# The Trade-Off between the Controller Effort and Control Quality on Example of an Electro-Pneumatic Final Control Element

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Controller Effort

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Remark.**

## 3. Research Environment

^{2}. The liquid level is measured by means of level sensor. Level signal is fed back to the main proportional-integral controller (${k}_{p}=10,{T}_{i}=50\mathrm{s})$ which governs the final control element. The liquid is free discharged from the tank. The liquid outflow is modeled by the equation:

^{3}—specific gravity of the liquid; $g$ = 9.81 m/s

^{2}—gravitational constant; and $h$—liquid level in [m].

^{3}/s. The final control element is intended for controlling liquid inflow to the tank.

## 4. Control Quality Factors

- cumulative effort of the controller ${Q}_{K}$ according to formula (6) for T = 100 s;
- normalized, average absolute tracking error ${e}_{Kr}$ according to the formula (9). The value of this factor is determined in test when applying a standardized trapezoidal setpoint shape with the constant slope equal to 0.025 s
^{−1}.$${e}_{Kr}=\frac{{f}_{s}}{K}{\displaystyle \sum}_{k=1}^{K}\left|CV\left(k\right)-X\left(k\right)\right|,$$ - normalized, average absolute tracking error ${e}_{Ks},$ according to Formula (9). The value of this factor is determined in a test in which a rectangular set-point with the 50% amplitude and period equal to 40 s is applied.
- overshoots ${\kappa}_{rise}$ and ${\kappa}_{fall}$ obtained respectively for applying positive and negative 60% stepwise set-points. Overshoot is defined as the ratio of the amplitude of the first transitional control error ${e}_{1}$ to the setpoint change ${e}_{0}$ and is expressed as a percentage.$$\kappa =\frac{{e}_{1}}{{e}_{0}}$$
- settling times: ${T}_{Rrise}$ and ${T}_{Rfall}$ for the 60% stepwise setpoint changes appropriately in positive and negative direction settling time is defined as the time that elapses from the moment of the set-point change until a positioner’s stem position X settles within $\pm 0.05{\mathrm{e}}_{0}$ tolerance band around the steady state value.

## 5. Positioner Controller

#### 5.1. Methodology

- the all controllers should be applied successively in the same control system. This allows a meeting of the necessary condition of comparability.
- the settings of all investigated controllers should be optimized by means of the same cost function. This satisfies the sufficient condition of comparability.

#### 5.2. Proportional and Proportionl-Derivative Controller

#### 5.3. Fuzzy PD Controller

#### 5.4. Neural Network Controller

## 6. The Choice of Simulation Tests

- chemically aggressive and corrosive environment,
- leakages of the controlled aggressive media,
- high operating temperature,
- environmental pollution,
- control loop oscillations,
- mechanical impacts,
- vibrations.

- nominal values of friction and actuator’s spring elasticity;
- friction varying within the range [−50%, +50%];
- actuator’s spring elasticity varying within the range [−50%, +50%].

## 7. Discussion of Obtained Results

#### 7.1. Control Effort

^{−1}. In contrast, Figure 15 depicts the best results achieved for classic proportional-and-derivative and fuzzy logic controllers.

#### 7.2. The Low Pass Filter Versus Control Effort

_{Kr}and e

_{Ks}versus the filter time constant.

_{Kr}and e

_{Ks}. Inversely, the usage of low pass filter evidently worsen tracking properties of the control system. It is very interesting also note that approximately proportional increase of tracking errors versus filter time constant. As the trends of cumulative effort and tracking errors versus the filter time constant are contrary, it is clear that a space for a trade-off between controller effort and tracking error exists.

## 8. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Simplified block diagram of the structure of the closed loop automatic control system. Notion: SP—setpoint; PV—process variable; e—control error; and CV—control value.

**Figure 3.**Block diagram of an electro-pneumatic final control element. Notions: CV—output of external controller; CVI—output of internal controller; PC—internal controller; E/P—electro-pneumatic transducer; PS—compressed air supply; p

_{z}—supply air pressure; p

_{s}—the output pressure of the E/P transducer; X—displacement of the valve plug; and XT—displacement transducer of the actuator’s stem [6].

**Figure 4.**An example of two control strategies with a strongly differentiated effort. Value of effort for aggressive strategy Q

_{K}= 0.32 and for a conservative strategy Q

_{K}= 0.029 [19].

**Figure 5.**Simplified simulation model of the liquid level control system. Notions: SP—liquid level setpoint; PV—measured liquid level; CV—output of the main process controller; CVI—output of the internal positioner controller; X—position of the valve stem; F—liquid flow rate; and PV—liquid level in the tank.

**Figure 7.**Normalized static characteristics of pneumatic actuator of the electro-pneumatic final control element.

**Figure 8.**Positive (blue line) and negative (red line) step response of the valve stem displacement of pneumatic actuator.

**Figure 10.**(

**a**) Membership functions representing fuzzy values of the control error. (

**b**) Membership functions representing fuzzy values of the derivative of the control error. (

**c**) Membership functions representing the fuzzy control value output. (

**d**) Block diagram of the fuzzy reasoning applied for fuzzy PD controller.

**Figure 13.**(

**a**) Snap shot of the final control element installed in the liquid level control loop. The effects of a highly corrosive environment are visible by naked eye. The ambient temperature varies in the range 15–50 °C, while relative humidity in the range 50–95%; (

**b**) An example of the application of the positioner for the control of air to fuel ratio in one of the black coal power stations. The final control element has been installed in extremely hot and dusty environments.

**Figure 16.**The comparison of the mean time effort of the four investigated controllers $\left(\Delta T=10\mathrm{s}\right)$.

**Figure 19.**The cumulative effort of the four investigated controllers in 1000s time horizon by chirp set-point.

**Figure 21.**Cumulative effort and tracking errors e

_{Kr}and e

_{Ks}versus filter time constant for: (

**a**) P controller, (

**b**) PD controller, (

**c**) FPD controller, and (

**d**) NN controller.

**Figure 22.**Control quality factors versus filter time constant. (

**a**) settling time T

_{rise}, (

**b**) settling time T

_{fall}, (

**c**) overshoot κ

_{rise}, and (

**d**) overshoot κ

_{fall}.

Controller | ${\mathbf{k}}_{\mathbf{p}}$ | ${\mathbf{T}}_{\mathbf{i}}\text{}\left[\mathbf{s}\right]$ | ${\mathbf{T}}_{\mathbf{d}}\text{}\left[\mathbf{s}\right]$ |
---|---|---|---|

$P$ | $100$ | $inf$ | $0$ |

$PD$ | $100$ | $inf$ | $0.244$ |

Control Error | Derivative of Control Error | Output |
---|---|---|

Negative | Negative | Large Negative |

Negative | Zero | Negative |

Negative | Positive | Zero |

Zero | Negative | Negative |

Zero | Zero | Zero |

Zero | Positive | Positive |

Positive | Negative | Zero |

Positive | Zero | Positive |

Positive | Positive | Large Positive |

Experiment | Quality Factor | |||||||
---|---|---|---|---|---|---|---|---|

${\mathit{Q}}_{\mathit{K}}$ | ${\mathit{e}}_{\mathit{K}\mathit{r}}$ | ${\mathit{e}}_{\mathit{K}\mathit{s}}$ | ${\mathit{\kappa}}_{\mathit{r}\mathit{i}\mathit{s}\mathit{e}}[\%]$ | ${\mathit{\kappa}}_{\mathit{f}\mathit{a}\mathit{l}\mathit{l}}[\%]$ | ${\mathit{T}}_{\mathit{R}\mathit{r}\mathit{i}\mathit{s}\mathit{e}}\left[\mathit{s}\right]$ | ${\mathit{T}}_{\mathit{R}\mathit{f}\mathit{a}\mathit{l}\mathit{l}}\left[\mathit{s}\right]$ | ||

P controller | Nominal parameters | 0.397 | 4.03 | 9.38 | 8.67 | 0.79 | 8.67 | 29.2 |

Reduced friction | 0.402 | 3.98 | 9.36 | 2.62 | 0.77 | 8.65 | 29.2 | |

Increased friction | 0.381 | 4.04 | 9.50 | 2.69 | 0.79 | 8.75 | 29.4 | |

Reduced elasticity | 0.428 | 3.92 | 8.84 | 3.41 | 0.99 | 7.43 | 27.3 | |

Increased elasticity | 0.344 | 4.34 | 10.2 | 2.12 | 0.64 | 10.1 | 31.0 | |

PD controller | Nominal parameters | 0.127 | 3.26 | 8.81 | 0.06 | 0.00 | 7.14 | 29.2 |

Reduced friction | 0.190 | 3.23 | 8.79 | 0.06 | 0.00 | 7.14 | 29.2 | |

Increased friction | 0.055 | 4.79 | 9.37 | 0.04 | 0.00 | 7.20 | 29.6 | |

Reduced elasticity | 0.288 | 2.94 | 7.86 | 0.05 | 0.00 | 5.79 | 27.3 | |

Increased elasticity | 0.104 | 3.72 | 9.78 | 0.05 | 0.00 | 8.62 | 30.9 | |

Fuzzy PD controller | Nominal parameters | 0.107 | 3.25 | 8.83 | 0.05 | 0.20 | 7.14 | 29.2 |

Reduced friction | 0.155 | 3.22 | 8.81 | 0.04 | 0.19 | 7.14 | 29.2 | |

Increased friction | 0.046 | 4.51 | 9.34 | 0.02 | 0.00 | 7.20 | 29.3 | |

Reduced elasticity | 0.150 | 2.91 | 7.90 | 0.04 | 0.29 | 5.78 | 27.3 | |

Increased elasticity | 0.086 | 3.71 | 9.76 | 0.04 | 0.13 | 8.62 | 30.9 | |

Neural NN controller | Nominal parameters | 0.048 | 7.06 | 14.77 | 0.39 | 0.00 | 7.37 | >50 |

Reduced friction | 0.043 | 7.05 | 14.73 | 0.36 | 0.00 | 7.37 | >50 | |

Increased friction | 0.049 | 7.11 | 14.84 | 0.43 | 0.00 | 7.38 | >50 | |

Reduced elasticity | 0.122 | 6.64 | 14.45 | 1.27 | 0.00 | 5.99 | >50 | |

Increased elasticity | 0.041 | 7.64 | 15.06 | 0.30 | 0.00 | 8.91 | >50 |

Controller | Overshoot | $\mathit{a}$ | b |
---|---|---|---|

$P$ | ${\kappa}_{rise}$ | −7.33 | 10.3 |

${\kappa}_{fall}$ | −3.35 | 4.24 | |

$PD$ | ${\kappa}_{rise}$ | −7.47 | 7.63 |

${\kappa}_{fall}$ | −3.14 | 2.88 | |

$FPD$ | ${\kappa}_{rise}$ | −7.37 | 7.23 |

${\kappa}_{fall}$ | −3.17 | 3.45 | |

$NN$ | ${\kappa}_{rise}$ | −8.03 | 8.61 |

${\kappa}_{fall}$ | −4.25 | 8.36 |

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**MDPI and ACS Style**

Bartyś, M.; Hryniewicki, B.
The Trade-Off between the Controller Effort and Control Quality on Example of an Electro-Pneumatic Final Control Element. *Actuators* **2019**, *8*, 23.
https://doi.org/10.3390/act8010023

**AMA Style**

Bartyś M, Hryniewicki B.
The Trade-Off between the Controller Effort and Control Quality on Example of an Electro-Pneumatic Final Control Element. *Actuators*. 2019; 8(1):23.
https://doi.org/10.3390/act8010023

**Chicago/Turabian Style**

Bartyś, Michał, and Bartłomiej Hryniewicki.
2019. "The Trade-Off between the Controller Effort and Control Quality on Example of an Electro-Pneumatic Final Control Element" *Actuators* 8, no. 1: 23.
https://doi.org/10.3390/act8010023