# Discrete-Time Angle Constraint Interception with Model-Assisted Target Maneuver Estimator

Authors: Y. I. Sheng; W. A. N. G. Jiang; Pei Pei; Zhou Yongjia
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

This paper investigates the problem of designing angle constraint guidance law against unknown maneuvering targets based on discrete-time sliding mode control theory. Invoking the fact that the future course of action of the target, an independent entity, cannot be predicted beforehand due to its complexity and unpredictability, a model-assisted discrete-time disturbance observer in cooperation with a singularity-free strategy is proposed first to estimate the target maneuver. Based on the reconstructed signal and a fast convergence time-varying sliding surface, a new chattering-mitigated super-twisting-like discrete-time impact angle constraint guidance law is then synthesized. Stability analysis shows that the closed-loop system trajectory can be forced to enter into a small region around the sliding surface. Simulations and comparisons with classical discrete-time sliding mode guidance law validate the effectiveness of the proposed guidance law.


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## Body

## 1. Introduction

Proportional navigation guidance (PNG) law has been widely used in missile guidance system in the past few decades due to its easy implementation and might suffice for many typical cases of applications [1]. The baseline concept of PNG lies in that it generates a latex command proportional to the line-of-sight (LOS) rate to nullify the LOS rate so as to guide a pursuer onto the collision triangle for target capture. However, during realistic engagement for intercepting maneuvering targets, the performance of PNG degrades drastically and the assumption for linearized collision triangle may be violated [1–4]. To the overall guidance performance, the most well-known extension of the PNG law is the augmented PNG (APNG) law, which augments a biased term to the original PNG formulation to counteract the effect of target maneuver [1]. Compared to PNG law, the APNG law reduces the required missile latex at the time of impact as well as the total required control effort throughout the homing engagement. But, unfortunately, an accurate target maneuver model is required to bring such an improvement for APNG law, which greatly limits the application of APNG.During the development of modern guidance law, the concept of imposing hard constraint on intercept angle or impact angle provides engineers a new view to design guidance laws for tactical missiles to exploit the weak points of the target and increase the overall kill probability [5–9]. In the literature, a considerable number of efforts that address the problem of impact angle control are based on optimal control theory with small-angle assumption around the collision triangle [5–9]. The reason why used linearized kinematics for optimal guidance law design lies in the fact that analytically solving Hamilton-Jacobi-Bellman partial differential equation for nonlinear systems is usually intractable [10]. The small-angle assumption itself, however, is usually not valid when considering large intercept angles since the pursuer trajectory may greatly deviate from the desired collision triangle to cater for large impact angle constraint [9].With the development of modern control theory, sliding mode control (SMC) methodology was widely accepted in angle constraint guidance law design for intercepting maneuvering targets [11–23]. In SMC guidance law design, most of the existing works viewed target maneuver as an external disturbance with known upper bound and using discontinuous sign function to counteract it [11–14]. Usually, the maximum maneuverability of a predesignated target is chosen as this upper bound in real applications. However, due to the unpredictability and complicity of target maneuvers, highly conservative bound requires excessive control energy consumption and severe chattering phenomenon will be excited, while lower upper bound will result in guidance performance degradation and even system instability. To address this dilemma, some elegant solutions were reported by using estimation algorithms to reconstruct the target maneuver profile online and augmenting the estimated signal to obtain a so-called add-on controller [4, 16–23]. Unfortunately, all the above mentioned SMC guidance laws were designed for continuous-time guidance systems, while the real interceptor can only change its guidance command at sampling instants, the command cannot respond to the designed continuous-time controller. When applying continuous-time controller to discrete system, firstly, the system can hardly reach and maintain on the sliding surface due to the discrete form; secondly, the continuous stable system may be unstable, the stability theory of discrete system is different; finally, the discrete system will result in time delay, chattering phenomenon, and invariable control command during sampling time [24]. These are the reasons why designing guidance law in discrete-time manner is more desirable for practical interceptors. Some invariance properties of continuous-time sliding mode control are not held in discrete-time sliding mode control, such as the system cannot stay on the sliding surface but move about the sliding manifold [25]. Based on the sampled-date systems, the sliding mode controller needs to be reconstructed.Motivated by the above analysis, this paper considers designing discrete-time SMC impact angle guidance law for unknown maneuvering target interception. A robust discrete-time guidance SMC guidance law with a time-delay observer was designed by Wang [26] for maneuvering target interception as well as actuator faults, but impact angle constraints were not taken into account. To the best of our knowledge, this may be the first time to address the problem of angle constraint interception within discrete-time control framework. Nonlinear numerical studies show that the proposed guidance law can achieve accurate interception with wide range of impact angles. The key features and contributions of this paper are summarized as follows.
(1)
An exponentially stable model-assisted disturbance observer is proposed to estimate the target maneuver online. Different from the existing target maneuver estimators [4, 15–22], the proposed design method is naturally a discrete-time and singularity-free strategy(2)
A discrete-time time-varying sliding surface is designed to accelerate the convergence speed of the LOS angle tracking error(3)
Based on the reconstructed target maneuver information, a new chattering-mitigated super-twisting-like discrete-time guidance law is proposed for accurate impact angle control. Theoretical analysis shows that the closed-loop system trajectory can converge into a small region around the designed sliding surface under the proposed guidance lawThe rest of the paper is organized as follows. The problem formulation stated in Sec. II. In Sec. III, the proposed guidance law and convergence analysis are presented in details, followed by the simulation results provided in Sec. IV. Finally, some conclusions are offered in Sec. V.


## 2. Backgrounds and Preliminaries

In this section, engagement kinematics of maneuvering target interception with impact angle constraint is formulated for the analysis and the design of guidance law. The following general assumptions in guidance law design are considered for convenient analysis.Assumption 1. (point mass interceptor and target).
both the interceptor and the target are considered as point masses, e.g., only kinematics is considered in this paper.Assumption 2. (ideal interceptor).
the interceptor is considered as an ideal point mass such that the achieved acceleration is the same as the command signal generated by guidance law.Assumption 3. (aerodynamically controlled interceptor).
only aerodynamically controlled missiles are considered in this paper, which means that the missile acceleration along the LOS usually cannot be adjusted.Assumption 4. (constant speed).
both the interceptor and targets fly with constant speeds.


## 3. Model Derivation

Consider a typical two-dimensional engagement scenario shown in Figure1, where the subscripts M and T denote the missile and the target, γM and γT the missile and the target flight path angle, λ and R the LOS angle and the missile-target relative range, VM and VT the missile and the target velocity, aM and aT the missile and the target acceleration, which are normal to their corresponding velocities, respectively.Figure 1
Planar homing engagement geometry.The corresponding equations describing the missile-target relative kinematics are formulated as(1)Ṙ=VTcosγT−λ−VMcosγM−λ,(2)λ̇=1RVTsinγT−λ−VMsinγM−λ,(3)γ̇M=aMVM,(4)γ̇T=aTVT.LetVr=Ṙ, Vλ=Rλ̇, differentiating Eqs. (1) and (2) with respect to time yields
(5)V̇r=Vλ2R+aTr−aMr,(6)V̇λ=−Ṙλ̇+aTλ−aMλ,where aTr=aTsinλ−γT, aMr=aMsinλ−γM represent the target and the missile accelerations along the LOS, respectively; aTλ=aTcosλ−γT, aMλ=aMcosλ−γM denote the target and the missile accelerations normal to the LOS, respectively.Compared with other variables, the relative range and its rate are slow-varying variables [1]. With this in mind and let T be the sampling period, a simple Euler discretization of Eqs. (5) and (6) at the kT time instant can be formulated as
(7)Vrk+1−Vrk=TVλ2kRk+TaTrk−TaMrk,Vλk+1−Vλk=−TṘkλ̇k+TaTλk−TaMλk.Assumption 5.
The change rate of the target maneuver between two sampling intervals is bounded by a constantδ∗, i.e., δk=aTk+1−aTk≤Tδ∗. It should be noted that Assumption 5 is required in stability proof of the proposed disturbance observer. Actually, in real applications, the change rate of the acceleration of the target is very limited between two time sampling intervals due to physical limitations, and therefore, Assumption 5 is reasonable.


## 4. Guidance Strategy

To ensure successful target interception and kill probability, the pursuer should have to impose hard constraints on terminal miss distance as well as terminal impact angle. In this regard, the following two general definitions are introduced for the convenience of analysis.Definition 1. (Zero-Effort-Miss-Distance) [27].
The term zero-effort miss, denoted byZEM, at any time instant t is defined as the closest miss distance if both the pursuer and the target do not perform any maneuver from the time instant t onward. The definition of ZEM is [28].
(8)ZEMt=rVλṙ2+Vλ2.According to Eq. (8), one can conclude that zeroing Vλ leads to perfect interception with zero miss distance.Definition 2. (Impact angle).
The quantity impact angle, denoted asθimp, is defined as the angle between target velocity vector and pursuer velocity vector, i.e., θimp=γT−γM.
Assuming a perfect interception is achieved, i.e.,Vλ=0, it follows from Eq. (2) that
(9)VTsinγT−λ=VMsinγM−λ.Based on Eq. (9) and assume that the interceptor has advantageous velocity, e.g., VT/VM<1, then one can imply that the impact angle and LOS angle have one-to-one correspondence, that is [12–14]
(10)λF=γT−tan−1sinθimpcosθimp−VT/VM,where λF denotes the desired LOS angle. It follows from Eq. (10) that the impact angle constraint can be satisfied by imposing a one-to-one correspondent LOS angle constraint. Considering this, the control interest here is to design a guidance law in such a way that the missile can capture the unknown maneuvering target with desired LOS angle λF.


## 5. Composite Guidance Law Design and Convergence Analysis

In this section, the proposed composite guidance law is derived in details, and the convergence analysis is also presented. A discrete-time model-assisted disturbance observer is firstly proposed to estimate the target maneuver. Then, a new discrete-time chattering-mitigated super-twisting-like guidance law is synthesized based on the designed sliding variable.


### 5.1. Disturbance Observer Design

Generally, by viewing the target maneuver as an external disturbance for the guidance system, classical disturbance observer can then be applied for target maneuver estimation, for instance, continuous time-delay estimator [4, 19], second-order sliding mode observer [20], inertial-delay control [21], higher-order sliding mode observer [17, 22], and extended state observer [18, 23]. Although the target maneuver can be estimated by these different types of observers, most of them are designed for continuous-time guidance laws. Different from these formulations, the proposed model-assisted observer is naturally based on discrete-time control approach, which is defined as
(11)a^Trk=KVrk−zk,(12)zk+1=zk+KTVλ2kRk+Ta^Trk−TaMrk,where K>0 is an observer gain to be designed. Let ek=aTrk−a^Trk be the disturbance estimation error, then one can obtain the error dynamics as
(13)ek+1=aTrk+1−a^Trk+1=aTrk+1−KVrk+1−zk+1=aTrk+1−KVrk+TVλ2kRk+TaTrk−TaMrk+zk+KTVλ2kRk+Ta^Trk−TaMrk=aTrk+1−KVrk−zk−KTaTrk−a^Trk=aTrk+1−aTrk⏟ϑk+aTrk−a^Trk−KTek=ϑk+1−KT⏟Mek=ϑk+Mek,Theorem 1.
Consider error dynamics (13), if M<1, then the estimation error can be bounded by
(14)limk→∞ek≤δ∗T1−M.Proof.
With error dynamics (13), one can imply that
(15)ek+1=ϑk+Mek≤δk+Mek≤δ∗T+Mek.
Solving inequality (15) gives
(16)ek+1≤Mke0+δ∗T∑i=1kMi.
If the upper limit whenk goes to infinity is considered, one can conclude that
(17)limk→∞ek≤δ∗T1−M.This completes the proof. It follows from Eq. (17) that the estimation error dynamics is exponentially stable with a small upper bound, and the bound can be lowered by increasing the observer gain K under the condition M<1. If the rough value of target maneuver is known, one can initialize disturbance observer (12) with z0=KVr0−a~Tr0, where a~Tr0 is the rough guess of target maneuver. From the practical point of view, however, it is better to initialize disturbance observer (12) with z0=KVλ0 in terms of smoothing the transient control input, that is, from zero to a certain value to avoid the peaking phenomenon.Similarly, the target maneuver perpendicular to the LOS can also be estimated via(18)a^Tλk=KVλk−z¯k,z¯k+1=z¯k+K−TṘkλ̇k+Ta^Tλk−TaMλk.With the above two target maneuver projection estimations, the real target maneuver estimation can be obtained by the following singularity-free strategy(19)a^Tk=a^Tλ2k+a^Tr2ksgna^Tr/sinλ−γTora^Tk=a^Tλ2k+a^Tr2ksgna^Tλ/cosλ−γT,where sgn⋅ is the sign function. In [29], the authors used a simple strategy a^Tk=a^Tλ/cosλ−γT as target maneuver estimation. However, it should be noted that when λ−γT is close to ±π/2, such estimation may suffer from the problem of singularity. With a simple modification by using the property of trigonometric function, estimation strategy (19) completely avoids the singular issue. Figure 2 presents the comparison results of these two different strategies, where the simulation conditions can be found in Sec. IV.Figure 2
Comparison of two target maneuver estimation strategy: (a) estimation algorithm [27]; and (b) estimation algorithm (19).
(a)
(b)


### 5.2. Sliding Surface Design

To begin with, letλek=λk−λFk denote the LOS angle tracking error, then. Based on Euler discretization concept and Eq. (6), the discrete-time angle error dynamics can be obtained as
(20)λek+1=λek+Tλ̇ekλ̇ek+1=λ̇k−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−λ̇Fk−δkVT.Lettf and N be the final impact time and impact step, respectively. Then, the relationship between tf and N can be obtained as N=roundtf/T, where round⋅ denotes the round operator for ⋅ to obtain the nearest integer. Based on the concept of time-to-go, the final impact step N can be estimated as
(21)N=roundtf/T=round−R/Ṙ+k.Next, consider the following time-varying sliding surface(22)sk=λ̇ek+fkλek,where fk=c/N−kT and c>0 is a user-designed parameter. It can be noted that with the increasing of k, the convergence speed of the angle tracking error also increases. Generally, using −R/Ṙ to approximate time-to-go in (21) would generate smaller value during the initial flight phase, but this approximation error will just speed up the convergence phase and not affect the angle tracking accuracy. The characteristics of sliding surface (22) are presented in the following theorem.Theorem 2.
If the closed-loop system enters a small region around the sliding manifold, e.g.,sk≤ε, both LOS angle tracking error and its rate will be bounded by some small positive constants.Proof.
Using Euler discretization approach, one can obtain that(23)sk=λek+1−λekT+fkλek≤ε.
Following the preceding equation, it can be deduced that(24)λek+1≤1−cN−kλek+εT≤1−1N−kcλek+εT≤1−kNcλe0+εT∑i=1k1−k−jN−i⏟M¯.Inequality (24) shows that λek+1 is ultimately upper bounded by M¯, and therefore, λ̇ek is also upper bounded by λ̇ek=λek+1−λek/T≤2M¯/T. This completes the proof.Theorem2 shows that if the closed-loop system trajectory can be forced into a small region around the sliding surface, both the LOS angle tracking error and its rate can also be bounded by some small positive constants, leading to successful interception with impact angle constraint.


### 5.3. Guidance Law Design

For guidance law design, deriving the sliding manifold dynamics as(25)sk+1=λ̇ek+1+fk+1λek+1=λ̇k−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−λ̇Fk−δkVT+cN−k−1Tλek+1=λ̇k−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−λ̇Fk−δkVT+cN−kN−k−1Tλek+1+cN−kTλek+1=sk−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−δkVT+fkN−k−1λek+TfkN−k−1λ̇k−aTkVT+Tfkλ̇k−aTkVT=sk−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−δkVT+fkN−k−1λek+cN−k−1λ̇k−aTkVT.For sliding variable dynamics (25), the proposed super-twisting-like guidance law is defined as
(26)aMλk=−2Ṙkλ̇k+a^Tλk+fkRkλekN−k−1T+cRkλ̇kN−k−1T−cRka^TkN−k−1TVT+ρ1RkskT−Rkξk+k1Rksig0.5skξk+1=ρ2ξk−Tk2sgnsk,where 0<ρ1<1, 0<ρ2<1, k1>0, and k2>0, sig0.5sk=sgnsksk0.5 is a nonsmooth but continuous function.It follows from Eq. (26) that the key features of the proposed guidance law are two folds. The first one lies in its chattering-mitigation property due to the fact that sig0.5⋅ in guidance command (26) is a nonsmooth but continuous function, and the second one is that the presented guidance law requires no information on target maneuvers by using a model-assisted discrete-time disturbance observer.Theorem 3.
For discrete-time sliding mode dynamics (25) under Assumptions 1-5, if the guidance law is designed as (26) with target maneuver estimation (12), (19), then the system trajectory can be forced to enter a small region around the sliding surface, and therefore, precise interception with impact angle constraint is ensured.Proof.
Substituting guidance law (26) into (25) yields
(27)sk+1=1−ρ1sk+Tξk−Tk1sig0.5sk+TΔk,(28)ξk+1=ρ2ξk−Tk2sgnsk,where Δk=aTλ−a^Tλ/Rk−caT−a^T/N−k−1TVT−δk/TVT. For convenience of analysis, defining ρ¯1=1−ρ1∈0,1, ζk=ξk+δk, then system (28) can be rewritten as
(29)sk+1=ρ¯1sk+Tζk−Tk1sig0.5sk,(30)ζk+1=ρ2ζk−Tk2sgnsk+Tηk,where ηk=1−ρ2Δk/T. For a realistic interception, the missile-target relative range is always bounded. With this in mind and considering Theorem 1 as well as Assumption 5, one can imply that ηk is also upper bounded, i.e., ηk≤η∗ with η∗ being a positive constant.Letxk=sk,ζkT, then, system (30) can be represented as the following matrix form
(31)xk+1=Axk+Bksignsk,where A∈ℝ2×2 and Bk∈ℝ2 are defined as
(32)A=ρ¯1T0ρ2,Bk=−Tk1sig0.5sk−Tk2sgnsk+Tηk.Suppose that the gainsρ1 and ρ2 are properly chosen such that the following discrete-time Lyapunov equation
(33)ATP+PΛPA−1−ℓP+Q≤0,with 0<ℓ<1, has a positive-definite solution P=PT>0 for a given Q=QT>0 and Λ=ΛT>0.For system (31), consider Vk=xTkPxk as a Lyapunov function candidate, then one has
(34)ΔVk=Vk+1−Vk=xTk+1Pxk+1−xTkPxk.Substituting Eq. (31) into (34) yields
(35)ΔVk=xTkATPA−Pxk+2xTkATPBksignVλk+BTkPBk.Applying lambda inequalityXTY+YTX≤XTΛ−1X+YTΛY, one has
(36)2xTkATPBksignVλk≤xTkATPΛPAxk+BTkΛ−1Bk.Substituting inequality (36) into (35) gives
(37)ΔVk≤xTkATP+PΛPA−1−ℓPxk+BTkΛ−1+PBk−ℓVk≤xTkQxk+BTkZBk−ℓVk,where Z=Λ−1+P. Expanding the term BTkZBk leads to
(38)BTkZBk=δ1sk+δ2sk1/2+δ3,(39)δ1=z11k12T2,δ2=2z12k1k2T2−2z12k1T2ηk,(40)δ3=z22k22T2+z22T2η2k−2z22T2k2ηksgnsk.Note that∀x∈ℝ, 0<ρ<1, the inequality xρ<1+x holds. Then, it follows from Eq. (38) that
(41)BTkZBk≤δ1+δ2sk+δ2+δ3.By choosingk¯2=k2−η∗>0, inequality (41) can be further reduced to
(42)BTkZBk≤δ¯1sk+δ¯2,δ¯1=δ1+2z12k1k¯2T2,δ¯2=2z12k1k¯2T2+z22T2k22+η∗2+2k2η∗.Following the previous result, one can imply that(43)ΔVk≤xTkQxk+δ¯1sk+δ¯2−ℓVk≤xTkQxk+δ¯1xk+δ¯2−ℓVk≤xTkQxk+δ¯1QQ−1xk+δ¯2−ℓVk≤xTkQxk+δ¯1QQ−1xk+δ¯2−ℓVk≤−Q1/2xk−12δ¯1Q−1TQ1/2xk−12δ¯1Q−1+δ¯2−ℓVk+14δ¯12Q−12≤−ℓVk+c,where c=δ¯2+1/4δ¯12Q−12. Then,
(44)Vk+1≤1−ℓVk+c.Solving inequality (44) yields
(45)Vk+1≤1−ℓkV0+c∑i=1k1−ℓi.If the upper limit whenk goes to infinity is considered, one can conclude that
(46)limk→∞Vk≤c/ℓ.Inequality (46) shows that the sliding variable sk is upper bounded by c/ℓ, which also implies that the LOS tracking error and its rate are also upper bounded by small constants, and therefore, accurate interception with impact angle is satisfied. This completes the proof.


## 5.1. Disturbance Observer Design

Generally, by viewing the target maneuver as an external disturbance for the guidance system, classical disturbance observer can then be applied for target maneuver estimation, for instance, continuous time-delay estimator [4, 19], second-order sliding mode observer [20], inertial-delay control [21], higher-order sliding mode observer [17, 22], and extended state observer [18, 23]. Although the target maneuver can be estimated by these different types of observers, most of them are designed for continuous-time guidance laws. Different from these formulations, the proposed model-assisted observer is naturally based on discrete-time control approach, which is defined as
(11)a^Trk=KVrk−zk,(12)zk+1=zk+KTVλ2kRk+Ta^Trk−TaMrk,where K>0 is an observer gain to be designed. Let ek=aTrk−a^Trk be the disturbance estimation error, then one can obtain the error dynamics as
(13)ek+1=aTrk+1−a^Trk+1=aTrk+1−KVrk+1−zk+1=aTrk+1−KVrk+TVλ2kRk+TaTrk−TaMrk+zk+KTVλ2kRk+Ta^Trk−TaMrk=aTrk+1−KVrk−zk−KTaTrk−a^Trk=aTrk+1−aTrk⏟ϑk+aTrk−a^Trk−KTek=ϑk+1−KT⏟Mek=ϑk+Mek,Theorem 1.
Consider error dynamics (13), if M<1, then the estimation error can be bounded by
(14)limk→∞ek≤δ∗T1−M.Proof.
With error dynamics (13), one can imply that
(15)ek+1=ϑk+Mek≤δk+Mek≤δ∗T+Mek.
Solving inequality (15) gives
(16)ek+1≤Mke0+δ∗T∑i=1kMi.
If the upper limit whenk goes to infinity is considered, one can conclude that
(17)limk→∞ek≤δ∗T1−M.This completes the proof. It follows from Eq. (17) that the estimation error dynamics is exponentially stable with a small upper bound, and the bound can be lowered by increasing the observer gain K under the condition M<1. If the rough value of target maneuver is known, one can initialize disturbance observer (12) with z0=KVr0−a~Tr0, where a~Tr0 is the rough guess of target maneuver. From the practical point of view, however, it is better to initialize disturbance observer (12) with z0=KVλ0 in terms of smoothing the transient control input, that is, from zero to a certain value to avoid the peaking phenomenon.Similarly, the target maneuver perpendicular to the LOS can also be estimated via(18)a^Tλk=KVλk−z¯k,z¯k+1=z¯k+K−TṘkλ̇k+Ta^Tλk−TaMλk.With the above two target maneuver projection estimations, the real target maneuver estimation can be obtained by the following singularity-free strategy(19)a^Tk=a^Tλ2k+a^Tr2ksgna^Tr/sinλ−γTora^Tk=a^Tλ2k+a^Tr2ksgna^Tλ/cosλ−γT,where sgn⋅ is the sign function. In [29], the authors used a simple strategy a^Tk=a^Tλ/cosλ−γT as target maneuver estimation. However, it should be noted that when λ−γT is close to ±π/2, such estimation may suffer from the problem of singularity. With a simple modification by using the property of trigonometric function, estimation strategy (19) completely avoids the singular issue. Figure 2 presents the comparison results of these two different strategies, where the simulation conditions can be found in Sec. IV.Figure 2
Comparison of two target maneuver estimation strategy: (a) estimation algorithm [27]; and (b) estimation algorithm (19).
(a)
(b)


## 5.2. Sliding Surface Design

To begin with, letλek=λk−λFk denote the LOS angle tracking error, then. Based on Euler discretization concept and Eq. (6), the discrete-time angle error dynamics can be obtained as
(20)λek+1=λek+Tλ̇ekλ̇ek+1=λ̇k−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−λ̇Fk−δkVT.Lettf and N be the final impact time and impact step, respectively. Then, the relationship between tf and N can be obtained as N=roundtf/T, where round⋅ denotes the round operator for ⋅ to obtain the nearest integer. Based on the concept of time-to-go, the final impact step N can be estimated as
(21)N=roundtf/T=round−R/Ṙ+k.Next, consider the following time-varying sliding surface(22)sk=λ̇ek+fkλek,where fk=c/N−kT and c>0 is a user-designed parameter. It can be noted that with the increasing of k, the convergence speed of the angle tracking error also increases. Generally, using −R/Ṙ to approximate time-to-go in (21) would generate smaller value during the initial flight phase, but this approximation error will just speed up the convergence phase and not affect the angle tracking accuracy. The characteristics of sliding surface (22) are presented in the following theorem.Theorem 2.
If the closed-loop system enters a small region around the sliding manifold, e.g.,sk≤ε, both LOS angle tracking error and its rate will be bounded by some small positive constants.Proof.
Using Euler discretization approach, one can obtain that(23)sk=λek+1−λekT+fkλek≤ε.
Following the preceding equation, it can be deduced that(24)λek+1≤1−cN−kλek+εT≤1−1N−kcλek+εT≤1−kNcλe0+εT∑i=1k1−k−jN−i⏟M¯.Inequality (24) shows that λek+1 is ultimately upper bounded by M¯, and therefore, λ̇ek is also upper bounded by λ̇ek=λek+1−λek/T≤2M¯/T. This completes the proof.Theorem2 shows that if the closed-loop system trajectory can be forced into a small region around the sliding surface, both the LOS angle tracking error and its rate can also be bounded by some small positive constants, leading to successful interception with impact angle constraint.


## 5.3. Guidance Law Design

For guidance law design, deriving the sliding manifold dynamics as(25)sk+1=λ̇ek+1+fk+1λek+1=λ̇k−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−λ̇Fk−δkVT+cN−k−1Tλek+1=λ̇k−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−λ̇Fk−δkVT+cN−kN−k−1Tλek+1+cN−kTλek+1=sk−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−δkVT+fkN−k−1λek+TfkN−k−1λ̇k−aTkVT+Tfkλ̇k−aTkVT=sk−2TṘkλ̇kRk+TaTλkRk−TaMλkRk−δkVT+fkN−k−1λek+cN−k−1λ̇k−aTkVT.For sliding variable dynamics (25), the proposed super-twisting-like guidance law is defined as
(26)aMλk=−2Ṙkλ̇k+a^Tλk+fkRkλekN−k−1T+cRkλ̇kN−k−1T−cRka^TkN−k−1TVT+ρ1RkskT−Rkξk+k1Rksig0.5skξk+1=ρ2ξk−Tk2sgnsk,where 0<ρ1<1, 0<ρ2<1, k1>0, and k2>0, sig0.5sk=sgnsksk0.5 is a nonsmooth but continuous function.It follows from Eq. (26) that the key features of the proposed guidance law are two folds. The first one lies in its chattering-mitigation property due to the fact that sig0.5⋅ in guidance command (26) is a nonsmooth but continuous function, and the second one is that the presented guidance law requires no information on target maneuvers by using a model-assisted discrete-time disturbance observer.Theorem 3.
For discrete-time sliding mode dynamics (25) under Assumptions 1-5, if the guidance law is designed as (26) with target maneuver estimation (12), (19), then the system trajectory can be forced to enter a small region around the sliding surface, and therefore, precise interception with impact angle constraint is ensured.Proof.
Substituting guidance law (26) into (25) yields
(27)sk+1=1−ρ1sk+Tξk−Tk1sig0.5sk+TΔk,(28)ξk+1=ρ2ξk−Tk2sgnsk,where Δk=aTλ−a^Tλ/Rk−caT−a^T/N−k−1TVT−δk/TVT. For convenience of analysis, defining ρ¯1=1−ρ1∈0,1, ζk=ξk+δk, then system (28) can be rewritten as
(29)sk+1=ρ¯1sk+Tζk−Tk1sig0.5sk,(30)ζk+1=ρ2ζk−Tk2sgnsk+Tηk,where ηk=1−ρ2Δk/T. For a realistic interception, the missile-target relative range is always bounded. With this in mind and considering Theorem 1 as well as Assumption 5, one can imply that ηk is also upper bounded, i.e., ηk≤η∗ with η∗ being a positive constant.Letxk=sk,ζkT, then, system (30) can be represented as the following matrix form
(31)xk+1=Axk+Bksignsk,where A∈ℝ2×2 and Bk∈ℝ2 are defined as
(32)A=ρ¯1T0ρ2,Bk=−Tk1sig0.5sk−Tk2sgnsk+Tηk.Suppose that the gainsρ1 and ρ2 are properly chosen such that the following discrete-time Lyapunov equation
(33)ATP+PΛPA−1−ℓP+Q≤0,with 0<ℓ<1, has a positive-definite solution P=PT>0 for a given Q=QT>0 and Λ=ΛT>0.For system (31), consider Vk=xTkPxk as a Lyapunov function candidate, then one has
(34)ΔVk=Vk+1−Vk=xTk+1Pxk+1−xTkPxk.Substituting Eq. (31) into (34) yields
(35)ΔVk=xTkATPA−Pxk+2xTkATPBksignVλk+BTkPBk.Applying lambda inequalityXTY+YTX≤XTΛ−1X+YTΛY, one has
(36)2xTkATPBksignVλk≤xTkATPΛPAxk+BTkΛ−1Bk.Substituting inequality (36) into (35) gives
(37)ΔVk≤xTkATP+PΛPA−1−ℓPxk+BTkΛ−1+PBk−ℓVk≤xTkQxk+BTkZBk−ℓVk,where Z=Λ−1+P. Expanding the term BTkZBk leads to
(38)BTkZBk=δ1sk+δ2sk1/2+δ3,(39)δ1=z11k12T2,δ2=2z12k1k2T2−2z12k1T2ηk,(40)δ3=z22k22T2+z22T2η2k−2z22T2k2ηksgnsk.Note that∀x∈ℝ, 0<ρ<1, the inequality xρ<1+x holds. Then, it follows from Eq. (38) that
(41)BTkZBk≤δ1+δ2sk+δ2+δ3.By choosingk¯2=k2−η∗>0, inequality (41) can be further reduced to
(42)BTkZBk≤δ¯1sk+δ¯2,δ¯1=δ1+2z12k1k¯2T2,δ¯2=2z12k1k¯2T2+z22T2k22+η∗2+2k2η∗.Following the previous result, one can imply that(43)ΔVk≤xTkQxk+δ¯1sk+δ¯2−ℓVk≤xTkQxk+δ¯1xk+δ¯2−ℓVk≤xTkQxk+δ¯1QQ−1xk+δ¯2−ℓVk≤xTkQxk+δ¯1QQ−1xk+δ¯2−ℓVk≤−Q1/2xk−12δ¯1Q−1TQ1/2xk−12δ¯1Q−1+δ¯2−ℓVk+14δ¯12Q−12≤−ℓVk+c,where c=δ¯2+1/4δ¯12Q−12. Then,
(44)Vk+1≤1−ℓVk+c.Solving inequality (44) yields
(45)Vk+1≤1−ℓkV0+c∑i=1k1−ℓi.If the upper limit whenk goes to infinity is considered, one can conclude that
(46)limk→∞Vk≤c/ℓ.Inequality (46) shows that the sliding variable sk is upper bounded by c/ℓ, which also implies that the LOS tracking error and its rate are also upper bounded by small constants, and therefore, accurate interception with impact angle is satisfied. This completes the proof.


## 6. Simulation Studies

In this section, the effectiveness of the proposed guidance law is demonstrated through numerical simulations, in which a surface-to-air missile is considered to intercept a maneuvering head-on target during its terminal guidance phase.


### 6.1. Simulation Setup

The simulations are performed in Matlab/Simulink platform with a sampling rate 0.01 s. Here, the missile is supposed to be equipped with an active radar seeker, providing LOS angle, LOS angular rate, range, and range rate measurement. The required simulation data for Eqs. (1)–(4) are summarized in Table 1. Since the undesirable noise always exists in seeker measurement and will affect the performance of the closed-loop guidance system, simulations are performed in a noise-corrupted homing environment to approximate the practical engineering for the most.Table 1
Required data for initializing simulations.
Parameter
Value
Initial relative range
R
0
=
10000
m
Initial flight path angle
γ
M
0
=
π
/
4
r
a
d, γT0=πrad
Initial LOS angle
λ
0
=
π
/
6
r
a
d
Flying velocity
V
M
=
400
m
/
s, VT=200m/s
Target maneuver
Case 1:aT=20m/s2Case 2: aT=20sin0.1tm/s2


### 6.2. Simulation Results for Different Impact Angles

The desired impact angles are set asθimp=0∘,45∘,90∘ for both Cases 1 and 2. The mean miss distances and impact angle errors for these two cases with 50 runs Monte-Carlo simulations are summarized in Table 2. The simulation results, including interception trajectory, achieved acceleration, LOS angle profile, target maneuver projection estimation performance, impact angle profile as well as target maneuver estimation for Cases 1 and 2 are provided in Figures 3 and 4, where Figure 3 is for Case 1 and Figure 4 is for Case 2. The desired values for LOS angle in these figures are calculated using Eq. (10).Table 2
Mean miss distances and impact angle errors of 50 runs Monte-Carlo simulations.
θ
i
m
p
=
0
∘
θ
i
m
p
=
45
∘
θ
i
m
p
=
90
∘
Case 1
Miss distance
0.749 m
0.659 m
0.713 m
Impact angle error
0.455
deg
0.391
deg
0.317
deg
Case 2
Miss distance
0.758 m
0.788 m
0.695 m
Impact angle error
0.501
deg
0.389
deg
0.426
degFigure 3
Simulation results under the proposed guidance law for Case 1: (a) interception trajectory; (b) missile achieved acceleration; (c) LOS angle profile; (d) target maneuver projection estimation; (e) impact angle profile; and (f) target maneuver estimation.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Simulation results under the proposed guidance law for Case 2: (a) interception trajectory; (b) missile achieved acceleration; (c) LOS angle profile; (d) target maneuver projection estimation; (e) impact angle profile; and (f) target maneuver estimation.
(a)
(b)
(c)
(d)
(e)
(f)One can note from these two figures that accurate interception is achieved under the proposed guidance law whatever the desired impact angle is, that is, the miss distances in these scenarios are less than0.8m. These figures also demonstrate that the LOS angles converge to their corresponding desired values during the homing engagement and therefore proves that the proposed guidance law exhibits satisfactory performance in impact angle control with the small acceptable miss distance for different impact angles, that is, the impact angle errors in these scenarios are less than 0.5deg. These results also show that the proposed model-assisted discrete-time disturbance observer can accurately track the real target maneuver in noise-corrupted guidance environment.


### 6.3. Comparison Results with Existing Discrete-Time Sliding Mode Guidance Law

To illustrate the effectiveness of the proposed guidance law, the existing discrete-time sliding mode control guidance (DSMCG) law [30] is also performed in simulations in this subsection for the purpose of comparison. In view of the idea of classical DSMC, the DSMCG law is formulated as
(47)aMk=−2Ṙkλ̇k+βsgnλ̇k,where β>0 is the guidance gain to be designed. To effectively counteract the effect of target maneuver, the guidance gain β needs to be larger than the maximum magnitude of target maneuver. It can be noted that the discontinuous function sgnλ̇k would result in high-frequency chattering in real applications.The simulation results, including interception trajectory, achieved acceleration, target maneuver estimation performance, withθimp=0∘ for Cases 1 and 2 are shown in Figures 5 and 6, which clearly demonstrate the chattering-mitigation property of the proposed guidance law. Moreover, as the main concern of DSMCG law is to nullify the LOS rate, this guidance law cannot be used for impact angle control. It should be noted that implementing the DMSCG law requires the information on the upper bound of target maneuver to choose the switching gain. However, such assumption on the knowledge of the upper bound of target maneuver would result in a design dilemma as discussed in Sec. I. As a comparison, the proposed guidance law adopts a disturbance observer to estimate the target maneuver, and therefore, it requires no priori knowledge on target maneuvers, bringing some advantages for practical implementation.Figure 5
Simulation results under the proposed guidance law for Case 1: (a) interception trajectory; (b) achieved acceleration; and (c) target maneuver estimation.
(a)
(b)
(c)Figure 6
Simulation results under the proposed guidance law for Case 2: (a) interception trajectory; (b) achieved acceleration; and (c) target maneuver estimation.
(a)
(b)
(c)


## 6.1. Simulation Setup

The simulations are performed in Matlab/Simulink platform with a sampling rate 0.01 s. Here, the missile is supposed to be equipped with an active radar seeker, providing LOS angle, LOS angular rate, range, and range rate measurement. The required simulation data for Eqs. (1)–(4) are summarized in Table 1. Since the undesirable noise always exists in seeker measurement and will affect the performance of the closed-loop guidance system, simulations are performed in a noise-corrupted homing environment to approximate the practical engineering for the most.Table 1
Required data for initializing simulations.
Parameter
Value
Initial relative range
R
0
=
10000
m
Initial flight path angle
γ
M
0
=
π
/
4
r
a
d, γT0=πrad
Initial LOS angle
λ
0
=
π
/
6
r
a
d
Flying velocity
V
M
=
400
m
/
s, VT=200m/s
Target maneuver
Case 1:aT=20m/s2Case 2: aT=20sin0.1tm/s2


## 6.2. Simulation Results for Different Impact Angles

The desired impact angles are set asθimp=0∘,45∘,90∘ for both Cases 1 and 2. The mean miss distances and impact angle errors for these two cases with 50 runs Monte-Carlo simulations are summarized in Table 2. The simulation results, including interception trajectory, achieved acceleration, LOS angle profile, target maneuver projection estimation performance, impact angle profile as well as target maneuver estimation for Cases 1 and 2 are provided in Figures 3 and 4, where Figure 3 is for Case 1 and Figure 4 is for Case 2. The desired values for LOS angle in these figures are calculated using Eq. (10).Table 2
Mean miss distances and impact angle errors of 50 runs Monte-Carlo simulations.
θ
i
m
p
=
0
∘
θ
i
m
p
=
45
∘
θ
i
m
p
=
90
∘
Case 1
Miss distance
0.749 m
0.659 m
0.713 m
Impact angle error
0.455
deg
0.391
deg
0.317
deg
Case 2
Miss distance
0.758 m
0.788 m
0.695 m
Impact angle error
0.501
deg
0.389
deg
0.426
degFigure 3
Simulation results under the proposed guidance law for Case 1: (a) interception trajectory; (b) missile achieved acceleration; (c) LOS angle profile; (d) target maneuver projection estimation; (e) impact angle profile; and (f) target maneuver estimation.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Simulation results under the proposed guidance law for Case 2: (a) interception trajectory; (b) missile achieved acceleration; (c) LOS angle profile; (d) target maneuver projection estimation; (e) impact angle profile; and (f) target maneuver estimation.
(a)
(b)
(c)
(d)
(e)
(f)One can note from these two figures that accurate interception is achieved under the proposed guidance law whatever the desired impact angle is, that is, the miss distances in these scenarios are less than0.8m. These figures also demonstrate that the LOS angles converge to their corresponding desired values during the homing engagement and therefore proves that the proposed guidance law exhibits satisfactory performance in impact angle control with the small acceptable miss distance for different impact angles, that is, the impact angle errors in these scenarios are less than 0.5deg. These results also show that the proposed model-assisted discrete-time disturbance observer can accurately track the real target maneuver in noise-corrupted guidance environment.


## 6.3. Comparison Results with Existing Discrete-Time Sliding Mode Guidance Law

To illustrate the effectiveness of the proposed guidance law, the existing discrete-time sliding mode control guidance (DSMCG) law [30] is also performed in simulations in this subsection for the purpose of comparison. In view of the idea of classical DSMC, the DSMCG law is formulated as
(47)aMk=−2Ṙkλ̇k+βsgnλ̇k,where β>0 is the guidance gain to be designed. To effectively counteract the effect of target maneuver, the guidance gain β needs to be larger than the maximum magnitude of target maneuver. It can be noted that the discontinuous function sgnλ̇k would result in high-frequency chattering in real applications.The simulation results, including interception trajectory, achieved acceleration, target maneuver estimation performance, withθimp=0∘ for Cases 1 and 2 are shown in Figures 5 and 6, which clearly demonstrate the chattering-mitigation property of the proposed guidance law. Moreover, as the main concern of DSMCG law is to nullify the LOS rate, this guidance law cannot be used for impact angle control. It should be noted that implementing the DMSCG law requires the information on the upper bound of target maneuver to choose the switching gain. However, such assumption on the knowledge of the upper bound of target maneuver would result in a design dilemma as discussed in Sec. I. As a comparison, the proposed guidance law adopts a disturbance observer to estimate the target maneuver, and therefore, it requires no priori knowledge on target maneuvers, bringing some advantages for practical implementation.Figure 5
Simulation results under the proposed guidance law for Case 1: (a) interception trajectory; (b) achieved acceleration; and (c) target maneuver estimation.
(a)
(b)
(c)Figure 6
Simulation results under the proposed guidance law for Case 2: (a) interception trajectory; (b) achieved acceleration; and (c) target maneuver estimation.
(a)
(b)
(c)


## 7. Conclusions

Under the framework of sliding mode control, the problem of intercepting unknown maneuvering targets with impact angle constraint is investigated by a new discrete-time guidance law enhanced by a disturbance observer. A model-assisted discrete-time disturbance observer is first proposed to estimate the target maneuver, and theoretical analysis shows that the estimation error is exponentially stable with a small upper bound determined by the sampling rate. Based on the reconstructed target maneuver information, a chattering-mitigated super-twisting-like discrete-time guidance law is proposed to force the system trajectory to converge into a small region around the sliding surface, and therefore, accurate interception with impact angle constraint is satisfied. Simulation and comparison studies of a surface-to-air missile have been carried out to show the efficiency of the proposed approach.



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*Source: 1764510-2020-08-05.xml*