# PID Controller Parameter Tables for Time-Delayed Systems Optimized Using Hill-Climbing

## Abstract

**:**

## 1. Introduction and Related Research

## 2. PTn Systems and ITAE, IAE and ISE Criteria

## 3. The Hill-Climbing Method for Calculating the PID Controller Parameters

^{3}= 1,000,000 simulations would be necessary if the parameters were increased from 0.1 to 10 in steps of 0.1.

## 4. Results: Calculated PID Parameters for the Minimized IAE, ITAE and ISE Criteria

_{1}is the time constant and n is the number of PT1 elements connected in series. Ks, T

_{1}and n can be calculated, for example, from Figure 2 and Table 1, or by using other methods.

## 5. Applications for the Use of the Table: PID-Controlled PT3 and PT5

_{1}= 1 s and Ks = 1, as well as an assumed controller output limitation of +/−2, the following PID parameters are read from the table.

_{1}≠ 1, as well as the correct use of the table with the controller output limitation. At this point, attention is also drawn to the special feature that the table values can be scaled with T

_{1}and Ks. This makes them usable for many applications. A thermal actuator heats a small heating chamber. For a biological experiment, the chamber should be heated from an initial temperature of 20 °C (outside temperature) to 40 °C. The temperature sensor in the chamber measures differentially in comparison to the outside temperature and has a linear characteristic with 1 V/4° K. The actuator is a coil, the power of which is linearized with an internal circuit and an amplifier. At the stationary end value, it heats the chamber at 6° K/1 V. The block diagram is shown in Figure 8.

_{1}= 16 s/5.12, equal to approximately 3 s. To read off the correct table values, the controller output limitation is required. In this system this is +/−10 V. For a step of ΔT = 20° K, the controller output at the stationary end value is 3.3 V. This results in a controller output limitation of +/−10 V/3.3 V = +/−3. Ks is calculated as 5 V/3.3 V = 6° K/4° K = 1.5.

## 6. Discussion and Outlook

## Funding

## Conflicts of Interest

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**Figure 3.**Illustration of the control error e(t) in the closed loop, as a basis for the IAE, ITAE and ISE criteria.

**Figure 6.**Example of convergence of parameter Ti, for a PT2 system and ITAE criterion with a controller output limitation of +/−5. The initial random values 7.7 and 4.8 represent the two lines and converge both to the end value 5.4.

**Figure 10.**PID control of the PT5 heating chamber, according to the optimal parameters for the IAE and ITAE criteria.

**Table 1.**Table values, order n and T1 of the n PT1 systems, as a function of the delay time Tu and the rise time Tn [21].

PTn | PT2 | PT3 | PT4 | PT5 | PT6 |
---|---|---|---|---|---|

Tg/Tu | 9.71 | 4.61 | 3.14 | 2.44 | 2.03 |

Tg/T1 | 2.72 | 3.69 | 4.46 | 5.12 | 5.70 |

Tu/T1 | 0.28 | 0.8 | 1.42 | 2.10 | 2.81 |

PT1 | +/−2 | +/−3 | +/−5 | +/−10 |

IAE | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 10 |

IAE | Ti = 3.1·T1 | Ti = 2·T1 | Ti = 1.3·T1 | Ti = 1·T1 |

IAE | Td = 0 (PI) | Td = 0 (PI) | Td = 0 (PI) | Td = 0 (PI) |

ITAE | Kp·Ks = 9.3 | Kp·Ks = 9.5 | Kp·Ks = 9.1 | Kp·Ks = 10 |

ITAE | Ti = 2.9·T1 | Ti = 1.9·T1 | Ti = 1.2·T1 | Ti = 1·T1 |

ITAE | Td = 0 (PI) | Td = 0 (PI) | Td = 0 (PI) | Td = 0 (PI) |

ISE | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 9.8 | Kp·Ks = 10 |

ISE | Ti = 2.7·T1 | Ti = 1.6·T1 | Ti = 1.5·T1 | Ti = 0.2·T1 |

ISE | Td = 0 (PI) | Td = 0 (PI) | Td = 0 (PI) | Td = 0 (PI) |

PT2, Tg/Tu = 9.71 | +/−2 | +/−3 | +/−5 | +/−10 |

IAE | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 10 |

IAE | Ti = 9.6·T1 | Ti = 7.3·T1 | Ti = 5.6·T1 | Ti = 3.7·T1 |

IAE | Td = 0.3·T1 | Td = 0.3·T1 | Td = 0.3·T1 | Td = 0.2·T1 |

ITAE | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 9.6 | Kp·Ks = 9.8 |

ITAE | Ti = 9.6·T1 | Ti = 7.3·T1 | Ti = 5.4·T1 | Ti = 4.7·T1 |

ITAE | Td = 0.3·T1 | Td = 0.3·T1 | Td = 0.3·T1 | Td = 0.3·T1 |

ISE | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 10 | Kp·Ks = 10 |

ISE | Ti = 9.7·T1 | Ti = 7.3·T1 | Ti = 5.1·T1 | Ti = 4.6·T1 |

ISE | Td = 0.2·T1 | Td = 0.2·T1 | Td = 0.2·T1 | Td = 0.1·T1 |

PT3, Tg/Tu = 3.61 | +/−2 | +/−3 | +/−5 | +/−10 |

IAE | Kp·Ks = 5.4 | Kp·Ks = 7 | Kp·Ks = 8.4 | Kp·Ks = 10 |

IAE | Ti = 9.4·T1 | Ti = 10·T1 | Ti = 9.8·T1 | Ti = 9.7·T1 |

IAE | Td = 0.7·T1 | Td = 0.7·T1 | Td = 0.7·T1 | Td = 0.7·T1 |

ITAE | Kp·Ks = 5.4 | Kp·Ks = 7 | Kp·Ks = 8.2 | Kp·Ks = 10 |

ITAE | Ti = 9.4·T1 | Ti = 10·T1 | Ti = 9.6·T1 | Ti = 9.7·T1 |

ITAE | Td = 0.7·T1 | Td = 0.7·T1 | Td = 0.7·T1 | Td = 0.7·T1 |

ISE | Kp·Ks = 6.1 | Kp·Ks = 8.1 | Kp·Ks = 10 | Kp·Ks = 10 |

ISE | Ti = 10·T1 | Ti = 9.8·T1 | Ti = 10·T1 | Ti = 7.8·T1 |

ISE | Td = 0.6·T1 | Td = 0.6·T1 | Td = 0.6·T1 | Td = 0.6·T1 |

PT4, Tg/Tu = 3.14 | +/−2 | +/−3 | +/−5 | +/−10 |

IAE | Kp·Ks = 2 | Kp·Ks = 2.9 | Kp·Ks = 3.3 | Kp·Ks = 3.3 |

IAE | Ti = 5.2·T1 | Ti = 6.5·T1 | Ti = 7.1·T1 | Ti = 6.9·T1 |

IAE | Td = 1.1·T1 | Td = 1.2·T1 | Td = 1.3·T1 | Td = 1.3·T1 |

ITAE | Kp·Ks = 1.9 | Kp·Ks = 2.4 | Kp·Ks = 2.3 | Kp·Ks = 2.1 |

ITAE | Ti = 5·T1 | Ti = 5.9·T1 | Ti = 5.7·T1 | Ti = 5·T1 |

ITAE | Td = 1.1·T1 | Td = 1.2·T1 | Td = 1.2·T1 | Td = 1.1·T1 |

ISE | Kp·Ks = 2.8 | Kp·Ks = 3.6 | Kp·Ks = 4.9 | Kp·Ks = 5.2 |

ISE | Ti = 6.6·T1 | Ti = 7·T1 | Ti = 7.1·T1 | Ti = 7·T1 |

ISE | Td = 1.2·T1 | Td = 1.2·T1 | Td = 1.4·T1 | Td = 1.4·T1 |

PT5, Tg/Tu = 2.44 | +/−2 | +/−3 | +/−5 | +/−10 |

IAE | Kp·Ks = 1.7 | Kp·Ks = 1.8 | Kp·Ks = 1.8 | Kp·Ks = 1.7 |

IAE | Ti = 5.8·T1 | Ti = 5.9·T1 | Ti = 5.8·T1 | Ti = 5.5·T1 |

IAE | Td = 1.6·T1 | Td = 1.6·T1 | Td = 1.6·T1 | Td = 1.6·T1 |

ITAE | Kp·Ks = 1.4 | Kp·Ks = 1.4 | Kp·Ks = 1.4 | Kp·Ks = 1.4 |

ITAE | Ti = 5.3·T1 | Ti = 5.2·T1 | Ti = 5.2·T1 | Ti = 5.0·T1 |

ITAE | Td = 1.4·T1 | Td = 1.4·T1 | Td = 1.4·T1 | Td = 1.4·T1 |

ISE | Kp·Ks = 1.9 | Kp·Ks = 2.6 | Kp·Ks = 2.5 | Kp·Ks = 2.5 |

ISE | Ti = 5.9·T1 | Ti = 6.5·T1 | Ti = 6.3·T1 | Ti = 6.1·T1 |

ISE | Td = 1.7·T1 | Td = 1.8·T1 | Td = 1.8·T1 | Td = 1.8·T1 |

PT6, Tg/Tu = 2.03 | +/−2 | +/−3 | +/−5 | +/−10 |

IAE | Kp·Ks = 1.3 | Kp·Ks = 1.3 | Kp·Ks = 1.3 | Kp·Ks = 1.3 |

IAE | Ti = 5.9·T1 | Ti = 5.8·T1 | Ti = 5.8·T1 | Ti = 5.6·T1 |

IAE | Td = 1.9·T1 | Td = 1.9·T1 | Td = 1.9·T1 | Td = 1.9·T1 |

ITAE | Kp·Ks = 1.1 | Kp·Ks = 1.1 | Kp·Ks = 1.1 | Kp·Ks = 1.1 |

ITAE | Ti = 5.5·T1 | Ti = 5.5·T1 | Ti = 5.4·T1 | Ti = 5.3·T1 |

ITAE | Td = 1.7·T1 | Td = 1.7·T1 | Td = 1.7·T1 | Td = 1.7·T1 |

ISE | Kp·Ks = 1.8 | Kp·Ks = 1.8 | Kp·Ks = 1.8 | Kp·Ks = 1.8 |

ISE | Ti = 6.8·T1 | Ti = 6.5·T1 | Ti = 6.5·T1 | Ti = 6.3·T1 |

ISE | Td = 2.1·T1 | Td = 2.1·T1 | Td = 2.1·T1 | Td = 2.1·T1 |

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**MDPI and ACS Style**

Büchi, R.
PID Controller Parameter Tables for Time-Delayed Systems Optimized Using Hill-Climbing. *Signals* **2022**, *3*, 146-156.
https://doi.org/10.3390/signals3010010

**AMA Style**

Büchi R.
PID Controller Parameter Tables for Time-Delayed Systems Optimized Using Hill-Climbing. *Signals*. 2022; 3(1):146-156.
https://doi.org/10.3390/signals3010010

**Chicago/Turabian Style**

Büchi, Roland.
2022. "PID Controller Parameter Tables for Time-Delayed Systems Optimized Using Hill-Climbing" *Signals* 3, no. 1: 146-156.
https://doi.org/10.3390/signals3010010