Leonardo Acho

Escola d'Enginyeria de Barcelona Est-EEBE,Universitat Polit`ecnica de Catalunya-UPC,Spain

Iterative Learning Control for homing guidance design of missiles☆

Leonardo Acho

Escola d'Enginyeria de Barcelona Est-EEBE,Universitat Polit`ecnica de Catalunya-UPC,Spain

1.Introduction

Basically,missile homing guidance is used to account for a steering system that can sense a moving target and then guide the missile to get it.This is a task accomplished by following well designed missile control commands[1-4].From the national security point of view,this technology has acquired an increasingly important role in warfare and defense since the end of World War II.Nowadays,there are many forms of missile homing guidance systems able to intercept targets that may maneuver unpredictable as an evasion strategy[1].

For missile homing guidance control design,it is usually assumed that the missile is on a reasonable near-collision to its target[5];where,essentially,to obtain the motion equation of the system missile-target pursuit dynamic behavior,a line-of-sigh(LOS)coordinate frame is adopted.Antecedently,many missile guided systems employ a kind of edition of the proportional navigation guidance(PNG)law(see,for instance,[6]).Usually,these guidance missiles have been extensively utilized for decades due to theirs relatively simplicity for implementation;and employed,for instance,in air-to-air,surface-to-air,and air-to-surface military missions,as well as on aerospace rendezvous applications[4,7].However,theirs performance can be further improved by using new missile control strategies[2,3].

Iterative Learning Control theory(ILC)1Surprisingly,the staple of ILC theory can be found in a U.S.patent filled in 1967 and available in 1971[8].is basically a control strategy to improve transient response,or similarly,the controller performance,of dynamic systems that operate repetitively.This is realized by re-adjusting the system control input(s)during the system cycle operations[9,10,8,11].Due to ILC-systems execute the same task multiple times,also called multi-pass processes,the control law may learn from the previous system action,or iteration,to improve its performance over the next task loop.Hence,the central challenger of ILC theory is to learn from every task iteration to go further improvement onto the next one.Moreover,the ILC control scheme has been widely employed in many industrial applications such as manufacturing,robotics,chemical process,etc.(see,for instance,[8]),This because the main benefit of ILC lecture is its low transient tracking error despite large model uncertainty and disturbances[8,12,13].

Among different control techniques based on ILC theoretical account include:modeling,two-dimensional systems theory,linear matrix inequalities,adaptive methods,and robust approaches(see Refs.[13]and[12]and references there in).In the meantime,some other techniques are mainly based on Lyapunov framework[14-16].

Aside,many of the ILC approaches require identical resetting initial conditions,at the beginning of each iteration:the well known resetting condition;but,in real applications,the perfect resetting condition may be not realizable[14,13].Therefore,and under differenttestson resetting initial conditions,the dynamic

Nomenclature

boundedness along the time evolution,and asymptotic stability on each iteration of a ILC system,were well proven in Ref.[14].Actually,this is an important robust property of ILC framework to design new engineering developments,including control of mechanisms in noisy environments[14].

On this paper,and motivated by the canonical ILC structure given in Ref.[14],an ILC design is here developed for homing guidance control of missiles.The rest of this work is organized as follows.Section 2 encloses the homing engagement motion equation of the missile-target system,and states the philosophy of the missile homing guidance control objective too.Section 3 gives our ILC design.Section 4 shows numerical experiments displaying the performance of the proposed ILC design and compared with respect to a homing missile controller recently published;and here named as the base controller.Finally,in Section 5 the conclusions are written.

2.3-D missile-target pursuit motion equation

The 3-D pursuit geometry is sketched in Fig.1.It is described by using spherical coordinates and represents a tracking missile in its terminal phase in persecution to a(trying evasive)maneuver target.The mathematical dynamic model of this system is given by Refs.[2,3]

where each variable is described in the Nomenclature of this paper.Then,by employing

Fig.1.Three-dimensional homing engagement geometry outline.

the dynamic equations(1)-(3)can be further expressed as[2,3]

The philosophy of the missile homing guidance control objective is to decrease the relative distance between the missile and the target,and keeping the pitch and yaw LOS angular rates as small as possible[2,3].When this happens,physically,it is said that the missile is on head-on condition to the target hit.According to[2,3],this condition is so important to minimize the missile lateral-force level to fulfill engaging in hyper-sonic target persecutions.

3.Missile homing guidance Iterative Learning Control realization

Taking into account the previously missile homing guidance control objective philosophy,when Vθand Vφgo to zero,it means that the missile and target are in the head-on status leading the relative velocity along the LOS,Vr,decreasing and the relative distance between them too.Therefore,just for control design,the next input-out plant dynamic relation can be invoked[3]

Let us consider the next auxiliary control terms

where u1i,u2iare sequences of appropriate ILC laws on t∈[0,T],for some T,and the i-task iteration tends to∞.Hence the notations uθ=uθiand uφ=uφi,respectively.Then,the closed-loop system(8)-(9)yields

To go on our ILC design,let us further assume that the acceleration components of the target have the following relationships

where θ(t),φ(t)∈C[0,T].The above assumptions can be interpreted as the required gains for Vθiand Vφito go on target hit.2In order to keep some kind of written homogeneity,we intentionally avoid using wθiand wφi.Therefore,the system(10)yields

Now,we are going to use the following reference generator systems(for y1(t)and y2(t),respectively)

Above,Vθiand Vφiare seen as reference inputs.Allowing(13)-(15),and by invoking the main result stated in Appendix A,we finally arrive to the following ILC laws

Fig.2.KTt()and its histogram.

Fig.3.Simulation results:Scenario 1.Red line is the ILC and the blue one is the base controller.

4.Numerical experiments

The performance of the proposed ILC control is here analyzed numerically.So,the controllers(9)and(14)-(21)are applied to the missile dynamic(7).We set k1=k2=10,θ*= φ*=100,T=0.5 s,and y1(0)=y2(0)=10.3These values were selected by the trail and error technique.To comparison,the base controller stated in Ref.[3]was programmed too.We employ the same target maneuver scenarios(cases of study)as in Refs.[2,3],and briefly described as follow(see Nomenclature to data description).

Scenario 1:Step target maneuver

Fig.4.Simulation results:Scenario 1.Red line is the ILC and the blue one is the base controller.

r(0)=6 Km,θ(0)= π/3 RAD,φ(0)= π/3 RAD,Vr(0)=-1000 m/s,Vθ(0)=300 m/s,Vφ(0)=500 m/s,and

Scenario 2:Ramp target maneuver

r(0)=12 Km,θ(0)= π/3 Rad,φ(0)= π/3 Rad,Vr(0)=-1400 m/s,Vθ(0)=300 m/s,Vφ(0)=500 m/s,and

Fig.5.Simulation results:Scenario 2.Red line is the ILC and the blue one is the base controller.

Fig.6.Simulation results:Scenario 2.Red line is the ILC and the blue one is the base controller.

Scenario 3:Sinusoidal target maneuver

r(0)=6 Km,θ(0)= π/3 Rad,φ(0)= π/3 Rad,Vr(0)=-1000 m/s,Vθ(0)=300 m/s,Vφ(0)=500 m/s,and wr=KTsin(wdt),

Fig.7.Simulation results:Scenario 3.Red line is the ILC and the blue one is the base controller.

Fig.8.Simulation results:Scenario 3.Red line is the ILC and the blue one is the base controller.

Fig.9.Parametric learning evolution for each scenario(a:Scenario 3;b:Scenario 2;c:Scenario 1).

Fig.10.Simulation results:Scenario 1(noisy event).Red line is the ILC and the blue one is the base controller.

Fig.11.Simulation results:Scenario 1(noisy event).Red line is the ILC and the blue one is the base controller.

where KTis the target's navigation gain[2,3],and wd=20 Rad/s.For more representative numerical evaluation tests,usually,the value of KTis assumed as a time-varying parameter.At this respect,in our experiments,this employed gain is displayed in Fig.2 along with its histogram evidencing a random behavior of it as time goes on.4According to[2,3],KTis usually a random variable.This variable is designed to devise a kind of a random evasive target maneuvering.In our Matlab numerical experiments,this gain was programmed by using the randn Matlab command.Simulation results are shown in Figs.3-9.According to the Figs.3-8,we can observe that the time to get the head-on condition is reduced by ours ILC strategy;although,with respect to the base controller performance,it is observed an increase of our controllers'energy.From Fig.3,besides the reduction time to headon condition,the relative distance between the missile and its target keep the same performance than the base stage.This is true for the other events too.On the other hand,the ILC law signals shown in Figs.4,6 and 8 have different transient behaviors than the base setting.These in response to reduce the time to head-on condition.In order to capture a noisy case scenario,we added uncorrelated standard normal distribution noises with variance of 250 and zero mean values to each of our ILC control laws and to the base-line ones too.Figs.10-16 show the respectively obtained numerical experiments.From these results,we can appreciate a performance improvement of our ILC-controller design.

5.Conclusions

Fig.12.Simulation results:Scenario 2(noisy event).Red line is the ILC and the blue one is the base controller.

Fig.13.Simulation results:Scenario 2(noisy event).Red line is the ILC and the blue one is the base controller.

Fig.14.Simulation results:Scenario 3(noisy event).Red line is the ILC and the blue one is the base controller.

Fig.15.Simulation results:Scenario 3(noisy event).Red line is the ILC and the blue one is the base controller.

Fig.16.Noisy event:Parametric learning evolution for each scenario(a:Scenario 3;b:Scenario 2;c:Scenario 1).

In this paper,an Iterative Leaning Control design to terminal guidance of missiles against maneuvering targets was presented.This design involved a correct interpretation of the ILC-leaning parameters to the missile homing guidance control objective.Moreover,the resultant controllers seem simple and then easy to realize them physically.On the other hand,and in favor of ILC theory principle versus adaptive control one,for instance,is that the drift parameter phenomenon[17]seems to be absented when invoking ILC design;both methods are based on time-parametric estimations.

From the numerical experiments point of view,we have used a simple missile model.But by employing the gain KTas an random time-varying parameter,the induced dynamic behavior may simulate a more realistic situation.Obviously,in order to go further,we require a more realistic missile model,or a benchmark platform commonly used in other research fields.For instance,in control design of wind turbines,there exists a well supported and free numerical benchmark platform named FAST(and certificated by the National Renewable Energy Laboratory from the United States)to simulate a closely real wind turbine[18,19].

Appendix A

The Iterative Learning Control(ILC)theory is based on a tracking task that ends in a finite-time interval and repeated almost for ever.

Basically,ILC law is able to improve the system performance on each iterative cycle job by learning from the previous one.In this Appendix,the ILC technique stated in Ref.[14]is resumed.We think that this technique is easy to follow and simple to realize,specially,to our application on hand.

Consider the next first-order system in the ith-iteration

where ξi= ξ(xi,t)is a known function which can be locally Lipschitz,and θ(t)∈C[0,T]is the unknown time-varying parameter.T represents the task duration.Now conceive that the reference trajectory is dynamically produced by

where fr(.,.,.)is a known smooth function being r the setting trajectory yielding a bounded solution to xr(t)over each cycle-time interval[0,T].The tracking error is stated as ei(t)=xr(t)-xi(t).Next is the ILC problem statement.

The objective of ILC consists to find a sequence of control input ui(t)for t∈[0,T]such that state dynamic xitracks the reference signal xras i→∞,in someway acceptable.

Theorem-1[14]:Suppose that ei(0)is random and bounded bya constant C.Then the ILC

and

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A R T I C L E I N F O

Article history:

25 July 2016

in revised form

30 October 2016

Accepted 5 January 2017

Available online 31 January 2017

Terminal guidance law

Missiles

Iterative learning control

This paper presents an Iterative Learning Control design applied to homing guidance of missiles against maneuvering targets.According to numerical experiments,although an increase of the control energies is appreciated with respect to a previous published base controller for comparison,this strategy,which is simple to realize,is able to reduce the time to reach the head-on condition to target destruction.This fact is important to minimize the missile lateral force-level to fulfill engaging in hyper-sonic target persecutions.

?2017 The Author.Published by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

☆This work was partially supported by the Spanish Ministry of Economy and Competitiveness under grant number DPI2015-64170-R(MINECO/FEDER).

E-mail address:leonardo.acho@upc.edu.

Peer review under responsibility of China Ordnance Society.