In wind tunnel tests, cantilever stings are often used as modelmount in order to reduce flow interference on experimental data. In this case, however, largeamplitude vibration and lowfrequency vibration are easily produced on the system, which indicates the potential hazards of gaining inaccurate data and even damaging the structure. This paper details three algorithms, respectively, Classical PD Algorithm, Artificial Neural Network PID (NNPID), and Linear Quadratic Regulator (LQR) Optimal Control Algorithm, which can realize active vibration control of sting used in wind tunnel. The hardware platform of the firstorder vibration damping system based on piezoelectric structure is set up and the realtime control software is designed to verify the feasibility and practicability of the algorithms. While the PD algorithm is the most common method in engineering, the results show that all the algorithms can achieve the purpose of over 80% reduction, and the last two algorithms perform even better. Besides, selftuning is realized in NNPID, and with the help of the Observer/Kalman Filter Identification (OKID), LQR optimal control algorithm can make the control effort as small as possible. The paper proves the superiority of NNPID and LQR algorithms and can be an available reference for vibration control of wind tunnel system.
In recent years, modern aircraft are gradually developing to light weight, high mobility, high speed, and high angle of attack, which apparently puts forward higher requirements for wind tunnel tests. And the quality of cantilever sting used in wind tunnel tests is the key point to obtain reliable data. However, when the model is exposed to aerodynamic loads from flow dynamic pressure, this flexible support system will easily produce lowfrequency vibration and largeamplitude vibration, which will dramatically affect the accuracy of tests and even damage the structure. Therefore, in order to enhance the safety and accuracy of experiments, it is of vital importance to reduce the vibration of the sting.
National Transonic Facility (NTF) of NASA and the European Transonic Wind tunnel (ETW) were the first to start the research in related fields. The study by NTF began around 1990. In 1994, applying piezoelectric actuators as damping device, Wimmel [
Besides, some other organizations are also concerned about the vibration of wind tunnel models. Up to now, a great many of convincing results has been obtained in thousands of attempts [
Some other studies have illustrated different controllers for active vibration control purpose [
In wind tunnel tests, the support system which is composed of a cantilever sting, a test balance and a model is exposed to wide frequency aerodynamic load, including static load and dynamic load, and the load produces dynamic bending along any section of the sting. In fact, the bending moment in pitch direction is much larger than that of the other two directions, so the primary target is to suppress the vibration in the direction of pitching.
The system can be regarded as a mass spring system with multiple degrees of freedom and low damping. Under aerodynamic loads, the vibration equation of the system can be expressed as
A piezoelectric actuator is applied to realize vibration control, the principle of the structure is shown in Figure
Diagram of the cantilever sting system.
Figure
Diagram of vibration damping system.
In order to realize PD control, the rational fraction polynomial fitting (levy) method is used to fit the transfer function according to the frequency response function of the vibration system, and the modal parameters such as frequency, damping ratio, and mode shape are obtained. Then, the PD control algorithm is proposed based on the fitted system function.
Rational fraction polynomial method, also known as levy method, is one of the most widely used frequency domain identification methods in engineering. The basic idea is to fit the theoretical transfer function to the measured frequency response curve and minimize the error. In the process of recognition, a target function
As for the experiment, firstorder mode of the sting contributes most to vibration. According to the theory of single mode vibration, the model is a secondorder system:
The system transfer function can be obtained by calculating the parameters in the matrix. And final result for the damping system is
PID (proportion, integral, and derivative) controller, as the earliest practical controller, has been used for nearly a hundred years and is still the most widely used industrial controller. Figure
Diagram of digital PID.
The mathematical expression of the PID controller is
Its transfer function is
Due to the characteristics of the vibration system, the structure will turn back to the original equilibrium point after attenuation, so there is no need to consider the steadystate error. Therefore, the proportional derivative (PD) control is adopted and the transfer function can be simplified as
Since the characteristic signal controlled by the system is a sinusoidal signal, it is necessary to calibrate the phase, especially the phase at the natural frequency. In engineering, a common method of phase correction is to set
Among a large number of neural networks, backpropagated network (BPNN), whose structure is clear and the learning method is simple, has the ability to fit any nonlinear function and is gradually developing to mature nowadays. Therefore, this paper adopts selflearning PID control algorithm based on BP neural network, and its structure is shown in Figure
Schematic diagram of a BPNN controller.
This paper uses adaptive linear neuron model, and the supervised error correction learning is realized by the gradient descent (Delta rule). The formula for the classical PID is
Selftuning is realized according to delta learning rules so that the recursive formula of weight value is
All the three weights use the same learning rate, and according to the chain rule, Equation (
The optimal control method, as shown in Figure
Diagram of a LQR controller.
In order to design the effective LQR controller, the statespace model of the system is essential. This paper adopts the Observer/Kalman Filter Identification (OKID) method together with Eigensystem Realization Algorithm (ERA) to get the statespace equation. OKID, which is proposed by Juang et al. [
Finally, the ERA is implemented. The algorithm begins by forming the block
Defining a
Define the quadratic performance index of the system:
The solution of the optimal control is
In order to verify the effectiveness of the control algorithm, a cantilever sting used in wind tunnel is processed and a measurement and control system is built. As shown in Figures
(a) The composition of the active vibration control system: (b) wiring diagram of the active vibration control system and (c) the figure of the cantilever sting with piezoelectric stacks.
When the sting begins to vibrate, the strain signal in the balance emerges and starts to transmit by the wires. After handling by a strain gauge and an antialiasing filter, the signal is sent to the controller, where the control signal is calculated according to the algorithm formula. Then, the power amplifier works to amplify the control signal. At last, the final control signal is transferred to the piezoelectric stack at the bottom of the sting to suppress vibration.
Figure
The control algorithm is programmed in LabVIEW2012. And a National Instrumentsproduced PXI7841R board card is used to collect and release signal, which possesses 8 analog input and output channels and has a FPGA module. Thus, the hardware platform can achieve high speed acquisition so that it ensures the least delay of time.
In the experiment, a vibration exciter is used to generate the vibration near the resonance frequency of the sting. By manually changing the parameters of the controller, namely
Comparison of vibration damping results with changing



Average pp value (mV)  Percentage of damping 

1  0  0  503.28  0 
0.5  −0.964  −3.62766  257.17  48.9% 
0.4  −1.446  −5.44149  210.23  58.23% 
0.3  −2.2493  −8.46455  167.8  66.7% 
0.2  −3.856  −14.5107  96.32  79.33% 
0.1  −8.676  −32.649  Fail  — 
Table
Comparison of vibration damping results with changing



Average pp value (mV) 

30  −3.3829  −45.3795  108.21 
20  −3.67067  −31.0414  94.02 
10  −3.84691  −15.7601  98.23 
5  −3.90002  −5.12393  100.03 
0  −3.90625  0  101.65 
+5  −3.90002  +5.12393  106.41 
+10  −3.84691  +15.7601  Fail 
+20  −3.67067  +31.0414  Fail 
+30  −3.3829  +45.3795  Fail 
After all, the results prove the stability and reliability of the algorithm. And the best result was obtained when
The most appealing advantage of NNPID algorithm is the ability of selftuning. As shown in Figure
Selftuning process of PID parameters.
Comparison of vibration damping results with changing

Time of selftuning (s)  pp value (mV) 

0.001  15.4  90.12 
0.01  3.4  65.78 
0.016  1.2  58.24 
0.02  0.5  70.09 
0.05  0.1  208.19 
0.1  Fail  Fail 
In the experiment, the best result is gained when the learning rate
Comparison of timedomain signal with NNPID control on/off.
Comparison of power spectrum with NNPID control on/off.
As far as LQR controller is considered, in order to get the optimal result in the experiment, the selection of parameter
Comparison of vibration damping results with changing

pp value (mV) 

0.02  66.49 
0.04  65.29 
0.06  65.31 
0.08  66.90 
0.1  67.89 
When
Figures
Comparison of timedomain signal with LQR control on/off.
Comparison diagram of power spectrum with LQR control on/off.
As shown in Figure
The experiments prove the feasibility and advancement of NNPID algorithm. By finding the most appropriate learning rate, NNPID can realize the selftuning of the control parameters. Different from the traditional PD algorithm, there is no need to launch system identification processing in NNPID, which means a significant convenience for different engineering occasions. Besides, the NNPID algorithm achieves the largest percentage of vibration damping (88%) in 1.2 s, which is also faster than PD algorithm (Table
Comparison of three controllers.
Types  Maximum damping percentage in experiments  Minimum response time  Most notable features 

PD  81%  1.5 s  Common and simple 
NNPID  88%  1.2 s  Fast and selftuning 
LQR  86%  1.7 s  Saving energy 
And on the premise of guaranteeing the effect of control, LQR optimal control algorithm shows the highest costsaving ability, though the one of the most obvious drawbacks of LQR is that the modeling process is complicated and many of the parameters need to be set. However, it has the potential to work in large power consumption conditions or powerlimited situations, like in highpower wind tunnels, which saves a huge amount of energy.
In this paper, a wind tunnel cantilever sting, with a pair of piezoelectric actuators embedded at the end is processed, and a measurement and control system based on LabVIEWFPGA module is built to verify the effectiveness of three algorithms. A classical PD, a NNPID, and a LQR controller have been designed in detail. In active damping evaluation tests, it is shown that, for all the controllers, over than 80% of displacement response of the sting mode can be eliminated. Typically, the NNPID and LQR even achieve nearly 90% reduction of vibration. The success of the experiments proves new application area of the algorithms, and it foreshadows innovative methods for wind tunnel test, which can be an available reference for wind tunnel study.
Need to say, the experiments in this paper contribute to reduce the vibration of only the first order of the system. Results could be better when the order increases, which would be the deeper research in the future.
The experimental data used to support the findings of this study may be released upon application to the Yuke Dai, who can be contacted at
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was funded by the National Natural Science Foundation of China (No. 11872207) and Jiangsu Innovation Program for Graduate Education (No. KYLX15_0245).