A guidance law design method for optimizing observability

Abstract

The application discloses a guidance law design method for optimizing observability, and comprises the following steps: establishing a relative motion model to obtain a measurement system; obtaining an observability measurement index based on a local nonlinear observability matrix of the system; constructing an optimization problem based on the observability measurement index and control energy, and solving the optimization problem to obtain a guidance instruction; and controlling the flight of an aircraft according to the obtained guidance instruction. The guidance law design method for optimizing observability disclosed by the application has better distance observability, reduces the estimation error of a passive sensor to the motion state of a target, thereby improving guidance precision, and the obtained acceleration instruction is smooth.

Description

A guidance law design method for optimizing observability Technical Field

[0001] This invention relates to a guidance law design method for optimizing observability, belonging to the field of aircraft guidance and control technology. Background Technology

[0002] Onboard sensors are used to extract the information needed for guidance, and they are divided into active sensors and passive sensors.

[0003] Compared to complex active sensors, passive sensors are widely used in various anti-tank and anti-ship missiles due to their simple mechanical structure and low cost.

[0004] However, the problem with passive sensors is that they can only measure the target's line-of-sight angle relative to the observer, lacking observability for distance. According to target estimation theory, this leads to weak observability of the target's motion state, increasing the estimation error and making passive sensors difficult to apply to tactical missiles with advanced guidance laws.

[0005] Extensive research has shown that increasing the relative normal maneuver between the target and the observer is beneficial for improving the range observability of pure azimuth measurement systems. However, traditional proportional guidance methods attempt to monotonically control the line-of-sight angular velocity to 0, which means that the relative normal maneuver between the target and the observer will rapidly decrease to 0, resulting in the unobservability of range information.

[0006] To optimize target observability during guidance, the inventors introduced a suitable performance criterion to measure it. According to classical control theory, the rank of the observability matrix can be used to analyze whether a system is observable. However, the rank criterion cannot quantitatively analyze the magnitude of observability. Common quantification metrics are based on the Fisher Information Matrix (FIM) or Cramer-Rao Lower Bound (CRLB), and some papers utilize the determinant of the error covariance matrix to calculate the optimal observer maneuver. However, these metrics are formally very complex and difficult to apply to optimal guidance laws. Therefore, these methods typically employ numerical computation to obtain the optimal trajectory.

[0007] Traditional guidance law design methods to enhance target observability achieve this by intentionally increasing oscillations during guidance. However, this approach not only results in excessive energy consumption, but the overly curved trajectory also increases the risk of missile stall. An improved approach, AIM (Aspect-Initiated Maneuvering), periodically turns guidance logic on and off to prevent the line-of-sight rate from rapidly converging to zero. This strategy improves target observability, but its guidance performance is relatively poor.

[0008] For the reasons mentioned above, the inventors have conducted extensive research on existing guidance methods in order to propose a guidance law design method that can solve the above problems. Summary of the Invention

[0009] To overcome the above problems, the inventors conducted in-depth research and designed a guidance law design method for optimizing observability, comprising the following steps:

[0010] S1. Establish a relative motion model and obtain the measurement system;

[0011] S2. Obtain observability metrics based on the local nonlinear observability matrix of the system;

[0012] S3. Construct an optimization problem based on the observability metric and control energy, and solve the optimization problem to obtain guidance commands;

[0013] S4. Control the aircraft to fly according to the obtained guidance commands.

[0014] In a preferred embodiment, in S1, the motion model is represented as:

[0015]

[0016]

[0017] Where r represents the relative distance between the aircraft and the target, V R γ represents the relative velocity between the aircraft and the target. s V represents the relative velocity vector between the aircraft and the target. TO With relative position vector r TO The included angle, γ TO V represents the relative velocity vector between the aircraft and the target. TO The angle between the aircraft and the x-axis, q represents the relative position vector r between the aircraft and the target. TO The angle with the x-axis, This represents the relative normal acceleration between the aircraft and the target.

[0018] In a preferred embodiment, the state quantity of the measurement system is the relative distance between the aircraft and the target.

[0019] The observations of the measurement system are the line-of-sight angles between the aircraft and the target.

[0020] In a preferred embodiment, the state equation of the measurement system is expressed as:

[0021]

[0022] Where x is the state variable of the measurement system, f(x) represents the state variable function, and v x The x-component of the relative velocity between the aircraft and the target, v y This represents the y-axis component of the relative velocity between the aircraft and the target.

[0023] The observable Z of the measurement system is represented as:

[0024]

[0025] Where h(x) represents the observable function.

[0026] In a preferred embodiment, in S2, the local nonlinear observability matrix of the system Represented as:

[0027]

[0028] in, For gradient operators, Let h be the i-th Lie derivative of the observable function h with respect to the state function f, where i = 0, 1, ..., n. (i) (x) represents the i-th derivative of the function h.

[0029] In a preferred embodiment, in S2, an observability index is obtained based on the local nonlinear observability matrix of the system, and an observability metric is determined based on the observability index.

[0030] In a preferred embodiment, in S2, the observability index is represented by the minimum and maximum singular values ​​of the observability matrix.

[0031] In a preferred embodiment, the observability metric is η = sinγ s .

[0032] In a preferred embodiment, in S3, the objective function J of the optimization problem is:

[0033]

[0034]

[0035] Where ω1(r) is the distance weighting function, u(r) is the virtual control quantity, ω3 is the penalty coefficient, and the function g(r) satisfies:

[0036]

[0037] g′(r)>0, r∈(0,r0)

[0038] r0 represents the initial distance.

[0039] In a preferred embodiment, the obtained guidance command is:

[0040]

[0041] Where N represents the proportional navigation gain term, and k represents a positive constant.

[0042] The beneficial effects of this invention include:

[0043] (1) The guidance law design method with optimized observability provided by the present invention has better distance observability, thereby reducing the target motion state estimation error, enabling passive sensors to be used in engineering applications on aircraft with advanced guidance laws, and improving the guidance accuracy of aircraft with passive sensors.

[0044] (2) The guidance law design method for optimized observability provided by the present invention has smooth acceleration commands and high aircraft stability;

[0045] (3) The guidance law design method for optimizing observability provided by the present invention has low energy consumption during the guidance process, which is conducive to improving the total range of the aircraft. Attached Figure Description

[0046] Figure 1 shows a schematic flowchart of a guidance law design method for optimizing observability according to a preferred embodiment of the present invention;

[0047] Figure 2 shows the relative motion trajectories of Example 1 and Comparative Examples 1 and 2.

[0048] Figure 3 shows the observability metrics of Example 1 and Comparative Examples 1 and 2;

[0049] Figure 4 shows the acceleration commands of Example 1 and Comparative Examples 1 and 2;

[0050] Figure 5 shows the controlled energy changes of Example 1 and Comparative Examples 1 and 2. Detailed Implementation

[0051] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Through these descriptions, the features and advantages of the present invention will become clearer and more apparent.

[0052] The term “exemplary” as used herein means “serving as an example, embodiment, or illustration.” Any embodiment illustrated herein as “exemplary” is not necessarily to be construed as superior to or better than other embodiments. Although various aspects of embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless specifically indicated otherwise.

[0053] A guidance law design method for optimizing observability according to the present invention includes the following steps:

[0054] S1. Establish a relative motion model and obtain the measurement system;

[0055] S2. Obtain observability metrics based on the local nonlinear observability matrix of the system;

[0056] S3. Construct an optimization problem based on the observability metric and control energy, and solve the optimization problem to obtain the guidance law;

[0057] S4. Control the flight of the aircraft according to the obtained guidance law.

[0058] According to the present invention, in S1, the motion model is represented as:

[0059]

[0060] Where r represents the relative distance between the aircraft and the target, V R γ represents the relative velocity between the aircraft and the target. s V represents the relative velocity vector between the aircraft and the target. TO With relative position vector r TO The included angle, γ TO V represents the relative velocity vector between the aircraft and the target. TO The angle between the aircraft and the x-axis, q represents the relative position vector r between the aircraft and the target. TO The angle with the x-axis, This represents the relative normal acceleration between the aircraft and the target.

[0061] Furthermore, the state variable X of the measurement system is the relative distance between the aircraft and the target, expressed as x = [r x ,r y ] T , where r x r represents the x-component of the relative distance between the aircraft and the target along the x-axis. y This represents the component of the relative distance between the aircraft and the target along the y-axis.

[0062] The observation Z of the measurement system is the line-of-sight angle between the aircraft and the target, expressed as z = q.

[0063] Preferably, the state equation of the measurement system is expressed as:

[0064]

[0065] Where X is the state variable of the measurement system, f(x) represents the state variable function, and v x The x-component of the relative velocity between the aircraft and the target, v y This represents the y-axis component of the relative velocity between the aircraft and the target.

[0066] Preferably, the observations of the measurement system are represented as follows:

[0067]

[0068] Where h(x) represents the observable function.

[0069] In S2, the observability index is obtained based on the local nonlinear observability matrix of the system, and the observability metric is determined based on the observability index. The observability index is used to characterize the observability of the system. The larger the observability index, the stronger the observability of the system.

[0070] According to a preferred embodiment of the present invention, in S2, the local nonlinear observability matrix of the system Represented as:

[0071]

[0072] in, For gradient operators, Let h be the i-th Lie derivative of the observable function h with respect to the state function f, where i = 0, 1, ..., n. (i) (x) represents the i-th derivative of the observable function h.

[0073] In a preferred embodiment, n is 2, meaning the first two orders of the observability matrix are retained as the final observability matrix to simplify the calculation process. The simplified observability matrix... Represented as:

[0074]

[0075] in, Let h be the 0th Lie derivative of the observable function h with respect to the state function f. Let h be the first-order Lie derivative of the observable function h with respect to the state function f.

[0076] In a preferred embodiment, in S2, the observability index is characterized by the minimum and maximum singular values ​​of the observability matrix, expressed as:

[0077]

[0078] in, It is a matrix The minimum singular value, It is a matrix The maximum singular value of OD is one value in the interval [0, 1]. The larger the OD value, the stronger the observability of the system.

[0079] The observability index OD is represented as:

[0080]

[0081] And there is,

[0082] According to the present invention, an observability metric η is obtained based on the observability index OD. Specifically, let... ρ2=2(r x v y -r y v x ), have

[0083]

[0084] According to the monotonicity of the function, when t∈[1,+∞), OD decreases monotonically with t. To find the maximum value of OD, we need to find the minimum value of t. According to the definition of t, t can be written in the following form:

[0085]

[0086] variable and γ s Independent and uncorrelated, taking the partial derivatives with respect to them separately yields:

[0087]

[0088]

[0089] Based on the expression after taking the partial derivative, we can obtain:

[0090] When V R When <r, t with respect to Monotonically decreasing; VR When > r, t with respect to Monotonically increasing;

[0091] when When t is about γ s Monotonically decreasing, when When t is about γ s Monotonically increasing.

[0092] Therefore, we can conclude that:

[0093] When r = V R and When t reaches its minimum value of 1, based on this, under the condition that... At that time, the observability metric can be obtained as η = sinγ. s .

[0094] According to the observability metric, system observability increases with |η|, and reaches its minimum value when |η| reaches its minimum value. min When = 0, the system is unobservable.

[0095] In S3, an optimization problem is set with the goal of minimizing the total control energy of the guidance process.

[0096] Based on the relative motion model, we can obtain:

[0097]

[0098] Substituting the observability metric, we obtain the state equation:

[0099]

[0100] Preferably, the Set as a virtual control variable.

[0101] In S3, the objective function J of the optimization problem is:

[0102]

[0103] Where u(r) is the virtual control variable. ω3 is the penalty coefficient, ω3=sgn(η)ω2, ω2 is a positive constant;

[0104] g(r) is a function that satisfies:

[0105]

[0106] g′(r)>0,r∈(0,r0] (16).

[0107] Furthermore, the constraints of the optimization problem are:

[0108]

[0109] Solving this optimization problem yields the guidance law, specifically...

[0110] The Hamiltonian function for the optimality problem is:

[0111]

[0112] According to the principle of minimum value Right now

[0113] ω1(r)u(r)-λ=0 (19)

[0114] The co-state equation is Right now

[0115]

[0116] The solution to equation (20) is:

[0117]

[0118] Substituting equation (21) into equation (19) yields

[0119]

[0120] Substituting equation (22) into the state equation, we can obtain

[0121]

[0122] When g(r) = r k At that time, Sometimes,

[0123]

[0124] From η(r0)=η0, we can obtain the constant C2.

[0125]

[0126] From η f =0 gives the constant C2=0

[0127]

[0128] Substituting equation (26) into equation (24) yields

[0129]

[0130] Substituting equation (26) into equation (22) yields

[0131]

[0132] The normal acceleration command is

[0133]

[0134] Replacing the initial state in equation (29) with the current state, we obtain the closed-loop solution of the normal acceleration command, which gives us the guidance command:

[0135]

[0136] Where N represents the proportional navigation gain, and k represents a positive constant.

[0137] Furthermore, equation (30) can be transformed into:

[0138]

[0139] Wherein, OD(r,V) R ) represents the observability compensation term, expressed as:

[0140]

[0141] Where N takes the value of 3 and k takes the value of 1.

[0142] Example

[0143] Example 1

[0144] A simulation experiment was conducted, simulating a missile's tail-chase interception of an air-to-ground target. The initial conditions for the missile and the target were as follows:

[0145] Table 1

[0146] Parameter values: Missile initial position (0, 4500), Missile initial velocity (300, 0), Missile initial trajectory angle -40 degrees, Target initial position (4000, 0), Target initial velocity (20, 0). surface

[0147] The simulation method includes the following steps:

[0148] S1. Establish a relative motion model and obtain the measurement system;

[0149] S2. Obtain observability metrics based on the condition number of the system's local nonlinear observability matrix;

[0150] S3. Construct an optimization problem based on the observability metric and control energy, and solve the optimization problem to obtain guidance commands;

[0151] S4. Control the aircraft to fly according to the obtained guidance commands.

[0152] In S1, the motion model is represented as:

[0153]

[0154]

[0155] In S2, an observability index is obtained based on the system's local nonlinear observability matrix, and an observability metric is determined based on the observability index. The system's local nonlinear observability matrix... Represented as:

[0156]

[0157] The observability index OD is represented as:

[0158]

[0159] The observability metric is η = sinγ s .

[0160] In S3, the objective function J of the optimization problem is:

[0161]

[0162] subject to:

[0163] The function g(r) satisfies:

[0164]

[0165] g′(r)>0, r∈(0,r0)

[0166] The constraints of the optimization problem are:

[0167]

[0168]

[0169] η f =0

[0170] The guidance commands obtained are:

[0171]

[0172] Comparative Example 1

[0173] The same experiment as in Example 1 was conducted, except that the guidance law was obtained by using the Improved Adaptive Intermittent Maneuvering (RAIM) strategy.

[0174] The acceleration command of the RAIM guidance law is determined by the following two conditions.

[0175] (1)

[0176]

[0177] (1)

[0178]

[0179] in, N r N and t1 are the design parameters of the RAIM guidance law. First time reaching At time t, the time-varying variable S(t) is defined as

[0180] S0 = 1

[0181]

[0182] S(t)=S i ,t i ≤t<t i+1 .

[0183] Comparative Example 2

[0184] The same experiment as in Example 1 was conducted, except that the guidance law was obtained using the conventional proportional guidance law (PNG), where the navigation gain of the PNG was set to 3.

[0185] The simulation results of Example 1, Comparative Example 1, and Comparative Example 2 are shown in Figures 2-5 and Table 2.

[0186] Table 2. Determinants of the Fisher information matrix (|FIM|) for different guidance laws

[0187] Guidance Law | FIM | Example 11.2×10 45 RAIM (Comparative Example 1) 1.1 × 10 44 PNG (Comparative Example 2) 2×10 33 surface

[0188] in:

[0189] Figure 2 shows the relative motion trajectories of Example 1 and Comparative Examples 1 and 2;

[0190] Figure 3 shows the observability metrics of Example 1 and Comparative Examples 1 and 2;

[0191] Figure 4 shows the acceleration commands of Example 1 and Comparative Examples 1 and 2;

[0192] Figure 5 shows the control energy changes of Example 1 and Comparative Examples 1 and 2;

[0193] Table 2 shows the determinants of the Fisher information matrix for different guidance laws.

[0194] The charts above reveal that the traditional proportional guidance law (Comparative Example 2) results in a lower |FIM|, signifying lower observability. This leads to a rapid decrease in the accuracy of target state information estimated from the line-of-sight angle measured by the missile-borne seeker. While the RAIM method (Comparative Example 1) offers better observability, its acceleration command exhibits abrupt changes, potentially reducing flight stability and guidance accuracy. In contrast, the guidance method in Example 1 consumes less energy than the RAIM method (Comparative Example 1) while achieving a higher |FIM| value, indicating better observability. Furthermore, the acceleration command generated by the method in Example 1 is smoother, facilitating engineering implementation.

[0195] In the description of this invention, it should be noted that the terms "upper," "lower," "inner," "outer," "front," and "rear," etc., indicate the orientation or positional relationship based on the orientation or positional relationship in the working state of this invention, and are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this invention. Furthermore, the terms "first," "second," "third," and "fourth" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0196] The present invention has been described above with reference to preferred embodiments; however, these embodiments are merely exemplary and illustrative. Various substitutions and modifications can be made to the present invention based on these embodiments, all of which fall within the scope of protection of the present invention.

Claims

1. A guidance law design method for optimizing observability, characterized in that, Includes the following steps: S1. Establish a relative motion model to obtain the measurement system; S2. Obtain an observability metric based on the system's local nonlinear observability matrix; S3. Construct an optimization problem based on the observability metric and control energy, and solve the optimization problem to obtain guidance commands; S4. Control the aircraft to fly according to the obtained guidance commands. In S1, the state equation of the measurement system is expressed as: Where x is the state variable of the measurement system, f(x) represents the state variable function, and v x The x-component of the relative velocity between the aircraft and the target, v y The relative velocity between the aircraft and the target along the y-axis is represented by the following: Z, the observable of the measurement system, is denoted as... Where h(x) represents the observable function, and in S2, the local nonlinear observability matrix of the system is... Represented as: in, For gradient operators, Let h be the i-th Lie derivative of the observable function h with respect to the state function f, where i = 0, 1, ..., n. (i) (x) represents the i-th derivative of the function h.

2. The guidance law design method for optimizing observability according to claim 1, characterized in that, In S1, the motion model is represented as: Where r represents the relative distance between the aircraft and the target, V R γ represents the relative velocity between the aircraft and the target. s V represents the relative velocity vector between the aircraft and the target. TO With relative position vector r TO The included angle, γ TO V represents the relative velocity vector between the aircraft and the target. TO The angle between the aircraft and the x-axis, q represents the relative position vector r between the aircraft and the target. TO The angle with the x-axis, This represents the relative normal acceleration between the aircraft and the target.

3. The guidance law design method for optimizing observability according to claim 1, characterized in that, The state quantity of the measurement system is the relative distance between the aircraft and the target, and the observation quantity of the measurement system is the line-of-sight angle between the aircraft and the target.

4. The guidance law design method for optimizing observability according to claim 1, characterized in that, In S2, the observability index is obtained based on the local nonlinear observability matrix of the system, and the observability metric is determined based on the observability index.

5. The guidance law design method for optimizing observability according to claim 4, characterized in that, In S2, the observability index is represented by the minimum and maximum singular values ​​of the observability matrix.

6. The guidance law design method for optimizing observability according to claim 1, characterized in that, The observability metric is η = sinγ s .

7. The guidance law design method for optimizing observability according to claim 6, characterized in that, In S3, the objective function J of the optimization problem is: Where ω1(r) is the distance weighting function, u(r) is the virtual control quantity, ω3 is the penalty coefficient, and the function g(r) satisfies: g′(r)>0, r∈(0,r0]r0 represents the initial distance.

8. The guidance law design method for optimizing observability according to claim 1, characterized in that, The guidance commands obtained are: Where N represents the proportional navigation gain term, and k represents a positive constant.

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