Variable gain inversion method for intercepting maneuvering targets considering attack angle constraints

Abstract

This invention relates to a variable-gain inversion-based maneuvering target interception guidance method considering attack angle constraints, belonging to the field of aircraft guidance and control technology. The method involves considering the necessary conditions for successful interception and terminal attack angle constraints. Based on trajectory shaping, a polynomial function representing the line-of-sight angle reference profile with the distance already flown as the independent variable is constructed to reduce initial and terminal constraints. This profile utilizes only the relative information between the missile and the target, without relying on the target's motion information, thus making it suitable for intercepting targets with various motion patterns. To accurately track the constructed line-of-sight angle reference profile, considering the adverse effects of aircraft control input saturation constraints and target maneuvering interference on the guidance system, an input saturation compensator and a finite-time interference estimator are introduced. Based on the inversion control architecture, a variable-gain interception guidance law is designed, enabling the aircraft to accurately track the profile while intercepting the maneuvering target at the desired angle. This invention has the advantages of low required overload, high interception accuracy, and good robustness.

Description

Technical Field

[0001] This invention relates to a variable gain inversion method for intercepting maneuvering targets that takes into account the constraints of the attack angle, and belongs to the field of aircraft guidance and control technology. Background Technology

[0002] In recent years, with the development of guidance technology and demand, incoming targets have become increasingly maneuverable, requiring interception guidance laws to maximize aircraft interception effectiveness while achieving zero misses. Therefore, advanced guidance laws constrained by attack angles have received increasing attention. Attack angle-constrained guidance laws enhance the lethality of aircraft by guiding them to strike the target's weak points at a specific angle, making them an effective means of increasing the probability of intercepting one-on-one maneuvering targets.

[0003] For the attack angle-constrained guidance problem, existing methods include proportional guidance, optimal control theory, and sliding mode control. However, these methods generally suffer from drawbacks such as requiring large overloads, dependence on linear or small angle assumptions, and discontinuous guidance commands, significantly impacting their practical value. In addition to the aforementioned methods, a novel guidance method combining polynomial trajectory shaping technology has also been used to address the attack angle-constrained guidance problem. This type of method eliminates the need for linear or small angle assumptions, constraining the aircraft's flight trajectory by constructing a polynomial function containing fractional and integer polynomials to meet constraints such as the attack angle. However, existing trajectory shaping guidance methods are generally based on assumptions about stationary or non-maneuvering targets and do not consider the saturation constraints of the guidance system's control input caused by the limitations of the actual aircraft's actuators. This makes it difficult to guarantee their interception effectiveness against maneuvering targets and prevents direct application to interception scenarios involving maneuvering targets. Summary of the Invention

[0004] To address the low success rate of intercepting maneuvering targets at the desired angle, this invention aims to provide a variable-gain inversion-based maneuvering target interception guidance method that considers attack angle constraints. Based on trajectory shaping, a line-of-sight (LAS) reference profile is constructed to reduce constraints such as the attack angle, providing the aircraft with a reference trajectory that guarantees successful interception. Considering the saturation constraints of the aircraft's control input, a variable-gain interception guidance law is designed based on an inversion control architecture, enabling the aircraft to accurately track the LAS reference profile while intercepting the maneuvering target. This invention employs trajectory shaping and variable-gain inversion control to achieve target interception guidance, offering advantages such as low required overload, high interception accuracy, and good robustness.

[0005] The objective of this invention is achieved through the following technical solution:

[0006] This invention discloses a variable-gain inversion-based maneuvering target interception guidance method that considers attack angle constraints. Addressing the low success rate of intercepting maneuvering incoming targets at the desired angle, this method considers the necessary conditions for successful interception and terminal attack angle constraints. Based on trajectory shaping, it constructs a line-of-sight angle reference profile using a polynomial function with the distance already flown as the independent variable, thereby reducing initial and terminal constraints. This profile utilizes only the relative information between the missile and the target, without relying on the target's motion information, making it suitable for intercepting targets with various motion patterns. To accurately track the constructed line-of-sight angle reference profile, considering the adverse effects of aircraft control input saturation constraints and target maneuvering interference on the guidance system, an input saturation compensator and a finite-time interference estimator are introduced. Based on the inversion control architecture, a variable-gain interception guidance law is designed, enabling the aircraft to accurately track the profile while successfully intercepting the maneuvering target at the desired angle.

[0007] The variable gain inversion maneuvering target interception guidance method considering attack angle constraints disclosed in this invention includes the following steps:

[0008] Step 1: Considering the kinematic and dynamic characteristics of the aircraft and target in a two-dimensional plane, establish dynamic models of the aircraft and target, and a relative line-of-sight angle dynamic model. Analyze and construct the constraints of the interception mission, and construct the interception and guidance problem for maneuvering targets. The constraints include initial constraints, terminal constraints, and control input saturation constraints.

[0009] Considering an inertial coordinate system xoy in a two-dimensional plane, the initial positions of the aircraft and the maneuvering target are (x... M ,y M ) and (x T ,y T The flight speed is constant, denoted as V. M and V T And satisfy V M >V T The initial trajectory inclination angles were γ. M and γ T The magnitude of the acceleration is a M and a T The direction is perpendicular to the velocity direction; the line-of-sight distance between the aircraft and the target is r, and the line-of-sight angle is λ. All angles are defined as positive counterclockwise.

[0010] The dynamic models of the aircraft and the target are established as follows:

[0011]

[0012] The relative dynamic model of the aircraft and the target is established as follows:

[0013]

[0014] Where, ηM and η T The velocity lead angles of the aircraft and the target are respectively calculated using the following formula:

[0015]

[0016] Differentiating equation (2), the line-of-sight angle dynamic model of relative motion is established as follows:

[0017]

[0018] The objective of the interception guidance problem under consideration is to successfully intercept a maneuvering target under a specified attack angle constraint. Therefore, the relative distance and line-of-sight angle need to satisfy the initial and terminal constraints shown in the following equation.

[0019]

[0020]

[0021] In this context, the subscripts 0 and f represent the initial and terminal states, respectively.

[0022] Considering the limited capabilities of the aircraft's actuators, the guidance command u needs to satisfy the control input saturation constraint shown in the following equation.

[0023]

[0024] Among them, u max This indicates the maximum allowed value for the control input, and sign(·) is the sign function.

[0025] Equations (1), (4), and (5) to (7) are the problem of intercepting and guiding maneuvering targets considering the attack angle constraint.

[0026] Step 2: Consider the initial and terminal constraints of the aircraft intercepting the maneuvering target in Step 1. Use a polynomial function with the distance already flown as the independent variable to characterize the line-of-sight reference profile, and transform the maneuvering target interception and guidance problem into a profile tracking problem.

[0027] Define the distance already flown r e The new independent variable is calculated as follows:

[0028] r e =r0-r (8)

[0029] The initial and terminal constraints in equations (5) and (6) are re-expressed as follows:

[0030]

[0031] Use the line-of-sight reference profile λ dConstruct it into a cubic polynomial as shown in the following equation:

[0032]

[0033] Combining equations (9) and (10), the calculation expressions for each coefficient are obtained as follows:

[0034]

[0035] If the aircraft can fly with the line-of-sight reference profile in equation (11), it can intercept maneuvering targets at the desired attack angle, and the original maneuvering target interception and guidance problem is transformed into the problem of precise tracking with the line-of-sight reference profile.

[0036] Step 3: Convert the relative line-of-sight dynamic model from Step 1 into a strictly feedback system, and construct an input saturation compensator and a finite-time disturbance estimator respectively. The input saturation compensator reduces the adverse effects of the control input saturation constraint guidance system, and the finite-time disturbance estimator reduces the adverse effects of target maneuvering on the guidance system.

[0037] To facilitate the design of guidance laws in the inversion control architecture, the line-of-sight dynamics model (4) in step one is written as a strictly feedback system as follows:

[0038]

[0039] in, Let v(u) be the system state variable, v(u) be the control input under saturation constraints, and y be the system output variable. The system state matrix, Let d be the system control input matrix, and d be the disturbance received by the system.

[0040] The input saturation compensator consists of a saturation approximation function and an auxiliary control system.

[0041] The saturation approximation function l(u) is constructed as a modified hyperbolic tangent function to approximate the control input signal v(u) in equation (7) which is subject to input saturation constraints, eliminating its inflection point and smoothing it out:

[0042]

[0043] Therefore, the control input signal v(u) subject to input saturation constraint becomes:

[0044] v(u)=l(u)+ε(u) (15)

[0045] Where ε(u)=v(u)-l(u) is the approximation error.

[0046] The auxiliary control system is used to compensate for control input errors caused by saturation constraints, and its structure is as follows:

[0047]

[0048] Where l is the input compensation signal, κ is the positive parameter to be designed, and Δ(u) is the error caused by saturation constraint.

[0049] Since the target's maneuvering acceleration is unknown in advance, a finite-time interference estimator is constructed to estimate the target's maneuvering interference value in order to effectively reduce the interference of the target's maneuvering on the guidance and control system, as shown in the following formula:

[0050]

[0051] Where α1, α2, and L are all positive parameters to be designed. The estimated value of the target maneuvering interference d is given. It will be achieved within a limited timeframe.

[0052] Step 4: Considering the input saturation compensator and finite-time disturbance estimator in Step 3, design a variable gain interception guidance law based on the inversion control architecture to enable the aircraft to accurately track the line-of-sight reference profile in Step 2 and achieve successful interception of maneuvering targets at a specified angle.

[0053] To accelerate error convergence and enhance system robustness, a variable gain function G(Ξ) is designed as follows:

[0054]

[0055] Here, k1 and k2 are both positive real numbers. Ξ refers to the independent variable of the function G(Ξ).

[0056] To accurately track the line-of-sight reference profile, even with a strictly feedback system, the output y = y d =λ d The tracking error surface is designed as follows:

[0057]

[0058] Where, x 2d =x 2d,1 For the virtual control quantity to be designed The filtered virtual control quantity is obtained after passing through the finite-time filter shown in equation (20).

[0059]

[0060] Combining the input saturation compensator and finite-time disturbance estimator from step three, a guided virtual control quantity is designed based on the inversion control architecture. The actual control quantity u is as follows:

[0061]

[0062] Where 0 < γ < 1 are the parameters to be designed, and η1 and η2 are the filtering error compensation signals. and This is the error surface after compensation.

[0063] Under the guidance and control command u in step four, the aircraft will accurately track the line-of-sight reference profile in step two and achieve successful interception of the maneuvering target if the constraints of equations (5) to (7) are met.

[0064] Beneficial effects:

[0065] 1. This invention discloses a variable gain inversion maneuvering target interception guidance method considering attack angle constraints. Addressing the interception guidance problem of intercepting maneuvering targets at a specific angle, it considers the initial and terminal constraints of the aircraft's interception process. Based on trajectory shaping, a polynomial function representing the line-of-sight angle reference profile is established with the distance already flown as the independent variable. By solving for the coefficients of the line-of-sight angle reference profile under the conditions of initial and terminal constraints, the initial and terminal constraints are reduced, transforming the maneuvering target interception guidance problem into a profile tracking problem, thus guiding the aircraft to successfully intercept the target.

[0066] 2. This invention discloses a variable-gain inversion-based maneuvering target interception guidance method considering attack angle constraints. It addresses the adverse effects of aircraft input saturation constraints and target maneuvering interference on system control performance by designing an input saturation compensator and a finite-time interference estimator. The input saturation compensator reduces the adverse effects of control input saturation constraints on the guidance system, while the finite-time interference estimator reduces the adverse effects of target maneuvering on the guidance system. Based on the inversion control architecture, a variable-gain guidance control command is designed. The variable-gain function accelerates error convergence and enhances system robustness, achieving precise tracking of the line-of-sight angle reference profile, thereby successfully intercepting the maneuvering target at a predetermined angle. Attached Figure Description

[0067] Figure 1 This is a schematic diagram illustrating the geometric relationship between the aircraft and the target of this invention;

[0068] Figure 2 This invention discloses a flowchart of a variable gain inversion maneuvering target interception and guidance method that considers attack angle constraints;

[0069] Figure 3 The flight trajectories in the simulation results of intercepting maneuvering targets under different angular constraints according to the present invention;

[0070] Figure 4The figure shows the aircraft acceleration variation curves in the simulation results of intercepting maneuvering targets under different angular constraints according to the present invention.

[0071] Figure 5 The figures show the relative distance variation curves in the simulation results of intercepting maneuvering targets under different angle constraints according to the present invention.

[0072] Figure 6 The image shows the line-of-sight profile tracking curves in the simulation results of intercepting maneuvering targets under different angular constraints according to the present invention. Detailed Implementation

[0073] To make the objectives, technical solutions, and advantages of this invention clearer, the design process of this invention will be described in detail below with reference to the accompanying drawings. Throughout the description, the same or similar symbols represent the same or similar functions.

[0074] Example 1:

[0075] like Figure 2 As shown in the figure, this embodiment discloses a variable gain inversion maneuvering target interception and guidance method considering attack angle constraints. The specific implementation steps are as follows:

[0076] Step 1: Considering the kinematic and dynamic characteristics of the aircraft and target in a two-dimensional plane, establish dynamic models of the aircraft and target, and a relative line-of-sight angle dynamic model. Analyze and construct the constraints of the interception mission, and construct the interception and guidance problem for maneuvering targets. The constraints include initial constraints, terminal constraints, and control input saturation constraints.

[0077] The initial positions, speeds, initial trajectory angles, and target accelerations of the aircraft and maneuvering targets are shown in the table below:

[0078] Table 1 Initial State Settings

[0079]

[0080]

[0081] Establish the dynamic model of the aircraft and the target as shown in Equation (1), establish the relative dynamic model of the aircraft and the target as shown in Equation (2), and establish the line-of-sight angle dynamic model of the relative motion as shown in Equation (3).

[0082] The objective of the interception guidance problem considered in this invention is to successfully intercept a maneuvering target under a specified attack angle constraint. The initial and terminal constraints of relative distance and line-of-sight angle in equations (5), (6) and (7), and the maximum allowable values ​​of the aircraft control input are shown in the following table:

[0083] Table 2 Constraint Settings

[0084]

[0085] Step 2: Consider the initial and terminal constraints of the aircraft intercepting the maneuvering target in Step 1. Use a polynomial function with the distance already flown as the independent variable to characterize the line-of-sight reference profile, and transform the maneuvering target interception and guidance problem into a profile tracking problem.

[0086] Line of sight reference section λ d The structure is as follows:

[0087]

[0088] If the aircraft can fly with the line-of-sight reference profile in equation (23), it can intercept maneuvering targets at the desired attack angle, and the original maneuvering target interception and guidance problem is transformed into the problem of precise tracking with the line-of-sight reference profile.

[0089] Step 3: Convert the relative line-of-sight dynamic model from Step 1 into a strictly feedback system, and construct an input saturation compensator and a finite-time disturbance estimator respectively. The input saturation compensator reduces the adverse effects of the control input saturation constraint guidance system, and the finite-time disturbance estimator reduces the adverse effects of target maneuvering on the guidance system.

[0090] The line-of-sight angle dynamics model (4) is transformed into a strictly feedback form system as shown in equation (13), where Let y be the system state variable and y be the system output variable. The system state matrix is... System control input matrix The disturbance received by the system is d = a T cos(γ T -x1) / r represents the target maneuver component in the line-of-sight direction.

[0091] The input saturation compensator consists of a saturation approximation function (14) and an auxiliary control system (16), where parameter κ = 1.

[0092] The finite-time disturbance estimator is constructed as shown in equation (17), where α1 = 1, α2 = 0.05, and L = 1. Under the action of the disturbance estimator (17), the disturbance estimation error... It will converge to zero within a finite amount of time.

[0093] Step 4: Considering the input saturation compensator and finite-time disturbance estimator in Step 3, design a variable gain interception guidance law based on the inversion control architecture to enable the aircraft to accurately track the line-of-sight reference profile in Step 2 and achieve successful interception of maneuvering targets at a specified angle.

[0094] First, to accelerate error convergence and enhance system robustness, a variable gain function G(Ξ) is designed as shown in equation (18), where k1 = 10 and k2 = 2. This variable gain function increases rapidly with the increase of the absolute value of Ξ, thus generating a large gain when the error is large, enabling the error to converge quickly; similarly, when subjected to unknown disturbances, the gain function will generate a large gain to suppress the influence of unknown disturbances on the error.

[0095] Based on the tracking error surface in equation (19), the design flow of the guidance control command u in the inversion control architecture is as follows:

[0096] Step 4.1: Differentiate the tracking error z1 to obtain:

[0097]

[0098] in, Let x be the virtual control variable to be designed. 2d =x 2d,1 This is the filtered virtual control quantity obtained after passing through the finite-time filter shown in equation (20). The parameters in the finite-time filter equation (20) are set to β1 = 1 and β2 = 0.01.

[0099] To compensate for filtering errors The compensation variable η1 is defined as follows:

[0100]

[0101] Where γ = 0.6. Therefore, the compensated tracking error is:

[0102]

[0103] For the compensated tracking error Differentiate,

[0104]

[0105] Construct Lyapunov functions as Differentiation yields

[0106]

[0107] Then virtual control quantity Designed for

[0108]

[0109] Step 4.2: Combining the auxiliary control system (16) from Step 3, differentiate the error surface z2 to obtain...

[0110]

[0111] Where κ = 1.

[0112] The compensation variable η2 is designed as follows:

[0113]

[0114] Differentiate the compensated tracking error z2.

[0115]

[0116] Construct Lyapunov functions as Differentiation yields

[0117]

[0118] Therefore, the actual control instructions are designed as follows:

[0119]

[0120] Through the above steps, under the guidance and control command u in step four, the aircraft will accurately track the line-of-sight reference profile in step two, and successfully intercept the maneuvering target while satisfying all the constraints in step one.

[0121] Table 3 shows the interception and guidance performance in the numerical simulation results.

[0122]

[0123] The simulation step size was set to 0.01s, and numerical simulation experiments were conducted under different terminal line-of-sight angle constraints. The simulation results are shown in Table 3. The rendezvous flight trajectories of the aircraft and the target are as follows: Figure 3 As shown, under different terminal line-of-sight constraints, the aircraft can successfully intercept the target with a small miss distance and angle error. However, when the initial angle error is large, the aircraft intercepts the target with a relatively curved flight trajectory, and the interception time increases accordingly. But thanks to the trajectory shaping technology and variable gain inversion control technology in this invention, the aircraft can achieve... Figure 6 The reference profile shown accurately tracks the line-of-sight angle, guiding the approach to the target with a relatively gentle trajectory; simultaneously, Figure 4 The acceleration variation curve of the aircraft shown indicates that the acceleration of the aircraft satisfies the saturation constraint and the overall required acceleration is relatively small, which is similar to the magnitude of the target maneuvering acceleration interference. This demonstrates the effectiveness of the variable gain inversion maneuvering target interception guidance method disclosed in this invention.

[0124] Based on the simulation results and analysis of the aforementioned examples of intercepting maneuvering targets under different terminal line-of-sight constraints, it can be seen that the variable-gain inversion maneuvering target interception guidance method described in this example can enable the aircraft to resist target maneuvering interference, achieve successful interception of maneuvering targets with a small miss distance and angle error, and maintain a low required acceleration while satisfying the input saturation constraint condition. Therefore, this invention has strong engineering applicability and can achieve the intended purpose of the invention.

[0125] The above detailed description is a further explanation of the purpose, technical solution and beneficial effects of the invention. It should be understood that the above description is only a specific implementation example of the present invention and is only used to explain the present invention. It is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A variable-gain inversion-based maneuvering target interception and guidance method considering attack angle constraints, characterized in that: Includes the following steps, Step 1: Considering the kinematic and dynamic characteristics of the aircraft and target in a two-dimensional plane, establish the dynamic model of the aircraft and target, and the dynamic model of the relative line-of-sight angle. Analyze and construct the constraints of the interception mission, and construct the interception and guidance problem of the maneuvering target. The constraints include initial constraints, terminal constraints, and control input saturation constraints. The implementation method for step one is as follows: Considering an inertial coordinate system xoy in a two-dimensional plane, the initial positions of the aircraft and the maneuvering target are (x... M ,y M ) and (x T ,y T The flight speed is constant, denoted as V. M and V T And satisfy V M >V T The initial trajectory inclination angles were γ. M and γ T The magnitude of the acceleration is a M and a T The direction is perpendicular to the velocity direction; the line-of-sight distance between the aircraft and the target is r, and the line-of-sight angle is λ; all angles are defined as positive in the counterclockwise direction. The dynamic models of the aircraft and the target are established as follows: The relative dynamic model of the aircraft and the target is established as follows: Where, η M and η T The velocity lead angles of the aircraft and the target are respectively calculated using the following formula: Differentiating equation (2), the line-of-sight angle dynamic model of relative motion is established as follows: The objective of the interception guidance problem under consideration is to successfully intercept a maneuvering target under a specified attack angle constraint. Therefore, the relative distance and line-of-sight angle need to satisfy the initial and terminal constraints shown in the following equation. In this context, the subscripts 0 and f represent the initial and terminal states, respectively. Considering the limited capabilities of the aircraft's actuators, the guidance command u needs to satisfy the control input saturation constraint shown in the following equation. Among them, u max This indicates the maximum allowed value for the control input, and sign(·) is the sign function; Equations (1), (4) and (5) to (7) are the problem of intercepting and guiding maneuvering targets considering the attack angle constraint; Step 2: Consider the initial and terminal constraints of the aircraft intercepting the maneuvering target in Step 1. Use a polynomial function with the distance already flown as the independent variable to characterize the line-of-sight angle reference profile, and transform the maneuvering target interception guidance problem into a profile tracking problem. The second step is implemented as follows: Define the distance already flown r e The new independent variable is calculated as follows: r e ＝r0-r (8) The initial and terminal constraints in equations (5) and (6) are re-expressed as follows: Use the line-of-sight reference profile λ d Construct it into a cubic polynomial as shown in the following equation: Combining equations (9) and (10), the calculation expressions for each coefficient are obtained as follows: If the aircraft can fly with the line-of-sight reference profile in Equation (11), it can intercept maneuvering targets at the desired attack angle, and the original maneuvering target interception and guidance problem is transformed into the problem of precise tracking with the line-of-sight reference profile. Step 3: Convert the relative line-of-sight angle dynamic model in Step 1 into a strictly feedback system, and construct an input saturation compensator and a finite-time disturbance estimator respectively. The input saturation compensator reduces the adverse effects of the control input saturation constraint guidance system, and the finite-time disturbance estimator reduces the adverse effects of target maneuvering on the guidance system. The method for implementing step three is as follows: The line-of-sight dynamics model (4) in step one is written as a rigorous feedback system as follows: in, Let v(u) be the system state variable, v(u) be the control input under saturation constraints, and y be the system output variable. The system state matrix, Here, d represents the system control input matrix, and d represents the disturbance received by the system. The input saturation compensator consists of a saturation approximation function and an auxiliary control system; The saturation approximation function l(u) is constructed as a modified hyperbolic tangent function to approximate the control input signal v(u) in equation (7) which is subject to input saturation constraints, eliminating its inflection point and smoothing it out: The control input signal v(u) subject to input saturation constraints becomes: v(u)=l(u)+ε(u) (15) Where ε(u)=v(u)-l(u) is the approximation error; The auxiliary control system is used to compensate for control input errors caused by saturation constraints, and its structure is as follows: Where l is the input compensation signal, κ is the positive parameter to be designed, and Δ(u) is the error caused by saturation constraint; A finite-time interference estimator is constructed to estimate the target maneuvering interference value, as shown in the following equation: Where α1, α2, and L are all positive parameters to be designed. The estimated value of the target maneuvering interference d. It will be achieved within a limited timeframe; Step 4: Considering the input saturation compensator and finite-time disturbance estimator in Step 3, design a variable gain interception guidance law based on the inversion control architecture to enable the aircraft to accurately track the line-of-sight reference profile in Step 2 and achieve successful interception of maneuvering targets at a specified angle. Step four is implemented as follows: To accelerate error convergence and enhance system robustness, a variable gain function G(Ξ) is designed as follows: Where k1 and k2 are both positive real numbers; Ξ refers to the independent variable of the function G(Ξ); To accurately track the line-of-sight reference profile, even with a strictly feedback system, the output y = y d =λ d The tracking error surface is designed as follows: Where, x 2d =x 2d,1 For the virtual control quantity to be designed The filtered virtual control quantity obtained after passing through the finite-time filter shown in equation (20); Where β1 and β2 are the parameters of the finite-time filter; Combining the input saturation compensator and finite-time disturbance estimator from step three, a guided virtual control quantity is designed based on the inversion control architecture. The actual control quantity u is as follows: Where 0 < γ < 1 are the parameters to be designed, and η1 and η2 are the filtering error compensation signals. and The error surface after compensation; Under the guidance and control command u in step four, the aircraft will accurately track the line-of-sight reference profile in step two and achieve successful interception of the maneuvering target if the constraints of equations (5) to (7) are met.

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