LIU Junjie,SUN Mingwei,CHEN Zengqiang,and SUN Qinglin

College of Artificial Intelligence,Nankai University,Tianjin 300350,China

Abstract: In this paper,a practical decoupling control scheme for fighter aircraft is proposed to achieve high angle of attack (AOA)tracking and super maneuver action by utilizing the thrust vector technology. Firstly, a six degree-of-freedom (DOF) nonlinear model with 12 variables is given. Due to low sufficiency of the aerodynamic actuators at high AOA, a thrust vector model with rotatable engine nozzles is derived. Secondly, the active disturbance rejection control (ADRC) is used to realize a three-channel decoupling control such that a strong coupling between different channels can be treated as total disturbance,which is estimated by the designed extended state observer. The control surface allocation is implemented by the traditional daisy chain method. Finally,the effectiveness of the presented control strategy is demonstrated by some numerical simulation results.

Keywords: high angle of attack (AOA), decoupling control, linear extended state observer(LESO),active disturbance rejection control(ADRC),thrust vector technology,control allocation.

1.Introduction

With larger flight envelope and higher agility,higher flight performance is urgently needed for advanced fighter aircraft [1]. Super maneuverability or higher agility at high angle of attack(AOA)can not only provide fighter aircraft with priority attack, but also increase its living opportunity. Nowadays, super maneuverability or higher agility has already been one of the important features for advanced fighter aircraft at high AOA[2].However,the maneuver action at high AOA can make the fighter aircraft become unstable due to high coupling and complicated unsteady aerodynamics[3].When maneuvering at high AOA,the air which flows through the aircraft will keep changing,that is,the attached flow becomes the vortex flow,and then changes into the separated flow.During this process,the unsteady aerodynamic force will make aircraft fighters have high nonlinearity,strong coupling and unstable characteristic with uncertainty[4].The control difficulty brings great challenges to flight controller designing. In past decades, many control methods were researched, such as the nonlinear dynamic inversion method(NDI)[5,6],variable structure control[7],and robust control[8].However,these methods are almost sensitive to model uncertainty,which is usually called model based. Furthermore, these traditional control strategies usually need special dynamic decoupling controllers by utilizing the multivariable control idea to realize decoupling.However,for the moment,it is difficult to find a systematic or mature tool to analyze the robustness of the multivariable system,which is a controversial issue[9].For example,the stability margin is often analyzed for a single-input-single-output (SISO) system.As a new control technology,active disturbance rejection control(ADRC)proposed by Han is not sensitive to model uncertainty [10]. To simplify parameters adjustment, the nonlinear part in ADRC is transformed to the linear part and becomes the linear ADRC(LADRC)[11].On account of simplicity and effectiveness, the novel method ADRC has been studied and applied to a lot of practical problems in recent years[12–17].

Based upon LADRC, a decoupling control strategy for high AOA tracking is presented. The independent controllers for pitch, yaw and roll state variables are constructed respectively to reject the strong couplings between channels.The aerodynamic uncertainty and channels coupling are treated as total disturbance, which is estimated in real time by the linear extended state observer(LESO).Ignoring the estimation error in generalized disturbance,the plant can be reduced to a unit integrator.The proportional derivative(PD)control method can then be utilized to realize reference tracking.The corresponding numerical simulations results are carried out to validate the performance.

The remaining parts are arranged as follows.Section 2 presents the aircraft nonlinear model and the thrust vector model.Section 3 shows the proposed control strategy.Some simulation results are given in Section 4. Section 5 comes to a conclusion.

2.Aircraft model

The aircraft nonlinear dynamic model comes from a classical model[18].Because of the singularity coming from high AOA or the high pitch angle, a mathematical model for fighter aircraft is derived from the body axis system to the track axis system.Moreover,the thrust vector with rotatable nozzles also needs to be utilized to cover the shortage of aerodynamic surfaces at high AOA. Thus, a nonlinear fighter aircraft model with the thrust vector model derived in the track axis system is given in this section.

2.1 Nonlinear dynamic model

The nonlinear model of the aircraft can be expressed as

where m is the mass of aircraft; α, β represent AOA and the sideslip angle respectively;V is the flight velocity;γ,χ, μ represent the flight path angle, velocity heading angle and bank angle around the velocity,respectively;p,q,r denote roll, pitch and yaw angular rates, respectively;xE, yE, zEare the position coordinates of the aircraft fighter; Iij(i = x,y,z;j = x,y,z) is the inertia moment; Ti(i = x,y,z) represents the thrust component along three-axis; D represents the aerodynamic drag, Y is the lateral force and L is the lift force. D,Y,L can be calculated as follows:

wherebdenotes the wing span, andcrepresents the wing mean aerodynamic chord,Cl tot,Cm tot,Cn totrepresent the total aerodynamic torque coefficients.

2.2 Thrust vector model

At the end of the aircraft, there are two rotatable nozzles installed symmetrically. Every nozzle can deflect in the yaw and pitch directions.The deflection angles can be described byδyiandδzi(i=l,rdenotes left and right),respectively.A pair of nozzles can produce required three axes torques through multiple combinations of deflections.Then, we can obtain the total deflections along the roll,yaw and pitch directions as

Under the body coordinate system,the thrust along the three axes can be derived as

whereζfirepresents the loss coefficient of the thrust. Ignoring installation errors,we assumeζfr=ζfl,Tr=Tl,δyr=δyl=δy.Then,we can obtain

whereTx,Ty,Tzdenote the thrust components about the three-axis.Trepresents the total thrust.When the nozzles deflection is within the deflection range(less than 20?),we can re-express(17)approximatively as follows:

LetxT,yT,zTrepresent the engine position coordinates respectively.The torque generated by the thrust vector can be expressed as follows:

3.Control strategy

To eliminate the strong coupling between different channels at high AOA, a decoupling rejection control strategy based on ADRC with the thrust vector technology is proposed.Considering the designing convenience,we choose the AOAα,the sideslip angleβand the roll angular ratepas controlled variables.Inα,βandpchannels,we design independent SISO controllers respectively. Fig. 1 shows the whole control structure. The reference commands are defined asαd,βd,pdand the thrust command is represented byTc.δe,δa,δrrepresent the deflection angles of the aerodynamic control surfaces, respectively.δx,δy,δzrepresent the deflection angles of the thrust vector along the three axes. The detailed controllers design process is shown in Fig.1.

Fig.1 Proposed decoupling control scheme

3.1 AOA channel controller design

We reformulate(2)as

Differentiate(20),and substitute(8)and(19)into it,then

wherev1is virtual control,which represents the expected pitch manipulation torque,and it depends on the pitch deflection of the aerodynamic elevator and the vector nozzles.According to(22),we can define

where x3represents the extended state or the total disturbance for the AOA channel,which consists of strong coupling,external disturbance and model uncertainty.Furthermore,we realize the re-description of(22)as

where b0αis related to the system which is an adjustable parameter, Hαis the derivative of the total disturbance.Then the original dynamics of the AOA channel will be simplified as a second order integral system with a generalized disturbance or total disturbance which causes deviation from the typical integrator system.The key of(24)is that x3is treated as a system state or a signal regardless of its original form. The state space expression for (24) can be described as

Then,we can establish the corresponding state observer as

where zj(j = 1,2,3) represents the estimation results of xj(j = 1,2,3) respectively. If the observer gains,β01,β02,β03can be adjusted appropriately,zj(j =1,2,3)can track the states xj(j = 1,2,3)approximately.In particular, the third state x3can be approximated by z3. An observer with such capability is usually defined as the extended state observer(ESO)[10].When the observer gains β01=3ωo,β02=are used,the observer is known as the bandwidth parameterized LESO[11,19]. ωoknown as the bandwidth of the LESO is the only parameter that needs to be adjusted.When z3≈x3is obtained by using the LESO,the disturbance compensation signal is used as follows:

where u0represents the control law which needs to be designed.Finally, the original dynamic for AOA can be approximated to

Thus,the dynamic of α becomes a second order integrator system.For such a typical system, the simple control law such as PD can be adopted.

Then we can obtain the final control law for the AOA channel

where αdis the command signal for the AOA channel.To generate the proper reference signal[20],a fastest tracking differentiator(TD)is employed,and its specific form can be described as

where the output signal sα1can track αd; sα2represents the differential signal of sα1;ε determines the TD convergence speed and is usually called the speed factor,and η is called the filtering factor;fhan(·)is an optimal synthetic function and its specific form is expressed as

For the convergence of the ESO, we consider the AOA channel and let ej(t) = xj(t)?zj(t)(j = 1,2,3).Combing (25) and (26), we describe the observation error dynamics as

where Hα(t) indicates the total disturbance in the AOA channel.DefineEquation(33)can be reformulated as

Theorem 1 Assuming Hα(t)is bounded,there exists a positive constant σi> 0,ωo> 0,and finite time TΘ> 0,such thatfor all

Proof The solution of(34)can be expressed as

Let

Since AΘis Hurwitz,there exists a setting time TΘ> 0,and for all(i,j =1,2,3),such that

TΘdepends on ωoAΘ.Letandit follows

Thus,from(37),(38)and(40),we can obtain

In summary,from(43),the estimation error of the ESO is bounded and its upper bound decreases when the observer bandwidth increases.

3.2 Sideslip angle controller designing

We can reformulate(3)as

Considering the symmetry of the AOA and the sideslip angle, we can directly give the virtual control law for the β channel as

where z3βcan be obtained by

sβ1is produced by TD as

3.3 Roll angular rate controller designing

In practice,it is difficult to measure the bank angle μ accurately.Therefore,we select roll angular rate p as the controlled state to realize the proper curve of μ.We rewrite(7)as

Thus,the corresponding LESO can be designed as

Finally,the whole control law is

wheresp1is generated by the TD

4.Control surfaces allocation

The controller outputs of three channels characterize the corresponding need of the expected moment of the force about the three axes. The goal of control surfaces allocation is to finish calculation for the deflection angle values of aerodynamic surfaces and the thrust vector. Thus,we definex1= [p,q,r]T, u= [δe,δa,δr,δx,δy,δz]T,Then,we can reformulate(7),(8)and(9),that is

The expressions ofcan be described as

wheredenotes control derivatives with respect to aerodynamic deflections and thrust vector deflections, respectively. In addition, we can definev= [v1,v2,v3]T,then the control surfaces allocation can be explained as follows.We letv(t)∈R3as the expected virtual control command andu(t)∈R6as the control surface deflection. In a word, the goal of control surfaces allocation is to obtain the solution of theequation under the following mapping relation:R6→R3,

with the control input constraintwhereumin(max)andΓmin(max)denote the lower bound (upper bound) of the control surface deflection and its rate of change. In order to extend the lifespan of the engine nozzles, the principle of control surfaces allocation is that the deflection of the thrust vector can be reduced as far as possible,that is,aerodynamic control has a higher priority than the thrust vector.Based on the above principle,we divide the control input into two parts

whereuaero= [δe,δa,δr]T,utv= [δx,δy,δz]T,Gaerorepresents the left three columns ofandGtvdenotes the right three columns.Moreover,we letrepresent the inverse matrices ofGaeroandGtv, respectively. Table 1 shows the limitations of the position and the rate of change. Fig. 2 gives the overall description of the control allocation process. The aerodynamic control deflections should try to satisfy

Table 1 Deflection limitations

Fig.2 Control allocation diagram

If the aerodynamic control deflections calculated by (60)are all in the limitation,then the control allocation is over.Or else,whenuaeroexceeds the limitation,the thrust vectorutvis required to eliminate the control allocation errorE.

The expression ofSat(x,u)is

5.Simulation verification

Some numerical simulations are conducted to demonstrate the effectiveness and robustness of the presented control strategy.Herbst maneuver is a typical super maneuvering action.Thus,the controller performance is validated by selecting a Herbst-type maneuver. Moreover, we select the initial flight speed,flight height and AOA asV= 90 m/s,h0=?zE0=?1 200 m andα0=10?,respectively.The controllers parameters are adjusted asωo=10,kpα=50,kdα= 0.02,kpβ= 20,kdβ= 0.02,kp= 30,b0α= 8,b0β=?2.5,b0p=?2. During the maneuver,the maximum thrust value is utilized.

5.1 Performance verification

The Herbst maneuver is to realize the rapid turn with a small radius. Thus, we consider a Herbst-type maneuver,and its end condition is that the flight direction of the fighter aircraft changes 180?.Figs.3–6 present the simulation results.From these figures,the AOA and roll angular rateptrack the desired value well,and the actual value of sideslip angleβkeeps almost zero.Forα,βandp, an appropriate reference is used to go through the fastest TD,and the outputs of TD are chosen as the final reference signals which are shown in Fig.3(a),Fig.3(b)and Fig.3(c),respectively.From Fig. 3(a), att= 1.5 s, AOA begins to change and then reaches 62.5?after about 2 s.AOA keeps 62.5?for a little while,and meanwhile the roll angular rate changes as Fig. 3(c)such that the aircraft can roll around the velocity vector. After rolling, the aircraft swoops for a while and then the Herbst-type maneuver is over. During this maneuver action, the velocity changes as shown in Fig.3(d),and it decreases at first and then increases.At high AOA,whenpchanges properly,μcan change as the expected curve shown in Fig. 3(f). From Fig. 3(g), it can be seen that the heading angle of the flight speed changes 180?. The aerodynamic deflections are shown in Fig. 4.From Fig. 4, we can see that the elevator reaches the deflection limitation,indicating that the aerodynamic surface is insufficient.The thrust vector deflections are illustrated in Fig. 5. Moreover, all the thrust vector controls do not exceed the saturation limitation. From Fig. 6, the turning diameter is about 150 m.

Fig.3 Herbst-type maneuver

Fig.4 Aerodynamic control surfaces deflection

Fig.5 Thrust vector nozzles deflection

Fig.6 Trajectory of the aircraft

5.2 Monte Carlo test

Monte Carlo simulation is conducted to realize the robustness verification. We select 43 aerodynamic parameters and perturb them randomly within±30%limits.Five hundred simulations are carried out totally.The simulation test results are shown in Fig.7.It can be easily known that the response indicates good robustness.

Fig.7 Monte Carlo simulation test

6.Conclusions

In this paper, a decoupling control scheme is proposed to implement the super maneuver with the thrust vector at high AOA. Independent controllers for three channels are designed based upon the linear active disturbance rejection control instead of the traditional multivariate control idea.LESOs are constructed to get the estimation of the total disturbance including strong coupling from channels and aerodynamic uncertainties,which are compensated in real time.The traditional daisy chain method is utilized to complete the control surface allocation.Simulation results are carried out to validate the effectiveness and robustness of the proposed coupling rejection scheme.