An Adaptive Integrated Guidance and Control Design against Ground Fixed Targets

CHEN Bo，LIAO Fei，JI Haibo

(1. Department of Automation, University of Science and Technology of China, Hefei 230027, China)(2. Department No.1, Bengbu Naval Petty Officer Academy, Bengbu 233012, China)

?

·控制技術(shù)·

An Adaptive Integrated Guidance and Control Design against Ground Fixed Targets

CHEN Bo1,2，LIAO Fei1，JI Haibo1

(1. Department of Automation, University of Science and Technology of China, Hefei 230027, China)(2. Department No.1, Bengbu Naval Petty Officer Academy, Bengbu 233012, China)

An adaptive integrated guidance and control (IGC) approach is proposed based on backstepping method and input-to-state stability for homing missiles against ground fixed targets with actuator failures. A model of the IGC system in the pitch plane is formulated. The IGC laws are designed via backstepping method, which can provide desired impact attitude angle to hit the fixed targets in the presence of external disturbances and actuator failures. In order to deal with the uncertainties, an adaptive control law is developed. The numerical simulation results demonstrate the effectiveness of the proposed design scheme.

adaptive; integrated guidance and control; ground fixed targets; actuator failures

陳 勃1，2，廖 飛1，季海波1

(1. 中國科學(xué)技術(shù)大學(xué) 自動(dòng)化系, 合肥 230027； 2. 海軍蚌埠士官學(xué)校1系， 安徽 蚌埠 233012)

0 Introduction

Guidance and control systems of homing missiles are traditionally designed separately followed by combining them. However, accuracy requirements of missiles lead to intense maneuvering during the last phase of the engagement, thus violating the spectral separation assumption of the conventional two-loop design. This has led to the birth of integrated guidance and control (IGC) designs that demonstrate superior hitting accuracies by accounting for the coupling between the kinematics and dynamics of the problem.

IGC scheme was introduced by Williams et al[1]firstly and was addressed in application to a homing missile by Lin and Yueh[2]. Since then, IGC design has become an emerging trend in missile technology and beendeveloped by being combined with different control theories.Palumbo et al[3]used the state-dependent Riccati equation to deal with a more comprehensive model characterized by nonlinear motion in three dimensions; Menon and Ohlmeyer[4]used the feedback linearization method associated with the linear quadratic regulator technique to formulate a nonlinear IGC laws; Sharam and Richards[5]used an IGC design for homing missiles based on the backstepping strategy; Shkolnikov et al[6]and Shima et al[7]introduced an IGC design by using sliding mode control; Choi and Chwa[8]proposed an adaptive nonlinear guidance law considering target uncertainties and control loop dynamics by way of sliding mode control approach; still, there were other interesting and effective methods, such as subspace stabilization method[9]andθ-Dmethod[10]; Guo and Zhou[11]adoptedH∞control method to design an adaptive nonlinear control law; in[12], the missile attitude angles were considered as state variables and the linear quadratic regulator (LQR) was adopted for designing the IGC law; more recently, the input-to-statestability is introduced to deal with the guidance law design sign problem[13-15]. Therefore, it stands to reason that the IGC system design will be one of the most noticed subjects in missile technology. However, in the guidance law design field against the ground fixed targets, there exists another important problem encountered in practice for complex systems, that is control failures. Control failures include bias, loss of effectiveness or thrusts failures, they can cause control system performance deterioration, and lead to instability and even catastrophic accidents. Therefore, fault-tolerant control (FTC) is introduced to increase availability by specifically designing control algorithms capable of maintaining stability and performance despite the occurrence of faults. In order to increase the safety and reliability of system[16], FTC is used properly to manage the redundancies of control and guidance law system in the event of component failures.

This paper focuses on designing the scheme of IGC for homing missiles against ground fixed targets. To achieve successful interception and better killing effects, impact attitude angle is required. The exact contributions of this paper are concluded as follows. First, an integrated guidance and control error model is given, and the actuation effectiveness phenomenon is described. Compared with existing approaches used to deal with the integrated guidance and control design problem, ground targets and actuator failures are explicitly taken into consideration. Next, FTC guidance laws based on input-to-state stability and backstepping for ground targets are proposed, which not only are robust against external disturbance, but also accommodate actuator failures. Finally, the backstepping approach is utilized to design the integrated guidance and fault-tolerant control (IGFTC) for missiles against ground fixed targets to make the state variables be input-to-state stable with respect to uncertainties and actuator failures. An adaptive law is designed to estimate the unknown parameter with respect to the actuation effectiveness components. And the stability analysis shows that the interception with desired impact angle can be gotten with the proposed law, and the IGFTC scheme enables stability and performance of the overall system.

The paper is organized as follows. Section 1 formulates the model of the IGC system in the pitch plane. The process of designing adaptive IGFTC law is given in Section 2 and the numerical simulation results and analysis are presented in Section 3. Finally, conclusions are given in Section 4.

1 Missile Dynamics

The model derivation of the IGC system in the pitch plane is presented in this section.

Linearized missile dynamics in the pitch plane is described as follows[17]

(1)

whereαis the attack angle,ωis the angular pitch rate,Pis the thrust of the missile,Qis the dynamic pressure,Sis the aerodynamic reference area,cαis the lift force derivative with the angle of attack,mis the missile mass,Vis the velocity of the missile,Lis the reference length,Jzis the moment of inertia aboutz-axis,δzis the deflection angle for pitch control,mω,mα,mδrepresent the pitch moment derivatives with respect to the angle of attack, the nondimensional angular pitch rate and the deflection angle for pitch control respectively,Δα,Δωare unknown bounded uncertainties.

Fig.1 Angular relationship

As shown in Fig.1, dynamics of the range along the light-of-sight (LOS) and the LOS angle are given by

(2)

whereris the range along the LOS,qis the LOS angle andθis the flight path angle of the missile. By transforming Eq.(2), we get

(3)

As shown in Fig.1, the missile's pitch attitude angle is related with the angle of attack and path angle as follow

?=α+θ

(4)

We may assume that the angle of attack is very small at the engagement time, so it is easy to obtain that

?d≈θd

(5)

where ?dandθdare the missile's desired impact attitude angle and the desired impact path angle. The following equation and inequality also hold in the terminal game

(6)

We can obtain that

qd=θd

(7)

Applying Eq.(5) into Eq.(7), we get

qd+?d

(8)

Hence, the desired impact attitude angles in the interception endgame can be satisfied by the constraint of the desired impact LOS angle. Letσ=q=qd, then differentiation yields

(9)

According to the analysis above, we obtain the model of IGC close loop of the pitch plane as follows

(10)

ThispaperaimstodesignanappropriateIGClawforhomingmissilesagainstgroundfixedtargets,sothatthemissilecanhitthetargetaccuratelywithadesiredimpactattitudeangle,andthestabilityoftheoverallsystemisalsorequired.Wesupposethatthespeedisconstantandthemissileaimsatthetargetbyadjustingitsflightdirection,andeventuallyzeroingtheLOSrateleadstosuccessfulinterception.TheterminalimpactattitudeangularconstraintcanbesatisfiedbythenullLOSangleerrorσ=q=qd.

2 Adaptive IGFTC Design

ThissectionpresentstheprocessofdesigningtheIGClawforthesystemdescribedbyEq.(10).Firstly,atransformationisemployed

(11)

The IGC system given by Eq.(10)can be transformed to the equivalent system as follows

(12)

The fault-tolerant model is

(13)

whereτis the actuation effectiveness components and 0<τ≤1. It assumes that the inequality 0<τm≤|τ|≤1 holds. 0<τ≤1 implies the situation in which the control partially fails.τ=0 means the actuator completely fails. When this case appears, it will make a complete failure of target interception. The control objective is to design an ISS-based fault-tolerant guidance law for interception such that the law keeps the LOS rate within a small neighborhood of zero. Then an IGFTC law is designed for system by recursive application of backstepping.

The basic concept of input-to-state stability is introduced as follows:

(14)

Eq.(14) guarantees that for any initial statex(t0) and any bounded inputu(t), the statex(t) will be bounded. Furthermore, with a zero inputu(t) (usually regarded as a disturbance input), the system will be uniformly asymptotically stable.

(15)

(16)

Proof.ConsidertheequivalentsystemEq.(13),choosevisualcontrolsasfollows

(17)

(18)

where 0<μ≤τmandδ>0. The derivation ofValong the trajectories of system Eq.(13) is given by

(19)

Consider the inequalities

(20)

Applying the inequalities into Eq.(19), obtain

(21)

Substituting the visual control law, we have

(22)

Define

(23)

and

(24)

For 0<μ<τm≤τ≤1，consider the worst caseτ=μ，obtain

(25)

(26)

Substituting the adaptive law

(27)

into Eq.(26)，we get

(28)

Consider the inequality

(29)

We get

(30)

(31)

Multiplying both sides by e2kt, then integrate them over [0,t], we get

(32)

(33)

3 Simulation Results

Inthissection,thefeasibilityandapplicabilityoftheproposedIGClawisverifiedbythenumericalsimulationsforsomepassivehomingmissile'snonlineardynamicmodelinthepitchplane.Theabbreviationsofunitsusedinthissectionaremformeter,sforsecondandradforradian.

Fig.2 LOS rate ωq

Fig.3 Pitch rate ω

Fig.4 Angle of attack α

Fig.5 LOS angle q

Fig.6 Relative range r

Fig.7 Deflection angle δz

Fig.8 Adaptive parameter estimate v

4 Conclusions

This paper develops a scheme of IGC for homing missiles against ground fixed targets to improve the performance of the system with respect to external disturbances and actuator failures. An integrated model of guidance and control loop in the pitch plane is firstly formulated and the adaptive control law is designed by adopting the backstepping method and input-to-state theorem. Numerical simulation results have validated the usefulness of the IGC design to the homing missiles.

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陳 勃 男，1984年生，碩士研究生。研究方向?yàn)閷?dǎo)航制導(dǎo)與控制。

廖 飛 男，1987年生，博士研究生。研究方向?yàn)閷?dǎo)航制導(dǎo)與控制。

季海波 男，1964年生，教授，博士生導(dǎo)師。研究方向?yàn)榉蔷€性控制、導(dǎo)航制導(dǎo)與控制等。

國家自然科學(xué)基金專項(xiàng)資助項(xiàng)目(61273090)

陳勃 Email：447001023@qq.com

2015-10-21

2015-12-22

一種針對(duì)地面固定目標(biāo)自適應(yīng)導(dǎo)控一體化設(shè)計(jì)

基于反步設(shè)計(jì)法和輸入-狀態(tài)穩(wěn)定理論, 針對(duì)地面固定目標(biāo)，提出了一種執(zhí)行器失效時(shí)自適應(yīng)導(dǎo)引控制一體化(IGC)設(shè)計(jì)方法。首先，描述了一體化導(dǎo)引控制的數(shù)學(xué)模型；然后，基于反步設(shè)計(jì)法提出一種導(dǎo)引控制一體化算法，該算法能夠在額外擾動(dòng)和執(zhí)行器失效的情況下以設(shè)計(jì)的沖擊姿態(tài)角打擊地面固定目標(biāo)；同時(shí)，提出一種自適應(yīng)律來處理由于執(zhí)行器部分失效所引起的不確定性。仿真結(jié)果表明該算法是行之有效的。

自適應(yīng)；導(dǎo)引控制一體化；地面固定目標(biāo)；執(zhí)行器失效

TN957.52

A

1004-7859(2016)03-0078-08

10.16592/ j.cnki.1004-7859.2016.03.017