Smooth tracking control for conversion mode of a tilt-rotor aircraft with switching modeling*

Kebi LUO, Shuang SHI, Cong PENG

College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

†E-mail: shishuang@nuaa.edu.cn

Abstract: This paper investigates the state-tracking control problem in conversion mode of a tilt-rotor aircraft with a switching modeling method and a smooth interpolation technique.Based on the nonlinear model of the conversion mode, a switched linear model is developed by using the Jacobian linearization method and designing the switching signal based on the mast angle.Furthermore, an H∞state-tracking control scheme is designed to deal with the conversion mode control issue.Moreover, instead of limiting the amplitude of control inputs, a smooth interpolation method is developed to create bumpless performance.Finally, the XV-15 tilt-rotor aircraft is chosen as a prototype to illustrate the effectiveness of this developed control method.

Key words: Tilt-rotor aircraft; State-tracking control; Switched linear systems; Time-scheduled multiple Lyapunov function approach; Smooth interpolation https://doi.org/10.1631/FITEE.2300266 CLC number: TP18; V279

1 Introduction

A tilt-rotor aircraft can switch between helicopter mode and airplane mode by installing two rotating nacelles to the tips of wings,which has the advantages of both fixed-wing aircrafts and helicopters(Thompson, 1990).The mode in which a tilt-rotor aircraft transforms between these two modes is called conversion mode, and it includes the characteristics of full flight modes (Rysdyk RT and Calise, 1999).Modeling a tilt-rotor aircraft in conversion mode is arduous, because it is difficult to reconcile the accuracy of the mathematical model and the difficulty of control design.For instance, a simplified nonlinear mathematical model was built based on the Euler equations in Kleinhesselink (2007).However, this simplified model is inaccurate because of the rotor effects.Simplifying the model can make the control design easier, but the model might be inaccurate.Designing control laws based on a nonlinear model is also a challenge, due to complex changes in dynamic characteristics and other problems (Rysdyk R and Calise, 2005).Although the transition process lasts only for a couple of seconds, it is still the most complicated and significant part(Li Z and Xia,2018).Consequently,many scholars have focused on the modeling and control issues of conversion mode(Wang YE et al.,2016;Barra et al.,2019;Abà et al.,2020).

The aerodynamic features change violently as the nacelles tilt.Thus, it is difficult to describe the transition process with a single mathematical model.It is also difficult to design control schemes directly for nonlinear models (Chen et al., 2017; Li YL et al., 2022; Tang et al., 2022; Zhao H et al.,2023).Modeling by the switched system and designing the switching control scheme can effectively resolve this problem.In Wang XH and Cai(2015),a switching control method was adopted in controller design based on a nonlinear model of conversion mode.A novel tilt-rotor was studied in Kong and Lu(2018),and a switching control law was designed by developing the back-stepping method.However,the tilting features were eliminated in designing the back-stepping controller.The methods above apply switching method to complete the switching controller design,but the nonlinear aerodynamic model is adopted for the modeling.A switched system generally consists of a number of subsystems and a switching law representing the switching order(Zhao XD et al., 2012; Ye et al., 2021; Hou et al., 2022;Roy et al., 2022).Modeling the tilting process by a switched linear model can obtain more tilting features, and the violent changes of aerodynamic parameters can be depicted more precisely.Moreover,the control design can be simplified based on the switched linear model.

Due to the switching characteristic,abrupt controller bumps will appear when switching occurs at the switching instants, which may destroy the performance and even the stability of the system.To reduce controller bumps, a bumpless transfer method is necessary to make the control input smooth at switching instants, which can further improve the transient performance of the system.In Daafouz et al.(2012), an additional global condition was introduced to restrain the controller gain.However,finding a feasible solution to avoid this global limitation is difficult (Yang and Zhao, 2019).To release this constraint, a local condition was proposed instead,and this bumpless transfer method was widely used(Yang and Zhao,2019;Zhao Y and Zhao,2020;Zhao Y et al.,2020).This approach requires a reference input to limit the amplitude range of the actual control input.As a result,the disparity of the control signal between adjacent sides of a switching instant can be maintained at a small level.Unfortunately,there is still no effective way to obtain a proper reference input.

An admissible solution to the previous problem is to develop a bumpless control architecture without reference control inputs.Recently, the timescheduled multiple Lyapunov function (MLF) approach was exploited for analysis and synthesis of switched systems, which can achieve better performance than traditional time-independent MLF methods(Allerhand and Shaked,2011;Xiang,2015;Yuan et al., 2018; Shi et al., 2021; Fei et al., 2023).However, the time-scheduled MLF is discontinuous at switching instants, which leads to controller bumps.In this paper, the time-scheduled method is further developed to construct an improved MLF.An additional subinterval is introduced after each switching (called a transition interval), and the linear interpolation method is adopted in transition intervals.By this manipulation, the revised MLF is continuous and non-increasing at the switching instants.Accordingly, the designed control scheme can decrease controller bumps without introducing an additional reference input.

In this paper,the conversion mode of a tilt-rotor aircraft is built as a switched linear system.On this foundation, an improved time-scheduled MLF technique is proposed to develop a smoothH∞statetracking control scheme, which can accomplish a smooth transition between helicopter mode and airplane mode.The main improvements of this article are twofold:

1.A switched linear model of a tilt-rotor aircraft in conversion mode is built.As a result, the nonlinear control problem of the conversion mode is transformed into the state-tracking control issue of the switched linear system.

2.A time-scheduled method is developed to construct an improved MLF.On this basis,the linear interpolation method is adopted in transition intervals to resolve the bumpless transfer issue.Compared with the conventional bumpless transfer technique,the reference input is no longer essential.

2 Switching modeling for the longitudinal dynamic model of a tilt-rotor aircraft

The partonomy is used in the modeling process,and the model is simplified into a longitudinal dynamic model.Some assumptions are essential in the modeling process (Harendra et al., 1973).For instance,each component of a tilt-rotor aircraft is rigid body and the rotation speed of the rotor is regarded as fixed during the process.

Take the XV-15 tilt-rotor aircraft as an example.Based on the Euler equations(Kleinhesselink,2007),the longitudinal dynamic model for conversion mode is

whereu,w,q, andθdenote the forward speed, longitudinal speed, pitch rate, and pitch angle, respectively.mgis the mass of the aircraft,gis the gravitational acceleration, andIyis the pitch rotational inertia.Fxdenotes the resultant aerodynamic force of each part in thexdirection, whileFzdenotes the force in thezdirection.Myrepresents the resultant aerodynamic moment of each component in theydirection.These forces and moments are related to the collective pitchθb,the longitudinal periodic pitchθa,the elevator deflectionδe,and the mast angleβM.

Maisel et al.(2000) presented the conversion corridor with respect to the mast angleβMand the air speedV, which can guarantee a safe tilting process.By selectingNoperating points in the conversion corridor, trimming the nonlinear model of conversion mode(1),and taking the Jacobian linearization,a cluster of linear models are obtained.Furthermore, introducing a switching signalσ(t), nonlinear model (1) can be transformed to a switched linear system:

wherex(t)=[Δu,Δw,Δq,Δθ]Tis the state,u(t)=[Δθb,Δθa,Δδe]Tis the input, andω(t) is the external disturbance.σ(t) : [0,∞)→ RN≜{1,2,···,N}is the switching signal.Ai,Bi, andHi(σ(t) =i,i ∈RN) are matrices with proper dimensions, and these matrix parameters are derived based on the trimming results.

Definition 1 (Zhao XD et al.,2012) The switching lawσ(t) =i,i ∈RN, can be called modedependent dwell-time(MDDT) switching if

whereτdi ＞0 is an existing scalar,Ni(tx,ty) andTi(tx,ty)represent the overall number that switches to theithmode and the total activated time of theithsubsystem (i ∈RN) in the interval [tx,ty), respectively.Then,τdiis called MDDT.

Remark 1 The mast angleβMis closely related to the inclination angle of the nacelles.In conversion mode, the variation ofβMfrom 0°to 90°represents the flight mode of the tilt-rotor aircraft changing from helicopter mode to airplane mode.By linearizing nonlinear model (1) based on the range ofβM,the switched linear system (2) is obtained, and thus the runtime of each subsystem in Eq.(2)has a lower bound.This MDDT switching can further describe the tilting characteristics.

3 Smooth H∞state-tracking control

In this section, the interval partitioning and smooth interpolation methods are further extended to develop an improved MLF.On this basis, anH∞state-tracking control scheme is proposed for the switched linear system(2).

The reference model can be scrupulously extracted from the tilting course based on the state characteristics and trimming results:

wherexr(t)∈Rnxandr(t)∈Rnrrepresent the state and input of the reference system,respectively.This reference system generates desirable state trajectories to be tracked.To ensure favorable tracking performance,Arshould have the same matrix structure asAi,i ∈RN.

Considering the switched system(2)with reference model (3), define the tracking error ase(t) =x(t)-xr(t).The control input is designed as

whereΛei=Ai+BiKei(t),Λci=Ai+BiKci-Ar, andΛri=BiKri-Br,∀σ(t) =i,i ∈RN.SelectKciandKrisuch thatΛci= 0 andΛri=0.Hence, the trajectory tracking issue becomes the stabilization issue of tracking error system(5).

Definition 2 Considering the switched system(2) with reference model (3), if there exist switching signalσ(t) and controller(4)satisfying

(1) whenω(t)≡0 and system (5) is globally uniformly exponentially stable (GUES),

(2) whenω(t)/= 0 and under the zero initial condition,

whereν ＞0,∈＞0, and ˜γ ＞0 are constants, then system (5) is said to achieve anH∞state-tracking performance.

The control objective here is to co-design controller (4) and an MDDT switching law such that tracking error system (5) has anH∞tracking performance.

Inspired by interpolation approaches in several works (Allerhand and Shaked, 2011; Xiang, 2015;Yuan et al., 2018), here we introduce a scalarτhand let [ts,ts+τh) denote the transition interval,wheretsdenotes thesthswitching instant.The interpolation process is developed during the interval to design time-scheduled controllers, which can prevent abrupt bumps appearing at switching instants.Additionally, let the interval [ts+τh,ts+1)be partitioned intoL+1 portions.Define a time sequence{ts0,ts1,···,tsL}, wherets0=ts+τh.Eventually, the interval [ts,ts+1) is divided into[ts,ts0)∪[ts0,ts1)∪···∪[ts(L-1),tsL)∪[tsL,ts+1).Let the length of each portion except[tsL,ts+1)beτliforσ(ts) =i,i ∈RNandl ∈Z[0,L].To realize this partitioning approach,τdi ＞τh+Lτliis obviously required fori ∈RN.Based on such a partitioning, a smoothH∞state-tracking control scheme is developed.

Theorem 1 Consider the switched linear system(2) with reference model (3).Letςsi ≥ςui ＞0,τli ＞0,τh＞0 be given constants fori ∈RN,andγ ＞0,and suppose that there exist matricesTik ＞0 andUikfor (i,j)∈RN×RN,i/=j,k ∈Z[0,L],l ∈Z[0,L-1],

with He{X}≜X+XT.

Then,for the MDDT switching signal satisfyingτdi ＞τh+Lτli,i ∈RN, the closed-loop system (5)has anH∞state-tracking performance no greater than

wherePi(t) =T-1i(t) fort ∈[0,∞).It can be verified that the constructed Lyapunov function is continuous and differential fort ∈[0,∞).

Fort ∈[ts,ts0), one can deduce that

can be guaranteed by inequalities(8)and(9).Meanwhile, inequality (10) can guarantee inequality (17)fort ∈[tsL,ts+1).

First, considerω(t)≡0.By integrating inequalities (16) and (17) fromtstots+1, one can obtain that

whereςi=ςsi-ςuifori ∈ RN.Becauseτdi＞τh, one can deduce thatV(e(t))≤ αexp(-ςu(t-t0))V(e(t0)), whereςu= mini∈RN(ςui) as mentioned.Furthermore, it is observed thatV(e(t)) converges to 0 ast →∞.Then one can deduce that the system is GUES.

Next, considerω(t)/= 0.Integrating inequalities (16) and (17) fromtstots+1and lettingF(t)=eT(t)e(t)-γ2ωT(t)ω(t), it holds that

whereThi(ϑ,t)denotes the total length of transition intervals during [ϑ,t) of theithsubsystem fori ∈RN.It holds thatThi(ϑ,t)≤τhNi(ϑ,t).

Consider the zero initial condition.BecauseV(e(t))≥0, one can deduce that

which demonstrates that tracking error system(5)achieves a non-weightedH∞performance as Eq.(11)according to Definition 2.

The following corollary can be derived based on Theorem 1 without considering interval partitioning or smooth interpolation.In this situation, the controller gainKei(t) in control law (4) becomes a time-invariant matrixKei,i ∈RN.

Corollary 1 Consider the switched linear system(2) with reference model (3).ςi ＞0 andμi ＞1 are given constants fori ∈RN.Supposing that there exist matricesTi ＞0 andUifori ∈RN,(i,j)∈RN×RN,i/=j,

for the MDDT switching law satisfyingτdi ＞ln(μi/ςi),i ∈RN, system (5) is GUES with anH∞performance no greater than

where ¯ς= maxi∈RN(ςi) and ¯φ=maxi∈RN(ln(μi/τdi)-ςi).In addition, the controller gains satisfyKeiTi=Ui,Ai+BiKci=Ar,andBiKri=Br, fori ∈RN.

The proof is omitted here.

4 An example based on the XV-15 tiltrotor aircraft

In this section, the XV-15 tilt-rotor aircraft is taken as an illustrative example to verify the effectiveness of the developed control method in Theorem 1.

Table 1 provides trimming results of each selected trimming point according to the conversion corridor,and the tilting process starts withβM=0°andV=101.21 ft/s(1 ft=0.3048 m).On this basis,the linearized Jacobian matrices of each trimming point can be obtained:

Table 1 Trimming results of the selected operating points

The whole tilting process is represented by the five subsystems provided above.

The parameters of the reference system are given by

and the reference input is selected asr(t) =22 sin(0.1146t).External disturbance input, additionally,is considered asω(t)=2 cos(0.53t)e-0.05t.

Setςu1=ςu2=ςu3=ςu4=ςu5=ςs1=ςs2=ςs3=ςs4=ςs5=0.1,τh=0.8, andτl1=τl2=τl3=τl4=τl5=0.5,and selectγ=1.1615 andL=1.

Dealing with the conditions in Theorem 1, one can obtain

Setγ= 1.0595,ς1=ς2=ς3=ς4=ς5=0.1, andμ= 1.3, and deal with the conditions in Corollary 1.By similar manipulation,KciandKrishare the same values in Theorem 1, andKeifori ∈RNare

The total runtime of tilting is 15 s, and the selected switching signal is displayed in Fig.1, which depends on the intervals between adjoining trimming points.The MDDT constraints of Theorem 1 and Corollary 1 are both satisfied.

The comparison of Theorem 1 and Corollary 1 is displayed in one figure to highlight the difference.Figs.2-5 track performances of forward speedu,longitudinal speedw,pitch rateq,and pitch angleθ,respectively.Both control schemes of Theorem 1 and Corollary 1 are capable of accomplishing state tracking and control.However,the control scheme of Theorem 1 has better tracking control performance than the scheme of Corollary 1, particularly at switching instants.

Fig.1 Switching signal σ(t)

Fig.2 Forward speed u (1 ft=0.3048 m)

Fig.3 Longitudinal speed w (1 ft=0.3048 m)

Fig.4 Pitch rate q

5 Conclusions

Fig.5 Pitch angle θ

In this paper, a switched linear modeling method is provided for the conversion mode of a tiltrotor aircraft.AnH∞state-tracking control law is designed that transforms the nonlinear control issue of conversion mode into a stabilization problem of the switched linear system.Moreover, the linear interpolation and interval partitioning methods are further developed to restrain the abrupt controller bumps at switching instants.Finally, the effectiveness and advantages of the proposed control scheme are validated by using the XV-15 aircraft as an example.It is worth mentioning that the linearization process leads to inevitable error between the switched linear model and the original nonlinear model,which has a certain impact on the performance of the system.To deal with this issue, a switched Takagi–Sugeno fuzzy modeling method can be further pursued to improve the accuracy of modeling in future studies.

Contributors

Shuang SHI designed the research.Kebi LUO processed the data and drafted the paper.Shuang SHI helped organize the paper.Cong PENG revised and finalized the paper.

Compliance with ethics guidelines

Kebi LUO, Shuang SHI, and Cong PENG declare that they have no conflict of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.