LI Ran,WEN Qiuqiu,TAN Wangchun,and ZHANG Yijie

1.School of Aerospace Engineering,Beijing Institute of technology,Beijing 100081,China

2.China North Industries Group Corporation,Beijing 100081,China

3.Beijing Institute of Aerospace Automatic Control,Beijing 100081,China

1.Introduction

Impact angle control has been widely used to increase the warhead effect for homing missiles.For example,anti-ship missiles rapidly climb and attack the deck near the target ship after sea-skimming to avoid radar and infrared detection.In this condition,air-to-ground missiles or guided bombs with penetration warheads need a high impact angle to increase the penetration depth.To solve the impact angle control problem,the optimal guidance law with impact angle constraint(OGL/IAC)has been developed[1–12].However,from a practical standpoint,the accurate ordering of impact angles is not as serious as that of miss distance in terminal guidance.In a generation,the impact angle only needs to not be lower than a minimal value(e.g.,60°),which can ensure the full effectiveness of a warhead[13].Meanwhile,the other constraints must be considered within the impact angle guidance process.

The prime problem comes from the missile’s seeker.For a homing missile,the seeker supplies necessary target information to implement the guidance law.A typical homing missile uses an inertial stabilized gimbaled seeker.The gimbal-type seeker is installed on a platform that is stabilized by the gimbaled system,which uses servo motors and rate sensors[14].As a result,the field-of-view(FOV)angle of the gimbaled seeker is limited by the maximum gimbal angle and is not more than±50°[14].When using a strap down seeker,which is rigidly fixed to the missile body,the available FOV angle becomes even narrower..On the contrary,while implying the typical OGL/IAC,an extreme maneuvering command will generate and significantly change the flight trajectory,which leads to an FOV angle that apparently increases[15–17].Therefore,the maximum seeker’s FOV angle constraint(FOVC)should be limited to ensure the normal functioning of a seeker.

There are few studies examining guidance problems with FOVC.Xing[18]designed an FOV angle control guidance law by minimizing the velocity component perpendicular to the line-of-sight(LOS)to reduce the seeker look angle.Lee et al.[19]derived a sliding mode guidance law to keep the look angle within the FOV angle by employing a Lyapunov-like function with a sliding mode control methodology.Sang and Tahk[20]proposed a guidance law switching logic between the original guidance law and the constant look angle guidance law using Lyapunov theory.By considering the look angle as an inequality constraint,Park et al.[21,22]proposed an optimal guidance law with FOVC.By adding the feedback of time-to-go error and the biased item composed of the seeker FOV limit,the proportional navigation guidance law with FOV angle limited was proposed[23–25].In[26],a switched-gain guidance scheme,based on proportional navigation guidance law(PN),was developed for impact angle control.Therein,navigation gains are determined by numerical solutions for handling the look angle and acceleration constraints.A two-phase guidance method with PN or biased proportional navigation guidance law(BPN)as guidance options for the first phase was presented in[27].In[28],implementing BPN for the first phase,missile trajectories close to the optimal ones are generated.

However,notice that prior research has mainly tried to derive specific guidance laws to control a seeker’s FOV angle in certain phases of guidance,bringing problems such as the complex switching operation between different guidance laws and negative influence to the accuracy of the original guidance law.Moreover,previous works on the impact angle guidance problem with FOVC are rare.

In this paper,focusing on the addition of the maximum seeker’s FOV angle constraint to the impact angle guidance process,we investigate an adaptive weighting optimal guidance law(OGL/AIAW).The concept of OGL/AIAW is based on the prediction-control principle[11]and varying coefficient of weighted optimal guidance theory[29].To this end,a new generalized optimal guidance law with a changeable impact angle weighting(IAW)coefficient is derived.The introduced IAW supplies an available way to modify the performance weighing of the guidance law to adjust multi-form guidance constraints.After integrating the closed-form solution of a guidance command with linearized engagement kinematics,the analytic predictive models of various constraints are built,and the available valued range of IAW corresponding to constraints is therefore certain.A minimal impact angle limitation is introduced to ensure that OGL/AIAW satisfies the minimal impact angle order even if IAW is too small.According to the given scheme,the suitable value of IAW is calculated in real time and further deters coefficients of the guidance law.

2.Generalized optimal guidance law

2.1 Derivation of guidance law

Consider the two-dimensional homing guidance for a stationary or a slowly moving target shown in Fig.1.In the figure,Vmand amare the missile’s velocity and acceleration.R,q,θ and qFare the missile-target relative distance,LOS angle,heading angle and desired impact angle,respectively.

The equations of motion in this homing guidance problem are given by

Fig.1 Two-dimensional homing geometry

Under the assumption that:

i)The dynamics of the control system can be neglected and have am=ac,where acdenotes the acceleration com-mand of the guidance law.

ii)Vmis constant and θ is small.Then we have

As a result,the linearized equations of motion is given by the state-space representation

where

and

To consider the following optimal problem,we find the control input u(t)that minimizes cost function J,defined by

where SFand R are terminal constraints weighting and control weighting matrices,respectively,which are positive semi-definite.tFpresents the total flight time.xFdenotes desired terminal values of state and is expressed as follows:

where yFdenotes the desired terminal position and has yF=0 if ordered to hit the target.˙yFpresents the terminal vertical velocity.According to the geometry definition in Fig.1,we have

According to(4),notice that˙yFis approximate to qF;thus,the realization of˙yFis equivalent to the control impact angle.

The weighting matrices SFand R are chosen as

Here,cIPWis the impact position weighting coefficient and used to control the terminal miss-distance of the guidance law.cIAWis the impact angle weighting coefficient and used to control the impact angle.

The proposed optimal control problem is to find a typical linear quadratic optimal terminal controller[30,31].The state feed-back solution is given by

where Φ(tF,t)denotes the state transition matrix to propagate the state from t to tF.

Let Φ(tF,t)be the Laplace transform of the matrix A in(1);thus,we obtain

The terminal states xF(t)can be expressed in terms of the conditions at any initial time and shown in(8).

Substituting(1),(3),(5),(6)and(7)into(8)yields

where tgopresents time-to-go and can be calculated in method given in[8].In the derivation of the guidance law in this paper,we use approximate expression tgo=tF-t for simplification.

Substituting(1),(3),(5)and(7)into(6),an alternative expression of the optimal control input is given as

Substituting(9)into(10)and replacing u(t)with the acceleration command ac(t),we obtain the generalized formulation of the varying weighting optimal guidance law.

Let cIPWof(11)equal in finity to ensure that the missdistance is low enough and keep cIAWas a variable;then,the basic expression of OGL/AIAW is obtained as

Referring to Fig.1,if considering that the angle is small enough,the LOS angle and its rate can be expressed as follows:

Rearranging(13)yields

Let t=tfof(14)and tgo=0;thus,the terminal constraint˙yFcan be expressed as

where qFcmeans the standard impact angle constraint input.

Substituting(14)–(15)into(12),OGL/AIAW can be rewritten as

The OGL/AIAW includes two parts:the former part contains the rate of the LOS angle,which is similar to the normal PN and used to guarantee miss-distance,and the latter part contains the LOS angle and impact angle constraints,which are responsible for controlling the impact angle.The coefficients of the two parts contain the variable cIAW;thus,the guidance law and its performance will change if we set different cIAW.

LetcIAWandcIPWof(11)both be infinity,then we have

or,expressed in a form similar to(13),

Note that the OGL/AIAW has degenerated into the classic OGL/IAC[1,2].Thereby,in a sense,OGL/IAC can be con-sidered as a special case of desiring impact angle accuracy to be maximum in OGL/AIAW.

2.2 Closed-form solution of guidance command

To build predictive modes related to guidance constraints,the closed-form solution of the guidance command is necessary.In other word,we need express acceleration command as the form of initial and terminal states.

Relating to initial time t=0,(7)and(8)can be expressed as

And

where x0denotes the initial values of state and x0=

Substituting(1),(3),(5),(6)and(15)into(16)yields

Substituting(21)into(10)and replacing u(t)with the acceleration command ac(t)yield

At the initial and terminal time,we have

Here,ε0presents the initial heading angle related to the LOS.Substituting(23)into(22)and replacing time t with non-dimension timeˉt=t/tF,the non-dimension guidance command is obtained as

According to the definition ofˉt,we haveˉt∈[0,1].Then,the velocity and position along the y-axis can be obtained through the integration of(24)

Substituting(24)into(25)and solving the equations yield

From(24)and(26),the analyticmodesdescribingthe variation of acceleration,velocity and position over the whole guidance process are given,making a foundation for further building predictive models of other flight states.

3.Predictive models of various constraints

3.1 Impact angle constraint

As shown in Fig.1,the impact angle is approximated as the LOS angle at the terminal time.

Substituting(24)into(27)and lettingˉt=1,the impact angle is obtained as

A new coefficient k=ε0/qFcis defined to simplify(28);thus,we obtain the relationship between the actual impact angle and its constraint input

It is obvious that cIAWqFpresents the low bound of cIAWdecided by the minimal impact angle constraint.Whenever the real cIAWis bigger than cIAWqF,the actual impact angle is accepted and approaches qFcgradually with increasing cIAW.

3.2 Seeker’s FOV angle constraint

To address directly the seeker’s FOV angle constraint attached to the missile body,the look angle φ can be defined to present the FOV angle[13].

Substituting(27)into(31)and replacing ε0with ε0=kqFc,we have

Differentiation ofˉt yields

The time corresponding to the maximum FOV angle is obtained by solving the function˙φ(ˉt)=0.

According to(34),the condition foris

If(35)is true,a maximum seeker’s FOV angle can be obtained through the substitution of(34)into(32)

Here,φmaxmeans the maximum seeker’s FOV angle.

Instead of φmaxby FOVC input φcin(36),the corresponding cIAWis given in(37)and denoted as cIAWφfor brevity.

Consider the case where(35)is invalid;then,we haveThis means that the FOV angle will reach its maximum value at the initial time.Thus,the value of φmaxequals ε0directly.

As discussed above,the impact angle control accuracy will be the highest when cIAWis large enough.However,the FOV angle will increase as well.If we let cIAWin(36)be infinite,the maximum value of φmax,denoted as φcmax,is obtained as

On the contrary,if we let cIAWequal zero,then the minimum value of φmax,denoted as φcmin,is obtained as

Combining(38)and(39),once the guidance conditions are given,a controllable range to φmaxcan be calculated to check whether the FOVC input is achievable.

4.Adaptive IAW calculation scheme

According to the predictive models,a cIAWcalculation scheme is designed and shown in Fig.2.

In the scheme,the first step is to calculate φcmaxand φcminaccording to(38)and(39).Then,based on the result of comparison with φc,three cases will be dealt with separately.

Case 1If,it means that the FOV angle must be more than φcin the following guidance process with out any cIAW.In this case,a new IAC or FOVC should be chosen.

Fig.2 Work flow of IAW calculation scheme

Case 2Ifthe FOVC will be satisfied always for all cIAW;thus,we just need to set cIAWto be in finite.

Case 3Ifit is necessary to further con firm the real value of cIAW.Then,the upper bound cIAWφand lower bound cIAWqFare calculated according to(30)and(37).In theory,any cIAWwithin the scope of[cIAWqFcIAWφ]is available.However,considering the errors in the model assumption and the probable guidance disturbances,a sufficient margin is necessary in engine practice.At the initial time,we hope cIAWis low and far from the upper bound because the trajectory will climb up and improve the FOV angle as usual.However,at the final time,the flight path has been moved toward the target with a small FOV angle.Thus,a large cIAWis needed to improve the impact angle accuracy.As a result,cIAWis designed to be a function of time and is expressed as follows:

where τcdenotes the time constant used to control the increasing speed of cIAW.The smaller the value of τc,the closer to the upper bound cIAWφat the terminal time of guidance.We usually set τc=ctFwith c=0.5-1.0 in practice.

Once cIAWis known,the final step is to re-calculate the coefficients of OGL/AIAW according to(18).As a result,OGL/AIAW is adaptive for current flight status and always guarantees the guidance constraints.

The work flow of OGL/AIAW in a close-loop guidance system is shown in Fig.3.The required input variables include velocity V,heading angle ε0,estimation of tgo,which comes from the measurements of the inertial navigation system(INS)on board.Under the given calculation scheme,cIAWis first obtained in real time and later used to renew the guidance law.The updated OGL/AIAW receives guidance information from the seeker and INS and then generates the acceleration command to control missile flight.

Fig.3 Guidance system with OGL/AIAW

5.Simulation results

In this section,the performance of the guidance strategy is verified through simulation.The following simulation example is considered in this paper:the relative spatial motions of the missile and the target are described only in the vertical plane,and the target is stationary.Therefore,the simplified pursuit model between the missile and the target in the non-rotating reference frame (inertial coordinate)can be described as follows:

where(xr,yr)denotes the relative positions between the missile and the target and R0represents the initial range along the x-axis of the inertial coordinate.

The simulation assumes that the autopilots and seeker dynamics can be described by a first-order linear time invariant(LTI)system.

where τayanddenote the time constants of the pitch autopilot and seeker,respectively,denotes the true rate of pitch of the LOS angles and˙qyrepresents measurement from the seeker.The relevant parameters used in the simulation are summarized in Table 1.Here,the initial range equals

Table 1 Simulation initial conditions and parameters

In the simulation,as a comparison,the OGL/AIAW and OGL/IAC in(14)are considered in the simulation.Table 2 gives the values of all guidance constraints input,and the coefficient c of(40)equals0.5.From the compared profiles of the simulation results in Figs.4–7,although the impact angles of the two guidance laws arrive at the desired values,the trajectory shape and FOV angle are quite different.The trajectory of OGL/AIAW does not climb up sharply at the initial phase of guidance,thereby reducing the maximum seeker’s FOV angle and not exceeding the constraint,as with OGL/IAC.However,when getting close to the target,more negative acceleration command is generated by OGL/AIAW,which makes the trajectory dive to increase the final impact angle.

Table 2 Guidance constraints input

Fig.4 Trajectory profiles of OGL/AIAW and OGL/IAC

Fig.5 Seeker FOV angle profiles

Fig.6 Heading angle profiles

Fig.7 Acceleration command profiles

The detailed reasons causing such difference can be found in the cIAWprofile shown in Fig.8.At the initial time,cIAWis limited by a small upper bound,which means the FOV angle must be controlled strictly.Then,cIAWbecomes larger with increasing upper and lower bounds,and thus the impact angle turns into the main constraint of the guidance law and more acceleration commands are used to acquire a high impact angle.The calculation scheme let cIAWapproach the lower bounds when we hope it would be small and approach the upper bounds for a large value order.

Fig.8 cIAWφ,cIAWqFand cIAWprofiles

In this case,different initial conditions and noises are adopted.First,the guidance range varying from 4 km to 10 km is considered to further test the robustness of OGL/AIAW.From the results given in Figs.9–11,the OGL/AIAW can always satisfy the FOVC with low miss-distance and high impact angle accuracy in spite of the varying guidance range.Among this,the maximum seeker’s FOV angle in all conditions is not over 27°and has a 3°margin distance from the constraint.Thus,a good adaptation of OGL/AIAW for varying guidance conditions is supported.On the contrary,although the maximum FOV angle of OGL/IAC possesses high impact angle accuracy,as well,it cannot satisfy the constraint in most simulation conditions.

Fig.9 Miss-distance profiles with different initial guidance ranges

Fig.10 Impact angle profiles

Fig.11 Maximum seeker FOV angle profiles

Next,we assume white-noise-type noises exist in the LOS angle,LOS angle rate and time-to-go estimation,the variances of which are given in Table 3.Applying the Monte Carlo simulation theory,Table 4 gives the statistical results of a total of 200 simulations,and the detailed distributions of miss-distance,impact angle and maximum seeker’s FOV angle are shown in Figs.12–14.Although much noise exists,both OGL/IAC and OGL/AIAW can achieve low miss-distance and realize the impact angle well,which presents high robustness of the optimal guidance law to guidance disturbance.Moreover,OGL/AIAW is always able to control the FOV angle below the constraint order, and the maximum FOV angle is not more than 29°.This finding is due to use of an adaptive calculation scheme.As a result,a high robustness of OGL/AIAW towards perhaps guidance noises and disturbances is proved.

Table 3 Disturbances and their values

Table 4 Means and standard deviations of all guidance constraints

Fig.12 Miss-distance distribution

Fig.13 Impact angle distribution

Fig.14 Maximum seeker’s FOV angle distribution

6.Conclusions

In this paper,by including the changeable impact angle weighted coefficient in the cost of terminal constraint control,a new optimal guidance law to control the maximum seeker’sFOV angle,impact angle and miss-distance is pro-posed.Through the analytic predictive modes given,,which contain certain initial conditions,constraint inputs and the parameter cIAWas a variable,some flight states can be predicted directly to determine whether the guidance constraints are satisfied.A calculation scheme is given to acquire a suitable value of cIAWin real time during the whole guidance process, hence ensuring that OGL/AIAW is adaptive for varying flight states and able to juggle guidance constraints and accuracy.As a result,the proposed guidance law is able to produce different acceleration command profiles and trajectories,depending on the choice of cIAWindependently,to satisfy guidance constraints simultaneously.

Results of numerical simulation demonstrate that OGL/AIAW can effectively solve the limitations of a seeker’sFOV angle in impact angle guidance, even in cases of arbitrary guidance initial conditions.The low cIAWlimits the trajectory climb-up and avoids a high FOV angle in the initial guidance phase;then,it increases when getting close to the target, finally resulting in the acquisition of the desired impact angle.Because the impact position item is directly set to infinity,as for the PN,the priority of miss-distance is guaranteed.Thus,all guidance constraints and accuracy orders are well met by OGL/AIAW.

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