Adaptive Integral Sliding Mode Saturation Guidance Law for Intercepting High Velocity Maneuvering Target

Chen Baowen; Sun Jingguang

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Aerospace Control ›› 2022, Vol. 40 ›› Issue (5) : 8-14.

Guidance, Navigation and Control

Adaptive Integral Sliding Mode Saturation Guidance Law for Intercepting High Velocity Maneuvering Target

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Abstract

In this paper, the problem of three-dimensional guidance for intercepting high velocity maneuvering target is studied and analyzed. Firstly, According to the head pursuit, the three-dimensional guidance system model is introduced. Secondly, an adaptive integral sliding mode guidance law is designed, which is based on the integral sliding mode and adaptive algorithm. Thirdly, an anti-saturation adaptive integral sliding mode based three dimensional guidance law is designed by introducing an auxiliary system. Finally, the stability of the designed guidance strategy is proved by using Lyapunov stability theory, and numerical simulations show the effectiveness of the proposed guidance scheme.

Key words

Saturation guidance law / Input constraint / Integral sliding mode control / Adaptive control

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Chen Baowen , Sun Jingguang. Adaptive Integral Sliding Mode Saturation Guidance Law for Intercepting High Velocity Maneuvering Target[J]. Aerospace Control, 2022, 40(5): 8-14

0 引言

高速飞行器技术的快速发展,对拦截导弹的飞行速度和机动性能提出了更高的要求^[1⇓-3]。目前导弹末端的主要拦截方式不再适用于拦截高速飞行的机动目标^[4-5]。

目前防御武器的拦截方式是拦截者的速度大于目标的速度时,采用尾追拦截,相反采用迎头拦截。与高速机动目标相比,拦截导弹不再具有速度上的优势,因此,传统的制导方式很难保证高拦截概率。为拦截高速大机动目标,文献[6]首次提出前向制导方式,使得弹目运动速度相对较小,末端制导时间变长,增大攻击区域。滑模控制对系统内部参数的摄动和外界的干扰具有强的鲁棒性,因此,在飞行器制导控制系统设计中得到了广泛应用^[7⇓-9]。文献[10-11] 针对二维平面内拦截高速机动目标,采用前向制导方式设计了自适应滑模制导律。文献[12]以拦截高速飞行器为目标,通过联合滑模控制和自适应技术,设计了自适应滑模三维制导律,但没有考虑执行器输入受限情况。文献[13]针对拦截高速机动目标,利用逆轨拦截方式,结合最优控制理论和双曲正弦函数,设计了带有角度约束的三维最优制导律。

在实际制导过程中,导弹的加速度指令具有一定的物理约束条件,若在制导律设计过程中不考虑输入受限,则可能导致制导系统性能的下降,甚至引起整个制导闭环系统的不稳定^[14]。文献[15-16]在制导律中引入双曲正切函数解决了导弹执行器输入受限问题。文献[17-18]基于指令滤波反步控制方法,分别设计了满足输入饱和约束的二维和三维制导律。

为了拦截高速飞行目标,本文采用前向制导方式,基于自适应方法、积分滑模控制理论和辅助系统,分别设计了自适应滑模三维制导律和抗饱和的自适应滑模三维制导律。

1 问题描述

采用前向制导方法,三维的弹目相对运动几何关系如图1所示。

图1 三维弹目相对运动几何示意图

Full size|PPT slide

则拦截机动目标的三维弹目相对运动学模型^[12]如下:

\dot{R} = V_{m} c o s θ_{m} c o s ϕ_{m} - V_{t} c o s θ_{t} c o s ϕ_{t}

(1)

R {\dot{θ}}_{L} = V_{m} s i n θ_{m} - V_{t} s i n θ_{t}

(2)

{\dot{ϕ}}_{L} R c o s θ_{L} = V_{m} c o s θ_{m} s i n ϕ_{m} - V_{t} c o s θ_{t} s i n ϕ_{t}

(3)

{\dot{θ}}_{t} = \frac{a_{z t}}{V_{t}} - {\dot{ϕ}}_{L} s i n θ_{L} s i n ϕ_{t} - {\dot{θ}}_{L} c o s ϕ_{t}

(4)

{\dot{ϕ}}_{t} = \frac{a_{y t}}{V_{t} cos θ_{t}} + {\dot{ϕ}}_{L} s i n θ_{L} c o s ϕ_{t} t a n θ_{t} - {\dot{θ}}_{L} s i n ϕ_{t} t a n θ_{t} - {\dot{ϕ}}_{L} c o s θ_{L}

(5)

{\dot{θ}}_{m} = \frac{a_{z m}}{V_{m}} - {\dot{ϕ}}_{L} s i n θ_{L} s i n ϕ_{m} - {\dot{θ}}_{L} c o s ϕ_{m}

(6)

{\dot{ϕ}}_{m} = \frac{a_{y m}}{V_{m} cos θ_{m}} + {\dot{ϕ}}_{L} s i n θ_{L} c o s ϕ_{m} t a n θ_{m} - {\dot{θ}}_{L} s i n ϕ_{m} t a n θ_{m} - {\dot{ϕ}}_{L} c o s θ_{L}

(7)

其中,T为目标,M为拦截弹。V_t和V_m分别是目标的速度和拦截弹的速度;θ_L和ϕ_L分别是视线关于参考坐标系的仰角和方位角。θ_t和ϕ_t为目标速度矢量前置角;θ_m和ϕ_m为拦截弹速度矢量前置角。a_yt和a_zt是目标的加速度,a_ym和a_zm是拦截弹的加速度。

在设计过程中要求下式成立

θ_m=n₁θ_t

(8)

ϕ_m=n₂ϕ_t

(9)

其中,n₁和₂均为大于1的常数。保证式(8)和(9)中θ_m和ϕ_m分别随着θ_t和ϕ_t的衰减而衰减。

为方便制导律设计,令

x = [\begin{array}{l} x_{1} \\ x_{2} \end{array}] = [\begin{array}{l} θ_{m} \\ ϕ_{m} \end{array}]

y = [\begin{array}{l} y_{1} \\ y_{2} \end{array}] = [\begin{array}{l} θ_{t} \\ ϕ_{t} \end{array}]

,对系统模型式(8)和(9)可整理为:

{\dot{x}}_{i} = B_{i} u_{i} + F_{i}

(10)

{\dot{y}}_{i} = M_{i} + E_{i}

(11)

其中,

u = [\begin{array}{l} u_{1} \\ u_{2} \end{array}] = [\begin{array}{l} a_{z m} \\ a_{y m} \end{array}]

为控制输入,

M = [\begin{array}{l} M_{1} \\ M_{2} \end{array}] = [\begin{array}{l} \frac{a_{z t}}{V_{t}} \\ \frac{a_{y t}}{(V_{t} c o s θ_{t})} \end{array}]

为外部干扰,

B = [\begin{array}{l} B_{1} \\ B_{2} \end{array}] = [\begin{array}{l} \frac{1}{V_{m}} & 0 \\ 0 & \frac{1}{(V_{m} c o s θ_{m})} \end{array}]

E = [\begin{array}{l} E_{1} \\ E_{2} \end{array}] = [\begin{array}{l} - {\overset{\cdot}{ϕ}}_{L} s i n θ_{L} s i n ϕ_{t} - {\overset{\cdot}{θ}}_{L} c o s ϕ_{t} \\ {\overset{\cdot}{ϕ}}_{L} s i n θ_{L} c o s ϕ_{t} t a n θ_{t} - {\overset{\cdot}{θ}}_{L} s i n ϕ_{t} t a n θ_{t} - {\overset{\cdot}{ϕ}}_{L} c o s θ_{L} \end{array}]

F = [\begin{array}{l} F_{1} \\ F_{2} \end{array}] = [\begin{array}{l} - {\overset{\cdot}{ϕ}}_{L} s i n θ_{L} s i n ϕ_{m} - {\overset{\cdot}{θ}}_{L} c o s ϕ_{m} \\ {\overset{\cdot}{ϕ}}_{L} s i n θ_{L} c o s ϕ_{m} t a n θ_{m} - {\overset{\cdot}{θ}}_{L} s i n ϕ_{m} t a n θ_{m} - {\overset{\cdot}{ϕ}}_{L} c o s θ_{L} \end{array}]

。

2 制导律设计

针对制导模型式(10)和(11),利用积分滑模控制理论、自适应方法和辅助系统,分别设计了自适应积分滑模制导律和自适应饱和积分滑模控制律。

2.1 自适应积分滑模制导律设计

为满足前向制导条件,令

e_i=x_i-n_iy_i(i=1,2)

(12)

设计带有自适应增益的积分滑模面为

s_{i} = e_{i} - e_{i} (0) + {\hat{k}}_{1} \underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ

(13)

其中,

{\hat{k}}_{1}

为自适应增益,0<α₁<1。

对式(13)求导可得

\begin{matrix} {\dot{s}}_{i} = x_{i} - n_{i} y_{i} + {\hat{k}}_{1} s i g^{α_{1}} (e_{i}) + {\overset{\cdot}{\hat{k}}}_{1} \underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ = \\ B_{i} u_{i} + F_{i} - n_{i} M_{i} - n_{i} E_{i} + {\hat{k}}_{1} s i g^{α_{1}} (e_{i}) + {\overset{\cdot}{\hat{k}}}_{1} \underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ \end{matrix}

(14)

其中,n_iM_i为系统干扰项。

为方便制导律的设计,给出以下相关假设和引理。

假设1:假设系统干扰项n_iM_i存在未知上界,即

|n_{i} M_{i}|

≤d_Mi,且d_Mi为未知正常数。

引理1^[8]:考虑系统

\dot{ξ}

=f(ξ),f(0)=0,ξ(0)=ξ₀,ξ∈Rⁿ,若存在正定连续函数V(x),满足

\overset{\cdot}{V}

(x)≤-αV^p(x)+σ,其中p∈(0,1),α>0, 0<σ<∞,则系统状态为实际有限时间稳定的。

为有效处理外部扰动,利用自适应算法对干扰上界进行估计,根据式(14),设计自适应积分滑模三维前向制导律。

u_{i} = - B_{i}^{- 1} (F_{i} - n_{i} E_{i} + {\hat{k}}_{1} s i g^{α_{1}} (e_{i}) + k_{2} s_{i}) - B_{i}^{- 1} (k_{3} s i g {(s_{i})}^{β} - {\hat{k}}_{M i} s i g n (s_{i}))

(15)

{\overset{\cdot}{\hat{k}}}_{1} = - ℓ_{1} s \underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) (τ) d τ

(16)

{\overset{\cdot}{\hat{d}}}_{M i} = γ_{i} (| s_{i} | - p_{i} {\hat{d}}_{M i})

(17)

其中,k₂,k₃,ℓ₁,γ_i和p_i为正常数,0<β<1。

定理1:针对系统式(11),在假设1条件下,当外部扰动上界未知时,在制导律(15)和自适应律式(16)~(17)的作用下,滑模变量s_i是实际有限时间稳定的,则e_i在有限时间内收敛到任意小区域内。

证明:选取Lyapunov函数

V_{1} = \frac{1}{2} s_{i}^{2} + \frac{1}{2 γ_{i}} {\dot{d}}_{M i}^{2}

(18)

其中,

{\dot{d}}_{M i} = {\hat{d}}_{M i}

-d_Mi

对式(18)求导可得

\begin{matrix} {\overset{\cdot}{V}}_{1} = s_{i} {\dot{s}}_{i} + \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} = \\ s_{i} (B_{i} u_{i} + F_{i} - n_{i} M_{i} - n_{i} E_{i}) + \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} + \\ + s_{i} (+ {\hat{k}}_{1} s i g^{α_{1}} (e_{i}) + {\dot{\hat{k}}}_{1} \underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ) \leq \\ - k_{2} s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} - ℓ_{1} s_{i}^{2} {(\underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ)}^{2} - \\ s_{i} (d_{M i} - {\hat{d}}_{M i} s i g n (s_{i})) + \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} \end{matrix}

(19)

根据

s_{i} (d_{M i} - {\hat{d}}_{M i} s i g n (s_{i})) \leq - {\tilde{d}}_{M i} | s_{i} |

(20)

将式(20)代入(19)中可得

\begin{matrix} {\dot{V}}_{1} \leq - k_{2} s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} - ℓ_{1} s_{i}^{2} {(\underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ)}^{2} + \\ {\tilde{d}}_{M i} | s_{i} | - \frac{1}{γ_{i}} {\tilde{d}}_{M i} (γ_{i} (| s_{i} | - p_{i} {\hat{d}}_{M i})) - k_{2} s_{i}^{2} - \\ k_{3} s_{i} s i g {(s_{i})}^{β} + {\tilde{d}}_{M i} | s_{i} | - \frac{1}{γ_{i}} {\tilde{d}}_{M i} (γ_{i} (| s_{i} | - p_{i} {\hat{d}}_{M i})) \leq \\ - k_{2} s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} + p_{i} {\tilde{d}}_{M i} {\hat{d}}_{M i} \end{matrix}

(21)

对满足δ₁>1/2的正常数δ₁,可使得下列不等式成立

p_{i} {\tilde{d}}_{M i} {\hat{d}}_{M i} \leq p_{i} (- {\tilde{d}}_{M i}^{2} + \frac{1}{2 δ_{1}} {\tilde{d}}_{M i}^{2} + \frac{δ_{1}}{2} d_{M i}^{2}) \leq - \frac{p_{i} (2 δ_{1} - 1)}{2 δ_{1}} {\tilde{d}}_{M i}^{2} + \frac{p_{i} δ_{1}}{2} d_{M i}^{2}

(22)

将式(22)代入(21)中,整理可得

\begin{matrix} {\dot{V}}_{1} \leq - k_{2} s_{i}^{2} - \frac{p_{i} (2 δ_{1} - 1)}{2 δ_{1}} {\tilde{d}}_{M i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} + \\ \frac{p_{i} δ_{1}}{2} d_{M i}^{2} \leq - m i n (2 k_{2}, \frac{p_{i} γ_{i} (2 δ_{1} - 1)}{δ_{1}}) V_{1} + \frac{p_{i} δ_{1}}{2} d_{M i}^{2} \end{matrix}

(23)

由式(23)可得s_i和为

{\tilde{d}}_{M i}

最终一致有界的,进一步可得是有界的。

选择李雅普诺夫函数

V_{2} = \frac{1}{2} s_{i}^{2} + \frac{1}{2 γ_{i}} {({\bar{d}}_{M i} - {\hat{d}}_{M i})}^{2}

(24)

其中,

{\bar{d}}_{M i}

为一个正常数,并且满足

{\bar{d}}_{M i} > {\hat{d}}_{M i}

和

{\bar{d}}_{M i}

>d_Mi。

利用式(15)和(17),对式(24)求导整理得

\begin{matrix} V_{2} = s_{i} {\dot{s}}_{i} - \frac{1}{γ_{i}} ({\bar{d}}_{M i} - {\hat{d}}_{M i}) {\dot{\hat{d}}}_{M i} \leq - k_{2} s_{i}^{2} \\ - k_{3} s_{i} s i g {(s_{i})}^{β} - ℓ_{1} s_{i}^{2} {(\underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ)}^{2} \\ - s_{i} (d_{M i} - {\dot{\hat{d}}}_{M i} s i g n (s_{i})) - \frac{1}{γ_{i}} ({\bar{d}}_{M i} - {\hat{d}}_{M i}) {\dot{\hat{d}}}_{M i} \end{matrix}

(25)

根据

s_{i} (d_{M i} - {\dot{\hat{d}}}_{M i} s i g n (s_{i})) \leq ({\bar{d}}_{M i} - {\hat{d}}_{M i}) | s_{i} |

(26)

将式(26)代入(25)中,整理可得

\begin{matrix} V_{2} \leq - k_{2} s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} + p_{i} ({\bar{d}}_{M i} - {\hat{d}}_{M i}) {\hat{d}}_{M i} \leq \\ - k_{2} s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} - ({\bar{d}}_{M i} - {\hat{d}}_{M i}) - \frac{p_{i}}{2} {\hat{d}}_{M i}^{2} + \\ \frac{p_{i}}{2} {\bar{d}}_{M i}^{2} + ({\bar{d}}_{M i} - {\hat{d}}_{M i}) \leq - m i n (2^{(1 + β) / 2} k_{2}, {\sqrt{2 γ_{i}}}^{(1 + β) / 2}) \\ min V_{2}^{(1 + β) / 2} + \frac{p_{i}}{2} {\bar{d}}_{M i}^{2} + ({\bar{d}}_{M i} - {\hat{d}}_{M i}) \leq - ε_{1} V_{2}^{β} + C_{1} \end{matrix}

(27)

其中,ε₁=min

\{2^{\frac{1 + β}{2}} k_{3}, {(\frac{P_{i} (2 δ_{1} - 1)}{δ_{1}})}^{(β + 1) / 2}\}

,C₁=

\frac{p_{i}}{2} {\bar{d}}_{M i}^{2} + ({\bar{d}}_{M i} - {\hat{d}}_{M i})

,因为

{\bar{d}}_{M i} - {\hat{d}}_{M i}

是有界的正数,因此,C₁为有界量。根据式(27)和引理1可得,系统是实际有限时间稳定的,因此,滑模变量s_i是实际有限时间稳定的。由文献[15]可知,通过选取合适的k₂,k₃,ℓ₁,γ_i和p_i等制导参数,可使得则e_i在有限时间内收敛到任意小区域内。

2.2 自适应饱和积分滑模制导律设计

考虑输入受限,前向制导模型可重写为

e_{i} = B_{i} s a t (u_{i}) + F_{i} - n_{i} M_{i} - n_{i} E_{i}

(28)

根据式(28)对(13)求导可得

\begin{matrix} {\dot{s}}_{i} = x_{i} - n_{i} y_{i} + k_{1} s i g^{α_{1}} (e_{i}) = \\ B_{i} s a t (u_{i}) + F_{i} - n_{i} M_{i} - n_{i} E_{i} + \\ {\hat{k}}_{1} si g^{α_{1}} (e_{i}) + {\overset{\cdot}{\hat{k}}}_{1} \underset{0}{\int^{t}} s i g^{α_{1}} (e_{i}) d τ \end{matrix}

(29)

为了处理饱和约束,引入如下的辅助系统

η_{i} {\begin{matrix} - k_{η} η_{i} - \frac{η_{i}}{{| η_{i} |}^{2}} (| s_{i} B_{i} Δ u_{i} | + \frac{1}{2} Δ u_{i}^{2}) + \\ Δ u_{i} - k_{η 1} sig {(η_{i})}^{β} | η_{i} | \geq σ \\ 0 | η_{i} | < σ \end{matrix}

(30)

其中,Δu_i=u_i-u_ic, u_ic为需要设计的控制输入,η_i为辅助系统状态量,σ,k_η和k_η₁为正常数。

设计自适应抗饱和积分滑模三维制导律

\begin{matrix} u_{i c} = - B_{i}^{- 1} (F_{i} - n_{i} E_{i} + {\hat{k}}_{1} s i g^{α_{1}} (e_{i}) + k_{2} s_{i}) - \\ B_{i}^{- 1} (k_{3} s i g {(s_{i})}^{β} - k_{η} η_{i} - {\hat{d}}_{M i} s i g n (s_{i})) \end{matrix}

(31)

{\dot{\hat{k}}}_{1} = - ℓ_{1} s \underset{0}{\int^{t}} si g^{α_{1}} (e_{i}) (τ) d τ

(32)

{\dot{\hat{d}}}_{M i} = (γ_{i} | s_{i} | - p_{i} {\hat{d}}_{M i})

(33)

定理2:针对三维制导系统式(28),在假设1条件下,当外部扰动上界未知时,在制导律式(31)的作用下,滑模面s_i为实际有限时间稳定的,可使得则e_i在有限时间内收敛任意小区域内。

证明:选取Lyapunov函数

V_{3} = \frac{1}{2} s_{i}^{2} + \frac{1}{2} η_{i}^{2} + \frac{1}{2 γ_{i}} {\tilde{d}}_{M i}^{2}

(34)

对式(34)求导并代入(31)和(33)整理得

\begin{matrix} {\dot{V}}_{3} = s_{i} {\dot{s}}_{i} + η_{i} {\dot{η}}_{i} + \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} \leq - k_{2} s_{i}^{2} \\ - k_{3} s_{i} s i g {(s_{i})}^{β} - ℓ_{1} s_{i}^{2} {(\underset{0}{\int^{t}} si g^{α_{1}} (e_{i}) d τ)}^{2} \\ - s_{i} (d_{M i} - {\hat{d}}_{M i} s i g n (s_{i})) + s_{i} B_{i} Δ u_{i} + k_{η} s_{i} η_{i} \\ + η_{i} {\dot{η}}_{i} + \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} \leq - k_{2} s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} \\ - ℓ_{1} s_{i}^{2} {(\underset{0}{\int^{t}} si g^{α_{1}} (e_{i}) d τ)}^{2} - s_{i} (d_{M i} - {\hat{d}}_{M i} s i g n (s_{i})) \\ + s_{i} B_{i} Δ u_{i} + k_{η} s_{i} η_{i} - k_{η} η_{i}^{2} - (| s_{i} B_{i} Δ u_{i} | + \frac{1}{2} Δ u_{i}^{2}) \\ + η_{i} Δ u_{i} {k_{η}}_{1} η_{i} s i g {(η_{i})}^{β} + \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} \end{matrix}

(35)

根据

s_{i} B_{i} Δ u_{i} - | s_{i} B_{i} Δ u_{i} | \leq 0

(36)

k_{η} s_{i} η_{i} + η_{i} Δ u_{i} \leq \frac{1}{2} k_{η} s_{i}^{2} + \frac{1}{2} (k_{η} + 1) η_{i}^{2} + \frac{1}{2} Δ u_{i}^{2}

(37)

将式(36)和(37)代入(35)整理可得

\begin{matrix} {\dot{V}}_{3} \leq - (k_{2} - \frac{k_{η}}{2}) s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} - \\ (\frac{k_{η}}{2} - \frac{1}{2}) η_{i}^{2} - ℓ_{1} s_{i}^{2} {(\underset{0}{\int^{t}} si g^{α_{1}} (e_{i}) d τ)}^{2} + \\ \frac{1}{γ_{i}} {\tilde{d}}_{M i} {\dot{\tilde{d}}}_{M i} - {k_{η}}_{1} η_{i} s i g {(η_{i})}^{β} + s_{i} (d_{M i} - {\hat{d}}_{M i} s i g n (s_{i})) \end{matrix}

(38)

根据式(20)和(22),则式(38)可整理为

\begin{matrix} {\dot{V}}_{3} \leq - (k_{2} - \frac{k_{η}}{2}) s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} - (\frac{k_{η}}{2} - \frac{1}{2}) η_{i}^{2} - \\ {k_{η}}_{1} η_{i} s i g {(η_{i})}^{β} - \frac{p_{i} (2 δ_{1} - 1)}{2 δ_{1}} {\tilde{d}}_{M i}^{2} + \frac{p_{i} δ_{1}}{2} d_{M i}^{2} \leq - \\ min (2 (k_{2} - \frac{k_{η}}{2}), 2 (\frac{k_{η}}{2} - \frac{1}{2}), \frac{p_{i} γ_{i} (2 δ_{1} - 1)}{δ_{1}}) V_{3} + \frac{p_{i} δ_{1}}{2} d_{M i}^{2} \end{matrix}

(39)

由式(39)可以得s_i和

{\tilde{d}}_{M i}

是有界的。进一步可以得到是有界的。

选择李雅普诺夫函数

V_{4} = \frac{1}{2} s_{i}^{2} + \frac{1}{2} η_{i}^{2} + \frac{1}{2 γ_{i}} {({\bar{d}}_{M i} - {\hat{d}}_{M i})}^{2}

(40)

其中,

{\bar{d}}_{M i}

为一个正常数,并且满足

{\bar{d}}_{M i}

{\hat{d}}_{M i}

和

{\bar{d}}_{M i}

>d_Mi。

利用式(32)和(33),对式(40)求导并整理可得

\begin{matrix} {\dot{V}}_{4} \leq - (k_{2} - \frac{k_{η}}{2}) s_{i}^{2} - k_{3} s_{i} s i g {(s_{i})}^{β} - (\frac{k_{η}}{2} - \frac{1}{2}) η_{i}^{2} \\ - {k_{η}}_{1} η_{i} s i g {(η_{i})}^{β} - ({\bar{d}}_{M i} - {\hat{d}}_{M i}) - \frac{p_{i}}{2} {\hat{d}}_{M i}^{2} + \frac{p_{i}}{2} {\bar{d}}_{M i}^{2} \\ + ({\bar{d}}_{M i} - {\hat{d}}_{M i}) \leq - m i n (2^{(1 + β) / 2} k_{3}, 2^{(1 + β) / 2} {k_{η}}_{1}, \\ {\sqrt{2 γ_{i}}}^{(1 + β) / 2}) V_{4}^{(1 + β) / 2} + \frac{p_{i}}{2} {\bar{d}}_{M i}^{2} + ({\bar{d}}_{M i} - {\hat{d}}_{M i}) \leq - ε_{2} V_{4}^{β} + C_{2} \end{matrix}

(41)

其中,ε₂=min{min(2⁽¹⁺^β⁾^/²k₃,2⁽¹⁺^β⁾^/²k_η₁,

{\sqrt{2 γ_{i}}}^{(1 + β) / 2}

)},C₂=

\frac{p_{i}}{2} {\bar{d}}_{M i}^{2} + ({\bar{d}}_{M i} - {\hat{d}}_{M i})

,且C₂为有界量。根据式(41)和引理1可得系统是实际有限时间稳定的。因此,滑模变量s_i是实际有限时间稳定的,即可得e_i在有限时间内收敛到任意小区域内。

3 仿真校验

为了验证制导律的有效性。初始参数为:弹目相对距离为5000m,目标位置为:(0m,0m,0m),拦截弹位置为(4820m,1020m,-870m),视线角值为θ_L=-10°和ϕ_L=-12°,导弹前置角为θ_m(0)=-20°和ϕ_m(0)=-15°,目标前置角为θ_t=-20°和ϕ_t=-15°,导弹的速度为1500m/s,目标的速度为2100m/s,目标的加速度为2g。

3.1 自适应积分滑模制导律仿真分析

为表明本文制导策略的鲁棒性,与文献[7]中的简称为SMGL的制导律进行仿真对比。为对比简便,自适应积分滑模制导律简写为ATSMGL。制导参数选取:k₂=0.2、k₃=0.5、ℓ₁=0.02、γ_i=0.05、p_i=0.01、β=0.62和α₁=0.7,其仿真结果如图2~3所示。表1给出了2种制导律所产生的脱靶量和拦截时间。

图2 视线倾角和视线偏角曲线

Full size|PPT slide

图3 导弹加速度曲线

Full size|PPT slide

图2给出的是视线倾角θ_m,θ_t曲线和视线偏角ϕ_m,ϕ_t曲线,从图中可看出,在ATSMGL作用下,θ_m与θ_t快速保持倍数关系且迅速随着θ_t收敛到0,而在SMGL作用下,θ_t收敛速度较慢。图3给出导弹加速度的曲线,从图中可看出,在ATSMGL作用下的加速度曲线变化连续光滑且抖振小。从表1可以看出,这两种制导律均能使导弹成功拦截目标。

3.2 自适应饱和积分滑模制导律仿真分析

为了验证所设计的抗饱和制导律式(31)的有效性,对以下2种目标机动进行仿真分析:

情况1:常值机动a_zt=a_yt=2g;

情况2:余弦机动a_zt=a_yt=2cos(2t)g。

制导律参数选取σ=0.01,k_η₁=1.25和k_η=0.5。其仿真结果如图4~5所示。表2给出了2种目标机动情况下的脱靶量和拦截时间。

图4 视线倾角和视线偏角曲线

Full size|PPT slide

图5 导弹加速度曲线

Full size|PPT slide

从表2可知,针对不同目标运动形式,导弹都能成功拦截。图4给出的是视线倾角θ_m,θ_t曲线和视线偏角θ_m和θ_t曲线,从图中可以看出,视线倾角和视线偏角在较短时间内能够快速收敛到,0。从导弹加速度曲线图5可以看出,针对不同目标机动形式,加速度值被限制在合理的范围内,充分证明了制导律的有效性。

4 结论

针对高速机动飞行器的拦截问题,利用前向制导方法,对拦截高速机动目标的制导问题进行了研究分析,主要结论如下

1) 在设计新型的自适应积分终端滑模面基础上,结合自适应方法设计了自适应积分滑模制导律;

2) 通过引入辅助系统处理输入饱和问题,设计了抗饱和的自适应积分制导律,能够保证系统滑模面为实际有限时间稳定的;

3) 利用李雅普诺夫函数对所设计制导律给出了稳定性证明,并利用数字仿真验证了所设计制导律的有效性。

References

Publishing order | Descend order by publishing year | Descend order by cited within

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Abstract

基于平面内的目标-导弹相对运动方程，采用指令滤波backstepping方法设计了一种过载指令约束下的平面导引律，同时考虑了导弹自动驾驶仪的二阶动态特性。依据制导系统主要状态变量响应特性要求将制导系统分为两个子系统。采用指令滤波backstepping方法基于零化视线角速率原则和目标-导弹相对速度小于一个负常数的原则分别对两个子系统进行导引律设计。指令滤波backstepping方法不仅能够有效地处理过载指令饱和约束而且克服了传统backstepping方法中“项数爆炸”的缺陷。在过载指令饱和且导弹自动驾驶仪存在较大滞后的情况下，针对视线方向施加控制和不施加控制两种情况进行了仿真，结果表明设计的导引律在拦截大机动目标时具有优良的性能。

(Meng

Kezi

, Zhou

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https://doi.org/10.3969/j.issn.1000-1093.2014.09.014

Cited in this article [1] Abstract

Based on target-missile relative motion equation in plane, a two-dimensional guidance law subject to acceleration command constraint is designed using command filtered backstepping approach, and the second-order dynamics characteristics of missile autopilot is considered. The guidance system is divided into two sub-systems in the light of stability requirement of the main states. Then the command filtered backstepping approach is used to design the guidance laws for two sub-systems based on the principles of zeroing line-of-sight(LOS) angular rate and letting the relative velocity between missile and target be less than a negative constant, respectively. The command filtered backstepping approach can not only address saturation constraint on acceleration command but also overcome the deficiency of “explosion of terms” in the conventional backstepping method. Finally, the simulations are performed for controlling and non-controlling along LOS in the case of acceleration command saturation and missile autopilot with big lag. The results show that the guidance law has excellent performance for intercepting a highly maneuvering target.

[18]

, Meng

, Zhou

. Design of three-dimensional nonlinear guidance law with bounded acceleration command[J]. Aerospace Science and Technology, 2015, 46(1): 168-175.

https://doi.org/10.1016/j.ast.2015.07.010

https://linkinghub.elsevier.com/retrieve/pii/S1270963815002175

Cited in this article [1]

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Abstract
Key words
Cite this article
0 引言
1 问题描述
图1 三维弹目相对运动几何示意图
2 制导律设计
2.1 自适应积分滑模制导律设计
2.2 自适应饱和积分滑模制导律设计
3 仿真校验
3.1 自适应积分滑模制导律仿真分析
图2 视线倾角和视线偏角曲线
图3 导弹加速度曲线
3.2 自适应饱和积分滑模制导律仿真分析
图4 视线倾角和视线偏角曲线
图5 导弹加速度曲线
4 结论
References

Received	Published
2022-01-10	2023-06-13
Issue Date
2023-06-13

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Abstract

Key words

Cite this article

0 引言

1 问题描述

图1 三维弹目相对运动几何示意图

2 制导律设计

2.1 自适应积分滑模制导律设计

2.2 自适应饱和积分滑模制导律设计

3 仿真校验

3.1 自适应积分滑模制导律仿真分析

图2 视线倾角和视线偏角曲线

图3 导弹加速度曲线

3.2 自适应饱和积分滑模制导律仿真分析

图4 视线倾角和视线偏角曲线

图5 导弹加速度曲线

4 结论

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References

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模态框（Modal）标题

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Abstract

Key words

Cite this article

0 引言

1 问题描述

图1 三维弹目相对运动几何示意图

2 制导律设计

2.1 自适应积分滑模制导律设计

2.2 自适应饱和积分滑模制导律设计

3 仿真校验

3.1 自适应积分滑模制导律仿真分析

图2 视线倾角和视线偏角曲线

图3 导弹加速度曲线

3.2 自适应饱和积分滑模制导律仿真分析

图4 视线倾角和视线偏角曲线

图5 导弹加速度曲线

4 结论

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References

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