Design method of guidance law with precise time specification based on delayed periodic sliding mode

Abstract

The invention solves the problem of accurate specified time guidance of missiles under the condition of terminal angle constraint based on the design method of accurate specified time guidance law of hysteresis period sliding mode, and the method comprises establishing the missile-target state space equation; designing a specified time expansion state observer to observe the target maneuvering acceleration; designing an accurate specified time guidance law based on the hysteresis period sliding mode method and combining the observation value obtained by the specified time expansion state observer. The invention is applied to the field of missile guidance.

Description

Technical Field

[0001] The invention relates to the technical field of guided weapon guidance control, and in particular to a design method of an accurate time-specified guidance law based on a hysteresis period sliding mode. Background Art

[0002] In recent years, as a new round of arms race among countries has intensified, new high-speed maneuvering targets such as ballistic missiles and near-space hypersonic vehicles have continued to emerge. In the traditional low-speed interception problem, the missile has advantages in flight speed and maneuverability over the target, so the classical guidance law can achieve a better guidance effect. However, the emergence of high-speed maneuvering targets has put forward higher requirements on the rapidity, maneuverability and robustness of the missile. In the high-speed target interception problem, since the speed of the missile is equivalent to that of the target, or even less than the target speed, the terminal guidance interception time is extremely short and the interception opportunity is fleeting. At this time, the traditional guidance theory has been difficult to adapt to the needs of the high-speed interception problem. It is necessary to study a guidance law that can converge the missile-target line-of-sight angular velocity to zero or near zero in a limited time to improve the interception accuracy of high-speed maneuvering targets.

[0003] In the missile guidance problem, a small miss distance can be guaranteed by the finite time convergence of the line of sight angular rate. However, the existing finite time guidance law has certain problems: the line of sight angular rate convergence time depends on the initial state, and when the initial state is far from the equilibrium point, the convergence time will be very long. To address this defect, people have proposed a fixed time guidance law, whose upper bound of the line of sight angular rate convergence time is a constant that does not depend on the initial state. However, the estimation of the upper bound of the convergence time is often a complex function determined by multiple parameters, which makes the design process very complicated. In view of the above shortcomings, designing a specified time guidance law has important theoretical and engineering significance. Summary of the invention

[0004] The present invention provides a method for designing a guidance law with precise specified time based on a hysteresis period sliding mode to achieve the task of convergence of the missile target sight angle rate and sight angle specified time, thereby solving the problem of precise specified time guidance of missiles under the condition of terminal angle constraint.

[0005] The design method of the guidance law with precise specified time based on the hysteresis period sliding mode includes the following steps:

[0006] S1: Establish the missile-target state space equation;

[0007] S2: Design a specified time dilation state observer to observe the target maneuvering acceleration;

[0008] S3: Based on the hysteresis periodic sliding mode method and combined with the observations obtained by the specified time extended state observer, the precise specified time guidance law is designed;

[0009] First, design the sliding surface:

[0010] s(t)=x 2 (t)+ax 1 (t)+K 2 (t)x 1 (th 2 )

[0011] In the formula: Let q d is the desired terminal sight angle, let q e is the angular error, q e =qq d , q is the sight angle, x 1 (t) = q e ,

[0012] Where K 2 (t) is defined as:

[0013]

[0014] Among them, a, h 1 ,h 2 are all constants greater than zero, T d Specify convergence time for the observer;

[0015] Second, design a guidance law that precisely specifies the time:

[0016] Any specified time T>0, select h 1 ,h 2 , T d So that T = 2(h 1 +h 2 )+T d , design the following guidance law:

[0017]

[0018] The constant τ∈(1 / 2, 1), R is the distance between the projectile and the target, is the relative velocity between the projectile and the target, is the observed value of the disturbance d, d = w q , w q Represents the component of target acceleration in the normal direction of the line of sight; where K 1 (t) is defined as:

[0019]

[0020] where sig 2τ-1 (s) is defined as sig 2τ-1 (s)=|s| 2τ-1 sgn(s); sgn(s) is defined as

[0021]

[0022] Specifically, the time-expanded state observer is designed as follows:

[0023]

[0024] where p≥0, λ≥0 is a constant and satisfies λ≥pΔ, i=1, 2 is the observation error, Yes 1 The observed value of is the observed value of the disturbance d, and its initial value satisfies The time-varying function η(t) is defined as

[0025]

[0026] The constant τ∈(1 / 2, 1),

[0027] The observation error can be written as:

[0028]

[0029] in

[0030] Compared with the prior art, the present invention has the following beneficial effects:

[0031] Compared with the finite time guidance law, the method proposed by the present invention solves the problem that the convergence time of the line of sight angular rate depends on the initial value, and compared with the fixed time guidance law, it solves the problem that the upper limit estimation of the convergence time is too complicated. The convergence time of the line of sight angular rate of the method proposed by the present invention can be arbitrarily specified, so it is more flexible and convenient, and more practical in actual engineering.

[0032] Target maneuvers are often regarded as unknown disturbances in missile guidance systems. In response to unknown disturbances, the present invention proposes an extended state observer to observe the unknown disturbances of the system and use the observed values ​​to compensate the system, thereby eliminating the adverse effects of the disturbances. Existing guidance laws often use sign functions to suppress system disturbances, which will bring about greater jitter and have an adverse effect on the actuator; the method proposed in the present invention does not need to introduce sign functions and can effectively suppress disturbances, so it has practical value in engineering.

[0033] The present invention will be further described below in conjunction with the accompanying drawings and embodiments: BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 It is a design flow chart of the precise time-specified guidance law based on the hysteresis period sliding mode of the present application;

[0035] Figure 2 It is the diagram of the relative motion between the projectile and the target;

[0036] Figure 3 is a curve diagram of missile sight angle error of some embodiments;

[0037] Figure 4 is a graph of projectile sight angles of some embodiments;

[0038] Figure 5 is a graph of missile-to-eye sight angular rate of some embodiments;

[0039] Figure 6 is a missile acceleration command curve diagram of some embodiments;

[0040] Figure 7 It is a flight trajectory graph of the projectile and target in some embodiments. DETAILED DESCRIPTION

[0041] The embodiments of the technical solution of the present invention will be described in detail below in conjunction with the accompanying drawings. Unless otherwise specified, the technical terms or scientific terms used in this application are generally understood by those skilled in the art.

[0042] Specific implementation method one: Figure 1 A method for designing an accurately timed guidance law based on a hysteresis periodic sliding mode is presented, which is implemented in the following steps:

[0043] S1: Establish the missile-target state space equation;

[0044] S2: Design a specified time dilation state observer to observe the target maneuvering acceleration;

[0045] S3: Based on the hysteresis periodic sliding mode method and combined with the observations obtained by the specified time extended state observer, the precise specified time guidance law is designed;

[0046] First, design the sliding surface

[0047] s(t)=x 2 (t)+ax 1 (t)+K 2 (f)x 1 (th 2 ) (1)

[0048] In the formula: Let q d is the desired terminal sight angle, let q e is the line of sight angle error, q e =qq d , q is the sight angle, x 1 (t) = q e ,

[0049] Where K 2 (t) is defined as:

[0050]

[0051] Among them, a, h 1 ,h 2 are all constants greater than zero, T d Specify convergence time for the observer;

[0052] Second, design a guidance law that precisely specifies the time:

[0053] Any specified time T>0, select h 1 ,h 2 , T d So that T = 2(h 1 +h 2 )+T d , design the following guidance law:

[0054]

[0055] The constant τ∈(1 / 2, 1), R is the distance between the projectile and the target, is the relative velocity between the projectile and the target, is the observed value of the disturbance d, d = w q , w q Represents the component of target acceleration in the normal direction of the line of sight; where K 1 (t) is defined as:

[0056]

[0057] where sig 2τ-1 (s) is defined as sig 2τ-1 (s)=|s| 2τ-1 sgn(s); sgn(s) is defined as

[0058]

[0059] Specific implementation method 2: The difference between this implementation method and specific implementation method 1 is step 1; specifically, it is:

[0060] In order to obtain the state space equation, a mathematical model of the missile guidance problem is established in advance: Considering the in-plane guidance situation, the relative motion relationship between the missile and the target is as follows: Figure 2 As shown, the relative motion equation of the projectile and the target is established as:

[0061]

[0062] Where R is the distance between the projectile and the target. is the relative velocity between the projectile and the target, q is the sight angle, is the line of sight angular velocity, w R 、u R They represent the components of target acceleration and missile acceleration in the line of sight direction, w q 、u q They represent the target acceleration and the missile acceleration normal component, namely:

[0063]

[0064]

[0065]

[0066]

[0067] Where V t 、V m are the flight speeds of the target and missile respectively, a t 、a m are the normal accelerations of the target and missile, respectively, are the direction angles of the target and the missile respectively.

[0068] Furthermore, let q d is the desired terminal sight angle, let q e is the line of sight angle error, i.e. q e =qq d ;

[0069] Define x 1 =q e , u=u q , d = w q , then the state space equation is established in the following form:

[0070]

[0071] Where u is the control input and d is the disturbance;

[0072] The following assumptions are made for the above equations: Assume that the disturbance d and its derivatives are bounded, that is, they satisfy:

[0073]

[0074] Where Δ and δ are known positive constants.

[0075] Specific implementation method three: This implementation method further defines step two; specifically, it is:

[0076] Design a specified time-dilated state observer;

[0077] In order to decouple R from d, the second equation of equation (3) can be rewritten as

[0078]

[0079] Definition 1 =Rx 2 ,ξ 2 =d, then formula (5) can be expanded into a second-order system of the following form:

[0080]

[0081] where ε(t)∈[-δ,δ].

[0082] Specify the convergence time T d >0, and define h=T d / 4. Define K (a，h) (t) are as follows:

[0083]

[0084] in

[0085]

[0086] Where a is a constant greater than zero.

[0087] Define two time-varying functions K 3 (t), K 4 (t):

[0088]

[0089]

[0090] Then the time-expanded state observer is designed as follows:

[0091]

[0092] where p≥0, λ≥0 is a constant and satisfies λ≥pΔ, i=1, 2 is the observation error, Yes 1 The observed value of is the observed value of the disturbance d, and its initial value satisfies The time-varying function η(t) is defined as

[0093]

[0094] The constant τ∈(1 / 2, 1),

[0095] The observation error can be written as:

[0096]

[0097] The following verifies that the specified time expansion state observer achieves the specified time convergence of the observation error:

[0098] Step 2.1: Verification Boundedness

[0099] Since all terms in equation (12) are greater than or equal to zero, η(t)≥0 always holds. The initial value satisfies Below is just to verify Will not leave [-Δ, Δ].

[0100] when have So the second equation of equation (11) can be written as

[0101]

[0102] Similarly, when The following formula holds

[0103]

[0104] So it can be concluded that hour Heng is established, which means So we have:

[0105]

[0106] Step 2.2: Verify 2 Convergence of (t)

[0107] Consider the Lyapunov function Then we have:

[0108]

[0109] The above formula shows that:

[0110] Step 2.3: Verify 1 Convergence of (t)

[0111] From the first equation of (13), we know

[0112]

[0113] The formula is further written as

[0114]

[0115] Where ω=t-2h, θ 1 (ω) = θ 1 (t-2h) = e 1 (t). Formula (18) shows that:

[0116] Further:

[0117] Combining the above steps, it can be verified that the designed specified time extended state observer can achieve the specified time convergence of the interference observation error.

[0118] Specific implementation method 4: This implementation method further defines step 3, which is specifically:

[0119] Design a sliding surface of the following form

[0120] s(t)=x 2 (t)+ax 1 (t)+K 2 (t)x 1 (th 2 ) (19)

[0121] Where K 2 (t) is defined as:

[0122]

[0123] where a, h 1 ,h 2 are all constants greater than zero.

[0124] Design of precisely timed guidance laws

[0125] Specify time T>0, select h 1 ,h 2 , T d So that T = 2(h 1 +h 2 )+T d . Design the following formal guidance law:

[0126]

[0127] The constant τ∈(1 / 2, 1), In formula (11), K 1 (t) is defined as

[0128]

[0129] The following verifies that the guidance law with precise specified time can achieve the specified time convergence of the line of sight angular rate and line of sight angle:

[0130] Step 3.1: Verify the convergence of the sliding mode variable s(t)

[0131] Select Lyapunov function V(t) = s 2 (t) / 2, its derivative can be written as

[0132]

[0133] When t∈[0, T d ), since all terms in (23) are bounded, it can be seen that V(t) is in the range [0, T d ) has boundaries within it.

[0134] When t∈[T d ,∞), due to e 2 (t) = 0 always holds true, then equation (23) can be written as:

[0135]

[0136] The above formula can be further written as

[0137]

[0138] where μ = tT d ,

[0139] The above formula shows that:

[0140] Further:

[0141] Step 3.4: Verify x 1 Convergence of (t)

[0142] From s(t)=0, we know

[0143]

[0144] This formula can be further written as

[0145]

[0146] where κ = t-2h 1 -T d , χ 1 (κ) = χ 1 (t-2h 1 -T d )=x 1 (t).

[0147] The above formula shows that:

[0148] Further:

[0149] Step 3.5: Verify x 2 Convergence of (t)

[0150] From formula (26), we can know

[0151]

[0152] in So there is

[0153] For t ≥ T + h 2 , considering x 1 (th 2 )=0, so x 2 (t)=0.

[0154] According to the above steps: That is, the designed precise specified time guidance law can achieve the specified time convergence of the line of sight angular rate and the line of sight angle.

[0155] Example

[0156] Assume that the initial relative distance between the projectile and the target is 37 km, and the initial velocities of the projectile and the target are The initial flight direction angle of the projectile is The initial sight angle is 15°, and the maximum acceleration of the missile is set to 200m / s 2 In the specified time dilation state observer of equation (11), the parameters are selected as a = 0.1, τ = 0.8, p = 1, L = 15, λ = 30, and h = 1.25. In the precise specified time guidance law of equation (21), the parameters are selected as τ = 0.8, a = 0.1, and h 1 =6s,h 2 =9s, specifying the convergence time T=35s.

[0157] Consider a target performing a sinusoidal maneuver of the form

[0158] a t =30cos(0.5t)m / s 2 (29)

[0159] Select 4 expected terminal sight angles q d =10°, 15°, 20°, 25°. Figure 3-Figure 5 The curves of the missile-target line-of-sight angle error, the missile-target line-of-sight angle, and the missile-target line-of-sight angle rate changing with time during the interception process are recorded. It can be seen that under the target's sinusoidal maneuver, the line-of-sight angle rate and the line-of-sight angle error can converge to zero within the specified time. Figure 6The missile acceleration command was recorded, and it can be observed that there was no jitter in the missile acceleration command and the curve was always continuous. Figure 7 The flight trajectory curves of the missile and the target are recorded (the horizontal and vertical coordinates represent the XY coordinates in the plane), and it can be seen that the missile can hit the target with a small miss amount and the desired sight angle. This shows that the precise time-specified guidance law proposed in this embodiment is very effective for intercepting maneuvering targets.

Claims

1. A method for designing a guidance law with precise time specification based on a hysteresis periodic sliding mode, characterized by: The following steps are involved: S1: Establish the missile-target state space equation; Define x1 = q e , u=u q , d = w q , then the state space equation is established in the following form: Let q d is the desired terminal sight angle, let q e is the angular error, i.e. q e =qq d , x1=q e , u=u q , d = w q , where q is the sight angle, R is the distance between the projectile and the target, is the relative velocity between the projectile and the target, w q 、u q They represent the components of target acceleration and missile acceleration in the normal direction of the line of sight respectively; Where u is the control input and d is the disturbance; The following assumptions are given: Assume that the disturbance d and its derivatives are bounded, that is, they satisfy: Where Δ and δ are known positive constants; S2: Design a specified time dilation state observer to observe the target maneuvering acceleration; The specific process of designing a specified time dilation state observer is as follows: To decouple R from d, the second equation of equation (3) is rewritten as; Define ξ1=Rx2,ξ2=d, then equation (5) can be expanded into a second-order system of the following form: where ε(t)∈[-δ,δ]; Specify the convergence time T d >0, and define h = T d / 4, define K (a，h) (t) are as follows: in: Where a is a constant greater than zero; Define two time-varying functions K3(t), K4(t): Then the time-expanded state observer is designed as follows: where p≥0, λ≥0 is a constant and satisfies λ≥pΔ, i=1, 2 is the observation error, is the observed value of ξ1, is the observed value of the disturbance d, and its initial value satisfies The time-varying function η(t) is defined as: where the constant τ∈(1 / 2,1), The observation error can be written as: S3: Based on the hysteresis periodic sliding mode method and combined with the observations obtained by the specified time extended state observer, the precise specified time guidance law is designed; First, design the sliding surface: s(t)=x2(t)+ax1(t)+K2(t)x1(t-h2) In the formula: Let q d is the desired terminal sight angle, let q e is the line of sight angle error, q e =qq d , q is the sight angle, x1(t)=q e , Where K2(t) is defined as: Among them, a, h1, h2 are all constants greater than zero, T d Specify convergence time for the observer; Second, design a guidance law that precisely specifies the time: Any specified time T>0, select h1, h2, T d So that T = 2 (h1 + h2) + T d , design the following guidance law: The constant τ∈(1 / 2, 1), R is the distance between the projectile and the target, is the relative velocity between the projectile and the target, is the observed value of the disturbance d, d = w q , w q represents the component of the target acceleration in the normal direction of the line of sight; where K1(t) is defined as: where sig 2τ-1 (s) is defined as sig 2τ-1 (s)=|s|2 τ-1 sgn(s); sgn(s) is defined as 2. The method for designing a precise time-specified guidance law based on a hysteresis periodic sliding mode according to claim 1 is characterized in that: Step S2 also includes specifying the time expansion state observer to achieve the specified time convergence of the observation error, specifically: Step 2.1: Verification Boundedness Since all terms in equation (12) are greater than or equal to zero, η(t)≥0 always holds, and The initial value satisfies Below is just to verify will not leave [-Δ, Δ]; when have So the second equation of equation (11) can be written as: Similarly, when The following formula holds: So it is concluded that hour Heng is established, which means So we have: Step 2.2: Verify the convergence of e2(t) Consider the Lyapunov function Then we have: The above formula shows that: Step 2.3: Verify the convergence of e1(t) From the first equation of (13), we know that The formula is further written as: Where ω=t-2h, θ1(ω)=θ1(t-2h)=e1(t), and equation (18) shows that: θ1(ω)=0, Further, we have: e1(t) = 0, 3. The method for designing a precise time-specified guidance law based on a hysteresis periodic sliding mode according to claim 2 is characterized in that: Step 3 also includes accurately specifying the time guidance law to achieve the specified time convergence of the line of sight angular rate and the line of sight angle, specifically: Step 3.1: Verify the convergence of the sliding mode variable s(t) Select Lyapunov function V(t) = s 2 (t) / 2, its derivative can be written as: When t∈[0, T d ), since all terms in (23) are bounded, it can be seen that V(t) is bounded in [0, T d ) is bounded within; When t∈[T d ,∞), since e2(t)=0 always holds, equation (23) can be written as: The above formula can be further written as: where μ = tT d , The above formula shows that: Further, we have: s(t) = 0, Step 3.2: Verify the convergence of x1(t) From s(t)=0, we know This formula can be further written as where κ = t-2h1-T d , χ1(κ)=χ1(t-2h1-T d )=x1(t), the above formula shows that: χ1(κ)=0, Further, we have: x1(t)=0, Step 3.3: Verify the convergence of x2(t) From formula (26), we can know that: in: So x2(t)=0, For t≥T+h2, considering that x1(t-h2)=0 holds, we have x2(t)=0.

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