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Active flutter suppression and subspace identification for a flexible wing model using micro fiber composite actuator were experimentally studied in a low speed wind tunnel. NACA0006 thin airfoil model was used for the experimental object to verify the performance of identification algorithm and designed controller. The equation of the fluid, vibration, and piezoelectric coupled motion was theoretically analyzed and experimentally identified under the open-loop and closed-loop condition by subspace method for controller design. A robust pole placement algorithm in terms of linear matrix inequality that accommodates the model uncertainty caused by identification deviation and flow speed variation was utilized to stabilize the divergent aeroelastic system. For further enlarging the flutter envelope, additional controllers were designed subject to the models beyond the flutter speed. Wind speed was measured online as the decision parameter of switching between the controllers. To ensure the stability of arbitrary switching, Common Lyapunov function method was applied to design the robust pole placement controllers for different models to ensure that the closed-loop system shared a common Lyapunov function. Wind tunnel result showed that the designed controllers could stabilize the time varying aeroelastic system over a wide range under arbitrary switching.

Aeroelastic problem has been always attracting extensive attentions in past few decades. Along with the rapid development of aeronautical science, the requirements of endurance and maneuverability are constantly growing, leading to lightweight and flexible characteristics so that the aeroelastic problem is more noticeable. Flutter is a self-excited phenomenon that occurs when fluid couples energy into structure motion causing instability and excessive vibration. So flutter suppression is one of the major objective problems in aeroelastic control.

Over the past decades, many research results have been obtained by using different actuator and algorithm to suppress flutter. In early 1980s, NASA Landley center launched active flexible wing plan (AFW) and Benchmark technology projects to suppress the wing model flutter under subsonic and transonic case by using flap [

Piezoelectric material is also a major topic of discussion due to its advance of broadband frequency range and ease to be embedded. Guan et al. [

In addition to the experimental research, control algorithm is also playing an important role in active flutter suppression. Due to the variation of flying condition and model error, uncertainty of system model is inevitable. In order to cope with the impact of uncertain, robust control algorithms are applied to ensure the stability and performance of the closed-loop system. Han et al. thought over the parameter uncertainty and structure uncertainty of aeroelastic system and designed the LQG and controller synthesis, respectively, by comparing two schemes implemented on a vertical thin plate, exhibiting a better performance and scope of action to the flutter control which improve the closed-loop critical speed [

Although the corresponding controllers designed for different models are mentioned above, the switching process between each controller has not been fully considered. Even if every controller ensures the performance under the static condition, the stability and performance under arbitrary switching could not been guaranteed [

The stability of arbitrary switching control is so far attributed to the common Lyapunov function method [

In this study, system analysis and controller design are all based on the common Lyapunov theory since the dynamic model of the aeroelastic system is time varying and uncertain so that the certain conditions based on exact models are not available. The contents of this paper are as follows: dynamic model of flutter is obtained by system identification, and then multicontrollers are designed by robust pole placement. Dynamic model beyond the critical wind speed could be obtained by using frequency domain subspace method under closed-loop conditions. The common Lyapunov function and liner matrix inequality are used to correlate with each controller to ensure the arbitrary switching stability. Wind tunnel test is implemented to indicate that the flutter of the wing under a specific wind speed can be suppressed by each robust controller and stabilization of arbitrary switching among controllers can be ensured perfectly. The main contributions of this research are that the switching stability is completely insured subjecting to the wind speed variation and the robust switching control method is effective significantly in suppressing the wing flutter.

A wing model is designed to verify the control algorithm in wind tunnel test. The model is made of elastoplastic with elastic modulus 1.1 Gpa, density 1430

Wind tunnel mode.

The configurations of actuator and sensor are shown in Figure

Configuration of the wing model for wind tunnel test.

Modal parameter of the wing can be obtained via FEM analysis or system identification test. The modal analysis is implemented in Nastran FEM software subjected to the first four modes. In system identification test, a low-pass white noise is applied to the MFC as input and the accelerometer signal is measured as output. The natural frequency of the wing model can be identified from the input and output signals via subspace method that will be mentioned later. Comparison between the numerical and experimental results is shown in Table

Comparison of numerical and experimental natural frequency.

Mode | Numerical | Experimental | Mode shape |
---|---|---|---|

1 | 1.99 Hz | 2.42 Hz | First bending |

2 | 8.60 Hz | 11.46 Hz | Second bending |

3 | 23.47 Hz | 25.93 Hz | Third bending |

4 | 59.24 Hz | 67.11 Hz | First torsion |

The difference between the numerical and experimental results is contributed to the deviation of the material parameters and the neglection of MFC actuation. Analysis and test results later show that the flutter occurs in the third mode at wind speed 28 m/s.

MFC actuator shown in Figure

Macro fibers composite.

The equivalent stress

To determine the aeroelastic equation, Hamilton principle is used derived from the virtual work of potential energy, kinetic energy, and generalized external force. The total potential energy

In order to obtain the equations of motion in modal space, coordinate transformation of the displacement variable is carried out. Displacement of the wing model is expressed in terms of generalized coordinates:

In this study, frequency domain subspace identification method is used to obtain the open-loop and closed-loop aeroelastic system. The identification algorithm is implemented based on the frequency response data to estimate system matrix of state space model and it exhibits a reliable performance for open-loop system and closed-loop system identification [

Computation simplicity is the major advantage over other method. Frequency domain estimation ensures that the identified data be selectable near the flutter frequency that provides an accurate information. Sufficient accuracy can be obtained if the outside disturbance is small enough. The state space description of multi-input and multi-output system in frequency domain can be expressed as:

To estimate

Flutter is caused by the instability of the aeroelastic system in general and the poles of the system should be located in the left half of complex plane for the linear stability system. Now, pole placement controllers are designed to make the closed-loop system poles in the left half of the complex plane beyond the flutter speed. Since aeroelastic system is time varying according to the change of the Mach number, attack angle, dynamic pressure, and other factors, single controllers cannot match well with the model and would be invalid if the factors vary in large scope. So single controller is limited in suppressing the wing flutter. In this research, several robust pole placement controllers are designed based on the aeroelastic model under several specific wind speed range; therefore each controller can theoretically place the closed-loop system poles within the designated area in complex plane.

After the state space model of aeroelastic system is established, linear fractional model of uncertain system shown in Figure

Linear fractional model of uncertain system.

The linear fractional model in form of additive uncertainty is consider as a feed forward dynamic uncertainty which is added to the model that represents the dynamic deviation caused by the model identification and wind speed variation. The state space with additive uncertainty can be expressed as

Consider a dynamic feedback controller which can be expressed as

The uncertain system (

The sufficient and necessary conditions for the robust

A output feedback controller

The symmetric positive definite matrix

Pre-and postmultiply (

Since

Similarly, pre- and postmultiply (

Calculate the matrices

Solve (

Equations (

Switching controller model of uncertain system.

The uncertain switched system of aeroelastic model in state space form can be expressed as follows:

System equations (

That satisfied

It can be seen that if the closed-loop system (

The aeroelastic control system, shown in Figure

Block diagram of flutter control system of wing model for wind tunnel tests.

In order to verify the dynamic characteristic and stability under arbitrary switching of the closed-loop system, wind speed is manually regulated by the knob on the control cabinet of wind tunnel, shown in Figure

Experimental apparatus of wind tunnel test

Control cabinet of wind tunnel speed

From Section

Power spectrum can be estimated based on the time domain data via Welch method. Three frequency response models are estimated under the condition with wind speeds 27, 31, and 33 m/s, respectively. The identification results of three frequency response function of input and output signals are shown in Figure

FRF of identified aeroelastic model.

FRF under the condition with 27 m/s wind speed

FRF under the condition with 31 m/s wind speed

FRF under the condition with 33 m/s wind speed

From Figure

From the comparison of frequency response function between identified and test models, we find that the modulus error and phase error caused by identification deviation and variation of flow speed are existence. It will be compensated in the process of robust controller design.

Vibration divergence of the wing model occurs when wind speed exceeds the critical value shown in Figure

Divergence signal.

The robust pole placement controllers are designed offline based on above three identified models, and each controller is solved by LMI.

By assigning the pole position to a specific region and estimating the uncertainty according to the frequency response, the robust pole placement controller under the condition witch 27 m/s, 31 m/s and 33 m/s is designed by setting

Three identified state space models with different wind speed are as follows:

Before flutter suppression experiment, the designed controller should be discretized. Zero-order hold transformation is used to convert the analog controller to the form of difference equation. The discrete transfer functions of designed controller in three wind speed are

To demonstrate the effectiveness of designed controller, three tests are implemented with 28 m/s, 31 m/s, and 33 m/s single wind speed. During the test, controllers are switched off at first and then vibration is converged quickly after the controllers were switched on, shown in Figure

Time domain response of closed-loop system.

Time domain response with 28 m/s wind speed

Time domain response with 31 m/s wind speed

Time domain response with 33 m/s wind speed

The closed-loop frequency response functions are shown in Figure

FRF of closed-loop system.

FRF under the condition with 28 m/s wind speed

FRF under the condition with 31 m/s wind speed

FRF under the condition with 33 m/s wind speed

The comparisons between the power spectrum of the controlled and uncontrolled response signal illustrate that the major vibration energy of flutter and turbulence is efficiently alleviated by the robust controllers shown in Figure

Comparison of power spectrum with different speed.

Power spectrums with 28 m/s wind speed

Power spectrums with 28 m/s wind speed

Power spectrums with 33 m/s wind speed

Figure

Comparison between the root locus of open-loop and closed-loop systems.

The switching control signal is obtained by the wind pressure transducer placed on the wall of closed tunnel. Switching rule is expressed by the following expression:

By setting the parameters

The wind speed of wind tunnel is varied from the range of 28 m/s to 35 m/s which was controlled by the knob of the panel. The controller is switched dynamically over three candidate controllers and the time domain of the switching signal is shown in Figure

Since the wind pressure signal was disturbed by high frequency noise generated by wind tunnel, there is a serious risk that the switching signal could be changed quickly. The existence of common Lyapunov function for each subsystem ensures the unconditional stability of arbitrary switching. From Figure

Subspace identification is used to model the aeroelastic system in this research. Aeroelastic model under different wind speeds 27 m/s, 31 m/s, and 33 m/s is identified though experiments. The model under 27 m/s is below the critical speed and controller designed based on this model can efficiently enlarge the critical speed and increase the Signal-to-Noise ratio for the identification under 31 m/s, and it has the same effect under 31 m/s. When the flow speed is closed to the critical speed, flutter mode plays a dominant role of vibration, so a two-order model is used to approximate the unstable mode. The instability of vibration can be represented by “negative damping” in vibration. The characteristic equation of second-order transform function is expressed as

Damping ratio of open and closed-loop systems.

Time domain signal of switching control.

Switching control test results.

The uncertainty of the system is caused by two aspects: change of wind speed and the error of identification. The

In practice, if each controller is individually designed, a global Lyapunov function would not be shared by each closed-loop system so that the convergence can only be satisfied under no switching or slow switching condition. At switching time, total energy of the system is still possible to increase. If the flow speed variation and switching time satisfy a specific condition, vibration is still possible to divergence. The major advantage of common Lyapunov function method is absolutely alleviating the risk of switching instability and ensures the safety of flutter control.

In this paper, macro fiber composite is used to suppress the flutter of wing model under the variation of wind speed. The frequency subspace identification and robust pole placement switching controller design method are discussed and the efficiency of this method is validated though flutter suppression test. The main contributions of the paper are the following two aspects:

The subspace identification method in frequency domain is utilized to obtain the dynamic model under open and closed-loop condition. Identified model effectively reflects the instability of aeroelastic system. The robust pole placement controllers are designed based on the identified model and solved by the LMIs. Theoretical analysis and wind tunnel test are implemented to prove that the time varying system within a range of uncertainty can be stabilized by the designed controllers.

To enlarge the stability envelope, multiple controllers are applied to suppress the flutter under different wind speed range. Three switching controllers are designed by the common Lyapunov method which ensure that the closed-loop system is stable under arbitrary switching. Theoretical analysis is implemented to prove that the designed controller can stabilize the time varying system within a certain range of uncertainty. To verify the effective of control algorithm, the switching control test is implemented in wind tunnel. Wind speed is manually adjusted to simulate the random change of flow. Experimental result shows that the switching control algorithm can successfully suppress the flutter over a wide range of speed under the arbitrary switching.

System matrix of the state space model

Input matrix of the state space model

Output matrix of the state space model

Feedthrough matrix of the state space model

Modal coordinate

State vector of the state space model

Control signal of aeroelastic system

Laplace variable

Stiffness constant

Displacement of the wing model

Symbol of variation

Modal eigenvector

Output vector of the state space model

Abbreviation of symmetric matrix

Max singular value of system matrix.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is supported by funding of Jiangsu Innovation Program for Graduate Education, Grant no. KYLX15_0237 (the Fundamental Research Funds for the Central Universities) and Priority Academic Program Development of Jiangsu Higher Education Institutions.

_{∞}design with pole placement constraints: an LMI approach