# Hybrid Adaptive Control for Tiltrotor Aircraft Flight Control Law Reconfiguration

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## Abstract

**:**

## 1. Introduction

## 2. Reconfigurability Evaluation

#### 2.1. Flight Dynamics Model of the Tiltrotor Aircraft

**F**

_{b}and

**M**

_{b}denote the resultant forces and moments acting on the aircraft’s center of mass, produced by the rotors, the fuselage, the empennage, and the control deflections; ${\dot{U}}_{b}={[\dot{u},\dot{v},\dot{w}]}^{\mathrm{T}}$ is the center of mass translational velocity derivative vector; ${\dot{\Omega}}_{b}={[\dot{p},\dot{q},\dot{r}]}^{\mathrm{T}}$ is the airframe angular velocity derivative vector; m denotes the aircraft mass;

**J**denotes the inertia tensor; ${g}_{e}={[0,0,g]}^{\mathrm{T}}$ is the gravitational acceleration vector in the earth frame; and $\dot{\Theta}={[\dot{\phi},\dot{\theta},\dot{\psi}]}^{\mathrm{T}}$ is the vector of derivatives of the Euler angles. ${T}_{be}^{\mathrm{\Theta}}$ is the transformation matrix from the earth frame to the aircraft body frame, and ${T}_{Eb}^{\mathrm{\Theta}}$ is that from the body frame to the Euler frame, where the Euler angles are defined:

#### 2.1.1. Rotor Forces and Moments

_{b}, K, r

_{0}, and r

_{1}are the number of blades, azimuth stations, blade root cut, and tip loss. Angles $\beta $ and $\psi $ are blade flapping and azimuth angle. Components dF

_{p}and dF

_{t}are the blade element perpendicular and tangential force elements, which can be represented by the element lift and drag as follows:

_{p}and U

_{t}denote the velocity in-plane and normal components seen by the rotor. These components can be evaluated by the advance ratio and inflow ratio. The rotor dimensionless induced velocity is governed by the Pitt–Peters’ dynamic inflow model:

#### 2.1.2. Airframe Forces and Moments

_{r}, the lateral reference length b

_{r}(the wingspan), the longitudinal reference length ${\overline{c}}_{r}$ (mean aerodynamic cord, MAC), and the aerodynamic coefficients:

_{D}, side force C

_{Y}, lift C

_{L}, rolling moment C

_{l}, pitching moment C

_{m}, and yawing moment C

_{n}are nonlinearly interpolated by the corresponding flight state variables and control surface deflections:

#### 2.2. Model Trimming and Control Effectiveness

**Rolling Axis**. As the airspeed increases (during the conversion from the helicopter to airplane mode), the effectiveness of differential collective pitch (${\delta}_{dcol}$) on rolling moment decreases rapidly. The differential collective transitions from controlling rolling to that of yawing (see Figure 3a,c). In the meantime, the rolling effectiveness of lateral cyclic pitch (${\delta}_{lat}$) also decreases with the forward tilting of the rotor nacelle. On the contrary, the rudder (${\delta}_{rud}$) contributes to an increasing rolling moment as airspeed increases, given the buildup of dynamic pressure. Also, it is worth noting the cross-coupling effect of differential longitudinal cyclic pitch (${\delta}_{dlon}$), which is the primary yawing control during helicopter mode, on both rolling and yawing channels. Coupling on the rolling axis is mainly because of the unsymmetric thrust variation during differential longitudinal flapping, which causes the difference in the effective blade element angle of attack on each rotor.**Pitching Axis**. Pitching controls are of great importance for tiltrotor aircraft conversion, during which the primary longitudinal controls are the longitudinal cyclic pitch (${\delta}_{lon}$) and the elevator (${\delta}_{ele}$), see Figure 3b. The elevator pitching moment increases significantly with airspeed as dynamic pressure builds up, whereas the decrease in longitudinal cyclic pitching moment is much less obvious with respect to the airspeed. The collective pitch (${\delta}_{col}$) also has some influence on the pitching moment because of the relative position of the thrust vector with respect to the aircraft’s center of gravity.**Yawing Axis**. Primary controls are differential collective pitch (${\delta}_{dcol}$) and the rudder, among which the former presents a much more controlling moment than the latter, see Figure 3c. The collective pitch, however, is coupling in both heading control and lateral control.

**Figure 3.**Effectiveness on each of the control axes: (

**a**) control effectiveness on the rolling axis, (

**b**) control effectiveness on the pitching axis, and (

**c**) control effectiveness on the yawing axis.

#### 2.3. Redundant Control Allocation Based on Optimal Control Effectiveness

**x**, Equation (13) can be written in the linear form as follows:

_{0}**x**can be set to zero in the above equation. In this case, given the desired angular velocity derivatives ${\dot{x}}_{ang}={\omega}_{dd}$, the linear system

**u**, given

_{min}**s**being the solution of:

_{0}**u**by

_{min}**u**is that it is composed of the minimum combined control surface deflections, which produce the desired angular derivatives ${\omega}_{dd}$.

_{min}**u**is the optimal mixed close-loop control variables that affect the three angular channels, in this case, ${\left[{\delta}_{dcol},{\delta}_{lat},{\delta}_{lon},{\delta}_{dlon},{\delta}_{ail},{\delta}_{ele},{\delta}_{rud}\right]}_{\mathrm{min}}^{\mathrm{T}}$, based on the so-called optimal control effectiveness, and ${\omega}_{dd}$ should be the angular velocity acceleration commands produced by the most inner-loop angular rate controller. For controller implementation, which will be discussed in the next section, the ${B}_{m}^{-}$ is the optimal effectiveness control transformation matrix that allocates three angular motion channel commands to actual vehicle controls. In this case, we have mixed the vehicle’s control surfaces into three generalized controls, and the angular equations of motion under change of variables of controls become

_{min}## 3. Basic Flight Controller for Normal Conditions

#### 3.1. Feedback Linearization Control for the Inner Angular Rate Loop

**h**along

**f**and

**g**.

**u**exists in the $\rho $th derivative of the output equation:

**u**as follows:

**v**.

#### 3.2. Mode-Conversion Controller and Forward/Vertical Speed Decoupling

#### 3.3. Simulation of the Mode-Conversion Flight

## 4. Fault-Tolerant Flight Control Design

#### 4.1. Multi-Model Switching Adaptive Control for Predictable Faults

#### 4.1.1. Fault Model and Controller Set

**C**:

#### 4.1.2. Reference Model Performance Index of the Supervisory Mechanism

_{i}denotes the performance index of the ith model ${\mathbb{M}}_{i}$, ${\epsilon}_{i}(t)={x}_{i}(t)-x(t)$ is the state error between reference model ${\mathbb{M}}_{i}$ and the aircraft, ${\Vert \xb7\Vert}_{2}$ denotes the Euclidean norm, and $\alpha $ and $\beta $ are the weights of the instant error term and the integral error term, respectively. The adopted index has the ability to balance the transient and steady performance of the switching mechanism. The integral term ${\int}_{0}^{t}{e}^{-\lambda (t-\tau )}}\cdot {\Vert {\epsilon}_{i}(\tau )\Vert}_{2}\mathrm{d}\tau $ requires a certain amount of time until it can provide significant information about the fault system. Thus, at the beginning of the adaptation process, the instant error term ${\Vert {\epsilon}_{i}(t)\Vert}_{2}$ is more reliable; moreover, this term is also able to reflect the transient characteristics of the adaptive system. If the transient term is used solely, however, the adaptation system will experience an excessively rapid switching and cause the close-loop system unstable. Therefore, an integral term has to be introduced. As the system reaches a steady state, the transient term approaches zero, and the integral term is able to provide accurate information about the system. The parameter $\lambda $ is the factor of the long-term forgetting term. For engineering practice, a forgetting factor is always desirable because, with long-term integration, the state measuring error can cause the state error term quadratically non-integrable. In the meantime, the integral term can also effectively suppress the influence of the output of the previously switched-out model.

#### 4.1.3. Mode-Conversion Flight Simulation under Predictable Fault Condition

#### 4.2. Direct Adaptive FTC for Unpredictable Faults

#### 4.2.1. Direct Adaptive Control for Unpredictable Faults

**K**

_{x}(t),

**K**

_{u}(t), and

**K**

_{e}(t) are time-varying gain matrices, and these factors are to be adjusted online. The adaptive law of the gain matrices is

#### 4.2.2. Hybrid Control Scheme for Tiltrotor Aircraft Control Law Reconfiguration

- The model set of the MMAC scheme is derived a priori, and controllers are also designed for each of the failure models to maintain the flight performance to the maximum extent. When the actuator floating or saturation at minimum or maximum position occurs, the MMAC controller is switched to by the online system switching logic. Through the comparison of the actual aircraft states and each of the failure model states, the supervisory mechanism of the MMAC controller will allocate the control authority to one of the MMAC controllers corresponding to the proper fault model.
- When the aircraft encounters a failure such as control surface damage and stuck, the switching logic may not match the current fault plant to any of the modeled failures. In this case, the SAC controller for unpredictable faults is activated. Utilizing the direct adaptive methodology, the aircraft is forced to track the reference model and does not need any pre-established actuator failure model.

#### 4.2.3. Simulation

## 5. Conclusions and Discussion

- (1)
- For the baseline controller under normal situations, an inner angular rate loop controller is adopted based on the model inversion technique. Outer attitude and velocity loops are designed considering the decoupling of the height/velocity channel. During model conversion flight simulation, the aircraft is able to track the velocity command and thus the prescribed conversion path with ignorable static error. The maximum height loss during conversion is no more than 2 m. Simulation results have shown good performance of the full-envelope controller of the healthy system.
- (2)
- Predictable flight control law reconfiguration strategy is derived based on a multiple-model switching adaptive control scheme, which is the inner loop controller of the layered hybrid direct adaptive reconfigurable controller. The possible actuator faults of tiltrotor aircraft are analyzed and fault modeling is carried out. The controller of the fault model is designed and the recognition of the predictable fault and the smooth switching of the controller are implemented by using the appropriate performance index and the switching logic of the controller. Simulation is performed, taking the elevator floating as an example of a predictable fault. The elevator floating fault is injected into the system during the late phase of the conversion mode to verify the effectiveness of the MMAC-FTC. Results show that the presented scheme can detect the fault rapidly and switch to the corresponding controller. Performance during conversion can be recovered, and the aircraft is able to track the conversion path with a minimum height loss similar to that of the healthy plant.
- (3)
- The outer layer of the hybrid adaptive scheme is derived by simple adaptive control. Under unpredictable fault conditions, the degree of damage cannot be modeled a priori. Thus, a reference model of the healthy aircraft is prescribed. The fault plant is forced to track the state trajectory of the healthy reference model, as long as there is sufficient surplus control effectiveness. Simulation is performed under the condition of 80% elevator damage during the late stage of conversion. Results show that the fault can be detected and the controller is able to stabilize the aircraft and complete the conversion phase. Although performance reduction indeed exists as evidenced by a much larger height variation than in a healthy case, the aircraft can still track the correct conversion path.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Trimming results: (

**a**) trimmed nacelle angle, (

**b**) trimmed control deflections, and (

**c**) trimmed pitching angle.

**Figure 7.**Mode-conversion flight simulation results: (

**a**) nacelle tilting angle command, (

**b**) airspeed response, (

**c**) descent rate response, (

**d**) relative height response, (

**e**) longitudinal states responses, and (

**f**) controller outputs.

**Figure 9.**Mode-conversion flight simulation results under an elevator floating condition with MMAC fault-tolerant controller: (

**a**) airspeed responses, (

**b**) descent rate responses, (

**c**) relative height responses, (

**d**) longitudinal states responses, (

**e**) collective controller outputs, (

**f**) longitudinal controller outputs, and (

**g**) model indexes and switching logic decision.

**Figure 13.**Nonlinear mode-conversion flight simulation under unpredictable elevator failure: (

**a**) airspeed responses, (

**b**) descent rate responses, (

**c**) relative height responses, (

**d**) longitudinal states responses, (

**e**) collective controller outputs, and (

**f**) longitudinal controller outputs.

Forward/Vertical | Lateral | Longitudinal | Heading | |
---|---|---|---|---|

Helicopter mode | ${\delta}_{lon}/{\delta}_{col}$ | ${\delta}_{dcol}$ | ${\delta}_{lon}$ | ${\delta}_{dlon}$ |

Conversion mode | ${\delta}_{lon},{\delta}_{col}/{\delta}_{col},{\delta}_{lon}$ | ${\delta}_{dcol},{\delta}_{ail}$ | ${\delta}_{lon},{\delta}_{ele}$ | ${\delta}_{dlon},{\delta}_{rud}$ |

Airplane mode | ${\delta}_{col}$ | ${\delta}_{ail},{\delta}_{dlon}$ | ${\delta}_{ele},{\delta}_{lon}$ | ${\delta}_{rud},{\delta}_{dcol}$ |

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## Share and Cite

**MDPI and ACS Style**

Wen, J.; Song, Y.; Wang, H.; Han, D.; Yang, C.
Hybrid Adaptive Control for Tiltrotor Aircraft Flight Control Law Reconfiguration. *Aerospace* **2023**, *10*, 1001.
https://doi.org/10.3390/aerospace10121001

**AMA Style**

Wen J, Song Y, Wang H, Han D, Yang C.
Hybrid Adaptive Control for Tiltrotor Aircraft Flight Control Law Reconfiguration. *Aerospace*. 2023; 10(12):1001.
https://doi.org/10.3390/aerospace10121001

**Chicago/Turabian Style**

Wen, Jiayu, Yanguo Song, Huanjin Wang, Dong Han, and Changfa Yang.
2023. "Hybrid Adaptive Control for Tiltrotor Aircraft Flight Control Law Reconfiguration" *Aerospace* 10, no. 12: 1001.
https://doi.org/10.3390/aerospace10121001