# Robust Liquid Level Control of Quadruple Tank System: A Nonlinear Model-Free Approach

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## Abstract

**:**

## 1. Introduction

- i.
- The proposed tracking differentiator is used in the control unit to provide the error signal and its derivative.
- ii.
- The proposed super twisting sliding mode controller (STC-SM), nonlinear proportional derivative (NLPD), and modified nonlinear state error feedback (MNLSEF) are used as nonlinear state error feedback (NLSEF) instead of the conventional NLSEF proposed by [18].
- iii.
- The modified nonlinear extended state observer (MNLESO) and fal function ESO are used instead of the linear extended state observer (LESO) [19].

## 2. Problem Statement

## 3. Mathematical Modeling of Nonlinear Four-Tank System

_{1}(M

_{1}) draws water from the source tank and distributes it to tank

_{1}and tank

_{4}via valve

_{1}. Similarly, the water pump

_{2}(M

_{2}) draws water from the source tank and distributes it to tank

_{2}and tank

_{3}via valve

_{2}. It is important to note the amount of water delivered to the tanks depends on the valve constant (i.e., ${\gamma}_{1}$ and ${\gamma}_{2}$). The nonlinear mathematical model of the four-tank system is given in [1]:

_{1}and M

_{2,}respectively;$g$ is the acceleration of gravity; ${K}_{c}$ is the calibrated constant;${k}_{F{T}_{1}}$ and ${k}_{F{T}_{2}}$ are pump proportionality constants; and ${d}_{1}$ and ${d}_{2}$ are the causes of the exogenous disturbances by the flow rate.

## 4. The proposed Modified Active Disturbance Rejection Control (MADRC)

#### 4.1. The First MADRC Scheme

- (a)
- The proposed tracking differentiator (TD)

- (b)
- The proposed nonlinear controllers

Nonlinear Controller | Mathematical Expression | |
---|---|---|

STC-SM | $\left\{\begin{array}{c}{\varsigma}_{i}=\kappa {\tilde{e}}_{i1}+{\dot{\tilde{e}}}_{i1}\\ {u}_{0{i}_{STC-SM}}={\kappa}_{i}{\left|{\varsigma}_{i}\right|}^{{\U0001d4c5}_{i}}sign\left({\varsigma}_{i}\right)+{\xi}_{i}\mathrm{tank}\left(\frac{{\varsigma}_{i}}{\delta}\right)\end{array}\right.$ | (4) |

NLPD | $\left\{\begin{array}{c}{u}_{0{i}_{NLPD}}={u}_{i1}+{u}_{i2}\\ {u}_{i1}=\frac{{k}_{i1}}{1+\mathrm{exp}\left({\tilde{e}}_{i1}{}^{2}\right)}{\left|{\tilde{e}}_{i1}\right|}^{{\mathsf{\alpha}}_{i1}}sign\left({\tilde{e}}_{i1}\right)\\ {u}_{i2}=\frac{{k}_{i2}}{1+\mathrm{exp}\left({\dot{\tilde{e}}}_{i1}{}^{2}\right)}{\left|{\dot{\tilde{e}}}_{i1}\right|}^{{\mathsf{\alpha}}_{i2}}sign\left({\dot{\tilde{e}}}_{i1}\right)\end{array}\right.$ | (5) |

- (c)
- The proposed Modified Nonlinear Extended State Observer (MNLESO)

MNLESO Schemes | Mathematical Expression | |
---|---|---|

$1\mathrm{st}$ scheme | $\left\{\begin{array}{c}\left\{\begin{array}{c}{\dot{z}}_{i1}\left(t\right)={z}_{i2}\left(t\right)+{\beta}_{i1}{\widehat{e}}_{i1}\left(t\right)\\ {\dot{z}}_{i2}\left(t\right)={\beta}_{i2}{\widehat{e}}_{i2}\left(t\right)\end{array}\right.\\ \left\{\begin{array}{c}{\widehat{e}}_{i1}\left(t\right)=sign\left({e}_{i}\right){\left|{e}_{i}\right|}^{{\U0001d4b6}_{i}}+{e}_{i}\\ {\widehat{e}}_{i2}\left(t\right)=sign\left({e}_{i}\right){\left|{e}_{i}\right|}^{2{\U0001d4b6}_{i}-1}+{e}_{i}\end{array}\right.\end{array}\right.$ | (6) |

$2\mathrm{nd}$ scheme | $\left\{\begin{array}{c}\left\{\begin{array}{c}{\dot{z}}_{i1}\left(t\right)={z}_{i2}\left(t\right)+{\beta}_{i1}{\widehat{e}}_{i1}\left(t\right)\\ {\dot{z}}_{i2}\left(t\right)={\beta}_{i2}{\widehat{e}}_{i2}\left(t\right)\end{array}\right.\\ \left\{\begin{array}{c}{\widehat{e}}_{i1}\left(t\right)=sign\left({e}_{i}\right){\left|{e}_{i}\right|}^{{\U0001d4b6}_{i}}+{\mathcal{A}}_{i}{e}_{i}\\ {\widehat{e}}_{i2}\left(t\right)=sign\left({e}_{i}\right){\left|{e}_{i}\right|}^{\raisebox{1ex}{${\U0001d4b6}_{i}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+{\mathcal{A}}_{i}{e}_{i}\end{array}\right.\end{array}\right.$ | (7) |

#### 4.2. The Second MADRC Scheme

- (a)
- The proposed Tracking Differentiator TD

- (b)
- The modified NLSEF (MNLSEF)

- (c)
- The proposed $fal$-function ESO

$\mathit{f}\mathit{a}\mathit{l}$ ESO Schemes | Mathematical Expression | |
---|---|---|

$Symmetricalfal$(S-$fal$ADRC) | $\left\{\begin{array}{c}\left\{\begin{array}{c}{\dot{z}}_{i1}\left(t\right)={z}_{i2}\left(t\right)+{\beta}_{i1}{\widehat{e}}_{i1}\left(t\right)\\ {\dot{z}}_{i2}\left(t\right)={\beta}_{i2}{\widehat{e}}_{i1}\left(t\right)\end{array}\right.\\ {\widehat{e}}_{i1}\left(t\right)=fal\left({e}_{i},{\alpha}_{{i}_{ESO}},{\delta}_{{i}_{ESO}}\right)\end{array}\right.$ | (11) |

$Differentfal$(D-$fal$ADRC) | $\left\{\begin{array}{c}\left\{\begin{array}{c}{\dot{z}}_{i1}\left(t\right)={z}_{i2}\left(t\right)+{\beta}_{i1}{\widehat{e}}_{i1}\left(t\right)\\ {\dot{z}}_{i2}\left(t\right)={\beta}_{i2}{\widehat{e}}_{i2}\left(t\right)\end{array}\right.\\ \left\{\begin{array}{c}{\widehat{e}}_{i1}\left(t\right)=fal\left({e}_{i1},{\alpha}_{i{1}_{ESO}},{\delta}_{i{1}_{ESO}}\right)\\ {\widehat{e}}_{i2}\left(t\right)=fal\left({e}_{i2},{\alpha}_{i{2}_{ESO}},{\delta}_{i{2}_{ESO}}\right)\end{array}\right.\end{array}\right.$ | (12) |

## 5. The Convergence of the Proposed STC-SM

**Assumption**

**1.**

**Theorem**

**1.**

**Proof.**

## 6. Simulation Results and Discussion

**Table 4.**Sampled parameters of the four-tank system [1].

Parameters | Description | Value | Unit | Reference |
---|---|---|---|---|

${h}_{{1}_{des}}$ | The water level of ${\mathrm{tank}}_{1}$ | $16$ | $\mathrm{cm}$ | Estimated |

${h}_{{2}_{des}}$ | The water level of ${\mathrm{tank}}_{2}$ | $13$ | $\mathrm{cm}$ | Estimated |

${h}_{3}$ | The water level of ${\mathrm{tank}}_{3}$ | $9.5$ | $\mathrm{cm}$ | Estimated |

${h}_{4}$ | The water level of ${\mathrm{tank}}_{4}$ | $6$ | $\mathrm{cm}$ | Estimated |

${a}_{1}$ | The cross-section area of the outlet hole of ${\mathrm{tank}}_{1}$ | $0.071$ | ${\mathrm{cm}}^{2}$ | [1] |

${a}_{2}$ | The cross-section area of the outlet hole of ${\mathrm{tank}}_{2}$ | $0.056$ | ${\mathrm{cm}}^{2}$ | [1] |

${a}_{3}$ | The cross-section area of the outlet hole of ${\mathrm{tank}}_{3}$ | $0.071$ | ${\mathrm{cm}}^{2}$ | [1] |

${a}_{4}$ | The cross-section area of the outlet hole of ${\mathrm{tank}}_{4}$ | $0.056$ | ${\mathrm{cm}}^{2}$ | [1] |

${A}_{1}$ | The cross-section area of ${\mathrm{tank}}_{1}$ | $28$ | ${\mathrm{cm}}^{2}$ | [1] |

${A}_{2}$ | The cross-section area of ${\mathrm{tank}}_{2}$ | $32$ | ${\mathrm{cm}}^{2}$ | [1] |

${A}_{3}$ | The cross-section area of ${\mathrm{tank}}_{3}$ | $28$ | ${\mathrm{cm}}^{2}$ | [1] |

${A}_{4}$ | The cross-section area of ${\mathrm{tank}}_{4}$ | $32$ | ${\mathrm{cm}}^{2}$ | [1] |

${\gamma}_{1}$ | The ratio of the flow in the ${\mathrm{valve}}_{1}$ | $0.7$ | $\mathrm{unitless}$ | [1] |

${\gamma}_{2}$ | The ratio of the flow in the ${\mathrm{valve}}_{2}$ | $0.6$ | $\mathrm{unitless}$ | [1] |

${k}_{F{T}_{1}}$ | Pump proportionality constant | $3.33$ | $\frac{{\mathrm{cm}}^{3}}{\mathrm{volt}.\mathrm{s}}$ | [1] |

${k}_{F{T}_{2}}$ | Pump proportionality constant | $3.35$ | $\frac{{\mathrm{cm}}^{3}}{\mathrm{volt}.\mathrm{s}}$ | [1] |

$g$ | Gravity constant | $981$ | $\mathrm{volt}/\mathrm{cm}$ | [1] |

${K}_{c}$ | The calibrated constant | $1$ | $\mathrm{cm}/{\mathrm{s}}^{2}$ | Estimated |

${h}_{max}$ | The maximum height | $25$ | $\mathrm{cm}$ | Estimated |

Scheme | TD | SEF | ESO | ||
---|---|---|---|---|---|

Linear active disturbance rejection control LADRC | - | LPID that can be given as | Linear extended state observer (LESO) [19] | ||

${u}_{{0}_{PID}}={k}_{p}{\tilde{e}}_{i}+{k}_{i}{{\displaystyle \int}}_{0}^{T}{\tilde{e}}_{i}dt+{k}_{d}\frac{d{\tilde{e}}_{i}}{dt}$ | (24) | $\left\{\begin{array}{c}{\dot{z}}_{i1}\left(t\right)={z}_{i2}\left(t\right)+{\beta}_{i1}\left({e}_{i}\right)\\ {\dot{z}}_{i2}\left(t\right)={\beta}_{i2}\left({e}_{i}\right)\end{array}\right.$ | (25) | ||

ADRC | - | NLSEF [18] | |||

$\left\{\begin{array}{c}{u}_{0{i}_{NLSEF}}=fal\left({\tilde{e}}_{i},{\alpha}_{i1},{\delta}_{i1}\right)\\ fal\left({\tilde{e}}_{i},{\alpha}_{i1},{\delta}_{i1}\right)=\left\{\begin{array}{c}\frac{{\tilde{e}}_{i}}{{\delta}_{i1}{}^{1-{\alpha}_{i1}}},\left|x\right|{\delta}_{i1}\\ {\left|{\tilde{e}}_{i}\right|}^{{\alpha}_{i1}}sign\left({\tilde{e}}_{i}\right),\left|x\right|\ge {\delta}_{i1}\end{array}\right.\end{array}\right.$ | (26) | ||||

Improved active disturbance rejection control (IADRC) | - | Improved nonlinear state error feedback (INLSEF) [29] | Sliding mode extended state observer (SMESO) [30] | ||

$\left\{\begin{array}{c}{u}_{i1}={k}_{i11}+\frac{{k}_{i12}}{1+\mathrm{exp}\left({\mu}_{i1}{\tilde{e}}_{i}{}^{2}\right)}{\left|{\tilde{e}}_{i}\right|}^{{\alpha}_{i1}}sign\left({\tilde{e}}_{i}\right)\\ {u}_{INLPID}={u}_{i1}\end{array}\right.$ | (27) | $\left\{\begin{array}{c}{\dot{z}}_{i1}\left(t\right)={z}_{i2}\left(t\right)+{\beta}_{i1}\left(k\left({e}_{i}\left(t\right)\right){e}_{i}\left(t\right)\right)\\ {\dot{z}}_{i2}\left(t\right)={\beta}_{i1}(k\left({e}_{i}\left(t\right)\right){e}_{i}\left(t\right)\\ k\left({e}_{i}\left(t\right)\right)={k}_{{\alpha}_{i}}{\left|{e}_{i}\right|}^{{\alpha}_{i}-1}+{k}_{\beta}{\left|{e}_{i}\right|}^{{\beta}_{i}}\end{array}\right.$ | (28) | ||

$\mathrm{where}{\tilde{e}}_{i}$$\mathrm{is}\mathrm{the}\mathrm{error}\mathrm{and}({k}_{i11},{k}_{i12},{\mu}_{i1},{\alpha}_{i1}$) are the controller parameters | $\mathrm{where}k\left({e}_{i}\left(t\right)\right)$ is a nonlinear function | ||||

S fal-ADRC | - | NLSEF Equation (26) | Symmetrical fal ESO Equation (11) | ||

D fal-ADRC | - | Different fal ESO Equation (12) | |||

$1\mathrm{st}$ MADRC scheme (NLP-ADRC) and (STC-ADRC) | Equation (3a) | Proposed NLPD Equation (5) | $\mathrm{MNLESO}1\mathrm{st}$ scheme Equation (7) | ||

Equation (3b) | Proposed STC-SM Equation (6) | $\mathrm{MNLESO}2\mathrm{nd}$ scheme Equation (8) | |||

$2\mathrm{nd}$ MADRC scheme (D fal-ADRC-TD) | Equation (8) | Proposed MNLSEF Equations (9) and (10) | Different fal ESO Equation (12) |

- A.
- Case study 1. Exogenous disturbance

- B.
- Case study 2. Parameter uncertainty

- C.
- Case study 3. Reference tracking

**Remark**

**1.**

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The water level when using the $1\mathrm{st}$ MADRC scheme. (

**a**) $1\mathrm{st}$ subsystem. (

**b**) $2\mathrm{nd}$ subsystem.

**Figure 5.**The water level when using the $2\mathrm{nd}$ MADRC scheme. (

**a**) $1\mathrm{st}$ subsystem. (

**b**) $2\mathrm{nd}$ subsystem.

**Figure 6.**The control signal when using the $1\mathrm{st}$ MADRC scheme. (

**a**) $1\mathrm{st}$ subsystem. (

**b**) $2\mathrm{nd}$ subsystem.

**Figure 7.**The control signal when using the $2\mathrm{nd}$ MADRC scheme. (

**a**) $1\mathrm{st}$ subsystem. (

**b**) $2\mathrm{nd}$ subsystem.

**Figure 8.**The water level in tank

_{1}with uncertainty in the outlet hole ${\mathrm{a}}_{1}$. Using the $2\mathrm{nd}$ MADRC scheme.

**Figure 9.**The water level under different time-varying references using the $1\mathrm{st}$ MADRC scheme. (

**a**) $1\mathrm{st}$ subsystem. (

**b**) $2\mathrm{nd}$ subsystem.

**Figure 10.**The water level under different time-varying references using the $2\mathrm{nd}$ MADRC scheme. (

**a**) $1\mathrm{st}$ subsystem. (

**b**) $2\mathrm{nd}$ subsystem.

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

LPID | ${k}_{{p}_{1}}$ | $18.6300$ | ${k}_{{p}_{2}}$ | $26.6550$ |

${k}_{{i}_{1}}$ | $0.0002$ | ${k}_{{i}_{2}}$ | $0.0024$ | |

${k}_{{d}_{1}}$ | $2.5300$ | ${k}_{{d}_{2}}$ | $3.0500$ | |

LESO | ${\omega}_{01}$ | $43.130000$ | ${\omega}_{02}$ | $15.910000$ |

${b}_{01}$ | $0.124875$ | ${b}_{02}$ | $0.094219$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

NLSEF | ${\alpha}_{1}$ | $0.7763$ | ${\alpha}_{2}$ | $0.4167$ |

${\delta}_{1}$ | $0.0140$ | ${\delta}_{2}$ | $1.8958$ | |

LESO | ${\omega}_{01}$ | $149.345000$ | ${\omega}_{02}$ | $173.005000$ |

${b}_{01}$ | $1.706625$ | ${b}_{02}$ | $1.287656$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

INLP (INSEF) | ${k}_{111}$ | $6.2650$ | ${k}_{212}$ | $7.0400$ |

${k}_{121}$ | $1.4124$ | ${k}_{222}$ | $0.0142$ | |

${\mu}_{11}$ | $8.5790$ | ${\mu}_{22}$ | $5.6130$ | |

${\alpha}_{11}$ | $0.6812$ | ${\alpha}_{22}$ | $0.6625$ | |

SMESO | ${k}_{{\alpha}_{1}}$ | $0.3675$ | ${k}_{{\alpha}_{2}}$ | $0.8579$ |

${\alpha}_{1}$ | $0.9733$ | ${\alpha}_{2}$ | $0.6265$ | |

${k}_{{\beta}_{1}}$ | $0.6713$ | ${k}_{{\beta}_{2}}$ | $0.6812$ | |

${\beta}_{1}$ | $0.2221$ | ${\beta}_{2}$ | $0.7062$ | |

${\omega}_{01}$ | $133.200000$ | ${\omega}_{02}$ | $163.840000$ | |

${b}_{01}$ | $0.666000$ | ${b}_{02}$ | $0.502500$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

NLPD | ${k}_{11}$ | $12.676500$ | ${k}_{12}$ | $24.414000$ |

${\mathsf{\alpha}}_{11}$ | $0.351000$ | ${\mathsf{\alpha}}_{12}$ | $0.453100$ | |

${k}_{21}$ | $22.057500$ | ${k}_{22}$ | $21.553500$ | |

${\mathsf{\alpha}}_{21}$ | $0.931500$ | ${\mathsf{\alpha}}_{22}$ | $0.69130$ | |

TD | $R$ | $55.380000$ | ${a}_{2}$ | $7.842000$ |

${a}_{1}$ | $0.142000$ | $-$ | $-$ | |

$1st$ MNLESO scheme | ${\omega}_{01}$ | $341.190000$ | ${\omega}_{02}$ | $538.050000$ |

${\U0001d4b6}_{1}$ | $0.696800$ | ${\U0001d4b6}_{2}$ | $0.587100$ | |

${b}_{01}$ | $2.414250$ | ${b}_{02}$ | $1.821562$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

STC-SM | ${\kappa}_{1}$ | $0.552000$ | ${\kappa}_{2}$ | $0.587800$ |

${\xi}_{1}$ | $3.405000$ | ${\xi}_{2}$ | $3.889500$ | |

${\U0001d4c5}_{1}$ | $0.704480$ | ${\U0001d4c5}_{2}$ | $0.695040$ | |

$\delta $ | $7.631000$ | $-$ | $-$ | |

TD | $R$ | $188.580000$ | ${a}_{2}$ | $3.896000$ |

${a}_{1}$ | $3.052000$ | $-$ | $-$ | |

NLESO | ${\omega}_{01}$ | $103.000000$ | ${\omega}_{02}$ | $80.300000$ |

${\U0001d4b6}_{1}$ | $0.905498$ | ${\U0001d4b6}_{2}$ | $0.873169$ | |

${\mathcal{A}}_{1}$ | $0.524300$ | ${\mathcal{A}}_{2}$ | $0.102500$ | |

${b}_{01}$ | $3.230100$ | ${b}_{02}$ | $2.688375$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

NLSEF | ${\alpha}_{1}$ | $0.962100$ | ${\alpha}_{2}$ | $0.542400$ |

${\delta}_{1}$ | $0.532800$ | ${\delta}_{2}$ | $0.693800$ | |

S-$fal$ | ${\alpha}_{{1}_{ESO}}$ | $0.097200$ | ${\alpha}_{{2}_{ESO}}$ | $0.547200$ |

${\delta}_{{1}_{ESO}}$ | $0.765600$ | ${\delta}_{{2}_{ESO}}$ | $0.369000$ | |

LESO | ${\omega}_{01}$ | $261.300000$ | ${\omega}_{02}$ | $224.220000$ |

${b}_{01}$ | $1.914750$ | ${b}_{02}$ | $1.444687$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

NLSEF | ${\alpha}_{1}$ | $0.962900$ | ${\alpha}_{2}$ | $0.968200$ |

${\delta}_{1}$ | $0.910600$ | ${\delta}_{2}$ | $0.139200$ | |

D-$fal$ | ${\alpha}_{{11}_{ESO}}$ | $0.151200$ | ${\alpha}_{{21}_{ESO}}$ | $0.443600$ |

${\delta}_{{11}_{ESO}}$ | $0.903800$ | ${\delta}_{{21}_{ESO}}$ | $0.217900$ | |

${\alpha}_{{12}_{ESO}}$ | $0.066200$ | ${\alpha}_{{22}_{ESO}}$ | $0.045300$ | |

${\delta}_{{12}_{ESO}}$ | $0.485300$ | ${\delta}_{{22}_{ESO}}$ | $0.024700$ | |

LESO | ${\omega}_{01}$ | $230.430000$ | ${\omega}_{02}$ | $266.220000$ |

${b}_{01}$ | $2.289375$ | ${b}_{02}$ | $2.167031$ |

ADRC Parts | Parameter | Value | Parameter | Value |
---|---|---|---|---|

MNLSEF | ${\alpha}_{11}$ | $0.802800$ | ${\alpha}_{21}$ | $0.646400$ |

${\delta}_{11}$ | $0.277500$ | ${\delta}_{21}$ | $0.090400$ | |

${\alpha}_{12}$ | $0.170300$ | ${\alpha}_{22}$ | $0.698500$ | |

${\delta}_{12}$ | $0.457400$ | ${\delta}_{22}$ | $0.153900$ | |

TD | $R$ | $6.050000$ | ${a}_{2}$ | $13.404000$ |

${a}_{1}$ | $0.887000$ | $-$ | $-$ | |

D-$fal$ | ${\alpha}_{{11}_{ESO}}$ | $0.843900$ | ${\alpha}_{{21}_{ESO}}$ | $0.724800$ |

${\delta}_{{11}_{ESO}}$ | $0.521700$ | ${\delta}_{{21}_{ESO}}$ | $0.634300$ | |

${\alpha}_{{12}_{ESO}}$ | $0.231600$ | ${\alpha}_{{21}_{ESO}}$ | $0.141500$ | |

${\delta}_{{12}_{ESO}}$ | $0.023300$ | ${\delta}_{{21}_{ESO}}$ | $0.086500$ | |

LESO | ${\omega}_{01}$ | $183.570000$ | ${\omega}_{02}$ | $292.230000$ |

${b}_{01}$ | $4.620375$ | ${b}_{02}$ | $3.548906$ |

PI | $\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{1}$ | $\mathit{I}\mathit{T}\mathit{A}{\mathit{E}}_{2}$ | $\mathit{I}\mathit{A}{\mathit{U}}_{1}$ | $\mathit{I}\mathit{A}{\mathit{U}}_{2}$ | $\mathit{I}\mathit{S}{\mathit{U}}_{1}$ | $\mathit{I}\mathit{S}{\mathit{U}}_{2}$ | $\mathit{O}\mathit{P}\mathit{I}$ |
---|---|---|---|---|---|---|---|

LADRC | $5.0488$ | $7.5142$ | $13115.098$ | $16124.001$ | $16194.336$ | $20840.561$ | $23.2190$ |

ADRC | $10.7318$ | $13.4633$ | $976.4132$ | $1021.5292$ | $54.0156$ | $46.700165$ | $2.48116$ |

IADRC | $2.6097$ | $2.6842$ | $2518.480536$ | $2695.5036$ | $678.3466$ | $658.7566$ | $1.578022$ |

$\mathrm{S}fal-\mathrm{ADRC}$ | $3.8810$ | $5.5178$ | $850.1411$ | $908.2580$ | $75.9051$ | $64.6534$ | $0.993720$ |

$\mathrm{D}fal-\mathrm{ADRC}$ | $2.5716$ | $2.4849$ | $682.9920$ | $528.7538$ | $59.2170$ | $29.5555$ | $0.675591$ |

$\mathrm{D}fal-\mathrm{ADRC}-\mathrm{TD}$ | $2.2750$ | $2.5949$ | $350.4789$ | $328.2163$ | $21.8594$ | $12.0197$ | $0.546157$ |

$\mathrm{NLPD}-\mathrm{ADRC}$ | $3.7407$ | $2.2867$ | $635.1552$ | $696.2649$ | $30.7188$ | $37.6227$ | $0.893849$ |

$\mathrm{STC}-\mathrm{ADRC}$ | $0.714658$ | $1.8739$ | $289.7189$ | $426.9655$ | $32.3728$ | $22.6835$ | $0.213131$ |

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## Share and Cite

**MDPI and ACS Style**

Hashim, Z.S.; Khani, H.I.; Azar, A.T.; Khan, Z.I.; Smait, D.A.; Abdulwahab, A.; Zalzala, A.M.; Jawad, A.J.M.; Ahmed, S.; Ibraheem, I.K.;
et al. Robust Liquid Level Control of Quadruple Tank System: A Nonlinear Model-Free Approach. *Actuators* **2023**, *12*, 119.
https://doi.org/10.3390/act12030119

**AMA Style**

Hashim ZS, Khani HI, Azar AT, Khan ZI, Smait DA, Abdulwahab A, Zalzala AM, Jawad AJM, Ahmed S, Ibraheem IK,
et al. Robust Liquid Level Control of Quadruple Tank System: A Nonlinear Model-Free Approach. *Actuators*. 2023; 12(3):119.
https://doi.org/10.3390/act12030119

**Chicago/Turabian Style**

Hashim, Zahraa Sabah, Halah I. Khani, Ahmad Taher Azar, Zafar Iqbal Khan, Drai Ahmed Smait, Abdulkareem Abdulwahab, Ali Mahdi Zalzala, Anwar Ja’afar Mohamad Jawad, Saim Ahmed, Ibraheem Kasim Ibraheem,
and et al. 2023. "Robust Liquid Level Control of Quadruple Tank System: A Nonlinear Model-Free Approach" *Actuators* 12, no. 3: 119.
https://doi.org/10.3390/act12030119