖ2. Thus, we should select small enough to guarantee the power rate reaching law property, meanwhile avoid the singularity problem.
Define the state error ofx3 as(33)s3=x3−x3d.
Then, substituting (33) and (30) into (29), one can obtain that(34)s˙2=f2+g2x3d+s3+d2+λ11sigx1α1+λ12sigx2α2=f2+g2s3+g2x3d+d2+λ11sigx1α1+λ12sigx2α2=−k2F2s2−λ^2s2+d2−d^2+g2s3.


## 3.2.2. Backstepping Strategy

The backstepping scheme contains a step-by-step construction of the new subsystem by the state errorsi=xi−xidi=3,4, where xid is the desired control command for state xi [29]. Based on this strategy, calculating the derivative of s3 yields(35)s˙3=x˙3−x˙3d=x4−x˙3d.It is obvious that the derivative ofx3d should be compensated in the subsequent design step. However, it is difficult to calculate the exact value of x˙3d, and the complexity will increase as the order of the system increases. In order to solve this “differential explosion” problem, x˙id will be considered as an uncertain term with the following reasonable assumption and finally resolved by the predesigned adaptive laws.Assumption 3.
(see [29]). x3d and x4d are bounded, satisfying xid≤χii=3and4 where χ3 and χ4 are all positive constants.
Thus, the virtual controlx4d can be designed as(36)x4d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2,where k3>0 is the designed parameter and F3s3 is constructed as(37)F3s3=sigs3β,s3≥ϖ3,τ31s3+τ32s32signs3,s3<ϖ3,with ϖ3 is a small enough positive constant, where τ31=2−βϖ3β−1, τ32=1−βϖ3β−2, and λ^3 is an adaptive gain designed as(38)λ^˙3=κ3s32−σ3λ^3,s3≥δ3,0,else,with κ2>0, σ2>0, and δ2>0. ρ^3 is an adaptive parameter designed as(39)ρ^˙3=γ3s322ε32−ξ3ρ^3,with ε3>0,γ3>0,andξ3>0.
Substituting (36) into (35), one obtains(40)s˙3=x4−x˙3d=x4d+s4−x˙3d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2−x˙3d+s4.
Similarly, the state error forx4 is defined as(41)s4=x4−x4d.
Calculating the time derivative ofs4 yields(42)s˙4=x˙4−x˙4d=f4+g4u+d4−x˙4d.
Then, the actual control law is designed as(43)u=g4−1−f4−s3−ρ^4s42ε42−k4F4s4−λ^4s4,where k4>0 is the designed parameter and F4s4 is constructed as(44)F4s4=sigs4β,s4≥ϖ4,τ41s4+τ42s42signs4,s4<ϖ4,with ϖ4 is a small enough positive constant, where τ41=2−βϖ4β−1 and τ42=1−βϖ4β−2. λ^4 is an adaptive gain designed as(45)λ^˙4=κ4s42−σ4λ^4,s4≥δ4,0,else,with κ4>0, σ4>0, and δ4>0. ρ^4 is an adaptive parameter designed as(46)ρ^˙4=γ4s422ε42−ξ4ρ^4,with ε3>0,γ3>0,andξ3>0.
Substituting (43) into (42), one has(47)s˙4=f4+g4u+d4−x˙4d=−s3−ρ^4s42ε42−k4F4s4−λ^4s4−x˙4d.
Based on the above design steps, the proposed FTDO-based guidance law can be summarized as(48)s2=x2−x20+∫λ11sigx1α1+λ12sigx2α2dτ,x3d=g2−1−f2−λ11sigx1α1−λ12sigx2α2−k2F2s2−λ^2s2−d^2,s3=x3−x3d,x4d=−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2,s4=x4−x4d,u=g4−1−f4−s3−ρ^4s42ε42−k4F4s4−λ^4s4,with functions Fisii=2,3,and4 defined as(49)Fisi=sigsiβ,si≥ϖi,τi1s2+τi2si2signsi,si<ϖi,and adaptive laws:(50)λ^˙i=κisi2−σiλ^i,si≥δi,0,else,i=2,3,and4,(51)ρ^˙j=γjsj22εj2−ξjρ^j,j=3and4.Theorem 2.
For guidance system (11) with proposed AISMB guidance law (48), FTDO (21), as well as adaptive laws (50) and (51), the closed-loop system can achieve finite time stability and the sliding modes sii=2,3,and4 and states x1 and x2 are guaranteed to converge to the small neighborhood around the origin in finite time.Proof.
Selecting the Lyapunov candidate as(52)V1=12∑i=24si2+12∑i=24λ˜i2κi+12∑j=3,4ρ˜j2γj,where λ˜i=λ¯i−λ^, ρ˜j=ρ¯j−ρ^j, λ¯i>0,andρ¯i>0.
Its time derivative is(53)V˙1=∑i=24sis˙i−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj.
Substituting equations (34), (40), and (47) into (53), one obtains(54)V˙1=s2−k2F2s2−λ^2s2+d2−d^2+g2s3+s3−ρ^3s32ε32−k3F3s3−λ^3s3−g2s2−x˙3d+s4+s4−s3−ρ^4s42ε42−k4F4s4−λ^4s4−x˙4d−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj,where e2=d2−d^2.
Sinceϖi are sufficient small positive constants, (54) can be rewritten as(55)V˙1=−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2−ρ^3s322ε32−x˙3ds3−ρ^4s422ε42−s4x˙4d−∑i=24λ˜iλ^˙iκi−∑j=3,4ρ˜jρ^˙jγj.
With adaptive laws (50) and (51), (55) becomes(56)V˙1=−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2−ρ^3s322ε32−x˙3ds3−ρ^4s422ε42−s4x˙4d−∑i=24λ˜iκiκisi2−σiλ^i−∑j=3,4ρ˜jγjγjsj22εj2−ξjρ^j=−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4−xjdsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
According to Assumption3, one has(57)V˙1≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4−xjdsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+s2e2+∑j=3,4χjsj−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
Notingsjχj≤sj2/2εj2χj2+εj2/2 and s2e2≤1/2ν2s22+ν2/2e22 (ν2 is a positive constant which is only used for analysis), one can get(58)V˙1≤−∑i=24kisiβ+1+∑i=24−λ¯isi2+σiκiλ˜iλ^i+12ν2s22+ν22e22+∑j=3,4sj22εj2χj2+εj22−ρ¯jsj22εj2+ξjγjρ˜jρ^j.
By the definition ofλ˜i and ρ˜j, it can be obtained that(59)λ˜iλ^i≤−12λ˜i2+12λ¯i2,ρ˜jρ^j≤−12ρ˜j2+12ρ¯j2.
Substituting (59) into (58) gives(60)V˙1≤−∑i=24kisiβ+1−λ¯2s22−λ¯3+ρ¯32ε32s32−λ¯4+ρ¯42ε42s42−∑i=24σi2κiλ˜i2−∑j=3,4ξj2γjρ˜j2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422.
Applyingxμyθ≤μ/μ+θxμ+θ+θ/μ+θι−μ/θyμ+θ and taking x=si2, y=1, μ=β+1/2,θ=1−β/2,andι=2/β+1 into account, one has(61)si2β+1/2≤β+12si2+1−β2β+121+β/1−β.
Similarly, one can obtain that(62)∑i=24σi2κiλ˜i2β+1/2≤β+12∑i=24σi2κiλ˜i2+1−β2β+121+β/1−β,(63)∑j=3,4ξj2γjρ˜j2β+1/2≤β+12∑j=3,4ξj2γjρ˜j2+1−β2β+121+β/1−β.
Substituting equations (61)–(63) into (60) yields(64)V˙1≤−∑i=24kisi2β+1/2−λ¯22β+1s22β+1/2−ι−λ¯3+ρ¯32ε322β+1s32β+1/2−ι−λ¯4+ρ¯42ε422β+1s42β+1/2−ι−2β+1∑i=24σi2κiλ˜i2β+1/2−ι−2β+1∑j=3,4ξj2γjρ˜j2β+1/2−ι+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422,where ι=1−β/2β+1/22β/1−β>0.
With Lemma3, (64) becomes(65)V˙1≤−k2+2λ¯2β+12β+1/212s22β+1/2−k3+2β+1λ¯3+ρ¯32ε322β+1/212s32β+1/2−k4+2β+1λ¯4+ρ¯42ε422β+1/212s42β+1/2−2β+1σ0β+1/212∑i=241κiλ˜i2β+1/2−2β+1ξ0β+1/212∑j=3,41γjρ˜j2β+1/2+ιλ¯2+λ¯3+ρ¯32ε32+λ¯4+ρ¯42ε42+2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422,where σ0=minσi>0 and ξ0=minξj>0.
By choosing appropriate parameters, following conditions will hold:(66)Ψ1=mink2+2λ¯2β+12β+1/2,k3+2β+1λ¯3+ρ¯32ε322β+1/2,k4+2β+1λ¯4+ρ¯42ε422β+1/2>0,Ψ2=2β+1σ0β+1/2>0,Ψ3=2β+1ξ0β+1/2>0,Ψ0=ιλ¯2+λ¯3+ρ¯32ε32+λ¯4+ρ¯42ε42+2+∑i=24σi2κiλ¯i2+∑j=3,4ξj2γjρ¯j2+12ν2s22+χ322ε32s32+χ422ε42s42+ν22e22+ε322+ε422>0.
Then, (65) can be rewritten as(67)V˙1≤−Ψ112∑i=24si2β+1/2−Ψ212∑i=241κiλ˜i2β+1/2−Ψ312∑i=341γiρ˜i2β+1/2+Ψ0≤−ΨV1β+1/2,where Ψ=minΨ1,Ψ2,Ψ3>0.
According to Lemma1, the system will converge to the equilibrium point in finite time and the settling time is given by(68)tr≤V11−βx0Ψ1−β.
Furthermore, all the state in the closed-loop guidance system are uniformly ultimately bounded.
Then, consider another Lyapunov function:(69)Vs=∑i=2412si2.
With (34), (40), and (47), the time derivative of Vs becomes(70)V˙s=∑i=24sis˙i≤−∑i=24kisiβ+1−∑i=24λ^isi2+s2e2.
Sinces2e2≤1/2ν2s22+ν2/2e22 with ν2 is a positive constant, we have(71)V˙s≤−∑i=24kisiβ+1−∑i=24λ^isi2+12ν2s22+ν22e22≤−∑i=24kisiβ+1−λ^2−12ν2s22−λ^3s32−λ^4s42+ν22e22≤−2β+1/2mink2,k3,k4∑i=24si22β+1/2−2minλ^2−12ν2,λ^3,λ^4∑i=24si22+ν22e22.
From (71), if we choose appropriate adaptive laws for λ^ii=2,3,and4 and large enough ν2 to guarantee λ^2−1/2ν2>0, λ^3>0, and λ^4>0, one can obtain that(72)V˙s≤−Γ1Vsβ+1/2−Γ2Vs+Γ0≤−Γ1Vsβ+1/2−Γ2V,where Γ1=21+β/2minkii=2,3,4>0 and Γ2=2minλ^2−1/2ν2,λ^3,λ^4>0.
Thus, according to Lemma2, si will converge to zero in finite time bounded by(73)ts≤2Γ21−βlnΓ2Vs01−β/2+Γ1Γ1.
Whens2=0 and s˙2=0 are achieved, the following equation can be obtained:(74)s˙2=x˙2+λ11sigx1α1+λ12sigx2α2=0.
Namely,(75)x˙2=−λ11sigx1α1−λ12sigx2α2.
Sincex˙1=x2, (75) can be rewritten as(76)x¨1+λ12sigx˙1α2+λ11sigx1α1=0.
According to Lemma4, once parameters λ12 and λ11 are selected to make polynomial p2+λ12pα2+λ1pα1 Hurwitz, x1 and x2 can converge to the equilibrium point in finite time which can be calculated as(77)tx≤3+α1V01−α1/3+α1x10,x20K1−α1,where K is a positive constant.
Therefore, the designed composite guidance law can guarantee the finite-time stability of the closed-loop system and drive the sliding modessi and states x1 and x2 to converge to zero in finite time with the total settling time tc=to+tr+ts+tx.
The proof of Theorem2 has been completed.Remark 7.
In most existing literature, the “differential explosion problem” of the traditional backstepping method is solved by introducing the first-order filter or tracking differentiator. In this study, the proposed adaptive laws are applied to estimate the derivative of virtual control inputs.Remark 8.
For the global guidance system,λ11,λ12,α1,β1,κi,σi,εi,γj,εj,andξji=2,3,and4;j=3and4 are all positive parameters to be regulated. Specifically, ki can be selected large enough to stabilize each subsystem. For subsystem x2, κ2 is selected large enough and σ2 is selected small enough, while for the subsystems x3 and x4, regulation laws of κ3,κ4 and σ3,σ4 are the opposite. γj and ξj can be selected small enough, while εi can be selected large enough. With the satisfactory parameters, the convergence characteristic of the closed-loop system can be guaranteed. In other words, the state errors can converge to any small interval of the origin. Besides, when the system gets convergent, λ11,λ12, α1,andα2 will further balance the LOS error and miss distance. As λ11 and λ12 increase, the miss distance will increase and the LOS error decreases. Besides, oversized λ11 and λ12 will result in the failure of the mission. Thus, appropriate small λ11 and λ12 should be determined. The parameters α1 and β mainly affect the convergence rate. Large α1 and β will lead to the oscillation of the transient process, while small α1 and β will degrade the convergence speed and bring about the interception failure. Therefore, when tuning these parameters, selecting proper λ11,λ12, α1, and β first so as to guarantee the relatively fast convergence rate with acceptable miss distance and LOS error. Then, choosing appropriate ki, κi, σi, εi, γj,εj,andξj to stabilize each subsystem. Finally, these parameters can be fine-tuned to improve comprehensive performance.
To further explain the tuning process, the corresponding main tuning steps can be described as Table1.Table 1
The main parameter tuning steps.
(i) Selectλ11 and λ12 to make the polynomial p2+λ12pα2+λ1pα1 Hurwitz and set α1 and β as 0.5(ii) Fixλ11, λ12, α1, and β, set κ2, σ2, and ε2as zero, and then tune the parameter k2 to stabilize the subsystem x2; if this subsystem looks stable or has a stable trend, go to step iii; otherwise, continue this step(iii) Adjust the parameterκ2, σ2, and ε2 to further stabilize the subsystem x2; if this subsystem looks stable or has a stable trend, go to step iv; otherwise, continue this step(iv) Fixλ11, λ12, α1, β, k2, κ2, σ2, and ε2, set κ3, σ3, and ε3 as zero, and then tune the parameters k3,γ3,ε3,andξ3to stabilize the subsystem x3; if this subsystem looks stable or has a stable trend, go to step v; otherwise, continue this step(v) Adjust the parameterκ3, σ3, and ε3 to further stabilize the subsystem x3; if this subsystem looks stable or has a stable trend, go to step vi; otherwise, continue this step(vi) Fixλ11, λ12, α1, β, k2, κ2, σ2, ε2, k3, κ3, σ3, ε3, γ3,ε3,andξ3, set κ4, σ4, and ε4 as zero, and then tune the parameters k4, γ4,ε4,andξ4 to stabilize the subsystem x4; if this subsystem looks stable or has a stable trend, go to step vii; otherwise, continue this step(vii) Adjust the parameterκ4, σ4, and ε4 to further stabilize the subsystem x4; if this subsystem looks stable or has a stable trend, go to step viii; otherwise, continue this step(viii) In this step, the stability of closed-loop is guaranteed; then, re-adjust the parametersλ11, λ12, α1, and β to balance the miss distance, LOS error, and convergence speed; these parameters will be further fine-tuned so as to achieve better overall performance


## 4. Simulation Results and Discussions

Numerical examples are performed in this section to verify the effectiveness of the proposed composite guidance law. The simulation conditions are listed in Table2. The maximal saturation acceleration of the interceptor missile is 40 g g=9.8.Table 2
Initial conditions for the simulation examples.
State (unit)Valuexmm0ymm0Vmm/s600γm°60ξ0.707θf°20xtm5000ytm2500Vtm/s300γt°10ωn10We assume that the relative disturbancer, the LOS angle θ, and their rates (r˙ and θ˙) can be provided by the seeker. The interception performance is evaluated by the interception time, miss distance, LOS error, and control energy (denoted as ∫amc2dτ). The designed parameters for the proposed guidance scheme (denoted as AISMB-FTDO) are chosen as λ11=0.1,λ12=0.05,α1=α=4/5,β=4/5,k2=6,κ2=10,σ2=0.1,δ2=0.005,k3=0.5,κ3=0.00001,σ3=1000,δ3=1000,ε3=100,γ3=0.0001,ξ3=0.1,k4=0.5,κ4=0.0001,σ4=1000,δ4=4000,ε4=1000,γ4=0.01,ξ4=0.01,k21=10,k22=5,k23=4,p=0.6,q=1.4,andϖ2=ϖ3=ϖ4=0.00001.To make a better showcase, the terminal sliding mode backstepping (SMB) guidance law [22] with the ESO [10] (denoted as SMB-ESO) and the finite time convergent (FTC) guidance law with SMO [31] (denoted as FTC-SMO) are also introduced into the simulation for comparison.The SMB-ESO is defined as(78)SMB:s2=x2+a1x1sigx1p1,x3d=g2−1−f2−α1x1p−1x2−λ21s2−λ22sigs2p2+z22,τx˙3c+x3c=x3d,x3c0=x3d0,s3=x3−x3c,x4d=−λ31sigs3p3−λ32s3−g2s2−x˙3c,τx˙4c+x4c=x4d,x4c0=x4d0,s4=x4−x4c,u=g4−1−f3−s3−λ41sigs4p4−λ42s4−x˙4c+z42,ESO:e21=z21−x2,z˙21=f2+g2x3d−β21e21+z22,z˙22=−β22fale21,α2,δ2,where fal⋅ is a nonlinear function defined as(79)fale,α,δ=sigeα,e>δ,eδ1−α,otherwise,with δ is a positive constant. The parameters of SMB-ESO can be chosen as a1=0.9,p1=0.9,λ21=10,λ22=2,p2=1.15,τ=0.02,λ31=0.01,λ32=0.3,p3=1.5,p4=1.5,λ41=0.9,λ42=0.5,β21=10,β22=100,α2=0.91,andδ2=0.00001.The FTC-SMO is defined as(80)FTC:u=−N1r˙q˙+N2sigq˙q+z1,SMO:z˙0=−2r˙rz0+g2amc+v0,v0=−l2L1/3sige212/3+z1e21=z0−x2,z˙1=v1,v1=−l1L1/2sigz1−v02/3+z2z˙2=l0Lsignz2−v1.The parameters of FTC-SMO are selected asN1=4,N2=30,q=0.5,l2=2,l1=1.5,l0=1.1,andL=400.Remark 9.
To alleviate the chattering phenomenon, the sign functionsignx is replaced by the following continuous function:(81)tanhxη=ex/η−e−x/ηex/η+e−x/η,where η>0 is a sufficient small constant.
In order to illustrate the effectiveness of the proposed guidance law, the simulation examples are divided into two kinds of target maneuverings, constant-maneuvering and cosine maneuvering. Detailed simulation results and relevant discussions are shown as below.


### 4.1. Scenario 1

In this scenario, the simulation results for the constant-maneuvering target are presented, in which the target acceleration is chosen asat=10g. The corresponding simulation curves of this scenario are shown in Figures 3–9 . The trajectories of the missile and target are shown in Figure 3, which can reveal that these three guidance laws could guarantee the successful interception. Besides, it is obvious that the curves of AISMB-FTDO and FTC-SMO are smoother than that of SMB-ESO. Nevertheless, the trajectory of AISMB-FTDO is flatter and easier to implement in practical engineering, while FTC-SMO does not have this property. Figure 4 reveals that both AISMB-FTDO and SMB-ESO can achieve the certain LOS angle, while FTC-SMO cannot. The reason for this result is that FTC-SMO does not consider the LOS angle constraint. The LOS rate curves are shown in Figure 5. As seen from Figures 4 and 5, although SMB-ESO can achieve the certain LOS angle, the LOS angle curve displays some chattering caused by the divergence of the LOS angular rates during the end phase. Howbeit, AISMB-FTDO could ensure little LOS angle error without chattering. The actual guidance command curves are shown in Figure 6. Apparently, the control input curves of AISMB-FTDO and FTC-SMO are much smoother, while SMB-ESO exhibits serious saturation. This saturation phenomenon will degrade the guidance performance and produce extremely large control energy. It can be seen from Figure 7 that the tracking error curves of SMO and FTDO can converge to zero in shorter time, while ESO need longer time. However, it is obvious that the curve of SMO causes a serious chattering phenomenon, while FTDO does not have this problem. This is because the sign function in FTDO is hidden in the integral term and the continuity of estimation value can be guaranteed. Therefore, this result further indicates that FTDO has better disturbance observation ability. The adaptive gains and adaptive parameters are shown in Figures 8 and 9, respectively. Both of these parameters can vary with respect to time and converge to certain values finally. These curves can further confirm the effectiveness of the designed adaptive laws. The interception results obtained from the guidance laws are summarized in Table 3. Compared with FTC-SMO and SMB-ESO, AISMB-FTDO possesses less interception time, smaller miss distance, less LOS error, and a relatively small control energy. It is worth noting that although FTC-SMO exhibits the largest LOS error, the required control energy is the smallest. The reason for this phenomenon is that FTC-SMO does not consider the LOS angle constraint. Besides, compared with SMB-ESO, the control energy of AISMB-FTDO is much smaller since the gain regulating laws work well during the interception process. Thus, we can draw the conclusion that AISMB-FTDO has better comprehensive performance than FTC-SMO and SMB-ESO.Figure 3
Trajectories of the missile and target in scenario 1.Figure 4
Curves of the LOS angle in scenario 1.Figure 5
Curves of the LOS angular rate in scenario 1.Figure 6
Curves of missiles’ actual accelerations in scenario 1.Figure 7
Curves of observation errors from the disturbance observer in scenario 1.Figure 8
Curves of adaptive gainsλ^ii=2,3,and4 in scenario 1.Figure 9
Curves of adaptive parametersρ^jj=3and4 in scenario 1.Table 3
Interception result data of scenario 1.
Guidance lawInterception time (s)Miss distance (m)LOS error (°)Control energym2/s4FTC-SMO10.0010.4336−17.84791.7734×105SMB-ESO9.8660.3968−2.07102.5870×108AISMB-FTDO9.5140.2763−0.29372.2498×105


### 4.2. Scenario 2

To further verify the capability of the proposed AISMB-FTDO guidance law in a more complicated situation, the target acceleration is described asat=10gcosπt/8 in this scenario. The design requirement and the guidance parameters are unchanged. It is shown from Table 4 that the AISMB-FTDO guidance law still possesses less interception time, smaller miss disturbance, less LOS error, and relatively small control energy than FTC-SMO and SMB-ESO. Figure 10 depicts the trajectories of the missile and target in scenario 2. It is obvious that the trajectories of AISMB-FTDO and FTC-SMO can still be smooth, while SMB-ESO exhibits slow chattering. The LOS angle and LOS angular rate are shown in Figures 11 and 12, respectively. The results in these two figures present that the proposed AISMB-FTDO guidance law can still guarantee the accuracy of the LOS angle, whereas FTC-SMO and SMB-ESO have worse performance. Figure 13 exhibits the accelerations of the missiles. It can be concluded that the curves of AISMB-FTDO and FTC-SMO are smoother. Besides, the curve of SMB-ESO is pretty difficult for the actual implement. Figure 14 shows that the estimation accuracy of FTDO is still favorable in the presence of high maneuverability, while SMO still causes severe chattering problem. The adaptive gains and adaptive parameters are shown in Figures 15 and 16. These variables determine the satisfactory guidance performance.Table 4
Interception result data of scenario 2.
Guidance lawInterception time (s)Miss distance (m)LOS error (°)Control energym2/s4FTC-SMO18.8570.1321−8.46661.2877×105SMB-ESO17.9390.93626.96544.6918×107AISMB-FTDO17.6750.1921−0.05791.7983×105Figure 10
Trajectories of the missile and target in scenario 2.Figure 11
Curves of the LOS angle in scenario 2.Figure 12
Curves of the LOS angular rate in scenario 2.Figure 13
Curves of missiles’ actual acceleration in scenario 2.Figure 14
Curves of observation errors from the disturbance observer in scenario 2.Figure 15
Curves of adaptive gainsλ^ii=2,3,and4 in scenario 2.Figure 16
Curves of adaptive parametersρ^jj=3and4 in scenario 2.Remark 10.
The observation errors in Figures7 and 14 both show large sudden changes at the ending of the interception, which will degrade the actual control effect. This may result from the rapidly decreasing of r. Thus, the disturbance observers are disabled if r<r∗, where r∗ is a predesigned parameter by trial and error.Remark 11.
Based on the above simulation results and analysis, it is apparent that the proposed AISMB-FTDO guidance law can provide better performance. Firstly, the comparisons between these three methods can verify the superiority of proposed FTDO. As shown in Figures7 and 14, it is obvious that the convergence time of SMO and FTDO is smaller than that of ESO. Besides, we can see from Figures 7 and 14 that the curves of FTDO are smoother than SMO since SMO causes serious chattering problem. Secondly, as shown in Tables 3 and 4, the control energy of SMB-ESO is much larger than that of AISMB-FTDO. For this phenomenon, one of the main reasons is that the proposed adaptive gain regulating laws exhibit well during the interception process. Finally, compared with traditional backstepping method in SMB-ESO, the adaptive rules are employed to handle the “differential explosion” problem of the virtual control input in the design process. Therefore, above analysis will further imply the contributions of our study.


## 4.1. Scenario 1

In this scenario, the simulation results for the constant-maneuvering target are presented, in which the target acceleration is chosen asat=10g. The corresponding simulation curves of this scenario are shown in Figures 3–9 . The trajectories of the missile and target are shown in Figure 3, which can reveal that these three guidance laws could guarantee the successful interception. Besides, it is obvious that the curves of AISMB-FTDO and FTC-SMO are smoother than that of SMB-ESO. Nevertheless, the trajectory of AISMB-FTDO is flatter and easier to implement in practical engineering, while FTC-SMO does not have this property. Figure 4 reveals that both AISMB-FTDO and SMB-ESO can achieve the certain LOS angle, while FTC-SMO cannot. The reason for this result is that FTC-SMO does not consider the LOS angle constraint. The LOS rate curves are shown in Figure 5. As seen from Figures 4 and 5, although SMB-ESO can achieve the certain LOS angle, the LOS angle curve displays some chattering caused by the divergence of the LOS angular rates during the end phase. Howbeit, AISMB-FTDO could ensure little LOS angle error without chattering. The actual guidance command curves are shown in Figure 6. Apparently, the control input curves of AISMB-FTDO and FTC-SMO are much smoother, while SMB-ESO exhibits serious saturation. This saturation phenomenon will degrade the guidance performance and produce extremely large control energy. It can be seen from Figure 7 that the tracking error curves of SMO and FTDO can converge to zero in shorter time, while ESO need longer time. However, it is obvious that the curve of SMO causes a serious chattering phenomenon, while FTDO does not have this problem. This is because the sign function in FTDO is hidden in the integral term and the continuity of estimation value can be guaranteed. Therefore, this result further indicates that FTDO has better disturbance observation ability. The adaptive gains and adaptive parameters are shown in Figures 8 and 9, respectively. Both of these parameters can vary with respect to time and converge to certain values finally. These curves can further confirm the effectiveness of the designed adaptive laws. The interception results obtained from the guidance laws are summarized in Table 3. Compared with FTC-SMO and SMB-ESO, AISMB-FTDO possesses less interception time, smaller miss distance, less LOS error, and a relatively small control energy. It is worth noting that although FTC-SMO exhibits the largest LOS error, the required control energy is the smallest. The reason for this phenomenon is that FTC-SMO does not consider the LOS angle constraint. Besides, compared with SMB-ESO, the control energy of AISMB-FTDO is much smaller since the gain regulating laws work well during the interception process. Thus, we can draw the conclusion that AISMB-FTDO has better comprehensive performance than FTC-SMO and SMB-ESO.Figure 3
Trajectories of the missile and target in scenario 1.Figure 4
Curves of the LOS angle in scenario 1.Figure 5
Curves of the LOS angular rate in scenario 1.Figure 6
Curves of missiles’ actual accelerations in scenario 1.Figure 7
Curves of observation errors from the disturbance observer in scenario 1.Figure 8
Curves of adaptive gainsλ^ii=2,3,and4 in scenario 1.Figure 9
Curves of adaptive parametersρ^jj=3and4 in scenario 1.Table 3
Interception result data of scenario 1.
Guidance lawInterception time (s)Miss distance (m)LOS error (°)Control energym2/s4FTC-SMO10.0010.4336−17.84791.7734×105SMB-ESO9.8660.3968−2.07102.5870×108AISMB-FTDO9.5140.2763−0.29372.2498×105


## 4.2. Scenario 2

To further verify the capability of the proposed AISMB-FTDO guidance law in a more complicated situation, the target acceleration is described asat=10gcosπt/8 in this scenario. The design requirement and the guidance parameters are unchanged. It is shown from Table 4 that the AISMB-FTDO guidance law still possesses less interception time, smaller miss disturbance, less LOS error, and relatively small control energy than FTC-SMO and SMB-ESO. Figure 10 depicts the trajectories of the missile and target in scenario 2. It is obvious that the trajectories of AISMB-FTDO and FTC-SMO can still be smooth, while SMB-ESO exhibits slow chattering. The LOS angle and LOS angular rate are shown in Figures 11 and 12, respectively. The results in these two figures present that the proposed AISMB-FTDO guidance law can still guarantee the accuracy of the LOS angle, whereas FTC-SMO and SMB-ESO have worse performance. Figure 13 exhibits the accelerations of the missiles. It can be concluded that the curves of AISMB-FTDO and FTC-SMO are smoother. Besides, the curve of SMB-ESO is pretty difficult for the actual implement. Figure 14 shows that the estimation accuracy of FTDO is still favorable in the presence of high maneuverability, while SMO still causes severe chattering problem. The adaptive gains and adaptive parameters are shown in Figures 15 and 16. These variables determine the satisfactory guidance performance.Table 4
Interception result data of scenario 2.
Guidance lawInterception time (s)Miss distance (m)LOS error (°)Control energym2/s4FTC-SMO18.8570.1321−8.46661.2877×105SMB-ESO17.9390.93626.96544.6918×107AISMB-FTDO17.6750.1921−0.05791.7983×105Figure 10
Trajectories of the missile and target in scenario 2.Figure 11
Curves of the LOS angle in scenario 2.Figure 12
Curves of the LOS angular rate in scenario 2.Figure 13
Curves of missiles’ actual acceleration in scenario 2.Figure 14
Curves of observation errors from the disturbance observer in scenario 2.Figure 15
Curves of adaptive gainsλ^ii=2,3,and4 in scenario 2.Figure 16
Curves of adaptive parametersρ^jj=3and4 in scenario 2.Remark 10.
The observation errors in Figures7 and 14 both show large sudden changes at the ending of the interception, which will degrade the actual control effect. This may result from the rapidly decreasing of r. Thus, the disturbance observers are disabled if r<r∗, where r∗ is a predesigned parameter by trial and error.Remark 11.
Based on the above simulation results and analysis, it is apparent that the proposed AISMB-FTDO guidance law can provide better performance. Firstly, the comparisons between these three methods can verify the superiority of proposed FTDO. As shown in Figures7 and 14, it is obvious that the convergence time of SMO and FTDO is smaller than that of ESO. Besides, we can see from Figures 7 and 14 that the curves of FTDO are smoother than SMO since SMO causes serious chattering problem. Secondly, as shown in Tables 3 and 4, the control energy of SMB-ESO is much larger than that of AISMB-FTDO. For this phenomenon, one of the main reasons is that the proposed adaptive gain regulating laws exhibit well during the interception process. Finally, compared with traditional backstepping method in SMB-ESO, the adaptive rules are employed to handle the “differential explosion” problem of the virtual control input in the design process. Therefore, above analysis will further imply the contributions of our study.


## 5. Conclusions

This paper investigates the interception problem for a missile intercepting a target with impact angle constraint and autopilot dynamics. To solve this problem, we propose a FTDO-based adaptive finite time guidance law. The employed FTDO can achieve fast estimation for the lumped disturbance, and the proposed adaptive guidance lawsλ^ii=2,3,and4 can adjust the guidance gains accordingly. Besides, the “explosion of complexity” problem inherent in the conventional backstepping is avoided by utilizing the adaptive laws ρ^ii=3and4. Furthermore, the simulation results clearly demonstrate the effectiveness of the proposed composite guidance law for intercepting the highly maneuvering target. In addition, we will further consider the guidance law design with control faults, sensor noises, and LOS limitations in our future study.



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