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1.College of Engineering, Huazhong Agricultural University, Wuhan 430070, P. R. China;2.State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

Abstract: The large-scale morphing aircraft can change its shape dramatically to perform high flight performance. To ensure the transient stability of aircraft in the morphing process, a novel gain-scheduled control method is investigated numerically in this paper. Based on quasi-steady assumption, the linear parameter varying (LPV) model of the morphing vehicle is derived from its nonlinear equation. Afterwards, by solving a set of linear matrix inequalities along with the bound of the morphing rate via slowly varying system theory, the designed controller which considers the transition stability during the morphing process is obtained. Finally, the transition process simulations of the morphing aircraft are performed via the changes simultaneously in both span and sweep, and the results demonstrate the effectiveness of the proposed controller.

Key words: morphing aircraft; gain-scheduled; convex hull theory; slowly varying system theory; transition stability

0 Introduction

The conventional aircraft is a compromise that allows the flight vehicle to fly with a limited range of flight conditions, which is unable to complete different types of missions and perform extreme maneuvers with high performance. Morphing aircraft can change its shape or geometry during flight to accomplish specific mission better and provide control authority for maneuvering and the special requirements of the military[1-3]. The pattern of morphing are mainly classified into three types[4]: (1) Planform alteration involves span, sweep and chord; (2) Out-of-plane transformation involves twist, dihedral/gull and span-wise bending; (3) Airfoil adjustment involves camber and thickness.

The current morphing aircraft design has been focused on the investigations of modeling, dynamic behavior and flight control[5-8]. For instance, Seigler et al.[3]proposed the methodology for modeling the fight control of morphing aircrafts undergoing large-scale shape change. Zhao and Hu[9]developed a parameterized structural model via the substructure synthesis approach and doublet lattice method. Afterward, Zhao and Hu[10]established the model that govern the dynamic behavior of folding wing via the floating frame method and computational fluid dynamics(CFD) code, and predicted the transient responses during the morphing process. Yue et al.[11]presented the nonlinear dynamic model and the longitudinal dynamic simulation of a folding-wing morphing aircraft through the CFD method. Taking this one step further, Yue et al.[12]proposed a linear parameter varying (LPV) model and a H∞self-scheduled control strategy of the folding wing. In addition, He et al.[13]built a new LPV model by using tensor product method and designed a controller via the parallel-distributed compensation control approach, which did not need trip map and numerical calculation. Dong et al.[14]proposed a H∞control for switched system via the average dwell time method and linear matrix inequalities (LMIs)[15,16], in which the controller could track the angle of attack well during the morphing process. Shi and Peng[17]presented a novel control strategy that involved virtual morphing control surface by employing the active disturbance rejection control.After considering the morphing parameters as a part of the control input variable, they proved that the energy consumption during the morphing flight will be reduced dramatically. However, the above literatures paid less attention to the transition stability of large-scale morphing aircraft. To determine the bound of morphing rate in transient stage, Shi et al.[17,18]derived the state-space equation in the term of morphing rate explicitly and obtained the minimum and maximum values of the morphing rate via the Hurwitz criterion. On the assumptions that the aerodynamic forces depend solely on the instantaneous configuration of the aircraft, Seigler and Neal[19]analyzed the transient stability of large-scale morphing aircraft by slowing varying system theory (SVST)[20,21].

By considering the morphing variables as scheduled parameters, the morphing aircraft can be modeled as a parameter-varying system, which could be analyzed through the gain-scheduled method[22]and LMIs. In this paper, the LPV model of the morphing aircraft is established after introducing the equations of motion (EOM), and a novel gain-scheduled controller is designed through the combination of convex hull theory and SVST. The approach not only proposes an effective controller with satisfactory robustness, but also gives the bounds of the rates of variables and considers the transition stability during the morphing process.

1 LPV Model of Morphing Aircraft

The flight dynamics of a kind of morphing aircraft which can change its span and sweep is considered. The body-axis coordinate systemO-XbYbZband the wind-axis coordinate systemO-XaYaZaare showed in Fig.1, whereαis the angle of attack andβthe slip angle. The aerodynamics forces are usually specified for wind axis, and the two coordinate systems can be obtained from each other via rotation matrices. In the body frame, the nonlinear equations of the morphing aircraft can be written as[11]

(1)

Fig.1 Sketch of morphing aircraft and its coordinate system

The 6-DOF EOM of the morphing aircraft in the body coordinate system can be derived from Eq.(1) as follows

(2)

It is noted thatJ,Ji,So,Soiand their derivatives change significantly with the morphing parameters during the flight, leading to a complex model. Hence, some assumptions are adopted to simplify the nonlinear EOM[19]. Firstly, the inertial forces, time derivative of inertia and static moment owing to morphing are ignored. In addition, the unsteady influence on this kind of morphing aircraft can be negligible[11]and the quasi-steady assumption is adopted. In other words, the instantaneous aerodynamic forces are completely decided by the current shape of the aircraft. Therefore, one obtains the longitudinal nonlinear EOM in a general form

(3)

wherex=[αθqVT]Tis the collection of state variables,θis the pitch angle,qis the pitch angle rate andVTis the flight velocity.u=δeis the control input, whereδeis the aileron deflection.ξ= [ξ1ξ2]Tis a morphing vector consisted of span and sweep.

By taking the morphing variables as scheduled parameters, Eq.(3) becomes a parameter-varying system. Several methods, such as Jacobian linearization, quasi-LPV approach, linear fractional transformation and so on, could be used to establish the LPV model of the morphing aircraft. In this work, according to Jacobian linearization technique, one has

(4)

(5)

whereρis the density of air,Sandcare the equivalent area and chord of wing;L,DandMAare aerodynamic lift, drag and moment, respectively.CL,CDandCmare the corresponding aerodynamic coefficients, which can be expressed as the following form at a low angle of attack

(6)

2 Gain-Scheduled Control Based on Slowly Varying System Theory

2.1 Gain-scheduled control

Rewriting an LPV plant from Eq.(4) in general form yields

(7)

whereA(ξ),B(ξ),C(ξ), andD(ξ) depend affinely on the time-varying parameterξ, for example

A(ξ)=A0+ξ1A1+…+ξnAn

(8)

whereA0,A1, …,Anare the given matrices.ξ(t) varies in a polytopeΘof verticesω1,ω2,…,ωkas

(9)

and the LPV plantPcan be expressed as a ploytope of matrices, it reads

(10)

Lemma1[24]Consider an LPV system as Eq.(7), for a prescribedγ> 0, if there exist matricesS1,S2,YiandXi> 0 (i=1,…，k), such that the following inequalities holds

(11)

where

Then the gain-scheduled controlleru(t)=Kσx(t) could be obtained, where

(12)

and the upper bound of theL2-norm isγ.

2.2 Slowly varying system theory[21]

Considering the linear system of Eq.(4) and a state feedback controlleru=K(ξ)z, one has

(13)

(14)

(15)

In addition,P(ξ) satisfies the following conditions for allξ∈Ψ, whereΨis the domain of interest.

c1zTz≤zTP(ξ)z≤c2zTz

(16)

(17)

(18)

The gain-scheduled controller of Eq.(12) can be solved by combining Eq.(11) and SVST. Fig.2 shows the flow chart:

(2) The corresponding gain-scheduled controllerKσcan be obtained from Eqs.(11) and (12);

(3) The upper boundεof the morphing rate can be determined via SVST;

(5) Otherwise, try different initial morphing rates to go through the above procedures until a proper solution is found.

Fig.2 Flow chart to solve the bound of morphing rate

3 Simulation

A steady andwings-level fight conditions with constant flight speed are employed, i.e.

(19)

where

These aerodynamic derivatives are the functions of morphing variables. In this paper, the vortex lattice solver Tornado[26]is used to calculate the steady aerodynamic forces for different shapes of morphing aircraft, whose spanLand sweepδvaries from 5.2 m to 7.2 m and 0° to 30°, respectively.

Note that the aerodynamic derivatives can be fitted as functions of normalized morphing variables using the generalized leasts quares method, namely

(20)

It is noted that different convex hull models can be derived by choosing different coefficients, if the convex coordinates satisfy Eq.(9)[27]. The vertices ofAiare the values ofA(ξ) at four vertices of the parameter box

ω1:=(Lmin,δmin),ω2:=(Lmin,δmax),

ω3:=(Lmax,δmin),ω4:=(Lmax,δmax)

Fig.3 Response of the close-loop system

Consider the state-space equation with an external disturbance

Another external disturbance square signalω(t) is given, reads

The responses of the close-loop system with different external disturbances are shown in Figs.4, 5.

Fig.5 Response of the close-loop system with external disturbance square signal

In Figs.4, 5, the effects of different disturbances can be eliminated quickly to make the close-loop system stable, which indicates a good capacity of resisting disturbance.

From Figs.4,5, it is indicated that the designed controller makes the systemt end to zero in different cases of disturbance.The primary usefulness of this approach is that it proposes a method to solve Eq.(11) with the consideration of transition stability. Although the introduced SVST reduces the conservatism of controller, it could provide criteria to ensure the transition stability of the slowly varying system, which is a concerned problem for morphing aircraft.In this paper, the minimum required time for large-scalemorphing is less than 1 min, which could give a reference value for flight testing.

4 Conclusions

An LPV model of a kind of morphing aircraft which can change its span and sweep is derived via convex hull theory. To reduce the conservative, the parameter-dependent Lyapunov functions are introduced along with the differential term of the variables, whose bounds are determined by slowly varying system theory. The gain-scheduled controller proposed via convex hull theory and slowly varying system theory is able to eliminate the influence of initial error and external disturbances quickly, which shows a good robustness of the close-loop system. Furthermore, the controller can guarantee the transition stability of the slowly varying system in the morphing process.

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2662018QD029), the National Natural Science Foundation of China (Nos. 91016022, 61503354), and in part by the Priority Academic Program of Jiangsu Higher Education Institutions.