IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 25, NO. 1, JANUARY 2017 351

Control Design for Nonlinear Flexible Wings of a Robotic AircraftWei He, Senior Member, IEEE, and Shuang Zhang, Member, IEEE

Abstract— In this brief, the control problem for flexible wingsof a robotic aircraft is addressed by using boundary controlschemes. Inspired by birds and bats, the wing with flexibilityand articulation is modeled as a distributed parameter systemdescribed by hybrid partial differential equations and ordinarydifferential equations. Boundary control for both wing twistand bending is proposed on the original coupled dynamics, andbounded stability is proved by introducing a proper Lyapunovfunction. The effectiveness of the proposed control is verified bysimulations.

Index Terms— Boundary control, distributed parametersystem, flexible wings, microaerial vehicle (MAV), robot, vibra-tion control.

I. INTRODUCTION

W ITH the increasing focus on the development ofaerospace engineering, the investigations for microaer-

ial vehicles (MAVs), which belong to a class of unmanned aer-ial vehicles, have gained much attention [1]–[3]. MAVs can beused for inspection, surveillance, and reconnaissance in someconstrained environments, which are more suitable for theunmanned aircraft. The earlier research focuses on the fixed-rigid wings of MAVs. However, driven by military and civilianapplications, designers strive to reduce weight and increasemaneuverability of the system, and thereby lightweight flexiblewings are widely used in modern aircraft. Different from therigid ones [4], [5], the wings with flexibility are believed tooffer major advantages such as cost-effective, fight agility,lower price, and high performance. The flexible property willlead to changes of the intended aerodynamics of the wings,thus potentially resulting in undesired aircraft responses [1],[6]. Consequently, there is a need to analyze the vibrationcontrol problem of the flexible wing, which is describedby hybrid partial differential equations-ordinary differentialequations (PDEs-ODEs). In recent years, increasing attentionhas been paid on the control design of various distributedparameter systems [7]–[18].

Over the last decades, great progress has been witnessed inMAVs, and the control is becoming an integral part. Due to theactuator and sensor limitations, boundary control is considered

Manuscript received January 30, 2015; revised May 27, 2015 andSeptember 11, 2015; accepted February 13, 2016. Date of publication May 23,2016; date of current version December 14, 2016. Manuscript received in finalform February 26, 2016. This work was supported in part by the National Nat-ural Science Foundation of China under Grant 61522302 and Grant 61403063,in part by the National Basic Research Program of China (973 Program)under Grant 2014CB744206, and in part by the Fundamental ResearchFunds for the China Central Universities under Grant FRF-TP-15-005C1 andGrant ZYGX2015J120. Recommended by Associate Editor H. R. Pota.

W. He is with the School of Automation and Electrical Engineering,University of Science and Technology Beijing, Beijing 100083, China(e-mail: hewei.ac@gmail.com).

S. Zhang is with the School of Aeronautics and Astronautic, University ofElectronic Science and Technology of China, Chengdu 611731, China (e-mail:zhangshuang@nus.edu.sg).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2016.2536708

to be a more practical method in various flexible-structuresystems [19]–[26]. In [1], elegant boundary control is designedfor a robotic aircraft with the flexible wings, where the effec-tiveness of the proposed control is verified by both simulationsand experiments. In [27], a novel robust controller is designedbased on minimax linear-quadratic-Gaussian (LQG) controlfor the control of vibrations in a flexible cantilever beam. In[28], an interdisciplinary project is discussed to set up a pro-totype flexible robotic link. Using a robust adaptive boundarycontrol, an axially moving string with nonlinear behavior isdiscussed in [29]. In [30], using hydraulic actuators at thetop end, a 3-D marine riser system is stabilized by boundarycontrollers. Using the Laplace transform to derive the exactsolution of the wave equation, boundary impedance controlfor a string system is investigated in [31]. A control schemeis proposed for an axially moving membrane system to track adesired axial transport velocity in [32], where it is shown thattransverse vibrations can also be suppressed with the proposedcontrol. In [33], three control inputs including one controlforce and two control torques at boundaries are proposed for anaxially moving string system to regulate both the longitudinaland transverse vibrations with velocity tracking.

In this brief, the deformation of the flexible wing hastwo elastic degrees of freedom: 1) twist and 2) bending. Thestructural dynamics of the flexible wing are coupled [3], [34],as shown in Section II. The system is transversely loadedwith spatiotemporally varying distributed loads. Due to theeffects of external loads, the governing equations of the systemare described as two coupled nonhomogeneous PDEs withthe unknown disturbance terms. The control and control-related issues are presented through theoretical analysis andsimulations. Boundary control laws are proposed on the basisof the original distributed parameter system to control thedeformation of the flexible wing. The exact values of externalloads are not required. With the proposed control, the closed-loop system is uniformly bounded via the Lyapunov directmethod. The control performance of the system is guaranteedby suitably tuning the control parameters.

The rest of this brief is organized as follows. In Section II,the dynamic model of the flexible wing and some lemmas aregiven for the subsequent development. Based on the Lyapunovdirect method, boundary control schemes are proposed tocontrol the deformation of the wing in Section III, whereit is shown that the uniform boundedness of the closed-loopsystem can be achieved by the proposed control. Simulationsare carried out to illustrate the performance of the proposedcontrol in Section IV. The conclusion of this brief is presentedin Section V.

II. PROBLEM FORMULATION

Remark 1: For clarity, notations (·)′ = ∂(·)/∂x and˙(·) = ∂(·)/∂ t are used throughout this brief.

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352 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 25, NO. 1, JANUARY 2017

Fig. 1. Flexible wings of a robotic aircraft.

Let w(x, t) and θ(x, t) denote the bending and twist dis-placements, respectively. Consider the following structuraldynamics of the right wing [1] with the governing equa-tions as:

mw(x, t) − mxecθ (x, t) + ηEIbw′′′′(x, t)

+ EIbw′′′′(x, t) = Fb(x, t) (1)

Ip θ (x, t) − mxecw(x, t) − ηG J θ ′′(x, t)

− G Jθ ′′(x, t) = −xacFb(x, t) (2)

∀(x, t) ∈ (0, L) × [0,∞), and the boundary conditions fora tip-based actuation as

w(0, t) = w′(0, t) = w′′(L, t) = θ(0, t) = 0 (3)

EIbw′′′(L, t) = F(t) (4)

GJθ ′(L, t) = M(t) (5)

∀t ∈ [0,∞), where L is the length of the wing, m is themass per unit span, Ip is the polar moment of inertia ofthe wing cross section, EIb is the bending stiffness, GJ istorsion stiffness, xec denotes the distance between the centerof mass and the shear center of the wing, xac denotes thedistance between the aerodynamic center and shear center,the term η denotes the Kelvin–Voigt damping coefficient, andF(t) and M(t) are the applied tip force and twisting moment,respectively. The wing in Fig. 1 is transversely loaded with aspatiotemporally varying distributed load Fb(x, t).

Remark 2: The boundary condition EIbξyyy = F(t) in (4)is simplified from EIb(ξyyy + ηξt yyy) = F(t) when theKelvin–Voigt damping coefficient η �= 0. EIb(ξyyy +ηξt yyy) =F(t) essentially creates a low-pass filter for the control inputF(t) [35], [36]. As long as the dominant frequencies in F(t)are smaller than the cutoff frequency 1/η, the boundarycondition in (4) is sufficiently accurate for control design.What is more, the Kelvin–Voigt damping term is physicallyakin to distributed load [Fb(x, t) in Eq. (1)] on the wing, whichdoes not affect the boundary conditions [1].

Assumption 1: For the unknown spatiotemporally varyingdistributed load Fb(x, t), we assume that there exists a con-stant Fb max ∈ R

+ such that |Fb(x, t)| ≤ Fb max, ∀(x, t) ∈[0, L] × [0,∞). This is a reasonable assumption as thespatiotemporally varying load Fb(x, t) has finite energy andthus is bounded, i.e., Fb(x, t) ∈ L∞.

Lemma 1 [37], [38]: Let φ(x, t) ∈ R be a function definedon x ∈ [0, L] and t ∈ [0,∞) that satisfies the boundarycondition

φ(0, t) = 0 ∀t ∈ [0,∞) (6)

then the following inequalities hold:

φ2(x, t) ≤ L∫ L

0[φ′(x, t)]2dx ∀x ∈ [0, L] (7)

∫ L

0φ2(x, t)dx ≤ L2

∫ L

0[φ′(x, t)]2dx ∀x ∈ [0, L]. (8)

III. CONTROL DESIGN

In this section, boundary control F(t) and M(t) aredesigned to suppress the bending and twist displacementsw(x, t) and θ(x, t) of the flexible wing system, i.e., to ensurethat both w(x, t) and θ(x, t) are bounded by a constant,respectively. First, we give the following definition [39].

Definition 1: Let f (t) : Rn → Rm be a continuousfunction or piecewise continuous function. The p-norm of fis defined by

‖ f ‖p =(∫ ∞

0| f (t)|pdt

)1/p

, p ∈ [0,∞) (9)

‖ f ‖∞ = supt∈[0,∞)

| f (t)|, p = ∞. (10)

By letting p = ∞, the corresponding spaces be called L∞.More precisely, let f (t) be a function on [0,∞) of the signalspaces, then they are defined as

L∞ �{

f : R+ → R, ‖ f ‖∞ = supt∈[0,∞)

| f | < ∞}

. (11)

From a signal point of view, the ∞-norm is its absolutemaximum amplitude or peak value. The definitions of thenorms for vector functions are not unique [40].

Consider the Lyapunov function candidate as

V (t) = V1(t) + �(t) (12)

where V1(t) and �(t) are defined as

V1(t) = β

2m

∫ L

0[w(x, t)]2dx + β

2E Ib

∫ L

0[w′′(x, t)]2dx

+ β

2Ip

∫ L

0[θ (x, t)]2dx + β

2GJ

∫ L

0[θ ′(x, t)]2dx

(13)

�(t) = αm∫ L

0w(x, t)w(x, t)dx + α Ip

∫ L

0θ (x, t)θ(x, t)dx

− αmxec∫ L

0[w(x, t)θ(x, t) + w(x, t)θ (x, t)]dx

− βmxec∫ L

0w(x, t)θ (x, t)dx (14)

where α and β are two small positive weighting constants.Define a new function as

κ(t) =∫ L

0{[w(x, t)]2 + [θ (x, t)]2 + [w′′(x, t)]2

+ [θ ′(x, t)]2}dx (15)

HE AND ZHANG: CONTROL DESIGN FOR NONLINEAR FLEXIBLE WINGS OF A ROBOTIC AIRCRAFT 353

then we obtain the upper and lower bounds of V1(t) as

γ2κ(t) ≤ V1(t) ≤ γ1κ(t) (16)

where γ1 = (β/2) max(m, Ip, EIb, GJ) and γ2 =(β/2) min(m, Ip, EIb, GJ).

Applying inequality (8), we obtain

|�(t)| ≤ (αm + αmxec + βmxec)∫ L

0[w(x, t)]2dx

+ (α Ip + αmxec + βmxec)∫ L

0[θ (x, t)]2dx

+ (αm + αmxec)L4∫ L

0[w′′(x, t)]2dx

+ (α Ip + αmxec)L2∫ L

0[θ ′(x, t)]2dx

≤ γ3κ(t) (17)

where

γ3 = max{αm + αmxec + βmxec, α Ip + αmxec + βmxec,

(αm + αmxec) L4, (α Ip + αmxec)L2}. (18)

Consider a positive β satisfying that β > (2γ3/min(m, Ip, EIb, GJ)), then we have

0 ≤ λ2κ(t) ≤ V (t) ≤ λ1κ(t) (19)

where λ2 = γ2 − γ3 and λ1 = γ1 + γ3.Differentiating V (t) leads to

V (t) = V1(t) + �(t) (20)

where V1(t) is given as

V1(t) = βm∫ L

0w(x, t)w(x, t)dx

+ β Ip

∫ L

0θ (x, t)θ (x, t)dx

+ βGJ∫ L

0θ ′(x, t)θ ′(x, t)dx

+ βEIb

∫ L

0w′′(x, t)w′′(x, t)dx . (21)

Substituting governing equations (1) and (2), using integrationby parts and boundary conditions (3) and (4), and applyinginequality (8), we have

V1(t) ≤ −(

βηEIb

2L4 − σ1β

) ∫ L

0[w(x, t)]2dx

−(

βηGJ

2L2 − σ2βxac

)∫ L

0[θ (x, t)]2dx

− βηEIb

2

∫ L

0[w′′(x, t)]2dx

− βηGJ

2

∫ L

0[θ ′(x, t)]2dx

+ βmxec∫ L

0[w(x, t)θ (x, t) + w(x, t)θ (x, t)]dx

− βw(L, t)[F(t) + η F(t)]+ βθ(L, t)[M(t) + ηM(t)]+

(β

σ1+ βxac

σ2

)L F2

b max (22)

where σ1 and σ2 are positive constants. Similarly, we have thederivative of �(t) as

�(t) = αm∫ L

0w(x, t)w(x, t)dx + αm

∫ L

0[w(x, t)]2dx

+ α Ip

∫ L

0θ (x, t)θ(x, t)dx + α Ip

∫ L

0[θ (x, t)]2dx

− βmxec∫ L

0[w(x, t)θ (x, t) + w(x, t)θ (x, t)]dx

− αmxec∫ L

0[w(x, t)θ(x, t) + 2w(x, t)θ (x, t)

+ w(x, t)θ (x, t)]dx . (23)

Substituting governing equations (1) and (2), using boundaryconditions (3) and (4), we obtain

�(t) ≤ −(

αEIb − αηEIb

σ3− σ6αL4

) ∫ L

0[w′′(x, t)]2dx

−(

αGJ − αηGJ

σ4− σ7αL2xac

) ∫ L

0[θ ′(x, t)]2dx

+ (αm + 2αmxecσ5)

∫ L

0[w(x, t)]2dx

+(

α Ip + 2αmxec

σ5

) ∫ L

0[θ (x, t)]2dx

+ σ3αηEIb

∫ L

0[w′′(x, t)]2dx

+ σ4αηGJ∫ L

0[θ ′(x, t)]2dx

− βmxec∫ L

0[w(x, t)θ (x, t) + w(x, t)θ (x, t)]dx

+ αθ(L, t)[M(t) + ηM(t)]− αw(L, t)[F(t) + η F(t)]+

(α

σ6+ αxac

σ7

)L F2

b max (24)

where σ3–σ7 are positive constants. Therefore, we have thederivative of the Lyapunov function candidate as

V (t) ≤ −[αw(L, t) + βw(L, t)][F(t) + η F(t)]+ [αθ(L, t) + βθ(L, t)][M(t) + ηM(t)]−

(αEIb − αηEIb

σ3− σ6αL4

) ∫ L

0[w′′(x, t)]2dx

−(

αGJ − αηGJ

σ4− σ7αL2xac

)∫ L

0[θ ′(x, t)]2dx

−(

βηEIb

2L4 − σ1β − αm − 2αmxecσ5

)

×∫ L

0[w(x, t)]2dx

−(

βηGJ

2L2 − σ2βxac − α Ip − 2αmxec

σ5

)

×∫ L

0[θ (x, t)]2dx

−(

βηEIb

2− σ3αηEIb

) ∫ L

0[w′′(x, t)]2dx

−(

βηGJ

2− σ4αηGJ

) ∫ L

0[θ ′(x, t)]2dx

+(

β

σ1+ βxac

σ2+ α

σ6+ αxac

σ7

)L F2

b max. (25)

354 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 25, NO. 1, JANUARY 2017

Let U(t) = F(t) + η F(t) and W (t) = M(t) + ηM(t) be thenew control variables, and we design the control laws as

U(t) = k1[αw(L, t) + βw(L, t)] (26)

W (t) = −k2[αθ(L, t) + βθ(L, t)] (27)

where k1 ≥ 0 and k2 ≥ 0 are the control parameters.Remark 3: In [1], the model of the flexible wing is decou-

pled as two separate variables, the bending displacementw(x, t) and the twist displacement θ(x, t). The nonlinearPDE model of the flexible wing (1), (2) is transferred to twosubsystems, and two independent controllers are designed fortwist and bending. In this brief, we design the boundary controlbased on the original PDE model without any discretizationor simplification of the dynamics in time and space. Thecouplings between the bending displacement w(x, t) and thetwist displacement θ(x, t) are considered in the dynamicanalysis, which make the control design more complicated.

Let (βηEIb/2)−σ3αηEIb ≥ 0 and (βηGJ/2)−σ4αηGJ ≥ 0,then we further have

V (t) ≤ −μ1

∫ L

0[w(x, t)]2dx − μ2

∫ L

0[θ (x, t)]2dx

− μ3

∫ L

0[w′′(x, t)]2dx − μ4

∫ L

0[θ ′(x, t)]2dx + ε

≤ −λ3κ(t) + ε (28)

where

μ1 = βηEIb

2L4 − σ1β − αm − 2αmxecσ5 > 0 (29)

μ2 = βηGJ

2L2 − σ2βxac − α Ip − 2αmxec

σ5> 0 (30)

μ3 = αEIb − αηEIb

σ3− σ6αL4 > 0 (31)

μ4 = αGJ − αηGJ

σ4− σ7αL2xac > 0 (32)

λ3 = min (μ1, μ2, μ3, μ4) > 0 (33)

ε =(

β

σ1+ βxac

σ2+ α

σ6+ αxac

σ7

)L F2

b max. (34)

Combining (19) and (28), we have

V (t) ≤ −λV (t) + ε (35)

where λ = λ3/λ1.From the above statement, the control design for the flexible

wing subjected to external loads can be summarized in thefollowing theorem.

Theorem 1: For the dynamical system described by gov-erning equations (1) and (2) and boundary conditions (3)–(5),under Assumption 1 and the proposed boundarycontrol (26), (27), if the initial conditions are bounded,then the closed-loop system is uniformly bounded.

Proof: Following the similar procedures in [41], integrat-ing inequality (35), we obtain:

V (t) ≤(

V (0) − ε

λ

)e−λt + ε

λ

≤ V (0)e−λt + ε

λ∈ L∞ (36)

TABLE I

PARAMETERS OF THE FLEXIBLE WING OF A ROBOTIC AIRCRAFT

which implies that V (t) is bounded. Utilizing inequalities(7) and (8), we have

1

L3 w2(x, t) ≤ 1

L2

∫ L

0[w′(x, t)]2dx ≤

∫ L

0[w′′(x, t)]2dx

≤ κ(t) ≤ 1

λ2V (t) ∈ L∞ (37)

1

Lθ2(x, t) ≤

∫ L

0[θ ′(x, t)]2dx ≤ κ(t) ≤ 1

λ2V (t) ∈ L∞.

(38)

Appropriately rearranging the terms of (36)–(38), we obtainthat w(x, t) and θ(x, t) are uniformly bounded as follows:

|w(x, t)| ≤√

L3

λ2

(V (0)e−λt + ε

λ

)(39)

|θ(x, t)| ≤√

L

λ2

(V (0)e−λt + ε

λ

). (40)

Furthermore, as t tends to infinity, we have

|w(x, t)| ≤√

L3ε

λ2λ∀x ∈ [0, L] (41)

|θ(x, t)| ≤√

Lε

λ2λ∀x ∈ [0, L]. (42)

IV. SIMULATIONS

In order to present the effectiveness of the proposed controlschemes, the finite-difference approximation method is intro-duced for approximating the system described by (1) and (2)in numerical simulations. Parameters of the system are listedin Table I.

Under the initial conditions, given as w(x, 0) = (x/L),θ(x, 0) = (πx/2L), w(x, 0) = 0 and θ (x, 0) = 0, the flexiblewing is excited by the inevitable spatiotemporally varyingdistributed load Fb(x, t), which is defined as Fb(x, t) =[1 + sin(π t) + 3 cos(3π t)]x .

HE AND ZHANG: CONTROL DESIGN FOR NONLINEAR FLEXIBLE WINGS OF A ROBOTIC AIRCRAFT 355

Fig. 2. Bending displacement of the system w(x, t) without control.

Fig. 3. Twist displacement of the system θ(x, t) without control.

Fig. 4. Bending displacement of the system w(x, t) with control.

From Figs. 2 and 3, it can be seen that the bending deflec-tion of the flexible wing w(x, t) is far beyond expectation,as large as 1 m. Considering the length of the wing withonly 2 m, it will result in system mechanical damage, which isunacceptable in practice. Besides, the twist deflection θ(x, t)can also be driven to reach the undesired position, which isharmful to the system physical structure and should be avoidedas well.

Fig. 5. Twist displacement of the system θ(x, t) with control.

Fig. 6. End-point bending displacement of the system w(L , t).

Fig. 7. End-point twist displacement of the system θ(L , t).

On account of the unsatisfactory system performances,the control target is to suppress the vibrations in bothw(x, t) and θ(x, t). With the proposed control (26), (27),the deflections of the wing system are reduced significantly,as shown in Figs. 4 and 5, where the parameters of thecontrollers are chosen as α = 50, β = 1, k1 = 1, andk2 = 1.5 × 10−2. Under the impact of the tip forcecontroller F(t), the bending deflection w(x, t) continues to

356 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 25, NO. 1, JANUARY 2017

Fig. 8. Control inputs.

decrease from the initial position and remains in a small rangearound zero about 60 s later. Meanwhile with the twistingmoment controller M(t), the twist deflection θ(x, t) can alsobe restrained and regulated to remain in an acceptable smallscope around zero after about 50 s.

To certify the effectiveness of the proposed control schemesmore clearly, we focus on the states of the end-point positionfor bending and twist deflection w(L, t) and θ(L, t) when thesystem is operated without and with controllers implemented.As shown in Figs. 6 and 7, it can be seen that comparedwith the freely vibrating situation, the designed controllers canmake the flexible wing system perform well, free from theinfluence of the disturbance Fb(x, t) and the vibration alongwith it, as the deflections can be controlled in small limitedextents. The control signals are depicted in Fig. 8.

In conclusion, the proposed control schemes are effectiveand feasible in dealing with the vibration problems existingin the system, and the stability of the closed-loop system canalso be ensured.

V. CONCLUSION

This brief has presented the control design and stabilityanalysis for flexible wings of a robotic aircraft. The flexiblewing is modeled as a coupled twist-bending system with theunknown spatiotemporally varying distributed load. Boundarycontrol schemes have been proposed to control the deformationof the wing. Based on the Lyapunov direct method, thestability has been proved. Numerical examples have illustratedthe performance of the control system.

ACKNOWLEDGMENT

The authors would like to thank the Editor-in-Chief, theAssociate Editor, and the anonymous reviewers for theirconstructive comments, which helped improve the quality andpresentation of this brief.

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