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## Journal of the Brazilian Society of Mechanical Sciences and Engineering

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*Print version* ISSN 1678-5878*On-line version* ISSN 1806-3691

### J. Braz. Soc. Mech. Sci. & Eng. vol.28 no.1 Rio de Janeiro Jan./Mar. 2006

#### https://doi.org/10.1590/S1678-58782006000100009

**TECHNICAL PAPERS**

**Active flutter suppression in a 2-D airfoil using linear matrix inequalities techniques**

**Samuel da Silva ^{I}; Vicente Lopes Júnior^{II}**

^{I}Department of Mechanical Design; State University of Campinas – UNICAMP; Rua Mendeleiev s/n, Cidade Universitária; 13083-970 Campinas, SP, Brazil; samsilva@fem.unicamp.br

^{II}Emeritus Member, ABCM; Department of Mechanical Engineering; Universidade Estadual Paulista – UNESP; Av. Brasil, n.º 56, Centro; 15385-000 Ilha Solteira, SP, Brazil; vicente@dem.feis.unesp.br

**ABSTRACT**

Flutter is an in-flight vibration of flexible structures caused by energy in the airstream absorbed by the lifting surface. This aeroelastic phenomenon is a problem of considerable interest in the aeronautic industry, because flutter is a potentially destructive instability resulting from an interaction between aerodynamic, inertial, and elastic forces. To overcome this effect, it is possible to use passive or active methodologies, but passive control adds mass to the structure and it is, therefore, undesirable. Thus, in this paper, the goal is to use linear matrix inequalities (LMIs) techniques to design an active state-feedback control to suppress flutter. Due to unmeasurable aerodynamic-lag states, one needs to use a dynamic observer. So, LMIs also were applied to design a state-estimator. The simulated model consists of a classical flat plate in a two-dimensional flow. Two regulators were designed, the first one is a non-robust design for parametric variation and the second one is a robust control design, both designed by using LMIs. The parametric uncertainties are modeled through polytopic uncertainties. The paper concludes with numerical simulations for each controller. The open-loop and closed-loop responses are also compared and the results show the flutter suppression. The perfomance for both controllers are compared and discussed.

**Keywords:** Flutter, active control, LMI, polytopic uncertainties, robustness

**Introduction**

Flutter occurs when the fluid surrounding a structure feeds back dynamic energy into the structure instead of absorbing it. Typically a structure will be stable up to a limiting velocity (the flutter velocity) for given conditions. Flutter is more likely to occur in wings, ailerons and other flexible parts of aircrafts with considerable aerodynamic loads. This aeroelastic phenomenon can cause increasing wing fatigue and limit aircraft flight velocities. So, it is necessary in the aeronautic industry to reduce or to suppress this effect, (Bisplinghoff et al., 1996).

In the last decades, this problem has been studied by many authors using different techniques. De Marqui Jr et al. (2001) conducted a complete historical review of flutter, showing the main methodologies and developments to suppress flutter in aircraft. In general, one can use either passive or active techniques.

Passive flutter supression techniques add weight to change the local or global stiffening and require redesign. Components can also be moved to perform a mass balancing, but this methodology may be not feasible in some situations. Another strategy is to operate below the flutter velocity, but this procedure reduces flight perfomance.

On the other hand, active flutter suppression control system suppresses flutter without redesign and adding mass. The idea is old and it was first tested in 1973 on a B-52-E aircraft that achieved flight velocity above the specified limit, besides some problems with model accuracy and robustness, (Garrick, 1976).

Nowadays, there are many control techniques that can be used for active flutter control. For instance, Olds (1997) used a flat plate in a 2-D flow and numerically simulated the uncontrolled model using MatlabÒ. Linear Quadratic Regulator (LQR) theory was used to design a state-feedback controller to maintain stability of the closed loop system at the flutter velocity. Despite good perfomance, the results are unrealistic because LQR controller requires all states to be known at all times in order to use state feedback. In real applications, it is not reasonable to expect that all states used are available.

Bail (1997), considering the same model of Olds (1997), used state estimators for compensate the lack of information. He investigated and compared two control methods by using a dynamic observer, namely the LQG control and H_{b} control, solved by Ricatti equations. Bail (1997) considered the disturbance-rejection problem assuming the external disturbance modeled by a wind gust in a control-flap. Norlander et al. (2000) also used simple LQG control techniques to evaluate a model on a wind-tunnel test. Haley and Soloway (1996) have made an experimental investigation in a transonic wind-tunnel to demonstrate the use of the generalized predictive control for flutter suppression of a subsonic airfoil.

Non-conventional techniques can also be used to suppress flutter. Belo et al. (2001) described an investigation on the application of fuzzy logic by using the method of Mamdani to establish control laws for flutter suppression. They simulated an aerolastic structure (NACA 0012 type rigid rectangular wing) with two-dimensional and non-linear aerodynamic behavior. However, non-conventional techniques, in special those based in fuzzy logic and neural networks, are not well defined in terms of passivity, robustness and stability. Many papers employ non-conventional techniques, but many omit stability and robustness proofs. Several researchers have studied these topics and managed to prove stability and robustness characteristics in this kind of controller. The development in this area is recent, (Lewis, 1999).

One of the most recent developments in active control uses convex optimization algorithms to solve problems described by linear matrix inequalities (LMI) requirements, (Ghaoui and Niculescu, 2000). There are many toolbox codes specially developed to solve this kind of problem, for example, LMI Toolbox Matlab', (Gahinet et al., 1995) and LMISol', with free code, (Oliveira et al., 1997). Once formulated in terms of an LMI, a problem can be solved efficiently through these algorithms, (Gahinet et al., 1995).

The LMI approaches contributed to overcome many difficulties in control design. In the last decade, LMI technique has been used to solve many problems that were unfeasible with other methodologies, (Boyd et al. 1994). The major advantage of LMI design is to enable specifications such as stability requirements, decay rate, input force limitation in the actuators and output peak bounder, (Silva et al., 2004). LMI also permits the consideration of model parameter uncertainties. It is a very useful tool for problems with constraints, where the parameters vary over a range of values.

Thus, the main aim of this article is to present a methodology for active flutter suppression with robustness to parametric uncertainties, based on LMI techniques. Harman and Liu (2002) demonstrated that it is necessary to consider robustness in flight control design. They have shown some typical reasons: a large envelope flight operation requires the controller to be robust (in combination with gain scheduling techniques); fly-by-wire control tends to provide poor handling quality; aircraft agility (e.g. rapid maneuver) requires robustness when the aerodynamic control is lost; and in hypersonic flight, high speed requires stability robustness as well. Harman and Liu discussed some popular control techniques in robust flight control applications and one of these is the possible application of LMI techniques (classified by the authors as *postmodern control*).

The present paper considers a linear system with polytopic uncertainty in some parameters of the system. The control procedure using LMI with polytopic uncertainty was first proposed by Geromel et al. (1991). The same model simulated by Bail (1997) and Olds (1997) is used. The paper is devoted to active flutter-supression by using LMI frameworks, so, the aerodynamic model is not discussed in detail. Two regulators were designed, the first one is a non-robust design for parametric variation and the second one is a robust control design. The numerical application compares the open-loop and closed-loop responses of each controller. The results show the flutter suppression obtained with the present methodology. The perfomance of both controllers are compared and discussed. Finally, conclusions are presented together with some future research directions for active flutter suppression by using LMI frameworks.

**Nomenclature**

A = dynamic matrix

A_{ai} = aerodynamic-lag states (i = 1,…,4)

b = semichord

B = input matrix

c = distance of control flap from shear center

C = output matrix

h = position with respect to plunge

I_{a} = moment of inertia of pitch angle

I_{b} = moment of inertia of flap angle

K = stiffness matrix

K_{a} = stiffness of pitch spring

K_{b} = stiffness of flap spring

K_{h} = stiffness of plunge

K_{c} = controller gain matrix

K_{e} = observer gain matrix

l = distance to trailing-edge flap center of gravity from c

L = lift

L_{1} = lift per unit span on main section

L_{2} = lift per unit span on trailing-edge control surface

m = mass of airfoil

m_{1} = mass of main body

m_{2} = mass of trailing-edge control surface

M = moment of external forces

M_{1} = pitching moment on main section

M_{2} = pitching moment on trailing-edge flap

M_{a} = mass matrix

S = static moment

S_{a} = static moment of pitch angle

S_{b} = static moment of flap angle

T = torque of flap spring

u = input control

V = velocity

V_{f} = flutter velocity

x = state vector

y = output vector

**Greek Symbols**

*a = pitch angle b = flap angle r = air density µ = controller decay rate g = observer decay rate. s = maximum value of input control amplitude W = convex space*

**A Two-Dimensional Aeroelastic Airfoil Model**

A typical airfoil is viewed as a flat plate suspended from a fixed point by a spring. The basic model is illustrated in fig. 1, (York, 1980).

The motion of the airfoil is described by three independent coordinates (degree of freedom): the plunge *h*, the pitch *a*, and the flap angle *b*. To provide the correct forces in order that the cross-section behaves like a part of the attached wing, one can use linear and torsional springs. The linear spring provides a restoring force for the plunge of the airfoil, and it is assumed to have constant stiffness *K _{h}*. Likewise, the torsional spring has constant stiffness

*K*and the flap spring has constant stiffness

_{a}*K*. The control flap is located at the trailing edge. The goal is to design a controller that produces an additional flap hing torque,

_{b}.*T*, used to control the flap.

_{s}The airfoil is subjected to three aerodynamic loads. The lift *L* is considered positive in the upward direction. The pitching moment *M* is assumed to be centered about the one-quarter chord of the airfoil. The flap torque *T* is applied to the flap hinge. A state space model is implemented that can be used for control design. Newton's second law and Euler equation ca be used to obtain the following equations of motion.

where *F* is force, *m* is mass, *i* is the number of rigid bodies, *a* is acceleration, *M* is momentum of external forces, *I* is the moment of inertia, and is the angular acceleration. The subscript c.m. specifies that the variable is described in the center of mass. The free body diagram is shown in fig. 2, (Olds, 1997).

Equations (1) are applied to the main body (body 1) and to the trailing edge control surface (body 2). Small angles *a* and *b* are assumed, so the equations of motion can be linearized about the trivial equilibrium point. Equations (2-5) show this dynamic model, (considering *T _{s} = 0* ):

Equations (2) and (3) are obtained from the main body and eqs. (4) e (5) from the trailing edge control surface. Here *q _{y}* is the vertical flap hinge force,

*q*is the horizontal flap hinge force,

_{x}*l*is the distance from the trailing edge flap to center of gravity from

*c, m*is the mass of body 1,

_{1}*m*is the mass of body 2,

_{2}*x*is the nondimensionalized distance of the main section center of gravity,

_{1}*I*is the moment of inertia per unit length of the total section,

_{G1}*I*is the moment of inertia per unit span of trailing edge flap about point

_{G2}*G*,

_{2}*b*is a normalizing constant, and

*c*is the nondimensionalized distance to the flap hinge line.

After some mathematical manipulation, eqs. (2) to (5) can be combined in a second order system given by, (Olds, 1997):

where *S _{a}* is the static moment of the airfoil per unit length,

*S*is the static moment of the control flap,

_{b}*I*is the moment of inertia for the airfoil, and

_{a}*I*is the moment of inertia for the control flap. The torque

_{b}*T*is an additional flap hinge torque used to control the flap. The uncontrolled system is defined by

_{s}*T*. Equation. (6) can be written in the form:

_{s}= 0where:

and

where *M*_{a} is the mass matrix and *K* is the stiffness matrix.

Following York (1980), equation (7) can be transformed to frequency domain and the motion of the airfoil is described by harmonic oscillations. Linearized equations and unsteady aerodynamic theory are considered. These considerations are applied to derive the motion equations, which give the pressure distribution over the wing and the aerodynamic responses of the oscillating hinges for any position of the hinge with respect to the leading edge. The linearization allows the total aerodynamic loads to be found by superposition of the forces and moments associated with each degree of freedom. Once the basic system has been derived, the inverse Fourier transform is used to construct a state space model. For more details about this topic see York (1980) and Olds (1997). For the present paper is considered the state space model of the form, (Bail, 1997):

where *A* is the dynamic matrix, *B* is the input control matrix, *C* is the output matrix, *y(t)* is the output vector, *u(t)* is the input control (applied torque to the flap), and *x(t)* is the state vector that is given by:

where *x _{a}(t)* are called aerodynamic-lag states, and it is used to describe the "states" of the fluid and to represent the aerodynamic load on the airfoil. The dynamic matrix

*A*is 10 x 10; it has the form:

where *A _{ij}* are derived in Olds (1997) and are given by:

where *r* is the air density, *Z _{i}* (

*i = 1,…,4*) and

*R*are constants,

_{i}(i = 1,…,10)*b*(

_{i}*i = 1,2*) are coefficients in exponent in two-term approximation to the Wagner function, and

*V*is the velocity of the airfoil.

The *R _{i}^{'}s* constants are given in table 1,

The *Z _{i}^{'}s* constants are given by, (Olds, 1997):

where *F( V t/b)* are Wagner functions (aerodynamic constants) that are derived in Olds (1997).

Now the input matrix *B* is given by:

In this study, the measurements of plunge, pitch, the flap angle, and the respectives velocities are assumed to be measurable. So, the output matrix *C* is given by:

Since a complete derivation of the model is not provided, once the aerodynamic-lag states are unmeasured, one must construct estimators for these states from the states that are measurable. The next section shows how to assure stability to closed-loop using LMI techniques to design a controller and a dynamic observer.

**Observer-Based State-Feedback Control Solved by LMI**

Rewriting eq. (10) and considering that the dynamic matrix has a parametric variation, so:

where *W* is a polytope that is described by a list of vertexes in a convex space. The dynamic matrix is described by polytopic linear differential inclusion (LDI):

where *v* is the number of vertexes of the polytopic system. The number of vertexes is givem by *2 ^{p}*, where

*p*is the number of uncertainty parameters. The operator

*Co*means that the matrices

*A*,

_{1}*A*,

_{2}, …*A*define a polyhedral convex bounded domain. A convex domain can have many different shapes, and Co precisely denotes the convex hull.

_{v}The problem to be investigated is state-feedback control, with the following linear control law:

where *K _{c}* must be found. The system described by eq. (26) can be rewritten in closed-loop form:

The system described by eq. (29) is quadratically stable if and only if there exists a symmetric matrix *Q=Q ^{T} > 0* such that:

where the symbols *>0* and *<0* means positive and negative definite, respectively, *A _{i}* is

*ith*vertex of the polytopic system,

*i*=1,2,..,

*v*is the number of vertexes of the polytope.

Inequality (30) is not convex, because the condition is not jointly convex in *K _{c}* and

*Q*. This constraint can be overcome by a simple trasformation of variables. We can obtain an equivalent LMI, defining

*Y=K*:

_{c}Qwhich is an LMI in *Q* and *Y*. This LMI problem (LMIP) consists of finding *Q > 0* and *Y* such that LMI (31) is feasible, or to determine if the LMI is infeasible, (Boyd et al., 1994). We can solve this LMIP by using interior-point methods, (Gahinet et al., 1995).

We can also impose a decay rate (or the largest Lyapunov exponent) to this problem. The decay rate is definied as the largest *µ* such that:

holds for all trajectories of *x(t)*.

We can use the quadratic Lyapunov function, *V _{Lyap}(x)=x^{T}Q^{-1}x* , to establish a lower bound to the decay rate of the system. If

*dV*for all trajectories, then

_{Lyap}(x)/dt__<__-2µV_{Lyap}(x)*V*and therefore the decay rate

_{Lyap}(x(t))__<__V_{Lyap}(x(0))e^{-2µ t}*µ*of the system is at least

*µ*. These conditions are equivalent to the following LMIP:

When the initial condition is known, it is also possible to find an upper bound on the norm of the control input, eq. (28). Given *Q*>0 and *Y*, which satisfy the quadratic stabilization condition, the inequalities (33) are limited in the ellipsoid given by, (Folcher and Ghaoui, 1994):

The ellipsoid is said to be invariant if:

where *x*(0), the initial state, is given.

where l_{max} is the maximum eigenvalue of the following matrix (*Q ^{-1/2}Y^{T}YQ^{-1/2}* )

^{1/2}.

Therefore, the constraints ||*u(t)* || <*s* is enforced at all times *t* >0 if the LMIs below hold:

where *s* is the maximum value of input control amplitude.

The state-feedback gain matrix is *K _{c}* =

*YQ*, where

^{-1}*Y*and

*Q*are solutions from LMIP (33) and (37). This problem can be solved using interior-point methods, (Gahinet et al., 1995). For each initial condition, the input

*u*assure:

Since the aerodynamic-lag states are not measured it is essential to design a dynamic observer. In this work, it is considered the design of a deterministic observer to estimate the aerodynamic states that are not available. So, the input control is:

where *(t)* is the estimated state vector. One can write the linear equation of the robust observer in the form, (Silva et al., 2004):

where *K _{e}* is the observer gain matrix, which can be obtained by different techniques. In the present work we have used LMI to obtain the observer gain matrix. It is possible to find an observer gain through the solution of the following LMI, (Boyd et al., 1994):

where *g* is the decay rate of the observer, with *g > > µ* , and *P = P ^{T}>0* a symmetric matrix. To every

*P*and

*W*satisfying these LMI, there corresponds a stabilizing dynamic observer. The observer gain is given by

*K*, where

_{e}= P^{-1}W*P*and

*W*are solutions to the LMI problem, as given by inequality (41).

The closed-loop system is given by:

**Open-Loop Simulations**

To verify the proposed methodology, the results of simulations of an open-loop system for diferentes flow velocities are presented. We consider the following velocities *V = 290 m/s*, *V = 298* m/s and *V = 305* m/s. The velocity *V = 298* m/s is the flutter velocity (*V _{f}*). It represents the velocity for which the open-loop system becomes marginally stable. So, for

*V < V*the system is asymptotically stable and for

_{f}*V > V*the system is unstable. In this case, the airfoil would become unstable and wing separation would occur. It is a dangerous situation in a real system.

_{f}Figures 3, 4 and 5 show the aerolastic time response for each flow velocity without considering uncertainty in any parameter. The list of constants are given in table 2. We considered, for these numerical applications, the following initial condition vector:

**Closed-Loop Simulations**

This section is devoted to the controller design for the flow velocity *V = 305* m/s (unstable case). The goal is to design an LMI regulator based on the solution of inequalities (33) and (37) for the controller and inequalities (41) for the observer. The values *µ = 1*, *g = 3* and *s =250* were chosen in order to represent practical values for constrains.

First the regulator is designed non-robust for uncertainties. The non-robust state-feedback gain matrix is obtained by solving the following LMIP:

The gain matrix *K _{c}=YQ^{-1}* is reached by solving inequalities (44). The non-robust observer gain matrix is obtained by

*K*, where

_{e}=P^{-1}W*P*and

*W*are solutions to LMIP given by:

Figure 6 shows the closed-loop response for this condition. The controller reached the requirements and the closed-loop simulation obtained is stable. The input control for this robust regulator, an additional flap hing torque, *T _{s}*, is computed by using eq. (39), figure 7.

However, when considering a parametric variation, it was found that the closed-loop system became unstable. The system was assumed to have a possible variation of ± 10 % in the values of stiffness of the flap spring (*K _{b}*), stiffness of pitch (

*K*) and stiffness of plunge (

_{a}*K*). So, it has three uncertainty parameters (

_{h}*p=3*):

where *K _{b}^{N}*,

*K*and

_{a}^{N}*K*are the nominal values of parameters of the respectives springs. For the above consideration, there are eight (

_{h}^{N}*2*) vertexes of the polytopic system. The uncertainties are shown in figure 8. The vertexes of the parameter box are a combination of the minimum and maximum values of uncertainties of the system. It is supposed that the system can assume any combination of values inside the box. The vertexes correspond to V1 (

^{3}*K*,

_{b}^{min}*K*,

_{a}^{min}*K*), V2 (

_{h}^{min}*K*,

_{b}^{min}*K*,

_{a}^{max}*K*), V3 (

_{h}^{min}*K*,

_{b}^{min}*K*,

_{a}^{min}*K*), V4 (

_{h}^{max}*K*,

_{b}^{min}*K*,

_{a}^{max}*K*), V5 (

_{h}^{max}*K*,

_{b}^{max}*K*,

_{a}^{min}*K*), V6 (

_{h}^{min}*K*,

_{b}^{max}*K*,

_{a}^{max}*K*), V7 (

_{h}^{min}*K*,

_{b}^{max}*K*,

_{a}^{m,in}*K*), and V8 (

_{h}^{max}*K*,

_{b}^{max}*K*,

_{a}^{max}*K*). These vertexes define the possible dynamic matrices

_{h}^{max}*A*,

_{1}*A*, … and

_{2}*A*.

_{8}

The non-robust regulator, designed by LMIPs (44) and (45), assures only the requirements to system in the nominal condition.

In order to test the system, the condition of vertexes V5 and V7 was considered. Figures 9 and 10 present the time responses of the closed-loop considering the system in vertexes V5 and V7, respectively. Clearly, this regulator is not robust to uncertainties in the considered parameters. The results were similar for all other vertexes tested.

So, it is required to design an LMI regulator to assure quadratic stability in the closed-loop considering parametric variation of the springs. To solve this problem, the model with parametric uncertainties can be quantified by ranges of parameter values. The parameters uncertainties ranges can be described as a parameter box, shown in figure 8. The controller that satisfies all systems described inside this convex space is said to be robust to parametric variations. In order to satisfy this requirement it is enough to solve the LMI problems from inequalities (33), (37) and (41) for all eight vertexes simultaneously. The controller can be found using the following LMIs:

The robust state-feedback gain matrix is *K _{c}=YQ^{-1}*, where

*Y*and

*Q*are solutions to LMIP (47), where

*x(0), s*and

*µ*are known. The observer gain matrix is obtained by

*K*, where

_{e}=P^{-1}W*P*and

*W*are solutions to LMIP given by inequality (45).

Figures 11, 12, 13 and 14 show the time responses to initial condition considering the robust controller to vertexes V1, V5, V7 and V8 for *V = 305* m/s (*V>V _{f}*). The results of all other vertexes were similar.

Analysing figures 11, 12, 13 and 14 it is possible to conclude that the regulator is robust to the considered parametric variations. However, nothing can be said about variation in other parameters. For any other parameter variation the procedure is the same, and it is necessary to consider all vertexes in a polytopic system.

Figure 15 shows the aeroelastic time response considering the system in the nominal condition (*K _{b}^{N}*,

*K*,

_{a}^{N}*K*). The controller designed satisfied the requirements and the closed-loop obtained was stable when feedback with robust regulator.

_{h}^{N}

The input control to the robust regulator (additional flap hing torque, *T _{s}*) is computed by using equation (39). Figure 16 shows

*T*considering the system in the nominal conditions. For the other conditions inside the uncertainty combinations the responses were similar.

_{s}

**Conclusions**

In this paper an alternative solution to suppress fluter in a 2-D airfoil by using active control were presented. We have chosen to use LMI techniques due to some advantages when compared with other techniques, as for instance, the facility to solve robust problems and formulation well defined in literature, (Boyd et al., 1994).

In the first part of this paper, a brief review was made of linear classical aeroelastic 2D-airfoil model and the state space realization. Following this, the strategy to design the regulator (controller and observer) was discussed. The proposal methodology was verified through two different designs of regulator, the first one is a non-robust to parametric variation and the other one is a robust design. The results showed that the system becomes unstable under feedback control with the non-robust regulator for parametric variation in the spring constants of the system. On the other hand, with robust regulator design the system was kept stable for flutter velocity conditions.

As a further study, we propose to use fuzzy Takagi-Sugeno models based on an LMI design to develop a non-linear regulator to suppress flutter in aircraft. So, in this case, it is possible to consider several non-linearities omited in the model used in this work or still to use a more advanced model. This kind of control design is well defined in literature and has many successful citations, as for instance, Tanaka et al. (1998) and Teixeira et al. (2001).

**Acknowledgements**

The authors acknowledge the support of the Research Foundation of the State of São Paulo (FAPESP-Brazil). The authors would also like to thank the Associate Editor and reviewers for the valuable comments.

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Paper accepted October, 2005.

Technical Editor: José Roberto de F. Arruda.