#
Thrust Vector Controller Comparison for a Finless Rocket^{ †}

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Thrust Vector Control

#### 2.1. Mechanical Deflection of the Nozzle

#### 2.2. Exhaust Flow Deflection

#### 2.3. Injection of Secondary Propellant

#### 2.4. Auxiliary Verner Thrusters

## 3. Rocket Stability

#### 3.1. Flight Dynamic Assumption for Controller Design

- Rigid body motion to describe the launcher system;
- Short-period flight dynamic, with small angle deviations from the reference trajectory;
- Linearization of the dynamics of the rocket plant obtained by time-invariant equations over a short-time period around a specific operating point and after the take-off phase;
- Linear-Time-Invariant (LTI) approach in the launcher design, assuming both mass and inertial properties of the launcher change slowly during the flight (no sloshing phenomena are considered).

#### 3.2. Rocket Model

#### 3.3. Stability Analysis

#### 3.4. Controller Design

- Overshoot < 10%
- Settling time < 10 s
- Rise time < 2 s
- Steady-state error < 2%

## 4. Simulations and Results

#### 4.1. LQR Simulation: System Analysis and Model Implementation

#### 4.2. LQR Tuning

#### 4.2.1. Tuning 1: R Varied, Q Unchanged

#### 4.2.2. Tuning 1: Q Varied, R Unchanged

#### 4.3. LQG Simulation

#### 4.4. PID Simulation

#### 4.5. System Robustness: Precompensator and PID Comparison

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

TVC | Thrust Vector Control |

LQR | Linear Quadratic Regulator |

LQG | Linear Quadratic Gaussian |

PID | Proportional, Integral, Derivative |

LEO | Low Earth Orbit |

SSTO | Single-stage-to-orbit |

EMA | Electro-mechanical actuator |

EHA | Electro-hydraulic actuator |

LTI | Linear Time Invariant |

SISO | Single Input Single Output |

MIMO | Multi Input Multi Output |

${I}_{sp}$ | Specific Impulse |

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**Figure 1.**Rocket main axes (

**left**), TVC system along pitch direction (

**right**). The gimbal action keeps $\theta $ as close as possible to 0, overcoming disturbance torques.

Data | Value | Unit |
---|---|---|

m | 567,718 | Kg |

${I}_{y}$ | 296.4 × 10^{6} | Kgm^{2} |

${T}_{0}$ | 10,506,450 | N |

${T}_{\delta}$ | 10,506,450 | N |

V | 410.56 | m/s |

${N}_{\alpha}$ | 3056.34 | kN/rad |

${M}_{\alpha}$ | 0.3807 | ${\mathrm{s}}^{-2}$ |

${M}_{\delta}$ | 0.5726 | ${\mathrm{s}}^{-2}$ |

${x}_{cg}$ | 16.21 | m |

${x}_{cp}$ | 39.94 | m |

M | 1.4 | – |

h | 10 | Km |

${D}_{d}$ | 903.3 | kN |

Parameter | LQR |
---|---|

Overshoot % | 2.58 |

Settling time (s) | 3.99 |

Rise time (s) | 1.59 |

Steady-state error | 2.4510 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

LQR Tuning Parameters | State Variables Response | ||
---|---|---|---|

R | Q | K | |

↑ | ↓ | Slower | |

↓ | ↑ | Faster | |

↑ | ↑ | Faster | |

↓ | ↓ | Slower |

Simulation | ${\mathit{Q}}_{\mathit{i}}$ = C′*C | ${\mathit{R}}_{\mathit{i}}$ | K |
---|---|---|---|

Weighting factor, p | 15 | 1.1 | [4.4168; 3.9275; 0.0016] |

Tuning 1 | 15 | 0.1 | [12.9302; 6.7201; 0.0016] |

2.1 | [3.4188; 3.4555; 0.0016] | ||

Tuning 2 | 5 | 1.1 | [2.8981; 3.1816; 0.0016] |

25 | [5.4781; 4.3740; 0.0016] |

Parameter | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{d}}$ |
---|---|---|---|

Zieger Nichols Method | 0.6${K}_{c}$ | 0.5${T}_{c}$ | 0.125${T}_{c}$ |

PID gain | 15.09 | 4.071 | 12.501 |

Tuned Gain | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{d}}$ |
---|---|---|---|

$PI{D}_{1}$ | 15.09 | 4.071 | 12.501 |

$PI{D}_{2}$ | 17 | 3.071 | 15 |

$PI{D}_{3}$ | 18.98 | 5.046 | 14.89 |

$PI{D}_{4}$ | 15.09 | 0 | 13 |

$PI{D}_{5}$ | 17.09 | 2.3 | 20 |

Parameter | ${\mathit{P}\mathit{I}\mathit{D}}_{1}$ | ${\mathit{P}\mathit{I}\mathit{D}}_{2}$ | ${\mathit{P}\mathit{I}\mathit{D}}_{3}$ | ${\mathit{P}\mathit{I}\mathit{D}}_{4}$ | ${\mathit{P}\mathit{I}\mathit{D}}_{5}$ |
---|---|---|---|---|---|

Overshoot (%) | 13.82 | 6.85 | 14.20 | 17.28 | 7.16 |

Settling time (s) | 3.19 | 15.43 | 4.13 | 34.12 | 4.43 |

Rise time (s) | 0.22 | 0.12 | 0.19 | 0.19 | 0.16 |

Steady-state error | 2.82$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 2.90$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 6.91$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.4$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 5.62$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

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## Share and Cite

**MDPI and ACS Style**

Sopegno, L.; Livreri, P.; Stefanovic, M.; Valavanis, K.P.
Thrust Vector Controller Comparison for a Finless Rocket. *Machines* **2023**, *11*, 394.
https://doi.org/10.3390/machines11030394

**AMA Style**

Sopegno L, Livreri P, Stefanovic M, Valavanis KP.
Thrust Vector Controller Comparison for a Finless Rocket. *Machines*. 2023; 11(3):394.
https://doi.org/10.3390/machines11030394

**Chicago/Turabian Style**

Sopegno, Laura, Patrizia Livreri, Margareta Stefanovic, and Kimon P. Valavanis.
2023. "Thrust Vector Controller Comparison for a Finless Rocket" *Machines* 11, no. 3: 394.
https://doi.org/10.3390/machines11030394