A Continuous Robust Control Strategy for the Active Aeroelastic Vibration Suppression of Supersonic Lifting Surfaces
 Author: Zhang K., Wang Z., Behal A., Marzocca P.
 Organization: Zhang K.; Wang Z.; Behal A.; Marzocca P.
 Publish: International Journal Aeronautical and Space Sciences Volume 13, Issue2, p210~220, 30 June 2012

ABSTRACT
The modelfree control of aeroelastic vibrations of a nonlinear 2D wingflap system operating in supersonic flight speed regimes is discussed in this paper. A novel continuous robust controller design yields asymptotically stable vibration suppression in both the pitching and plunging degrees of freedom using the flap deflection as a control input. The controller also ensures that all system states remain bounded at all times during closedloop operation. A Lyapunov method is used to obtain the global asymptotic stability result. The unsteady aerodynamic load is considered by resourcing to the nonlinear Piston Theory Aerodynamics (PTA) modified to account for the effect of the flap deflection. Simulation results demonstrate the performance of the robust control strategy in suppressing dynamic aeroelastic instabilities, such as nonlinear flutter and limit cycle oscillations.

KEYWORD
Nonlinear aeroelastic control , modelfree control , robust and neural control , Supersonic aerodynamic

Nomenclature
A, G, Gd State and input matrices
Ac, Bc State and input matrices of transformed system
A*c Matrix of the system zero dynamics
a∞, p∞, ρ∞ Sound speed ,the pressure and air density of the undisturbed flow respectively
B Nonlinear restoring moment
C0, C1, C, C3 Constants used in the bounded neural network composite weight matrix
e, r Tracking error and filtered tracking error, respectively
F, G Positive definite diagonal gain matrixes for update laws of ？ and V
g1, g2 Auxiliary saturation gains
Kz, Kv, Γ , Zb, Kd Controller gains
La(t), Ma(t) lifting and aerodynamic moment
p(y, t) Unsteady pressure
V Dimensionless flight speed
vz(t) Downwash velocity normal to the airfoil surface
W, V Ideal neural network interconnection weight matrices
？ ,Estimated neural network interconnection weight matrices
w(t) Transverse deflection
x, u System state and input, respectively
Z, ZB Ideal neural networks composite weight matrix and its bound
Estimated neural networks composite weight matrix and mismatch
Auxiliary control input
ξ Dimensionless plunging displacement ξ = h / b
γ Isentropic gas coefficient (γ = 1.4 for dryair)
τ Dimensionless time τ = Ut / b
λ Aerodynamic factor
η(t) Vector of system states for analysis of zero dynamics
tr{？ } Trace of a square matrix defined as the sum of the elements on the matrix main diagonal
？F Frobenius norm defined as AF =
< A, B> Inner product of two matrix, defined as tr{B*A}
I. Introduction
In recent years, aeroelastic control and flutter suppression of flexible wings have been extensively investigated by numerous researchers. There are two basic problems associated with the aeroelastic instability of lifting surfaces ? the determination of the flutter boundary and of its character,
i.e ., the identification of the presence of a stable or unstable Limit Cycle Oscillation (LCO) in the proximity of the flutter boundary. Classical flutter analysis is based on the linearized aeroelastic equations, while LCO analysis requires a nonlinear approach [1]. The goal of the control is to expand the flight envelope above the uncontrolled flutter instability speed without weight penalties and eventually convert the catastrophic nature of flutter, associated with an unstable LCO typical of a subcritical Hopfbifurcation behavior, into benign flutter, which conversely is associated with a stable LCO typical of a supercritical Hopfbifurcation. A great deal of research activity devoted to the aeroelastic active control and flutter suppression of flight vehicles has been accomplished,e.g ., see [2]. The model nonlinearities can help to stabilize the LCO or be detrimental by destabilizing the LCO [3]. The nonlinearities to be included in the aeroelastic model can be structural [4] (i.e ., arising from the kinematic equations); physical [5] (i.e ., those involving the constitutive equations); or aerodynamic appearing in the unsteady aerodynamic equations [1][3][6]. This issue is discussed in the context of panel flutter in [1][6][7].A plethora of techniques is available for dealing with the effect of nonlinear structural stiffness in the context of subsonic flow; linear control theory, feedback linearizing techniques, adaptive, and robust control techniques have been employed to account for these nonlinearities,
e.g ., [8][19]. Recently, neuralnetworkbased (i.e ., modelfree) control approaches have been proposed in [20] and [21] to stabilize a nonlinear aeroelastic wing section. However, there is very little work (e.g ., [3],[22]) dealing with the aeroelastic vibration suppression for a supersonic wing section in the presence of both structural and aerodynamic nonlinearities.Motivated by our previous work in [19][21] and [23][25], a novel neural network (NN) based robust controller has been designed to asymptotically stabilize a supersonic aeroelastic system with unstructured nonlinear uncertainties. The nonlinearity of the model depends on the plunging distance and pitching angle. If the nonlinearity is known and could be linearly parameterized, then adaptive control is often considered to be the method of choice. In this paper, we assume unstructured uncertainty in the sense that the structure of the system nonlinearity is considered to be unknown. In contrast to existing neural networkbased controllers that only achieve practical stability, the novel continuous control design in this paper is able to achieve asymptotic stability of the origin. A threelayer neural network is implemented to approximate the unknown nonlinearity of the system. While adaptive control relies on linear parameterizability of the system nonlinearity and the determination of a regression matrix, the universal approximation property of the NN controller enables approximation of the unstructured nonlinear system in a more suitable way. To compensate for the inevitable NN functional approximation error, an integral of a sliding mode term is introduced. Through a Lyapunov analysis, global asymptotic stability can be obtained for the tracking error in the pitching degree of freedom. Then, based on the fact that the system is minimum phase, the asymptotic stability of the plunging degree of freedom is also guaranteed. Simulation results show that this NNbased robust continuous control design can rapidly suppress the flutter and limit cycle oscillations of the aeroelastic system.
The rest of the paper is organized as follows. In Section II, the aeroelastic system dynamics are introduced. In Section III, the control objective is stated explicitly while zero dynamics of the system is analyzed. The openloop error system is developed in Section IV to facilitate the subsequent control design while the closedloop error system is developed in Section V. In Section VI, Lyapunovbased analysis of the stability of the closedloop system is presented while the simulation results are shown in Section VII. Appropriate conclusions are drawn in Section VIII.
II. Model Development
The aeroelastic governing equations of a supersonic wing section with plunging and twisting degreesoffreedom (graphically represented in Fig. 1), accounting for flap deflections, and constrained by a linear translational spring and a nonlinear torsional spring, are given as follows
The dimensionless plunging distance (positive downward) is expressed as ξ (≡
h / b ), whileα is the pitch angle (positive nose up),are derivatives with respect to dimensionless time τ =
U_{t} / b , and V =U /bω _{α} is the dimensionless flight speed. The parameterB represents the nonlinear restoring moment and is defined as the ratio between the linear and nonlinear stiffness coefficients, thus it measures of the degree of nonlinearity of the system;B > 0 corresponds to hard structural nonlinearities,B = 0 corresponds to a linear model, whileB < 0 corresponds to soft structural nonlinearities. In addition,l_{α } andm_{α } represent the dimensionless aerodynamic lift and moment with respect to the elastic axis.In order to account for flap deflections, some modifications need to be made to the nonlinear Piston Theory Aerodynamics (PTA) which is used here to produce the aerodynamic loads on the lifting surface. To keep the paper selfcontained, a short description of the PTA modified
to account for the flap deflection is presented next. Within the PTA, the unsteady pressure can be defined as follows
where
v_{z}(t) andα _{∞ } represent the downwash velocity normal to the airfoil surface and the undisturbed speed of sound respectively, and are defined as followsIn the definition of
v_{z}(t) ,denotes the upper and lower surfaces, respectively, while
U _{∞ } denotes the air speed of the undisturbed flow. In the expression (3),α_{∞}, p_{∞} andp_{∞} denote the pressure and air density of the undisturbed flow, respectively, whileγ is the isentropic gas coefficient (γ = 1.4 for dryair). The transverse deflectionw(t) in (3) can be expressed as [26]where
x _{0} andx _{1} denote the dimensionless location of the elastic axis and of the torsional spring of the flap from the leading edge respectively, whileβ(t) represents the flap displacement. In the binomial expansion of (PTA), the pressure formula for PTA in the thirdorder approximation can be obtained by retaining the terms up to and including (v_{z} / α _{∞ }) as follows [7], [27][29]The aerodynamic correction factor,
is used to correct the PTA to better approximate the pressure at low supersonic flight speed regime. It is important to note that (2) and (5) are only applicable as long as the transformation through contraction and expansion can be consider isentropic,
i.e ., as long as the induced show losses are negligible (lowintensity waves). For more details, see [1] [5][30]. PTA provides results in excellent accordance with those based on the Euler solution and the CFL3D code [31]. Considering that flow takes place on both the upper and lower surfaces of the airfoil,U^{+}_{∞}= U^{?}_{∞} = U ; from (3)(5), the aerodynamic pressuredifference can be expressed as
Notice that
δ_{p} also accounts for the deflection of the flap β. Here,M = U_{∞} / α_{∞} is the undisturbed flight Mach number, whileq_{∞} = ρ _{∞}U^{2}_{∞} / 2 is the undisturbed dynamic pressure as presented in [1] and [3]. The model can be simplified to account only for the nonlinearities associated withα and discarding those associated withβ . Even though this is an approximation, the magnitude of the nonlinearities associated withβ is much smaller than those associated withα and will thus be omitted in this paper. In addition, it is assumed in the following development that the nonlinear aerodynamic damping in (6),i.e ., the termsw^{3}_{t}, w^{2}_{t} w_{x}, andw_{t} w^{2}_{t} will be discarded and consequently, the cubic nonlinear aerodynamic term reduces tow^{3}_{t} only. Although nonlinear damping can be included in the model, this paper only considers linear damping and thus conservative estimates of the flutter speed are expected.Finally, the nonlinear aerodynamic lifting and moment can be obtained from the integration of the difference of pressure on the upper and lower surfaces of the airfoil
where δp+
_{x< bx1} and δp+_{x< bx1} are the aerodynamic pressure difference on the clean airfoil and on the flap. In the governing EOM presented in (1),l_{α} andm_{α} denote the counterpart of (7) and (8), which are defined asHere,
μ represent a the dimensionless mass ratio defined asm / 4ρb ^{2}. Given the definitions above, the governing EOM can be transformed into the following formwhere
is a vector of systems states,
β(t) is a flap deflection control input, whileA, G(z), G_{d}(z), andΦ (y ) are defined as followswhere the explicit definitions for the constants
c_{i}, k_{i} , ∀_{i} = 1, …, 4 as well asp _{2} andp _{4} are reported in the Appendix.III. Control Objective and Zero Dynamics
The explicit control objective of this paper is to design a modelfree aeroelastic vibration suppression strategy to guarantee the asymptotic convergence of the pitch angle
α using the flap deflectionβ as a control input. The secondary objective is to ensure that all system states remain bounded at all times during closedloop operation. It is assumed that the measurable variables available for control implementation are the pitch angleα , pitch angle velocityplunging displacement ξ and plunging displacement velocity
Since the proposed control strategy is predicated on the assumption that the system of (12) is minimum phase, the stability of the zero dynamics of the system needs to be assured. For that purpose, the system of (11) is transformed into the following statespace form
Where
=
U^{2}β is an auxiliary comtrol input,is a new vector of system states, while
are explicitly defined as follows
where
θ_{i} ∀_{i} = 1,2,3,4 are constants that are explicitly defined in the Appendix. In (13) above,denotes a nonlinearity that encodes the nonlinear structural stiffness. It is to be noted here Φ(0) = 0. The statespace system of (13) can be expanded into the following from
Here, the stability of the zero dynamics is studied for the case when the pitch displacement is
regulated to the origin. Mathematically, this implies that
which implies from the second equation of (15) that
Since Φ_{1}(0) = 0. The zero dynamics of the system then reduce to the reduce to the third order system given by
Substituting (16) into the above set of equations for
we obtain the linear system of equations
and A^{*}_{c} is given by
For the nominal system of (15), the eigenvalues of
A^{*}_{c} lie in the left half plane which implies that the zero dynamics of the system are asymptotically stable,i.e ., this is a minimum phase system. This implies that asymptotic convergence of the pitching variable α assures the asymptotic convergence of the plunging variablez .IV. OpenLoop Error System Development
Given the definitions of (13) and (14),
can be expressed as follows
The tracking error
is defined where
denotes the desired output vector which needs to be smooth in deference to the requirements of the subsequent control design. For the control objective, one can simply choose
α_{d} to be zero all the time or use another desirable smooth timevarying trajectoryα_{d}(t) along which the actual pitching variableα can be driven towards the origin. In order to facilitate the ensuing control design and stability analysis, we also define the tracking errorand the filtered tracking error signal
as follows
where
λ _{1},λ _{2} are positive constants. By utilizing the definitions above, one can obtainBy substituting (17) for
in the above expression, the openloop dynamics for
r can be obtained as followsAfter a convenient rearrangement of terms, the openloop dynamics can be rewritten as follows
In order to design a modelfree controller, we define an auxiliary nonlinear signal
N (·) as followsBy utilizing the definition of (22) above, the openloop dynamics of the system can be compactly
rewritten as follows
V. Control Design and ClosedLoop Error System
Since the structure of the model is assumed to be unknown in the control design, standard adaptive control cannot be applied. In its lieu, a neural network feedforward compensator
along with a robustifying term is proposed to compensate for the function
N as defined above in (22). By the universal function approximation property [32], the nonlinear function of the system N can be approximated as a threelayer network target function as followsas long
N is a general smooth function fromto
and the set of inputs to the function is restricted to a compact set
S of. In (24),
denotes the augmented input vector, vector
is the ideal first layer interconnection weight matrix between input layer and hidden layer,
denotes the sigmoidal activation function, while
denotes the ideal second layer interconnection weight matrix. In this work, the weight matrixes
W andV are assumed th be constant and bounded as ∥W ∥_{F} ≤W _{B} and ∥V ∥_{F} ≤V _{B}, whereW _{B} andV _{B} are positive constants. The approximation error is assumed to be bounded in compact set ∥ε∥ < ε_{N} where ε_{N} is an unknown positive constant related to the number of nodes in the hidden layer.After substituting the approximation from (24) into (23), one can rewrite the openloop dynamics as follows
where
Motivated by the openloop dynamics and the ensuing stability analysis, the control law is designed as follows
where
K_{v}, K_{d} > 0 are constant control gains,is typical threelevel neural network compensator for target function
defined as follows
v is a robustifying term which will be defined later while？ is an adaptive estimate forg . The dynamic update law for？ is designed as followswhere the parameter projection operator
proj {·} is designed to bound？ in a known compact set Ω such that sgn(g _{3})？(t) ≥ε > 0 for all time. The projection operator defined here is meaningul because the minimumphase nature of the system ensures that sgn(g _{3})g(t)=g _{3}^{?1}g _{1} is always positive. In (25),？ andare estimates for the neural network interconnection weight matrices that are dynamically generated as follows
where
and
are postive definite diagonal gain matrixes, while
k > 0 is a scalar design parameter. By substituting the expression for control law in (26) into the openloop dynamics of (25) and conveniently rearranging the terms, one can obtalin the closedloop system dynamics as followswhere
is a parameter estimation error. Also note that we can write
where the weight estimation errors are defined as
while
w is defined as followsTo facilitate the subsequent analysis, one can also obtalin a compact form representation for ∥
w ∥ followswhere
C _{0},C _{1} andC _{2} are all positive constants while the ideal composite weight matrixZ , estimated composite weight matrixand the composite weight mismatch matrix
are given as follows
Per the boundedness property for ∥
W ∥ _{F} and ∥W ∥ _{F} as described above, there exists a constantZ_{B} such thatZ_{B} > ∥Z ∥ _{F}. Based on the definitionZ_{B} , the robustifying termv can be designed asWhere
K_{z} is a positive constant. Finally, it is noted that the functional reconstruction erroris assumed to be bounded. Thus, the closedloop dynamics can be finally written as
VI. Stability Analysis
In this section, we provide the stability analysis for the proposed modelfree controller. We begin by defining a nonnegative Lyapunov function candidate
V _{2} as followsAfter differentiating
V _{2} along the closedloop dynamics of r(t) as well as (28), one can obtain the following expression forAfter applying the neural network weight update laws designed in (29), canceling out the matched terms and utilizing the definitions of (31), (35) can be upperbounded as
By substituting (30) and (32) into (36), it is possible to further upperbound
as
where the following relation has been used to derive
Based on the fact that
one can choose
K_{z} >C _{2} such that (37) can be cast asBy defining
C _{3} =Z_{B} +C _{1} /k and conveniently rearranging the terms, (40) yieldsBy choosing
K_{d} > [C_{0} ? kC^{2}_{3} / 4], one can obtain the following upperbound onFrom (34) and (40), it is easy to see that
r ∈ L_{2} ∩ L_{∞ } while？ ,The boundedness of
r implies thatα ,are bounded by virtue of the definitions of (18) and (19). Since the system is minimum phase and relative degree one, the boundedness of the output guarantees that any first order stable filtering of the input will remain bounded. This implies that all system states remain bounded in closedloop operation which further implies that
stays bounded. Since (26) defines a stable filter acting on a bounded input, it is easy to see that
β andstay bounded; furthermore, the flap deflection control input
β is continuous at all times. The boundedness ofβ implies in turn thatby virtue of the closedloop dynamics of r. Thus, using previous assertions, one can utilize Barbalat's Lemma [33] to conclude that
r → 0 ast → ∞ which further implies thatFrom the asymptotic stability of the zero dynamics, we can further guarantee that
x _{3},x _{4} → 0 ast → ∞ . Thus, both the pitching and plunging variables show asymptotic convergence to the origin.”. Simulation Result
In this section, simulation results are presented for an aeroelastic system controlled by the proposed continuous robust controller. The nonlinear aerodynamic model is simulated using the dynamics of (1), (7) and (10). The nominal model parameters are list as follows
and the controller parameters are listed in Table 1.
The desired trajectory variables
α _{d},are simply selected as zero. The inital conditions for pitching displacement
α(t) and plunging displacementξ(t) are chosen asα (0) = 5.729deg(about 0.1 radians) andξ (0) = 0 m, while all other state variables are initialized to zero. The initial parameter estimate？ (0) is set to be 1.20, which is a 10% shift from its nominal value. The flap deflectionβ(t) is constrained to vary between ± 15 deg.The effect of structural nonlinearities on LCO amplitude was analyzed before applying any control. As shown in [22], increase in structural stiffness factor denoted by B led to decrease in LCO amplitude provided the flutter speed
remains constant. Furthermore, we also explored the effect of the location of the elastic axis from the leading edge. It was shown in [22] that a decrease in
x _{0} leads to decrease inLCO amplitude while the flutter speed increases. It was also shown that increasing the damping ratios
ζ_{h} andζ_{α} resulted in decrease of the amplitude of the LCO.Fig 2 shows the dynamics of openloop pitching displacement
α and plunging displacementξ at preflutter speed. The simulation is carried out in subcritical flight speed regime,M = 2, below the flutter speed ofM_{flutter} = 2.15.Without the controller, it is obvious that the oscillation of pitching degreeoffreedom
α will converge within 3[s] while the plunging displacement is lightly damped and it takes over 3[s] to converge. In Fig 3, it is shown that the proposed robust controller suppresses the oscillation ofα in less than 1.5[s] while the plunging displacementξ is suppressed in 2.5[s]. The parameter estimateg is seen to converge to a constant value within less than 0.5[s].Another set of simulations is run for postflutter speed. As shown in Fig 4, when M is set to be 3, the system dynamics show sustained limit cycle oscillations in openloop operation. Such LCOs is experienced due to the nonlinear pitch stiffness and the aerodynamic nonlinearities. After applying the control to the plant, from Fig 5, it is shown that when the control is turned on at t=0[s], the oscillation of
α is suppressed within 1.5[s]. The dynamic oscillatory behavior of the plunging displacementξ is suppressed within 2.5[s].The control performance is very satisfactory when it start to work at t=0. Next simulation is for delay open of control. In Fig 6, control was turned on at t=4[s] after the system had gone into an LCO. It is seen that the oscillations of the pitching degree α and plunging displacement ξ are suppressed respectively in 1.5[s] and 3[s], while in [13] convergence time of the two states are 1.5[s] and 4[s]. The parameter estimate？ also converges to a constant in less than 0.5[s].These simulation results show that the proposed novel robust controller can effectively suppress the oscillation of both pitching and plunging degreesoffreedom of the airfoil in both preflutter and postflutter flight speed regimes.
VIII. Conclusions
A modular modelfree continuous robust controller was proposed to suppress the aeroelastic vibration characteristics (including flutter and limit cycle oscillations in pre and postflutter
condition) of a supersonic 2DOF lifting surface with flap. Differently from traditional adaptive control strategies, which strictly require the linear parameterization of the system, no prior knowledge of the system model is required for the method presented in this paper. A Lyapunov method based analysis was provided to obtain the global asymptotic stability result. Finally, the simulation results showed that this control strategy can rapidly suppress any aeroelastic vibration
in pre and postflutter flight speed regimes.
> Appendix

[Fig 1.] Supersonic wing section with flap

[Table 1] Controller Parmeters

[Fig 2.] Openloop dynamics of the aeroelastic system at preflutter speed M=2< Mflutter

[Fig 3.] Closedloop plunging, pitching, control deflection and parameterestimation at preflutter speed, M=2< Mflutter

[Fig 4.] Openloop dynamics of the aeroelastic system at postflutter speed M=3> Mflutter

[Fig 5.] Closedloop plunging,pitching,control deflections and parameterestimation at postflutter speed, M=3> Mflutter

[Fig 6.] Closedloop plunging, pitching, control deflection and parameterestimation at postflutter speed, M=3> Mflutter; control appliedat t=4 s.