# Design and Comparison of Strategies for Level Control in a Nonlinear Tank

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description and Modeling of the System under Study

^{3}/s, which is consistent with the mechanical capacity of the water pumps used for tanks of these dimensions.

#### 2.1. Equation Model

#### 2.2. Tank Linear Model

_{S}.

## 3. Research Design and Control Strategies Applied

- PID control is the most widely used strategy in industrial applications due to its feasibility and easy implementation. The PID gains can be obtained based on system parameters and whether they can be accurately achieved or estimated. However, if the system parameters are unknown, the PID gains can be obtained based solely on the system tracking error and treated as a “black box.”
- GS control facilitates process control when the gains and the time constants vary with the current value of the process variable. GS is particularly appropriate for processes that speed up or slow down as the process variable rises and falls.
- IMC allows a feedback regulator under external disturbances to regain regulation and stability as long as a suitably duplicated model of the disturbance signal is fitted into the feedback path.
- FL provides a high level of automation incorporating expert knowledge. It also offers robust nonlinear control and reduced development and maintenance time.

#### 3.1. Classic PID Control

#### 3.2. Gain Scheduling PID Control

_{p}, K

_{i}, and K

_{d}gains are invariable for all the operation range, which causes overshoot and oscillations when changing the operation point considered in the first tuning. An effective solution for this problem is to divide the operation range into several segments (in this case, three) and input these gains (obtained from tuning) into a table for each segment. Afterward, the PID controller will change its gains dynamically by reading them from the table, according to the water level.

#### 3.2.1. Division of Level Range into Segments

#### 3.2.2. GSPID Parameter Tuning

#### 3.3. IMC Controller

#### 3.4. Fuzzy Logic Control

- Input fuzzification that assigns degrees of membership to the input membership functions.
- Interaction between the inference engine and a set of linguistic rules, which generates a number of output fuzzy sets.
- Defuzzification of the output set. This generates one or more real numbers (controller outputs) that constitute the control action, whose most used method is the centroid one.

#### 3.4.1. Implementation of Fuzzy Controller in Simulink

- Input 1: Error, e(t); difference between the reference and the current level of the tank.
- Input 2: Derivative dh/dt of the tank’s level (the “dnivel” variable), which allows for more precise control of the system by rapid anticipation of changes.
- Output: Relative control action, c(t) (the “valve” output variable), which acts on a modulating valve.

- Fuzzy system: Mamdani
- Logic “Y”: Product
- Logic “O”: Probabilistic
- Defuzzification: Centroid
- Implication: Product
- Aggregation: Maximum

#### 3.4.2. Linguistic Rules of the Controller

- If (error is ok) then (valve is no change).
- If (error is neg) then (valve is close fast).
- If (error is pos) then (valve is open fast).
- If (error is ok) and (dnivel is neg) then (valve is open).
- If (error is ok) and (dnivel is pos) then (valve is close).

## 4. Results Based on Performance Indexes

- Integral of the Absolute Error (IAE), defined as:

- Integral of Time-Weighted Absolute Error (ITAE), defined as:

- Integral of the Square Error (ISE), defined as:

- Residual Mean Square (RMS), defined as:

- Residual Standard Deviation (RSD), defined as:

_{i}) and observed value sets (o

_{i}), respectively. Additionally, to evaluate the performance of the four controllers more accurately, the following criteria were considered:

- Rising time (considering from 0% to 90% of the reference step’s height).
- Overshoot (% of the reference step’s height).
- Stabilization time (considering a 2% proportional band centered on the desired stationary value).
- Stationary-State Error (% based on the reference level value).

#### 4.1. Step at the Reference Level

#### 4.2. Random Disturbance on Input Flow

#### 4.3. Transitory Load Disturbance at the Tank Level

## 5. Conclusions

- Graphs: as fast visual indicators of the controlled variable behavior.
- Overshoot and times: increase and stabilization.
- Integrative error indexes: IAE, ITAE, ISE.
- Statistical error indexes: RMS, RSD.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Transfer function in the Laplace that represents the linearized tank in the operation point (${h}_{S}$,${Q}_{s}$).

**Figure 11.**Random disturbance in the tank input flow (upper graph) and controllers’ response to this disturbance (lower graph).

**Figure 12.**Controllers’ response to a load disturbance at the tank level: (

**a**) GSPID and IMC controllers, (

**b**) PID and Fuzzy controllers; (

**c**) Level change ($\mathsf{\Delta}h)$ as a result of the load disturbance.

Variable | Symbol | Value (cm) |
---|---|---|

Maximum radium of the tank | R | 19.25 |

Minimum radium of the tank | r | 2 |

Tank height | H | 73 |

Discharge coefficient | k_{v} | 22 |

Variable | Symbol | Unit |
---|---|---|

Input flow (manipulated variable) | Q(t) | (cm^{3}/s) |

Output flow | Q_{o}(t) | (cm^{3}/s) |

Water level (controlled variable) | H(t) | (cm) |

Auxiliary output flow (for load purposes) | L(t) | (cm^{3}/s) |

Tank | $\mathbf{Plant}(\mathbf{Linearized}\mathbf{at}{\mathit{h}}_{\mathit{s}})$ | PID Parameters | |||||
---|---|---|---|---|---|---|---|

$h$range (cm) | ${h}_{s}$ (cm) | $K$ | $\tau $ | ${K}_{\mathrm{p}}$ | ${K}_{\mathrm{i}}$ | ${K}_{\mathrm{d}}$ | Tuning |

10 to 27 | 18.5 | 0.391 | 29.2 | 54.5 | 6.28 | 34.2 | T-L |

27 to 44 | 33.5 | 0.541 | 149.1 | 197.7 | 22.5 | 125.5 | T-L |

44 to 60 | 52.5 | 0.659 | 396.6 | 301.1 | 18.8 | 200 | SK |

Nominal level | 40 | 0.575 | 201 | 250 | 28.4 | 158.7 | T-L |

Initial Reference Level (cm) | Final Reference Level (cm) | Rising Level 90% (cm) | Tolerance Band (%) | Upper Limit, Tol. Band (cm) | Upper Limit, Tol. Band (cm) |
---|---|---|---|---|---|

35 | 45 | 44 | 2.00 | 45.9 | 44.1 |

**Table 5.**Results for transitory response and stationary state (s.s.) error, 90 s computational simulation.

Controller | S.S. Level (cm) | S.S. Error (%) | Level Max. (cm) | Overshoot (%) | Set Time (s) | Rise Time (s) |
---|---|---|---|---|---|---|

GSPID | 45.00 | 0.00 | 49.10 | 41.0 | 46.00 | 13.30 |

IMC | 44.96 | 0.00 | 44.96 | 0.00 | 15.00 | 14.40 |

PID | 45.00 | 0.00 | 50.14 | 51.4 | 40.20 | 13.30 |

Fuzzy | 45.35 | 0.78 | 45.64 | 6.40 | 14.40 | 14.00 |

Controller | IAE | ITAE | ISE | RMS | RSD |
---|---|---|---|---|---|

GSPID | 159.20 | 3100.00 | 747.10 | 2.9794 | 0.0660 |

IMC | 110.70 | 1979.00 | 560.20 | 2.6080 | 0.0596 |

PID | 159.00 | 2801.00 | 821.20 | 3.1147 | 0.0692 |

Fuzzy | 107.90 | 1665.00 | 647.10 | 2.7819 | 0.0628 |

Controller | IAE | ITAE | ISE | RMS | RSD |
---|---|---|---|---|---|

GSPID | 159.7 | 3155.0 | 747.1 | 1.4133 | 0.0314 |

IMC | 153.0 | 10,080.4 | 568.1 | 1.2450 | 0.0279 |

PID | 159.0 | 2802.0 | 821.2 | 1.4774 | 0.0328 |

Fuzzy | 215.7 | 27,980.5 | 684.6 | 1.3546 | 0.0300 |

Controller | IAE | ITAE | ISE | RMS | RSD |
---|---|---|---|---|---|

GSPID | 25.60 | 10,687.10 | 3.710 | 0.0681 | 0.0017 |

IMC | 179.3 | 39,708.10 | 84.37 | 5.4932 | 0.1380 |

PID | 20.10 | 8381.000 | 2.310 | 0.0537 | 0.0013 |

Fuzzy | 265.2 | 10,4710.5 | 93.43 | 0.3416 | 0.0085 |

Controller | IAE | ITAE | ISE | RMS | RSD |
---|---|---|---|---|---|

GSPID | 188.4 | 76,245.40 | 505.5 | 0.9491 | 0.0205 |

IMC | 421.9 | 155,371.5 | 488.4 | 0.9407 | 0.0205 |

PID | 179.3 | 73,940.40 | 479.3 | 0.9241 | 0.0200 |

Fuzzy | 369.3 | 150,198.7 | 453.4 | 0.9079 | 0.0195 |

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**MDPI and ACS Style**

Urrea, C.; Páez, F.
Design and Comparison of Strategies for Level Control in a Nonlinear Tank. *Processes* **2021**, *9*, 735.
https://doi.org/10.3390/pr9050735

**AMA Style**

Urrea C, Páez F.
Design and Comparison of Strategies for Level Control in a Nonlinear Tank. *Processes*. 2021; 9(5):735.
https://doi.org/10.3390/pr9050735

**Chicago/Turabian Style**

Urrea, Claudio, and Felipe Páez.
2021. "Design and Comparison of Strategies for Level Control in a Nonlinear Tank" *Processes* 9, no. 5: 735.
https://doi.org/10.3390/pr9050735