Variable gain inversion maneuvering target interception guidance method considering attack angle constraint

Abstract

The invention discloses a variable gain inversion maneuvering target interception guidance method considering attack angle constraint, and belongs to the technical field of aircraft guidance control. The implementation method comprises the following steps: considering necessary conditions for interception success and end attack angle constraints, based on trajectory shaping, representing a sight angle reference profile by constructing a polynomial function taking a flied distance as an independent variable, and realizing reduction of initial and end constraints; the profile only utilizes missile-target relative information and does not depend on motion information of the target, so that the profile is suitable for intercepting targets in various motion forms. In order to accurately track a line-of-sight reference profile, adverse effects of aircraft control input saturation constraint and target maneuvering interference on a guidance system are considered, an input saturation compensator and a finite time interference estimator are introduced, and a variable gain interception guidance law is designed based on an inversion control architecture. And the aircraft can accurately track the profile and can intercept the maneuvering target at an expected angle at the same time. The method has the advantages of low required overload, high interception precision and good robustness.

Description

Technical Field

[0001] The invention relates to a variable gain inversion maneuvering target interception guidance method considering attack angle constraints, belonging to the technical field of aircraft guidance control. Background Art

[0002] In recent years, with the development of guidance technology and requirements, incoming targets have become more and more maneuverable, requiring the interception guidance law to achieve zero miss interception while improving the interception efficiency of the aircraft as much as possible. Therefore, the advanced guidance law constrained by the attack angle has received more and more attention. The attack angle constrained guidance law improves the lethality of the aircraft by guiding the aircraft to hit the weak point of the target's defense at a specific angle, which is an effective means to increase the probability of one-on-one maneuvering target interception.

[0003] There are methods based on proportional guidance, optimal control theory, sliding mode control, etc. for the problem of attack angle constraint guidance. However, most of these methods have the disadvantages of large overload, linear or small angle assumption dependence, and discontinuous guidance instructions, which greatly affect their practical value. In addition to the above methods, a novel guidance method combined with polynomial trajectory shaping technology is also used to solve the problem of attack angle constraint guidance. This type of method does not require linear or small angle assumptions. It constrains the flight trajectory of the aircraft by constructing a polynomial function containing fractional and integer polynomials to satisfy the constraints such as the attack angle. However, the existing trajectory shaping guidance method research is generally based on assumptions such as stationary or non-maneuvering targets, and does not consider the control input saturation constraints of the guidance system caused by the capacity limitations of the actual aircraft actuators. It is difficult to guarantee its interception effectiveness against maneuvering targets and cannot be directly applied to the interception scenario of maneuvering targets. Summary of the invention

[0004] In view of the low success probability of intercepting a maneuvering incoming target at a desired angle, the purpose of the present invention is to provide a variable gain inversion maneuvering target interception guidance method that takes into account the attack angle constraint, constructs a sight angle reference profile based on trajectory shaping to achieve the reduction of constraints such as the attack angle, provides a reference trajectory for the aircraft to ensure successful interception, considers the aircraft control input saturation constraint, and designs a variable gain interception guidance law based on the inversion control architecture, so that the aircraft can accurately track the sight angle reference profile while achieving the interception of the maneuvering target. The present invention adopts trajectory shaping and variable gain inversion control to achieve target interception guidance, which has the advantages of low required overload, high interception accuracy and good robustness.

[0005] The objective of the present invention is achieved through the following technical solutions:

[0006] The variable gain inversion maneuvering target interception guidance method considering the attack angle constraint disclosed by the present invention aims at the problem of low success probability of intercepting a maneuvering incoming target at a desired angle, takes into account the necessary conditions for successful interception and the terminal attack angle constraint, and based on trajectory shaping, constructs a polynomial function with the flown distance as the independent variable to characterize the sight angle reference profile, thereby achieving the reduction of the initial and terminal constraints; the profile only uses the relative information between the missile and the target, and does not need to rely on the motion information of the target, so it is suitable for intercepting targets of various motion forms. In order to accurately track the constructed sight angle reference profile, considering the adverse effects of the aircraft control input saturation constraint and the target maneuvering interference on the guidance system, an input saturation compensator and a finite time interference estimator are introduced, and a variable gain interception guidance law is designed based on the inversion control architecture, so that the aircraft can accurately track the profile while achieving successful interception of the maneuvering target at the desired angle.

[0007] The variable gain inversion maneuvering target interception guidance method considering attack angle constraint disclosed by the present invention comprises the following steps:

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

[0009] Considering the inertial coordinate system xoy in the 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 by V M and V T , and satisfies V M >V T ; The initial ballistic inclination angles are γ M and γ T ; The magnitude of 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 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] Among them, ηM and η T are the velocity lead angles of the aircraft and the target, respectively, and are calculated using the following formula:

[0015]

[0016] By taking the derivative of equation (2), the dynamic model of the sight angle of relative motion is established as follows:

[0017]

[0018] The mission objective of the interception guidance problem considered 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 equations:

[0019]

[0020]

[0021] The subscripts 0 and f represent the initial and terminal respectively.

[0022] Considering the limited actual capability of the aircraft actuator, the guidance command u needs to satisfy the control input saturation constraint as shown below:

[0023]

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

[0025] Equations (1), (4), and (5) to (7) are the maneuvering target interception guidance problems considering attack angle constraints.

[0026] Step 2: Considering the initial and terminal constraints of the aircraft intercepting the maneuvering target in step 1, the line of sight angle reference profile is represented by a polynomial function with the flight distance as the independent variable, and the maneuvering target interception guidance problem is transformed into a profile tracking problem.

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

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

[0029] The initial and terminal constraints in (5) and (6) are reformulated as:

[0030]

[0031] The line of sight angle reference section λ dIt is constructed as a cubic polynomial as shown below:

[0032]

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

[0034]

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

[0036] Step 3: Convert the relative line of sight angle dynamics model in step 1 into a strict feedback form system, and construct an input saturation compensator and a finite-time interference estimator respectively. The input saturation compensator is used to reduce the adverse effects of the control input saturation constraint guidance system, and the finite-time interference estimator is used to reduce the adverse effects of target maneuvers on the guidance system.

[0037] In order to facilitate the design of the guidance law in the backstepping control framework, the line-of-sight angle dynamics model (4) in step 1 is written as the following strict feedback formal system:

[0038]

[0039] in, is the system state variable, v(u) is the control input under saturation constraint, y is the system output variable, is the system state matrix, is the system control input matrix, and d is the disturbance suffered by the system.

[0040] The input saturation compensator is composed of a saturation approximate 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) subject to input saturation constraints in equation (7), eliminating its inflection point and making it smooth:

[0042]

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

[0044] v(u)=l(u)+ε(u) (15)

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

[0046] The auxiliary control system is used to compensate for the control input error caused by saturation constraints. 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 the saturation constraint.

[0049] Since the target maneuvering acceleration cannot be known in advance, in order to effectively reduce the interference of target maneuverability to the guidance control system, a finite time interference estimator is constructed to estimate the target maneuvering interference value, as shown in the following formula:

[0050]

[0051] Among them, α1, α2 and L are all positive parameters to be designed. is the estimated value of the target maneuvering disturbance d, Will be available for a limited time.

[0052] Step 4: Considering the input saturation compensator and finite-time interference estimator in step 3, a variable gain intercept guidance law is designed based on the inverse control architecture so that the aircraft can accurately track the line-of-sight angle reference profile in step 2 and achieve successful interception of the maneuvering target at a specified angle.

[0053] In order to accelerate error convergence and enhance system robustness, the variable gain function G(Ξ) is designed as follows:

[0054]

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

[0056] In order to accurately track the sight angle reference profile, even if the output of the strict feedback system y = y d =λ d , the design tracking error surface is as follows:

[0057]

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

[0059]

[0060] Combining the input saturation compensator and the finite-time disturbance estimator in step 3, the guided virtual control quantity is designed based on the inverse control architecture. And the actual control quantity u is as follows:

[0061]

[0062] Among them, 0<γ<1 is the parameter to be designed, η1 and η2 are the filter error compensation signals, and is the error surface after compensation.

[0063] Under the guidance and control command u in step 4, the aircraft will accurately track the line of sight angle reference profile in step 2 and successfully intercept the maneuvering target while satisfying the constraints of equations (5) to (7).

[0064] Beneficial effects:

[0065] 1. The present invention discloses a variable gain inversion maneuvering target interception guidance method considering attack angle constraints. For the interception guidance problem of intercepting a maneuvering target at a specific angle, the initial constraints and terminal constraints of the aircraft intercepting the maneuvering target are considered. Based on trajectory shaping, a polynomial function with the flown distance as the independent variable is established to characterize the line of sight angle reference profile. By solving the coefficients of the line of sight angle reference profile based on the initial constraints and the terminal constraints, the initial constraints and the terminal constraints are eliminated, and the maneuvering target interception guidance problem is transformed into a profile tracking problem, so as to guide the aircraft to successfully intercept the target.

[0066] 2. The present invention discloses a variable gain inversion maneuvering target interception guidance method considering attack angle constraints. The method considers the adverse effects of aircraft input saturation constraints and target maneuvering interference on system control performance, designs an input saturation compensator and a finite time interference estimator respectively, reduces the adverse effects of the control input saturation constraint guidance system through the input saturation compensator, and reduces the adverse effects of target maneuvering on the guidance system through the finite time interference estimator; designs variable gain guidance control instructions based on the inversion control architecture, accelerates error convergence and enhances system robustness through variable gain functions, realizes precise tracking of the line of sight angle reference profile, and successfully intercepts the maneuvering target at a predetermined angle. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 A schematic diagram of the geometric relationship between the aircraft and the target of the present invention;

[0068] Figure 2 The present invention discloses a flow chart of a variable gain inversion maneuvering target interception guidance method taking into account attack angle constraints;

[0069] Figure 3 The flight trajectory in the simulation result of intercepting a maneuvering target under different angle constraints of the present invention;

[0070] Figure 4It is the aircraft acceleration variation curve in the simulation result of intercepting maneuvering target under different angle constraints of the present invention;

[0071] Figure 5 It is the relative distance variation curve in the simulation result of intercepting maneuvering target under different angle constraints of the present invention;

[0072] Figure 6 It is the line of sight angle profile tracking curve in the simulation result of intercepting maneuvering target under different angle constraints of the present invention. DETAILED DESCRIPTION

[0073] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the design process of the present invention is described in detail below in conjunction with the accompanying drawings. Among them, the same or similar symbols throughout represent the same or similar functions.

[0074] Embodiment 1:

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

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

[0077] The initial position, flight speed, initial trajectory inclination and target acceleration of the aircraft and maneuvering target are as follows:

[0078] Table 1 Initial state settings

[0079]

[0080]

[0081] The dynamic model of the aircraft and the target is established as shown in formula (1), the relative dynamic model of the aircraft and the target is established as shown in formula (2), and the line of sight angle dynamic model of relative motion is established as shown in formula (3).

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

[0083] Table 2 Constraint setting

[0084]

[0085] Step 2: Considering the initial and terminal constraints of the aircraft intercepting the maneuvering target in step 1, the line of sight angle reference profile is represented by a polynomial function with the flight distance as the independent variable, and the maneuvering target interception guidance problem is transformed into a profile tracking problem.

[0086] Sight angle reference profile λ d The construction is as follows:

[0087]

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

[0089] Step 3: Convert the relative line of sight angle dynamics model in step 1 into a strict feedback form system, and construct an input saturation compensator and a finite-time interference estimator respectively. The input saturation compensator is used to reduce the adverse effects of the control input saturation constraint guidance system, and the finite-time interference estimator is used to reduce the adverse effects of target maneuvers on the guidance system.

[0090] The sight angle dynamics model (4) is transformed into a strict feedback formal system as shown in formula (13), where is the system state variable, y is the system output variable, and the system state matrix System Control Input Matrix The interference to the system is d = a T cos(γ T -x1) / r is the component of the target maneuver in the line of sight angle direction.

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

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

[0093] Step 4: Considering the input saturation compensator and finite-time interference estimator in step 3, a variable gain intercept guidance law is designed based on the inverse control architecture so that the aircraft can accurately track the line-of-sight angle reference profile in step 2 and achieve successful interception of the maneuvering target at a specified angle.

[0094] First, in order to accelerate error convergence and enhance system robustness, a variable gain function G(Ξ) is designed as shown in formula (18), where k1 = 10 and k2 = 2. The variable gain function increases rapidly with the increase of the absolute value of Ξ, thereby generating a larger gain when the error is large, so that the error converges quickly; similarly, when subject to unknown interference, the gain function will generate a larger gain to suppress the influence of unknown interference on the error.

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

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

[0097]

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

[0099] To compensate for the filtering error The compensation variable η1 is defined as:

[0100]

[0101] Where γ = 0.6. Then, the tracking error after compensation is

[0102]

[0103] The tracking error after compensation Seeking guidance,

[0104]

[0105] Construct the Lyapunov function as The derivative is

[0106]

[0107] The virtual control quantity Designed for

[0108]

[0109] Step 4.2: Combined with the auxiliary control system (16) in step 3, the error surface z2 is derived to obtain

[0110]

[0111] Among them, κ=1.

[0112] The designed compensation variable η2 is

[0113]

[0114] Derivative of the compensated tracking error z2,

[0115]

[0116] Construct the Lyapunov function as The derivative is

[0117]

[0118] Then, the actual control instruction is designed as

[0119]

[0120] Through the above steps, the aircraft, under the action of the guidance control instruction u in step 4, will accurately track the line of sight angle reference profile in step 2, and successfully intercept the maneuvering target while satisfying various constraints in step 1.

[0121] Table 3 Interception and guidance performance in numerical simulation test results

[0122]

[0123] The simulation step size is set to 0.01s, and numerical simulation experiments are carried out under different terminal sight angle constraints. The simulation results are shown in Table 3. The rendezvous flight trajectory of the aircraft and the target is shown in Figure 3 As shown in the figure, under different terminal sight angle constraints, the aircraft can successfully intercept the target with a small miss distance and angle error. When the initial angle error is large, the aircraft intercepts the target with a relatively curved flight trajectory, and the interception time increases accordingly. However, thanks to the trajectory shaping technology and variable gain inversion control technology in the present invention, the aircraft can Figure 6 The reference profile of the sight angle is accurately tracked, and the target is approached with a relatively gentle trajectory; at the same time, Figure 4 The aircraft acceleration change curve shown indicates that the acceleration of the aircraft satisfies the saturation constraint, and the overall required acceleration is small, which is similar to the target maneuvering acceleration interference size, illustrating the effectiveness of the variable gain inversion maneuvering target interception guidance method disclosed in the present invention.

[0124] According to the simulation results and analysis of the above-mentioned examples of intercepting maneuvering targets under different terminal sight angle 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 smaller miss amount and angle error, and the acceleration of the aircraft meets the input saturation constraint condition while maintaining a lower required acceleration. Therefore, the present invention has strong engineering practicality and can achieve the expected invention purpose.

[0125] The above specific description is a further detailed explanation of the purpose, technical solutions and beneficial effects of the invention. It should be understood that the above is only a specific implementation example of the present invention, which is only used to explain the present invention and is not used 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 in the scope of protection of the present invention.

Claims

1. A variable gain inversion maneuvering target interception guidance method considering attack angle constraints, characterized by: The following steps are included: Step 1: Considering the kinematic and dynamic characteristics of the aircraft and the target in the two-dimensional plane, establish the aircraft and target dynamics model, the relative line of sight angle dynamics model, analyze and construct the constraints of the interception mission, and construct the interception guidance problem of the maneuvering target; the constraints include initial constraints, terminal constraints, and control input saturation constraints; Step 2: Considering the initial and terminal constraints of the aircraft intercepting the maneuvering target in step 1, the line of sight angle reference profile is represented by a polynomial function with the flight distance as the independent variable, and the maneuvering target interception guidance problem is transformed into a profile tracking problem; Step 3: Convert the relative line of sight angle dynamics model in step 1 into a strict feedback form system, construct an input saturation compensator and a finite time disturbance estimator respectively, reduce the adverse effect of the control input saturation constraint guidance system through the input saturation compensator, and reduce the adverse effect of the target maneuver on the guidance system through the finite time disturbance estimator; Step 4: Considering the input saturation compensator and finite-time interference estimator in step 3, a variable gain intercept guidance law is designed based on the inverse control architecture so that the aircraft can accurately track the line-of-sight angle reference profile in step 2 and achieve successful interception of the maneuvering target at a specified angle.

2. The variable gain inversion maneuvering target interception guidance method considering attack angle constraints as claimed in claim 1, characterized in that: The implementation method of step one is: Considering the inertial coordinate system xoy in the 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 by V M and V T , and satisfy V M >V T ; The initial ballistic inclination angles are γ M and γ T ; The magnitude of 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 counterclockwise as positive; 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: Among them, η M and η T are the velocity lead angles of the aircraft and the target, respectively, and are calculated using the following formula: By taking the derivative of equation (2), the dynamic model of the sight angle of relative motion is established as follows: The mission objective of the interception guidance problem considered 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 equations: Among them, the subscripts 0 and f represent initial and terminal, respectively; Considering the limited actual capability of the aircraft actuator, the guidance command u needs to satisfy the control input saturation constraint as shown below: Among them, u max represents the maximum value allowed for the control input, sign(·) is the sign function; Equations (1), (4), and (5) to (7) are the maneuvering target interception guidance problems considering attack angle constraints.

3. The variable gain inversion maneuvering target interception guidance method considering attack angle constraints as claimed in claim 2, characterized in that: The implementation method of step 2 is: Define the distance flown r e is a new independent variable, calculated as follows: r e ＝r0-r (8) The initial and terminal constraints in (5) and (6) are reformulated as: The line of sight angle reference section λ d It is constructed as a cubic polynomial as shown below: Combining equations (9) and (10), the calculation expressions of each coefficient are as follows: If the aircraft can fly with the line of sight angle reference profile in equation (11), it can intercept the maneuvering target at the desired attack angle, and the original maneuvering target interception guidance problem is transformed into the line of sight angle reference profile accurate tracking problem.

4. The variable gain inversion maneuvering target interception guidance method considering attack angle constraints as claimed in claim 3, characterized in that: The implementation method of step three is: The sight angle dynamics model (4) in step 1 is written as the following strict feedback formal system: in, is the system state variable, v(u) is the control input under saturation constraint, y is the system output variable, is the system state matrix, is the system control input matrix, d is the disturbance suffered by the system; The input saturation compensator is composed of a saturation approximate 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) subject to input saturation constraints in equation (7), eliminating its inflection point and making it smooth: The control input signal v(u) subject to input saturation constraint becomes: v(u)=l(u)+ε(u) (15) Among them, ε(u)=v(u)-l(u) is the approximation error; The auxiliary control system is used to compensate for the control input error caused by saturation constraints. Its structure is as follows in, is the input compensation signal, κ is the positive parameter to be designed, and Δ(u) is the error caused by the saturation constraint; Construct a finite time interference estimator to estimate the target maneuver interference value, as shown in the following formula: Among them, α1, α2 and L are all positive parameters to be designed. is the estimated value of the target maneuvering disturbance d, Will be available for a limited time.

5. The variable gain inversion maneuvering target interception guidance method considering attack angle constraints as claimed in claim 4, characterized in that: The implementation method of step 4 is: In order to accelerate error convergence and enhance system robustness, the variable gain function G(Ξ) is designed as follows: Wherein, k1 and k2 are both positive real numbers. Ξ refers to the independent variable of the function G(Ξ). In order to accurately track the sight angle reference profile, even if the output of the strict feedback system y = y d =λ d , the design tracking error surface is as follows: Among them, x 2d =x 2d,1 is the virtual control quantity to be designed The filtered virtual control quantity obtained after passing through the finite time filter shown in the following formula (20): Combining the input saturation compensator and the finite-time disturbance estimator in step 3, the guided virtual control quantity is designed based on the inverse control architecture. And the actual control quantity u is as follows: Among them, 0<γ<1 is the parameter to be designed, η1 and η2 are the filter error compensation signals, and is the error surface after compensation; Under the action of the guidance control command u in step 4, the aircraft will accurately track the line of sight angle reference profile in step 2 and successfully intercept the maneuvering target while satisfying the constraints of equations (5) to (7).

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