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Original Article

Proc IMechE Part I:J Systems and Control Engineering1–16� IMechE 2018Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/0959651818804377journals.sagepub.com/home/pii

Novel linear parameter–varyingmodeling and flutter suppressioncontrol of a smart airfoil

Ali MH Al-Hajjar1, Sean Shan-Min Swei2 and Guoming Zhu3

AbstractIn this article, two novel linear parameter–varying modeling and control techniques are proposed for active flutter sup-pression of a smart airfoil model. The smart airfoil model is instrumented with a moving mass that can be used toactively control the airfoil pitching and plunging motions. The first linear parameter–varying modeling approach makesuse of the moving mass position as a scheduling parameter, and the hard constraint at the boundaries is imposed byproper selection of the parameter-varying function. The second modeling technique utilizes nonlinear springs and dam-pers, which are added to both ends of the airfoil groove to confine the motion of the moving mass. A state-feedback-based linear parameter–varying gain-scheduling controller with the guaranteed H‘ performance is proposed by utilizingthe dynamics of the moving mass. In this study, both the position of the moving mass and the free-stream airspeed areconsidered as the scheduling parameters. The numerical simulations demonstrate the effectiveness of the proposed lin-ear parameter–varying control architectures by significantly improving the performance, while increasing the flutterspeed and reducing the control effort.

KeywordsLinear parameter–varying, linear parameter–varying modeling and control, gain-scheduling control, control with hardconstraints, flutter suppression, smart airfoil

Date received: 4 May 2018; accepted: 8 September 2018

Introduction

Linear parameter–varying (LPV) systems can be gener-ally classified, by their representation style,1 intogeneral LPV,2–5 linear-fractional-transformation (LFT)LPV,6–10 input–output LPV,11–14 and polytopic LPVsystems.15–19 Among these forms, the polytopic LPVrepresentation is theoretically and computationallyattractive as parameter-varying convex combination oflinear time-invariant (LTI) systems,1 which offers anelegant and convenient way of representing and analyz-ing LPV systems via convex optimization techniques.1

In addition, a wide class of LPV systems can be repre-sented in the polytopic LPV form, which is the mainreason it is adopted in this article.

Active flutter suppression has been a critical researchtopic in aerospace applications for many decades.Reducing the aircraft weight, improving the aerody-namic efficiency, and increasing the critical flight speedcontinue to be the main thrusts for future aeronauticalresearch, especially, as the emerging air vehicle struc-tures become highly flexible, active flutter suppressionbecomes a key technical design requirement. There is a

good body of design methods that are available in liter-ature concerning suppression of flutter phenomena.Passive methods have been used to solve this problemfor many years; however, these methods lead toincreased aircraft mass, which is undesirable.20–23 Onthe other hand, active control techniques can providecrucial and liable solutions that would increase the air-craft critical speed and suppress the oscillations, whileenhancing flight efficiency and performance.

There are many active control techniques in litera-ture for flutter suppression. Using piezoelectric

1Department of Mechanical Engineering, Michigan State University, East

Lansing, MI, USA2Intelligent Systems Division, NASA Ames Research Center, Mountain

View, CA, USA3Department of Mechanical Engineering and Department of Electrical and

Computer Engineering, Michigan State University, East Lansing, MI, USA

Corresponding author:

Guoming Zhu, Department of Mechanical Engineering and Department

of Electrical and Computer Engineering, Michigan State University, East

Lansing, MI 48823, USA.

Email: [email protected].

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actuation to control flutter was given by Han et al.,24

where experimental and numerical study were con-ducted for active flutter suppression of a sweptbackcantilever surface. The finite element analysis, panelaerodynamics, and the minimum statespace realizationwere used to develop the equations of motion, whichwere then used for system analysis and control designby utilizing the H2 and m-synthesis control techniques,and subsequently the flutter suppression performancewas evaluated through wind tunnel tests. De Marquiet al.25 introduced a flexible hanged arrangement forflutter investigation with hard wings in a wind tunnel.The wing model was a rectangular shape with the 0012NACA model and a rear-edge surface mechanism con-trol. They introduced an aeroelastic model to emulatethe aeroelastic behavior of the corresponding system. Afull-state feedback control was designed for this modelto cancel the flutter and retain the system stability.Zhang and Behal26 introduced a continuous-time con-troller to suppress the aeroelastic oscillations of thewing shape in an unsteady aerodynamic incompressibleflow environment. The flap connection force of thewing rear edge was used as the input and the pitchingangle as the output. The numerical simulation resultsdemonstrated the effectiveness in canceling aeroelasticvibrations in the region around the flutter travel speedsubjected to multiple external disturbances. Additionalactive flutter suppression techniques includes electro-hydraulic mechanical actuation of control surface,27

reaction jets,28 and micro-flaps,29 to just name a few.Special attentions were given to LPV modeling and

control of an airfoil. For example, Van Wingerdenet al.30 developed a system-identification algorithm foran aeroelastic LPV system outfitted with rear-edge con-trol surfaces, where a factorization was used to form asensor-like dependence on the old inputs, outputs, andgiven aeroelastic information. These sensor-like signalswere then used to calculate the state progression toform LPV aeroelastic matrices of the system. As thealgorithm can be utilized in a closed-loop arrangement,it can be used for dealing with flutter suppression prob-lems. Barker and Balas31 designed gain-scheduled con-trollers dependent on two LPV parameters for activeflutter suppression of the benchmark active controltechnology (BACT) wing section at NASA LangleyResearch Center. The BACT wing dynamics shiftsnotably relative to the dynamic pressure and Machnumber. The LPV gain-scheduled controllers depend-ing on two parameters incorporated these alternationsas well as limits on the rate of change of dynamic pres-sure and Mach number. The incorporation of the ratelimits in the design procedure resulted in improved per-formance over a vast range of operational conditionswhen compared to a former designed gain-scheduledcontroller based on linear fractional transformation.Lau and Krener32 utilized a regular linear system forcontrolling a slim airfoil under subsonic flow. The two-

dimensional (2D) airfoil was designed with 3 degrees offreedom (DOFs): flap, plunge, and pitch angles, result-ing a six-dimension linear system with a control inputat the flap hinge. The objective was to utilize feedbackfor stabilizing the airfoil at or above its flutter speedwith many control procedures. Chen et al.33 developedan LPV aeroservoelastic system utilizing adaptivemethod with non-linear aerodynamics. The LPV con-troller was able to suppress flutter with perfect preci-sion and validity; moreover, it provided a feasible toolfor practical flutter flight tests. Balas et al.34 appliedLPV and H‘ control strategy procedures to a body-freeflutter (BFF) airplane for designing aeroservoelasticcontrollers and compared the control performances inboth time and frequency domains. Though the perfor-mance from the designed LPV controller was satisfac-tory, it did not achieve the level of performance fromtheH‘ ones. It should be noted that, from the literaturereview, the conventional LPV model considers the sche-duling parameters as physical parameters that changeswith time,35–38 whereas the quasi-LPV model39–41 con-siders systems with one or more states as schedulingparameters in the system matrices.

In this article, we consider the smart airfoil modelproposed by Swei and Jiang42 (Figure 1), to investigatethe problem of active flutter suppression using thenovel LPV modeling and control techniques of scalingthe scheduling parameter. The smart airfoil is a 2D air-foil with a groove along its chord that contains a mov-ing mass. The mass is allowed to move along thegroove, and through its coupling with airfoil aerody-namics, it can control and suppress the pitching andplunging motion of the airfoil. The airspeed and posi-tion of the moving mass are considered as schedulingparameters in the LPV model and the mass position isutilized as a scheduling parameter in the control design.The smart airfoil considered in this article was first pro-posed by Swei and Jiang.42 The idea is that by utilizingthe dynamic coupling effect between the motion ofmoving mass and aeroelasticity, the airfoil flutter beha-vior can be suppressed; also refer Swei and Ayoubi.43

In this study, we propose to utilize the scheduling para-meter as part of a scaling factor for the smart airfoil

Figure 1. The smart airfoil model.

2 Proc IMechE Part I: J Systems and Control Engineering 00(0)

model. In particular, position of the moving mass isscaled and parametrized such that it is confined withinthe length of groove. Furthermore, we propose anothernovel LPV modeling technique by implementing a pairof nonlinear springs and dampers at both ends of thegroove to gracefully prevent the moving mass fromreaching the hard boundaries. To the best of ourknowledge, the integration of scheduling parameterwith the scaled control effector and the parametrizationof boundary conditions are novel approach that havenever been reported in the LPV control literature in thepast. In this article, a number of LPV control modelsare developed to best describe the effect of boundaryconstraints and also to reduce the level of conservative-ness. A full-state feedback LPV gain-scheduling con-troller with guaranteed H‘ output performance isproposed, in which the controller gains are obtained bysolving the numerically tractable Parameterized LinearMatrix Inequality (PLMI).

The article is organized as follows. Section ‘‘LPVmodeling of a smart airfoil’’ presents the nonlinearmodel of the smart airfoil, the baseline LPV model(LPV-0), the LPV model with parameter scaling (LPV-1), the LPV model with nonlinear springs and dampers(LPV-2) at both ends of groove, and the parameter-reduced (from three to two) LPV model (LPV-2A).Section ‘‘LPV controller design’’ contains the LPVproblem formulation and associated controller design.Simulation results are presented and compared in sec-tion ‘‘Numerical studies.’’ Conclusions and future workare provided in section ‘‘Conclusion.’’

LPV modeling of a smart airfoil

In this section, the mathematical model of the smart air-foil is presented. The linearized equations of motion ofthe airfoil aeroservoelastic model can be written as42

m+M Mxa

Mxa Ia

� � €h(t)€a(t)

" #+

Kh 0

0 Ka

� �h(t)

a(t)

� �

=0

mg

� �y(t)+Fa(t)

ð1Þ

m€y(t)=mga(t)+ u(t) ð2Þ

where Fa(t) denotes the aerodynamic loading; m and Mare the moving mass and airfoil mass, respectively;additional variables and parameters used in equations(1) and (2) can be found in Figure 1.42 It is importantto note that the position of moving mass y(t) in equa-tion (1) can be considered as the control input to theairfoil, whereas u(t) in equation (2) can be consideredas the control input to the moving mass m. The airfoilwith such a control device is called ‘‘Smart Airfoil.’’42

The following quasi-steady aerodynamic load model44

of Fa(t) is adapted in this study

Fa(t)=P�1V 0eV 0

� �_h(t)_a(t)

� �+

0 �10 e

� �h(t)a(t)

� �� ð3Þ

P= qpcCLa

Now, substituting equation (3) into equation (1) andperforming nondimensionalization for all the physicalparameters, we obtain the nondimensionalized equa-tions of motion for the smart airfoil model as follows

1+b �xa

�xa �r2a

� � €�h(t)€a(t)

" #+

2 �Vm 0

�2 �V�em 0

" #_�h(t)

_a(t)

" #

+

v2h

v2a

2 �V2

m

0 �2�V2�e

m + �r2a

24

35 �h(t)

a(t)

" #=

0

b�g

� ��y(t)

ð4Þ

€�y(t)= �ga(t)+ �u(t) ð5Þ

where t =vat is the nondimensional time. Note that tosimplify the notation, the overhead ‘‘dot’’ in equations(4) and (5) represents the time derivative with respect tot. When the flutter occurs, the plunging displacement hand pitching angle a are fed back in order to properlyposition the moving mass m, which generates a damp-ing effect to the airfoil, hence reducing the flutter vibra-tion and increasing the critical flutter speed.

LPV plant model: LPV-0

Rearranging equations (4) and (5) yields the following

€�hðtÞ€aðtÞ

� �þ

�2�r2 �Vqm

� 2�V�e �xaqm

0

2�xa �V

qmþ 2

�V�eð1þ bÞqm

0

2664

3775 _�hðtÞ_aðtÞ� �

þ

��r2av2hqv2a

�2�ra �V2

qm� 2

�V2�e �xa

qmþ �r

2�xaq

�xav2h

qv2a

2�xa �V2

qmþ 2

�V2�eð1þ bÞqm

þ �r2ð1þ bÞ

q

26664

37775

�hðtÞaðtÞ

� �¼

��xabgq

ð1þ bÞb�gq

2664

3775�yðtÞ

ð6Þ€�y(t)= �ga(t)+ �u(t) ð7Þ

where q=� ½�r2a(1+b)� �x2a�. Now, we define the aug-mented state x as

x= �xT, xTu� �T ð8Þ

where

�x= �h,a, _�h, _a

h iTand xu = �y, _�y

� �T ð9ÞThen, equations (6) and (7) can be described in thestate-space representation as follows

_x(t)=A(u(t))x(t)+Bu(u(t))�u(t)y(t)=C(u(t))x(t)+Du(u(t))�u(t)

�ð10Þ

Al-Hajjar et al. 3

where y(t) is the controlled output, and the systemmatrices (A(u),Bu(u),C(u),Du(u)) are affine in u andgiven by

A(u)=

0 0 1 0 0 00 0 0 1 0 0

�r2av2h

qv2a

2�r2au3qm +

2u3�e �xaqm �

�r2a �xaq

2�r2au2qm +

2u2�e �xaqm 0

��xabq 0

��xav2hqv2a

�2�xau3qm �

2u3�e(1+b)qm �

�r2a(1+b)q

�2�xu2qm �

2u2�e(1+b)qm 0

b(1+b)q 0

0 0 0 0 0 10 �g 0 0 a65 a66

266666664

377777775, Bu(u)=

00000b6

26666664

37777775

C(u)= 0 0 0 0 1 0½ �, Du(u)=0

ð11Þ

where u : = ½u1, u2, u3�T, and u1 is the scheduling para-meter utilized to properly scale and constrain the posi-tion of the moving mass at the boundaries, u2 and u3are the scheduling parameters representing, respec-tively, the normalized airspeed �V and its square �V2. Inthis study, we consider u2 2 ½0:5, 2:92�; hence,u3 2 ½0:25, 8:52�. Furthermore, the parameters a65 anda66 are used to represent nonlinear spring and dampingeffects at the boundaries. It should be emphasized thatu2 and u3 are not independent parameters; therefore,inevitably it would introduce certain conservativenessin the modeling process. However, a less conservativeapproximation approach can be adopted.

To formulate the baseline LPV-0 plant model, we setthe parameters b6 =1 and a65 =0, a66 =0. This LPV-0model will be used to assess the baseline closed-loopsystem performance.

LPV control design model: LPV-1

The proposed LPV-1 model is based on the LPV-0model described in equations (10) and (11) witha65 =0, a66 =0, but b6 = u1, where u1 is a function of�y (nondimensinalized mass displacement), that is,u1 = f(�y), which is used to constrain the moving mass.In particular, u1 is devised such that

u1 =1, if �y 2 ½�0:35, 0:35�u1�y=0:35, if �y. 0:35u1�y= � 0:35, if �y\ � 0:35

8<:

It is clear from the first condition above that when�y 2 ½�0:35, 0:35�, u1 =1 and the LPV-1 model will beequivalent to the LPV-0 model. The purpose of the sec-ond and third conditions is to impose a constraint onthe moving mass when it travels beyond 60:35, bymodulating the control gains through u1. The variation

of u1 as function of the moving mass position is illu-strated in Figure 2. To constrain the mass within 60:5the damping function must be active starting at 60:35.This approach may introduce a slight conservativenessto the control design, it is intuitively appealing andproved to be effective.

LPV control design model with nonlinear springs anddampers: LPV-2

The second proposed LPV control design model con-tains nonlinear springs and dampers at both ends ofthe groove. They are to gradually stop and reverse themotion of the control mass. Figure 3 shows an illustra-tion of the proposed boundary constraints with non-linear springs and dampers, and as indicated earlier,the scheduling parameter u1, which is a function of �y, isused to scale the control effector.

Augmenting the nonlinear spring and damperdynamics into the equations of motion, a new LPVmodel, called LPV-2, can be formed in the form ofequation (11), where the spring and damping para-meters are given by a65 = � Ksu1 and a66 = � Csu1;and Ks and Cs denote the elastic and damping coeffi-cients, respectively. In this study, we have chosenKs =214 and Cs =16. Moreover, the scheduling para-meter u1 is chosen as follows; also refer Figure 3

u1 =0, if �y 2 ½�0:35, 0:35�u1 = �y� 0:35, if �y. 0:35u1 = � 0:35� �y , if �y\ � 0:35

8<:

It should be emphasized that the choice of Ks and Csdepends on the location from which the spring anddamping effect are expected to engage, that is, thecloser to 60:5, the higher the Ks and Cs value.

Two-parameter LPV-2 model: LPV-2A

A third LPV control design model is modified fromLPV-2, in which the parameter u3 is approximated as afunction of u2; hence, the number of scheduling para-meters is reduced down to two. Since u2 and u3 repre-sent the velocity and its squared value, they are notindependent and will introduce conservativeness due toover bounding both parameters. Kwiatkowski andFigure 2. A saturation function.


Werner45 suggested a method to reduce conservative-ness by taking the intersection between the two poly-topes. In this section, we propose to approximate u3as a linear function of u2 with slope b1, as shown inFigure 4. Therefore, the resulting LPV model, denotedas LPV-2A, contains two-scheduling parameters: u1and u2.

Recall the affine matrix A(u) (equation (11)) is of theform

A(u)=A0 + u1A1 + u2A2 + u3A3 ð12Þ

From Figure 4, u3 can be approximated as

u3 = u30 + u2 tan (b1) ð13Þ

Substituting equation (13) into equation (12) yields

A(u)= A0 + u30A3|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}A0new

+ u1A1 + u2 (A2 + tan (b1)A3)|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A2new

ð14Þ

Therefore, the affine matrix A(u) in the LPV-2Amodel can be described by

A(u)=A0new+ u1A1 + u2A2new ð15Þ

LPV controller design

In this section, we develop state-feedback-based gain-scheduling LPV controllers based on LPV-0, LPV-1,LPV-2, and LPV-2A models. The goal of LPV controldesign is to achieve active flutter suppression with guar-anteed H‘ output performance for the given range ofairspeed by controlling the moving mass in the smartairfoil model, subject to actuation constraints andexternal disturbances. In particular, consider the fol-lowing LPV systems

_x(t)=A(u(t))x(t)+Bu(u(t))�u(t)+Bw(u(t))w(t)y(t)=C(u(t))x(t)+Du(u(t))�u(t)+Dw(u(t))w(t)

�ð16Þ

where w(t) denotes the disturbance input.

Problem formulation

The LPV model described in equation (16) is assumedto have affine parameters. For instance, A(u) matrixcan be represented by

A(u(t))=A0 +Xqi=1

ui(t)Ai ð17Þ

where A0 and Ai are constant matrices, and q denotesthe number of scheduling parameters. Scheduling para-meter vector u(t) is defined as

u(t)= ½u1(t), u2(t), u3(t), . . . , uq(t)�T ð18Þ

and each ui is bounded by

h1, i ł ui ł h2, i ð19Þ

where h1, i and h2, i denote the upper and lower bounds,respectively. In addition, each scheduling parameter, ui,also has its rate bound given by

m1, i ł _ui ł m2, i ð20Þ

The proposed LPV control law is a state-feedbackcontroller

�u(t)=K(u)x(t) ð21Þ

where K(u) asymptotically stabilizes the closed-loopsystem subject to the H‘-norm constraint from the exo-genous input w to the measured output y over the entireparameter variation range. Substituting controllerequation (21) into equation (16) yields the closed-loopsystem representation given by

_x(t)=Ac(u)x(t)+Bw(u)w(t)y(t)=Cc(u)x(t)+Dw(u)w(t)

�ð22Þ

where

Ac(u)=A(u)+Bu(u)K(u)Cc(u)=C(u)+Du(u)K(u)

ð23Þ

Figure 3. (a) Airfoil groove with nonlinear springs and dampersat the boundaries and (b) u1 as function of moving mass position.

Figure 4. Proposed parameter reduction approach.

Al-Hajjar et al. 5

It should be noted that although matricesA(u),Bu(u),C(u), and Du(u) are affine in u, the gain-scheduling feedback K(u) is not. Before proceeding fur-ther, we introduce the following definitions:

Definition 1.46,47. A unit simplex Yr is a polytope of rvertices defined as

Yr= a= ½a1, . . . , ar� :Xri=1

ai =1, aiø0, i=1, 2, . . . , r

( )

Definition 2.48. The Cartesian product of a finite numberof simplexes is a multisimplex Y. For instance, if thereare q simplexes, then

Y=YN1 3YN2 3YN3 3 � � �YNq =Yqi=1

YNi

where the dimension of multisimplex Y is representedby the index vector N=(N1, . . . ,Nq), where parametervector is denoted by a=(a1, a2, . . . , aq) and ai 2 YNihas the form (ai(1), ai(2), . . . , ai(r)).

Transferring from affine to multisimplex. To formulate aconvex control design problem, we first need to per-form a transformation on the system matrices, fromthe affine parameter space u to the multisimplex convexspace Y. Each affine scheduling parameter ui is trans-ferred over a unit simplex ai of two vertices as follows

49

ai(1)=ui +h2, i2h2, i

! ui =2h2, iai(1)� h2, iai(2)=1� ai(1)=1� ui +h2, i2h2, i =

h2, i�ui2h2, i

ai =(ai(1), ai(2)) 2 Y2, 8 i=1, 2, � � � , qð24Þ

With this transformation, the affine scheduling para-meters in equation (16) can be converted into a systemwith multisimplex parameters, where the multisimplexvariables are defined as

a 2 Y= Y2 3Y2 3 � � �Y2|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}q

Moreover, for three-scheduling parameters, that is,q=3 as the case of our current consideration, thehomogeneous term in the multisimplex variables can bewritten as

a1=(a1(1), a1(2)), a2=(a2(1), a2(2)), a3=(a3(1), a3(2))

Hence, by utilizing the above transformation, the LPVsystem equation (16), which was described in affineparameter space u, can be equivalently described inmultisimplex convex space Y as follows

_x(t)=A(a)x(t)+Bu(a)�u(t)+Bw(a)w(t)y(t)=C(a)x(t)+Du(a)�u(t)+Dw(a)w(t)

�ð25Þ

Furthermore, the proposed state-feedback gain-sche-duling controller has the form

�u(t)=K(a)x(t) ð26Þ

where K(a) is to be determined. Therefore, the closed-loop system representation in multisimplex space isgiven by

_x(t)=Ac(a)x(t)+Bw(a)w(t)y(t)=Cc(a)x(t)+Dw(a)w(t)

�ð27Þ

where

Ac(a)=A(a)+Bu(a)K(a)Cc(a)=C(a)+Du(a)K(a)

ð28Þ

In addition, the rate changes of the scheduling para-meters in the unit simplex can also be described by

_ai(1)+ _ai(2)=0, i=1, 2, 3 ð29Þ

From equations (20) and (24) we can derive the ratebounds between affine scheduling parameters and mul-tisimplex variables, and they are given by

m1, i2h2, i

ł _ai(1)łm2, i2h2, i

ð30Þ

where m2, i and m1, i are rate bounds given in equation(20), and h2, i is the upper bound of ui given in equation(19). Furthermore, from equation (29), we obtain that_ai(2)= � _ai(1).

H‘ control problem. The proposed LPV control problemis to design a gain-scheduling state-feedback control-ler of the form (26) such that for a given g‘ . 0 andany (ai, i=1, 2, . . . , q) 2 Y with ( _ai, i=1, 2, . . . , q)satisfying (30), the closed-loop system (27) is stabi-lized and the following H‘ performance constraint issatisfied

sup(ai, _ai)

supw2l2w 6¼ 0

kzk2kwk2 \ g‘ ð31Þ

To synthesize the H‘ control problem, we utilize thefollowing theorem.

Theorem 1.1,35,50. For any (ai, i=1, 2, . . . , q) 2 Y with( _ai, i=1, 2, . . . , q) satisfying (30), the gain-schedulingcontroller K(a)=Z(a)G(a)�1 stabilizes the closed-loopsystem equation (27) with guaranteed H‘ performance(31) if there exist a scalar e . 0, a continuously differen-tiable positive definite and symmetric matrix P(a), andmatrices G(a) and Z(a) satisfying the following PLMI


Q(a) � � �P(a)� G(a)+ e(A(a)G(a)+Bu(a)Z(a))0 �e(G(a)G(a)0) � �

C(a)G(a)+Du(a)Z(a) eC(a)G(a)+ eDu(a)Z(a) �I �Bw(a)

0 0r3 n Dw(a)0 �g2‘I

2664

3775\ 0 ð32Þ

where � denotes symmetric entry and Q(a)=A(a)G(a)+Bu(a)Z(a)+G(a)

0A(a)0+Z(a)0Bu(a)0+

(∂P(a)=∂a) _a.It should be noted that although the H‘ control

problem is presented in multisimplex convex space Y,the actual gain-scheduling controllers are implementedin affine u domain. Roughly speaking, in order to com-pute the affine feedback gain matrix described in equa-tion (21), we first need to utilize Theorem 1 tosynthesize the solutions Z(a) and G(a) at the vertices ofmultisimplex space Y, followed by the inverse transfor-mation process35 to convert the multisimplex solutionsinto affine representations Z(u) and G(u). The detailprocedure is given below.

For three-scheduling parameter case, the parametersZ(u) and G(u) in affine u domain can be described by

Z(u)=Z0 + u1Z1 + u2Z2 + u3Z3G(u)=G0 + u1G1 + u2G2 + u3G3

ð33Þ

where (Zi, Gi), i=0, 1, 2, 3, are constant matrices to bedetermined in the sequel. Moreover, given Z(u) andG(u), the gain matrix K(u) in equation (21) will then begiven by

K(u)=Z(u)G�1(u) ð34Þ

Therefore, it is clear that the control gain K(u) is notan affine function of u. For three-scheduling parametercase, there are eight polytopic solutions for Z(a) andG(a) in equation (32). Let Zijk and Gijk, i, j, k=1, 2,denote these polytopic solutions for Z(a) and G(a),respectively. Take Z(a) as an example and it can beexpressed as follows

Z(a)= a1(1)a2(1)a3(1)Z111 + a1(1)a2(1)a3(2)Z112+ a1(1)a2(2)a3(1)Z121 + a1(1)a2(2)a3(2)Z122+ a1(2)a2(1)a3(1)Z211 + a1(2)a2(1)a3(2)Z212+ a1(2)a2(2)a3(1)Z221 + a1(2)a2(2)a3(2)Z222

=Z(u)

ð35Þ

Following the inverse transformation process givenin Al-Jiboory et al.35 results that

Z0 =164

P2j1 =1

P2j2 =1

P2j3 =1

Zj1j2j3

Zi =1

64h2, i

P2j1 =1

P2j2 =1

P2j3 =1

(� 1)ji + i Zj1j2j3ð36Þ

Note that Gi, i=0, 1, 2, 3, can be obtained in a simi-lar way. Now, substituting (Z0, Zi) and (G0, Gi) intoequation (33) yields Z(u) and G(u); hence, the feedbackgain matrix K(u) can be obtained from equation (34).The inverse transformation formulas for any number ofscheduling parameters q. 3 can be found in Al-Jiboory

et al.,35 Oliveira et al.46 and Lacerda et al.49 For thepurpose of demonstration, a list of (Z0, Zi) and (G0, Gi)matrices for various LPV models are presented in theAppendix 2.

It should be noted that PLMI is an infinite dimen-sional linear matrix inequality (LMI), which is in gen-eral difficult to solve. Many efficient solvers areavailable in dealing with such problem; forinstanceScherer,51 Oliveira and Peres,52 Scherer andHol,53 and Peaucelle and Sato.54 This article adapts therelaxation method for PLMIs relaxation46 to deal withPLMIs with multisimplex parameters. Numerically, theROLMIP package55 along with YALMIP56 usingSeDuMi57 solver are used to solve the convex optimiza-tion problem. An LPV-based control block diagram isdepicted in Figure 5.

Numerical studies

The smart airfoil parameters used in this study aregiven in Table 1. In addition, for the model in equation(16) Bw = ½0 0 0 0 0 0:1�T and Dw =0. Thebaseline LPV-0 controller and the proposed LPV-1 andLPV-2 controllers are designed based on their corre-sponding control design models with g‘ =0:0055. Forthe purpose of comparison, the scheduling parameteru2 is varied from 4–23.8m/s, and as a result u3 from16–566:44m2=s2. Note that the LPV-0 controller isdesigned using the (unsaturated) LPV-0 model. Hence,the resulting LPV-0 controller is in the form of equa-tion (34) with u1 =0 in equation (33).

For the two-scheduling parameters LPV-2A model,b1 =73:3

8 (Figure 4) and u30 = � 1:2 (equation (13)).The associated gain-scheduling gain K(u)=Z(u)G(u)�1 and the affine parameters Z(u) and G(u) aregiven by

Figure 5. An LPV control architecture.

Al-Hajjar et al. 7

Z(u)=Z0 + u1Z1 + u2Z2G(u)=G0 + u1G1 + u2G2

ð37Þ

where (Zi, Gi), i=0, 1, 2, are computed by followingthe similar procedure presented in section ‘‘H‘ controlproblem.’’ The main motivation of designing the LPV-2A controllers is to reduce the number of schedulingparameters from three to two by representing u3 interms of u2, leading to reduced control design conserva-tiveness. Note that g‘ is a measure of robustness per-formance. The minimum achievable g‘ for the LPV-2control is 0.00029, whereas for the LPV-2A control it is0.0002, a 45% improvement.

Simulation results are compared with those obtainedfrom the nonlinear controller presented by Swei andJiang.42 Figure 6 shows a comparison between the pro-posed LPV-1 controller and the nonlinear controller,42

with small initial pitch angle, that is, aS=0:01 rad. Itcan clearly be seen that the proposed LPV-1 controlcan significantly improve the overall closed-loop perfor-mance. In addition, the control effort, in terms of jjujj2,is reduced by 57%. As mentioned, the control mass m isconfined to move within the groove between -0.5 and

0.5. In the proposed LPV-1 control design, the schedul-ing parameter u1 is used to constrain the mass move-ment. Figure 7 shows the same comparisons but withlarger initial pitch angle at aL =0:6 rad. It is apparentthat the nonlinear controller42 cannot handle the largeinitial angle of attack, whereas the proposed LPV-1control handles the same initial condition effectivelywith fast convergence and small control effort. The con-trol effort is reduced by 92% when compared to thebaseline nonlinear controller.42 Additional simulationresults are presented next.

Figure 8 shows a comparison between the LPV-1control design and the baseline LPV-0 design. Recallthat in LPV-0 control design, no position limitation isimposed on the moving mass m. Therefore, it can beseen that in Figure 8 for LPV-0 controller to effectivelysuppress airfoil vibration, the control mass needs totravel much larger distance. Furthermore, Figure 9shows a comparison between LPV-1 and LPV-2 con-trollers when the initial pitch angle is large. Both con-trollers, as shown, are able to effectively suppress theairfoil vibration, though the LPV-2 controller, withnonlinear springs and dampers imposed at the bound-aries, produces less aggressive mass motion. Moreover,as shown in Figure 10, the mass motion of LPV-2Acontrol is less aggressive than that of LPV-2 control.Note that Figures 6–10 are generated with a fixed air-speed at �V=2:92 (or 23.8m/s).

For flutter analysis, Figure 11 shows the closed-loopsystem responses for LPV-1 control with varying air-speed. As shown, the system is unstable when �V=3:4.However, Figure 12 shows the responses of LPV-2Acontrol with initial airspeed of �V=2:92, then increasedto �V=3:08 (or 25.1m/s) at t =250 and to �V=3:4(or 27.7m/s) at t =450. As shown in Figure 12, the

Table 1. Parameters used in the article.42

Parameter Value Parameter Value

m 152 b 0.01�e 0.35 initial h 0�xa 0.25 aS (rad) 0.01�r2a 0.388 aL (rad) 0.6b (m) 0.127 initial _h 0va (rad/S) 64.1 initial _a 0vh (rad/S) 55.9 airspeed �V 2.92

Figure 6. Comparison between proposed LPV-1 control andSwei and Jiang42: aS.

Figure 7. Comparison between proposed LPV-1 control andSwei and Jiang42: aL.


LPV-2A control can effectively suppress the airfoilvibration beyond the previously identified flutter air-speed of �V=2:96 (or 24.1m/s).42

A quantitative study of the simulation results, interms of jj:jj‘ and jj:jj2 norms of the signals, are pre-sented in Tables 2–4. Table 2 presents a comparisonbetween the jj:jj‘ norms of the proposed design tech-niques and that by Swei and Jiang42 with small andlarge initial conditions. It is clear that the proposeddesign techniques render more than 50% improvementfor most of the system performance–related norms with

nearly 50 times less control effort. A comparison ofjj:jj‘ norms between LPV-1 control and baseline LPV-0control is presented in Table 3, which reveals the benefitof modeling the boundary conditions. Table 4 shows acomparison of jj:jj2 norms between the proposed designtechniques and the one in Swei and Jiang42 with smalland large initial conditions. It is clear that the proposedLPV design techniques provide much improved closed-loop responses, with faster convergent rate as indicatedin the percentage improvement in the last column ofTable 4.

Figure 9. Comparison between LPV-1 control and LPV-2control: aL.

Figure 10. Comparison between proposed LPV-2 control andLPV-2A control: aL.

Figure 11. LPV-1 control at �V = 3:08 and �V = 3:4.

Figure 8. Comparison between LPV-1 control and LPV-0control: aL.

Al-Hajjar et al. 9

Finally, to demonstrate the nonlinear nature of thegain-scheduling feedback matrix K(u), the first, second,and fifth entries of the LPV-2A control gain matrixK(u) are plotted (Figures 13–15). It is clear that K(u) isnot an affine function of u. Furthermore, these figuresshow that the control gain is clearly more affected bythe boundary condition parameter u1 than the airspeedparameter u2.

Conclusion

This article presented two novel LPV modeling andcontrol design techniques for a smart airfoil model thatutilizes a moving mass for flutter suppression. The LPVgain-scheduling state-feedback controllers based on thecorresponding models were proposed. In the first pro-posed technique, the moving mass position was used asthe scaled scheduling parameter, whereas in the secondtechnique, nonlinear springs and dampers were addedto the ends of the groove to constrain the moving mass.In addition to the baseline unconstrained plant modelLPV-0, three LPV-based control design models,namely, LPV-1, LPV-2, and LPV-2A, were developedby utilizing the two proposed model constrain tech-niques. The simulation results clearly demonstrated theadvantages and effectiveness of the proposed LPVmodeling and control techniques in flutter suppression,when compared to the results generated using anearlier developed nonlinear controller in the literature.Furthermore, the proposed LPV-1 controller was ableto effectively suppress the airfoil vibration while stillmeeting the actuator constraint, whereas the uncon-strained LPV-0 controller failed to meet the constraint.

Figure 12. LPV-2A control performance at a defined velocityprofile, �V 2 ½2:92, 3:4�.

Table 2. jj:jj‘ comparison; LPV-1 and LPV-2 controllers versusnonlinear controller.42

Case jj:jj‘ Swei and Jiang42 Proposed LPV

LPV-1 at aS �h 0.0040099 0.0029746a 0.01 0.01�y 0.24058 0.083381�u 0.23176 0.18602

LPV-1 at aL �h 0.24444 0.29736a 0.6 0.6�y 0.86839 0.4705�u 619.82 11.347

LPV-2 at aL �h 0.24444 0.23804a 0.6 0.6�y 0.86839 0.48297�u 619.82 5.5392

Table 3. LPV-1 controller versus LPV-0 controller: aL.

jj:jj‘ LPV-0 LPV-1

�h 0.14294 0.29736a 0.6 0.6�y 7.8091 0.4705�u 47.979 11.347

Table 4. jj:jj2 comparison; LPV-1 and LPV-2 controllers versusnonlinear controller.42

Case jj:jj2 Swei andJiang42

Pro. LPV1 Con.rate%2

LPV-1 at aS �h 0.4946 0.20804 57a 0.98788 0.38593 60�y 28.36 4.0848 85�u 26.744 11.359 57

LPV-1 at aL �h 30.216 27.771 8a 60.868 45.592 25�y 254.07 115.57 54�u 6802.3 519.59 92

LPV-2 at aL �h 30.216 22.143 27a 60.868 41.641 31�y 254.07 65.663 74�u 6802.3 502.97 92

Figure 13. First entry of K(u).


The performance of the proposed LPV-2 controller wasprogressively improved over that of the LPV-1 control-ler, when a pair of fictitious spring and damper wasintroduced at the boundaries. These fictitious mechan-isms were devised to become active when the movingmass gets closer to the boundaries. A less conservative,two-scheduling parameter LPV-2A controller was pro-posed, which was derived from the three-schedulingparameter LPV-2 controller. The simulation resultsconfirmed the effectiveness of the LPV-2A controllerover LPV-2 controller, especially when the initial angleof attack is large. The implementation of the proposedLPV controllers involves the construction of the full-state feedback gains in real-time based on the schedul-ing parameter value and the constant coefficientmatrices given in Appendix 2. Depending on the designpreference, the corresponding proposed LPV controllercan be chosen for implementation by following thesame process as implementing a conventional full-statefeedback controller. Future research includes experi-mental verification of the proposed concept and imple-mentation of the switched LPV control techniques.

Acknowledgements

A.M.H.A.-H. would like to thank the funding supportof the Higher Committee For Education DevelopmentIraq and the University of Kufa. The authors wouldalso like to thank Dr Ali Al-jiboory for providing LPVcontrol design code used in this paper.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship, and/or publi-cation of this article.

Funding

The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This research was supported byNASA ARMD Convergent Aeronautics Solutions(CAS) project.

ORCID iD

Guoming Zhu https://orcid.org/0000-0002-2101-2698

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Appendix 1

Notation

b typical section semi-chord (the length ofthe groove)

c typical section chorde elastic axis (e.a.) from elastic aerodynamic

center, aft positive�e nondimensional e, e=bg gravity constant�g nondimensional g, g=v2abh plunging displacement�h nondimensional plunging displacement,

h=bm mass of the control deviceqp dynamic pressure, rV

2=2ra radius of gyration about e.a.,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIa=Mb2

pt timeu control input�u nondimensional control input, u=Mv2abxa distance from e.a. to inertia axis, aft

positive�xa nondimensional static unbalance, xa=by traveling distance of control mass, (m)�y nondimensional displacement, y=bCL lift coefficientCLa dCL=da, (CLa =2p)Fa(t) aerodynamic load on the airfoilIa mass moment of inertia about e.a. per unit

spanKh spring constant for plunging modeKa spring constant for pitching modeM per unit span mass of the typical sectionV free-stream airspeed�V nondimensional free-stream airspeed,

V=vab

a angel of attack, positive nose upaS small initial, aaL large initial, ab mass ratio, m=Mu1 first scheduling parameter, a function of �yu2 second scheduling parameter, �Vu3 third scheduling parameter, �V

2

m mass ratio of the typical section to theapparent mass, M=prb2

r air densityt nondimensional time, vatvh natural frequency of uncoupled plungingva natural frequency of uncoupled pitching

Al-Hajjar et al. 13

Appendix 2

Scheduling parameter matrices

In this section, the constant matrices (Z0, G0) and (Zi, Gi) for constructing LPV-0, LPV-1, LPV-2, and LPV-2Acontrollers are presented.

For LPV-0 controller: (Z00, G00) and (Z

0i , G

0i )

Z00 = ½�0:051788000, � 0:2853700, � 0:261390, � 0:476160, � 0:045888, � 8:81260�Z02 = ½�0:000028579, � 0:0095015, � 0:036397, � 0:083472, � 0:018931, � 0:63590�Z03 = ½�0:008960500, � 0:1393600, � 0:006209, � 0:042522, � 0:117880, � 0:31722�

ð38Þ

G00 =

1:0292e� 05 �7:0926e� 06 �2:0329e� 07 3:6702e� 06 7:1505e� 09 �9:4204e� 05�7:032e� 06 3:6004e� 05 �2:4069e� 06 �2:6686e� 06 �4:575e� 06 3:9664e� 04�4:1334e� 07 �2:1419e� 06 1:0487e� 05 �1:3243e� 05 9:1849e� 05 4:0045e� 053:9355e� 06 �3:5161e� 06 �1:3181e� 05 4:2533e� 05 �3:7746e� 04 �1:4701e� 04�1:6558e� 06 2:3264e� 06 9:6502e� 05 �3:961e� 04 4:9923e� 02 �3:2631e� 01�9:4264e� 05 3:9815e� 04 1:1691e� 05 �3:0011e� 05 �3:288e� 01 104:92e� 01

26666664

37777775ð39Þ

G02 =

1:0241e� 05 �6:8766e� 06 �1:9139e� 07 3:642e� 06 �7:7403e� 08 �9:4036e� 05�6:8168e� 06 3:54e� 05 �2:4457e� 06 �2:648e� 06 �3:9416e� 07 3:8971e� 04�4:0415e� 07 �2:17e� 06 1:063e� 05 �1:3519e� 05 9:2068e� 05 2:5043e� 053:9103e� 06 �3:5054e� 06 �1:3456e� 05 4:2983e� 05 �3:792e� 04 �9:1409e� 05�2:0406e� 06 7:7154e� 06 9:6801e� 05 �3:982e� 04 5:1528e� 02 �2:3771e� 01�9:4179e� 05 3:9065e� 04 4:2126e� 06 �5:6076e� 06 �2:52e� 01 29:561e� 01

26666664

37777775ð40Þ

G03 =


26666664

37777775ð41Þ


1i , G

1i )

Z10 = ½�0:548420, � 0:00019372, � 0:15522000, � 0:71419000, � 1:08060000, � 109:9000�Z11 = ½�0:553410, � 0:00019562, � 0:13229000, � 0:70671000, � 1:08980000, � 110:8800�Z12 = ½�0:014035, � 0:03191200, � 0:08397600, � 0:05061400, � 0:00059914, � 0:048978�Z13 = ½�0:287940, � 0:00022911, � 0:00022877, � 0:00082886, � 0:02613200, � 0:121390�

ð42Þ

G10 =

7:8896e� 06 �4:88e� 06 �2:1003e� 07 3:0824e� 06 �4:144e� 05 �3:9649e� 05�4:8329e� 06 2:7367e� 05 �1:6313e� 06 �3:1912e� 06 1:3371e� 04 2:3919e� 04�3:6471e� 07 �1:4518e� 06 7:9214e� 06 �9:6416e� 06 8:1299e� 05 �5:8287e� 053:2593e� 06 �3:784e� 06 �9:5948e� 06 3:2516e� 05 �3:6529e� 04 7:8805e� 05�4:2384e� 05 1:3887e� 04 8:1301e� 05 �3:6761e� 04 1:8855e� 02 �5:0645e� 01�2:8195e� 05 1:7853e� 04 �5:6427e� 05 1:6936e� 04 �5:1254e� 01 507:95e� 01

26666664

37777775ð43Þ

G11 =


26666664

37777775ð44Þ


G12 =


26666664

37777775ð45Þ

G13 =


26666664

37777775ð46Þ


2i , G

2i )

Z20 = ½�2:4213e� 06, � 4:2664e� 06, 6:1342e� 06, � 3:2752e� 05, � 1:9581e� 02, � 5:5480e� 01�Z21 = ½�5:9946e� 06, � 2:3481e� 05, 4:3778e� 05, � 1:8995e� 04, � 5:1250e� 02, � 4:0719e� 01�Z22 = ½�2:2156e� 06, � 2:9162e� 05, 1:1126e� 04, � 3:9439e� 04, � 4:1207e� 02, � 5:2469e� 01�Z23 = ½�2:7218e� 06, � 8:0204e� 06, 2:2373e� 04, � 8:4153e� 04, � 8:6814e� 02, � 6:8411e� 01�

ð47Þ

G20 =

8:6375e� 06 �5:8541e� 06 �1:7052e� 07 3:061e� 06 1:9876e� 06 �8:5609e� 05�5:8037e� 06 2:9948e� 05 �1:9898e� 06 �2:3014e� 06 �1:1892e� 05 3:8407e� 04�3:4681e� 07 �1:7695e� 06 8:7657e� 06 �1:0947e� 05 8:0774e� 05 1:3285e� 043:2827e� 06 �3:0092e� 06 �1:0893e� 05 3:5244e� 05 �3:3881e� 04 �4:2294e� 043:2859e� 07 �4:7744e� 06 8:6082e� 05 �3:6101e� 04 5:3732e� 02 �4:4081e� 01�8:6032e� 05 3:9208e� 04 2:2815e� 04 �8:0485e� 04 �3:9729e� 01 71:034e� 01

26666664

37777775ð48Þ

G21 =

8:6356e� 06 �5:8486e� 06 �1:6711e� 07 3:0549e� 06 3:458e� 06 �9:3032e� 05�5:7989e� 06 2:9927e� 05 �1:989e� 06 �2:3008e� 06 �1:6895e� 05 4:168e� 04�3:4266e� 07 �1:7703e� 06 8:7587e� 06 �1:0942e� 05 8:0278e� 05 2:285e� 043:2751e� 06 �3:0071e� 06 �1:0891e� 05 3:5264e� 05 �3:3025e� 04 �9:0546e� 042:0057e� 06 �9:9109e� 06 8:5962e� 05 �3:5333e� 04 5:539e� 02 �3:4871e� 01�9:6388e� 05 4:2388e� 04 1:476e� 04 �5:6966e� 04 �3:7782e� 01 67:63e� 01

26666664

37777775ð49Þ

G22 =

8:6022e� 06 �5:6961e� 06 �1:4859e� 07 2:9851e� 06 3:8124e� 07 �7:9024e� 05�5:6459e� 06 2:951e� 05 �2:0785e� 06 �2:0486e� 06 8:2653e� 07 3:3505e� 04�3:2917e� 07 �1:8423e� 06 8:9527e� 06 �1:1358e� 05 6:6592e� 05 8:7906e� 053:2152e� 06 �2:7843e� 06 �1:13e� 05 3:6103e� 05 �2:8264e� 04 �3:1086e� 04�1:4043e� 06 7:7799e� 06 7:1544e� 05 �3:0434e� 04 2:7715e� 02 �1:6798e� 01�7:7589e� 05 3:3782e� 04 1:6989e� 04 �6:2933e� 04 �1:3177e� 01 30:6e� 01

26666664

37777775ð50Þ

G23 =

8:6002e� 06 �5:6906e� 06 �1:4519e� 07 2:9789e� 06 1:8516e� 06 �8:6448e� 05�5:6411e� 06 2:9489e� 05 �2:0777e� 06 �2:048e� 06 �4:1759e� 06 3:6778e� 04�3:2503e� 07 �1:8431e� 06 8:9457e� 06 �1:1354e� 05 6:6096e� 05 1:8355e� 043:2076e� 06 �2:7822e� 06 �1:1298e� 05 3:6122e� 05 �2:7409e� 04 �7:9338e� 042:7281e� 07 2:6434e� 06 7:1424e� 05 �2:9666e� 04 2:9374e� 02 �7:5872e� 02�8:7945e� 05 3:6962e� 04 8:9333e� 05 �3:9413e� 04 �1:123e� 01 27:196e� 01

26666664

37777775ð51Þ

For LPV-2A controller: (Z2A0 , G2A0 ) and (Z

2Ai , G

2Ai )

Z2A0 = ½9:5650e� 05, 7:7015e� 05, � 1:1704e� 04, � 1:4560e� 04, � 1:3835e� 03, � 28:959e� 01�Z2A1 = ½5:2380e� 05, 9:1385e� 06, � 1:0587e� 04, � 5:9202e� 04, � 5:5273e� 02, � 3:0144e� 01�Z2A2 = ½1:6166e� 05, 4:9277e� 05, � 1:2832e� 04, � 7:2469e� 04, � 1:9366e� 01, � 1:6593e� 01�

ð52Þ

Al-Hajjar et al. 15

G2A0 =


26666664

37777775ð53Þ

G2A1 =


26666664

37777775ð54Þ

G2A2 =


26666664

37777775ð55Þ


Date post:	01-Feb-2021
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