FU Bin,GUO Hang,CHEN Kang,FU Wenxing,WU Xingyu,and YAN Jie

1.School of Astronautics,Northwestern Polytechnical University,Xi’an 710072,China;

2.Xi’an Microelectronic Technology Institute,Xi’an 710065,China

Abstract:This paper proposes a multiple-constraints-guaranteed midcourse guidance law for the interception of the hypersonic targets.In traditional midcourse law design,the constraints of the aero-thermal heating are rarely taken into consideration.The performance of the infrared detection system may be degraded and the instability of the flight control system may be induced.To address this problem,a state-constrained model predictive static programming method is introduced such that both terminal constraints(position and angle)and optimal energy consumption can be ensured.As a result,a sub-optimal midcourse guidance,guaranteeing the aforementioned multiple-constraints to be never violated,is synthesized.Simulation results demonstrate the effectiveness of the proposed method.

Keywords:midcourse guidance,model predictive control,optimal control,state constraint.

1.Introduction

Hypersonic weapons with speed higher than 5 Ma and a cruise range more than thousands miles will surely play a key role in the future warfare[1–3].To tackle this threat,massive work on the defense system has been done[4–8].Dwivedi et al.[9]developed a lead angle constraint optimal midcourse guidance which can generate the desired optimal terminal condition.Sahu et al.[10]developed an optimal terminal impact angle constrained midcourse guidance law with near-zero lateral acceleration in the terminal phase.Indig et al.[11]developed a near-optimal spatial midcourse guidance law.In this design,the terminal angular constraint is formulated in terms of a required difference between the interceptor and the target flight directions.Tekin et al.[12]developed a two-phased proportional navigation guidance scheme.The look angle and acceleration constraints are handled by online designed navigation gains.Hu et al.[13]developed a trajectory shaping guidance law based on the virtual angle with the constraints of terminal impact angles and their time derivatives.

Model predictive static programming(MPSP)is a recently proposed method to handle the optimal control problem with terminal constraints[14].It can be perfectly implemented in a midcourse guidance problem.Oza et al.[15]developed an air-to-ground missile guidance law based on the MPSP method considering the nonlinear kinemics of the missile as well as the nonlinear aerodynamic characteristics of the missile.By reconstructing the performance index of the original system,the adjusted system could achieve a terminal hard constraint with an optimal energy consumption.Tripathi[16]applied the idea of MPSP into a reference trajectory tracking system.The author first designed a reference trajectory and then the tracking error of the reference trajectory was introduced into the performance index to be optimized.Joshi et al.[17]developed a generalized MPSP technique which could be directly implemented to the continuous-time system without any discretization process.With the generalized MPSP method,any higher-order numerical scheme can easily be incorporated in the solution process.

State constraints like the aero-thermal heating rate are also very important in a midcourse design.It is known that in the high speed cruise phase,the unlimited heating rate will cause a serious aero-thermal heating problem[18].It will lead to severe image distortion of the infrared detection and guidance system.To avoid this problem,the aerothermal heating rate constraint should be taken into consideration.The traditional MPSP method is able to handle the terminal constraints but lacks the ability to handle state constraints.Numerical solutions like Gauss pseudo-spectral methods can deal with multiple constraints well[19,20].However,the calculation process is in nature a time-consuming one.In view of the limited on board calculation capability,such a complex algorithm is not acceptable in practice.

The main objective of this work is to design a computationally efficient multiple constrained optimal guidance law.First,a slack variable method is introduced.This method could transcribe the state inequality constraint into an equality constraint by adding a set of slack variables.The original system together with the slack variable dynamic forms an augmented system.In the augmented system,a virtual control signal will ensure the original system moves within a constraints-guaranteed region.Secondly,a state-constrained MPSP(SC-MPSP)method is designed.The discrete form state update formula is established using the virtual control signal.And then,the Jacobi matrix of the augmented system is derived.And revision of the virtual control signal can be calculated from the residual of the terminal constraints.Finally,the real control will be resolved according to the slack variable dynamic.Thus,the main contributions in this paper are:

(i)An optimal midcourse guidance law with the heating rate constraint is proposed to deal with the hypersonic interception problem.

(ii)Compared with[21,22],the proposed method provides better performances in the aspect of energy consumption,calculation efficiency as well as state-constrained ability.

The rest of this paper is organized as follows.In Section 2,the midcourse guidance problem is introduced.In Section 3,by using the slack variable approach,a state variable inequality transformation method is given.In Section 4,the SC-MPSP method is implemented to provide a midcourse trajectory with the terminal constraint and the heating rate constraint.In Section 5,numerical simulation is carried out to demonstrate the effectiveness of the proposed method.

2.Problem description

In a typical midcourse guidance problem,the interceptor is guided to a predicted intercept point(PIP)[23]with terminal angle constraints.Thus,the dynamic model of the interceptor will be given at first.And then,the state constraints and the terminal constraints will be discussed in detail.

2.1 Model description

The mass point model of the missile is given and the meaning of each parameter is illustrated in Fig.1.

Fig.1 Engagement model

States of the missile dynamics areVM,γM,ψM,xM,yM,zM,which represent the missile velocity,path angle,heading angle and positions on theX,Y,Zaxis.TM,DMare the thrust and the drag force of the missile.Srefis the reference surface,CDis the drag coefficient of the drag force,mMis the mass,andazandayare the acceleration of the missile in the pitch plane and the yaw plane.gis the gravity constant.

Considering a large scale interception scenario,the air density varies with the altitude,so the air densityρis a function of altitudezM,the experience fitting formula is as follows:

where,k1=1.226,k2=-1/7 315.

In the practical sense,the missile aerodynamic varies with the Mach number and the altitude,but in the midcourse guidance problem,the variation of the drag force coefficientCDis not obvious.Thus in this paper,the drag force coefficientCDis treated as a constant number.

2.2 Max heating rate design

During the supersonic flight,extremely high speed flow may cause serious problems to the infrared detection system:(i)High speed flow may lead to a severe aero-thermal heating problem to the infrared guidance system.(ii)High speed flow may cause irregular flow field compression which will result in light field distortion.Therefore,it is necessary to constrain the aero-thermal heating rate.

wherernis the nose radius,ρ∞is the flow density,ρslis the sea level air density,V∞is the flow speed,andVcois the earth orbit speed.When analyzing the midcourse guidance problem,rn,ρsl,Vcocan all be treated as constants.Therefore,a normalized heating rate can be given as

whereρ,Vrepresent the air density and the missile velocity,respectively.

2.3 Terminal constraints design

The midcourse guidance method will have to satisfy the following terminal constraints:

(i)The miss distance should be less thanε1;

(ii)The residual error of the final path angle and the desired path angle must be within a limit as follows:

(iii)In the guidance process,the normalized heating rate must be within the limit as follows:

3.Transformation of state inequality constraints

Jacobson and Lele proposed a method by introducing a slack variable method to transform the system with the state inequality constraint to a higher order system with the sate equality constraint[25].

Consider a nonlinear system like this:

The state inequality constraint is as follows:

wherex∈Rn,u∈R and the relative degree of the state constraintS(x)to the control input isq.And then,a slack variableαsv(t)can be introduced to transform the system.

If∀t∈[t0,tf],αsv(t)is subject to the equation:

Knowing thatis permanently non-negative,the constraint(9)will be satisfied naturally.

By introducing the slack variable method,the constraint(5)can be transformed into the following equation:

As mentioned above in the former section,we need to keep doing derivation to(11)until the control signalsazandayare explicitly presented in a certain order of the derivative.

The first order of derivative of(11)provides

It does not explicitly contain a control signal,so we keep on doing derivation.A second order derivation is as follows:

Equation(13)can be presented as

The new control signalα2can be expressed with the original control signalaz:

In the meantime,the original control signalazcan be expressed with the new control signalα2as

Substitute(17)into the original system(1),and the augmented system is given:

4.SC-MPSP method

Basic principles of the model predictive static programing method are not presented in detail.Only the main results are introduced.

Consider a discrete system as below:

whereX∈Rn,U∈Rm,Y∈Rp,andk=1,2,...,N-1 represents the steps.

Under the necessary condition of the optimal control theory,the revised controlUkcan be calculated as follows:

whereis the last control history being optimized.dUkis the correction term to modify the control history.Rk＞0 is the weight coefficient.

Using Taylor series expanding to expandYN,the high order of the series is neglected:

The calculation of everyBkis very efficient in an iterative way as shown below.

First,theis defined as

Fork=N-2,N-3,...,1,all theBkcan be calculated literately as

To implement the SC-MPSP method,another concern is the need for a specified ending step.However,in the scenario of the interception,the ending time point is not easy to be predicted.Thus we have to choose another variable other than time to describe the equations.

Considering a certain interception mission,the ending step of the downrangexis always determined,and the downrange is also a monotonic variable.Therefore,it is more convenient to use downrangexas an independent variable instead of the timetto define the prediction interval and the steps.

Hence(18)can be rewritten as

Present the system into discrete form

where the state and the control are

The state transform function is

The initial value ofis as follows:

wherehis the step length.

After the dynamic model is determined,we can apply this method into the midcourse guidance problem,according to the following steps:

Step 1Initialize the first control history ofayandα2.In order to provide a reasonable control history,a PN guidance law with the impact angle constraint is introduced here.The guidance law is as follows:

whereazandayare the acceleration commands in the pitch plane and the yaw plane;kγ1andkψ1are the navigation ratio;are the line of sight rate;Ris the relative range;γdandψdare the desired path angle and the heading angle.The initial value ofα2is calculated using(16).

Step 2Calculate the step lengthhbased on the initial valuex1,the final valuexk,and the total step numbersN:

Step 3Use the initial control history to integrate the dynamic system,and to get the predicted output at the final step.By comparing the desired output and the predicted output,the error dYNis given as follows:

whereyandzare the positions,γandψare the path angle and the heading angle,the subscriptsfandddenote the final value and the desired value of each output.

Step 4 If any element of dYNis greater than the designed residual limit,then dYNwill be used as a feedback term to modify the control history by using(20).Then,the modified control history will be substituted into the third step to recalculate the final step and the residual error,until the residual error is within the limit.

5.Simulation and analysis

To validate the effectiveness of the proposed method,this part of the paper will present a numerical simulation in a typical interception scenario.To show the advantage of this proposed method,another two guidance laws are introduced as baseline design.

5.1 Baseline design:nonsingular terminal sliding mode(NTSM)guidance

In[22],Kumar proposed a finite time convergent sliding mode guidance law which is able to intercept targets at desired impact angles with negligible miss distance.

5.2 Baseline design:Radau pseudospectral method(RPM)

The Pseudospectral method has been widely used in the optimal control problem recent years.In the RPM[21],a continuous time optimal control problem is transcribed to a finite-dimensional nonlinear programming(NLP)problem.The resulting NLP problem will be solved by a sparse nonlinear programming solver named SNOPT.

5.3 Simulation scenario

The simulation scenario is set in the midcourse part of a hypersonic target interception mission.And the initial conditions are shown in Table 1.

Table 1 Initial condition

5.4 Simulation results and analysis

The simulation results are shown in the following figures and tables.However,before any further discussion about the guidance performances,the time efficiency should be compared first.

In Table 2,it can be seen that the RPM needs more than 8 000 iterations in a single guidance problem.As a result,the guidance planning costs about 10 s.While the SC-MPSP method needs only a four times iteration,and the total time cost is only 0.255 s.The NTSM is not an iteration method,so it costs only 0.023 s.

Table 2 Iteration numbers and total time cost

To see from Fig.2 and Fig.3,the trajectory and the overload command of the SC-MPSP method and the radau pseudospectral method look very alike.The overload command using the NTSM guidance law fluctuates drastically in the ending phase.This ending phase fluctuation is a general defect of the proportional navigation(PN)type guidance law which is all based on the line of sight rate.Though the NTSM guidance law is deduced using the sliding mode control theory,it can still be seen as a PN type method.The NTSM guidance law can be divided into two parts:the LOS rate part and the terminal error adjustment part.As the missile approaching the target point,both parts diverge.Consequently,the overload command rises up quickly.

Fig.2 Missile trajectory

Fig.3 Overload command profile

However,the overload command provided by the SCMPSP guidance law is calculated under the necessary condition of the optimal control theory.In the SC-MPSP frame,the task of the elimination of the terminal constraint error dYNis disassembled into every previous step of the control sequences.Control commands of each step will be revised by the Jacobi matrixBk.Therefore,the optimization process is a coordinate action in the predictive horizon.Asa result,the control command adjustment is arranged all through the time series.The sudden rising of the demanded control effort in the terminal phase is perfectly avoided.

To compare the overload commands from the aspect of energy consumption,an energy cost functionP(N)is given as

whereuiis the control command of the stepi,and the sum of therepresents the total energy consumption.In Fig.4,the energy cost is in line with the overload command profile.The energy cost trend of the RPM is very close to that of the SC-MPSP,expect for the very initial part.The NTSM guidance law sees an even lower cost than the RPM in the first few steps,but the adjustment in the terminal phase gives a sudden surge in the total cost.

Fig.4 Energy consumption

At the end,the NTSM guidance law costs 1 615g2,the RPM costs 908g2,and the state-constrained MPSP method costs only 711g2.

In Fig.5,it can be seen that the path angles of all the three methods are well constrained and finally reach the desired value.The final errors of the flight path angle are 0.62°for the NTSM guidance law and 0.04°for the SCMPSP,0.02°for the RPM.The NTSM guidance law shows a gentle fluctuation and the convergence is slow,while the other two curves turn out to be simpler and more effective.

Fig.5 Path angle profile

In Fig.6,both the SC-MPSP method and the RPM enable the required max heating rate constraint.Detailed results are shown in Table 3.The max heating rate of the SC-MPSP is 4.83%lower than that of the NTSM guidance scheme.The final heating rate of the SC-MPSP scheme is 31.25%,lower than that of the RPM scheme.After comparing the simulation results of the three methods,it can be concluded that the NTSM guidance could achieve the terminal constraints but lacks the ability to restrain the heating rate.Both RPM and the SC-MPSP method could handle the multiple constraints problem but provide a much more efficient calculation process.

Fig.6 Normalized heating rate

Table 3 Heating rate comparison

6.Conclusions

In this paper,the design of an aero-thermal heating rate constrained midcourse guidance law is motivated by the fact that serious aero-thermal heating will cause damage to the detection system.To solve this problem,the slack variable method is introduced to transcribe the heating rate inequality constraints into equality constraints.Together with the original missile model,an augmented system is established.By using the SC-MPSP method,the original system dynamic is only able to move within the constrained region.The proposed method does not need to solve the large scale NLP problem,which also makes it a computationally efficient algorithm.Simulation results show that this method could satisfy multiple constraints in the midcourse guidance.