Zhenxing GAO,Jun FU

College of Civil Aviation,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

KEYWORDS Function substitution;Linear parameter varying;LPV control;Robust control;Wind shear

Abstract This article deals with the disturbance attenuation control of aircraft flying through wind shear via Linear Parameter Varying(LPV)modeling and control method.A Flight Dynamics Model(FDM)with wind shear effects considered was established in wind coordinate system.An LPV FDM was built up based on function substitution whose decomposing function was optimized by Genetic Algorithm(GA).The wind disturbance was explicitly included in the system matrix of LPV FDM.Taking wind disturbance as external uncertainties,robust LPV control method with the LPV FDM was put forward.Based on ride quality and flight safety requirements in wind disturbance,longitudinal and lateral output feedback robust LPV controllers were designed respectively,in which the scheduling flight states in LPV model were actually dependent parameters in LPV control.The results indicate that LPV FDM can reflect the instantaneous dynamics of nonlinear system especially at the boundary of aerodynamic envelope.Furthermore,the LPV FDM also can approach nonlinear FDM's response in wind disturbance special flight. Compared with a parameter-invariant LQR controller designed with a small-disturbance FDM,the LPV controllers show preferable robustness and stability for disturbance attenuation.

1.Introduction

Low-altitude wind shear makes aircraft face deadly dangers during takeoff and landing.According to the statistic report from Boeing Company,flight phases of takeoff and landing only occupy 6%of the total flight time of civil aircraft.However,61%flight accidents do take place during takeoff and landing.Furthermore,66%accidents during takeoff and landing are caused by wind shear or pilots'improper manipulation induced by wind shear.1Nowadays,flight accidents induced by aircraft's subsystem failure have decreased greatly.On the other hand,accidents due to wind disturbance are increasing.Airlines'commercial aircraft would inevitably encounter moderate or even severe wind shear in routine flights.Once encountering severe wind shear,in the premise of keeping stable,the crew's best choice is to escape.2Federal Aviation Administration(FAA)has recommended three kinds of escape strategies for escaping from strong wind shear.3

It is a difficult problem to estimate the wind effects on aircraft.At one time,the small-disturbance Flight Dynamics Model(FDM)was adopted for flight performance analysis in wind shear.4However,research from National Aeronautics and Space Administration(NASA)indicates that flight states under wind disturbance probably deviate far away from trim states,and small-disturbance FDM cannot describe the special flight states accurately.5Frost and Bowles firstly proposed an FDM with wind shear effects considered.Under body coordinate system,wind shear effects on airspeed and flight trajectory were analyzed.6In real-time flight simulation such as flight simulator,the model can be numerically integrated to indicate wind effects.However,due to its complexity,the model should be simplified for nonlinear control law design.For example, primary aerodynamic derivatives should be abstracted from the sophisticated aerodynamic coefficient expressions and described as continuous functions.Besides,another simplified FDM with wind effects appears in Refs.7,8.Built up under flight path coordinate,the model is based on constant wind hypothesis and mass-point assumption of aircraft.Consequently,flight path angle can be calculated by numerical integration, after that the angle of attack was acquired by geometry relation.This modeling method does not accord with the principle that wind disturbance should firstly have effects on aerodynamic angle.

To maintain steady flight in wind shear,there are two research methods.The first is to model the wind shear as a kind of deterministic disturbance,and then put the wind shear model into FDM.Some control methods including optimal and intellectual control can be adopted to study wind shear escape.The second is to take wind shear as a kind of uncertain disturbance and deal with it via robust control method.In terms of optimal control law with deterministic wind model,some flight trajectory parameters and their deviations are taken as optimization objective,while the constraint conditions are safety objectives such as energy altitude and Ffactor.For example,Ref.8is aimed at takeoff in wind shear,and a Bolza optimization problem was built up with the target of vertical speed minimum.Refs.9,10studied optimal control for longitudinal escape by taking altitude change minimization as optimization objective.Furthermore,feedback linearization method(e.g.nonlinear dynamic inversion)based on deterministic wind model and simplified nonlinear FDM can be used for wind shear escape.11

Actually,wind shear taking place in neighboring airports is often accompanied with stochastic turbulence.Therefore,it is more reasonable to build an uncertain wind field model and study it by robust control theory.External disturbance and internal parametric uncertainties can be easily addressed by robust control.12Recently,model recovery antiwindup compensator was further put forward to counteract the effects of control saturation.13When it comes to flight control,H∞,14μ-synthesis15and LQG/LTR16technologies were adopted in the augmented linear state-space FDM.Aimed at flight states variation for commercial aircraft in a wide range,a gain scheduling controller was designed to interpolate among different trim points based on a group of linear FDMs.17However,traditional gain scheduling cannot guarantee the global stability of nonlinear system.Once the system deviates far away from trim states,the performance of gain scheduling controller would reduce,and controller may even become unstable.Linear Parameter Varying(LPV)method,initially applied in hypersonic vehicle control,was designed with time varying parameters considered.By LPV control method,the stability and performance objectives can be satisfied in a larger working area.Robust LPV control,with system parametric uncertainties and external disturbance considered,is widely used in advanced flight control. For example, a gainscheduled controller synthesis approach is proposed based on common Lyapunov stability theory for switched systems evolving on locally overlapped switching law.18In terms of polytopic LPV system,the vertex model can be extracted by High-Order Singular Value Decomposition(HOSVD).In this way, the design complexity of LPV controller can be reduced.19Ref.20designed a multi-loop controller for morphing aircraft.The proposed controller uses a set of inner-loop gains to provide stability using classical techniques,whereas a gain self-scheduled H∞outer-loop controller is devised to guarantee a specific level of robust stability and performance for time-varying dynamics.When it comes to the modeling method,initially there are three methods for LPV modeling including Jacobian linearization, state transformation and function substitution.21Moreover,there are several revised LPV models by selecting different ways of optimization algorithm22and mismatch measurement between models.23

This article will firstly study the modeling of LPV FDM with wind effects considered. Differing from Refs.6-8, an FDM with wind effects under wind coordinate will be built.On this basis,function substitution LPV modeling techniques will be imported to describe wind disturbance effects.After that,wind disturbance will be deemed as external uncertainties,a robust LPV control infrastructure will be designed for wind disturbance attenuation.Based on LPV FDM and combined with ride quality and flight safety requirements,the longitudinal and lateral robust LPV controller will be designed successively. Finally, by simulation comparison, both the LPV FDM and robust LPV control law will be examined and verified.

2.LPV flight dynamics modeling with wind effects

2.1.Nonlinear flight dynamics modeling with wind effects

Aircraft's aerodynamic forces are calculated under wind coordinate system.Wind disturbance would firstly have effects on airspeed V,angle of attack α and angle of sideslip β,and then make aerodynamic forces change.As a result,dynamics equations under wind coordinate can indicate wind effects directly.It is more reasonable to study wind disturbance under wind coordinate.In wind-free situation,force equations under wind coordinate is17

In the above equations,FTis the engine thrust;αTis the engine installation angle;L,D and C are lift,drag and side aerodynamic force respectively.This article adopts transition matrix Cbe to describe the transition from ground coordinate system to body frame,whileis used to transit state from body frame to wind frame.Both of the matrices can be easily found in some literatures.In this way,airspeed and angular velocity can be transformed freely between body frame and wind frame.It is worth mentioning that there is a triangular relationship among ground speed VE,airspeed V and wind speed WE:

Gravity acceleration g also needs to be transformed from ground to wind frame.In fact,g1,g2and g3in Eq.(1)are the components of gravity acceleration in wind coordinate.Consequently,the force equation with wind effects can be obtained:

In Eq.(4),m stands for aircraft mass,p,q and r are the body angular rates.Compared with the force equation described in body frame,17Eq.(4)shows the influence of wind disturbance on[V,α,β]Tdirectly.Besides,wind effects can be directly inserted into navigation equations:

In Eq.(5),x,y and z are the position components and Vx,Vxand Vxare the velocity components in ground coordinated system.The B737-800 aircraft is selected as the research object since this article actually studies the flight safety problem of commercial aircraft in wind disturbance.Taking the lift coefficient as an example,there are 14 aerodynamic derivatives.24,25Complete expressions can be used for hi-fi flight simulation.For dynamic analysis and control law design,it is common to simplify the aerodynamic model by selecting some key derivatives.As far as the lift aerodynamic derivatives are concerned,the selected derivatives in this article are

2.2.LPV modeling by function substitution

As previously mentioned,there are three basic methods for LPV modeling.Similar to small-disturbance model,Jacobian linearization is actually one-order approximation at trim points. In state transformation method, non-scheduling parameter and control inputs are expressed with continuous differentiable function via partial derivative calculation.Both methods need to build up a trim map according to internal states,while system states can be interpolated in trim map21.However,once flight states further exceed the envelope range made by trim map,the poor extrapolation ability will lead to model inaccuracy.18,20In function substitution,system's nonlinear characteristics are described by decomposing function which can be solved by optimization.Function substitution LPV model possesses better local extrapolation ability.Furthermore,it can simulate the system dynamics much better than the former two methods.21In function substitution LPV modeling,the nonlinear system can be described by

where z(t)∈Rnzis scheduling states, w(t)∈Rnwis nonscheduling states,and u(t)∈Rnuis control inputs.Eq.(7)can be modified as the following equation by distinguishing trim value and offset:

where F is the decomposing function for(ηz,ztrim,wtrim,utrim)whose complete form is

The key procedure of function substitution LPV modeling is to rebuild Eq.(9)as a function of scheduling parameters.By inserting Eq.(9)into Eq.(8),we have

2.3.LPV flight dynamics modeling with wind effects

In previous robust control studies,an input matrix of external disturbance was usually built for linear robust control.14Another method is to consider the parametric uncertainties by evaluating the wind effects on aerodynamics in a general way.15This article tries to explicitly insert the wind effects into the LPV system.Taking the longitudinal LPV flight dynamics modeling as an example,[V,α,h]Tare selected as scheduling parameters which have main effect on aerodynamic derivatives.Some non-scheduling parameters,such as pitch angle θ,are one-order approximated.The longitudinal state-space model is described by the scheduling parameter-dependent in Eq.(11)as an example,scheduling parameters[V,α,h]Tcan be divided by an i×j×k grid.According to Ref.21,it is feasible to solve F1α,F1Vand F1hby Linear Programming(LP) with absolute value constraints. However, with the increase of the scheduling parameters and the refining of the grid,especially when the system is strongly nonlinear,it is difficult to obtain the optimal solution only by linear programming.In this article,Genetic Algorithm(GA)is adopted to solve the optimization problem as follows:

form with remnants as shown in Appendix A.The matrix elements related to wind disturbance are also listed.In Eq.(A1),f(x,h)and f′(x,h)refer to the wind field and its first-order derivatives.Wind field,related to flight path and altitude,can be calculated by a wind disturbance model.

According to function substitution method,the remnants in Eq.(A1)will be substituted and re-described by scheduling parameters.Furthermore,an optimization objective function should be found to minimize the error between LPV FDM and nonlinear system.When the C,D matrixes in Eq.(A1)are substituted by[F1,F2,F3,F4,F5]T,the complete equation is shown as follows:

2.4.Heuristic optimization by genetic algorithm

Function substitution method is to approximate Eq.(11)by reconstructing a new equation,which is shown at the end of Eq.(11).There are several kinds of realizations of this new equation.It is necessary to select a group of optimal matrix elements in order to make the LPV model approach the nonlinear system as much as possible.Taking the solution of F1

In Section 4, simulation comparison between GAoptimized LPV model (GA-LPV) and LP-optimized LPV model(LP-LPV)will prove the above statement.

3.Robust LPV control for wind disturbance attenuation

The GA-LPV FDM forms a good foundation for further LPV control research.No matter keeping the aircraft fly stably,or commanding it escape from wind field,it is necessary to study the performance objective of LPV control. Moreover, a parameter dependent LPV controller should be designed to satisfy the requirement of both stability control and command response.

3.1.Robust stability and control theory of parameter-dependent LPV model

In this section,the parameter-dependent robust stability and control theory will be imported for wind disturbance attenuation.Stability control problem of flying through wind shear is transformed into scheduling parameter-dependent LPV model's stability analysis.An LPV model Gz(A,B,C,D)can be described as follows:

For simplification,z(t)is omitted by z in the following analysis.To describe a performance measurement of an LPV model from wind disturbance d to error e in terms of norm values,an induced L2-norm of an LPV model can be defined as follows:26

Definition 1.For zero initial conditions,the induced L2-norm of a parameter-dependent stable LPV model Gzis defined as

With respect to parameter-dependent LPV system stability,there is a lemma for judging the stability:

Lemma 126Given the LPV model Gzand scalar γ ＞0,if there exists an X ∈Rn×n, X=XT＞0, such that for all z ∈Rnz,the following Linear Matrix Inequality(LMI)exists:

Fig.1 Closed-loop LPV controller structure.

Then,the matrix function A(z)is parameter-dependent stable over Rnz, and there exists m ＜γ such thatThe research object is to design an output feedback controller K(z(t)),as shown in Fig.1.

In terms of Fig.1,the state-space expression of K(z(t))is defined as follows:

LPV model built in Section 2.3 can be modified by

Using Lemma 1,we are able to check whether the closedloop LPV system achieves the performance requirements or not.

Theorem 1.Given a compact set Rnz＞0,the performance level γ ＞0 and the LPV system in Eq.(19),the parameterdependent γ performance problem is solvable if and only if there exist matrix functions X=XT＞0,Y=YT＞0 for all z(t)∈Rnz,and the following LMIs exists:

In Fig.1,d(t)can be further divided by wind disturbance and command input d1(t)and measuring noise input d2(t).For the purpose of simplification,the following assumptions are made on the above LPV model:

(1)D22(z)=0ny×nu.

(2)D12(z)is full column rank for all z ∈Rnz.

(3)D21(z)is full row rank for all z ∈Rnz.

Assumption(1)can be relaxed easily by including a feed through term to the controller for the modified plant which has D22term equal to zero.Under assumptions(2)and(3),the D12and D21terms of the above LPV model can be rewritten as[0,I]Tand[0,I]by norm-preserving transformations on disturbance/error and invertible transformations on input/output signal.Thereby,Eq.(17)can be changed as

According to the LPV robust control structure in Fig.1,an output feedback parameter dependent LPV controller can be achieved as mentioned above.What needs to be pointed out is that the scheduling flight states in LPV model are also dependent parameters in LPV control.In the subsequent simulation studies,both the LPV model and controller can vary harmoniously according to scheduling parameters.

3.2.Performance objective and controller structure for wind disturbance attenuation

Fig.2 Longitudinal robust LPV controller structure for wind attenuation.

As far as flying through wind shear is concerned,the comprehensive objective is to maintain the desired flight path and velocity while keeping laterally stable and avoiding extreme longitudinal and lateral load.In this article,the performance objectives are designed according to civil aircraft's ride quality requirements in wind disturbance.27The control problem is transformed to look for an output feedback controller K which can stabilize the closed-loop LPV system and achieve the desired performance specification.

An LPV robust controller structure for wind disturbance attenuation during longitudinal escape is shown in Fig.2.For aircraft flying through wind shear,the primary control objective is to track velocity and flight path angle commands within 3 m/s and 0.6°error range in steady-state conditions.Taking the tracking error requirement of flight path angle as an example,the bandwidth of the ideal model from the flight path angle command to actual measurement is 4 rad/s,while the roll-off frequency of the weighting function is chosen as 1.2 rad/s to specify the tracking error less than 0.6°at the low frequency region(＜1.2 rad/s).The performance weighting functions WPVand WPγare used to represent the desired performance objectives.In Fig.2,sensor and actuator models are given as Linear Time Invariant(LTI)models whose performance objectives and actuator limits can be represented by weighting functions respectively.

For lateral escape,the objective for the lateral-directional axes is to achieve a good low-frequency tracking performance of φ commands while keeping lateral acceleration small.The primary performance objective for control synthesis of the lateral-directional axes is for the aircraft to track roll angle command inputs within 10%error range in steady state.The ride quality requirement is that the lateral acceleration remains less than±5 g in±30°bank angle command.To improve the ride quality,a turn coordination error defined as r-0.037φ is developed,which should be less than 3%in steady state flight.

The above design performance objectives are accounted for by minimizing weighted closed-loop norms.The weighting functions are described as follows.For example,the weighting function WPφon difference between the command roll angle and measuring responses is derived by the primary performance objective to keep less than 10%tracking error in steady state flight.All the performance requirement and weighted function of LPV controller is given in Table 1.

Table 1 Performance weighted function design for LPV control.

Fig.3 Lateral robust LPV controller structure for wind attenuation.

Table 2 Sensor and actuator model selection.

In the above Figs.2 and 3,the sensors and actuators of a commercial aircraft are chosen as Table 2.

As mentioned earlier,V,α,β and H are selected as LPV FDM's scheduling parameters since they do have effects on aerodynamic derivatives.Contrary to hypersonic vehicle LPV control problem,effects on aerodynamics due to flight altitude change can be neglected during wind shear escape procedure. Consequently, scheduling parameters V and α will be taken into longitudinal controller design;β will be added to the lateral controller design in addition to V and α.On the other hand,reduction of numbers in scheduling parameters will do good to decomposing function optimization in LPV modeling and LMI solution in LPV controller synthesis.

4.LPV FDM and robust LPV control performance analysis

4.1.LPV FDM performance analysis

A direct method to check the performance of LPV FDM is to compare the dynamic response of LPV FDM and nonlinear FDM.In this section,the dynamic response will be firstly examined.Then,the response of flying through microburst wind field will be compared between the nonlinear FDM and LPV FDM.

Based on the B737-800 nonlinear FDM,a stable flight state has been achieved by trim. By selecting the grid by α ∈ {0°, 5°,10°}and V ∈ {1 60 m/s,200 m/s,240 m/s},LPV FDM can be acquired according to Section 2.3 and 2.4.

Fig.4 Elevator deflection response comparison between LPV FDMs and nonlinear FDM.

Fig.5 Eigenvalue variation of GA-LPV FDM.

Fig.6 Aileron deflection response comparison between LPV FDMs and nonlinear FDM.

4.1.1.Dynamic response comparison between LPV FDM and nonlinear FDM

In this section,the dynamic response of nonlinear FDM,GALPV and LP-LPV FDM are compared. During aircraft's approach and landing,elevators are frequently controlled by pilot or flight control system.The elevator δeis deflected according to Fig.4.The dynamic response of the two LPV FDMs and nonlinear FDM is shown as follows.

Simulation results show that GA-LPV FDM can approach the instantaneous dynamics of nonlinear FDM much better than LP-LPV FDM. Particularly in high angle of attack domain,GA-LPV FDM can approach the nonlinear FDM preferably while it is on the boundary of aerodynamic envelope. Another interesting simulation result is shown in Fig.5.During the simulation,two sets of varying eigenvalues of GA-LPV FDM system matrices are plotted.The varying eigenvalues of the GA-LPV FDM show that the short-period and phugoid characteristics change accordingly with scheduling parameters.This advantage provides the GA-LPV FDM with better performance than small-disturbance FDM.

The following Fig.6 shows the dynamic response of the aircraft to aileron deflection.Since lateral control is relatively simple during landing,Fig.6 shows a test of-15°deflection of aileron.Benefitted by genetic algorithm's convergence precision,the steady-state error of GA-LPV FDM is much less than that of LP-LPV FDM.

In addition to the above simulation results,horizontal stabilizer deflection,thrust lever deflection and rudder deflection are tested respectively. Comparison analysis shows that GA-LPV FDM can approach the nonlinear system perfectly.Since the nonlinear FDM selected in this article is complex,the computation time of genetic algorithm will be extended with the increase of grid points.However,Eq.(12)can be solved off-line.Once the LPV FDM was built up,it does not needto solve the optimization problem repeatedly.The following simulation studies are all based on GA-LPV FDM.

4.1.2.Dynamic response comparison of flying through wind shear

Fig.7 Wind shear response comparison after grid optimization.

This section will test the dynamic response of LPV FDM and nonlinear FDM for flying through wind field.A microburst wind field was generated by ring vortex and Rankine vortex according to Ref.28.Here the wind strength is set by 10 m/s.A real-time wind shear vector can be acquired by look-up algorithm.The nonlinear system was trimmed to approach and landing state,in which the aircraft glides constantly from H=1000 m with reference airspeed V=74 m/s.If the grid was chosen the same as in Section 4.1.1,LPV FDM cannot simulate the instantaneous state change precisely once encountering wind shear.Reselect the grid by α ∈ 0°,2°,5°{ },a new simulation result shows in Fig.7.Simulation results show that LPV FDM can further approach the nonlinear FDM by refining the grid of scheduling parameters.The airspeed increased firstly when encountering headwind,and then a downburst.Finally,a strong tailwind will cause the aircraft lose airspeed.A strong wind shear will lead to air crash.In contrast to hypersonic vehicle LPV modelling which pay more attention to wide range flight state changes,the LPV FDM built in this article shows preferable ability to simulate the instantaneous dynamics.

4.2.LPV control simulation for wind disturbance attenuation

After the validation of LPV FDM,the performance of LPV controller will be further examined in this section.Aircraft's approach and landing phase is still taken as an example.The scheduling-parameter varied LPV controller will be compared to a parameter-invariant LQR controller.Linear Quadratic Regulator(LQR)with small-disturbance model has been built up for performance comparison.Once the nonlinear FDM is trimmed for approach and landing, a small-disturbance FDM has been generated at the trim point.A constant state feedback matrix can be acquired based on LQR theory.In wind-free condition,LQR controller shows good performance.

Two kinds of circumstances will be simulated. In Section 4.2.1,the ability of disturbance attenuation of longitudinal LPV controller will be tested while the aircraft is commanded to fly down the glideslope and forced to pass right through the wind shear center. In Section 4.2.2, a more sophisticated LPV controller for lateral escape from wind field will be tested.LPV controllers will respond to command signal input while attenuate wind shear disturbance at the same time.It should be noted that in LPV control,the performance objectives should be adjusted slightly if the LMI optimization solution is not satisfactory.

4.2.1.Wind disturbance attenuation during routine approach and landing

Fig.8 Wind disturbance attenuation during flying through center of wind field.

The same simulation condition as in Section 4.1.2 was given but a stronger wind field(wind strength rises up to 15 m/s)was exerted.Simulation results of LQR and LPV control are shown in Fig.8.The controller commands the aircraft to descend with a constant glide slope angle from t=20 s,and the two controllers show little difference.However,once encountering strong wind shear from t=60 s,the LQR controller cannot maintain aircraft's glide attitude and made the aircraft touch ground in advance.The reason is that after obtaining enough airspeed increase in headwind,the angle of attack limit is reached by the successive downdraft.The aircraft is forced down quickly.In essence,a small-disturbance FDM does not accord with the nonlinear FDM, and furthermore, the parameter-invariant controller cannot adapt to the rapid change in wind speed and direction.On the contrary,the LPV controller shows better glide path tracking ability in wind field.It can be observed that the control deflection is significantly less than LQR controller's.The value of δtand δeand their derivatives have been restricted into a safe range.Simulation results show that the performance objectives of LPV controller are |ΔV |≤1.21 ft/s and |Δγ |≤0.35°,while those of the LQR controller are |Δ V|≤3.5 m/s and |Δ γ|≤13.3°.

The performance objectives of longitudinal LPV controller,velocity ΔV and flight path angle Δγ are studied.Since the real wind field is sophisticated,the wind strength changes randomly.In the article,a varied wind strength ranging from 10 m/s to 20 m/s is simulated to test the stability and robustness of LPV controller.Fig.9 shows the simulation results of performance objectives.It can be concluded that although the performance deteriorates seriously in wind shear,it recovers immediately once leaving the wind field.The LPV model can follow the changes of nonlinear FDM,while the LPV controller shows preferable stability and robustness.On the contrary,LQR controller would lose stability once the wind strength is bigger than 13.6 m/s.

4.2.2.Wind disturbance attenuation during lateral escape

If the aircraft flied through the center of the wind field,it would experience the procedure of headwind, downburst,and tailwind in sequence.At this time,longitudinal escape would be adopted irrespective of the lateral wind.Obviously,it is feasible to escape wind shear from weak wind side.This article compared the performance of lateral escape of both LQR and LPV controllers.The center of the wind shear is located 304.8 m offset to the glide slope.

Fig. 9 Longitudinal performance objective variation under varied wind strength.

As shown in Fig.10,the aircraft was commanded to deviate from the weak wind side since t=10 s.Once the lateral escape initiates,there exists prominent lateral displacement and the aircraft begins to steer steadily.Obviously,lateral escape is much safer under this circumstance. Nevertheless, once encountering the wind shear,the lateral deviation of LQR controller is much more serious than LPV controller.It would do harm to the neighboring aircraft.As can be easily seen,large control surface deflections are required for lateral escape.Similar to the former simulation,the control deflections of LQR controller is bigger than that of LPV controller in wind shear.It is disadvantageous for safe flight.Simulation results show that,for lateral escape,the performance of LPV controller iswhile the performance of LQR is

The performance objectives of lateral LPV controller are studied here.A more intuitional x-y space change is used to substitute Δetc.Similarly,a varied wind strength scope from 10 m/s to 20 m/s is simulated to test the stability and robustness of LPV controller.Fig.11 shows the simulation results of performance objectives.It can be concluded that although the performance objectives still deteriorate seriously in wind shear,they recover immediately once the wind becomes weak.Simulation of LQR controller shows much poorer performance than LPV controller.It should be pointed out that,compared to LQR controller,the accumulated lateral offset because of strong crosswind is much less by LPV controller.It is of great significance for flight safety.

Based on the above simulation,it can be concluded that the LPV controllers can not only make the aircraft respond to the reference command effectively but also attenuate wind disturbance preferably.

5.Conclusions

With the progress of aircraft's reliability and automation,accident rate due to aircraft system failure has continuously reduced.On the contrary,flight accidents induced by wind disturbance have been highlighted.In recent years,many Loss of Control(LoC)flight accidents of civil aircraft are attributed to wind disturbance.29Since it is difficult to estimate and attenuate the wind shear effects on aircraft,in this article,some attempts are made to solve this problem under the framework of LPV modeling and control.Based on the dynamics model with wind effects under wind coordinate,an LPV FDM with wind effects has been derived.Furthermore,an LPV robust control structure for wind attenuation is put forward.Some innovative work in this article involves:

(1)Differing from common robust control in wind shear,the wind shear effects are neither regarded as an input matrix of external disturbance nor parametric uncertainties of FDM.Instead,based on the dynamics model with wind effects under wind coordinate,wind disturbance enters the system matrix of LPV model explicitly.After that,genetic algorithm was adopted to solve the decomposing function in function substitution LPV modeling.The dynamic response analysis shows that GA-LPV model can approximate the nonlinear FDM because it can reflect the instantaneous dynamics especially at the boundary of aerodynamic envelope.By the simulation of flying through wind field,GA-LPV FDM also shows similar performance compared to nonlinear FDM.The current achievements can be applied to analyze the safety problems in special flight case induced by wind disturbance.Based on aerodynamic data,the scheduling parameters can be specifically selected to design the LPV FDM for special flight.What's more,refining grid partition in LPV FDM can be used to deal with local strong nonlinearities.

(2)Based on the LPV FDM,the LPV control method was studied to attenuate wind disturbance for the first time.In contrast to the switching LPV control for wide-range flight envelope,this article pays more attention on the disturbance attenuation ability of LPV controller in wind disturbance so as to keep the aircraft in the safe envelope.In consideration of ride quality and safe performance objectives in wind disturbance,longitudinal and lateral robust LPV controllers were designed respectively.Command response and stability control problem were solved by parameter-dependent output feedback LPV control method.The scheduling flight states in LPV model are actually dependent parameters in LPV control.It could facilitate the design and operation of LPV controller. Compared to parameter-invariant LQR controller designed with a small-disturbance FDM,LPV controllers show preferable robustness and stability for disturbance attenuation.In the near future,better controller synthesis method should be considered to reduce the design conservatism of robust LPV control.

Fig.10 LPV control for lateral escape.

Fig.11 Lateral performance objective variation under varied wind strength.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China(Nos.U1533120 and U1733122)and the Fundamental Research Funds for the Central Universities of China(No.NS2015066).

Appendix A.Longitudinal FDM in scheduling parameter dependent form with remnants:

The matrix elements related to wind disturbance are shown as follows: