# Sliding Mode Control of Active Trailing-Edge Flap Based on Adaptive Reaching Law and Minimum Parameter Learning of Neural Networks

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Aeroelastic Equations

#### 2.1. Equations of Motions

_{1}, with the middle-line equation (the green dotted line) of the section fitted in terms of NACA 6315 profile and expressed as specific equations [23]; Figure 1c shows the equivalent cross-section with parameters at position l

_{1}.

_{1}in Figure 1a. Two piston rods of the two cylinders are connected to the fixed joint P

_{2}(in the rigid trailing-edged flap) through two connecting rods. P

_{1}is another hinge point, which is connected to the host structure of the blade through a shaft. The position point P

_{1}is above the midline of the blade section with slight deviation so that the swing angle of trailing-edge flap $\beta $ is within the range of –15 to 18 degrees. When the two piston rods synchronously move, in reverse, the trailing-edge flap structure can deflect. Note that the other two cylinders with the same function are installed in position l

_{2}.

_{0}), where the tip speed ratio λ = 1. The variable chord length c can be fitted as the form of Fourier Series Models according to an optimization design for the blade [24]. The composite properties are as follows: the sectional thickness is h; the ply thickness is 6.35 × 10

^{–4}m; the ply angle is denoted by θ

_{p}; the sectional element mass is m

_{b}decided by density ρ

_{b}= 1672 kg/m

^{3}; and the other elastic parameters are marked as G

_{12}= 3.5GPa, E

_{1}= 24.8 GPa, E

_{2}= 8.7 Gpa, and v

_{12}= 0.34.

_{p}]

_{6}in the top (above the chord and perpendicular to the chordwise direction) flange and [–θ

_{p}]

_{6}in the bottom flange.

_{r}of the trailing-edge flap in the spanwise direction, it is exactly a specified value l

_{r}= c/6. In addition, the expressions of Lift ${F}_{z}$ and Moment ${M}_{\theta}$ in Equations (3) and (4) are only used for the aerodynamic calculation of the trailing-edge flap structure, while the Beddoes–Leishman model based on fitted aerodynamic coefficients [23] is used to calculate the stall-induced aerodynamics of the other structural parts along the spanwise direction in this study.

#### 2.2. Calculation of Structural Damping

_{d}(s), B

_{d}(s), and C

_{d}(s) in $\overline{C}$, respectively [25].

_{s}) versus ply angles, ranging from 0° to 90°, are displayed in Figure 2.

_{s}into Equation (7) and implement the indicated integration in the Galerkin method, resulting in the matrix equations of motions:

## 3. Vibration Control based on Adaptive SMC

#### 3.1. Theory of Discrete Exponential Reaching Law

- when $\left|s\left(k\right)\right|>\frac{\epsilon T}{2-qT}$, we can obtain$$p>1-qT-\frac{\epsilon T\left(2-qT\right)}{\epsilon T}=1-qT-\left(2-qT\right)=-1$$
- when $\left|s\left(k\right)\right|<\frac{\epsilon T}{2-qT}$, we can obtain$$p<1-qT-\frac{\epsilon T\left(2-qT\right)}{\epsilon T}=1-qT-\left(2-qT\right)=-1$$
- when $\left|s\left(k\right)\right|=\frac{\epsilon T}{2-qT}$, we can obtain$$p=1-qT-\frac{\epsilon T\left(2-qT\right)}{\epsilon T}=1-qT-\left(2-qT\right)=-1$$

#### 3.2. Theory of SMC based on Adaptive Reaching Law (SMC/ARL)

_{e}. The system Equation (13) can be transformed as:

## 4. Analysis and Discussion

#### 4.1. The Controlled Displacements and Control Signals

#### 4.2. Pneumatic Transmission System

_{1}in Figure 1a, is mainly composed of a gas source processing unit, two proportional valves, a set of pneumatic pipeline systems, a four-bar linkage mechanism, and two pneumatic pressure cylinders. Please refer to the experiment section in the present study. It can be modeled as a fifth-order transfer function [31]; then, the Optimal Hankel Norm [32] is used to make a model reduction, so a reduced second-order model can be obtained, and with the Laplace Inverse Transform applied, the equation of the pneumatic driving behavior can be deduced as:

#### 4.2.1. Adaptive SMC based on Minimum Parameter Learning of Neural Networks

#### 4.2.2. The Superiority of ASMC/MPLNN Algorithm

## 5. Experimental Platform based on Pneumatic Transmission and Control System

_{1}illustrated. The platform consists of Siemens PLC controller hardware system and the execution unit of pneumatic driving. The PLC system is composed of a CPU module, an analog module, a Human Machine Interface (HMI), and displacement sensors. The pneumatic execution unit in position of l

_{1}is composed of a gas source processing unit, two proportional valves, a pneumatic pipeline system, pneumatic cylinders, and other accessory valve structures. The implementation scheme of the project is as follows:

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The circumferentially asymmetric stiffness (CAS)-based blade system with the rigid trailing-edged flap: (

**a**) The CAS-based blade system with the trailing-edged flap; (

**b**) The actual cross-section at position l

_{1}; (

**c**) The equivalent cross-section with parameters in position l

_{1}.

**Figure 3.**The uncontrolled twist/flap-wise displacements under conditions of ply angles ${\theta}_{0}=\mathsf{\pi}/6,\mathsf{\pi}/4,\mathsf{\pi}/3,\mathsf{\pi}/2$, respectively: (

**a**) The uncontrolled twist displacement; (

**b**) The uncontrolled flap-wise displacement.

**Figure 4.**The controlled displacements and control signals: (

**a**) The controlled displacements under conditions of ply angles with ${\theta}_{0}=\mathsf{\pi}/4,\mathsf{\pi}/3$, respectively; (

**b**) The control inputs (trailing-edge flap angles) and sliding mode functions.

**Figure 5.**The real-time tracking of trailing-edge flap angles and the real-time diaplacements controlled by pneumatic cylinders under conditions of ${\theta}_{0}=\mathsf{\pi}/4,\mathsf{\pi}/3$, respectively: (

**a**) The real-time tracking of trailing-edge flap angles; (

**b**) The real-time diaplacements controlled by pneumatic cylinders.

**Figure 6.**The real-time tracking of trailing-edge flap angles and the real-time displacements controlled by pneumatic cylinders using optimal proportional–integral–derivative (OPID) theory: (

**a**) The real-time tracking of trailing-edge flap angles; (

**b**) The real-time displacements controlled by pneumatic cylinders.

**Figure 7.**The noise contrast results: sound pressure level of noise between uncontrolled cases and the controlled cases with trailing-edge flap structure.

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**MDPI and ACS Style**

Liu, T.; Gong, A.; Song, C.; Wang, Y.
Sliding Mode Control of Active Trailing-Edge Flap Based on Adaptive Reaching Law and Minimum Parameter Learning of Neural Networks. *Energies* **2020**, *13*, 1029.
https://doi.org/10.3390/en13051029

**AMA Style**

Liu T, Gong A, Song C, Wang Y.
Sliding Mode Control of Active Trailing-Edge Flap Based on Adaptive Reaching Law and Minimum Parameter Learning of Neural Networks. *Energies*. 2020; 13(5):1029.
https://doi.org/10.3390/en13051029

**Chicago/Turabian Style**

Liu, Tingrui, Ailing Gong, Changle Song, and Yuehua Wang.
2020. "Sliding Mode Control of Active Trailing-Edge Flap Based on Adaptive Reaching Law and Minimum Parameter Learning of Neural Networks" *Energies* 13, no. 5: 1029.
https://doi.org/10.3390/en13051029