ZOU Yiru,LIU Chunsheng,*,and LU Ke,2

1.School of Automation Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 211100,China;2.Science&Technology on Rotorcraft Aeromechanics Laboratory,China Helicopter Research and Development Institute,Jingdezhen 333001,China

Abstract:A tilt-rotor aircraft has three flight modes:helicopter mode,airplane mode and conversion mode.Unlike the traditional aircraft,the tilt-rotor aircraft,which combines the characteristics of helicopters and fixed-wing aircraft,is a complex multi-body system with the violent variation of the aerodynamic parameters.For these characteristics,a new smooth switching control scheme is provided for the tilt-rotor aircraft.First,the reference commands for airspeed and nacelle angles are calculated by analyzing the conversion corridor and the conversion path.Subsequently,based on the finite-time switching theorem,an average dwell time condition is designed to guarantee the stability in the switching process.Besides,considering the state vibrations and bumps may appear in switching points,the fuzzy weighted logic is employed to improve the system transient performance.For disturbance rejection,three extended state observers are designed separately to estimate the disturbances in the switched systems.Compared with the traditional auto disturbance rejection control and proportion integration differentiation control,this method overcomes the conservatism of wasting the whole model information.The control performances of robustness and smoothness are verified with simulation,which shows that the new smooth switching control scheme is more targeted and superior than the traditional design method.

Key words:tilt-rotor aircraft,switching control,extended state observer(ESO),smooth switching.

1.Introduction

Unlike the traditional aircraft,a tilt-rotor aircraft can change its configuration by tilting the nacelle,and it provides a variety of flight characteristics as it converts from the helicopter mode to the aircraft mode[1,2].It is for this reason that it can enhance civil or military transportation.However,the disadvantages are also unavoidable.First,the flight dynamic of the tilt-rotor aircraft is much more complex than that of the traditional flight vehicles.Meanwhile,the system suffers from high interference when the aerodynamic configuration changes[3].

There are so many control schemes applied to tilt-rotor aircraft.In[4],the flight control law of a tilt-rotor aircraft was designed with the help of the inner/outer loop control structure.In[5],the nonlinear trajectory tracking controller for a tilt-rotor unmanned aerial vehicle(UAV)was designed using the dynamicmodel inversion technique integrated with the adaptation of neural networks.In[6],based on an optimal control concept,an online optimization control method was developed for the tilt-rotor UAV.

However,due to the violent aerodynamic change in tiltrotor aircraft,a single controller may not satisfy the control precision,stability,reliability and other requirements of this complex flight system[7].To solve this problem,a common multi-modes control scheme was proposed as the gain scheduling method[8].The author developed a gain scheduling method for the tilt-rotor aircraft altitude control in[9],whereby the tilt-rotor frame pitch angle was selected as the scheduled variable of the off-line controller.Kong et al.[10]designed the classical proportion integration differentiation(PID)for the vertical flight stage and the backstepping controller for the transition flight.The two controllers are scheduled by the flight trajectory.Although the gain scheduling method is attractive,it may not guarantee the stability in the switching process,not to say the required performance over the full operating envelope of the system[11].

In such cases,drawing the lessons from the multi-model switching idea in[12,13],the author designed a multimodes adaptive control for the tilt-rotor aircraft in[7].However,this nonlinear switching control algorithm neglects the rotor inflow dynamic characteristic,which is computed by the iteration calculation in the practical flight process[14].For this reason,it is difficult to simply implement some traditional nonlinear control methods to the tilt-rotor aircraft,which are based on the exact feedback linearization technique.

To deal with the mentioned problem,recently,some researchers focus on utilizing some novel multi-modes control algorithms on tilt-rotor aircraft.Sun et al.[15]proposed the switching conditions to maintain the stability of the tilt-rotor aircraft switching process. To further facilitate the practical use of the gain scheduling method in the transition process of the tilt-rotor aircraft,Lu et al.[16]studied the developed gain scheduling algorithms based on the corrected generalized corridor.However,owing to the reason that the tilt-rotor aircraft needs to track different velocity signals with the nacelle tilting in the conversion corridor,the mentioned multi-modes control algorithms lead to two hidden troubles. First, the transition process of the tilt-rotor is unsmooth within the conversion corridor.Besides,the sudden switch may make the control signal jump in the switching point,which will bring the undesirable transient responses named bump[17].The bump will activate the high-frequency dynamic characteristics that we have neglected in the small perturbations linearized method, which will further lead to mechanical damage,fatigue loading,or signal saturation[18].

In such cases,this paper can be inspired by smooth switching methods[19,20].It is worth mentioning that,in these methods,the emergence of bumpless switching technique effectively reduces the control signal jumps and unsatisfactory performance caused by switching[21].The idea of it is to force the output of the activated controller to be equal to the plant input at the switching time[22].However,this method has a heavy computing burden,because it needs to calculate the system closed-loop dynamics between each off-line controller and bumpless transfer compensator in real time.

Besides,in the field of smooth switching,He et al.[19]developed the controllers for flexible airplane wing by minimizing the index ofH2output performance and smoothness.Jiang et al.[23]was concerned with a systematic method of linear parameter varying(LPV)controller design for a morphing aircraft.Jiang et al.[24]designed the smooth switching control law based on the interpolation approach.But regrettably,although there are much progress made by the above literature,these methods have not been introduced to the research field of tilt-rotor aircraft yet,which causes the smooth transition control problem of this complex,changeable aircraft far from being solved.

Motivated by those mentioned above,a smooth switching controller based on the extended state observer(ESO)is proposed in this paper.The scheme of the smooth switching controller is novel in the sense that,based on the analysis of the conversion corridor,the conversion path is given.In contrast to the normal case,as the nacelle tilts,the transition of the tilt-rotor aircraft is a switch process with the sequence constraints.Thus,we propose a family of feedback stabilizing controllers as local controllers by satisfying the average dwell time(ADT)conditions.To reduce the bumps in the switching points,the flight states based fuzzy weighted strategy is employed as a measure for‘smoothness’,which realizes the online calculation of the matching degree between the current dynamics and each sub-controller.In this way,it reconfigures the control law.For disturbance rejection,three ESO for each channel are respectively combined with this model-based control scheme,which avoids the waste of all the model information in the traditional auto disturbance rejection control(ADRC)technique[25,26].

By applying both the proposed control scheme and inner/outer loop based PID to the nonlinear tilt-rotor model,the simulation comparison results demonstrate that the stability,robustness and smoothness of the proposed method are more superior to those of the traditional PID.

2.Problem description

2.1Nonlinear model of tilt-rotor aircraft

A tilt-rotor aircraft has three flight modes:helicopter mode,airplane mode and conversion mode.Table 1 shows the relationship between the nacelle angleβMand different flight modes.During the conversion from the helicopter mode to the airplane mode,the flight mechanics(Fx,Fy,Mz)has complicated nonlinear mathematical relationships with the variables(Vx,Vy,ωz,θ,δc,δlong,δe,βM).For the details of aerodynamic forces calculation,one can refer to[27,28].According to the above-cited modeling method,the complete nonlinear flight dynamic mathematical model of a tilt-rotor aircraft is built in the simulated environment of Matlab/Simulink,of which the primary purposes are:(i)to calculate the conversion corridor and the conversion path of the tilt rotor;(ii)to acquire the linear models in the trim points;(iii)to apply the control law to the nonlinear aircraft to verify the effect of the control law.

Remark 1Considering the decoupling conditions in[29],this study decouples the longitudinal channel and the lateral channel for the convenience of design.

The formulation of the nonlinear model is briefly given as(1)for the completeness concern.

wherex=[VxVyωzθ]Tdenotes the state vector;u=[δcδlongδe]Tdenotes the input vector;Iyydenotes the rotational inertia;FxandFydenote the net forces along theX-axis andY-axis in the Soviet body coordinate system;Mzis the pitch moment.

Fx,FyandMzcan be described as

As a result,the whole flight longitudinal model is established.

Remark 2In contrast to[30],this simulation model considers the effect of the rotor wake on the other aerodynamic surfaces and the nacelle tilting dynamics,so it is better to reflect the characteristics of large parameter changes caused by mode transition.

Table 1Tilt-rotor aircraft at different flight modes

2.2Conversion corridor and conversion path

During the whole transition process,the desired motion is the one that can keep the altitude unchanged and drive the airspeed velocity to increase monotonically within the safe conversion corridor.As the nacelle tilts,the lower boundary of the conversion corridor is limited by wing stall,which means the tilt-rotor aircraft is not permitted flying with excessive low speed for fear of wing stall.The upper boundary of the conversion corridor is relative to engine power,the aircraft construction,etc.

All of the above limitations are considered in the modeling process.Thus,the conversion corridor can be obtained by the trim calculation,which is shown as Fig.1.

Fig.1Conversion corridor

Taking the conversion corridor into consideration,the conversion path is defined as below.

Definition 1The conversion pathScpis

wherer(t)(t∈[T0,Tf])denotes the scheduling rule of the nacelle angle,which will be given in the simulation section;V?is the corresponding airspeed,which can be divided into the horizontal componentVxand the vertical componentVy.

In general,there are so many possible tilting transition curves in the conversion corridor,as mentioned in[31],such as transition with a constant power or transition with a constant airspeed.The transition process in this paper aims at altitude maintenance,which demands that the reference value ofVyshould always be zero to prevent falling.In addition,the pitch angle of the aircraft should make slow changes at low speeds.Satisfying the above requirements,the Gaussian function(4)is used to fit the curve of the conversion path,and the corresponding results can be seen in Fig.2.

Fig.2Conversion path

Assume that the nonlinear system(1)is differentiable everywhere.We propose the linear model sets in the operation region by a perturbation method,which can be described as

wherexρa(bǔ)nduρdenote the state vector and input vector in equilibrium points.

Based on the above mathematic model,three issues will be discussed in this paper.

Question 1Construct the switching signalσ(t)and the corresponding controllers to ensure the stability of the closed-loop system and the switching process.

Question 2Improve the transient performance of the flight control system when the switching signals occur.

Question 3Reduce the influence of the disturbancefand enhance the robustness.

3.Controller design

The control structure is depicted in Fig.3.

Fig.3Control structure

3.1ESO design

As we know from the tilt-rotor nonlinear model(1),it is the airframe component that is disturbed by the external factors,which causes that the disturbance of the system is related to the aerodynamicsFx,FyandMz.

Based on these,the external disturbancefis added to the statesVx,Vyandωz,which have direct mathematical relations withFx,FyandMzin(1).In this paper,a class of disturbancefsatisfying the matching condition is considered,with the form ofBρdas(13).

Then,we design three ESOs to observe the states and disturbances.

It can be derived that the ESO[32]for observingVxis

The ESO for observingVyis

The ESO for observingωzis proposed as

ω01,ω02andω03denote the observer bandwidth of the ESO.By choosingω01,ω02andω03,the extended states can estimate the state variables,thus making the estimation errors converge to zero,i.e.,

Without loss of generality,we take the pitch angle ESO for example to analyze disturbance compensation.ωzcan be described as

whereek=1,andf3is the total disturbance to be observed by the ESO of the pitch channel.

Subtracting(11)from(10),it yields

With the definition of the observer errorεasε=is expressed asf3=z33?ε,the external interferencef3approaches the extended statez33when the ESO accurately estimates the aircraft state,namely,

Similarly,the ESO for the forward channel and vertical channel can estimate not only the external disturbancef2andf3,but also the non-measurable state variables of the aircraft.

With the prior assumption

the system can be rewritten as

Then,we design the control law as

It can be derived that the aircraft switching system under the disturbancefturns into(16)after the interference compensation.

3.2Switching control design

Considering that the switching sequence of the tilt-rotor aircraft is restricted by the conversion path,the whole working regionalXcan be divided into multiple overlapping sub-regions,in form of

In this study,a set of feedback controllers are designed with concept of finite-time switching,which are in form of

Adjacent controllers are switched according to the ADT logic.To reduce the states vibration in switching points, the control parameters scheduling module will be introduced later,which makes reference to the idea of fuzzy weighted.

The switching control module is depicted in Fig.4.

Fig.4Switching control module

In order to obtain the switching control law,the following definitions and lemmas are given.

Definition 2[33]For the given positive constantsc1andc2withc1

Remark 3The selection of the matrixRhas a close relation with the system performance,which will be explained in the simulation section.

Definition 3[34]For each switching signalσ(t),which satisfiesT0?t0,it denotes the set of all switching signals for which

The constantis called the ADT.

Lemma 1Let,giving a positive definite matrixPσ,positive constantsα,λandc2/c1>1,the continuous time system is finite-time stabilized when the dwell timeταfor each subsystem satisfies the ADT condition(23),and the switching controllers can be constructed by solving linear matrix inequations(LMIs)(21)and(22).

where

The ADT condition is

Remark 4In the design process of the tilt-rotor switching control,to guarantee the ADT condition,the dwell time should be limited within the transition time between adjacent command points,through adjusting control parameters.This procedure will be discussed in the simulation section.

ProofConstruct the piecewise Lyapunov function as

When the aircraft subsystem is activated by the switching signalσ(t),(25)can be derived.

When it satisfies the following condition(26),(x)can be expressed as(27).

It should be noted that Matlab/LMI toolbox cannot solve the inequality constraint(26)directly,so we transform the condition into(28)by the Schur theorem.

In the tilt-rotor aircraft system,the transition initial timeT0=0.

Thus the piecewise Lyapunov function is

Defining the initial switching time asN0=0,the switching time satisfies,thus

Due to the definition ofλ1andλ2,we have(31)and(32).

It can be derived that

According to(22),we can obtain

Thus(34)can be written as

It implies the system is finite time stable as long as the state trajectory is continuous in[0,Tf].?

In this paper,to reduce the state vibrations and bumps in the switching point,the fuzzy weighted strategy is employed as a measure of‘smoothness’for switching.Due to the fact that the tilt-rotor aircraft transits along the previously known conversion path,this paper employs the fuzzy weighted strategy to schedule the controller,the smooth control signals can be expressed as

Without loss of generality,this study takes nacelle tilting from 65?to 45?as an example to illustrate the control design process.The flight speed and nacelle angle,which determine the flight mode along the conversion corridor,are chosen as the fuzzy inputs to estimate the current plant dynamic.The fuzzy segmentations are given as below.

(i)Input1(V):fuzzy set=[V1,V2,V3],denotes states of flight velocities.

(ii)Input2(βM):fuzzy set=[βM1,βM2,βM3],denotes states of the nacelle angle.

(iii)Output(MD):fuzzy set=[MD1,MD2,MD3],corresponds to three flight modes.

The degree of membership is shown as Fig.5.

Fig.5Membership function

Remark 5The fuzzy rules and fuzzy segmentation are designed according to the conversion path.

The center-of-area method is employed for the defuzzification,and the corresponding result is shown as Fig.6.

Fig.6Fuzzy weighted surface

Thus,the weight coefficientshρcan be described as

4.Simulations and analysis

To demonstrate the effectiveness of the control law,three typical tilt-rotor aircraft working points are taken as an example(with nacelle angles of 45?,55?,65?)and the references of the command points are trimmed as Table 2.

Table 2Trim points at transition mode

It is worth noting that all these simulations are performed on the nonlinear model.The transition parameters and the simulation results are given respectively to prove that the system has good characteristics of smoothness and robustness.

The scheduling rule for the nacelle angle can be described as

whereT0?t?Tf.

As is shown in(38),the aircraft tilting from 65?to 45?is described as a constant rotation process with the conversion time of 4 s.

Remark 6In[35],the whole tilting process of the nacelle can be divided into three stages:initial acceleration stage,tilting in a constant speed,slowing down to stop.This simulation concentrates on the second stage.

It should be noted that,not only the tilting process,but also the selection of the matrixRhas a close relation with the transition performance,which can be shown as below.(InRselection,to clearly observe the system performances after the last switching,the aircraft keeps in 45?for 2 s(from 4 s to 6 s)).

As can be seen from Table 3,the selection of the matrixRhas a close relation with the system dynamic performance and ADT.ADT needs to be shortened within the transition time of adjacent points,so the switching time is chosen ast=2 s.The results are shown as below.

Table 3Relation between R and ADT

As can be seen from Table 3 and Fig.7,the control system with a larger matrixRhas a shorter ADT time,but worse dynamic performance.

Fig.8Structure of PID control

Fig.7Flight states under different R

For the purpose of altitude maintenance,the pitch angle should track the command signal in a short time;otherwise,the lack of pitch angle makes the wing unable to provide enough lift force to balance the rotor thrust loss in the vertical direction,which will make the altitude fall.

Thus,we adjust the parameter of the controller as:R=diag([40,25,48,97]),α=0.79,λ=0.01,c2/c1=200.

The classic PID is common for the tilt-rotor aircraft in control design.To show the design difference and system performance between PID and the proposed method,this paper makes a simple comparison.

The control structure of PID is shown as Fig.8.The control allocation strategy and the control channel switching strategy are both employed to solve the problem of the control surface redundancy.We will not cover the details of them here.

The corresponding simulation results are shown in Fig.9 and Fig.10.

As is shown above,due to the huge variation of aerodynamic characteristics,the traditional PID control,which only uses one set of parameters in the transition process,is hard to satisfy the requirement of control precision.Due to the changeable dynamics of the tilt-rotor,a single controller does not work very well either.In addition,when PID controls the tilt-rotor aircraft with the characteristic of multivariable,strong coupling and control surface redundancy,the control channel switching strategy should also be considered,which further increases the complexity of the design.

Compared with switching directly, the smooth switching by the fuzzy weighted strategy reduces the states vibrations and bumps at the switching point.Meanwhile,the smooth switching scheme also reduces the jumps of the control signal.Thus,the proposed smooth switching method enhances the transient performance in the transition process.

Fig.9System states responses

Fig.10System control responses

To further validate the robustness of the proposed method,the external sinusoidal interference signald(with the amplitude of 15 and the frequency of 10)is respectively added to the forward speedVx,the vertical speedVy,and the pitch rateωz.

The following simulations compare two systems,one applying the smooth switching controller with ESO,and the other applying the smooth switching controller without ESO.The corresponding results are shown in Fig.11.

Fig.11System states responses(under disturbance)

As can be seen from Fig.11,without applying the ESO to smooth switching control,the disturbance fluctuation toVxis reduced to 1.11%of the initial interference amplitude,Vyto 11.1%,ωzto 8.6%,which illustrates that the smooth switching control method(without ESO)has a certain degree of robustness.However,applying ESO to compensating for the disturbance,the disturbance toVxcan be furthered to 0.22%of the initial interference amplitude,Vyto 1.19%,ωzto 2.9%.

The two results show that the ESO based system has a better disturbance rejection ability,which could lead to a better response.The system robustness can be further analyzed that,the switching parameters gradually change along the conversion trajectory,which suggests that the state matrix of each subsystem is perturbed actually.However,it can be seen from the above figures that,although there exists both external disturbance and parameter perturbation in the system,the transition is stable.This result further supports the conclusion that the proposed control can achieve better results with a strong robustness.

5.Conclusions

The flight dynamic of a tilt-rotor aircraft is much more complex than the traditional aircraft,because its configuration changes by tilting.

Based on its characteristic of violent aerodynamic parameter variation,the smooth switching control scheme is designed to deal with the complex flight dynamics problem.Furthermore,to diminish the influence of disturbance,the proposed control structure combines the ESO and the model information-based control strategy.

The simulation comparison between the classic PID and the proposed method demonstrates that,for the tilt-rotor aircraft,the new smooth switching control scheme is more targeted and superior to the traditional design method.