# Improved Adaptive Augmentation Control for a Flexible Launch Vehicle with Elastic Vibration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the Launch Vehicle with Second-Order Vibration Modes

## 3. Adaptive Augmentation Controller Design

#### 3.1. Reference Model

#### 3.2. Baseline PD Controller

_{T}are shown in Figure 3. From this, the critical value of the octave forward gain in the adaptive augmentation control can be determined.

#### 3.3. Multiplicative Adaptive Control Law Based on Spectral Damping

_{max}is the upper bounds of the adjustment gain, k

_{0}is the lower bounds of the adjustment gain, k

_{e}is the adjustment gain of tracking error term, y

_{e}is the output signal of the tracking error signal through the high- and low-pass filters, k

_{s}is the adjustment gain of control command error term, and y

_{s}is the output signal of the control command error signal through the spectrum dampener.

#### 3.4. Spectrum Dampers

_{e}and k

_{s}of the two spectrum dampers adjust the spectrum output signals y

_{e}and y

_{s}of the two channels, which are formed by the tracking error signal and the controller command error signal, as follows:

_{T}is to be enhanced by increasing the error suppression gain k

_{e}, thus improving the overall forward gain of the system to reduce the system tracking error and improve system performance. In this channel, we set the control frequency near the shear frequency of the rigid-body system (0.87 rad/s), considering the need to compensate the control of the system tracking error and improve the overall gain of the system [5,28]. The shear frequency of the low-pass filter is near this frequency, and the shear frequency of the high-pass filter is one octave above this frequency, so the transfer function of the high and low-pass filters is given accordingly as follows:

_{T}by setting a specific frequency to adjust the elastic rejection gain k

_{s}, thus reducing the overall gain (excessive gain) of the system to suppress the elastic vibration of the system and reduce its instability. The input of the high- and low-pass filters is the additional instruction error generated by the rigid-body controller instruction and the elastic vibration excitation, where the high-pass filter is used to obtain the elastic vibration signal from the control instruction. The analysis in the previous section showed that the system was prone to modal vibration at 7.69 rad/s, so we designed the control frequency at this frequency point. The cut-off frequency ${\omega}_{hp}$ of the high-pass filter should be taken slightly higher than this frequency. The low-pass filter is used to eliminate the high-frequency components of the signal, which is squared before entering the low-pass filter, so the value of the cut-off frequency ${\omega}_{lp}$ of the low-pass filter should be taken near this frequency. The corresponding parameters of the spectrum dampers in the elastic rejection channel are as follows:

## 4. Simulation Results and Analysis

_{max}= 2 and K

_{min}= 0.5.

_{e}and k

_{S}of two channels in the AAC were close to 0, as shown in Figure 6c, and the overall adaptive gain was always kept at a stable value k

_{T}= 1, which meant that the AAC did not produce any effect in the normal state. This was in line with our original design requirement that the AAC not be involved in control activities when the baseline PD controller was able to achieve a good performance output.

_{s}produced elastic suppression due to the disturbance in 25 s, and then the gain for suppressing elastic vibration fell back to 0; while k

_{e}produced error suppression mainly after 60 s due to the tracking error. The overall gain k

_{T}of the AAC control was less than 1 at 25 s in the elastic suppression channel k

_{s}, and then gradually increased (>1) due to the error suppression gain k

_{e}generated, and the overall AAC gain was always maintained at a saturated value due to the long-term presence of steady-state errors.

_{e}, k

_{s}and the overall gain k

_{T}of the AAC control are shown in Figure 8c,d. We observed that in the case in which the AAC was involved in the control, the baseline PD controller was not able to achieve a good tracking effect due to the ingress of the elastic mode, then the AAC controller generated a corresponding gain value k

_{s}(in this case mainly for the suppression of elastic vibration), and then set the AAC gain to less than 1 to reduce the overall gain of the system and meet the requirements. When the baseline controller can achieve the tracking effect better, then the value of AAC gain KT will fall back to 1. It is obvious from the above analysis that the adaptive control designed in this paper had a good robust stability to the ingress of the elastic mode, and under the adjustment of the AAC control, the launch vehicle could adjust the control gain online and in real time to set the engine swing angle of the servo to keep the rocket stable.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Ting, C. Development of LM-9 heavy rocket has made great progresses. China Aerosp.
**2018**, 11, 29–31. [Google Scholar] - Wu, Y.; He, L. Attitude control technology of new-generation launch vehicles. J. Beijing Univ. Aeronaut. Astronaut.
**2009**, 35, 1294–1297. [Google Scholar] - Orr, J.; Van Zwieten, T. Robust, practical adaptive control for launch vehicles. In Proceedings of the AIAA Guidance, Navigation, and Control Conference, Minneapolis, MN, USA, 13–16 August 2012. [Google Scholar]
- Gaylor, R.; Schaeperkoetter, R.L.; Cunningham, D.C. An adaptive tracking filter for bending-mode stabilization. J. Spacecr. Rocket.
**1967**, 4, 573–577. [Google Scholar] [CrossRef] - Orr, J.S. Optimal recursive digital filters for active bending stabilization. In Proceedings of the 2013 American Astronautical Society (AAS) Guidance, Navigation, and Control Conference, Breckenridge, CO, USA, 1–6 February 2013. [Google Scholar]
- Trotta, D.; Zavoli, A.; De Matteis, G.; Neri, A. Adaptive attitude control of launch vehicles in atmospheric flight. In Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Pittsburgh, PA, USA, 9–12 August 2020. [Google Scholar]
- Jang, J.W.; Hall, R.; Bedrossian, N.; Hall, C. Ares-I bending filter design using a constrained optimization approach. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, HI, USA, 18–21 August 2008. [Google Scholar]
- Khoshnood, A.M.; Moradi, H.M. Robust adaptive vibration control of a flexible structure. ISA Trans.
**2014**, 53, 1253–1260. [Google Scholar] [CrossRef] [PubMed] - Yang, F.; Wei, C.; Cui, N.; Xu, J. Adaptive generalized super-twisting algorithm based guidance law design. In Proceedings of the 14th International Workshop on Variable Structure Systems (VSS), Nanjing, China, 1–4 June 2016. [Google Scholar]
- Cui, N.G.; Xu, J.; Mu, R.; Han, P. Gain-scheduled reusable launch vehicle attitude controller design. In Proceedings of the International Conference on Mechatronics and Automation, Changchun, China, 9–12 August 2009. [Google Scholar]
- Guo, Z.; Zhao, J.; Zhou, M.; Zhou, J. On a new adaptive multivariable twisting sliding mode control approach and its application. In Proceedings of the 3rd International Conference on Control and Robotics Engineering (ICCRE), Nagoya, Japan, 20–23 April 2018. [Google Scholar]
- Liu, J.; Yu, X.; Jin, S.; Hou, Z. Model free adaptive attitude control for a launch vehicle. In Proceedings of the Chinese Control Conference (CC(C), Guangzhou, China, 27–30 July 2019. [Google Scholar]
- Navarro-Tapia, D.; Marcos, A.; Bennani, S.; Roux, C. Robust-control-based design and comparison of an adaptive controller for the VEGA launcher. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7–11 January 2019. [Google Scholar]
- Trotta, D.; Zavoli, A.; De Matteis, G.; Neri, A. Opportunities and limitations of adaptive augmented control for launch vehicle attitude control in atmospheric flight. In Proceedings of the AAS/AIAA Astrodynamics Specialist Conference, Portland, MA, USA, 11–15 August 2019. [Google Scholar]
- Wall, J.; Orr, J.; VanZwieten, T. Space launch system implementation of adaptive augmenting control. In Proceedings of the 2014 American Astronautical Society (AAS) Guidance, Navigation, and Control Conference, Breckenridge, CO, USA, 31 January–5 February 2014. [Google Scholar]
- Orr, J.S.; Wall, J.H.; VanZwieten, T.S.; Hall, C.E. Space launch system ascent flight control design. In Proceedings of the AAS Guidance, Navigation, and Control Conference, Breckenridge, CO, USA, 31 January–5 February 2014. [Google Scholar]
- VanZwieten, T.S.; Gilligan, E.T.; Wall, J.H.; Orr, J.S.; Miller, C.J.; Hanson, C.E. Adaptive augmenting control flight characterization experiment on an F/A-18. In Proceedings of the 2014 American Astronautical Society (AAS) Guidance Navigation and Control Conference, Breckenridge, CO, USA, 31 January–5 February 2014. [Google Scholar]
- VanZwieten, T.S.; Gilligan, E.T.; Wall, J.H.; Miller, C.J.; Hanson, C.E.; Orr, J.S. In-flight suppression of an unstable F/A-18 structural mode using the space launch system adaptive augmenting control system. In Proceedings of the AIAA Guidance, Navigation, and Control Conference, Kissimmee, FL, USA, 5–9 January 2015. [Google Scholar]
- Brinda, V.; Narayanan, A.; Smrithi, U.S.; Kishore, W.A. Classical adaptive augmentation control for a typical second-generation launch vehicle. In Proceedings of the 4th IFAC Conference on Advances in Control and Optimization of Dynamical Systems (ACODS 2016), Tiruchirappalli, India, 1–5 February 2016. [Google Scholar]
- Smrithi, U.S.; Brinda, V. Augmentation of Classical and Adaptive Control for Second Generation Launch Vehicles. Int. J. Eng. Res. Technol.
**2016**, 5, 432–439. [Google Scholar] [CrossRef] - Zhang, L.; Wei, C.; Jing, L.; Cui, N. Heavy lift launch vehicle technology of adaptive augmented fault tolerant control. In Proceedings of the 2016 IEEE Chinese Guidance, Navigation and Control Conference, Nanjing, China, 12–14 August 2016. [Google Scholar]
- Cui, N.; Cheng, C.; Zhe, P.; Wei, C.; He, F. Adaptive augmented disturbance rejection and load-relief control for launch vehicle. Missile Space Veh.
**2017**, 6, 1–6. [Google Scholar] - Trotta, D.; Zavoli, A.; De Matteis, G.; Neri, A. Optimal tuning of adaptive augmenting controller for launch vehicles in atmospheric flight. J. Guid. Control. Dyn.
**2020**, 43, 2133–2140. [Google Scholar] [CrossRef] - Navarro-Tapia, D.; Marcos, A.; Bennani, S. Envelope extension via adaptive augmented thrust vector control system. J. Guid. Control. Dyn.
**2021**, 44, 1044–1052. [Google Scholar] [CrossRef] - Zhao, X. Study on Attitude Control Method 0f Heavy Launch Vehicle Based on Adaptive H∞ Control. Master’s Thesis, Deptartment of Aerospace Engineering, Harbin Institute of Technology, Harbin, China, 2012. [Google Scholar]
- Tan, S.; Zhou, W.; Wu, Z.; Yang, Y. Adaptive control design of attitude control system for launch vehicle. In Proceedings of the 9th National Conference on Dynamics and Control, Xi’an, China, 18 May 2012. [Google Scholar]
- He, F. Research on Model Reference Adaptive Augmenting Control of Heavy lift Launch Vehicle. Master’s Thesis, Department of Aerospace Engineering, Harbin Institute of Technology, Harbin, China, 2018. [Google Scholar]
- Frost, S.A.; Balas, M.J.; Wright, A.D. Augmented adaptive control of a wind turbine in the presence of structural modes. In Proceedings of the 2010 American Control Conference, Baltimore, MD, USA, 30 June–2 July 2010. [Google Scholar]
- Balas, M.J.; VanZwieten, T.; Hannan, M. Nonlinear (Lyapunov) Stability of the Space Launch System Flight Control System with Adaptive Augmenting Control. In Proceedings of the AIAA Scitech 2019 Forum, San Diego, CA, USA, 7 January 2019. [Google Scholar]

**Figure 6.**Performance in the nominal state: (

**a**) pitching attitude; (

**b**) control commands; (

**c**) gain adjustment for the tracking error and control command error; (

**d**) total gain of adaptive augmentation control.

**Figure 7.**Performance in a state in which the baseline controller gain was misloaded and suffered external perturbations at t = 25 s: (

**a**) pitching attitude; (

**b**) control commands; (

**c**) gain adjustment for the tracking error and control command error; (

**d**) total gain of adaptive augmentation control.

**Figure 8.**Performance at an elastic vibration frequency of 40% perturbation and external perturbation at t = 25 s: (

**a**) pitching attitude; (

**b**) control commands; (

**c**) gain adjustment for the tracking error and control command error; (

**d**) total gain of adaptive augmentation control.

Notation | Identification |
---|---|

$m$ | Mass of the launch vehicle |

$g$ | Gravitational acceleration |

$P$ | Total engine thrust |

$V$ | Arrow speed |

${C}_{y}^{\alpha}$ | Lifting coefficient |

$q$ | Motive pressure |

${S}_{m}$ | Characteristic area of arrows |

$\theta $ | Trajectory inclination |

$\alpha $ | Attack angle |

$\sigma $ | Heading (angle of course) |

${J}_{z}$ | Moment of inertia |

${\omega}_{z}$ | Pitch velocity |

${m}_{z}^{{\overline{\omega}}_{z}}$ | Pitch damping torque factor |

${\delta}_{\phi}$ | Pitch channel motor pendulum |

${x}_{R}$ | Distance from thrust point to arrow tip |

${x}_{z}$ | Distance from mass centre to tip of arrow body |

${x}_{d}$ | Distance from pneumatic core to arrow tip |

$l$ | Arrow length |

${m}_{R}$ | Quality of each engine |

${l}_{R}$ | Distance from engine pendulum to pivot |

${\dot{W}}_{x}$ | Radial apparent acceleration of rocket |

${F}_{y}$ | Interference |

${M}_{z}$ | Interference torque |

${U}_{iy}$ | Micro-displacement of plane bending vibration |

${R}_{iy}$ | Micro-element corner of distortion elastic vibration |

${q}_{iy}$ | The ith-order oscillation pattern of the pitch channel |

${\omega}_{i}$ | The ith-order oscillation angle frequency |

${\xi}_{i}$ | The ith-order oscillation damping ratio |

Parameters | Values | Parameters | Values | Parameters | Values |
---|---|---|---|---|---|

${b}_{1}$ | 0.6348 | ${D}_{11}$ | 0.6407 | ${\omega}_{1}$ | 3.5906 |

${b}_{2}$ | −0.0286 | ${D}_{12}$ | −27.2466 | ${\omega}_{2}$ | 7.6941 |

${b}_{3}$ | 1.1530 | ${D}_{21}$ | −1.6713 | ${\xi}_{1}$ | 0.005 |

${b}_{11}$ | −3.2693 × 10^{−5} | ${D}_{22}$ | −3.6153 | ${\xi}_{2}$ | 0.005 |

${b}_{12}$ | 0.0015 | ${D}_{31}$ | −3.5752 | ${W}_{1}{}^{\prime}\left({X}_{T}\right)$ | −0.015 |

${b}_{21}$ | 7.2608 × 10^{−4} | ${D}_{32}$ | −142.71 | ${W}_{2}{}^{\prime}\left({X}_{T}\right)$ | 2 × 10^{−4} |

${b}_{22}$ | 0.0029 | ${W}_{1}{}^{\prime}\left({X}_{gT1}\right)$ | 0.01 | ${W}_{2}{}^{\prime}\left({X}_{gT1}\right)$ | 0.004 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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**

Pang, A.; Zhou, H.; Cai, W.; Zhang, J.
Improved Adaptive Augmentation Control for a Flexible Launch Vehicle with Elastic Vibration. *Entropy* **2021**, *23*, 1058.
https://doi.org/10.3390/e23081058

**AMA Style**

Pang A, Zhou H, Cai W, Zhang J.
Improved Adaptive Augmentation Control for a Flexible Launch Vehicle with Elastic Vibration. *Entropy*. 2021; 23(8):1058.
https://doi.org/10.3390/e23081058

**Chicago/Turabian Style**

Pang, Aiping, Hongbo Zhou, Wenjie Cai, and Jing Zhang.
2021. "Improved Adaptive Augmentation Control for a Flexible Launch Vehicle with Elastic Vibration" *Entropy* 23, no. 8: 1058.
https://doi.org/10.3390/e23081058