Integrated guidance and control design of a flight vehicle with side-window detection

Tinyu ZHENG,Yu YAO,*,Fenghu HE,Denggo JI

aDepartment of Astronautics,Harbin Institute of Technology,Harbin 150080,China

bBeijing Institute of Nearspace Vehicle’s Systems Engineering,Beijing 100083,China

1.Introduction

In order to satisfy the requirements of intercepting high-speed maneuvering targets,flight vehicles are generally flying at a high-speed and equipped with IR imaging seeker.1However,as the flight vehicle flying at a high-speed,the severe aerodynamic heating on the surface of the IR seeker deteriorates its detection performance.Therefore,the IR seeker has to be mounted to the side of flight vehicle’s head,namely using the side-window detection technique.As the side-window constrains the Line-of-Sight(LOS)and the attitude of the flight vehicle,the guidance problem under constraints is necessary to be studied.

The guidance problem under constraints is essentially a control problem for a class of nonlinear system with state or input constraints.On the research of state constraints,the terminal constraints are mainly considered,which is the state constraints at the end of the terminal guidance.Some methods are proposed based on the optimal control,2–4sliding mode control,5–9polynomial guidance,10,11and switched-gain Guidance.12However,rare research is available for state constraints during the whole terminal guidance process.Considering the impact angle constraint for antiship missile systems,Lee et al.2-developed a guidance law to weigh the guidance performance and the control energy costs.Considering the terminal impact angle constraints,Ryoo et al.3analyzed the intercept situations under different impact angles and developed a guidance law based on the optimal control to minimize the miss distance.However,the target maneuver and the model uncertainty are neglected in the research.Lee et al.4presented an optimal linear time-varying guidance laws for controlling impact angles to meet the impact angle constraints.The guidance problem was formulated into an inverse optimal problem and solved,and the relationship between the guidance coefficients and the guidance performance was analyzed.Li et al.5presented a guidance law to intercept non-maneuvering targets at a desired impact angle by combining the SMC method with the adaptive neuro-fuzzy inference system,which can enhance the robustness and reduce the chattering of the system.Considering the interception of maneuvering targets,Ref.6developed a guidance law for multiple constraints based on Sliding Mode Control(SMC).By separating the switching surfaces for the impact angle constraint and the homing constraint,the constraints were considered separately and the two surfaces were associated by introducing an appropriate virtual controller.Wu and Yang7developed an integrated guidance and control law for missile with terminal impact angle constraint based on variable structure control approach.The relationship between terminal impact angle and desired LOS is described and the integrated guidance and control law is designed based on the sliding mode control approach,whose design parameters can be attained directly with a linear matrix inequality.Considering the desired terminal impact angle and hit-to-kill interception,Wang and Wang8proposed a partial integrated guidance and control law based on SMC,which can get a minimum miss distance.Kumar9proposed a finite time convergent guidance law based on nonsingular terminal sliding mode control theory,which can intercept maneuvering targets at desired impact angles.Simulations with a realistic interceptor model show that the guidance law can adapt to aerodynamic variations and different initial engagement geometries and impact angles.Fu et al.13–15proposed a Sliding Mode Control method with Unidirectional Auxiliary Surfaces(UAS-SMC)for a class of nonlinear systems with state constraints.The state constraints are transformed into the unidirectional auxiliary surfaces and satisfied by designing the control law to ensure the states trajectory moving inside the unidirectional auxiliary surfaces.However,this method cannot be used for the timevarying state constraints or the mixture of state and input constraints.

As for input constraints,the constraints caused by the discrete input of reaction jet are considered,some methods are proposed based on the model predictive control16,17and virtual control.18,19Actually,the virtual control is complete control allocation by designing an appropriate distribution law,while the input constraint problem is ignored and avoided in the control design.

In brief,although there are some researches on the state and input constraints,the time-varying state constraints or the mixture of state and input constraints are not considered.In this paper,side-window detection would cause the state constraints in the whole terminal guidance course,while the reaction jet would cause the input constraints.Furthermore,to avoid constraints time-varying,the guidance and control problem is studied in the body-LOS coordinate system,leading to the coupling between the guidance control and the attitude control.Therefore,Integrated Guidance and Control(IGC)problems with constraints needs to be considered.Because the IGC problems has the multiple time scale characteristic,the singular perturbation theory is a possible way for the control design.

The study of nonlinear singular perturbation theory began in the 1990 s.In the stability analysis,the composite Lyapunov method is an effective way,20whose main idea is to decompose the original system into two subsystems,establish their Lyapunov respectively,and define the weighted sum of the two Lyapunov functions as the composite Lyapunov function.Grujic21studied on the existence of composite Lyapunov function in an early time.Saberi and Khalil22extended the composite Lyapunov function method to the general nonlinear system by choosing quadratic form Lyapunov functions and proved the exponential stability.In Ref.23,the exponential stability of singular perturbed systems was studied.These early studies considered the general nonlinear system,therefore the assumptions were harsh and difficult to be applied to the actual control design.In recent years,some singular perturbation methods were proposed for some particular forms of nonlinear system.In Ref.24,the stability of a class of time-varying nonlinear singularly perturbed systems was analyzed and a control method was proposed based on the gain scheduling approach.In Ref.25,the problem of semi globally practical stabilization was considered for a class of affine nonlinear singularly perturbed systems with uncertain parameters.Assuming that the fast subsystem dynamic could be asymptotically stabilized and the slow subsystem was partly input/output linearizable,the composite Lyapunov function was established and the robust state feedback controller was proposed based on the Lyapunov direct method.

The singular perturbation theory has wide applications in guidanceand controlproblems.Considering thenonminimum phase characteristics of the aerodynamic force controlled missile,Lee and Ha26proposed a control law based on the singular perturbation theory,which relaxes the angleof-attack constraints.In Ref.27,the trajectory optimization and guidance of the hypersonic vehicle was studied.Considering the kinematic and dynamic characteristics of the hypersonic vehicle,the control model was divided into three time scales,and the optimal control law is designed based on the singular perturbation theory.Bertrand28presented a hierarchical controller for unmanned aerial vehicles.Considering the time-scale separation between the translational dynamics and the orientation dynamics,the position and attitude control laws are designed successively based on singular perturbation theory.

In our previous work29,the relative motion equations in the body-LOS coordinate system are derived to describe the guidance and control problems with side-window constraints and the equivalence between the body-LOS rate convergence and the LOS rate convergence is proved.Based on the work in Refs.29,30,IGC model with side-window constraints is decomposed via singular perturbation method.Based on the IGC model in Ref.30,the singular perturbation-based design approach is extended to a general affine nonlinear system in this paper.The significance of time scales is analyzed and the lower bound of the time scale perimeter is provided.Then,the IGC problem is decomposed into the control design of the quasi-steady-state subsystem and the boundary-layer subsystem based on the proposed singular perturbation approach.The receding horizon control is applied to design the two subsystems.

The rest of this paper is organized as follows:In Section 2,the guidance and control problem for the flight vehicle with side-window detection is formulated into an IGC problem with constraints.In Section 3,a singular perturbation-based design approach is proposed and applied to decompose the IGC problem and the control laws of the two subsystems are designed based on the model predictive control.In Section 4,the simulation results for the proposed method are given and analyzed.In Section 5,some concluding remarks and future work are given.

2.Guidance and control problem formulation

In this section,the guidance and control problem of the flight vehicle with side-window detection is analyzed and the constraints caused by side-window detection are given.Then,the guidance and control problem of the flight vehicle is modeled and formulated as an IGC problem with constraints.

2.1.Preliminaries and background

Before studying the guidance and control problem of the flight vehicle with side-window detection,we first define coordinate systems,Euler angles and parameter description that will be adopted.We introduce a new coordinate system,called Body-LOS coordinate system and define Body-LOS rate to describe the guidance and control problem with constraints.Then,the background knowledge of the guidance and control problem is provided.

We denote the inertial coordinate system byAxyz,the body-axis coordinate system byOx1y1z1,the ballistic coordinate system byOx2y2z2,the velocity coordinate system byOx3y3z3,and the LOS coordinate system byOxsyszs.

definition 1(Body-LOS coordinate system(Oxbybzb)).Ois the mass center of the flight vehicle.Oxbis the LOS between the flight vehicle and the target whose positive direction is pointing to the target.Oybis perpendicular toOxbin the longitudinal symmetry plane of the flight vehicle and its direction pointing upward is defined to be positive.Ozbis defined conforming to Right-Hand Rule withOxbandOyb.

definition 2(Elevation angle of body-LOS).is the angle between the projection of the LOS in the planeOx1y1.

definition 3(Azimuth angle of body-LOS).is the angle between the LOS and the planeOx1y1.

Fig.1 illustrates the orientation of transformation matrix which defines the transformation between the body-axis coordinate system and the body-LOS coordinate system:

Fig.1 Body-LOS coordinate system with respect to body-axis coordinate system.

definition 4(Body-LOS rate(ωb)).ωbis the rotation rate of the body-LOS coordinate system relative to the inertial coordinate system.

definition 5(Roll angle of LOS(γb)). γbis the angle between axisOyband planeOxsys.

Fig.2 illustrates the orientation of transformation matrix which defines the transformation between the LOS coordinate system and the body-LOS coordinate system:

The side-window seeker of the flight vehicle under consideration is mounted to the side of its head,as shown in Fig.3,where the vision of the flight vehicle has to be constrained.Therefore,in order to guarantee the LOS inside the side window vision,the elevation angle of body-LOSand the azimuth angle of body-LOSshould be maintained inFurthermore,the attitude of the flight vehicle is also constrained due to the side-window:In order to avoid the influence of the eddy current,the angle-of-attack should be less than the half cone angle of the interceptor’s head.Otherwise,the angle-of-attack should be remained positive to reduce the influence of aerodynamic heating.

Fig.2 Body-LOS coordinate system with respect to LOS coordinate system.

Fig.3 Side-window vision.

When we consider the guidance problem in the LOS coordinate system,the side-window constraints are described as dynamic constraints related to the attitude of the flight vehicle which is hard to deal with in the guidance and control design.Otherwise,the side-window constraints are described as time invariant constraints if we consider the problem in the Body-LOS coordinate system.So we will study the guidance problem of the flight vehicle in the Body-LOS coordinate system,which will cause the coupling between the guidance control and the attitude control.The integrated design of guidance and control is thus necessary.In the following,the guidance and control problem of the flight vehicle is formulated as an IGC problem with constraints.

2.2.Guidance and control model

In this subsection,the guidance and control model is established.First,the relative motion equations in the body-LOS coordinate system is derived.Then,the attitude dynamics model of the vehicle is given.Finally,the IGC model is obtained by combining the relative motion equations and the attitude dynamics model.

In the body-LOS coordinate system,the relative velocity between the flight vehicle and the target is expressed as

The relative acceleration between the flight vehicle and the target is expressed as

Ignoring the relative acceleration along the direction of the LOS(ax),Eq.(4)can be rewritten as

According to the definition of the body-LOS coordinate system,ωbis expressed as

Substituting Eqs.(6)–(9)into Eq.(5)and ignoring higher order terms,the relative motion equations in the body-LOS coordinate system can be obtained:

Substituting Eq.(7)into Eq.(6),the body-LOS rate is simplified as

When considering the terminal guidance problem in the LOS coordinate system,the convergence of LOS rate should be guaranteed.In other words,the direction of the LOS in the inertial coordinate system should be maintained.If it is the case,the flight vehicle can intercept the target successfully while the closing speed is negative.Namely,the LOS rate convergence is the aim of the guidance and control design in the LOS coordinate system.Because of the constraints caused by the side-window detection,the guidance and control problem would be considered in the Body-LOS coordinate system.Therefore,the aim of the guidance and control design in the Body-LOS coordinate system is obtained by the following theorem.

Theorem 129.The LOS rate is convergent(limt→∞˙qε=0andlimt→∞˙qβ=0),if and only if the body-LOS rate is convergent(limt→∞ωby=0andlimt→∞ωbz=0).

The proof of Theorem 1 is given in the Appendix A.

Remark 1.Theorem 1 implies that the flight vehicle can intercept the target,if the body LOS is convergent.Thus,when considering the guidance problem in the LOS coordinate system,the flight vehicle can also intercept the target,if the body-LOS rate convergence is guaranteed in the terminal guidance process.

Remark 2.As shown in Eqs.(10)and(11),the relative motion equations in the body-LOS coordinate system contain the terms of attitude rate,which cause the coupling between the guidance control and the attitude control,hence the integrated design of guidance and control is necessary.

Next,we will give the attitude dynamics model.As the flight vehicle under consideration is equipped with the rail and attitude reaction jet,the thrusts of the reaction jets are considered in the attitude dynamics model.The centroid dynamic equations in body-axis coordinate system are expressed as

α ∈ ［0，μ］

Now,we are ready to establish the IGC model of the flight vehicle.Since the IGC problems of the pitch and yaw plane are similar,we take the pitch plane as an example to illustrate the IGC design process in this paper.Substituting Eqs.(12)–(14)into Eq.(10),we get the IGC model of the flight vehicle’s pitch channel:

Since the side-window’s lateral sight of view is small and the flight vehicle’s lateral overload is weak,the influence of the yaw plant can be ignored while researching the guidance problem of its pitch plant.Thus,the IGC model Eq.(15)is simplified as

2.3.IGC problem formulation

Next,we will give the formulation of the IGC problem.Moreover,the characteristics of the problem will be analyzed.

According to Theorem 1,the aim of the control design is to makey→0,namely,to make the body-LOS rate convergent.

As shown in Eq.(18),the IGC problem of the flight vehicle is a stabilization problem of an affine nonlinear system with state constraints. The problem has the following characteristics:

(1)Multi-time scale

In the actual guidance course,the states about the guidance change slowly,while the states about the attitude control change rapidly.Namely,the IGC model has multi time scale characteristic.Thus,the model can be decomposed and simplified based on singular perturbation theory in the control design.

(2)State constraints

The elevation angle of body-LOS and the angle-of-attack are constrained in the entire guidance course.Once the constraints are not satisfied,the target is outside the vision of the side-window and the flight vehicle cannot intercept the target.Thus,the state constraints need to be considered in the control design.

(3)Continuous and discrete mixed inputs

As the interceptor equipped with the rudder and the reaction jets,some control inputs are continuous,others are discrete.Namely,the IGC model also has hybrid characteristic.

3.Singular perturbation-based IGC design

In this section,a singular perturbation-based design approach is proposed and applied to decompose the IGC problem into a quasi-steady state subsystem and a boundary layer subsystem.Then,the control laws for the two subsystems are designed based on the receding horizon control.

3.1.Singular perturbation-based model decomposition

Because of the multi time scale characteristic,the IGC problem can be decomposed and simplified based on singular perturbation theory.However,existing time scale decomposition methods are not applicable to the model.Thus,a new time scale decomposition method is proposed based on the singular perturbation theory.

We consider the following affine nonlinear system

Selecting the slow states in X as x and the fast states in X as z,we can rewrite the system Eq.(19)in the following form

where x is the slow states of the system,x∈Rn,z is the fast states of the system,u is the control inputs,u∈Rc,the subscript s represents the corresponding variables for the slow subsystem,the subscript f represents the corresponding variables for the fast subsystem,κ is the time scale parameter,κ＞0.

We start by def i ning the quasi-stable state subsystem and its state-feedback control.Let κ=0,we propose the following equation

Assumption 1.Eq.(21)has solutions.We define a suitable solution z=w（x，z）as the quasi-stable state of the system zs,namely

Substituting the quasi-stable state zsinto Eq.(20),we obtain a slow subsystem as follows,called quasi steady state subsystem

where usis the control input for the slow subsystem which is defined as

usis an exponential stable control law for subsystem Eq.(23),whose unique exponential stable equilibrium point is at x=0.Thus,there is a Lyapunov functionVs:Rn→R+such that

Remark 3.Eq.(29)implies that the quasi steady state zsis a variable related to x when the quasi-steady state subsystem is close-loop,which can also be expressed as zs（x）.

Next,we will define the boundary layer subsystem and its state-feedback control.Setting a time scale parameter as τ=t/κ,Eq.(20)can be rewritten as

Let φf（z）=|z|(Namely,φf（·）is a scalar function,when and only when z=0 is φf（z）=0),Eqs.(34)and(35)can be written as

And the state-feedback control of the system Eq.(20)is defined as

The following assumptions is given,which imply the relationship between the two subsystems.

Assumption 2.The system Eq.(20)satisfies the following Lipschitz-like conditions:

Remark 4.These conditions determine the permissible interaction between the quasi steady state subsystem and the boundary layer subsystem,which imply the limits of the growth rate offs.

The inequality Eqs.(39)and(40)can be written as

The following Theorem provides the feed-back control of the system Eq.(20).

Remark 5.κ is a parameter related to the time scale of the system Eq.(20),which is the reciprocal of the time scale between the slow subsystem and the fast subsystem.Theorem 2 shows the upper bound of κ under a composite Lyapunov function determined byVsandVf.In other words,this upper bound implies the minimum time scale between the slow subsystem and the fast subsystem.If the time scale is less thanthe former control design cannot be applied.

Next,the time scale decomposition method will be applied to decompose the IGC problem.For singular perturbation systems,slow states and fast states are relative,which are selected according to the physical properties of the system in the actual control design.For the guidance and control system under consideration,the states about the guidance course are slow,while the states about the attitude control change fast.Thus,we choose the states about the guidanceandas slow states and the states about the attitude control(α, ωz)as fast states.Then,the IGC model Eq.(18)is rewritten as a canonical form

The control law Eq.(38)is an addition of two subsystems’control laws.From Eq.(18)we know the reaction jet inputs(Fy，Mz)are discrete,which cannot be added.However,the control lawusdetermines the guidance process and has small effect on flight vehicle’s attitude in the terminal guidance course.Meanwhile,the control lawufdetermines the interceptor’s attitude and has small effect on the guidance process.In order to avoid the addition of the discrete inputs,the control strategy is proposed as follows:

In the control design of quasi steady state system,the control inputsFyand ρzare designed,while assumingMz=0.Meanwhile,in the control design of boundary layer system,the control inputs ρzandMzare designed,while assumingFy=0.By setting= ［Fy，ρsz］T= ［ρfz，Mz］T,the control laws of two subsystems are expressed as

3.2.Quasi steady state subsystem control design

Considering the state constraints,we will use the receding horizon control31to design the control law of the quasi steady state system.We first disperse the model of the quasi steady state system,next rewrite the discrete model as a Mixed Logic Dynamic(MLD)model,then transform the control design problem into an optimization problem which can be solved by Mixed-Integer Linear Programming(MILP).

We start by model discretization.The approximate discretization of the system Eq.(50)is expressed as

Next,we will rewrite the discrete model as an MLD model.In order to describe the discrete input(Fy),Boolean variables δsi（k）∈ ｛0，1｝，i=1，2 are defined to satisfy the following conditions:

In other words,δs1=1 means the rail jet reaction generates a positive trust along the axisOy1,and δs2=1 means the rail jet reaction generates a negative trust along the axisOy1.

In order to avoid the jet reaction opening on both sides at the same time,we introduce a constraint as

Then,the system Eq.(52)is converted to the form of an MLD model,by describing the continuous and discrete control inputs separately:

Then,the control design problem is transformed into an optimization problem.According to the objective of guidance control,the control design of the quasi steady state system is to find appropriate control input making the control outputyconvergence,while saving the jet reaction’s energy.We obtain the following optimization problem:

By solving the optimization problem,we can get the control law of the quasi steady-state system.

3.3.Boundary layer subsystem control design

In this subsection,the control of the boundary layer system is designed based on the receding horizon control.31The design process is similar to the design of the quasi-stable state subsystem.

We start by model discretization.The discretization of the system(51)is expressed as

Then,the control design problem is transformed into an optimization problem.According to the objective of attitude control,the control design of the boundary layer system is to find appropriate control input,making the system stable,while saving the jet reaction’s energy.We obtain the following optimization problem:

Remark 6.Assuming that the quasi steady state subsystem Eq.(52)and the boundary layer subsystem Eq.(57)satisfy the controllability assumption in Ref.32,the stability of the two subsystems can be guaranteed by Theorem 9 in Ref.32 Since the quasi steady state subsystem and the boundary layer subsystem are both stable,the stability of the system Eq.(44)can be guaranteed according to Theorem 2.

4.Simulation experiments

In this section,we choose two simulation experiments.In the first one,we consider the target maneuver in different forms to illustrate the performance of the IGC law.In the second one,we consider the situation when the target is located in the edge of the vision to illustrate how the IGC law deal with constraints.

4.1.Simulation results under different forms of target maneuver

In this part,the influence of target maneuver is considered.The simulation set-up is given in Table 1 and the target maneuver forms are given in Table 2.

The side-window constraints are considered as

The body-LOS rate under different forms of target maneuver is shown in Fig.4 which shows that the control law can grantee the body-LOS rate to converge under different forms of target maneuver.Namely,the target can be intercepted.While the target does the CV or CA maneuver,the body-

By solving the optimization problem,we can get the control law of the boundary layer system.

According to Eq.(38),we obtain the control law of the IGC problem with state constraints by combining the control law of the two sub systems,which completed the control design.LOS rate is convergent in the similar way of the proportional navigation.While the target does the Sine maneuvers,the body-LOS rate is also convergent,although its convergence way is influenced by the target maneuver.

The angle-of-attack,the rudder angle and the rail reaction jet under different forms of target maneuver are shown inFigs.5–7.During the terminal guidance course,the terminal guidance deviation is fixed by establishing the angle-of-attack using aerodynamic force in most of time.The change of the angle-of-attack is related to the body-LOS rate and the target maneuver.Besides,the terminal guidance deviation caused by the target maneuver is fixed by the reaction jet rapidly in the endgame,since the response speed of the aerodynamic force cannot meet the dynamic response demand.i.e.While the target does the CV maneuver,the rail reaction jet does not work in the whole guidance course.While the target does the CA or the Sine maneuver,the rail reaction jet works in the endgame.

Table 1 Set-up of the simulation.

Table 2 Target maneuver forms.

Fig.4 Body-LOS rate under different forms of target maneuver.

The attitude reaction jet under different forms of target maneuver is shown in Fig.8.The attitude reaction jet is suggested to stabilize and adjust the attitude rapidly.When the angle-of-attack is established at the beginning of the terminal guidance course or the attitude is stabilized in the endgame,the attitude reaction jet will work.

The elevation angle of body-LOS under different forms of target maneuver is shown in Fig.9,which shows that the elevation angle of body-LOS meets the constraints in the whole terminal guidance course.However,this is because the target is located in the center of the side-window vision at the beginning of the terminal guidance,which could not reflect the influence of the side-window constraints.The simulation cases when the target locates in the edge of the side-window vision will be proposed and analyzed in the next part to illustrate how the IGC law deal with constraints.

Fig.5 Angle of attach under different forms of target maneuver.

Fig.6 Rudder angle under different target forms of maneuver.

4.2.Simulation results when target located in the edge of the vision

In this part,the influence of side-window constraints is considered.We consider the target under the CV maneuver and set the control parameters the same as the above.By setting the simulation initial conditions as Tables 1,3,and 4,we obtain three simulation cases with the same initial body-LOS rate and the different initial body-LOS angles,where the initial elevation angle of body-LOS is 20°in Case 1,10°in Case 2,and 7°in Case 3.

Fig.7 Rail reaction jet under different form soft arget maneuver.

The body-LOS rate,the elevation angle of body-LOS and the angle-of-attack are shown in Figs.10–12.Figs.10 and 11 show that the control law can make the body-LOS rate converge while the body-LOS angle meets the constraints during the whole terminal guidance course.Namely,the target can be intercepted.However,compared with Case 1 and Case 2,we can find that the control law cannot provide enough aerodynamic force by establishing the angle-of-attack in order to satisfy the side-window constraints in Case 2,because the target is located in the edge of the side window vision.This will slow down the convergence of the body-LOS rate.Meanwhile,the IGC law tend to maintain a greater angle-of-attack in this case.Case 3 is a worse situation:In order to satisfy the side-window constraints,the IGC law cannot provide enough aerodynamic force by establishing the angle-of-attack and the body-LOS rate is divergent at the beginning of the guidance course.While the body-LOS rate is large enough,the rail reaction jet will work to f i x the terminal guidance deviation.

Fig.9 Elevation angle of body-LOS under different forms of target maneuver.

Fig.10 Body-LOS rate when the target located in the edge of the vision.

Table 3 Set-up of simulation for Case 2.

Table 4 Set-up of simulation for Case 3.

Fig.11 Elevation angle of body-LOS when target located in edge of vision.

Fig.12 Angle-of-attack when target located in edge of vision.

Fig.13 Rudder angle when the target located in the edge of the vision.

Fig.14 Rail reaction jet when target located in edge of vision.

Fig.15 Attitude reaction jet when target located in edge of vision.

The rudder angle,the rail reaction jet and the attitude reaction jet are shown in Figs.13–15.In most of the time,the terminal guidance deviation is fixed by aerodynamic force and the rail reaction jet do not work while the target does the CV maneuver.However,when the aerodynamic force is not enough,the rail reaction jet will work.And this will consume more fuel.The attitude reaction jet is also suggested to stabilize and adjust the attitude rapidly.When the angle-of-attack is established at the beginning of the terminal guidance or the attitude is stabilized in the endgame,the attitude reaction jet will work.

In conclusion,the proposed IGC law can make the flight vehicle intercept the target under different forms of the target maneuver while satisfying the side-window constraints,even if the target is located in the edge of the side window vision.However,the interception effect needs to be improved,as the estimation and compensation of the target maneuver is not considered.

5.Conclusions

This paper considers the guidance and control problem of a flight vehicle with side-window detection.Firstly,the relative motion equations are derived in the body-LOS coordinate system.In this way,the side-window constraints are described as time-invariant constraints in the control design.Then,the guidance and control problem of the flight vehicle is formulated into an IGC problem with state constraints.Finally,based on the singular perturbation method,the IGC problem is decomposed into the control design of the quasi-steady-state subsystem and the boundary-layer subsystem individually.Receding horizon control is applied in the control design for the two subsystems to deal with the state constraints and the hybrid input problem.

In the current work,as the stability of the control design based on the receding horizon control for the two subsystems is difficult to be proved,the stability is simply analyzed based on the simulation experiments.Although the proposed IGC law shows a good performance in the actual guidance course,the stability of the control design need to be proved in the future work.

Acknowledgements

This study was supported by National Natural Science Foundation of China(Nos.61473099 and 61333001).

Appendix A

In this appendix,the proof of Theorem 1 is given.This proof is taken from our previous work.29

Proof.Firstly,we will prove the sufficiency,that is,the body-LOS rate is convergent(limt→∞ωby=0 and limt→∞ωbz=0),if the LOS rate is convergent(limt→∞˙qε=0 and limt→∞˙qβ=0).In the LOS coordinate system,the LOS rate can be expressed as

whereqεandqβare the elevation and azimuth angle of LOS,respectively.□

According to the definition of the body-LOS coordinate system,ωbcan be expressed as

Appendix B

In this appendix,the proof of Theorem 2 is given.

Proof.Firstly,some inequalities are established according to the assumptions and the conditions given by the control design of the subsystems.Considering Eq.(37),we have

Substituting Eqs.(27),(28),(36),(42)and(B3)into Eq.(B6)gives

Namely,the matrix P is positive definite.Thus,（x，z）is negative definite and the system Eq.(20)is asymptotically stable,which completes the proof of Theorem 2.□

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