# Balancing Control of an Absolute Pressure Piston Manometer Based on Fuzzy Linear Active Disturbance Rejection Control

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Modeling of an Absolute Pressure Piston Manometer

#### 2.1. The Working Principle of the Absolute Pressure Piston Manometer

#### 2.2. Kinetic Analysis of the Weight Combination Section

_{v}resulting from the change in mass of the gas in the piston cylinder (F

_{i}and F

_{o}depending on the inlet and outlet processes), the resistance F

_{f}of the gas preventing the change in the height of the piston, the pressure F

_{q}lost by the gas leakage and the elastic force F

_{z}resulting from the compression of the gas in the cylinder. Take vertical upwards as the positive direction.

_{w}is the total mass of the combined portion of weights and h is the real-time height at which the piston rises. The process by which the gas is compressed and thus expands is similar to that of a spring, and the process by which the gas in the gap in the side of the piston prevents the piston from rising is similar to that of a damping system, so the resistance F

_{f}can be calculated using Equation (2) and the elastic force F

_{z}can be calculated using Equation (3).

_{0}is the cylinder volume corresponding to the primary gas intake and is a fixed value that can be calculated from the flow rate of the intake valve, and E is the modulus of elasticity of the gas media. At this point, according to the calculation Equation of the elastic force generated by the compression of the gas (5), the calculation Equation of the piston cylinder volume (6), and Equation (3), the calculation formula of the elasticity coefficient (7) can be deduced.

_{n}− h, where h

_{n}is the ideal height of the piston rise after n gas inlets without taking into account the gas compression, which needs to be determined according to the actual mass of the intake gas.

_{z}in Equation (1) can be expressed by Equation (8).

_{v}in Equation (1) is the driving force for the entire combined part of the piston and can be calculated using the actual gas state Equation (9) as well as the pressure Equation (10) in the closed vessel together.

_{t}/M. Taking the inlet process as an example, m

_{t}is the mass of gas entering the piston cylinder through the inlet valve, and M is the molar mass of the gas media. V

_{2}is the volume in the cylinder after the piston has risen to an expected height and is a fixed value.

#### 2.3. Flow Analysis of Switching Valves

_{in}is the gas inlet mass of the single switch of the inlet valve body, m

_{out}is the outlet mass of the single switch of the gas outlet valve body, C

_{d}is the flow coefficient, A

_{1}is the flow area of the valve opening, ΔP

_{1}is the pressure difference between the left and right of the inlet valve body, ΔP

_{2}is the pressure difference between the left and right of the inlet valve body, and ρ is the density of the gas medium in the gas cylinder.

**Remark 1.**

#### 2.4. Analytical Correction of Other Uncertainties

#### 2.4.1. Analytical Correction of Piston Effective Area

_{c}is the thermal deformation coefficient of the piston cylinder, a

_{e}is the thermal deformation coefficient of the piston, θ is the temperature of the piston system during actual operation, λ is the elastic deformation coefficient of the piston at the bottom of the piston, and P is the working pressure at the bottom of the piston rod.

#### 2.4.2. Analytical Correction of Gas Leakage Volumes

_{t}is the real-time gas leakage, d is the diameter of the checking piston, ΔP

_{0}the pressure difference between the two ends of the piston gap leakage surface, L is the length of the flow path, and δ the width of the annular gap.

#### 2.5. Establishment of the Differential Equilibrium Equations of the System

_{1t}is the real-time inlet mass flow rate of the inlet valve, and m

_{2t}is the real-time inlet mass flow rate of the outlet valve, both of which are input signals to the system and the output signal is the real-time rise height h of the piston, the others are fixed system parameters.

## 3. Design of the Controller

#### 3.1. Design of the LADRC

#### 3.1.1. Structure of the LADRC

#### 3.1.2. Design of the LTD

_{f}is the given value input value, h

_{1}is the softened height given value, and h

_{2}is the differential signal of h

_{1}, which are used later in the design of the state error feedback controller. r is the tracking speed factor.

#### 3.1.3. Design of the LESO

_{0}is the system gain; $f(h\left(t\right),\dot{h}\left(t\right),t)$ is the generalized perturbation of the system, abbreviated as f() for neatness, expanding f() to a new state variable. The resulting state variables have been chosen as follows.

_{1}tracks the estimated height of the piston rise in the actual manometer, z

_{2}tracks the estimated speed of the piston rise, and z

_{3}is the estimate of the total perturbation f(); and ${\beta}_{i}={\left[\begin{array}{ccc}{\beta}_{1}& {\beta}_{2}& {\beta}_{3}\end{array}\right]}^{T}$ is the observer gain matrix. The bandwidth method was proposed to rectify the observer gain parameters [35], let the bandwidth of the observer be ω

_{0}and try to configure all the eigenvalues of the observer as −ω

_{0}, i.e.,

_{1}, β

_{2}, and β

_{3}all become functions with respect to ω

_{0}, i.e.,

**Remark 2.**

^{T}, since rank(N) = 3, from which it follows that the expansion system is fully observable.

#### 3.1.4. Design of the LESF

_{1}is the error between the LESO’s estimate of the piston rise height, and the preset height input value, e

_{2}is the error between the LESO’s estimate of the piston rise speed and the differential signal of the input value; k

_{1}and k

_{2}are the feedback control parameters; u

_{0}is the state error feedback control quantity. For the consideration of the dynamic performance of the LESF link, the same choice of the pole configuration method is used to rectify the two feedback control parameters so that the closed-loop transfer function has a pole of −ω

_{c}and is solved as.

_{c}is the controller bandwidth, for most common engineering objects, ω

_{c}and ω

_{0}have a multiplicative relationship, i.e., ω

_{0}= (3~5)ω

_{c}[44], in this paper ω

_{0}= 4ω

_{c}is taken.

#### 3.2. Design of the FLADRC

#### 3.2.1. Structure of the FLADRC

_{c}and Δb

_{0}in the LADRC online and in real time so that the two parameters meet the requirements of the errors e

_{1}and e

_{2}at different times, thus improving the system’s immunity to disturbances and stability performance. The specific control structure of the fuzzy LADRC is shown in Figure 4.

#### 3.2.2. Design of the Fuzzy Controller

_{1}between h

_{1}and z

_{1}and its corresponding rate of change of deviation e

_{2}, and the theoretical domains of both are selected to be [−2 × 10

^{−3}, 2 × 10

^{−3}] and [−1 × 10

^{−3}, 1 × 10

^{−3}], respectively. The outputs of the fuzzy controller are chosen to be Δω

_{c}and Δb

_{0}, where the theoretical domains of the two are assumed to be Range1 and Range2, respectively, the specific values of which are shown in the subsection on Experimental Parameter Settings later on. Seven subsets of the fuzzy language are defined on the respective theoretical domains of the inputs and outputs: {negative big (NB), negative medium (PM), negative small (NS), zero (ZO), positive small (PS), positive medium (NM), positive big (PB)}. Gaussian-type affiliation functions with smooth transitions are chosen for the inputs and high-sensitivity triangular affiliation functions for the outputs.

_{c}, Δb

_{0}was developed as in in Table 1 and Table 2.

_{c}and b

_{0}are the initial control parameters of the LADRC obtained by genetic algorithm search.

#### 3.3. Stability Analysis

**Theorem 1.**

_{1}> 0, k

_{2}> 0, such that the tracking error of the closed-loop system (33) tends to 0.

**Proof of Theorem 1.**

_{1}and λ

_{2}, and diagonalize the matrix, then one obtains

_{e}, and when the eigenvalue has been determined, ${\Vert p\Vert}_{2}{\Vert {p}^{-1}\Vert}_{2}$ is a constant here, denoted by C. When $t\to \infty $, ${e}^{-{\lambda}_{1}t}\to 0$, then ${\Vert {e}^{{A}_{e}t}\Vert}_{2}\to 0$.

_{a}, when t > t

_{a}, ${\Vert \epsilon \Vert}_{2}<a$. Using Equation (36), then there is

_{1}> 0, and k

_{2}> 0, and ${\epsilon}_{i}={y}_{i}-{z}_{i}$, then Equation (33) is bounded. This also means that for a bounded input h, the output of the system is also bounded, and the system satisfies BIBO stability. □

## 4. Simulation Results and Analysis

#### 4.1. Experimental Parameter Settings

#### 4.2. Experimental Analysis of Stability Performance

_{s}, the integral IAE of the absolute value of the actual error, and the maximum overshoot rate 6% are chosen as performance indicators in this paper. The real-time height profile of the piston is shown in Figure 5, and the experimental results of the selected performance metrics are shown in Figure 6 and Table 6.

#### 4.3. Experimental Analysis of Interference Immunity Performance

_{s(n)}and the integral IAE

_{(n)}of the absolute value of the steady-state error are taken as the steady-state performance indicators so that they are compared with the experimental results in the previous section, and the corresponding rates of change ζ% and δ% of the two are derived, which are the system interference immunity performance indicators chosen in this paper. The specific results for t

_{s(n)}and IAE

_{(n)}are shown in Table 7 and Figure 8, and the corresponding rates of change are calculated in Table 8 and Table 9.

**Remark 3.**

#### 4.4. Experimental Analysis of Engineering Energy Consumption

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Analysis of the forces on the combined weight section under different working conditions. (

**a**) Piston rise; (

**b**) Piston drop.

**Figure 5.**Real-time piston height curves at three different pressure operating points. (

**a**) 0.1 MPa; (

**b**) 3 MPa; (

**c**) 6 MPa.

**Figure 7.**Real-time piston height curves after disturbance at three different pressure operating points. (

**a**) 0.1 MPa; (

**b**) 3 MPa; (

**c**) 6 MPa.

**Figure 9.**Control volume input curves at three different pressure operating points. (

**a**) 0.1 MPa; (

**b**) 3 MPa; (

**c**) 6 MPa.

e_{1} | e_{2} | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | PB | PB | PM | PM | PS | PS | ZO |

NM | PB | PB | PM | PM | PS | ZO | ZO |

NS | PM | PM | PM | PS | ZO | NS | NM |

ZO | PM | PS | PS | ZO | NS | NM | NM |

PS | PS | PS | ZO | NS | NS | NM | NM |

PM | ZO | ZO | NS | NM | NM | NM | NB |

PB | ZO | NS | NS | NM | NM | NB | NB |

e_{1} | e_{2} | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | NS | NS | ZO | ZO | PS | ZO | NB |

NM | PS | PS | PS | PS | PS | ZO | PS |

NS | PB | PB | PM | PS | PS | ZO | NS |

ZO | PB | PM | PM | PS | ZO | ZO | NS |

PS | PB | PM | PS | PS | NS | ZO | NS |

PM | PM | PS | PS | PS | NS | ZO | NS |

PB | NS | ZO | ZO | ZO | NS | ZO | NB |

Parameters | Value | Parameters | Value |
---|---|---|---|

C_{d} | 0.9 | a_{c} | 4.5 × 10^{−5} °C^{−1} |

ρ | 1.25 kg/m^{3} | a_{e} | 4.5 × 10^{−5} °C^{−1} |

λ | 7.1 × 10^{−7} MPa^{−1} | θ | 21 °C |

$\delta $ | 6 × 10^{−7} m | T | 294 k |

R | 296.8 J/(kg∙K) | μ | 1.741 × 10^{−2} N·s·m^{−2} |

Z | 0.292 | A_{1} | 7.85 × 10^{−9} m^{2} |

V_{0} | 3.7 × 10^{−7} m^{3} | A_{0} | 5 × 10^{−5} m^{2} |

Parameters | Value | ||
---|---|---|---|

0.1 MPa | 3 MPa | 6 MPa | |

m_{1} | 2.94 × 10^{−9} kg | 3.53 × 10^{−10} kg | 1.12 × 10^{−9} kg |

m_{2} | 2.23 × 10^{−9} kg | 1.93 × 10^{−9} kg | 2.73 × 10^{−9} kg |

m_{w} | 0.5 kg | 16 kg | 32 kg |

Controllers | Value | ||
---|---|---|---|

0.1 MPa | 3 MPa | 6 MPa | |

Kp | 55 | 65 | 90 |

PID | K_{p} = 26, K_{i} = 0.55, K_{1d} = 7.5 | K_{1p} = 30, K_{1i} = 0.5, K_{1d} = 7 | K_{1p} = 40, K_{1i} = 0.9, K_{1d} = 8 |

LADRC | r = 4, ω_{c} = 46, b_{0} = 0.22 | r = 4, ω_{c} = 77, b_{0} = 0.37 | r = 4, ω_{c} = 65, b_{0} = 0.25 |

FLADRC | Range1 = [−4, 4] Range2 = [−0.022, 0.022] | Range1 = [−7, 7] Range2 = [−0.037, 0.037] | Range1 = [−6, 6] Range2 = [−0.025, 0.025] |

Conditions | Kp | PID | LADRC | FLADRC |
---|---|---|---|---|

0.1 MPa | 23.36 | 19.58 | 14.87 | 12.37 |

3 MPa | 18.72 | 14.76 | 11.84 | 8.54 |

6 MPa | 20.98 | 15.79 | 12.04 | 8.67 |

Conditions | Kp | PID | LADRC | FLADRC |
---|---|---|---|---|

0.1 MPa | 27.62 | 22.48 | 17.09 | 13.26 |

3 MPa | 21.67 | 16.28 | 14.85 | 9.87 |

6 MPa | 25.33 | 22.03 | 13.07 | 10.41 |

Conditions | Kp | PID | LADRC | FLADRC |
---|---|---|---|---|

0.1 MPa | 18.23% | 18.95% | 14.92% | 7.19% |

3 MPa | 15.76% | 12.06% | 11.74% | 10.77% |

6 MPa | 21.97% | 13.29% | 8.33% | 6.22% |

Conditions | Kp | PID | LADRC | FLADRC |
---|---|---|---|---|

0.1 MPa | 19.21% | 10.44% | 3.35% | 1.51% |

3 MPa | 15.34% | 9.16% | 7.63% | 2.68% |

6 MPa | 16.47% | 9.79% | 4.44% | 1.24% |

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## Share and Cite

**MDPI and ACS Style**

Wu, H.; Zhai, X.; Gao, T.; Wang, N.; Zhao, Z.; Pang, G.
Balancing Control of an Absolute Pressure Piston Manometer Based on Fuzzy Linear Active Disturbance Rejection Control. *Actuators* **2023**, *12*, 275.
https://doi.org/10.3390/act12070275

**AMA Style**

Wu H, Zhai X, Gao T, Wang N, Zhao Z, Pang G.
Balancing Control of an Absolute Pressure Piston Manometer Based on Fuzzy Linear Active Disturbance Rejection Control. *Actuators*. 2023; 12(7):275.
https://doi.org/10.3390/act12070275

**Chicago/Turabian Style**

Wu, Hongda, Xianyi Zhai, Teng Gao, Nan Wang, Zongsheng Zhao, and Guibing Pang.
2023. "Balancing Control of an Absolute Pressure Piston Manometer Based on Fuzzy Linear Active Disturbance Rejection Control" *Actuators* 12, no. 7: 275.
https://doi.org/10.3390/act12070275