# Vector-Coupled Flight Controller Design Based on Multivariable Backstepping Sliding Mode

^{*}

## Abstract

**:**

## 1. Introduction

- Cross-coupling that is inherent to flight dynamics. In previous studies [13,14,16,17,18,19,20,21], the models used to design flight control were decoupled into a scalar form, i.e., state equations with velocity, AOA, sideslip angle, roll rate, pitch rate, and yaw rate, and multiple single-channel controllers were designed. During the decoupling process, some coupling components terms are first calculated/estimated and then compensated for, while the other coupling terms may be disregarded owing to their negligible influence. However, this is a conservative method because the coupling terms may include the control input.
- Design of control laws that ensure the stability of the whole control system. In some studies [22,23], the design of the control systems is based on the assumption with “timescale separation”, wherein the slow attitude dynamics are separated from the fast angular-rate dynamics. The outer and inner controllers correspond to the slow and fast subsystems, respectively. They can be designed individually to simplify the complexity of the control system. However, the inherent weakness of this framework is that the overall system’s stability cannot be achieved theoretically.
- Uncertainties regarding various disturbances, including external disturbances, un-modeled dynamics, and parameter uncertainties. Balancing the robustness and the control performance of an aircraft control system during the design process is important. The robustness and control performance of a control system have an inverse relationship, i.e., a more robust controller implies more certain attenuation; however, the robustness is obtained by sacrificing the nominal control performance to a certain extent.

- We model the flight dynamics using a body-fixed frame in the vectorial form and consider both matched and unmatched disturbances. The attitude control of an aircraft, along with the total velocity, can be converted into a space-vector tracking equation. The triplet airspeed (${V}_{T}$), AOA ($\alpha $) and sideslip angle ($\beta $) are controlled simultaneously in a vectorial manner. This also allows us to deal with the cross-term actively and correctly. A key aspect of this study is the active use of the cross-coupling in terms of flight dynamics instead of decoupling and passive suppression and compensation.
- The control-oriented model has a lower-triangular form. Then, the structure of a Lyapunov-based backstepping approach [24,25] is used in this study to ensure the stability of the closed-loop system theoretically. This work establishes a combined multivariable backstepping sliding mode controller, along with nonlinear disturbance observers. The disturbance observers enhance the scheme of construction of the control law by combining the backstepping sliding mode control feedback with disturbance estimation, based on straightforward and feedforward compensation. Unlike similar works [12,16,17,18,19], the developed controller not only avoids solving complex matrix equations and treating inverse matrixes, but also fully realizes vector-coupled control.
- The developed control structure is concise and aesthetically appealing compared with traditional control structures that use a decoupled collection of single variables. The combined control scheme has a symmetry structure and each term is meaningful, and this feature is significant in that the control parameters can be adjusted in each term directively.

## 2. Modelling

- Body-fixed Frame: ${\Sigma}_{b}$, the reference frame with the origin at the gravity center and axes pointing forward, over the right wing, and down (relative to the pilot).
- Inertial Frame: ${\Sigma}_{i}$, the reference frame with a specific ground origin and axes pointing the North, East, and down to the Earth center.
- Wind Frame: ${\Sigma}_{w}$, the reference frame with the origin at the gravity center and the x-axis pointing to the velocity direction of the flight. The orientation of this frame relative to the body-fixed frame is determined by AOA ($\alpha $) and sideslip angle ($\beta $). The lift, drag, and side forces are defined naturally in this reference frame, respectively.

## 3. Vector-Coupled Flight Controller Design

#### 3.1. Control Objective

#### 3.2. Control Law Design

**Remark**

**1.**

**Assumption**

**1.**

**Remark**

**2.**

#### 3.2.1. Nonlinear Disturbance Observer

**Remark**

**3.**

**Proof.**

#### 3.2.2. Multivariable Backstepping Sliding Mode Controller

**Theorem**

**1.**

**Proof of Theorem 1.**

**Step 1.**Firstly, the angular velocity $\mathit{W}$ and ${u}_{V}$ are viewed as control input variables to control the velocity vector $\mathit{V}$ tracking the given command ${\mathit{V}}_{d}$.

**Step 2.**Second, the angular acceleration ${\mathit{u}}_{T}$ is the control input to maintain the angular-velocity vector $\mathit{W}$ tracking the desired virtual angular-velocity vector ${\mathit{W}}_{d}$.

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6.**

## 4. Simulations

#### 4.1. Effectiveness of the Proposed Controller

**Case**

**1.**

#### 4.2. Robustness of the Proposed Nonlinear Disturbance Observer Enhanced Controller

**Case**

**2.**

**Remark**

**7.**

#### 4.3. Comparison with a Decoupled Controller

**Case**

**3.**

**Remark**

**8.**

## 5. Conclusions

- The developed control scheme allowed the conversion of attitude and airspeed control of an aircraft into a space-vector tracking problem. The results showed that the triplet airspeed (${V}_{T}$), AOA ($\alpha $), and sideslip angle ($\beta $) could be controlled separately and simultaneously in the form of vectors. A key feature of this study is the active use of cross-coupling in the flight dynamics instead of decoupling and passive suppression and compensation.
- The use of the Lyapunov stability theory enabled the development of a flight control system that is semi-globally uniformly ultimately bounded. The simulation results were comparable to the theoretical results, and it was shown that the developed controller is effective and robust.
- The developed control scheme is concise and aesthetically more appealing compared with traditional control structures, which uses decoupled collections of single variables. The combined control scheme has a symmetry structure and each term is meaningful. The feature that the control parameters can be considered and adjusted in each term directively is significant.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

∼ | Error signal |

∧ | Estimation signal |

$\alpha $ | Angle of attack (AOA) (rad) |

$\beta $ | Sideslip angle (rad) |

$\gamma $ | Roll angle (rad) |

$\psi $ | Yaw angle (rad) |

$\rho $ | Atmospheric density (kg/${\mathrm{m}}^{3}$) |

$\phi $ | Pitch angle (rad) |

g | Acceleration of gravity ($\mathrm{m}/{\mathrm{s}}^{2}$) |

m | Flight mass (kg) |

$p,q,r$ | Roll, pitch and yaw rates (rad/s) |

${q}_{p}$ | Dynamic pressure $0.5\rho {V}_{T}^{2}$ ($\mathrm{p}a$) |

$\mathit{\delta}$ | Deflection vector ${\left(\begin{array}{cccc}{\delta}_{c}& {\delta}_{re}& {\delta}_{le}& {\delta}_{r}\end{array}\right)}^{T}$ (rad) |

$\mathit{B}$ | Distribution matrix |

$\mathit{F}$ | External force vector (N) |

$\mathit{G}$ | Gravity vector (N) |

$\mathit{J}$ | Inertia matrix $diag\left\{{J}_{x},{J}_{y},{J}_{z}\right\}$ ($\mathrm{kg}\xb7{\mathrm{m}}^{2}$) |

$\mathit{R}$ | Aerodynamics force vector (N) |

$\mathit{T}$ | Thrust force vector (N) |

${u}_{V}$ | Thrust control term ($\mathrm{m}/{\mathrm{s}}^{2}$) |

${\mathit{V}}_{0}$ | Unit vector of flight velocity |

$\mathit{V}$ | Flight velocity vector ${\left[\begin{array}{ccc}{v}_{x}& {v}_{y}& {v}_{z}\end{array}\right]}^{T}$ (m/s) |

$\mathit{W}$ | Flight angular velocity vector ${\left[\begin{array}{ccc}p& q& r\end{array}\right]}^{T}$ (rad/s) |

${\mathit{W}}_{d}$ | Virtual angular velocity control term ${\left[\begin{array}{ccc}{p}_{d}& {q}_{d}& {r}_{d}\end{array}\right]}^{T}$ (rad/s) |

$\mathit{M}$ | Control torque vector ${\left[\begin{array}{ccc}{m}_{x}& {m}_{y}& {m}_{z}\end{array}\right]}^{T}$ ($\mathrm{N}\xb7\mathrm{m}$) |

${\sum}_{b}$ | Body-fixed frame coordinates $O{x}_{b}{y}_{b}{z}_{b}$ |

${\sum}_{i}$ | Inertial frame coordinates $O{x}_{i}{y}_{i}{z}_{i}$ |

${\sum}_{w}$ | Wind frame coordinates $O{x}_{w}{y}_{w}{z}_{w}$ |

${V}_{T}$ | Magnitude of velocity (m/s) |

${x}_{i},{y}_{i},{z}_{i}$ | Flight position in the inertial frame (m) |

${\mathit{D}}_{1}$ | Lumped acceleration disturbances vector ${\left[\begin{array}{ccc}{D}_{11}& {D}_{12}& {D}_{13}\end{array}\right]}^{T}$ ($\mathrm{m}/{\mathrm{s}}^{2}$) |

${\mathit{D}}_{2}$ | Lumped angular acceleration disturbances vector ${\left[\begin{array}{ccc}{D}_{21}& {D}_{22}& {D}_{23}\end{array}\right]}^{T}$ ($\mathrm{deg}/{\mathrm{s}}^{2}$) |

## Appendix A

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**Figure 1.**Flight model with inertial frame ${\Sigma}_{i}$, body-fixed frame ${\Sigma}_{b}$, and wind frame ${\Sigma}_{w}$.

**Figure 3.**Case 1: flight states: (

**a**) airspeed ${V}_{T}$; (

**c**) angle of attack $\alpha $; (

**e**) angle of sideslip $\beta $; and their tracking errors: (

**b**) tracking error of ${V}_{T}$; (

**d**) tracking error of $\alpha $; (

**f**) tracking error of $\beta $.

**Figure 4.**Case 1: control variables: (

**a**) force control ${u}_{V}$; (

**b**) virtual angular velocity ${\mathit{W}}_{d}$; (

**c**) angular acceleration control ${\mathit{u}}_{T}$; and deflection vector $\mathit{\delta}$: (

**d**) canard wings ${\delta}_{c}$; (

**e**) right elevon ${\delta}_{re}$; (

**f**) left elevon ${\delta}_{le}$, and (

**g**) rudder ${\delta}_{r}$.

**Figure 5.**Case 1: (

**a**) the disturbance ${D}_{11}$ and its observer ${\widehat{D}}_{11}$; (

**b**) the estimation error ${\tilde{D}}_{11}$; (

**c**) the disturbance ${D}_{12}$ and its observer ${\widehat{D}}_{12}$; (

**d**) the estimation error ${\tilde{D}}_{12}$; (

**e**) the disturbance ${D}_{13}$ and its observer ${\widehat{D}}_{13}$; (

**f**) the estimation error ${\tilde{D}}_{13}$.

**Figure 6.**Case 1: (

**a**) the disturbance ${D}_{21}$ and its observer ${\widehat{D}}_{21}$; (

**b**) the estimation error ${\tilde{D}}_{21}$; (

**c**) the disturbance ${D}_{22}$ and its observer ${\widehat{D}}_{22}$; (

**d**) the estimation error ${\tilde{D}}_{22}$; (

**e**) the disturbance ${D}_{23}$ and its observer ${\widehat{D}}_{23}$; (

**f**) the estimation error ${\tilde{D}}_{23}$.

**Figure 7.**Case 2: flight states: (

**a**) airspeed ${V}_{T}$; (

**c**) angle of attack $\alpha $; (

**e**) angle of sideslip $\beta $; and their tracking errors: (

**b**) tracking error of ${V}_{T}$; (

**d**) tracking error of $\alpha $; (

**f**) tracking error of $\beta $.

**Figure 8.**Case 2: control variables: (

**a**) force control ${u}_{V}$; (

**b**) virtual angular velocity ${\mathit{W}}_{d}$; (

**c**) angular acceleration control ${\mathit{u}}_{T}$; and deflection vector $\mathit{\delta}$: (

**d**) canard wings ${\delta}_{c}$; (

**e**) right elevon ${\delta}_{re}$; (

**f**) left elevon ${\delta}_{le}$, and (

**g**) rudder ${\delta}_{r}$.

**Figure 9.**Case 2: (

**a**) the disturbance ${D}_{11}$ and its observer ${\widehat{D}}_{11}$; (

**b**) the estimation error ${\tilde{D}}_{11}$; (

**c**) the disturbance ${D}_{12}$ and its observer ${\widehat{D}}_{12}$; (

**d**) the estimation error ${\tilde{D}}_{12}$; (

**e**) the disturbance ${D}_{13}$ and its observer ${\widehat{D}}_{13}$; (

**f**) the estimation error ${\tilde{D}}_{13}$.

**Figure 10.**Case 2: (

**a**) the disturbance ${D}_{21}$ and its observer ${\widehat{D}}_{21}$; (

**b**) the estimation error ${\tilde{D}}_{21}$; (

**c**) the disturbance ${D}_{22}$ and its observer ${\widehat{D}}_{22}$; (

**d**) the estimation error ${\tilde{D}}_{22}$; (

**e**) the disturbance ${D}_{23}$ and its observer ${\widehat{D}}_{23}$; (

**f**) the estimation error ${\tilde{D}}_{23}$.

**Figure 11.**Case 3: flight states: (

**a**) angle of attack $\alpha $; (

**c**) angle of sideslip $\beta $; and their tracking errors: (

**b**) tracking error of $\alpha $; (

**d**) tracking error of $\beta $.

**Figure 12.**Case 3: (

**a**) canard wings ${\delta}_{c}$; (

**b**) right elevon ${\delta}_{re}$; (

**c**) left elevon ${\delta}_{le}$; and (

**d**) rudder ${\delta}_{r}$.

Control | Unit | Min. | Max. | Rate Limit |
---|---|---|---|---|

Canard wings | deg | −55 | 25 | ±50 deg/s |

Right elevon | deg | −25 | 25 | ±50 deg/s |

Left elevon | deg | −25 | 25 | ±50 deg/s |

Rudder | deg | −30 | 30 | ±50 deg/s |

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**MDPI and ACS Style**

Han, Y.; Li, P.; Ma, J.
Vector-Coupled Flight Controller Design Based on Multivariable Backstepping Sliding Mode. *Symmetry* **2019**, *11*, 1225.
https://doi.org/10.3390/sym11101225

**AMA Style**

Han Y, Li P, Ma J.
Vector-Coupled Flight Controller Design Based on Multivariable Backstepping Sliding Mode. *Symmetry*. 2019; 11(10):1225.
https://doi.org/10.3390/sym11101225

**Chicago/Turabian Style**

Han, Yang, Peng Li, and Jianjun Ma.
2019. "Vector-Coupled Flight Controller Design Based on Multivariable Backstepping Sliding Mode" *Symmetry* 11, no. 10: 1225.
https://doi.org/10.3390/sym11101225