Appointed-time Prescribed Performance Control for Missile Cold Launch Response Based on Nonlinear Filters and Dynamic Surface

doi:10.12382/bgxb.2024.0773

Abstract

Abstract:

Focusing on the unsupported random vertical launching process of a vehicular missile launching system, this paper proposes an idea of actively regulating the oscillation of launch canister and hence controlling the attitude of missile through the hydraulic actuator, which ensures the launching safety and the adaptability of vehicular missile launching system to the launching site. A 6-DOF dynamics model of the launching system is established, which takes into account the interactions among the launch canister, adapters and missile. The seventh-order state-space equation is derived by combining with the pressure dynamic equation of the hydraulic system. For avoiding the problem of “differential explosion” in a backstepping controller design of the seventh-order nonlinear system, the dynamic surface technique based on a smooth nonlinear filter is introduced to simplify the controller design. Meanwhile, in order to suppress the influence of hard-to-modelling factors on the launch dynamics, a time delay estimation control method is introduced for the estimation and attenuation of unmodeled errors and external disturbances. The fast convergence of missile tracking is achieved and the prescribed indicators of oscillation amplitude and angular velocity are met by using the appointed-time prescribed performance control method. The calculation results of mathematical model are compared with the results from the finite element model to verify the correctness of the proposed launch dynamics model. In addition, the launch dynamic response results under the influences of proportion-iIntegral-differential (PID) controller, dynamic surface controller, controller with dynamic surface and time delay estimation, and controller with the appointed-time prescribed performance are compared. The results show that the proposed controller can be used to constrain the angle of missile within 0.5° and the angular velocity within 1.5°/s when a missile comes out of the launch canister.

Key words: missile, launching system, unsupported vertical launch, appointed-time prescribed performance control, time delay estimation, dynamic surface control

CLC Number:

TJ812+.6

YU Xiaochuan, YANG Xiaowei, LIANG Xianglong, ZHU Weilin, YAO Jianyong. Appointed-time Prescribed Performance Control for Missile Cold Launch Response Based on Nonlinear Filters and Dynamic Surface[J]. Acta Armamentarii, 2024, 45(11): 4155-4174.

Figures/Tables 22

Fig.1 Multi-rigid body dynamic model of vehicular missile cold launching system

Fig.2 Geometric relationships among missile, adapter and laun chcanister

Fig.3 Block diagram of APPC control system based on DSC and TDC

Fig.4 Finite element model of launching system

Table 1 Parameters of launch dynamics model

参数	数值	参数	数值
m₁/kg	38000	$y ¯ 1$ /m	0.591
m₂/kg	14950	$x ¯ 2$ /m	1.198
m₃/kg	42470	$y ¯ 2$ /m	-6.065
J₁/(kg·m²)	1.42×10⁶	D/m	2
J₂/(kg·m²)	2.98×10⁵	z₁/m	5.324
J₃/(kg·m²)	1.11×10⁶	z₂/m	1.634
k_l/(N·m^-3)	1.6×10⁷	z₃/m	-2.826
c_a/(N·m^-1·s^-1)	10	z₄/m	-7.226
k_s₁/(N·m^-1)	6×10⁷	μ	0.16
c_s₁/(N·m^-1·s^-1)	6×10⁵	β_e/Pa	7×10⁸
k_s₁/(N·m^-1)	6×10⁷	p_s/Pa	2.8×10⁷
c_s₂/(N·m^-1·s^-1)	6×10⁵	p_r/Pa	0
x_A₀/m	2.504	A₁/m²	0.0283
y_A₀/m	-0.845	A₂/m²	0.017
x_B₀/m	-0.009	V_c1/m³	0.051
y_B₀/m	7.02	V_c2/m³	0.0119
$x ¯ 1$ /m	-10.580	k_q	0.7

Table 1 Parameters of launch dynamics model

参数	数值	参数	数值
m₁/kg	38000	$y ¯ 1$ /m	0.591
m₂/kg	14950	$x ¯ 2$ /m	1.198
m₃/kg	42470	$y ¯ 2$ /m	-6.065
J₁/(kg·m²)	1.42×10⁶	D/m	2
J₂/(kg·m²)	2.98×10⁵	z₁/m	5.324
J₃/(kg·m²)	1.11×10⁶	z₂/m	1.634
k_l/(N·m^-3)	1.6×10⁷	z₃/m	-2.826
c_a/(N·m^-1·s^-1)	10	z₄/m	-7.226
k_s₁/(N·m^-1)	6×10⁷	μ	0.16
c_s₁/(N·m^-1·s^-1)	6×10⁵	β_e/Pa	7×10⁸
k_s₁/(N·m^-1)	6×10⁷	p_s/Pa	2.8×10⁷
c_s₂/(N·m^-1·s^-1)	6×10⁵	p_r/Pa	0
x_A₀/m	2.504	A₁/m²	0.0283
y_A₀/m	-0.845	A₂/m²	0.017
x_B₀/m	-0.009	V_c1/m³	0.051
y_B₀/m	7.02	V_c2/m³	0.0119
$x ¯ 1$ /m	-10.580	k_q	0.7

Fig.5 Comparison of calculatrf results for missile angle

Fig.6 Comparison of calculated results for missile angular velocity

Fig.7 Comparison of calculated results for missile angular acceleration

Fig.8 Comparison of calculated results for the first adapter load

Fig.9 Comparison of calculationed results for the second adapter load

Fig.10 Comparison of calculated results for the third adapter load

Fig.11 Comparison of calculated results for the forth adapter load

Fig.12 Changes in missile angle with and without lumped disturbances in uncontrolled state

Fig.13 Changes in missile angular velocity with and without lumped disturbances in uncontrolled state

Fig.14 Changes in missile angular acceleration with and without lumped disturbances in uncontrolled state

Fig.15 Comparison of missile angle tracking performances using four kinds of controllers

Fig.16 Comparison of missile angular velocities using four kinds of controllers

Fig.1 Estimation of lumped disturbance d1

Fig.18 Estimation of lumped disturbance d2

Fig.19 Estimation of lumped disturbance d3

Fig.20 Estimation of lumped disturbance d4

Table 2 Comparison of control effects of four kinds of controllers

控制算法	指标(≥0.6s)
控制算法	e_1sm/(°)	e_1max/(°)	$θ · 3 s m$ / (°·s^-1)	$θ · 3 m a x$ / (°·s^-1)
NC	3.1175	3.5019	3.3015	6.2344
PID	0.2246	0.4376	1.6847	4.3430
DSC	0.3047	0.3301	0.7324	1.2855
DSC+TD	0.2145	0.2478	0.7440	1.4857
DSC+TD+PPC	0.0844	0.1001	0.3927	1.1833

Table 2 Comparison of control effects of four kinds of controllers

控制算法	指标(≥0.6s)
控制算法	e_1sm/(°)	e_1max/(°)	$θ · 3 s m$ / (°·s^-1)	$θ · 3 m a x$ / (°·s^-1)
NC	3.1175	3.5019	3.3015	6.2344
PID	0.2246	0.4376	1.6847	4.3430
DSC	0.3047	0.3301	0.7324	1.2855
DSC+TD	0.2145	0.2478	0.7440	1.4857
DSC+TD+PPC	0.0844	0.1001	0.3927	1.1833

References 30

[1]	张滨, 郝磊. 导弹无依托作战及其能力需求综述[J]. 飞航导弹, 2017 (4): 43-46,59.
	ZHANG B, HAO L. Overview of missile unsupported combat and its capability requirements[J]. Aerodynamic Missile Journal, 2017(4): 43-46,59. (in Chinese)
[2]	周晓和, 马大为, 胡建国, 等. 某导弹无依托发射场坪动态响应研究[J]. 兵工学报, 2014, 35(10): 1595-1603. doi: 10.3969/j.issn.1000-1093.2014.10.012
	ZHOU X H, MA D W, HU J G, et al. Research on dynamic response of launching site for missile unsuported random launch[J]. Acta Armamentarii, 2014, 35(10): 1595-1603. (in Chinese)
[3]	李昊临. 海上冷发射冲击载荷下火箭发射架动力响应及优化研究[D]. 哈尔滨: 哈尔滨工程大学, 2023.
	LI H L. Research on dynamic response and optimization of rocket launcher under impact load of cold launch at sea[D]. Harbin: Harbin Engineering Technology, 2023. (in Chinese)
[4]	缴天航, 姜毅, 王登, 等. 基于随动时变风载荷的车载导弹发射动力学研究[J]. 弹箭与制导学报, 2023, 43(1): 54-62.
	JIAO T H, JIANG Y, WANG D, et al. Research on the launching dynamics of vehicle-mounted missile based on following-up and time-varying wind load[J]. Journal of Projectiles, Rockets, Missiles and Guidance, 2023, 43(1): 54-62. (in Chinese)
[5]	励明君, 姜毅, 马立琦, 等. 基于神经网络算法的发射场坪承载能力预测方法[J]. 兵工学报, 2022, 43(5): 982-991. doi: 10.12382/bgxb.2021.0245
	LI M J, JIANG Y, MA L Q, et al. The prediction method based on neural network algorithm for the bearing capacity of launching site[J]. Acta Armamentarii, 2022, 43(5): 982-991. (in Chinese) doi: 10.12382/bgxb.2021.0245
[6]	张震东, 马大为, 仲健林, 等. 某型导弹冷发射装备场坪适应性研究[J]. 兵工学报, 2020, 41(2): 280-290. doi: 10.3969/j.issn.1000-1093.2020.02.009
	ZHANG Z D, MA D W, ZHONG J L, et al. Research on adaptability of a cold launching system of missile to launching site[J]. Acta Armamentarii, 2020, 41(2): 280-290. (in Chinese) doi: 10.3969/j.issn.1000-1093.2020.02.009
[7]	张震东, 马大为, 任杰, 等. 冷发射装备对地载荷作用下预设场坪的动力响应研究[J]. 兵工学报, 2015, 36(2): 279-286. doi: 10.3969/j.issn.1000-1093.2015.02.013
	ZHANG Z D, MA D W, REN J, et al. Research on the bearing capacity of unsupported random launching site[J]. Acta Armamentarii, 2015, 36(2): 279-286. (in Chinese)
[8]	周晓和, 王惠方, 马大为, 等. 导弹无依托待发射阶段场坪准静态响应研究[J]. 弹道学报, 2015, 27(2): 91-96.
	ZHOU X H, WANG H F, MA D W, et al. Study on unsupported launching site quasi-static response during missile standby phase[J]. Journal of Ballistics, 2015, 27(2): 91-96. (in Chinese)
[9]	杨春浩, 马大为, 王晓锋, 等. 某导弹无依托发射场坪动态响应影响因素敏感度分析[J]. 弹道学报, 2016, 28(2): 87-92.
	YANG C H, MA D W, WANG X F, et al. Sensitivity analysis of factors affecting launching-site dynamic response of missile with unsupported random launch[J]. Journal of Ballistics, 2016, 28(2): 87-92. (in Chinese)
[10]	孙船斌, 马大为, 任杰. 冷发射平台垂直弹射响应特性研究[J]. 南京理工大学学报, 2015, 39(5): 516-522,555.
	SUN C B, MA D W, REN J. Study on vertical launching response characteristics of cold launch platform[J]. Journal of Nanjing University of Science and Technology, 2015, 39(5): 516-522,555. (in Chinese)
[11]	闫攀运, 梁国柱, 吕永志, 等. 车载冷发射刚柔耦合动力学建模与仿真[J]. 兵工学报, 2017, 38(12): 2386-2394. doi: 10.3969/j.issn.1000-1093.2017.12.012
	PAN Q Y, LIANG G Z, LÜ Y Z, et al. Modeling and simulation of rigid-flexible coupling dynamics of vehicular cold launch system[J]. Acta Armamentarii, 2017, 38(12): 2386-2394. (in Chinese)
[12]	WANG X, RUI X T, YANG F F, et al. Launch dynamics modeling and simulation of vehicular missile system[J]. Journal of Guidance, Control, and Dynamics, 2018, 41(6): 1370-1379.
[13]	ZHOU A F, LI D K, ZHOU S M, et al. Semianalytical model for dynamic analysis of missile vertical cold launch[J]. Journal of Spacecraft and Rockets, 2022, 59(6): 1976-1986.
[14]	CHENG C, SHEN H K. Fractional-order dynamics and adaptive dynamic surface control of flexible-joint robots[J]. Asian Journal of Control, 2023, 25(4): 3029-3044.
[15]	YANG X H, DENG X F, TAO L, et al. Neuroadaptive dynamic surface asymptotic tracking control of a VTOL aircraft with unknown dynamics and external disturbances[J]. Mathematics, 2023, 11(12): 2725.
[16]	YANG X W, GE Y W, DENG W X, et al. Precision motion control for electro-hydraulic axis systems under unknown time-variant parameters and disturbances[J]. Chinese Journal of Aeronautics, 2024, 37(1): 463-471.
[17]	LEE J, CHANG P H, JIN M. Adaptive integral sliding mode control with time-delay estimation for robot manipulators[J]. IEEE Transactions on Industrial Electronics, 2017, 64(8): 6796-6804.
[18]	CHOI J, KWON W, LEE Y S, et al. Adaptive time-delay estimation error compensation for application to robot manipulators[J]. Control Engineering Practice, 2024, 151: 106029.
[19]	MA J Y, YU S D, HU W K, et al. Finite-time robust flight control of logistic unmanned aerial vehicles using a time-delay estimation technique[J]. Drones, 2024, 8(2): 58.
[20]	YUAN S S, DENG W X, YAO J Y, et al. Robust control for bidirectional stabilization system with time delay estimation[J]. International Journal of Control, Automation and Systems, 2024, 22(4): 1163-1175.
[21]	BECHLIOULIS C P, ROVITHAKIS G A. Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance[J]. IEEE Transactions on Automatic Control, 2008, 53(9): 2090-2099.
[22]	GUI M Z, DAI M Z, ZHANG C X, et al. Prescribed performance spacecraft attitude control with multiple convergence rates[J]. Symmetry, 2024, 16(7): 789.
[23]	DING S H, LI S H, LI Q. Stability analysis for a second-order continuous finite-time control system subject to a disturbance[J]. Journal of Control Theory and Applications, 2009, 7(3): 271-276.
[24]	HUANG D Q, HUANG T P, QIN N, et al. Finite-time control for a UAV system based on finite-time disturbance observer[J]. Aerospace Science and Technology, 2022, 129: 107825.
[25]	ZHANG L, LIU J F, CUI N G. Backstepping control for a two-link manipulator with appointed-time convergence[J]. ISA Transactions, 2022, 128: 208-219.
[26]	WU X J, ZHENG W Y, ZHOU X X, et al. Adaptive dynamic surface and sliding mode tracking control for uncertain QUAV with time-varying load and appointed-time prescribed performance[J]. Journal of the Franklin Institute, 2021, 358(8): 4178-4208.
[27]	ZHANG Y C, WU G Q, YANG X Y, et al. Appointed-time prescribed performance control for 6-dof spacecraft rendezvous and docking operations under input saturation[J]. Aerospace Science and Technology, 2022, 128: 107744.
[28]	GONG K, JIA Y M. Appointed-time velocity-free prescribed performance control for space manipulators[J]. Aerospace Science and Technology, 2024, 144: 108783.
[29]	王成罡. 弹道导弹无依托冷发射出筒姿态动力学建模与仿真分析[D]. 哈尔滨: 哈尔滨工业大学, 2019.
	WANG C G. Research on launching attitude of ballistic missile unsupported random cold launching[D]. Harbin: Harbin Institute of Technology, 2019. (in Chinese)
[30]	JIN M, LEE J, AHN K K. Continuous nonsingular terminal sliding-mode control of shape memory alloy actuators using time delay estimation[J]. IEEE/ASME Transactions on Mechatronics, 2014, 20(2): 899-909.