Composite Terminal Guidance Law for Supercavitating Torpedoes with Impact Angle Constraints

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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. Te latest researches indicate that a new kind of guidance based on the underwater electric feld of target warships [1,2] can be adopted to enhance SCT combat capability. By measuring the three components of the electric feld, an SCT can obtain the target's location, velocity, and heading angle with a detection range of more than 1 km. Te 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 difcult for SCT. Compared with traditional torpedoes, there are some new peculiarities of guided SCT: a signifcant increase in speed due to the supercavitation drag reduction efect, 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 classifed into three main categories, i.e., modifed 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 specifed 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. Te additional time-varying bias term compensates for target accelerations, sensor noises, and impact angle. Te 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-controlbased guidance law and a linear-quadratic diferential-gamebased guidance law are derived, which enable imposing predetermined impact angles on maneuvering targets with signifcant initial heading errors. Closed-form solutions of optimal impact-angle-control guidance for a frst-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 diferential-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 frst-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]. Tis 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 frst-order derivate, named the LOS angular rate error. In some literature, it is assumed that the fnal heading angle of the target can be acquired at the beginning [7] or that the desired LOS angle is fxed [15]. Tis 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 fnite-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 fnite time; meanwhile, in the sliding phase, the tracking errors achieve fnite-time convergence on the sliding surface. Te terminal sliding mode control (TSMC), with the sliding variable as a nonlinear function of tracking errors, can improve the performance hugely. Guidance law with fnite-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 sufers 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 fnite-time stable strategy. On the other hand, diferent ways have been taken to regulate the switching surfaces for a fnite reaching time, such as using adaptive sliding mode control [10], adopting the method of smooth second-order SMC [13,14], and defning a switching control command [12,[15][16][17][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 flters produce phase delays and amplitude distortions, whereas signal processing using a tracking diferentiator (TD) can avoid these defciencies. Te 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 diferent felds, such as constructing a transient profle according to the actuator's capacity [24]. However, only estimates of the input and its frst-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][26][27]. Nevertheless, these TDs, both the conventional version [25,26] and its improved version (TD-L) [27], all sufer from signifcant 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 signifcantly infuence guidance accuracy, especially in the presence of target maneuvers [12]. In some literature, the autopilot dynamic is treated as ideal, introducing signifcant control biases [29]. Te traditional design methodology for handling autopilot lag is the backstepping control, which may lead to the "explosion of items" problem. Te 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 frst-order flter, 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. Te 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 SCTcan 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 fnite-time stability is proved using homogeneous theory. Te 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. Te fnite-time convergence analysis of the tracking errors is carried out by Lyapunov theory under a novel framework. (4) To efectively account for the autopilot lag of the SCT, the variable gain sliding mode TD [27] is used to calculate the frst-order time derivative of the virtual control command, which improves the tracking accuracy and avoids the "explosion of items" problem at the same time.
Te rest of this paper is arranged as follows. In Section 2, the engagement dynamics are derived. In Section 3, a new sliding mode tracking diferentiator 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 ofered.

Problem Formulation
Te planar engagement and the relative motion dynamics are derived based on the vector derivative algorithm rather than diferentiating equations [15]. Te relationship between the impact angle and the LOS angle is derived, and the mistake in treating the desired LOS angle as a fxed 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 by T and M, respectively.
In the LOS coordinate system as shown in Figure 1, the origin is fxed at the torpedo, r represents the relative range of torpedo-to-target, θ M and θ T are the heading angles of the torpedo and the target, and q is the LOS angle. Moreover, V M and V T , respectively, denote the velocities of the torpedo and the target, both of which are assumed to be constant. Te a M and a T , respectively, refer to the accelerations of the torpedo and the target, which are perpendicular to their velocity directions. [16]. In the coordinate system, subscripts r and q represent the radial direction along the LOS and the axial direction normal to the LOS. Te base vector along the LOS is denoted by e r , while the one normal to the LOS is aligned with e r rotating π/2 counter-clockwise, denoted by e q .
Te derivatives of the base vectors are d dt where _ q refers to the LOS angular rate. It is evident from (1) that the velocity vector is Defning V r as the relative velocity between torpedo and target along the LOS, and V q as the one normal to the LOS, (2) can be rewritten as follows:

Mathematical Problems in Engineering
Identically, diferentiating (2) with respect to time yields Similarly, defning a r as the relative acceleration of the torpedo target along the LOS, and a q as the one normal to the LOS, (4) can be rewritten as follows: With a transposition, (5) can be transformed to Combining (3) and (6), one can imply that Te autopilot can be approximately described by the following frst-order dynamics [12,15]: where τ is the autopilot lag, and u is the acceleration command for the autopilot. As shown in Figure 2, 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., Te relative velocity component should be nullifed to form a collision triangle [6,7].
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: where the target-to-torpedo velocity ratio is It is evident that q f and θ T are linear with only one constant bias. For a maneuvering target, θ T is a time-varying quantity, and so is the q f . In practice, the terminal heading angle of the maneuvering target is generally unavailable at the beginning of guidance, implying that the q f 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 q f simultaneously. Furthermore, with θ imp and ] 1 predetermined, on diferentiating (11), one can imply that It can be seen from the above analysis that, to achieve the impact angle constraints, the frst-order and second-order time derivatives of the target heading angle, _ θ T and € θ T , are needed. Or else, the target lateral acceleration a Tq as well as its frst-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. Te frst method is to get the _ θ T and € θ T using a second-order TD, and the other way is to estimate the a Tq and _ a Tq based on a third-order DO. Here, the frst method is adopted in this paper. Let It follows from (6), (8), and (12) that In designing a guidance law with an impact angle constraint, the unknown external disturbance includes the target lateral acceleration a Tq , as well as the second-order time derivative of the target heading angle € θ T . Te design objective of control input u is to drive x 1 and x 2 to zero in fnite time, that is,

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 fnite-time stability of the proposed DO is proved using homogeneous degree theory. Firstly, some conclusions are recalled, which are indispensable in subsequent analysis.

Essential Preliminary Concepts
Lemma 1 (see [34]). Defne a smooth vector feld and the dilation where x 1 , · · · , x n are suitable coordinates on R n and r 1 , · · · , r n are positive real numbers. Te vector feld (14) is homogeneous of degree m ∈ R, if ∀η > 0, i � 1, 2, · · · , n, f i is a homogeneous function with respect to dilation (15) Suppose the origin is a stable equilibrium of f(x), if its degree of homogeneity, m⩾ − max r i , is negative, then the origin is fnite-time stable.

Tracking Diferentiator for Processing Measured Target Heading Angle
Lemma 3 (see [35]). Suppose the unknown function f 0 (t), t ≥ 0, has a noisy approximation f(t) � f 0 (t) + n(t) + n c (t) available in real time, where the derivative f (k) 0 (t) exists, and there is a known Lipschitz constant L > 0. Te frst noise term n(t) is Lebesgue-measurable and bounded by ∃δ > 0, |n(t)| < δ, and the second noise term n c (t) is integrable.
Te kth-order tracking diferentiator proposed is where sign(·) is signum function. λ i > 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 w i , i � 0, 1, · · · , k, is the estimate of f (i) 0 (t), with the estimation errors converging in fnite time. Increasing L contributes to the convergence rate, but it also degrades the noise-attenuating performance, so there is a trade-of between the convergence rate and the convergence precision.
A second-order TD for processing noisy measured target heading angle, denoted by θ T , is designed with Lemma 2 using TD (18): With the proper parameters selected, we have

Proposal of New Disturbance Observer and Its Application in Estimating Target Lateral Acceleration.
Consider the following frst-order SISO dynamics: where x is the system state, u 1 is the control input and is Lebesgue-measurable, and d is the lumped disturbance and is sufciently smooth. Adapted from TD (18), the proposed DO is where λ i > 0, i � 0, 1, · · · , k + 1, L > |d (k) |.Te z −1 is an auxiliary variable, while z 0 is the estimate of x, z i is the estimate of d (i− 1) , i � 1, · · · , k, i.e., Theorem 1. Te estimation errors of (22) converge in fnite time.
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 the a Tq based on (7) using DO (22): With a proper choice of the parameters L 2 and λ 2i , i � 0, 1, 2, z 1 approaches to a Tq in a short, fnite time. It should be noted that, in some literature, the DO for estimating a Tq is based on (6), with the item a Tq /r estimated. However, in this way, extensive tracking range is to be handled for a Tq /r change with r dramatically as the torpedo approaches the target, and difculties 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].

Composite Sliding Mode Guidance Law with Finite-Time Convergence
Firstly, a new nonsingular terminal sliding manifold (NNTSM) is designed. Ten, an adaptive super-twisting algorithm (ASTA) to deal with the unknown upper bounded disturbance is designed. Finally, using the variable gain sliding mode tracking diferentiator (TD-N), the control input for autopilot with a frst-order dynamic is obtained. Te TD (19) designed for processing noisy target heading angle is referred to as TD1. Te DO (29) designed for estimating the target lateral acceleration is referred to as DO1. Te structure of the guidance law is shown in Figure 3.

Design of Virtual Control Input Based on NNTSM and
ASTA. Traditional TSM can only guarantee asymptotical convergence rather than fnite-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: where α 2 ∈ (0, 1), α 1 � α 2 /(2 − α 2 ) ∈ (0, 1), c 1 , c 2 ensure that l 2 + c 2 l + c 1 � 0 is Hurwitz, and their choices can be referred to [39]. In addition, τ 1 � τ 2 > 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 frstorder time derivative are needed in choosing parameters. Inspired by [40,41], an adaptive super-twisting algorithm (ASTA) is designed as follows: where k 1 and k 2 are adaptive gains with the nonlinear adaptation law as follows [41]: where c, λ, ε, k m , and δ are all positive constants. A small ε can improve the controller accuracy, but it will also reduce the convergence rate of s [42]. Increasing the control parameters λ, c, and δ can shorten the convergence time and increase the control accuracy, but it also augments the acceleration magnitude. Tus, there is a trade-of needed in tuning parameters. (32), the gains k 1 and k 2 are upper bound, i.e., there exists k * 1 > 0 and k * 2 such that

Proposition 1. Given the sliding variable dynamics (31) and the gain-adaptation law
Sketch Proof of Proposition 1. Since there is a positive linear correlation between k 1 and k 2 , only k 1 is analyzed for clarity. If the sliding variable |s| > ε, k 1 and k 2 increases gradually enough to make the sliding variable s converge to zero. Once the sliding mode is established, i.e., |s| < ε, the proposed gainadaptation law (32) will keep k 1 approaching k m . If gains are not large enough to counteract the perturbations, resulting in |s| > ε, k 1 will increase again making the sliding variable converge to |s| < ε. Terefore, 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 Te prominent merits of the sliding manifold (30) and the ASTA can be inferred from the virtual control command (34). In the reaching phase, x 3c is attenuated with 1 − e − t/τ i < 1; wheras in the sliding phase there is t > t r , with Mathematical Problems in Engineering c i (1 − e − t/τ i ) approaching to c i , the convergence rate on the sliding surface will not be afected. In addition, adaptive gains k 1 and k 2 are gradually increasing, reducing x 3c at the start of guidance.

Design of Autopilot Control Input via New Tracking
Diferentiator. 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 One signifcant difculty in calculating u is to get the frst-order time derivative of x 3c . In dynamic surface control, x 3c is passed through a frst-order flter, getting x 3d and _ x 3d , and the "explosion of items" problem in the backstepping control is avoided. Nevertheless, the use of a low-pass flter may introduce a signifcant delay as well as a magnitude attenuation. Tus x 3c is not followed precisely by x 3d . 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 x 3 tracking x 3d . Tough the error between x 3 and x 3d can be removed by feedback control, there is still an error between x 3 and x 3c , resulting in reduced tracking accuracy. In some literature, TD proposed by Han [22] is used to replace the low-pass flter, yet it may lead to a phase lag in the transient period. In this paper, x 3c is passed through an improved sliding mode TD [28], getting new x 3d and _ x 3d . Te improved TD has slighter overshoot and faster response speed, which can guarantee the accuracy of _ x 3c − _ x 3d . Ten feedback control is carried out to improve the control accuracy of x 3 − x 3c . In this way, improved control accuracy is achieved, while the common "explosion of items" problem in the backstepping control design is avoided.
Te TD2 for dealing with x 3c can be constructed as where λ 3i > 0, i � 0, 1, 2, and can be set following [27]. Te c refects the average change rate of the input, while v and m −1 are auxiliary variables. m 0 � x 3d , m 1 � _ x 3d are the estimates of x 3c and _ x 3c , respectively. Moreover, χ > 0, κ > 0, 0 ≤ β < 1, and τ 0 > 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 efect of noise. Te stability proof and parameter selection rules of the variable gain sliding mode TD are presented in [28].
Confned by ε 3 . Tus x 3 can track the virtual control command x 3c precisely.

Finite-Time Convergence Analysis of the Tracking Errors
Theorem 2. Te tracking errors of the LOS angle x 1 and LOS angular rate x 2 will converge into a small region near zero in fnite time.
Te proof can be divided into two steps. Firstly, we prove that the sliding variable s will converge into a neighbourhood Ω 0 � s‖|s| < ε { } in fnite time under the control of the adaptive super-twisting algorithm (32). Secondly, we prove that the tracking errors are fnite-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 modifed version of the proof in [41]. 8 Mathematical Problems in Engineering To facilitate the stability analysis, a new state vector is introduced: (42) then combining (31) and (40) into System (42) can be rearranged as where _ Suppose there exists a positive constant σ > 0, such that In [41], to deal with (46), it set _ d � κϕ 2 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 difculties in algebraic manipulations.
Unlike the way in [41], the ingenious way here is that we get the following inequality by combining (44) with (46): Ten we defne To analyze the stability of the system (44), defne where P is a positive defnite symmetric matrix and the quadratic form V 1 � ξ T Pξ is positive defnite and radically unbounded by the standard inequality for quadratic forms: where ‖ξ‖ 2 is the Euclidean norm of ξ.
Consider the Lyapunov function candidate Te positive defnite symmetric matrix P is chosen as follows: where β > 0, then V is continuously derivable except the equilibrium. If the following conditions are satisfed: then the matrix Q defned as (54) is negative defnite.
where * is used to indicate a symmetric element. Tis conclusion can be easily deduced from (53) and (54) with the Schur complement. Using (50), the fact can be easily noted that Case a. If the adaptive gain k 1 meets condition (53), then substituting (44), (48), and (54) into (49) yields
Case b. When the adaptive gain k 1 < k m , k 1 and k 2 starts to increase until |s| < ε, the time derivative of (51) is which indicates the closed-loop system is uniformly bounded when k 1 < k m .
In summary, the system state s will converge to |s| < ε within a fnite reaching time T r .
On the sliding surface, it follows from (13) and (30) that Here, we introduce the vector feld h: Te trajectories of function (66) are invariant with respect to the transformation From (65)-(67) and α 1 � α 2 /(2 − α 2 ), we can get which shows the system (66) is homogeneous of degree α 2 − 1 < 0 with respect to dilation (2 − α 2 , 1). With Lemma 1 in mind, system (65) is fnite-time stable. Terefore, the tracking errors, x 1 and x 2 , converge to a small region near zero on the sliding surface within a fnite time T s . Since the reaching time T r and the sliding time T s are both fnite, the tracking errors of LOS angle x 1 and LOS angular rate x 2 will converge to a small region near zero within a fnite time T � T r + T s .

Simulation Results
In this section, some numerical simulations are performed to illustrate the performance of the proposed composite guidance law, TD and DO. Te simulations are performed using the implicit Euler discretization with a fxed step size of 0.01 s for updating the system states. Suppose the range r, LOS angle q, and noisy target heading angle θ T can be provided by the electric-feld detection device.
Te target is a kind of large warship with a turning angular rate less than 2°/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. Tis 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 are (0, 0) and (960, 720), with the initial relative range r(0) � 1200 m. Te constant velocities for the torpedo and the target are V M � 100 m/s and V T � 15 m/s. Te initial heading angles of the torpedo and the target are θ M (0) � 10°a nd θ T (0) � 0°. Te limited maneuverability of a torpedo can be described by _ θ M ≤ 12°or a M ≤ 21m/s 2 . Two intercepting scenarios with diferent escaping maneuvers are considered. In Case 1, 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.
Additionally, the positive coefcients λ 1i , λ 2i , and λ 3i can be set following the recursive sequences in [26], which are partly listed in Table 2.

Case 1. Constant Maneuvering Targets
According to the actual maneuverability of the target warship, we assume that the target is executing counterclockwise rotation maneuvers with an angular rate of ω � 0.04rad/s. Te impact angle constraint for SCT is not for a "hit to kill" interception to improve warhead efectiveness [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 diferent desired impact angles, −25°and −45°, and the results are shown in Figure 4.
Te interception geometries are shown in Figure 4(a). Under the control law (37), the interceptions of a constant maneuvering target at diferent impact angles, −25°and −45°, can be achieved. Diferent corresponding lateral acceleration profles are shown in Figure 4(b), indicating that the impact angle of −45°requires a larger magnitude of acceleration. Tat is the fact that with diferent desired impact angles, various tracking errors are to be eliminated, and a more signifcant initial LOS angle error demands a more considerable acceleration. As shown in Figures 4(c) and 4(d), both the LOS angle error x 1 and the LOS angular rate error x 2 converge to zero in fnite 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 Figure 4(e), the TD-N (TD2) for processing the virtual control command x 3c is compared with traditional TDs, with the impact angle selected as −45°. Te 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. Tis defciency will bring about the reduction of control accuracy in the design of autopilot control input, as discussed in part C of Section 5.
Te 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 a Tq precisely after an initial transient period. Te 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 difcult in application.

Case 2. Weaving Maneuver Target
It is assumed that the target weaving acceleration is chosen as a Tq � 0.6 sin (πt/10)m/s 2 .  Te 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 In STSMG, the standard super-twisting algorithm and nonlinear integral sliding manifold in [21] are employed, with the virtual control command chosen as Here, we set the fxed parameters k 5 � 0.2 and k 6 � 0.01 with the assumption of known upper bounds of d and _ d. Te simulation results are shown in Figure 5, with the desired impact angle selected as −40°. Figure 5(a) shows that the torpedo hits the target with desired impact angle under diferent guidance laws. All three guidance laws can ensure a fnite-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 x 1 and the LOS angular rate error x 2 converge to zero in fnite 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 x 3c is attenuated due to 1 − e − t/τ i < 1 and small adaptive gains k 1 , k 2 at the beginning. Figure 5(f ) shows the variation profles of adaptive gains k 1 , k 2 , which gradually increase in the reaching phase and decrease in the sliding phase.
Te performance of TD1, which is used to process the measured heading angle of the target, is evaluated under the following noisy measurement signal: where δ ∼ N(0, 1) is the unit Gaussian white noise. On the one hand, the TD1 acts as a flter, reconstructing a noiseless signal from the measurement signal, as shown in Figure 6(a). On the other hand, the TD1 works as a diferentiator, acquiring the frst-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 θ T , _ θ T , and the unknown external disturbance € θ T .

. Conclusions
Provided suitable parameters choice, the high-order sliding mode TD1 has a good denoising efect on the measurement signal θ T , which meets the need for exact θ T , _ θ T , and € θ T . Te 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 insufcient target maneuver information in guidance law design is solved. A novel composite guidance law with fnite-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 frst-order integral flter, 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 fnite time. In addition, utilizing the revised sliding manifold and the adaptive super-twisting algorithm, the control input of autopilot is signifcantly 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. Terefore, a larger detection range of SCT, which is becoming a reality with the development of electric-feld detection technology, will beneft 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. Tus, a theoretical analysis should be made on the initial conditions for better application of the new guidance law in future research.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.