# Design of PI Controllers for Irrigation Canals Based on Linear Matrix Inequalities

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## Abstract

**:**

## 1. Introduction

- -
- To provide a novel design method for PI controllers in irrigation canals.
- -
- To bring the benefits of LMI based design to irrigation canals, thus providing a design framework for linear controllers that can deal with issues as uncertainties in model parameters in a natural way.
- -
- To exploit subsystem synergies by using a centralized design to avoid undesired mutual interaction.
- -
- To make controller tuning closer to actual canal performance.
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- To validate and assess our results using a digital twin of ASCE Test Canal 1, which is implemented in a realistic state of the art software.

## 2. Materials and Methods

#### 2.1. Problem Setting

#### The Linear Canal Model

#### 2.2. Controller Design Procedure

#### 2.2.1. Lmi Based Controller Design

- $H(\xb7)$ is a symmetric matrix for every ${X}_{1},{X}_{2},\dots ,Xm$ and
- the dependence of the matrix function $H(\xb7)$ with respect to all variable decision ${X}_{1},{X}_{2},\dots ,Xm$ is affine.

#### 2.2.2. Tuning

- Penalties set by trial and error.
- Penalties proportional to the pool length of each section.
- Penalties proportional to the backwater surface area of each canal section.

#### 2.3. Other Methods

- -
- Method 1: the resonance frequency based on the canal pool celerity.
- -
- Method 2: the resonance frequency based on the canal pool celerity and integral constants adjusted based on downstream resonance.
- -
- Method 3: the resonance frequency determined based on the maximum cross-over frequency.
- -
- Method 4: the resonance frequency determined based on the maximum cross-over frequency and integral constants adjusted based on downstream resonance.

#### 2.4. Simulations

#### 2.4.1. Case Study

#### 2.4.2. Simulation Settings

## 3. Results

- -
- The mean value of all the mean absolute water level error values of each pool, named as mean of mean absolute error (MMAE).
- -
- The mean value of how dispersed the absolute water level error values are with respect to its mean value, named as mean of standard deviations (MSTD).
- -
- The total sum value of how dispersed the absolute water level error values are with respect to its mean value, named as the sum of standard deviations (SSTD).

## 4. Discussion

#### 4.1. Performance Indicators

#### 4.2. Tuning and Performance

- LMI Method 3 has the best performance, presenting the best values for all but one KPI. It has the second best value for the resilience (Table 4), i.e., its great performance is achieved by slightly sacrificing the total time required to recover. Its main characteristic is the lowest maximum value as shown in Figure 2, Table 3 (MAE and IAE), and Table 5 (vulnerability).
- LMI Method 2 has the worst performance of LMI methods, however it is sufficient when all methods are jointly considered as its KPI values are above average. In particular, its response is similar to the other two LMI methods, however the performance indicators are slightly worse.

#### 4.3. Overall Assessment

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Simulation results for step change of 0.2 ${\mathrm{m}}^{3}/\mathrm{s}$ in the non-lineal canal model for the LMI methods (described in Section 2.2.2): (

**a**) Q tuning approach 1 of LMI method; (

**b**) Q tuning approach 2 of LMI method; (

**c**) Q tuning approach 3 of LMI method.

**Figure 3.**Simulation results for step change of 0.2 ${\mathrm{m}}^{3}/\mathrm{s}$ in the non-lineal canal model for the proportional-integral filtered (PIF) methods [51]: (

**a**) Method 1 of PIF method; (

**b**) Method 2 of PIF method; (

**c**) Method 3 of PIF method; (

**d**) Method 4 of PIF method.

ASCE Test Canal 1 | ||||
---|---|---|---|---|

Bottom Slope (m/m) | Manning’s n (-) | Side Slopes (m/m) | Drop at Each Gate (m) | ${\mathit{y}}_{\mathit{target}}$/${\mathit{y}}_{\mathit{normal}}$(m/m) |

0.002 | 0.014 | 1.5 | 1.0 | 1.45 |

Pool Number | Pool Length (m) | Upstream Elevation (m) | Downstream Elevation (m) | Target Level (m) |

1 | 100 | 415.6 | 415.3 | 0.9 |

2 | 1200 | 408.0 | 405.6 | 0.9 |

3 | 400 | 403.4 | 402.6 | 0.8 |

4 | 800 | 399.7 | 398.1 | 0.9 |

5 | 2000 | 392.1 | 388.1 | 0.9 |

6 | 1700 | 387.4 | 384.8 | 0.8 |

7 | 1600 | 383.8 | 380.3 | 0.8 |

8 | 1700 | 379.0 | 376.1 | 0.8 |

**Table 2.**Proportional and Integral parameters of proportional-integral (PI) controllers based on the linear matrix inequality (LMI) Method.

No. Pool | LMI Method 1 | LMI Method 2 | LMI Method 3 | |||
---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | |

1 | 1.3079 | 0.0383 | 1.1542 | 0.0334 | 1.4104 | 0.0428 |

2 | 2.5170 | 0.0718 | 2.6483 | 0.0826 | 2.5172 | 0.0727 |

3 | 2.1069 | 0.0631 | 2.1260 | 0.0653 | 2.1411 | 0.0659 |

4 | 5.9515 | 0.1676 | 5.8530 | 0.1635 | 6.1181 | 0.1715 |

5 | 6.8866 | 0.1956 | 6.8934 | 0.1961 | 6.8971 | 0.1939 |

6 | 8.5417 | 0.2418 | 8.5433 | 0.2410 | 8.5357 | 0.2331 |

7 | 5.3328 | 0.1619 | 5.3329 | 0.1620 | 5.3329 | 0.1621 |

**Table 3.**Non-linear model simulation key performance indicators (KPIs): maximum absolute error (MAE), integral of absolute magnitude of error (IAE), mean of mean absolute error (MMAE), mean of standard deviations (MSTD), and sum of standard deviations (SSTD).

Methods | MAE (${10}^{-2}$) | IAE (${10}^{1}$) | MMAE (${10}^{-3}$) | MSTD (${10}^{-2}$) | SSTD (${10}^{-2}$) | ||
---|---|---|---|---|---|---|---|

Max | Mean | Max | Mean | ||||

LMI Method 1 | 2.76 | 0.90 | 0.38 | 0.13 | 0.15 | 0.31 | 2.20 |

LMI Method 2 | 3.09 | 0.93 | 0.48 | 0.16 | 0.18 | 0.35 | 2.42 |

LMI Method 3 | 2.63 | 0.86 | 0.37 | 0.13 | 0.15 | 0.30 | 2.12 |

PIF Method 1 | 7.60 | 3.67 | 15.04 | 4.51 | 5.21 | 3.14 | 21.96 |

PIF Method 2 | 5.21 | 3.21 | 9.22 | 3.03 | 3.51 | 2.43 | 17.02 |

PIF Method 3 | 3.38 | 2.47 | 5.21 | 1.85 | 2.14 | 1.69 | 11.83 |

PIF Method 4 | 2.64 | 2.18 | 2.31 | 0.85 | 0.98 | 1.12 | 7.85 |

RESILIENCE (${10}^{-1}$) | ||||||||
---|---|---|---|---|---|---|---|---|

Methods | Canal 1 | Canal 2 | Canal 3 | Canal 4 | Canal 5 | Canal 6 | Canal 7 | Mean |

LMI Method 1 | 3.33 | 3.33 | 3.33 | 10.00 | 0.01 | 0.01 | 0.01 | 2.86 |

LMI Method 2 | 3.33 | 3.33 | 3.33 | 5.00 | 0.01 | 0.01 | 0.01 | 2.15 |

LMI Method 3 | 2.00 | 3.33 | 3.33 | 10.00 | 0.01 | 0.01 | 0.01 | 2.67 |

PIF Method 1 | 3.33 | 2.50 | 2.00 | 2.00 | 1.43 | 1.11 | 1.11 | 1.93 |

PIF Method 2 | 5.00 | 2.00 | 1.11 | 1.67 | 1.25 | 1.00 | 1.11 | 1.88 |

PIF Method 3 | 5.00 | 3.33 | 1.25 | 1.11 | 1.67 | 1.43 | 1.43 | 2.17 |

PIF Method 4 | 3.33 | 2.50 | 1.43 | 2.50 | 1.67 | 1.43 | 1.25 | 2.02 |

VULNERABILITY (${10}^{-2}$) | ||||||||
---|---|---|---|---|---|---|---|---|

Methods | Canal 1 | Canal 2 | Canal 3 | Canal 4 | Canal 5 | Canal 6 | Canal 7 | Mean |

LMI Method 1 | 2.76 | 1.12 | 1.51 | 0.93 | 0.00 | 0.00 | 0.00 | 0.90 |

LMI Method 2 | 3.09 | 1.06 | 1.47 | 0.90 | 0.00 | 0.00 | 0.00 | 0.93 |

LMI Method 3 | 2.63 | 1.15 | 1.56 | 0.71 | 0.00 | 0.00 | 0.00 | 0.86 |

PIF Method 1 | 1.24 | 2.21 | 2.64 | 3.00 | 3.58 | 5.39 | 7.60 | 3.67 |

PIF Method 2 | 1.95 | 2.31 | 2.16 | 3.25 | 3.17 | 4.43 | 5.21 | 3.21 |

PIF Method 3 | 2.22 | 1.81 | 2.22 | 2.50 | 2.11 | 3.05 | 3.38 | 2.47 |

PIF Method 4 | 2.64 | 2.00 | 2.14 | 2.10 | 2.05 | 1.96 | 2.40 | 2.18 |

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**MDPI and ACS Style**

Arauz, T.; Maestre, J.M.; Tian, X.; Guan, G. Design of PI Controllers for Irrigation Canals Based on Linear Matrix Inequalities. *Water* **2020**, *12*, 855.
https://doi.org/10.3390/w12030855

**AMA Style**

Arauz T, Maestre JM, Tian X, Guan G. Design of PI Controllers for Irrigation Canals Based on Linear Matrix Inequalities. *Water*. 2020; 12(3):855.
https://doi.org/10.3390/w12030855

**Chicago/Turabian Style**

Arauz, Teresa, José M. Maestre, Xin Tian, and Guanghua Guan. 2020. "Design of PI Controllers for Irrigation Canals Based on Linear Matrix Inequalities" *Water* 12, no. 3: 855.
https://doi.org/10.3390/w12030855