# Composite Terminal Guidance Law for Supercavitating Torpedoes with Impact Angle Constraints

Authors: Dongxue Xu; Jingyuan Zhang; Peng Wang; Xiaojia Yan; Haidi Dong
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A novel composite terminal guidance law with impact angle constraints is proposed for supercavitating torpedoes to intercept maneuvering warships. Based on an adaptive super-twisting algorithm and nonsingular terminal sliding mode (NTSM), the proposed guidance law can guarantee the finite-time convergence of line-of-sight (LOS) angle error and the LOS angular rate error. The new guidance law is a combination of finite-time stability theory, sliding mode control (SMC), tracking differentiator (TD), disturbance observer (DO), and dynamic surface control. A high-order sliding mode TD is used for denoising, tracking, and differentiating the measured target heading angle. A novel DO, with its finite-time stability proved, is designed to estimate the target lateral acceleration for feedforward compensation to attenuate chattering in control input. In the case of a first-order-lag autopilot, a new kind of tracking differentiator is adopted to compute the first-order time derivative of the virtual control command, which can improve the accuracy of dynamic surface control and avoid the “explosion of items” problem encountered with the backstepping control. Finally, numerical simulation results are presented to validate the effectiveness and superiority of the proposed TD, DO, and the composite guidance law.


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## Body

## 1. Introduction

It is generally assumed that supercavitating torpedoes (SCT) can only be designed for straight running, which may miss the target at the terminal phase. The latest researches indicate that a new kind of guidance based on the underwater electric field of target warships [1, 2] can be adopted to enhance SCT combat capability. By measuring the three components of the electric field, an SCT can obtain the target’s location, velocity, and heading angle with a detection range of more than 1 km. The impact angle constraint refers to the interceptor attacking the target with a predetermined angle relative to the target’s velocity vector. As for an SCT with a contact fuse, achieving a proper predetermined impact angle can ensure a high killing probability using the longitudinal length of the target warship. However, hitting the target with an impact angle constraint is challenging for traditional torpedoes and is especially difficult for SCT. Compared with traditional torpedoes, there are some new peculiarities of guided SCT: a significant increase in speed due to the supercavitation drag reduction effect, a smaller detection range due to the limitation of the detecting method, and limited maneuverability on account of cavity shape stabilization [3]. For the high-speed SCT with limited maneuverability attacking against the maneuvering target with impact angle constraint in a short time, traditional torpedo guidance laws are no longer suitable. New guidance laws for guided SCT have yet to be developed and deserve special attention.Although there are no guidance laws with impact angle constraints for SCT in the public literature, an abundance of devotion has been made to study on high-speed interceptors, such as missiles and hypersonic aircraft. In recent years, guidance laws with impact angle constraints based on various fundamental theories have been proposed for different application scenarios. All the guidance laws can be classified into three main categories, i.e., modified proportional navigation (PN) guidance laws, modern guidance laws, and sliding mode control guidance laws.Variations of PN were thoroughly studied for their simple structure, mature theory, and ease of implementation. In [4], a three-dimensional adaptive PN guidance approach is presented for a hypersonic vehicle to impact a stationary ground target with a specified direction in the terminal phase. In [5], a biased PN guidance law for passive homing missiles intercepting stationary targets with constrained impact angles is designed and analyzed. Ratnoo and Ghose [6] proposed an impact angle constrained PN guidance law against moving but non-maneuvering targets, with the navigation constant N being a function of the initial engagement geometry. In [7], a biased PN guidance law against maneuvering targets with impact angular constraint is proposed. The additional time-varying bias term compensates for target accelerations, sensor noises, and impact angle. The above guidance laws all assume that the maneuverability of the interceptor is much greater than that of the target, while SCT doesn’t satisfy this assumption.In [8], an optimal guidance law is presented for missiles with constant speed against stationary targets by adopting a time-to-go weighted energy cost function. However, accurately estimating the time-to-go is a tough challenge when the target is maneuvering and the interceptor’s trajectory is highly curved. In [9], a linear-quadratic optimal-control-based guidance law and a linear-quadratic differential-game-based guidance law are derived, which enable imposing predetermined impact angles on maneuvering targets with significant initial heading errors. Closed-form solutions of optimal impact-angle-control guidance for a first-order lag system are derived in [10]. To reduce the complexity of the analytic solutions, ideal dynamics of the missiles are assumed, and an approximate time-to-go is used. However, the optimal-control-based and differential-game-based guidance laws are too complicated to derive and implement. Besides, they all rely on the accurate target motion model strictly, which cannot be obtained in practice.Traditional SMC guidance laws can deal with the maneuvering target and dynamic uncertainties, yet they may lead to excessive or high-frequency chattering in control input. In [11], a guidance law with adaptive estimation for target maneuver is designed using two separate switching surfaces, one for regulating the impact angle constraints and the other for regulating the homing constraint. In [12], a kind of first-order sliding-mode guidance law with autopilot lag is proposed both for planar and three-dimensional interceptions in the presence of target maneuvers. In [13, 14], a hit-to-kill guidance strategy in the presence of target evasive maneuvers and dynamic uncertainty is developed based on the smooth second-order SMC.In designing the guidance law for terminal interception, the general strategy is to nullify the relative velocity component perpendicular to the LOS [14]. This goal can be achieved by nulling the LOS rate for guidance laws without impact angle constraints. Nevertheless, when there is an impact angle constraint, it can only be achieved by nulling the LOS angle error and its first-order derivate, named the LOS angular rate error. In some literature, it is assumed that the final heading angle of the target can be acquired at the beginning [7] or that the desired LOS angle is fixed [15]. This assumption may be valid when the targets are nonmaneuvering but are improper when attacking maneuvering targets, and more details will be discussed in the subsequent section.Another concern is that the terminal interception process is very short; thus, a finite-time convergence of the tracking errors is required. Finite-time convergence based on SMC consists of two parts, i.e., in the reaching phase, the sliding variable, which is a function of the tracking errors and their time derivatives, reaches zero or near zero in a short finite time; meanwhile, in the sliding phase, the tracking errors achieve finite-time convergence on the sliding surface. The terminal sliding mode control (TSMC), with the sliding variable as a nonlinear function of tracking errors, can improve the performance hugely. Guidance law with finite-time convergent impact angle based on TSMC for missiles has been proposed and studied in [16]. However, the TSMC does not prevail over the linear counterpart in the convergence rate when the system state is far from the equilibrium. Moreover, the TSMC suffers from a singularity problem, which will lead to control saturation. To get rid of the singularity, some new guidance laws are proposed to deal with non-maneuvering targets [17] and maneuvering targets [12, 18] based on nonsingular terminal sliding-mode control (NTSMC) [19]. A nonsingular fast terminal sliding mode control (NFTSMC) is introduced to achieve fast convergence performance, especially when the system states are far from equilibrium [20]. In [21], a nonlinear integral SMC method is employed for a finite-time stable strategy. On the other hand, different ways have been taken to regulate the switching surfaces for a finite reaching time, such as using adaptive sliding mode control [10], adopting the method of smooth second-order SMC [13, 14], and defining a switching control command [12, 15–18], which may cause the chattering phenomenon.In addition, the guidance process requires real-time measurements of information such as LOS and target heading angle, which is inevitably contaminated by noise and can lead to large guidance deviations if unprocessed. Linear filters produce phase delays and amplitude distortions, whereas signal processing using a tracking differentiator (TD) can avoid these deficiencies. The TD-H proposed by Han [22] and rigorously proved by Guo and Zhao [23] has the advantage of minimum-time convergence and has been utilized in different fields, such as constructing a transient profile according to the actuator’s capacity [24]. However, only estimates of the input and its first-order derivative can be obtained by TD-H. Several other kinds of TDs, which have the merits of simple recursive structure, easily adjusting parameters, and high accuracy, have been proposed by Levant [25–27]. Nevertheless, these TDs, both the conventional version [25, 26] and its improved version (TD-L) [27], all suffer from significant undesirable overshoot during the transient. Recently, a new high-order adaptive sliding mode TD was presented (TD-N) [28], which adaptively changes its gains for improved tracking performances and reduced overshoot.An interceptor’s autopilot lag can significantly influence guidance accuracy, especially in the presence of target maneuvers [12]. In some literature, the autopilot dynamic is treated as ideal, introducing significant control biases [29]. The traditional design methodology for handling autopilot lag is the backstepping control, which may lead to the “explosion of items” problem. The dynamic surface control proposed by Swaroop et al. [30] can avoid this problem by calculating the time derivative of the virtual control command through the first-order filter, which is a standard method for dealing with similar issues at present [31, 32]. However, it will inevitably induce a phase delay, which reduces convergence speed and control precision.In guidance design, the lateral acceleration of the target can be regarded as an unknown bounded disturbance, which can be estimated by using a disturbance observer and cancelled by feedforward compensation simultaneously. As a result, the chattering caused by excessive switching control can be mitigated. In [13, 15, 16] and [21], sliding mode disturbance observers are used to estimate target lateral acceleration. In [31, 33], extended state observers (ESO) are used for the same purpose. The inherent drawbacks of these observers are that they are prone to overshoot during the transition period and do not work well under noise conditions.A novel composite guidance law is proposed to address the problems aforementioned in existing guidance laws. Under the novel guidance law, an SCT can hit a maneuvering warship target with an impact angle constraint. Compared with the existing methods, the main highlights of the proposed solution are outlined as follows:(1)
In the guidance law design, the second-order time derivative of the target heading angle is needed, which is usually regarded as zero in the present literature, for it is hard to get. A recently proposed high-order sliding mode TD [27], which performs well in handling noisy signals, is used to track the measured noisy target heading angle and obtain its second-order time derivative.(2)
A novel high-order sliding mode DO is designed based on TD [27], and its finite-time stability is proved using homogeneous theory. The novel DO is applied to estimate the target lateral acceleration, which is regarded as an unknown disturbance.(3)
A new fast nonsingular terminal sliding manifold is designed, and an adaptive super-twisting algorithm is applied, both of which help to lower the control input of autopilot at the initial time. The finite-time convergence analysis of the tracking errors is carried out by Lyapunov theory under a novel framework.(4)
To effectively account for the autopilot lag of the SCT, the variable gain sliding mode TD [27] is used to calculate the first-order time derivative of the virtual control command, which improves the tracking accuracy and avoids the “explosion of items” problem at the same time.The rest of this paper is arranged as follows. In Section2, the engagement dynamics are derived. In Section 3, a new sliding mode tracking differentiator is introduced, and a novel sliding mode disturbance observer is designed. In Section 4, the nonsingular terminal sliding manifold and an adaptive second-order SMC algorithm are designed. In Section 5, numerical simulations are carried out for performance evaluations of new TD, DO, and the proposed composite guidance law. Finally, in Section 6, some conclusions and discussions are offered.


## 2. Problem Formulation

The planar engagement and the relative motion dynamics are derived based on the vector derivative algorithm rather than differentiating equations [15]. The relationship between the impact angle and the LOS angle is derived, and the mistake in treating the desired LOS angle as a fixed value in the existing literature is analyzed.To simplify the analysis of the pursuer-target relative motion, the interception is assumed to occur in a planar engagement scenario, with the target and torpedo treated as point masses, and denoted byT and M, respectively.In the LOS coordinate system as shown in Figure1, the origin is fixed at the torpedo, r represents the relative range of torpedo-to-target, θM and θT are the heading angles of the torpedo and the target, and q is the LOS angle. Moreover, VM and VT, respectively, denote the velocities of the torpedo and the target, both of which are assumed to be constant. The aM and aT, respectively, refer to the accelerations of the torpedo and the target, which are perpendicular to their velocity directions. Let ηM=q−θM, and ηT=q−θT, and then, we have θ˙T=aT/VT, and θ˙M=aM/VM [16].Figure 1
Planar torpedo-target engagement.In the coordinate system, subscriptsr and q represent the radial direction along the LOS and the axial direction normal to the LOS. The base vector along the LOS is denoted by er, while the one normal to the LOS is aligned with er rotating π/2 counter-clockwise, denoted by eq.The derivatives of the base vectors are(1)ddtereq=0q˙−q˙0ereq,where q˙ refers to the LOS angular rate. It is evident from (1) that the velocity vector is(2)v=drdt=drerdt=drdter+rderdt=r˙er+rq˙eq.DefiningVr as the relative velocity between torpedo and target along the LOS, and Vq as the one normal to the LOS, (2) can be rewritten as follows:(3)Vr=VTr−VMr=−VMcosηM+VTcosηT=r,˙Vq=VTq−VMq=VMsinηM−VTsinηT=rq.˙Identically, differentiating (2) with respect to time yields(4)a=dr˙er+rq˙eqdt=dr˙dter+r˙derdt+drdtq˙eq+rdq˙dteq+deqdtq˙=r¨−rq˙2er+rq¨+2r˙q˙eq.Similarly, definingar as the relative acceleration of the torpedo target along the LOS, and aq as the one normal to the LOS, (4) can be rewritten as follows:(5)ar=aTr−aMr=aTsinηT−aMsinηM=r¨−rq˙2,aq=aTq−aMq=aTcosηT−aMcosηM=rq¨+2r˙q.˙With a transposition, (5) can be transformed to(6)r¨=rq˙2−aMr+aTr,rq¨=−2r˙q˙+aTq−aMq.Combining (3) and (6), one can imply that(7)V˙q=rq¨+r˙q˙=−r˙q˙+aTq−aMq.The autopilot can be approximately described by the following first-order dynamics [12, 15]:(8)a˙M=−aMτ+uτ,where τ is the autopilot lag, and u is the acceleration command for the autopilot.As shown in Figure2, the impact angle is described as the angle between the velocity vectors of the torpedo and the target at the time of interception [18], i.e.,(9)θimp=θT−θMf.Figure 2
The interception triangle.The relative velocity component should be nullified to form a collision triangle [6, 7].(10)Vqf=rfq˙f=0,VMsinθMf−qf=VTsinθT−qf.With (9) and the trigonometric equation in (10), the relationship between the desired LOS angle and the impact angle [6, 11] can be derived as follows:(11)qf=θT−arctansinθimpcosθimp−ν1,where the target-to-torpedo velocity ratio is ν1=VT/VM<1.It is evident thatqf and θT are linear with only one constant bias. For a maneuvering target, θT is a time-varying quantity, and so is the qf. In practice, the terminal heading angle of the maneuvering target is generally unavailable at the beginning of guidance, implying that the qf cannot be treated as a given constant as in [15]. In other words, it is impossible to regulate the q˙ at near zero while keep the q tracking a varying desired qf simultaneously. Furthermore, with θimp and ν1 predetermined, on differentiating (11), one can imply that(12)q˙f=θ˙T=aTVT=aTqVTcosηT,q¨f=θ¨T=a˙TVT=a˙Tq+aTqη˙TtanηTVTcosηT.It can be seen from the above analysis that, to achieve the impact angle constraints, the first-order and second-order time derivatives of the target heading angle,θ˙T and θ¨T, are needed. Or else, the target lateral acceleration aTq as well as its first-order time derivative a˙Tq are required. In the present literature, θ¨T or a˙Tq is usually regarded as zero because they are hard to get. Here, two equivalent solutions are proposed to deal with this problem. The first method is to get the θ˙T and θ¨T using a second-order TD, and the other way is to estimate the aTq and a˙Tq based on a third-order DO. Here, the first method is adopted in this paper.Letx1=q−qf, x2=q˙−q˙f, and x3=aM. It follows from (6), (8), and (12) that(13)x˙1=x2,x˙2=−2r˙q˙r−x3⋅cosηMr+aTqr−θ¨T,x˙3=−x3τ+uτ.In designing a guidance law with an impact angle constraint, the unknown external disturbance includes the target lateral accelerationaTq, as well as the second-order time derivative of the target heading angle θ¨T. The design objective of control input u is to drive x1 and x2 to zero in finite time, that is, q⟶qf, q˙⟶q˙f.


## 3. High-Order Sliding Mode Tracking Differentiator and Design of New Disturbance Observer

In this section, a high-order sliding mode TD for processing measurement signals with noise is introduced. A new kind of sliding mode DO adapted from the high-order sliding mode TD is designed, and the finite-time stability of the proposed DO is proved using homogeneous degree theory.Firstly, some conclusions are recalled, which are indispensable in subsequent analysis.


### 3.1. Essential Preliminary Concepts

Lemma 1 (see [34]).
Define a smooth vector field(14)fx1,⋯,xn=f1x,⋯,fnxT:Rn⟶Rn,and the dilation(15)Δη=r1,⋯,rn∈Rn,where x1,⋯,xn are suitable coordinates on Rn and r1,⋯,rn are positive real numbers. The vector field (14) is homogeneous of degree m∈R, if ∀η>0, i=1,2,⋯,n, fi is a homogeneous function with respect to dilation (15)(16)fiηr1x1,⋯,ηrnxn=ηri+mfix1,⋯,xn.Suppose the origin is a stable equilibrium offx, if its degree of homogeneity, m⩾−maxri, is negative, then the origin is finite-time stable.Lemma 2 (see [15]).
Finite-time convergence theory.On systemx˙=fx,t, f0,t=0, x∈Rn, suppose f:U0×R⟶Rn is a C1 smooth, positive definite function on U⊂Rn and V˙x+β1Vβ2x is a negative semidefinite function on U⊂Rn, where β1>0 and β2∈0,1, then there exists an area U0⊂Rn such that Vx starting from U0⊂Rn can reach Vx≡0 in finite time(17)Treach⩽V1−β2x0β11−β2.


### 3.2. Tracking Differentiator for Processing Measured Target Heading Angle

Lemma 3 (see [35]).
Suppose the unknown functionf0t, t≥0, has a noisy approximation ft=f0t+nt+nct available in real time, where the derivative f0kt exists, and there is a known Lipschitz constant L>0. The first noise term nt is Lebesgue-measurable and bounded by ∃δ>0, nt<δ, and the second noise term nct is integrable.Thekth-order tracking differentiator proposed is(18)w˙−1=−λk+1L1/k+2w−1k+1/k+2signw−1+w0−ft,w˙0=−λkL2/k+2w−1k/k+1signw−1+w1,⋮,w˙k−1=−λ1Lk+1/k+2w−11/k+1signw−1+wk,w˙k=−λ0Lw−10signw−1,where sign⋅ is signum function. λi>0, i=0,1,⋯,k+1, as well as the design parameter L can be set follows [25]. Moreover, w−1 is an auxiliary variable, while wi, i=0,1,⋯,k, is the estimate of f0it, with the estimation errors converging in finite time. Increasing L contributes to the convergence rate, but it also degrades the noise-attenuating performance, so there is a trade-off between the convergence rate and the convergence precision.A second-order TD for processing noisy measured target heading angle, denoted byθ˜T, is designed with Lemma 2 using TD (18):(19)w˙−1=−λ13L11/4w−13/4signw−1+w0−θ˜T,w˙0=−λ12L12/4w−12/4signw−1+w1,w˙1=−λ11L13/4w−11/4signw−1+w2,w˙2=−λ10L1w−10signw−1.With the proper parameters selected, we have(20)w0=θ^T,w1=θ˙^T,w2=θ¨^T.


### 3.3. Proposal of New Disturbance Observer and Its Application in Estimating Target Lateral Acceleration

Consider the following first-order SISO dynamics:(21)x˙=u1+d,where x is the system state, u1 is the control input and is Lebesgue-measurable, and d is the lumped disturbance and is sufficiently smooth. Adapted from TD (18), the proposed DO is(22)z˙−1=−λk+1L1/k+2z−1k+1/k+2signz−1+z0−x,z˙0=−λkL2/k+2z−1k/k+2signz−1+z1+u1,⋮z˙k−1=−λk−iLk+1/k+2z−11/k+2signz−1+zk,z˙k=−λ0Lz−10signz−1,where λi>0,i=0,1,⋯,k+1, L>dk.The z−1 is an auxiliary variable, while z0 is the estimate of x, zi is the estimate of di−1, i=1,⋯,k, i.e.,(23)z0=x^,z1=d^,⋯,zk=d^k−1.Theorem 1.
The estimation errors of (22) converge in finite time.Proof.
Here, we denote(24)σ−1=z−1L,σ0=z0−xL,σi=zi−di−1L,i=1,2,⋯,k.
Using (22) and (24) and dk/L∈−1,1, any solution of (22) is governed by the following differential inclusion in the Filippov sense:(25)σ˙−1=−λk+1σ−1k+1/k+2signσ−1+σ0,σ˙0=−λkσ−1k/k+2signσ−1+σ1,⋮,σ˙i=−λk−iσ−1k−i/k+2signσ−1+σi+1,i=1,2,⋯,k−1,⋮,σ˙k=−λ0signσ−1−dk/L∈−λ0signσ−1+−1,1.
Introduce the vector fieldg:Rk+2⟶Rk+2(26)gσ−1,⋯,σk=g1,⋯,gj⋯,gk+2T=σ˙−1,⋯,σ˙j−2,⋯,σ˙kT,where j=2,⋯,k+1. The trajectories of function (26) are invariant concerning the transformation [36](27)∀η>0,Gη:t,σi↦ηt,ηk−i+1σi,i=−1,0,1,⋯,k.
By combining (25)–(27), one can imply that(28)gjηt,ηk+2σ−1,ηk−j+3σj−2,ηk−j+2σj−1=ηk−j+3+−1gjt,σ−1,σj−2,σj−1,gk+2ηt,ηk+2σ−1,ησk=η1+−1gk+2t,σ−1,σk.
One can note that the differential inclusion (25) is homogeneous of degree −1 with respect to dilation k+2,⋯,k−j+3,⋯,1, j=2,⋯,k+1; therefore, it is finite-time stable by Lemma 1.
External disturbance increases the chattering in autopilot control input, yet it can be mitigated by feedforward compensation. With total disturbance compensated, the guidance law can be designed in the nominal system according to the required control accuracy. A new DO is constructed to estimate theaTq based on (7) using DO (22):(29)z˙−1=−λ22L21/3z−12/3signz−1+z0−rq˙,z˙0=−λ21L22/3z−11/3signz−1+z1−r˙q˙−aMcosηM,z˙1=−λ20L2z−10signz−1.
With a proper choice of the parametersL2 and λ2i, i=0,1,2, z1 approaches to aTq in a short, finite time. It should be noted that, in some literature, the DO for estimating aTq is based on (6), with the item aTq/r estimated. However, in this way, extensive tracking range is to be handled for aTq/r change with r dramatically as the torpedo approaches the target, and difficulties will be encountered in choosing parameters [37]. In addition, the proposed disturbance observer can also be utilized in other scenarios such as control of quadrotors [38].


## 3.1. Essential Preliminary Concepts

Lemma 1 (see [34]).
Define a smooth vector field(14)fx1,⋯,xn=f1x,⋯,fnxT:Rn⟶Rn,and the dilation(15)Δη=r1,⋯,rn∈Rn,where x1,⋯,xn are suitable coordinates on Rn and r1,⋯,rn are positive real numbers. The vector field (14) is homogeneous of degree m∈R, if ∀η>0, i=1,2,⋯,n, fi is a homogeneous function with respect to dilation (15)(16)fiηr1x1,⋯,ηrnxn=ηri+mfix1,⋯,xn.Suppose the origin is a stable equilibrium offx, if its degree of homogeneity, m⩾−maxri, is negative, then the origin is finite-time stable.Lemma 2 (see [15]).
Finite-time convergence theory.On systemx˙=fx,t, f0,t=0, x∈Rn, suppose f:U0×R⟶Rn is a C1 smooth, positive definite function on U⊂Rn and V˙x+β1Vβ2x is a negative semidefinite function on U⊂Rn, where β1>0 and β2∈0,1, then there exists an area U0⊂Rn such that Vx starting from U0⊂Rn can reach Vx≡0 in finite time(17)Treach⩽V1−β2x0β11−β2.


## 3.2. Tracking Differentiator for Processing Measured Target Heading Angle

Lemma 3 (see [35]).
Suppose the unknown functionf0t, t≥0, has a noisy approximation ft=f0t+nt+nct available in real time, where the derivative f0kt exists, and there is a known Lipschitz constant L>0. The first noise term nt is Lebesgue-measurable and bounded by ∃δ>0, nt<δ, and the second noise term nct is integrable.Thekth-order tracking differentiator proposed is(18)w˙−1=−λk+1L1/k+2w−1k+1/k+2signw−1+w0−ft,w˙0=−λkL2/k+2w−1k/k+1signw−1+w1,⋮,w˙k−1=−λ1Lk+1/k+2w−11/k+1signw−1+wk,w˙k=−λ0Lw−10signw−1,where sign⋅ is signum function. λi>0, i=0,1,⋯,k+1, as well as the design parameter L can be set follows [25]. Moreover, w−1 is an auxiliary variable, while wi, i=0,1,⋯,k, is the estimate of f0it, with the estimation errors converging in finite time. Increasing L contributes to the convergence rate, but it also degrades the noise-attenuating performance, so there is a trade-off between the convergence rate and the convergence precision.A second-order TD for processing noisy measured target heading angle, denoted byθ˜T, is designed with Lemma 2 using TD (18):(19)w˙−1=−λ13L11/4w−13/4signw−1+w0−θ˜T,w˙0=−λ12L12/4w−12/4signw−1+w1,w˙1=−λ11L13/4w−11/4signw−1+w2,w˙2=−λ10L1w−10signw−1.With the proper parameters selected, we have(20)w0=θ^T,w1=θ˙^T,w2=θ¨^T.


## 3.3. Proposal of New Disturbance Observer and Its Application in Estimating Target Lateral Acceleration

Consider the following first-order SISO dynamics:(21)x˙=u1+d,where x is the system state, u1 is the control input and is Lebesgue-measurable, and d is the lumped disturbance and is sufficiently smooth. Adapted from TD (18), the proposed DO is(22)z˙−1=−λk+1L1/k+2z−1k+1/k+2signz−1+z0−x,z˙0=−λkL2/k+2z−1k/k+2signz−1+z1+u1,⋮z˙k−1=−λk−iLk+1/k+2z−11/k+2signz−1+zk,z˙k=−λ0Lz−10signz−1,where λi>0,i=0,1,⋯,k+1, L>dk.The z−1 is an auxiliary variable, while z0 is the estimate of x, zi is the estimate of di−1, i=1,⋯,k, i.e.,(23)z0=x^,z1=d^,⋯,zk=d^k−1.Theorem 1.
The estimation errors of (22) converge in finite time.Proof.
Here, we denote(24)σ−1=z−1L,σ0=z0−xL,σi=zi−di−1L,i=1,2,⋯,k.
Using (22) and (24) and dk/L∈−1,1, any solution of (22) is governed by the following differential inclusion in the Filippov sense:(25)σ˙−1=−λk+1σ−1k+1/k+2signσ−1+σ0,σ˙0=−λkσ−1k/k+2signσ−1+σ1,⋮,σ˙i=−λk−iσ−1k−i/k+2signσ−1+σi+1,i=1,2,⋯,k−1,⋮,σ˙k=−λ0signσ−1−dk/L∈−λ0signσ−1+−1,1.
Introduce the vector fieldg:Rk+2⟶Rk+2(26)gσ−1,⋯,σk=g1,⋯,gj⋯,gk+2T=σ˙−1,⋯,σ˙j−2,⋯,σ˙kT,where j=2,⋯,k+1. The trajectories of function (26) are invariant concerning the transformation [36](27)∀η>0,Gη:t,σi↦ηt,ηk−i+1σi,i=−1,0,1,⋯,k.
By combining (25)–(27), one can imply that(28)gjηt,ηk+2σ−1,ηk−j+3σj−2,ηk−j+2σj−1=ηk−j+3+−1gjt,σ−1,σj−2,σj−1,gk+2ηt,ηk+2σ−1,ησk=η1+−1gk+2t,σ−1,σk.
One can note that the differential inclusion (25) is homogeneous of degree −1 with respect to dilation k+2,⋯,k−j+3,⋯,1, j=2,⋯,k+1; therefore, it is finite-time stable by Lemma 1.
External disturbance increases the chattering in autopilot control input, yet it can be mitigated by feedforward compensation. With total disturbance compensated, the guidance law can be designed in the nominal system according to the required control accuracy. A new DO is constructed to estimate theaTq based on (7) using DO (22):(29)z˙−1=−λ22L21/3z−12/3signz−1+z0−rq˙,z˙0=−λ21L22/3z−11/3signz−1+z1−r˙q˙−aMcosηM,z˙1=−λ20L2z−10signz−1.
With a proper choice of the parametersL2 and λ2i, i=0,1,2, z1 approaches to aTq in a short, finite time. It should be noted that, in some literature, the DO for estimating aTq is based on (6), with the item aTq/r estimated. However, in this way, extensive tracking range is to be handled for aTq/r change with r dramatically as the torpedo approaches the target, and difficulties will be encountered in choosing parameters [37]. In addition, the proposed disturbance observer can also be utilized in other scenarios such as control of quadrotors [38].


## 4. Composite Sliding Mode Guidance Law with Finite-Time Convergence

Firstly, a new nonsingular terminal sliding manifold (NNTSM) is designed. Then, an adaptive super-twisting algorithm (ASTA) to deal with the unknown upper bounded disturbance is designed. Finally, using the variable gain sliding mode tracking differentiator (TD-N), the control input for autopilot with a first-order dynamic is obtained.The TD (19) designed for processing noisy target heading angle is referred to as TD1. The DO (29) designed for estimating the target lateral acceleration is referred to as DO1. The structure of the guidance law is shown in Figure 3.Figure 3
Structure flowchart of compound guidance law.


### 4.1. Design of Virtual Control Input Based on NNTSM and ASTA

Traditional TSM can only guarantee asymptotical convergence rather than finite-time convergence. Inspired by [20, 34] and [39], a new nonsingular terminal sliding mode control is introduced to improve the convergence rate, with its sliding manifold designed as follows:(30)s=x2+∑1=12∫0tci1−e−t/τixiαisignxi∙dt,where α2∈0,1, α1=α2/2−α2∈0,1, c1, c2 ensure that l2+c2l+c1=0 is Hurwitz, and their choices can be referred to [39]. In addition, τ1=τ2>0 and can be set by trial and error considering the torpedo’s maneuverability limit and the reaching time.A main drawback in the standard super-twisting algorithm is that the upper bounds of disturbance and its first-order time derivative are needed in choosing parameters. Inspired by [40, 41], an adaptive super-twisting algorithm (ASTA) is designed as follows:(31)s˙=−k1ϕ1−∫0tdt+d,ϕ1=s1/2signs,ϕ2=12signs,where k1 and k2 are adaptive gains with the nonlinear adaptation law as follows [41]:(32)k˙1=γs,k1≤km,λssigns−ε,k1>km,,k10>0,k2=2δk1,where γ, λ, ε, km, and δ are all positive constants.A smallε can improve the controller accuracy, but it will also reduce the convergence rate of s [42]. Increasing the control parameters λ, γ, and δ can shorten the convergence time and increase the control accuracy, but it also augments the acceleration magnitude. Thus, there is a trade-off needed in tuning parameters.Proposition 1.
Given the sliding variable dynamics (31) and the gain-adaptation law (32), the gains k1 and k2 are upper bound, i.e., there exists k1∗>0 and k2∗ such that(33)k˜1=k1−k1∗≤0,k˜2=k2−k2∗≤0.Sketch Proof of Proposition 1.
Since there is a positive linear correlation betweenk1 and k2, only k1 is analyzed for clarity.
If the sliding variables>ε, k1 and k2 increases gradually enough to make the sliding variable s converge to zero. Once the sliding mode is established, i.e., s<ε, the proposed gain-adaptation law (32) will keep k1 approaching km. If gains are not large enough to counteract the perturbations, resulting in s>ε, k1 will increase again making the sliding variable converge to s<ε. Therefore, adaptive parameters are always globally bounded, and a detailed proof can be referred to [42].
Taking (13), (30) and (31) into account, the virtual control command can be chosen as(34)x3c=rcosηM∑i=12ci1−e−t/τixiαisignxi−θ¨^T−2r˙q˙r+a^Tqr+k1ϕ1+∫k2ϕ2dt.
The prominent merits of the sliding manifold (30) and the ASTA can be inferred from the virtual control command (34). In the reaching phase, x3c is attenuated with 1−e−t/τi<1; wheras in the sliding phase there is t>tr, with ci1−e−t/τi approaching to ci, the convergence rate on the sliding surface will not be affected. In addition, adaptive gains k1 and k2 are gradually increasing, reducing x3c at the start of guidance.


### 4.2. Design of Autopilot Control Input via New Tracking Differentiator

Considering the autopilot lag, dynamic surface control [30] is frequently used to design the control input. From (13), the theoretical command input for the autopilot should be(35)u=τx˙3c+x3c.One significant difficulty in calculatingu is to get the first-order time derivative of x3c. In dynamic surface control, x3c is passed through a first-order filter, getting x3d and x˙3d, and the “explosion of items” problem in the backstepping control is avoided. Nevertheless, the use of a low-pass filter may introduce a significant delay as well as a magnitude attenuation. Thus x3c is not followed precisely by x3d. Even more, in the dynamic surface method, with the accuracy of x˙3c−x˙3d not guaranteed, the controller can only be designed to regulate x3 tracking x3d. Though the error between x3 and x3d can be removed by feedback control, there is still an error between x3 and x3c, resulting in reduced tracking accuracy. In some literature, TD proposed by Han [22] is used to replace the low-pass filter, yet it may lead to a phase lag in the transient period. In this paper, x3c is passed through an improved sliding mode TD [28], getting new x3d and x˙3d. The improved TD has slighter overshoot and faster response speed, which can guarantee the accuracy of x˙3c−x˙3d. Then feedback control is carried out to improve the control accuracy of x3−x3c. In this way, improved control accuracy is achieved, while the common “explosion of items” problem in the backstepping control design is avoided.The TD2 for dealing withx3c can be constructed as(36)ct=κmaxz∈t−τ0,tm0z−minz∈t−τ0,tm0zt−τ0,L3t=χm−1t+ct,vt=−λ32L31/3tm−1t2/3signm−1t+m0t−x3ct,m˙−1t=vt−βvtsignm−1t,m˙0t=−λ31L32/3tm−1t1/3signm−1t+m1t,m˙1t=−λ30L3tm−1t0signm−1t,where λ3i>0, i=0,1,2, and can be set following [27]. The c reflects the average change rate of the input, while v and m−1 are auxiliary variables. m0=x3d, m1=x˙3d are the estimates of x3c and x˙3c, respectively. Moreover, χ>0, κ>0, 0≤β<1, and τ0>0. Increasing χ contributes to the tracking response speed but will also increase the overshoot magnitude in the transient phase. A big β can reduce overshoot, but it also restrains response speed at the same time. Increase of κ can improve tracking speed and reduce overshoot, but it can also augment the effect of noise. The stability proof and parameter selection rules of the variable gain sliding mode TD are presented in [28].The actual command input for the autopilot is(37)u=τx˙3d+x3c.Denote the errory3=x3−x3c, and then substitute (13) and (34) into (37) yields(38)y˙3=x˙3−x˙3c=−y3τ+x˙3d−x˙3c.With the proven finite-time convergence of the DO1, TD1, and TD2, there must exist smallε1,ε2,ε3>0 such that(39)a^Tq−aTq<ε1,θ¨^T−θ¨T<ε2,x˙3c−x˙3d<ε3.Sincey˙3<−y3/τ+ε3, τ is small, it follows that the tracking error y3 is ultimately uniformly.Confined byε3. Thus x3 can track the virtual control command x3c precisely.Substituting (13) and (34) into (30) yields(40)s˙=x˙2+∑i=12ci1−e−t/τixiαisignxi=−2r˙q˙r−x3c+y3cosηMr+aTqr−θ¨T+∑i=12ci1−e−t/τixiαisignxi=−k1ϕ1−∫k2ϕ2dt+d,where the lumped disturbance(41)d=aTq−a^Tqr+θ¨^T−θ¨T−cosηMry3.According to (38) and (39), d should be bounded.


### 4.3. Finite-Time Convergence Analysis of the Tracking Errors

Theorem 2.
The tracking errors of the LOS anglex1 and LOS angular rate x2 will converge into a small region near zero in finite time.The proof can be divided into two steps. Firstly, we prove that the sliding variables will converge into a neighbourhood Ω0=ss<ε in finite time under the control of the adaptive super-twisting algorithm (32). Secondly, we prove that the tracking errors are finite-time convergent on the sliding surface.Inspired by [43, 44], a novel proof of this theorem under a new framework is proposed, which can be regarded as a modified version of the proof in [41].To facilitate the stability analysis, a new state vector is introduced:(42)ξT=ξ1,ξ2T=ϕ1,−∫k2ϕ2dt+dT,then combining (31) and (40) into(43)s˙=−k1ϕ1+ξ2,ξ˙2=−k2ϕ2+d˙.System (42) can be rearranged as(44)ξ˙=ϕ˙1−k1ϕ1−∫k2ϕ2dt+d−k2ϕ2+d˙=ϕ˙1−k1ϕ1−∫k2ϕ2dt+d−k2ϕ1+d˙ϕ˙1=ϕ˙1Aξ+Bg˜,where ϕ˙1=dϕ1/ds=s−1/2/2=ϕ2/ϕ1, g˜=d˙/ϕ˙1=2d˙ϕ1, B=01T, and(45)A=−k11−k20.Suppose there exists a positive constantσ>0, such that(46)d˙≤σ2=σϕ2.In [41], to deal with (46), it set d˙=κ¯ϕ2 with κ¯≤σ. However, as we know, d˙ is a variable, which means κ¯ should not be a constant. However, κ¯ frequently appears in the derivations of [41], which will cause many difficulties in algebraic manipulations.Unlike the way in [41], the ingenious way here is that we get the following inequality by combining (44) with (46):(47)g˜=2d˙⋅ϕ1≤σϕ1=σξ1.Then we define(48)ωξ,g˜=σ2ξ1Tξ1−g˜2=ξg˜TR00−1ξg˜≥0,R=σ2000.To analyze the stability of the system (44), define(49)V1=ξTPξ,V2=0.5k˜12+k˜22,where P is a positive definite symmetric matrix and the quadratic form V1=ξTPξ is positive definite and radically unbounded by the standard inequality for quadratic forms:(50)V11/2xλmax1/2P≤ξ2≤V11/2xλmin1/2P,where ξ2 is the Euclidean norm of ξ.Consider the Lyapunov function candidate(51)V=V1+V2.The positive definite symmetric matrixP is chosen as follows:(52)P=β+4δ2−2δ−2δ1,where β>0, then V is continuously derivable except the equilibrium. If the following conditions are satisfied:(53)0<σ<4δ−1,k1>4δ2−2δ+β22β4δ−σ−1+4δ2+σ2+σ2β,then the matrix Q defined as (54) is negative definite.(54)Q=ATP+PA+R+σI+PBBTP=−2k1β−4δ2−σ2−σ∗−β+4δ2+2δ4δ−σ−1<0,where ∗ is used to indicate a symmetric element. This conclusion can be easily deduced from (53) and (54) with the Schur complement.Using (50), the fact can be easily noted that(55)s1/2=ξ1≤ξ2.Case a. If the adaptive gaink1 meets condition (53), then substituting (44), (48), and (54) into (49) yields(56)V˙1=ξ˙TPξ+ξTPξ˙=ϕ˙1ξTATP+PAξ+g˜BTPξ+ξTPBg˜=ϕ˙1ξg˜TATP+PAPBBTP0ξg˜≤ϕ˙1ξg˜TATP+PAPBBTP0ξg˜+ωξ,g˜=ϕ˙1ξg˜TATP+PA+RPBBTP−1ξg˜<ϕ˙1ξg˜T−σI000ξg˜=−ϕ˙1σξTξ=−s−1/2σξTξ2.Combing (50) and (55), (56) can be written as(57)V˙1≤−μV11/2,where μ is(58)μ=σλmin1/2P2λmaxP.Substituting (32) and (33) into (49) yields(59)V˙2=k˜1k˙1+k˜2k˙2=−ηk˜1+k˜2−λk˜1sα−2δλk˜2sα+ηk˜1+k˜2≤−ηk˜1+k˜2−λsα−ηk˜1−k˜22λδsα−η,whereη>0. Noting k˜1+k˜22≥k˜12+k˜22, if s>ε, k1>km and that(60)s>φ=maxηλ1/α,η2λδ1/α.one can imply that(61)V˙2≤−ηk˜1+k˜2≤−2ηV21/2.The following inequality is obtained by using (57) and (61):(62)V˙≤−μV11/2−2ηV21/2≤−minμ,2ηV1/2.With Lemma3, it can be concluded that the following region can be reached in finite time:(63)Ω1=ss≤maxε,φ.Ifε<s<φ, the adaptive gains k1 and k2 continue to increase until s<ε.Case b. When the adaptive gaink1<km, k1 and k2 starts to increase until s<ε, the time derivative of (51) is(64)V˙=V˙1+V˙2≤−μV11/2−γk˜1s−2δγk˜2s<0,which indicates the closed-loop system is uniformly bounded when k1<km.In summary, the system states will converge to s<ε within a finite reaching time Tr.On the sliding surface, it follows from (13) and (30) that(65)x˙1=x2,x˙2=−∑i=12ci1−e−t/τixiαisignxi.Here, we introduce the vector fieldh:R2⟶R2(66)hx1,x2=h1,h2T=x˙1,x˙2T.The trajectories of function (66) are invariant with respect to the transformation(67)∀η>0,Hη:x1,x2↦η2−α2x1,ηx2.From (65)–(67) and α1=α2/2−α2, we can get(68)h1η2−α2x1,ηx2=η2−α2+α2−1h1x1,x2,h2η2−α2x1,ηx2=η1+α2−1h2x1,x2,which shows the system (66) is homogeneous of degree α2−1<0 with respect to dilation 2−α2,1. With Lemma 1 in mind, system (65) is finite-time stable. Therefore, the tracking errors, x1 and x2, converge to a small region near zero on the sliding surface within a finite time Ts.Since the reaching timeTr and the sliding time Ts are both finite, the tracking errors of LOS angle x1 and LOS angular rate x2 will converge to a small region near zero within a finite time T=Tr+Ts.


## 4.1. Design of Virtual Control Input Based on NNTSM and ASTA

Traditional TSM can only guarantee asymptotical convergence rather than finite-time convergence. Inspired by [20, 34] and [39], a new nonsingular terminal sliding mode control is introduced to improve the convergence rate, with its sliding manifold designed as follows:(30)s=x2+∑1=12∫0tci1−e−t/τixiαisignxi∙dt,where α2∈0,1, α1=α2/2−α2∈0,1, c1, c2 ensure that l2+c2l+c1=0 is Hurwitz, and their choices can be referred to [39]. In addition, τ1=τ2>0 and can be set by trial and error considering the torpedo’s maneuverability limit and the reaching time.A main drawback in the standard super-twisting algorithm is that the upper bounds of disturbance and its first-order time derivative are needed in choosing parameters. Inspired by [40, 41], an adaptive super-twisting algorithm (ASTA) is designed as follows:(31)s˙=−k1ϕ1−∫0tdt+d,ϕ1=s1/2signs,ϕ2=12signs,where k1 and k2 are adaptive gains with the nonlinear adaptation law as follows [41]:(32)k˙1=γs,k1≤km,λssigns−ε,k1>km,,k10>0,k2=2δk1,where γ, λ, ε, km, and δ are all positive constants.A smallε can improve the controller accuracy, but it will also reduce the convergence rate of s [42]. Increasing the control parameters λ, γ, and δ can shorten the convergence time and increase the control accuracy, but it also augments the acceleration magnitude. Thus, there is a trade-off needed in tuning parameters.Proposition 1.
Given the sliding variable dynamics (31) and the gain-adaptation law (32), the gains k1 and k2 are upper bound, i.e., there exists k1∗>0 and k2∗ such that(33)k˜1=k1−k1∗≤0,k˜2=k2−k2∗≤0.Sketch Proof of Proposition 1.
Since there is a positive linear correlation betweenk1 and k2, only k1 is analyzed for clarity.
If the sliding variables>ε, k1 and k2 increases gradually enough to make the sliding variable s converge to zero. Once the sliding mode is established, i.e., s<ε, the proposed gain-adaptation law (32) will keep k1 approaching km. If gains are not large enough to counteract the perturbations, resulting in s>ε, k1 will increase again making the sliding variable converge to s<ε. Therefore, adaptive parameters are always globally bounded, and a detailed proof can be referred to [42].
Taking (13), (30) and (31) into account, the virtual control command can be chosen as(34)x3c=rcosηM∑i=12ci1−e−t/τixiαisignxi−θ¨^T−2r˙q˙r+a^Tqr+k1ϕ1+∫k2ϕ2dt.
The prominent merits of the sliding manifold (30) and the ASTA can be inferred from the virtual control command (34). In the reaching phase, x3c is attenuated with 1−e−t/τi<1; wheras in the sliding phase there is t>tr, with ci1−e−t/τi approaching to ci, the convergence rate on the sliding surface will not be affected. In addition, adaptive gains k1 and k2 are gradually increasing, reducing x3c at the start of guidance.


## 4.2. Design of Autopilot Control Input via New Tracking Differentiator

Considering the autopilot lag, dynamic surface control [30] is frequently used to design the control input. From (13), the theoretical command input for the autopilot should be(35)u=τx˙3c+x3c.One significant difficulty in calculatingu is to get the first-order time derivative of x3c. In dynamic surface control, x3c is passed through a first-order filter, getting x3d and x˙3d, and the “explosion of items” problem in the backstepping control is avoided. Nevertheless, the use of a low-pass filter may introduce a significant delay as well as a magnitude attenuation. Thus x3c is not followed precisely by x3d. Even more, in the dynamic surface method, with the accuracy of x˙3c−x˙3d not guaranteed, the controller can only be designed to regulate x3 tracking x3d. Though the error between x3 and x3d can be removed by feedback control, there is still an error between x3 and x3c, resulting in reduced tracking accuracy. In some literature, TD proposed by Han [22] is used to replace the low-pass filter, yet it may lead to a phase lag in the transient period. In this paper, x3c is passed through an improved sliding mode TD [28], getting new x3d and x˙3d. The improved TD has slighter overshoot and faster response speed, which can guarantee the accuracy of x˙3c−x˙3d. Then feedback control is carried out to improve the control accuracy of x3−x3c. In this way, improved control accuracy is achieved, while the common “explosion of items” problem in the backstepping control design is avoided.The TD2 for dealing withx3c can be constructed as(36)ct=κmaxz∈t−τ0,tm0z−minz∈t−τ0,tm0zt−τ0,L3t=χm−1t+ct,vt=−λ32L31/3tm−1t2/3signm−1t+m0t−x3ct,m˙−1t=vt−βvtsignm−1t,m˙0t=−λ31L32/3tm−1t1/3signm−1t+m1t,m˙1t=−λ30L3tm−1t0signm−1t,where λ3i>0, i=0,1,2, and can be set following [27]. The c reflects the average change rate of the input, while v and m−1 are auxiliary variables. m0=x3d, m1=x˙3d are the estimates of x3c and x˙3c, respectively. Moreover, χ>0, κ>0, 0≤β<1, and τ0>0. Increasing χ contributes to the tracking response speed but will also increase the overshoot magnitude in the transient phase. A big β can reduce overshoot, but it also restrains response speed at the same time. Increase of κ can improve tracking speed and reduce overshoot, but it can also augment the effect of noise. The stability proof and parameter selection rules of the variable gain sliding mode TD are presented in [28].The actual command input for the autopilot is(37)u=τx˙3d+x3c.Denote the errory3=x3−x3c, and then substitute (13) and (34) into (37) yields(38)y˙3=x˙3−x˙3c=−y3τ+x˙3d−x˙3c.With the proven finite-time convergence of the DO1, TD1, and TD2, there must exist smallε1,ε2,ε3>0 such that(39)a^Tq−aTq<ε1,θ¨^T−θ¨T<ε2,x˙3c−x˙3d<ε3.Sincey˙3<−y3/τ+ε3, τ is small, it follows that the tracking error y3 is ultimately uniformly.Confined byε3. Thus x3 can track the virtual control command x3c precisely.Substituting (13) and (34) into (30) yields(40)s˙=x˙2+∑i=12ci1−e−t/τixiαisignxi=−2r˙q˙r−x3c+y3cosηMr+aTqr−θ¨T+∑i=12ci1−e−t/τixiαisignxi=−k1ϕ1−∫k2ϕ2dt+d,where the lumped disturbance(41)d=aTq−a^Tqr+θ¨^T−θ¨T−cosηMry3.According to (38) and (39), d should be bounded.


## 4.3. Finite-Time Convergence Analysis of the Tracking Errors

Theorem 2.
The tracking errors of the LOS anglex1 and LOS angular rate x2 will converge into a small region near zero in finite time.The proof can be divided into two steps. Firstly, we prove that the sliding variables will converge into a neighbourhood Ω0=ss<ε in finite time under the control of the adaptive super-twisting algorithm (32). Secondly, we prove that the tracking errors are finite-time convergent on the sliding surface.Inspired by [43, 44], a novel proof of this theorem under a new framework is proposed, which can be regarded as a modified version of the proof in [41].To facilitate the stability analysis, a new state vector is introduced:(42)ξT=ξ1,ξ2T=ϕ1,−∫k2ϕ2dt+dT,then combining (31) and (40) into(43)s˙=−k1ϕ1+ξ2,ξ˙2=−k2ϕ2+d˙.System (42) can be rearranged as(44)ξ˙=ϕ˙1−k1ϕ1−∫k2ϕ2dt+d−k2ϕ2+d˙=ϕ˙1−k1ϕ1−∫k2ϕ2dt+d−k2ϕ1+d˙ϕ˙1=ϕ˙1Aξ+Bg˜,where ϕ˙1=dϕ1/ds=s−1/2/2=ϕ2/ϕ1, g˜=d˙/ϕ˙1=2d˙ϕ1, B=01T, and(45)A=−k11−k20.Suppose there exists a positive constantσ>0, such that(46)d˙≤σ2=σϕ2.In [41], to deal with (46), it set d˙=κ¯ϕ2 with κ¯≤σ. However, as we know, d˙ is a variable, which means κ¯ should not be a constant. However, κ¯ frequently appears in the derivations of [41], which will cause many difficulties in algebraic manipulations.Unlike the way in [41], the ingenious way here is that we get the following inequality by combining (44) with (46):(47)g˜=2d˙⋅ϕ1≤σϕ1=σξ1.Then we define(48)ωξ,g˜=σ2ξ1Tξ1−g˜2=ξg˜TR00−1ξg˜≥0,R=σ2000.To analyze the stability of the system (44), define(49)V1=ξTPξ,V2=0.5k˜12+k˜22,where P is a positive definite symmetric matrix and the quadratic form V1=ξTPξ is positive definite and radically unbounded by the standard inequality for quadratic forms:(50)V11/2xλmax1/2P≤ξ2≤V11/2xλmin1/2P,where ξ2 is the Euclidean norm of ξ.Consider the Lyapunov function candidate(51)V=V1+V2.The positive definite symmetric matrixP is chosen as follows:(52)P=β+4δ2−2δ−2δ1,where β>0, then V is continuously derivable except the equilibrium. If the following conditions are satisfied:(53)0<σ<4δ−1,k1>4δ2−2δ+β22β4δ−σ−1+4δ2+σ2+σ2β,then the matrix Q defined as (54) is negative definite.(54)Q=ATP+PA+R+σI+PBBTP=−2k1β−4δ2−σ2−σ∗−β+4δ2+2δ4δ−σ−1<0,where ∗ is used to indicate a symmetric element. This conclusion can be easily deduced from (53) and (54) with the Schur complement.Using (50), the fact can be easily noted that(55)s1/2=ξ1≤ξ2.Case a. If the adaptive gaink1 meets condition (53), then substituting (44), (48), and (54) into (49) yields(56)V˙1=ξ˙TPξ+ξTPξ˙=ϕ˙1ξTATP+PAξ+g˜BTPξ+ξTPBg˜=ϕ˙1ξg˜TATP+PAPBBTP0ξg˜≤ϕ˙1ξg˜TATP+PAPBBTP0ξg˜+ωξ,g˜=ϕ˙1ξg˜TATP+PA+RPBBTP−1ξg˜<ϕ˙1ξg˜T−σI000ξg˜=−ϕ˙1σξTξ=−s−1/2σξTξ2.Combing (50) and (55), (56) can be written as(57)V˙1≤−μV11/2,where μ is(58)μ=σλmin1/2P2λmaxP.Substituting (32) and (33) into (49) yields(59)V˙2=k˜1k˙1+k˜2k˙2=−ηk˜1+k˜2−λk˜1sα−2δλk˜2sα+ηk˜1+k˜2≤−ηk˜1+k˜2−λsα−ηk˜1−k˜22λδsα−η,whereη>0. Noting k˜1+k˜22≥k˜12+k˜22, if s>ε, k1>km and that(60)s>φ=maxηλ1/α,η2λδ1/α.one can imply that(61)V˙2≤−ηk˜1+k˜2≤−2ηV21/2.The following inequality is obtained by using (57) and (61):(62)V˙≤−μV11/2−2ηV21/2≤−minμ,2ηV1/2.With Lemma3, it can be concluded that the following region can be reached in finite time:(63)Ω1=ss≤maxε,φ.Ifε<s<φ, the adaptive gains k1 and k2 continue to increase until s<ε.Case b. When the adaptive gaink1<km, k1 and k2 starts to increase until s<ε, the time derivative of (51) is(64)V˙=V˙1+V˙2≤−μV11/2−γk˜1s−2δγk˜2s<0,which indicates the closed-loop system is uniformly bounded when k1<km.In summary, the system states will converge to s<ε within a finite reaching time Tr.On the sliding surface, it follows from (13) and (30) that(65)x˙1=x2,x˙2=−∑i=12ci1−e−t/τixiαisignxi.Here, we introduce the vector fieldh:R2⟶R2(66)hx1,x2=h1,h2T=x˙1,x˙2T.The trajectories of function (66) are invariant with respect to the transformation(67)∀η>0,Hη:x1,x2↦η2−α2x1,ηx2.From (65)–(67) and α1=α2/2−α2, we can get(68)h1η2−α2x1,ηx2=η2−α2+α2−1h1x1,x2,h2η2−α2x1,ηx2=η1+α2−1h2x1,x2,which shows the system (66) is homogeneous of degree α2−1<0 with respect to dilation 2−α2,1. With Lemma 1 in mind, system (65) is finite-time stable. Therefore, the tracking errors, x1 and x2, converge to a small region near zero on the sliding surface within a finite time Ts.Since the reaching timeTr and the sliding time Ts are both finite, the tracking errors of LOS angle x1 and LOS angular rate x2 will converge to a small region near zero within a finite time T=Tr+Ts.


## 5. Simulation Results

In this section, some numerical simulations are performed to illustrate the performance of the proposed composite guidance law, TD and DO. The simulations are performed using the implicit Euler discretization with a fixed step size of 0.01 s for updating the system states. Suppose the ranger, LOS angle q, and noisy target heading angle θ˜T can be provided by the electric-field detection device.The target is a kind of large warship with a turning angular rate less than2°/s. It should be pointed out that the interception with an impact angle can only be realized under some initial conditions of engagement. Moreover, large initial heading errors may result in a target missed or a predetermined impact angle not being obtained. This conclusion is consistent with practical application, for the fact mentioned above in the introduction that the SCT has limited maneuverability to maintain super-cavity stability. To achieve the desired impact angle, the torpedo launch platform needs to change the initial engagement by prelaunch maneuver.For brevity, only one kind of typical initial engagement condition is considered here. In the inertial coordinate system, the initial positions of the torpedo and the target are0,0 and (960, 720), with the initial relative range r0=1200m. The constant velocities for the torpedo and the target are VM=100m/s and VT=15m/s. The initial heading angles of the torpedo and the target are θM0=10° and θT0=0°. The limited maneuverability of a torpedo can be described by θ˙M≤12° or aM≤21m/s2.Two intercepting scenarios with different escaping maneuvers are considered. In Case1, the target is executing a constant maneuver. In Case 2, the target is assumed to execute a sinusoidal maneuver [42]. Several critical parameters need to be tuned, and the regulating rules of these parameters are discussed under each component’s description. Simulation parameters for the proposed guidance law are listed in Table 1.Table 1
Simulation parameters in guidance law.
ParametersBelongs toCase1Case2L1TD10.010.01χTD2600600κTD222τ0TD20.5 s0.5 sβTD20.50.5L2DO10.61.2c1NNTSM0.050.045τAutopilot0.05 s0.05 sc2NNTSM0.150.15τ1, τ2NNTSM1.6 s1.6 sγASTA55λASTA11kmASTA0.150.15εASTA0.10.1δASTA0.050.05Additionally, the positive coefficientsλ1i, λ2i, and λ3i can be set following the recursive sequences in [26], which are partly listed in Table 2.Table 2
Recursive sequences of 1st order and 3rd order.
Orderλ0λ1λ2λ3Component1st order1.11.52nd order1.12.122λ2i in DO1, λ3i in TD23rd order1.13.064.163λ1i in TD1Case 1.
Constant Maneuvering Targets
According to the actual maneuverability of the target warship, we assume that the target is executing counter-clockwise rotation maneuvers with an angular rate ofω=0.04rad/s.
The impact angle constraint for SCT is not for a “hit to kill” interception to improve warhead effectiveness [14, 18] but for making use of the target scale to increase hit probability, given that a larger impact angle corresponds to a larger interception width. Simulations are performed with two different desired impact angles, −25° and −45°, and the results are shown in Figure 4.
The interception geometries are shown in Figure4(a). Under the control law (37), the interceptions of a constant maneuvering target at different impact angles, −25° and −45°, can be achieved. Different corresponding lateral acceleration profiles are shown in Figure 4(b), indicating that the impact angle of −45° requires a larger magnitude of acceleration. That is the fact that with different desired impact angles, various tracking errors are to be eliminated, and a more significant initial LOS angle error demands a more considerable acceleration. As shown in Figures 4(c) and 4(d), both the LOS angle error x1 and the LOS angular rate error x2 converge to zero in finite time, and larger initial errors bring about longer convergence time. In addition, the interception with the impact angle of −45°, of which the initial tracking errors are more prominent, takes a little more attacking time.
As shown in Figure4(e), the TD-N (TD2) for processing the virtual control command x3c is compared with traditional TDs, with the impact angle selected as −45°. The TD-N can acquire the best tracking result, while a time delay can be found in the tracking responses of both TD-L and TD-H during the initial transient. This deficiency will bring about the reduction of control accuracy in the design of autopilot control input, as discussed in part C of Section 5.
The proposed DO1 is compared with ESO reported in [22] under the impact angle of −45°. From Figure 4(f), the two disturbance observers can all estimate the aTq precisely after an initial transient period. The transient time of proposed DO1 is approximately the same as that of ESO. However, the estimation result via ESO has a larger overshoot, which is undesirable and may bring about the saturation of autopilot control input. What’s more, there are several parameters that need to be prudently tuned in ESO [45], making it more difficult in application.Figure 4
Simulation results in Case1. (a) Interception trajectories. (b) Acceleration profiles of torpedo. (c) LOS angle error. (d) LOS angular rate error. (e) Tracking of the virtual control command. (f) Estimate of target lateral acceleration.
(a)(b)(c)(d)(e)(f)Case 2.
Weaving Maneuver Target
It is assumed that the target weaving acceleration is chosen asaTq=0.6sinπt/10m/s2.
The traditional discontinuous sliding mode control guidance law (TDSMG) and the super-twisting sliding mode control guidance law (STSMG) are also performed in the simulation for comparison.
In TDSMG, the adaptive super-twisting algorithm is substituted by the reaching law of constant plus proportional rate strategy, while the same DO1 for estimating target lateral acceleration and the same TD2 for processing the virtual control command are still adopted. As a result, the virtual control command changes into(69)x3c′=rcosηM∑i=12ci1−e−t/τixiαisignxi−θ¨^T−2r˙q˙r+a^Tqr+k3s+k4⋅signs.
In STSMG, the standard super-twisting algorithm and nonlinear integral sliding manifold in [21] are employed, with the virtual control command chosen as(70)x3c″=rcosηM∑i=12cixiαisignxi−θ¨^T−2r˙q˙r+a^Tqr+k5ϕ1+∫k6ϕ2dt.
Here, we set the fixed parametersk5=0.2 and k6=0.01 with the assumption of known upper bounds of d and d˙.
The simulation results are shown in Figure5, with the desired impact angle selected as −40°. Figure 5(a) shows that the torpedo hits the target with desired impact angle under different guidance laws. All three guidance laws can ensure a finite-time convergence of the sliding variable in reaching phase, which is shown in Figure 5(b). It can be inferred from Figures 5(c) and 5(d) that, similar to Case 1, the LOS angle error x1 and the LOS angular rate error x2 converge to zero in finite time. As can be seen from Figure 5(e), compared with TDSMG, the proposed guidance law and STSMG can eliminate the chattering caused by discontinuous terms in the sliding phase. Furthermore, compared with STSMG, the proposed guidance law can avoid the undesirable extremely large acceleration at the initial time, which is beyond the maneuverability limit of the SCT. By comparing the virtual control command (34) and (70), one can observe that x3c is attenuated due to 1−e−t/τi<1 and small adaptive gains k1, k2 at the beginning. Figure 5(f) shows the variation profiles of adaptive gains k1, k2, which gradually increase in the reaching phase and decrease in the sliding phase.
The performance of TD1, which is used to process the measured heading angle of the target, is evaluated under the following noisy measurement signal:(71)θ˜T=θT+0.03δ,where δ∼N0,1 is the unit Gaussian white noise. On the one hand, the TD1 acts as a filter, reconstructing a noiseless signal from the measurement signal, as shown in Figure 6(a). On the other hand, the TD1 works as a differentiator, acquiring the first-order and second-order derivatives of the measurement signal. As shown in Figures 6(b) and 6(c), the estimations are close to the theoretical ones, which meets the need for exact θT, θ˙T, and the unknown external disturbance θ¨T.Figure 5
Simulation results in Case2. (a) Trajectories of target and torpedo. (b) Occurrence of sliding mode. (c) LOS angle error. (d) LOS angular rate error. (e) Acceleration of torpedo. (f) Change of adaptive gains.
(a)(b)(c)(d)(e)(f)Figure 6
Output of TD1 with the noisy heading angle of the target. (a) Target heading angle with noise and its reconstruction. (b) 1st-order derivative estimate of target heading angle. (c) 2nd-order derivative estimate of target heading angle.
(a)(b)(c)


## 6. Conclusions

Provided suitable parameters choice, the high-order sliding mode TD1 has a good denoising effect on the measurement signalθ˜T, which meets the need for exact θT, θ˙T, and θ¨T. The novel DO1 proposed in this article improves the estimation of target lateral acceleration in real time. With the help of TD1 and the proposed DO1, the problem of insufficient target maneuver information in guidance law design is solved. A novel composite guidance law with finite-time convergence and impact angle constraint is presented based on a new nonsingular terminal sliding manifold and adaptive super-twisting algorithm. A new TD2 is applied for dynamic surface control, replacing the first-order integral filter, and high-accuracy tracking of the virtual control command can be realized.Simulation results verify that, under the proposed guidance law, an SCT can achieve interception with predetermined impact angle constraints under suitable initial conditions, ensuring the LOS angle error and the LOS angular rate error converge to zero in finite time. In addition, utilizing the revised sliding manifold and the adaptive super-twisting algorithm, the control input of autopilot is significantly reduced, and the chattering phenomenon is eliminated, making it more suitable for practical applications.One shortcoming of the proposed guidance law is that there is a short time for the TDs and DO to be stable, which makes it disabled when the target is found at a too-close distance and the time for interception is extremely short. Therefore, a larger detection range of SCT, which is becoming a reality with the development of electric-field detection technology, will benefit its performance at the terminal phase. Moreover, the capture region of the proposed guidance law, which refers to the initial engagement conditions that make the interception occur, has yet to be studied. Only with a known capture region can the launch platform occupy a favorable position by prelaunch maneuver. Thus, a theoretical analysis should be made on the initial conditions for better application of the new guidance law in future research.



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*Source: 6508811-2022-12-28.xml*