ZHAO Heweiand LI Rui

1.Shore Guard Institute,Naval Aviation University,Yantai 264001,China;2.Wenjing College,Yantai University,Yantai 264005,China

Abstract:A typical adaptive neural control methodology is used for the rigid body model of the hypersonic vehicle.The rigid body model is divided into the altitude subsystem and the velocity subsystem.The proportional integral differential(PID)controller is introduced to control the velocity track.The backstepping design is applied for constructing the controllers for the altitude subsystem.To avoid the explosion of differentiation from backstepping,the higher-order filte dynamic is used for replacing the virtual controller in the backstepping design steps.In the design procedure,the radial basis function(RBF)neural network is investigated to approximate the unknown nonlinear functions in the system dynamic of the hypersonic vehicle.The simulations show the effectiveness of the design method.

Keywords:hypersonic vehicle,adaptive neural control,higherorder filte,differential explosion.

1.Introduction

The study of the hypersonic vehicle is important for utilization of the Near Space,which has become a hot issue in recent years.However,there are many challenges to be addressed.One of the challenges is the control system design for the hypersonic vehicle.Many nonlinear control methods and intelligent control methods are used for the design of the control system,moreover,many achievements have been proposed in recent years.It is known that the adaptive backstepping control method can be systematically formulated in a recursive way to achieve the asymptotic stability for a strict feedback system from[1],which is used for the control system design of the hypersonic vehicle in many papers.The adaptive backstepping design based on the nonlinear disturbance observer was proposed in[2],the robustness can be guaranteed and the explosion of differential can be solved by a dynamic surface method.As discussed in[3],the adaptive discrete-time controller was designed via backstepping and the neural network(NN)was used for approximating the known nonlinear functions.The adaptive backstepping method cannot be directly applied when the nonlinear terms in the hypersonic vehicle models are totally unknown or partially unknown.The popular methodology is that the unknown terms are approximated by the NN.The NN methods used in control system design were proposed in many papers[4–7].However,there is a drawback associated with backstepping,the complexity of the control system arising from the repeated differentiations of virtual controllers is difficult to solve,i.e.,the so-called differentiation explosion problem.The popular scheme aims to solve such a problem by a filtering instead of the virtual controller differentiations in each backstepping step[8,9].However,as we know,the stability analysis of the whole closed-loop system consists of the original system and the filter system.It is easy to see that the boundedness of the input of filters and filter dynamics are so important and difficult to establish.So far,there is no systematic methodology to select the control gains and the time constants of the filters.In most papers,the first-order filter is established to solve the explosion problem,moreover,the input of the filter is the virtual controller,and hence the boundedness of filter dynamics is quite involved in the virtual control and the selection of time constants for the filter is important for the stability of the filter.In[10],the higher-order filter was proposed to solve the explosion problem of the backstepping design,and the design parameters are selected to render the polynomial Hurwitz.Therefore,the selection of design parameters can ensure the stability of the filter dynamics.

In this paper,problem formulation will be described in Section 2 which consists of the representation of the dynamic model of the hypersonic vehicle,the control law of the velocity subsystem,and the description of the reference flight path angle.The typical adaptive NN backstepping controller design based on higher-order filters will be pro-posed in Section3.The explosion problem arising from the backstepping method will be avoided by the higher-order filters,moreover,the stability of filter dynamics can be ensured.The stability analysis for the whole closed-loopsystem will be described in Section 4.In Section 5,to demonstrate its usefulness,simulation will be carried out to verify the effectiveness of the controller proposed.Finally,conclusions and future works are discussed in Section 6.

2.Problem formulation

The longitudinal dynamic model of the hypersonic vehicle in this study was given in[11].The detail form can be described as follows.There are five rigid-body state variables involved in the model,i.e.,V,h,γ,α,Q,while the four flexible state variables are not considered in this study.Moreover,the control inputs areδe,i.e.,elevator deflection andΦ,i.e.,the fuel equivalence ratio.

whereVis the velocity,his the altitude,γis the flight path angle,αis the attack angle,Qis the pitch rate,mis the mass of the aircraft,gis the acceleration due to gravity,θis the angle of pitch,moreover,T,D,L,Mrepresent the thrust,drag,lift-force,and pitching moment,respectively.AndIyyis the moment of inertia about the pitch axis.

The related expressions are described as follows:

where

The detail information of parameter values was described in[11].It is easy to see thatθ=α+γ,the equations(1)–(5)can be expressed as

The dynamics of model(10)can be divided into two subsystems.One is the velocity subsystem,and the other is the altitude subsystem.For the velocity subsystem,we have

where

For the velocity subsystem,the control method is not discussed in this study,and the controller is designed directly with the control algorithm form[12],which is in a form of

wherekpv,kiv,kdvare designed parameters for the proportional integral differential(PID)controller,zV=Vr?Vis the tracking error of velocity,andVris the reference velocity.

Based on the timescale conclusion from[13,14],it is obvious that the velocity can be considered as slow dynamic compared with the state variables of the altitude subsystem,therefore the velocity will be treated as constant during the controller design for the altitude subsystem.For the altitude subsystem,the flight path command[15]is designed as

wherekh,kiare positive constants,hris the reference altitude.If the flight path angle can followγd,then the altitude tracking errorh=h?hrcan be regulated to zero exponentially.According to[12],we have

Assumption 1In(10),it is easy to know that the termTsinαis generally much smaller thanL,which can be neglected.

De finex1=γ,x2=θ,x3=Q,andX=[x1,x2,x3]T,based on Assumption 1,model(10)can be written in a form of

where

whereis the elevator coefficient andceis the canard coefficient.

In(15),X=[x1,x2,x3]T∈R3,u∈R,andy∈R are the state,input and output of the attitude subsystem,respectively.It is obvious thatg1,g3are known constants.Without of generality,some assumptions are essential in the sequel.

Assumption 2The reference outputydis bounded and smooth.

Assumption 3The state vectorXis measurable.

Assumption 4The nonlinear functionsf1andf3are unknown and bounded.

For Assumption 1,it is very necessary for establishing the boundedness of signals in the closed-loop system[16,17].For Assumption 2,the state feedback can be guaranteed,which is needed for the control design.For model(15),it is a strict-feedback form[18],and the backstepping design algorithms can be used to design the controller of the attitude subsystem.

The control goal is that controllerudesigned in this study for the altitude subsystem and the given controllerΦof the velocity subsystem can steer the altitude and velocity of longitudinal system output dynamics to track the reference values.In this study,the radial basis function(RBF)NN is incorporated with the adaptive method,which will be used for approximating the unknown nonlinear functions.Moreover,the higher-order filter will be used for solving the explosion of the complexity problem from the backstepping design procedure.

3.Controller design

In this section,the classical backstepping method will be developed to design the controller with the higher-order filter solving the differential explosion problems from repeated differentiations of the virtual controllers in each backstepping procedures[19].The detail steps will be shown in the following.

Step 1Consider the first case,i.e.,˙x1=g1x2+f1(x1),then define the tracking errorz1=x1?yd.The direct differentiation forz1is

Forz1dynamic(16),x2can be treated as the virtual controller designed for stabilizing thez1dynamic.Moreover,the unknown nonlinear functioncan be approximated by RBF NN[20],which is in a form of

whereis the optimal weight vector,φ1(?1)is the radical basis function,?1=[x1,˙yd]Tis the input vector of NN,andε1is the construction error of NN with the supremeε1m.Based on(17),we have

De fine the second tracking errorz2=x2?x2v,and the virtual controllerx2vis proposed in a form of

wherek1>0 is the design parameter,δis a positive constant[21],is the estimation of,andis the estimation ofε1m,moreover,we can define the errorsBy substituting(19)into(18),it yields

whereΓ1,1are positive design parameters.By a direct differentiation of(21),it yields

A Lyapunov function is constructed in a form of

The updating laws forW1,ε1,respectively,can be designed as follows:

whereλ1,σ1are the positive design parameters.Substituting the updating algorithm(23)into(22),it yields

Step 2The differentiationz2is calculated as

Clearly,the differentiation of the virtual controllerx2vis so difficult to calculate and the explosion of the complexity problem is unfavorable to the practical implementation[22].Thus in this section,the differentiation ofx2vis replaced with filtering to conquer the explosion problem.Then letx2vpass through an(n?1)th-order filter in a form of

wherenis the order number of the system,η2,iis the positive number chosen to render the polynomialη2,n?1sn?1+

η2,n?2sn?2+···+η2,0Hurwitz,i.e.,η2,0q2=x2v,q2(0)=x2v(0).Based on the results from[6],x2vcan be replaced with the outputq2of filter(26),i.e.,x2v=q2,and

Remark 1The reason for using a higher-order filter instead of a typical general first-order filter is that the output of the filter(26)is involved in the next virtual controller.It is easy to see that the virtual controller(19)is bounded,and based on the higher-order design algorithm,stability of the filter(26)dynamics can be ensured,then the output of the filter(26)is bounded,which is an important issue for the boundedness of the next virtual controller.

Based on the result above,the explosion of the complexity problem arising from the differentiation of the virtual controller(19)can be avoided.To this end,we have

Likewise,forz2dynamic,x3can be treated as the virtual controller designed for stabilizing equilibriumz2=0.De fine the tracking errorz3=x3?x3v,the virtual controllerx3vcan be proposed as

wherek2is the positive design parameter.Substituting(28)into(27),it yields

Consider the following Lyapunov function

The differentiation of(30)is calculated as

Step 3The differentiation ofz3is attained as follows:

Likewise,letx3vpass a higher-order filter,which can be replaced with the output of the filter,i.e.,˙x3v=˙q3.The higher-order filter is in a form of

whereη3,iis the positive number chosen to render the polynomialη3,n?1sn?1+η3,n?2sn?2+···+η3,0Hurwitz,i.e.,

Remark 2Clearly,the virtual controllerx3vis bounded,moreover,it is the input of the filter(33),then the dynamics stability of the filter(33)can be ensured,i.e.,q3is bounded,which is involved in the next controller.

Substituting˙x3of(15)into(32),it yields

The unknown nonlinear functionf3(X)can be approximated by RBF NN,which is in a form of

whereis the optimal weight vector,φ3(?3)is the radical basis function,?3=[x1,x2,x3]Tis the input vector of NN,andε3is the construction error of NN with the supremeε3m.Based on(35),we have

Clearly,the overall controller of the system,i.e.,u,is proposed in a form of

wherek3is a positive design parameter,δis a positive constant,likewise,define errors.It is easy to see thatis in the overall controller,

Consider the following Lyapunov function:

The direct differentiation of(39)is calculated as

Substituting(38)into(40),it yields

The updating algorithms can be designed as follows:

whereλ3,σ3are the positive design parameters.Then substituting updating laws(42)into(41),it yields

4.Stability analysis

As mentioned,the notion of input-state-stability(ISS)has been utilized in the controller design of nonlinear systems in recent years.ISS means that the bounded input implies the bounded state[16].In this section,the stability of the control system designed is verified in two steps.Firstly,the ISS stability will be analyzed.Besides,the Lyapunov stability method[23–25]will be used for verifying that all the states of the closed-loop control system are uniformly ultimately bounded[26,27].

Theorem 1For the closed-loop control system consisting of the plant(15),virtual controllers(19)and(28),overall controller(37),and the NN adaptive tuning laws(23)and(42)are designed.If the control gain and tuning parameters can be selected reasonably,all the signals in the closed-loop system are uniformly ultimately bounded.Besides,tracking errors can converge uniformly to the following setΩ,and the radius can be controlled arbitrarily small with the sufficiently large design parameters.The relative definitions will be given later.

For the proposed control system consists of two subsystems,i.e.,the filter closely system,and the original error system,the stability of the whole closely system needs two sides.One is the output boundedness of filters,and the other is the boundedness of original error system signals.

ProofIt is easy to know that(23)is bounded,which in turn can ensure the boundedness of virtual control law(19).In(26),the virtual controller is the input of the filter(26),clearly,the boundedness of the virtual controller(19)implies the boundedness of output stateq2from the filter(26).Output stateq2of the filter(26)is the component of virtual controller(28).Thus.it is easy to see that the virtual controller(28)is bounded,which is the input of the filter(33),and can ensure the bounded stability of the constructed filter dynamics,in turnq3is bounded,which is the component of the overall controller(37).Besides,based on the boundedness of(42),the overall control input(37)is bounded,we can know that all state signals of the original error system are bounded,and the design filters dynamics are bounded,then the whole closed-loop system is ISS.The two systems are coupling through the above analysis,thus the boundedness needs to be verified for the two systems.Then the uniform ultimately bounded will be analyzed with Lyapunov stability methods.

Consider the Lyapunov function in a form of

where

Differentiating(44)along(24),(31)and(43),we have

For easy reference,we quote the following inequalities[28–30]:

The following inequalities can be attained:

Based on the results from[17],the following inequality holds:

wherecηis selected as 0.278 5.It is easy to verify that the above-mentioned inequality is very useful for the stability analysis.

Hence according to(48),we know

Likewise we have

Substituting(49),(50)into(47)yields

Considering(21),(30)and(39),we have

with

The conclusion is attained as follows:

withζ=min[ζ1,ζ2,ζ3].It is easy to know thatCis a bounded constant.

Based on(49),we have

Besides considering(44),we can obtain the conclusion that all signals of the closed-loop system are bounded,moreover,the tracking errors can be converted into a bounded invariant set as follows:

5.Simulations

In this section,the effectiveness of control system design strategies will be proved through simulation.The control gains are selected ask1=18,k2=7,k3=10,kh=0.3,ki=0.1,kpv=0.5,kiv=7,kdv=10.Moreover,the updating laws parameters are selected asΓ1=5,λ1=0.001,1=15,σ1=2.5,Γ3=2.5,λ3=0.001,3=0.01,σ3=0.002,δ=0.05.And the higher-order filters parameters can be designed asη2,2=2,η2,1=1,η2,0=1,η3,1=1,η3,0=0.05.The numbers of NN nodes are selected asN1=20 andN2=20,with their centers being evenly spaced in[?0.5,0.5]×[?0.5,0.5]and[?0.5,0.5]×[?9.7,9.7]×[?4.5,4.5],respectively.

In simulation,the reference altitudehrof altitude subsystem simulation is achieved through a filter presented as follows:

wherehcis the command signal,which climbs from 85 000 ft to 87 000 ft in 40 s,and then descends to 86 000 ft.The reference velocityVrof the velocity subsystem simulation is achieved through a filter presented as follows:

whereVcis the command signal,which climbs from 8 850 ft/s to 8 900 ft/s in 20 s,and then climbs to 9 150 ft/s.

The initial condition of simulation is shown in Table 1.The simulation results are shown in figures that follow.

Table 1 The initial values

In Fig. 1, we can see that the altitude tracks the reference signal well.The altitude tracking error is shown in Fig.2,and the results can illustrate that the altitude subsystem design is satisfied.As shown in Fig.3,the flight velocity can track the reference signal well,moreover,the result,i.e.,the velocity tracking error from Fig.4 can indicate that the PID contrller can fulfill the velocity tracking task.

Fig.1 Altitude tracking

Fig.2 Altitude tracking error

Fig.3 Velocity tracking

Fig.4 Velocity tracking error

The elevator deflection is shown in Fig.5.It is easy to see that the input of the altitude subsystem is bounded and smooth.The virtual controllers of the altitude subsystem are shown in Fig.6 and Fig.7.The input of the velocity subsystem,i.e.,the fuel equivalence raio is shown in Fig.8,which is bounded within(0,1.2).From Fig.9 to Fig.11,the altitude subsystem states,i.e.,the flight path angle,the angle of pitch,and the pitch rate,are bounded.The tracking errors of system states can be demonstrated in Fig.12.

Fig.5 Elevator deflection

Fig.6 Virtual controller x2v

Fig.7 Virtual controller x3v

Fig.8 Fuel equivalence ratio

Fig.9 Flight path angle

Fig.11 Pitch rate

Fig.12 System states tracking errors

The filter dynamicsq2andq3are represented in Fig.13 and Fig.14.It is obvious that the fitler dynamics are bounded and smooth,thus the filters design is effective.

Fig.13 Filter dynamic q2

Fig.14 Filter dynamic q3

The estimations of NN weights in Fig.15 are smooth,and the NN approximations for unknown nonlinear functionsf1andf3are shown in Fig.16 and Fig.17.The approximation performance could satisfy the requirement for the control system design.

Fig.15Norm ofand

Fig.16 NN approximation for f1

Fig.17 NN approximation for f3

6.Conclusions

The typical adaptive neural control involved in the backstepping method is applied for the control system design of the hypersonic vehicle in this paper.Moreover,the higherorder filters are used for avoiding the explosions of differentiation in backstepping steps.The important issue is that the design parameter of higher-order filter selection methodology is proposed.The NN can be improved in the future work,since the range of RBF function action is limited,and NN cannot approximate the unknown nonlinear functions when the dynamics are out of the range of RBF function action.Moreover,the flexible state stability control issue for the hypersonic vehicle should be considered in the controller design.