# ArticleGust Alleviation by Active–Passive Combined Control of the Flight Platform and Antenna Array for a Flying Wing SensorCraft

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

- Adopting a sensor–integrated design.

- The high–aspect–ratio wing is designed to improve the cruise lift–to–drag ratio and extend the flight time in the mission area.

_{∞}robust controller suitable for high–altitude long–endurance UAVs [6] which reduces the maximum gust–induced wing deformation by 54% under the harshest open–loop conditions. Federico et al. designed a wingtip device for maneuvering and gust load active alleviation, and compared two control strategies [7], i.e., static output feedback (SOF) controller and recursive neural network (RNN) controller. Castrichini studied a gust load alleviation (GLA) scheme of a folded wingtip [8], which was not coupled to aircraft attitude control but operated under gust disturbance, making the control design simple. However, the wingtip rotation needed to be driven by mechanical components, resulting in increased wingtip weight, which indicates a complex structural design.

- (1)
- Based on the Volterra series theory, an unsteady aerodynamic reduced–order model (ROM) is established, which ensures the flow field nonlinearity and reduces computational time. Coupled with the structural dynamics model, the aeroelastic model of the flying wing SensorCraft is obtained. The aeroelastic model of the AGARD445.6 wing verifies the accuracy and ef–ficiency of the aerodynamic reduced–order method.
- (2)
- The wing–conformal antenna array is designed. A method for calculating the far–field pattern of the antenna array based on mode superposition is proposed, which enables fast and quantitative analysis of the electromagnetic performance of antenna arrays under dynamic loads.
- (3)
- The passive wingtip device is designed, and the GLA control method is proposed by combining the LQG/LTR active controller with the passive wing–tip. Compared with the LQG/LTR active GLA method, the influence of the active–passiveactive–passive combined GLA method on the aircraft platform and the conformal antenna is analyzed in detail.

## 2. Physical Model

#### 2.1. Flying Wing SensorCraft Model

#### 2.2. Wing–Conformal Antenna Array Model

#### 2.3. Passive Wingtip Device Model

^{4}N∙m/rad, and the wingtip device was able to pitch around it. If gust disturbance was not detected, the wingtip device locked; when gusts did appear, the wingtip device was open and rotates around the shaft to achieve passive gust alleviation.

## 3. Calculation Method

#### 3.1. Reduced–Order Aerodynamic Model Based on the Volterra Series

_{i}is the i–th order kernel. Under the assumption of a minor disturbance, the unsteady aerodynamic force obtained from the N‒S equation was weakly nonlinear, and the first–order Volterra kernel was able to reflect the system’s characteristics. The first–order kernel could be identified through step signal input, and the unit step input was as follows.

**x**

_{a}(k) is the aerodynamic state variable,

**u**(k) is the generalized displacement of the structure, and

**F**

_{a}(k) is the generalized aerodynamic coefficient.

**A**

_{a},

**B**

_{a},

**C**

_{a}, and

**D**

_{a}are the system matrix, input matrix, output matrix, and feedforward matrix obtained by reducing order.

#### 3.2. Structural Dynamics Model with the Passive Wingtip Device

**x**

_{1}is the generalized displacement of the aircraft,

**M**

_{1}is the generalized mass of the aircraft,

**C**

_{1}is the generalized damping of the plane,

**K**

_{1}is the generalized stiffness of the plane,

**F**

_{1}is the generalized aerodynamic force of the plane, and

**F**

_{2}is the generalized aerodynamic force of the passive wingtip device. ${\mathit{F}}_{1}={\mathit{A}}_{1}\mathit{u}$,

**u**represents the input of the control surfaces, and

**A**

_{1}represents the aerodynamic influence matrix of the control surfaces.

**x**

_{2}is the generalized displacement of the passive wingtip device,

**M**

_{2}is the generalized mass of the passive wingtip device,

**C**

_{2}is the generalized damping,

**K**

_{2}is the generalized stiffness, and

**A**

_{12}and

**A**

_{2}are coefficient matrices corresponding to the aerodynamic model.

**F**

_{2}is the dynamic load acting on the aircraft by the passive wingtip in Equation (6), and it can be obtained as follows.

#### 3.3. Aeroelastic Coupled Model

**u**(k) is the generalized displacement of the structure,

**x**

_{a}(k) and

**x**

_{s}(k) are the discrete aerodynamic and structural state space equation state variables, and q is the dynamic pressure.

#### 3.4. Flight Dynamics Model

#### 3.5. Actuator Model

#### 3.6. Discrete Gust Model

_{m}of the gust. When using the gust model, the space domain was converted to the time domain, and the transformation relation was as follows.

_{m}is the time taken for the wind speed to reach the maximum value, and s represents the aircraft’s position.

#### 3.7. Gust Response Model

#### 3.8. Fast Method of Antenna Array Pattern Based on Modal Superposition

- (1)
- According to the modal method, the node displacement vector $\mathrm{\Delta}\mathit{r}(t)$ of the antenna carrier under dynamic load was obtained, and each array element’s distribution position and deflection angle were calculated. $\mathrm{\Delta}\mathit{r}(t)$ could be expressed as the linear combination of each order vibration mode ${\left\{P\right\}}_{i}$ of the antenna carrier.$$\mathrm{\Delta}\mathit{r}(t)={\displaystyle \sum _{i=1}^{I}{\left\{P\right\}}_{i}{x}_{i}(t)=\mathit{P}x}$$
- (2)
- The pattern of each array element was rotated according to its deflection angle.
- (3)
- The phase difference of each array element was calculated according to its distribution position.
- (4)
- Each array element’s pattern was superimposed to obtain the antenna array’s pattern.

## 4. Design of the Active Control System

#### 4.1. Active Control Scheme

#### 4.2. LQG/LTR Control Method

- (1)
- Solving the Riccati equation to determine the observer gain
**K**_{f}. - (2)
- Designing the optimal control gain
**K**_{c}and selecting the appropriate weighting matrices**Q**and**R**cause the system to approach the open–loop gain of the Kalman filter observer as closely as possible.

**K**

_{f}was obtained from the following equation.

**P**

_{f}satisfies the Riccati equation.

**P**

_{f}is the symmetric semidefinite matrix, and

**G**and

**F**are the covariance matrices of input noise and measurement noise.

**K**

_{c}could be obtained by solving the Riccati equation.

## 5. Verification of the Reduced–Order Aerodynamic Method

## 6. Analysis of the Active–Passive Combined Gust Load Alleviation

#### 6.1. Structural Modal of the SensorCraft

#### 6.2. The Effects of the Active–Passive Combined Method on the Flight Platform

^{4}N∙m/rad. The effects of the active–passive combined GLA system were investigated with a gust length of 95 m and a gust strength of 10 m/s. Time domain response curves are shown below.

#### 6.3. Influences of the Passive Wingtip Parameters on Gust Response

^{3}N∙m/rad and 10

^{4}N∙m/rad. The effects of shaft position and torsional stiffness on the gust response with a gust length of 95 m and a gust strength of 10 m/s were investigated.

#### 6.4. Gust Responses under Different Gust Conditions

^{4}N∙m/rad. An analysis was carried out for conditions with a gust length of 50~200 m and strength of 5~20 m to investigate the effects of gust length and strength on gust response.

## 7. Impact of Wing Deformation on the Electromagnetic Performance of the Antenna Array

#### 7.1. The Impact of Static Deformation

#### 7.2. The Impact of Dynamic Deformation

^{4}N∙m/rad, a gust length of 95 m, and a gust strength of 10 m/s. The effects of the active–passive combined GLA method were investigated. The time domain response curves of the conformal antenna array are shown below.

## 8. Conclusions

- (1)
- The unsteady aerodynamic reduced–order model based on the Volterra series is effective for predicting the modal response of the system, and can be used for rapid analysis of the aeroelastic response of the SensorCraft platform under gust. The far–field pattern method based on modal superposition can be used to rapidly evaluate the electromagnetic performance of conformal arrays under dynamic loads.
- (2)
- Compared with the LQG/LTR active GLA method, the active–passive combined GLA method of “LQG/LTR active controller + passive wingtip” can significantly reduce the peak response of wingtip displacement and pitch angle. The peak response of the inner control surface decreases, and the outer control surface increases under the active–passive combined GLA method. The reason for this is that the outer control surface needs to control both the attitude angle and wingtip displacement, causing a large surface deflection.
- (3)
- As the shaft position of the passive wing moves backward, the peak value of gravity overload reaches the minimum when the shaft position is at 0.3–0.4 c from the leading edge, which is located behind the aerodynamic center, where the wingtip is easier to deflect. The peak values of the wingtip displacement and pitch angle reach the minimum when the shaft of the passive wingtip is located at 0.25–0.35 c from the leading edge. When using a passive wingtip for GLA, it is necessary to conduct a trade–off evaluation of various targets and reasonably define the position of the shaft.
- (4)
- With the increase in gust length, the peak values of gravity center overload, wingtip displacement, and pitch angle increase and decrease. The wingtip displacement peaks when the gust length is approximately 150 m, and the peak value of the pitch angle is achieved when the gust length is about 100 m.
- (5)
- The 1st–order deformation has little impact on the pattern. The effects of the 2nd–4th order deformation on the pattern are mainly reflected in the main beam angle and sidelobe level, and the main lobe gain exhibits little change.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Cord, T.; Newbern, S. Unmanned air vehicles: New challenges in design. In Proceedings of the Aerospace Conference, Big Sky, MT, USA, 10–17 March 2001. [Google Scholar]
- Hall, J.K.; Clark, C.S. SensorCraft Mission Simulation Study; Air Force Research Laboratory, Wright–Patterson AFB OH Air Vehicles Directorate: Dayton, OH, USA, 2002. [Google Scholar]
- Genello, G.J.; Baldygo, W.J., Jr.; Callahan, M.J. Modeling and simulation for Sensor Craft multi–mission radar. In Proceedings of the Aerospace Conference, Big Sky, MT, USA, 10–17 March 2001. [Google Scholar]
- Smallwood, B.; Canfield, R.; Terzuoli, A. Structurally integrated antennas on a joined–wing aircraft. In Proceedings of the 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Norfolk, VA, USA, 7–13 April 2003. [Google Scholar]
- Vartio, E.; Shaw, E.; Vetter, T. Gust load alleviation flight control system design for a SensorCraft vehicle. In Proceedings of the 26th AIAA Applied Aerodynamics Conference, Honolulu, HI, USA, 18–21 August 2008. [Google Scholar]
- Yagil, L.; Raveh, D.E.; Idan, M. Elastic deformations control of highly flexible aircraft in trimmed flight and gust encounter. J. Aircr.
**2017**, 55, 829–840. [Google Scholar] [CrossRef] - Fonte, F.; Toffol, F.; Ricci, S. Design of a wing tip device for active maneuver and gust load alleviation. In Proceedings of the 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, USA, 8–12 January 2018. [Google Scholar]
- Castrichini, A.; Hodigere Siddaramaiah, V.; Calderon, D.E.; Cooper, J.E.; Wilson, T.; Lemmens, Y. Nonlinear folding wing tips for gust loads alleviation. J. Aircr.
**2016**, 53, 1391–1399. [Google Scholar] [CrossRef] - Perron, G.; Drela, M. Competition passive gust load alleviation through the bend–twist coupling of composite beams on typical commercial airplane wings. In Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Boston, MA, USA, 8–11 April 2013. [Google Scholar]
- Cooper, J.E.; Miller, S.; Vio, G.A.; Sensburg, O. Optimization of a scaled SensorCraft model with passive gust alleviation. In Proceedings of the 12th AIAA/ISSMO Multidisciplinary Analysis & Optimization Conference, Victoria, TX, USA, 10–12 September 2008. [Google Scholar]
- Cooper, J.E. Structural Design and Analysis of an Aeroelastic Tailoring and Passive Load Alleviation Concept for A Sensor Craft; Manchester University: Manchester, UK; EOARD: London, UK, 2007. [Google Scholar]
- Cooper, J.E.; Chekkal, I.; Cheung, R.; Wales, C.; Allen, N.J.; Lawson, S.; Peace, A.J.; Cook, R.; Standen, P.; Hancock, S.D.; et al. Design of a morphing wingtip. J. Aircr.
**2014**, 52, 1394–1403. [Google Scholar] [CrossRef] - Guo, S.; Li, D.; Sensburg, O. Optimal design of a passive gust alleviation device for a flying wing aircraft. In Proceedings of the 12th AIAA Aviation Technology, Integration, and Operations (ATIO) Conference and 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN, USA, 17–19 September 2012. [Google Scholar]
- Guo, S.; Sensburg, O. Passive Gust Alleviation for a Flying Wing Aircraft; Cranfield University: Bedford, UK; AFRL: Dayton, OH, USA, 2013. [Google Scholar]
- Guo, S.; Sensburg, O. Wind Tunnel Model and Test to Evaluate the Effectiveness of a Passive Gust Alleviation Device for a Flying Wing Aircraft; Cranfield University: Bedford, UK; AFRL: Dayton, OH, USA, 2016. [Google Scholar]
- Roberts, R.W. Sensor–Craft Analytical Certification; Air Force Institute of Technology, Wright–Patterson AFB OH School of Engineering and Management: Dayton, OH, USA, 2003; pp. 22–36. [Google Scholar]
- Knott, P. Deformation and Vibration of Conformal Antenna Arrays and Compensation Techniques; Fgan–Fhr Research Institute for High Frequency Physics and Radar Techniques Wachtberg: Waterberg, Germany, 2006. [Google Scholar]
- Lucia, D. The SensorCraft configurations: A non–linear aeroservoelastic challenge for aviation. In Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Austin, TX, USA, 18–21 April 2005. [Google Scholar]
- Silva, W. Identification of nonlinear aeroelastic systems based on the Volterra theory: Progress and opportunities. Nonlinear Dyn.
**2005**, 39, 25–62. [Google Scholar] [CrossRef] - Marzocca, P.; Librescu, L.; Silva, W. Nonlinear open/closed–loop aeroelastic analysis of airfoils via Volterra series. AIAA J.
**2004**, 42, 673–686. [Google Scholar] [CrossRef] - Vartio, E.; Shimko, A.; Tilmann, C.; Flick, P. Structural modal control and gust load alleviation for a SensorCraft concept. In Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Austin, TX, USA, 18–21 April 2005. [Google Scholar]
- Song, L.; Lu, S.; Han, C.; Zhou, J.; Huang, J.; Li, P.; Ghorbani, K. Efficient computation of real–time distorted conformal load–bearing antenna structure under dynamic mechanical load based on modal superposition. IEEE J. Multiscale Multiphys. Comput. Tech.
**2018**, 3, 246–254. [Google Scholar] [CrossRef] - Gao, C.; Liu, X.; Zhang, W. On the dispersion mechanism of the flutter boundary of the AGARD 445.6 wing. AIAA J.
**2021**, 8, 2657–2669. [Google Scholar] [CrossRef] - Yateo, E.C.J. AGARD Standard Aeroelastic Configurations for Dynamic Response Wing 445.6; NASA Langley Research Center: Hampton, VA, USA, 1988.
- Goura, G.; Badcock, K.J.; Woodgate, M.A.; Richards, B.E. Implicit Methods for the Time Marching Analysis of Flutter. Aeronaut. J.
**2001**, 105, 199–214. [Google Scholar] [CrossRef]

**Figure 5.**Coordinate transformation process of the element pattern. where $\left({\phi}_{p},{\theta}_{p},{f}_{p}\right)$ is the local spherical coordinate form of the pattern of each array element, and $\left({x}_{p},{y}_{p},{z}_{p}\right)$ is the local rectangular coordinate form. In step 1, the pattern was converted from local spherical to rectangular coordinates. In step 2, the pattern was rotated from the local rectangular coordinates and converted to the pattern of the global rectangular coordinates. In step 3, the pattern was converted from the global Cartesian coordinates to the pattern of the global spherical coordinates.

**Figure 7.**Sine response of AGARD 445.6 wing modes: (

**a**) sine response of the 1st−order mode; (

**b**) sine response of the 2nd−order mode; (

**c**) sine response of the 3rd−order mode; and (

**d**) sine response of the 4th−order mode.

**Figure 8.**Comparison of AGARD 445.6 wing flutter boundaries: (

**a**) normalized flutter velocity and (

**b**) flutter frequency ratio.

**Figure 10.**Modal parameters of the SensorCraft: (

**a**) 1st−order mode of the SensorCraft; (

**b**) 2nd−order mode of the SensorCraft; (

**c**) 3rd−order mode of the SensorCraft; and (

**d**) 4th−order mode of the SensorCraft.

**Figure 11.**Comparison between the active method and the combined method. (

**a**) Response curve of gravity center overload; (

**b**) response curve of wingtip displacement; and (

**c**) response curve of pitch angle.

**Figure 12.**Response curve of the modal displacement. (

**a**) Generalized displacement of 1st−mode; (

**b**) generalized displacement of 2nd−mode; (

**c**) generalized displacement of 3rd−mode; and (

**d**) generalized displacement of 4th−mode.

**Figure 13.**Response curve of the wingtip and control surfaces. (

**a**) response curve of the passive wingtip rotation angle; (

**b**) response curve of the control surface 1; (

**c**) response curve of the control surface 2; and (

**d**) response curve of the control surface 4.

**Figure 14.**Effects of shaft position and torsional stiffness on the maximum center of gravity overload.

**Figure 15.**Effects of shaft position and torsional stiffness on the maximum center of gravity overload.

**Figure 17.**Effects of gust length and gust strength. (

**a**) Maximum gravity center overload; (

**b**) maximum wingtip displacement; and (

**c**) maximum pitch angle.

**Figure 18.**Time domain response under different gust conditions. (

**a**) Response curve of gravity center overload; (

**b**) response curve of wingtip displacement; and (

**c**) response curve of pitch angle.

**Figure 19.**Influence of the 1st–4th order static deformation on the antenna array pattern. (

**a**) Comparison of the E–plane pattern; and (

**b**) comparison of H–plane pattern.

**Figure 20.**Comparison between the active method and the combined method. (

**a**) Comparison of the main beam angle; (

**b**) comparison of the gain loss; and (

**c**) comparison of the sidelobe level.

Compressive Modulus (MPa) | Tensile Modulus (MPa) | Shear Modulus (MPa) |

22,000 | 3000 | 3000 |

Compressive strength (MPa) | Tensile strength (MPa) | Shear strength (MPa) |

388 | 540 | 120 |

Mode Shape | Frequency (Hz) |
---|---|

1st bending mode | 0.763 |

2nd bending mode | 4.336 |

1st torsion mode | 8.502 |

3rd bending mode | 10.559 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hao, S.; Ma, T.; Wang, Y.; Li, H.; Zhao, S.; Tan, P.
ArticleGust Alleviation by Active–Passive Combined Control of the Flight Platform and Antenna Array for a Flying Wing SensorCraft. *Aerospace* **2023**, *10*, 511.
https://doi.org/10.3390/aerospace10060511

**AMA Style**

Hao S, Ma T, Wang Y, Li H, Zhao S, Tan P.
ArticleGust Alleviation by Active–Passive Combined Control of the Flight Platform and Antenna Array for a Flying Wing SensorCraft. *Aerospace*. 2023; 10(6):511.
https://doi.org/10.3390/aerospace10060511

**Chicago/Turabian Style**

Hao, Shuai, Tielin Ma, Yi Wang, Huadong Li, Shiwei Zhao, and Puxue Tan.
2023. "ArticleGust Alleviation by Active–Passive Combined Control of the Flight Platform and Antenna Array for a Flying Wing SensorCraft" *Aerospace* 10, no. 6: 511.
https://doi.org/10.3390/aerospace10060511