# Three-Surface Model with Redundant Longitudinal Control: Modeling, Trim Optimization and Control in a Preliminary Design Perspective

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## Abstract

**:**

## 1. Introduction

## 2. Flight Mechanics Models for a Three-Surface Aircraft

#### 2.1. Static Lumped Model for a Three-Surface Aircraft

#### 2.1.1. Lift for a Three-Surface Aircraft

#### 2.1.2. Drag and Polar for a Three-Surface Aircraft

#### 2.1.3. Pitching Moment for a Three-Surface Aircraft

#### 2.2. Model for the Longitudinal Dynamics of a Three-Surface Aircraft

#### 2.2.1. Derivatives with Respect to the Angle of Attack $\alpha $

#### 2.2.2. Derivatives with Respect to Forward Velocity U

#### 2.2.3. Derivatives with Respect to the Pitch Rate Q

#### 2.2.4. Derivatives with Respect to the Time Rate of the Angle of Attack $\dot{\alpha}$

#### 2.2.5. Control Derivatives

## 3. Optimization of Three-Surface Aircraft with Redundant Control

#### 3.1. Minimum Drag Solution and Optimal Trimmed Polar

**I**is the 3 × 3 identity matrix.

**y***, as

#### 3.2. Optimization of Three-Surface Configuration for Maximum Lift-To-Drag Ratio

#### 3.3. Updating a Twin-Engine, Two-Surface, Propeller Driven Airplane into an Optimal Three-Surface One

## 4. Flying Qualities and Control of a Three-Surface Aircraft: Assessment and Applications

#### 4.1. Emulating the Dynamics of a Back-Tailed Aircraft with a Three-Surface Aircraft via Automatic Control

#### 4.1.1. Control Scheme A

#### 4.1.2. Results of Dynamics Reassignment with Scheme A

#### 4.1.3. Control Scheme B

#### 4.1.4. Results of Dynamics Reassignment with Scheme B

#### 4.2. Linear-Quadratic Regulator Law for Dynamics Stabilization on a Three-Surface Aircraft

#### Results of LQR Control for Longitudinal Stabilization

#### 4.3. Comparison of Control Loads in Time Domain for Pole-Placement and Lqr Techniques

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Tail surface (

**top-left**), sum of tail and canard surface (

**bottom-left**), variation in main wing location $\Delta {x}_{{\mathit{AC}}^{w}}$ and in center of gravity position $\Delta {x}_{\mathit{CG}}$ (

**top-right**) and increase in mass $\Delta m$ (

**bottom-right**) as functions of the canard size ${S}^{c}$. ${S}^{c}=0$ indicates the traditional two-surface back-tailed configuration. The black thin dashed line at ${S}^{c}=2.38$ refers to the two-surface canard configuration.

**Figure 4.**Maximum lift-to-drag ratio as function of the canard surface. Circle markers: analyzed configurations; solid line: spline interpolation.

**Figure 5.**Sketch of the nominal DA42 airplane (

**left**) and optimal three-surface equivalent version (

**right**).

**Figure 7.**Eigenvalues of short period (

**a**) and phugoid (

**b**) for the two-surface testbed (triangles), three-surface testbed (circles) and three-surface with re-assigned dynamics via Scheme A (crosses). Lines represent limits for Level 1 flying qualities (black lines: phugoid, blue lines: Category B short period, red lines: Category A short period.

**Figure 9.**Eigenvalues of short period (

**a**) and phugoid (

**b**) for the two-surface testbed (triangles), three-surface testbed (circles) and three-surface with reassigned dynamics (crosses) via Scheme B. Lines represent limits for Level 1 flying qualities (black lines: phugoid, blue lines: Category B short period, red lines: Category A short period.

**Figure 10.**Eigenvalues of short period (

**a**) and phugoid (

**b**) for the three-surface testbed (circles) and three-surface with reassigned dynamics (crosses) via LQR. Lines represent limits for Level 1 flying qualities (black lines: phugoid, blue lines: Category B short period, red lines: Category A short period.

**Figure 11.**Pilot control set-points ${\delta}_{{e}_{pilot}}$ and ${\delta}_{{c}_{pilot}}$ for response comparison.

**Figure 12.**Comparison of the time histories of controls (first row,

**a**,

**b**), vertical control force coefficient (mid row,

**c**,

**d**) and control pitching moment (bottom row,

**e**,

**f**), for a control set-point history as in Figure 11, for the three-surface testbed conditioned by a Scheme B, pole placement designed controller (left) and an LQR controller (right).

**Table 1.**Main characteristics of the nominal aircraft model, loosely based on the Diamond DA42 Twin Star. Longitudinal axis is pointing forward and has origin in the aerodynamic center of the tail.

Variable | Value | Unit of Measure |
---|---|---|

Aircraft mass, m | 2000 | $\mathrm{kg}$ |

Wing mass, ${m}^{w}$ | 571.5 | $\mathrm{kg}$ |

Tail mass, ${m}^{t}$ | 20.0 | $\mathrm{kg}$ |

Wing area, S | 16.29 | ${\mathrm{m}}^{2}$ |

Tail area, ${S}^{t}$ | 2.35 | ${\mathrm{m}}^{2}$ |

Wing mean aerodynamic chord $\overline{c}$ | 1.1 | $\mathrm{m}$ |

Tail mean aerodynamic chord ${\overline{c}}^{t}$ | 0.55 | $\mathrm{m}$ |

Wing position, ${x}_{A{C}^{w}}$ | 4.6 | $\mathrm{m}$ |

Maximum forward fuselage extension | 7.35 | $\mathrm{m}$ |

Nominal center of gravity position, ${x}_{CG}$ | 4.11 | $\mathrm{m}$ |

Wing Oswald’s efficiency factor, ${e}^{w}$ | 0.8265 | - |

Tail Oswald’s efficiency factor, ${e}^{t}$ | 0.75 | - |

Wing incidence, ${i}^{w}$ | 0 | $\mathrm{deg}$ |

Tail incidence, ${i}^{w}$ | −1.1 | $\mathrm{deg}$ |

Slope of wing lift curve, ${C}_{{L}_{\alpha}}^{w}$ | 0.0585 | $1/\mathrm{deg}$ |

Slope of tail lift curve, ${C}_{{L}_{\alpha}}^{t}$ | 0.0775 | $1/\mathrm{deg}$ |

Tail lift-to-elevator derivative, ${C}_{{L}_{{\delta}_{e}}}^{t}$ | 0.051 | $1/\mathrm{deg}$ |

Constant drag coefficient of wing-body, ${C}_{{D}_{0}}^{w}$ | 0.03 | - |

Constant drag coefficient of tail, ${C}_{{D}_{0}}^{t}$ | 0.01 | - |

Wing aspect ratio, ${\lambda}^{w}$ | 11.06 | - |

Tail aspect ratio, ${\lambda}^{t}$ | 3.7 | - |

Wing pitching moment coefficient, ${C}_{{\mathcal{M}}_{{\mathit{AC}}^{w}}}$ | −0.03 | - |

Tail pitching moment coefficient, ${C}_{{\mathcal{M}}_{{\mathit{AC}}^{t}}}$ | −0.02 | - |

Constant downwash angle at tail, ${\epsilon}_{{D}_{0}}$ | 0.0 | $\mathrm{deg}$ |

Derivative of downwash angle respect $\alpha $ at tail, ${\epsilon}_{{D}_{{\alpha}^{w}}}$ | 0.33 | - |

**Table 2.**Characteristics of the canard surface common to all considered three-surface configurations.

Canard-Related Variable | Value | Unit of Measure |
---|---|---|

Slope of canard lift curve, ${C}_{{L}_{\alpha}}^{c}$ | 0.098 | $1/\mathrm{deg}$ |

Lift-to-elevator derivative, ${C}_{{L}_{{\delta}_{c}}}^{c}$ | 0.0654 | $1/\mathrm{deg}$ |

Incidence, ${i}^{c}$ | 0 | $\mathrm{deg}$ |

Constant drag coefficient, ${C}_{{D}_{0}}^{c}$ | 0.01 | - |

Oswald’s efficiency factor, ${e}^{c}$ | 0.85 | - |

Aspect ratio, ${\lambda}^{c}$ | 5.5 | - |

Pitching moment coefficient, ${C}_{{\mathcal{M}}_{{\mathit{AC}}^{c}}}$ | −0.02 | - |

Constant downwash angle at wing, ${\epsilon}_{{C}_{0}}$ | 0.0 | $\mathrm{deg}$ |

Derivative wrt. $\alpha $ of downwash angle at wing, ${\epsilon}_{{C}_{{\alpha}^{c}}}$ | 0.02 | - |

Derivative wrt. ${\delta}_{c}$ of downwash angle at wing, ${\epsilon}_{{C}_{{\delta}_{c}}}$ | 0.01 | - |

Wing-induced constant upwash, ${\epsilon}_{{U}_{0}}$ | 0.0 | $\mathrm{deg}$ |

Derivative wrt. $\alpha $ of wing-induced upwash, ${\epsilon}_{{U}_{{\alpha}^{w}}}$ | 0.001 | - |

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**MDPI and ACS Style**

Cacciola, S.; Riboldi, C.E.D.; Arnoldi, M.
Three-Surface Model with Redundant Longitudinal Control: Modeling, Trim Optimization and Control in a Preliminary Design Perspective. *Aerospace* **2021**, *8*, 139.
https://doi.org/10.3390/aerospace8050139

**AMA Style**

Cacciola S, Riboldi CED, Arnoldi M.
Three-Surface Model with Redundant Longitudinal Control: Modeling, Trim Optimization and Control in a Preliminary Design Perspective. *Aerospace*. 2021; 8(5):139.
https://doi.org/10.3390/aerospace8050139

**Chicago/Turabian Style**

Cacciola, Stefano, Carlo E.D. Riboldi, and Matteo Arnoldi.
2021. "Three-Surface Model with Redundant Longitudinal Control: Modeling, Trim Optimization and Control in a Preliminary Design Perspective" *Aerospace* 8, no. 5: 139.
https://doi.org/10.3390/aerospace8050139