# Application of the Robust Fixed Point Iteration Method in Control of the Level of Twin Tanks Liquid

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. On the Concept of Using of Fixed Point Iteration-Based Control Method

## 4. Study of Dual Tank Systems

^{2}. The water level in the upper tank is denoted by ${h}_{1}\left(t\right)$, which is the first state variable. Similarly, the water level in the lower tank 2 is expressed by ${h}_{2}\left(t\right)$ (i.e., the second state variable). Suppose that a pump can be adjusted that pumps the liquid into the first (upper) tank by a connected pipe. Variable ${q}_{1}\left(t\right)$ ${m}^{3}\xb7{\mathrm{s}}^{-1}$ expresses the input flow of the upper tank after pumping by the pump, ${q}_{2}\left(t\right)$ ${m}^{3}\xb7{\mathrm{s}}^{-1}$ is the output flow of the upper tank and ${q}_{3}\left(t\right)$ ${m}^{3}\xb7{\mathrm{s}}^{-1}$ denotes the output flow of the lower tank. In the steady state, the conservation of the total volume of water leads to ${q}_{1}\left(t\right)={q}_{3}\left(t\right)$.

#### Model Representation

## 5. Control Task of the Dual Tank System

## 6. Simulation Results for the Coupled Tank

## 7. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Trajectory tracking for $({\mathsf{\Lambda}}_{max}$ = 1 (upper Left Hand Side), 4 (upper Right Hand Side), 9 (bottom Left Hand Side), 12 (bottom Right Hand Side)).

**Figure 3.**For ${\mathsf{\Lambda}}_{max}=5.5$ with adaptivity: Trajectory tracking with its zoomed in excerpt (upper Left & Right Sides) and tracking errors with its zoomed in excerpt (bottom Left & Right Sides).

**Figure 4.**The adaptive control input and the second time-derivatives with their zoomed in excerpts for ${\mathsf{\Lambda}}_{max}=5.5$ with adaptivity.

**Figure 7.**For ${\mathsf{\Lambda}}_{max}=5.5$ and $T=5\mathsf{\Delta}t$ with adaptivity: Trajectory tracking with its zoomed in excerpt (upper Left & Right Sides) and tracking errors with its zoomed in excerpt (below Left & Right Sides).

**Figure 8.**For ${\mathsf{\Lambda}}_{max}=5.5$ and $T=5\mathsf{\Delta}t$ with adaptivity: The adaptive control input with its zoomed in excerpt (upper Left & Right Sides) and the second time-derivatives with its zoomed in excerpt (below Left & Right Sides).

Parameter | Exact Value | Approximate Value |
---|---|---|

S | $1.0\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ (model parameter) | $1.8\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ |

${k}_{1}$ | $0.02\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{5/2}\xb7{\mathrm{s}}^{-1}$ (model parameter) | $0.005\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{5/2}\xb7{\mathrm{s}}^{-1}$ |

${k}_{2}$ | $0.03\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{5/2}\xb7{\mathrm{s}}^{-1}$ (model parameter) | $0.0058\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{5/2}\xb7{\mathrm{s}}^{-1}$ |

${h}_{{1}_{0}}$ | $3.0\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (initial height) | Not Applicable |

${h}_{{2}_{0}}$ | $0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (initial height) | Not Applicable |

${\mathsf{\Lambda}}_{max}$ | $5.5\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$ (control parameter) | Not Applicable |

K | $40\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\xb7{\mathrm{s}}^{-2}$ (control parameter) | Not Applicable |

B | $-1$ (control parameter) | Not Applicable |

A | $\frac{{10}^{-1}}{K}$ (control parameter) | Not Applicable |

${e}_{max}$ | $0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$ (control parameter) | Not Applicable |

${\dot{e}}_{max}$ | $0.1\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\xb7{\mathrm{s}}^{-1}$ (control parameter) | Not Applicable |

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Khan, H.; Issa, H.; Tar, J.K. Application of the Robust Fixed Point Iteration Method in Control of the Level of Twin Tanks Liquid. *Computation* **2020**, *8*, 96.
https://doi.org/10.3390/computation8040096

**AMA Style**

Khan H, Issa H, Tar JK. Application of the Robust Fixed Point Iteration Method in Control of the Level of Twin Tanks Liquid. *Computation*. 2020; 8(4):96.
https://doi.org/10.3390/computation8040096

**Chicago/Turabian Style**

Khan, Hamza, Hazem Issa, and József K. Tar. 2020. "Application of the Robust Fixed Point Iteration Method in Control of the Level of Twin Tanks Liquid" *Computation* 8, no. 4: 96.
https://doi.org/10.3390/computation8040096