Shock and Vibration

Volume 2015, Article ID 348971, 11 pages

http://dx.doi.org/10.1155/2015/348971

## GPC-Based Gust Response Alleviation for Aircraft Model Adapting to Various Flow Velocities in the Wind Tunnel

School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China

Received 21 February 2015; Revised 27 April 2015; Accepted 17 May 2015

Academic Editor: Sakdirat Kaewunruen

Copyright © 2015 Yuting Dai and Chao Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A unified autoregressive (AR) model is identified, based on the wind tunnel test data of open-loop gust response for an aircraft model. The identified AR model can be adapted to various flow velocities in the wind tunnel test. Due to the lack of discrete gust input measurement, a second-order polynomial function is used to approximate the gust input amplitude by flow velocity. Afterwards, with the identified online aeroelastic model, the modified generalized predictive control (GPC) theory is applied to alleviate wing tip acceleration induced by sinusoidal gust. Finally, the alleviation effects of gust response at different flow velocities are estimated based on the comparison of simulated closed-loop acceleration with experimental open-loop one. The comparison indicates that, after gust response alleviation, the wing tip acceleration can be reduced up to 20% at the tested velocities ranging from 12 m/s to 24 m/s. Demonstratively, the unified control law can be adapted to varying wind tunnel velocities and gust frequencies. It does not need to be altered at different test conditions, which will save the idle time.

#### 1. Introduction

Dynamic response of aircrafts induced by gust or turbulence may reduce the ride quality, and it increases the structural load [1]. Researchers found that the gust loads can be successfully reduced when gust load alleviation systems are applied to aircrafts [2, 3]. In fact, gust load alleviation (GLA) active control is an effective tool to reduce the dynamic gust response, with a minor increasing of aircraft’s weight. For example, they are used in aircraft such as the B-52 and C5 [4, 5]. In order to guarantee real aircraft’s safety, gust response alleviation systems are designed and validated using many wind-tunnel tests [6, 7]. The control law design is an important part in the design process of GLA active technique. Most researches about GLA control law design were concentrated on PID method and linear quadratic Gauss (LQG) theory [8]. optimal control and synthesis are also effective robust control methods to account for variations in the mathematical model. However, all those control laws are based on a known theoretical continuous aeroelastic model. Model-based control laws may fail when the aeroelastic model is not accurate enough. Alternatively, a data-based control law design may be suitable for gust load alleviation. The data-based autoregressive model is a good tool to construct a discrete model without the theoretical continuous mathematical model [9]. It is found to be effective in aeroelastic modeling for online flutter prediction [10]. Therefore, the controller design can be conducted on a data-based AR model. This is the significant advantage of generalized predictive control (GPC) method. GPC can tackle not only with the theoretical continuous model but also for discrete AR model. Notably, it is demonstrated to be a useful controller for linear-parameter-varying nonlinear system [11]. Based on the above advantages, the data-based GPC is welcome to aeroelastic active control, for both gust load alleviation and flutter suppression [12, 13]. It is validated to be effective for gust load alleviation in the simulation [12].

Wu et al. conducted a gust response wind tunnel test and they designed a PID controller to alleviate the wing tip acceleration. The open-loop response data was measured at a specific velocity under sinusoidal gust. After all the open-loop responses are measured ranging from 12 m/s to 24 m/s, the closed-loop responses are also measured at these conditions by a PID controller. In this test, there are two challenged problems. Because there are some discrepancies of theoretical aeroelastic model and real aircrafts [14], the best PID parameters on theoretical model may not work well on real wind tunnel test model. They have to be trialed several times in the wind tunnel test. The other problem is that the amplitude of discrete sinusoidal gust disturbance was not measured in this test, which makes the comparison between theoretical results and testing ones difficult. Hence, the data-based GPC is designed to the wind gust response alleviation test. The basis for generalized predictive control is the identification of an autoregressive mode. While the gust input used for AR model identification is unknown, it varies with test velocity. Moreover, if we identify one AR model at one test condition, we will need to alter the control law for other different test conditions. It will be a waste of time to switch the control law manually in the wind tunnel test.

Hence, a unified GPC controller is developed. It is adapting to all the test conditions, used for gust response alleviation in the wind tunnel test. The sections of this paper are as follows. First, the standard GPC control law is introduced, and then it is modified to adapt to varying wind tunnel test velocities. Finally, a wind tunnel test of an aircraft model is employed to validate the alleviation effect.

#### 2. Control Law Design for Gust Response Alleviation

In this section, firstly, the standard GPC design method for gust response alleviation is derived at a specific flow velocity and at a specific gust frequency. Afterwards, the standard GPC method is modified to adapt to varying flow velocities.

##### 2.1. Gust Response Alleviation at a Fixed Flow Velocity

GPC is a data-based method for control law design. The design of it begins by identifying an autoregressive (AR) model, based on open-loop input and output data [12]. After the AR model is identified, a controller is acted on it to minimize the prediction of the system response in the future. In this design process, the control law is not represented by a state-space equation form but is described as a sequence of discrete input-output data.

When an external excitation exists, a time-invariant multi-input-multi-output AR model is written as [12]where integer is called the order of the AR model. is the output of the model and is the measured disturbance. Since the gust response in the wind tunnel test is excited by a sinusoidal-moving gust generator, the level of signal-to-noise ratio is large enough to omit noise . is the system input at the time of . It contains two parts. One is the deflection of control surfaces , the other is the gust velocity on the aircraft model in the wind tunnel test. As mentioned above, the discrete gust is generated by a biplane which moves as a sinusoidal function. Hence, the gust input has the following form:where is the gust amplitude; it is related with flow velocity. is the known gust frequency.

Substituting (2) to (1) and dividing the input signal into two parts, and , then (1) can be modified asWhen we keep the flow velocity fixed under one test condition, the gust amplitude can be regarded as a constant. Rearranging (3), we can getwhere and . By this rearranging, the gust disturbance is known to us. Hence, when the responses of wing tip acceleration and deflections of control surfaces are measured, the regressive relationship of input and output in the time steps can be written aswhere and is formed of the discrete time series of , , and . It is written aswhere the matrix element is the combination of gust disturbance signal, input signal, and output signal at the th time step. That is, . is the vector of observer Markov parameters to be identified. Associating with (4), we can getThe solution of is calculated by employing a least square algorithm. That is [12],After the parameter of the AR model at the th time step in (4) is identified according to the past time steps, the control system is switched on from the time step. It drives the control surface to move. In the closed-loop system, the deflection of control surface is still denoted as . Then in the future time steps, the response of the future time step th can be also represented as a linear combination of three parts. One part is the response of future time steps and the last time steps. The second part is the deflection of control surface in the future time steps and in the last time steps. The third part is the gust disturbance in the future time steps and the past time steps. The regressive relationship of future response is written asIn the above equation, the regressive coefficients are represented by combinations of the observer Markov parameters in (7). That is, elements in : , elements in and : , , elements in and : , .The goal for control law design is to alleviate the gust response in the wind tunnel test. Hence, in the framework of GPC strategy, it is required to minimize the predicted wing tip acceleration in the future time steps. Hence, the objective of the optimized control command in the next time steps is written as [12] whereIt is a typical optimal control problem. Equation (10) means that we want to find a control command to obtain a weighted minimized response in the future time steps, with relatively low control energy. By solving the optimal problem shown in (10), the optimal deflection of control surface at the time step is written aswhere , , , , and can be calculated in advance and be written into the control program for online gust load alleviation. and are the weighting matrices for predicted response and control effort, respectively. In the wind tunnel test, the wing tip acceleration has to be alleviated. And the aileron and elevator are selected as the control surfaces for gust alleviation.

##### 2.2. Gust Response Alleviation at Varying Flow Velocities

The basis of standard GPC is to identify an AR model according to the known input-output data. Hence, when GPC is employed to gust response alleviation, it is assumed that the sinusoidal gust input is already known, not only its frequency, but also its amplitude. The gust frequency is easy to get from measuring the angular velocity of the gust generator. When the flow velocity in the wind tunnel is fixed, the value of gust amplitude can be merged to the identified coefficients , as shown in (4). However, when the flow velocity varies, the corresponding gust input has been changed. In this case, another AR model should be identified. In this case, a different control law should be redesigned for this different test condition. Therefore, we have to switch the control law frequently from one test condition to another. This may waste a lot of time in a real wind tunnel test. The idea in this paper is to identify a unified AR model to apply only one GPC controller to all the test velocities in the wind tunnel.

As mentioned above, the gust velocity in the wind tunnel test is not measured. Moreover, it is difficult to deduce the gust velocity from the comparison between theoretical response and experimental one. Hence, the relationship between gust velocities and flow velocities is unknown for us. In order to solve this problem, the relationship between gust input and flow velocity is approximated in this current work.

A direct way is to employ a fitting function to represent the relationship between gust input and flow velocity. For the sake of simplicity, a second-order polynomial function is applied to represent it by the linear combinations of flow velocities. That is, where is the gust frequency at different test conditions. To combine this expression to the standard AR model, we increase the dimensions of gust disturbance from one to three. Therefore, the gust disturbance in (4) is written as a vector:

By this approximation, we can represent the gust disturbance by varying flow velocities. The predicted control law also needs to be modified to adjust to three gust inputs, shown in (12). Hence, a unified control law can be adapted to different flow velocities, and there is no need to switch the control laws from one test condition to another.

#### 3. Application Example

##### 3.1. The Open-Loop Gust Response Wind Tunnel Test

In 2011, the gust response and alleviation test for the half-span aircraft model was conducted in the FD-09 Wind Tunnel [8]. The dimension of the wind tunnel is 3 × 3 m in the test section. The pitch-plunge supporting mechanism was mounted below the wind tunnel floor without disturbance to the airflow. A feedback control law is acting on the actuator of the elevator, to stabilize the pitch and plunge motions. The responses are measured by the NI PXI-4472B NI data acquisition equipment, with the acquisition frequency of 200 Hz. The mounted aircraft model and gust response alleviation flowchart are shown in Figure 1. In this test, the open-loop gust response is measured first. And then a PI control law is acted on the aileron for gust load alleviation.