Active control laws for aeroelasticity

Alexandru Dumitrache∗1 and Ruxandra Botez∗∗2

1 Institute of Statistics and Applied Mathematics, P.O. Box 1-24, RO-010145, Bucharest, Romania2 Ecole de technologie superieure, 1100, rue Notre-Dame Ouest, H3C 1K3 Montreal, Canada

The objective of this paper is to describe a methodology to design control laws in the context of a computational aeroelasticityenvironment. The technical approach involves employing a systems identification technique to develop an explicit state-spacemodel for control law design from the output of a computational aeroelasticity code. As a control law design techniques, thestandard LQG technique is employed. The computational aeroelasticity code is modified to accept control laws and performclosed-loop simulations. Numerical results for flutter suppression of the BACT wind-tunnel model are given.

1 Introduction

Aeroelasticity has been and continues to be an extremely important consideration in many aircraft designs. The controlof aeroelastic response through feedback to control surfaces or to active materials, is an alternative to ”passive control”through increased stiffness. Within the last few decades, a significant increase in advancing methods to consider nonlinearaeroelasticity, especially nonlinear aerodynamics in the transonic region, has taken place.

Computational aeroservoelasticity involves coupling structural dynamics, CFD, and active control systems together. Batinaand Yang [1] were perhaps the first researchers to examine control of an aeroelastic system in a computational aeroelasticityenvironment for transonic flow. The effect of a simple constant gain control law utilizing displacement, velocity, and accelera-tion feedback on the time responses was determined. Comparison with linear theory indicated that the frequency and dampingvalues were significantly different for transonic and linear subsonic theory results. More recently another studies show thatfeedback control can be effective in suppressing transonic flutter. Guillot and Friedman [2] employed adaptive control theoryto design control laws using a CFD technique.

2 Description of technical approach

The overall methodology begins with performing a computational aeroelasticity simulation (CASM) (uncontrolled) with pre-scribed control surface inputs to obtain a set of corresponding output time histories. The next step is to employ a systemidentification technique, using the time histories of outputs and inputs from the first step, to determine an ”equivalent linearsystem” for use as a control law design model. Next, design of a control law design can be performed using any control lawdesign technique. Finally, the control law is evaluated in the CASM.

CASM involves integrating the structural, aerodynamic, and control equations simultaneously. For transonic flow case isemployed here the CAP-TSD [3] code. The primary outputs of the code are time histories of the pressures and the generalizedcoordinate displacements, velocities, and accelerations. The basic equations of motion implemented in CAP-TSD are:

[M ]{q} + [C]{q} + [K]{q} = {F} (1)

This code is a finite difference program that solves the general-frequency modified TSD potential equation. The pressurecoefficient is calculated at each time step and is employed to calculate the generalized force vector.

Equation 1 can be rewritten in state-space form as: X = [A]X + [B]u.Control design model development. A CASM provides time histories of the variables of an aeroelastic system, but does

not generate an explicit mathematical model of the system. Therefore a control design mathematical model of a ”compu-tational aeroelasticity system” must be developed. System identification techniques are widely employed for developing amathematical model given experimental data. The Observer/Kalman Filter Identification (OKID) technique [4] for identifyingmodels for control law design, is employed in this paper. One of the keys to the OKID algorithm is the introduction of anobserver into the identification process. The first step of the process is the calculation of the observer Markov parameters.Then the system Markov parameters are obtained. After the Markov parameters are computed using a least squares technique,a state-space model of the system is developed using Eigensystem Realization Algorithm (ERA) [5].

As a control law design techniques, the standard Linear Quadratic Gaussian (LQG) technique is employed. The resultingcontrol law is of the form: {ˆx} = [A]{x} + [L]{y}; {u} = [−G]{x}, where x is the estimate of the state vectorx, G is thestate feedback gain matrix and L is the Kalman filter gain matrix.

∗ Corresponding author: e-mail: dalex@gheorghita.ima.ro, Phone: +40 21 318 2433, Fax: +040 21 318 2439∗∗ e-mail: ruxandra.botez@etsmtl.ca

PAMM · Proc. Appl. Math. Mech. 5, 651–652 (2005) / DOI 10.1002/pamm.200510302

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3 Results and discussion

The example employed in this paper to demonstrate the overall control law design methodology is active flutter suppressionfor the Benchmark Active Controls Technology (BACT) wind-tunnel model. The BACT model is a rigid, rectangular wing,equipped with a trailing-edge control surface. Most of the control law design and evaluation process results will be demon-strated with the equivalent 2-d model. The results will begin with some basic steady aerodynamic data and then proceed touncontrolled flutter calculations, and finally to controlled flutter calculations.

The CAP-TSD model shows approximately a 2% greater flutter dynamic pressure than the experimental value. In orderto perform a system identification (ID), an exponential pulse provided a good input signal for identifying an equivalent linearmodel of the CAP-TSD outputs.

Five different cases have been investigated to illustrate the design methodology for a Mach number of 0.77 and a dynamicpressure of 5.75 kPa. The cases begin with the linear case at 0 degrees angle of attack and then proceed with nonlinear cases.For each angle of attack, a system-ID model is developed and the control law designed with the system ID model for thatangle of attack is compared with the control law for the linear case. The comparison is in terms of gain and phase margins,acceleration time history (using the exponential control input), and control surface deflection. Only last case will be describedhere. Case 5 employs the system ID model derived from the nonlinear CAP-TSD outputs at 0.6 degrees angle of attack todesign a control law. Figure 1 shows a comparison of CAP-TSD outputs for the uncontrolled and controlled case. There is asignificant reduction in the acceleration response with the controlled case. Figure 1 also shows the feedback control surfacecommand for Case 5. Similar to the previous cases, the maximum control surface displacement is approximately 0.3 degreesand occurs during the exponential pulse excitation. When using this control law clearly show much better results, in particularstability margins and damping, than using the control law designed using the system ID model of the linear CAP-TSD outputs.

Fig. 1 Case 5 controlled case.

4 Conclusion

Equivalent linear models developed by employing a system identification technique can represent the input-output relationshipof a computational aeroelasticity simulation very well. For the BACT model used in this study, the system ID model representsthe input-output relationship very well until the transonic flow conditions cause the shock on the upper surface to move aftof the 40% chord. A control law designed using a system ID model developed from a nonlinear simulation can control thenonlinear model better than a control designed using a system ID model developed from a linear CASM.

References

[1] J. T. Batina and T. Y. Yang, Transonic Calculation of Airfoil Stability and Response With Active Controls, AIAA 84-0873, (1984).[2] D. M. Guillot and P. P. Friedmann, A Fundamental Aeroservoelastic Study Combining Unsteady CFD With Adaptive Control, AIAA

94-1721-CP, (1994).[3] R. M. Bennett and J. W. Edwards, An Overview of Recent Developments in Computational Aeroelasticity, AIAA 98-2421, (1998).[4] G. P. Guruswamy, E. L. Tu and P. M. Goorjian, Transonic Aeroelasticity of Wings with Active Control Surfaces, AIAA 87-0709-CP,

(1987).[5] O. Bendiksen, G. Hwang and J. Piersol, Nonlinear Aeroelastic and Aeroservoelastic Calculations for Transonic Wings, AIAA 98-1898,

(1998).

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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