# Performance Comparison of Control Strategies for a Variable-Thrust Solid-Propellant Rocket Motor

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. Solid-Propellant Rocket Motor Specification

#### 2.2. Pressure Dynamic Modeling

#### 2.3. Thrust Dynamic Modeling

#### 2.4. Stability Analysis

## 3. Pressure Control

#### 3.1. Classical PID Control

#### 3.2. Feedback Linearization Control

#### 3.3. Fuzzy PID Control

## 4. Thrust Control

#### 4.1. Classical PID Control

#### 4.2. Fuzzy PID Control

## 5. Numerical Simulation and Analysis

#### 5.1. Pressure Control Results and Analysis

#### 5.2. Thrust Control and Analysis

#### 5.3. Combining Pressure Control and Thrust Control

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${A}_{b}$ | Burning surface area |

${A}_{e}$ | Nozzle exit area |

${A}_{t}$ | Nozzle throat area |

${C}^{*}$ | Characteristic velocity |

F | Thrust |

M | Accumulated mass in the free volume |

${M}_{e}$ | Nozzle exit Mach number |

${P}_{a}$ | Ambient pressure |

${P}_{c}$ | Combustion chamber pressure |

${P}_{e}$ | Nozzle exit pressure |

R | Combustion gas constant |

T | Combustion chamber temperature |

${V}_{fv}$ | Free volume of chamber |

${V}_{e}$ | Nozzle exit velocity |

a | Burn rate coefficient |

c | Speed of sound of combustion gas |

k | Specific heat ratio |

${\dot{m}}_{d}$ | Discharge mass flow rate from chamber |

${\dot{m}}_{g}$ | Generated mass flow rate from solid propellant |

n | Burn rate exponent |

${\rho}_{c}$ | Combustion gas density |

${\rho}_{p}$ | Solid propellant density |

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**Figure 1.**Pintle nozzle technology to control thrust by chamber pressure andthroat area [7].

**Figure 2.**Schematic of throttleableducted rocket [8].

**Figure 4.**Schematic of variable-thrust solid-propellant rocketmotor with pintle nozzle [33].

Actuator | 1st Settling Time | 2nd Settling Time | ||||
---|---|---|---|---|---|---|

Model | Classical PID | Feedback Linearization | Fuzzy PID | Classical PID | Feedback Linearization | Fuzzy PID |

0.01 s | 0.0713 | 0.0183 | 0.0552 | 0.0557 | 0.0222 | 0.0339 |

0.04 s | 0.0973 | 0.0362 | 0.0837 | 0.0793 | 0.0339 | 0.0488 |

0.2 s | 0.2335 | 0.1866 | 0.1943 | 0.2634 | 0.1751 | 0.2084 |

Actuator | Undershoot | Settling Time | ||
---|---|---|---|---|

Model | Classical PID | Fuzzy PID | Classical PID | Fuzzy PID |

0.01 s | 18.43 % | 6.69 % | 0.0954 | 0.0822 |

0.04 s | 2.57 % | 1.18 % | 0.1267 | 0.1254 |

0.2 s | 0.17 % | 0.11 % | 0.2825 | 0.2797 |

Thrust Control | Pressure Control | 1st Transient | 2nd Transient | |||
---|---|---|---|---|---|---|

Undershoot | Settling Time | Undershoot | Overshoot | Settling Time | ||

Classical PID | Classical PID | 1.98% | 0.2647 | 20.05% | 17.17% | 0.505 |

Feedback Linearization | 1.13% | 0.2319 | 18.71% | 18.42% | 0.417 | |

Fuzzy PID | 2.16% | 0.2759 | 21.36% | 14.27 % | 0.465 | |

Fuzzy PID | Classical PID | 1.10% | 0.2561 | 9.85 % | 21.37% | 0.474 |

Feedback Linearization | 0.63% | 0.2130 | 9.95% | 18.50% | 0.625 | |

Fuzzy PID | 1.11% | 0.2561 | 9.88% | 21.39% | 0.476 |

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**MDPI and ACS Style**

Cha, J.; de Oliveira, É.J.
Performance Comparison of Control Strategies for a Variable-Thrust Solid-Propellant Rocket Motor. *Aerospace* **2022**, *9*, 325.
https://doi.org/10.3390/aerospace9060325

**AMA Style**

Cha J, de Oliveira ÉJ.
Performance Comparison of Control Strategies for a Variable-Thrust Solid-Propellant Rocket Motor. *Aerospace*. 2022; 9(6):325.
https://doi.org/10.3390/aerospace9060325

**Chicago/Turabian Style**

Cha, Jihyoung, and Élcio Jeronimo de Oliveira.
2022. "Performance Comparison of Control Strategies for a Variable-Thrust Solid-Propellant Rocket Motor" *Aerospace* 9, no. 6: 325.
https://doi.org/10.3390/aerospace9060325