# Tuning of PID Control for the Double Integrator Plus Dead Time Model by Modified Real Dominant Pole and Performance Portrait Methods

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## Abstract

**:**

## 1. Introduction

## 2. Time and Shape Related Performance Measures

## 3. PD and PID Controller Tuning with the MRDP Method

**Remark**

**1**

**.**Due to the introduction of the I-action implemented in form of a positive feedback, the proportional gain of the PID controller has to be increased by $0.1248/0.079=1.58$ times and the derivative gain by $0.5045/0.461=1.1$ times. Nonetheless, the speed of the transients decreased by $0.586/0.416=1.4$ times with respect to the dominant poles and with respect to $IA{E}_{w}$, in the optimal case given by $IA{E}_{w2}$ by $5.513/4.1213=1.34$ times, $1.92$ times for $IA{E}_{w1}$, and by 2.5 times for $IA{E}_{w0}$. This velocity decrease, together with the increased sensitivity of the control loop, may be sufficient to completely eliminate the advantages of the controller I-action (as experimentally confirmed in [31]).

**Remark**

**2**

**.**The integration time constant of the QRDP PID controller ${T}_{i}=10.324{T}_{d}$ is much larger than the dominant time constant ${T}_{o}=1.707{T}_{d}$ of transients stabilised by the TRDP PD controller. This means that the signal at the output of the low-pass filter with the time constant ${T}_{i}$ settles much later than the output of the stabilising PD controller (14).

## 4. Modified Controller Tuning

#### 4.1. Tuning of the PID Controller According to a Modified Position of One Real Pole

#### 4.2. Tuning the PID Controller According to a Modified Position of Pair of Poles

#### 4.3. Tuning PID Controller According to Threefold Change in Pole Position

#### 4.4. Tuning the Controller with Equivalent Delay

## 5. Evaluation of the Performance Offered by the Analytical Design

#### Comparing the PD and PID Controllers

## 6. Analytical Tuning versus Performance Portrait Method

- PPM is the only optimisation method that, once a performance portrait (PP) has been generated in dimensionless variables, allows its easy and unlimited reuse for different circuit parameters and criteria to select the optimal solution in different combinations.
- Has no problems with convergence to the optimal solution, if the scope of PP generation and the point grid have been determined appropriately.
- The success of the method depends primarily on the choice of appropriate performance measures that can overcome the effectiveness of traditional constraints based on maximum sensitivity (${M}_{s}$).
- Similar methods for evaluating all relevant (admissible) solutions and selecting the best solution are often applied in other areas of social practice, not only to mention public procurement as a possible example.

#### 6.1. Parallel PID Tuning without Prefilter Optimisation

#### 6.2. Alternative Controller Tuning Optimised under Consideration of Sensitivity Constraints

**Remark**

**4**

**.**Also the PP-based analysis of transients satisfying shape-related constraints (46) confirms the property resulting from Remarks 2 and 3 that the value of ${T}_{i}$ cannot be arbitrarily reduced to accelerate the reconstruction of the disturbances and the speed of the transients.

## 7. PID Controllers as Stabilising Controllers with Disturbance Observer (DOB)

#### A Brief History of Series and Parallel PID Control

**Definition**

**1**

**.**The series PI and PID controllers are the first historically known disturbance counteracting controllers using DOB that complement the stabilising P and PD controllers by the positive feedback of their output. They can be designed by approximating the steady-state output values of the controller based on integral models, representing the negative values of constant input disturbances. To filter out stabilising transients, the (nearly) steady-state values of the controller output can be achieved using low-pass filters with sufficiently long (integral) time constant ${T}_{i}$.

**Remark**

**5**

**.**The reconstructed disturbance signal could also be obtained (see Figure 15) by observing the output of the parallel PID controller through a delay with a sufficiently long time constant ${T}_{i}$ (to filter out stabilising transients), but (in contrast to compact series PID controllers), such an observer would require an additional filter transfer function. Furthermore, the use of an integrator in parallel PI and PID controllers is a source of redundant integration called windup.

**Remark**

**6**

**.**Understanding PID functionality is crucial to understanding why we cannot arbitrarily reduce the integral time constant ${T}_{i}$ with respect to the dominant time constant of the loop stabilised by the PD controller (see Remarks 2–4). This assumption can be confirmed by all the above analytical and numerical calculations of the optimal PID controller settings.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1P | One-Pulse, response with 2 monotonic segments (1 extreme point) |

2P | Two-Pulse, response with 3 monotonic segments (2 extreme points) |

3D | Three-Dimensional |

ADRC | Active Disturbance Rejection Control |

DIPDT | Double Integrator Plus Dead-Time |

$IAE$ | Integral Absolute Error |

IPDT | Integrator Plus Dead-Time |

LESO | Linear Extended State Observer |

MFRP | Modified sets of Four Real Poles |

MRDP | Multiple Real Dominant Pole |

PID | Proportional-Integral-Derivative |

PP | Performance Portrait |

PPM | Performance Portrait Method |

QRDP | Quadruple Real Dominant Pole |

TRDP | Triple Real Dominant Pole |

TV | Total Variation |

TV${}_{0}$ | Deviation from Monotonicity |

TV${}_{1}$ | Deviation from 1P Shape |

TV${}_{2}$ | Deviation from 2P Shape |

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**Figure 5.**$IAE$ values of the 2DOF PID control with prefilter (24) and a first-order implementation filter tuned according to (41) with a triple real pole ${p}_{o}$ and a single pole ${p}_{o}/m$ according to Table 1 (

**left**) and with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$ according to Table 2 (

**right**), no noise; ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{f}={T}_{e}=0.1;n=N=1$.

**Figure 7.**Matlab/Simulink simulation scheme in performance evaluation with measurement noise generator by Uniform Random Number block; $Pn=\left[{T}_{f}\phantom{\rule{0.277778em}{0ex}}1\right]$ represents the PID filter denominator, $a=0$.

**Figure 8.**$IAE$ and $T{V}_{2}\left(u\right)$ values of the 2DOF PID control with prefilter (24) and a first-order implementation filter tuned according to (41) with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$ according to Table 2, noise amplitude $|{\delta}_{n}|<0.01$; ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{f}={T}_{e}=0.1;n=N=1$; ${T}_{s}=0.001$.

**Figure 9.**${J}_{k}\left(u\right)$ values (6) of the 2DOF PID control with prefilter (24) and a first-order implementation filter tuned according to (41) with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$ according to Table 2 for $k=1$ and $k=1/10$, noise amplitude $|{\delta}_{n}|<0.01$; ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{f}={T}_{e}=0.1;n=N=1$; ${T}_{s}=0.001$.

**Figure 10.**Matlab/Simulink simulation scheme used to generate PP with ${F}_{p}\left(s\right)=1/(1+{T}_{i}s)$.

**Figure 11.**Unit setpoint (

**left**) and disturbance responses (

**right**) corresponding to optimal tuning calculated by the PPM with the parameter grid (43) under the performance specifications (46) and (48), by the QRDP tuning (22) and by the $IAE$-based optimisation under ${M}_{s}$ constraints [11]; ${T}_{m}=1$; ${P}_{n}=\left[{T}_{f}\phantom{\rule{0.277778em}{0ex}}1\right];{T}_{f}=0.1$; ${K}_{s}=1$; ${T}_{s}=0.001$.

**Figure 14.**Performance measures for unit setpoint step responses corresponding to optimal tuning calculated for the PP achieved with (43) under the performance specifications (46) (

**left**), by means of PPM with decreased controller gains (48) (

**right**), by the QRDP tuning (22) and by the $IAE$-based optimisation under ${M}_{s}$ constraints [11]; ${T}_{d}=1$; ${K}_{s}=1$; ${T}_{s}=0.001$.

**Figure 15.**Matlab/Simulink simulation schemes of series PI and PID controllers (above) designed as stabilising controllers extended by the disturbance observer based on evaluating the steady-state values of the controller output and the parallel PID controller with additional disturbance observer filter with the time constant ${T}_{i}$ (below); ${T}_{f}$—the implementation and noise attenuation filter time constant.

**Table 1.**PID control according to the quasi-polynomial (28) with a triple real pole ${p}_{o}$ and a single pole ${p}_{o}/m$; ${K}_{d}={K}_{c}{T}_{D};{K}_{i}={K}_{c}/{T}_{i}$.

m | ${\mathit{p}}_{\mathit{o}}={\mathit{s}}_{\mathit{o}}{\mathit{T}}_{\mathit{d}}$ | ${\mathit{\tau}}_{\mathit{o}}=-1/{\mathit{p}}_{\mathit{o}}$ | $\mathit{\kappa}={\mathit{K}}_{\mathit{c}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}^{2}$ | ${\mathit{\tau}}_{\mathit{D}}={\mathit{T}}_{\mathit{D}}/{\mathit{T}}_{\mathit{d}}$ | ${\mathit{\tau}}_{\mathit{i}}={\mathit{T}}_{\mathit{i}}/{\mathit{T}}_{\mathit{d}}$ | $\mathit{\delta}={\mathit{K}}_{\mathit{d}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}$ | $\mathit{\eta}={\mathit{K}}_{\mathit{i}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}$ |
---|---|---|---|---|---|---|---|

$1/2$ | −0.33822 | 2.9567 | 0.1161 | 4.243 | 11.052 | 0.4926 | 0.01050 |

1 | −0.41578 | 2.4051 | 0.1248 | 4.043 | 10.324 | 0.5045 | 0.01209 |

2 | −0.47949 | 2.0855 | 0.1185 | 4.200 | 11.131 | 0.4977 | 0.01065 |

3 | −0.50810 | 1.9681 | 0.1115 | 4.401 | 12.513 | 0.4906 | 0.00891 |

4 | −0.52450 | 1.9066 | 0.1063 | 4.569 | 14.051 | 0.4856 | 0.00757 |

**Table 2.**PID control corresponding to the quasi-polynomial (28) with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$; ${K}_{d}={K}_{c}{T}_{D};{K}_{i}={K}_{c}/{T}_{i}$.

m | ${\mathit{p}}_{\mathit{o}}={\mathit{s}}_{\mathit{o}}{\mathit{T}}_{\mathit{d}}$ | ${\mathit{\tau}}_{\mathit{o}}=-1/{\mathit{p}}_{\mathit{o}}$ | $\mathit{\kappa}={\mathit{K}}_{\mathit{c}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}^{2}$ | ${\mathit{\tau}}_{\mathit{D}}={\mathit{T}}_{\mathit{D}}/{\mathit{T}}_{\mathit{d}}$ | ${\mathit{\tau}}_{\mathit{i}}={\mathit{T}}_{\mathit{i}}/{\mathit{T}}_{\mathit{d}}$ | $\mathit{\delta}={\mathit{K}}_{\mathit{d}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}$ | $\mathit{\eta}={\mathit{K}}_{\mathit{i}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}$ |
---|---|---|---|---|---|---|---|

1 | −0.41578 | 2.4051 | 0.1248 | 4.043 | 10.324 | 0.5045 | 0.01209 |

2 | −0.56336 | 1.7750 | 0.1148 | 4.292 | 11.354 | 0.4926 | 0.010109 |

3 | −0.64740 | 1.5446 | 0.1013 | 4.707 | 13.063 | 0.4768 | 0.007756 |

4 | −0.70300 | 1.4225 | 0.08978 | 5.164 | 14.932 | 0.4636 | 0.006012 |

**Table 3.**PID control corresponding to the quasi-polynomial (28) with a single real pole ${p}_{o}$ and a triple pole ${p}_{o}/m$; ${K}_{d}={K}_{c}{T}_{D};{K}_{i}={K}_{c}/{T}_{i}$.

m | ${\mathit{p}}_{\mathit{o}}={\mathit{s}}_{\mathit{o}}{\mathit{T}}_{\mathit{d}}$ | ${\mathit{\tau}}_{\mathit{o}}=-1/{\mathit{p}}_{\mathit{o}}$ | $\mathit{\kappa}={\mathit{K}}_{\mathit{c}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}^{2}$ | ${\mathit{\tau}}_{\mathit{D}}={\mathit{T}}_{\mathit{D}}/{\mathit{T}}_{\mathit{d}}$ | ${\mathit{\tau}}_{\mathit{i}}={\mathit{T}}_{\mathit{i}}/{\mathit{T}}_{\mathit{d}}$ | $\mathit{\delta}={\mathit{K}}_{\mathit{d}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}$ | $\mathit{\eta}={\mathit{K}}_{\mathit{i}}{\mathit{K}}_{\mathit{s}}{\mathit{T}}_{\mathit{d}}$ |
---|---|---|---|---|---|---|---|

1 | −0.41578 | 2.4051 | 0.1248 | 4.043 | 10.324 | 0.5045 | 0.01209 |

2 | −0.67643 | 1.4783 | 0.1161 | 4.292 | 11.052 | 0.49258 | 0.010504 |

3 | −0.87156 | 1.1474 | 0.1013 | 4.707 | 13.063 | 0.4716 | 0.0084570 |

4 | −1.0290 | 0.9718 | 0.09108 | 4.938 | 13.341 | 0.4498 | 0.006827 |

**Table 4.**$IAE$ values of the 2DOF PID control with prefilter (24) and a first-order implementation filter tuned according to (41) with a triple real pole ${p}_{o}$ and a single pole ${p}_{o}/m$ according to Table 1 (above), with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$ according to Table 2 (middle) and with a single real pole ${p}_{o}$ and a triple pole ${p}_{o}/m$ according to Table 3 (below); a measurement noise amplitude $\left|\delta \right|<0.01$; ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{e}=0.1;n=N=1$; ${T}_{s}=0.001$.

- | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ |
---|---|---|---|---|

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 1.1348 | 1.3169 | 1.5088 |

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 0.9019 | 0.8780 | 0.8741 |

$IA{E}_{d}/IA{E}_{d}\left(1\right)$ | 1.0000 | 1.1353 | 1.3546 | 1.5878 |

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 1.1969 | 1.4321 | 1.6725 |

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 0.9987 | 1.0875 | 1.1985 |

$IA{E}_{d}/IA{E}_{d}\left(1\right)$ | 1.0000 | 1.1953 | 1.5564 | 1.9993 |

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 1.2090 | 1.3931 | 1.5616 |

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 1.0224 | 1.1034 | 1.1938 |

$IA{E}_{d}/IA{E}_{d}\left(1\right)$ | 1.0000 | 1.1507 | 1.4289 | 1.7688 |

**Table 5.**The 2DOF PID control with prefilter (24) and a first-order implementation filter tuned according to (41) with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$ according to Table 2: $IAE$, $T{V}_{2}\left(u\right)$, ${J}_{1}\left(u\right)$ and ${J}_{1/10}\left(u\right)$ values ((6) for $k=1$ and $k=1/10$) normalised by numbers corresponding to $m=1$; a measurement noise amplitude $\left|\delta \right|<0.01$; ${T}_{m}=1$; ${K}_{s}=1$; ${T}_{e}=0.1;n=N=1$; ${T}_{s}=0.001$.

- | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ |
---|---|---|---|---|

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 1.1969 | 1.4321 | 1.6725 |

$IA{E}_{w}/IA{E}_{w}\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 0.9987 | 1.0875 | 1.1985 |

$IA{E}_{d}/IA{E}_{d}\left(1\right)$ | 1.0000 | 1.1953 | 1.5564 | 1.9993 |

$T{V}_{2}\left({u}_{w}\right)/T{V}_{2}\left({u}_{w}\right)\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 0.9766 | 0.9452 | 0.9190 |

$T{V}_{2}\left({u}_{w}\right)/T{V}_{2}\left({u}_{w}\right)\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 0.9766 | 0.9452 | 0.9189 |

$T{V}_{2}\left({u}_{d}\right)/T{V}_{2}\left({u}_{d}\right)\left(1\right)$ | 1.0000 | 0.9766 | 0.9452 | 0.9190 |

${J}_{1}\left({u}_{w}\right)/{J}_{1}\left({u}_{w}\right)\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 1.1689 | 1.3536 | 1.5370 |

${J}_{1}\left({u}_{w}\right)/{J}_{1}\left({u}_{w}\right)\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 0.9753 | 1.0278 | 1.1013 |

${J}_{1}\left({u}_{d}\right)/{J}_{1}\left({u}_{d}\right)\left(1\right)$ | 1.0000 | 1.1673 | 1.4712 | 1.8373 |

${J}_{1/10}\left({u}_{w}\right)/{J}_{1/10}\left({u}_{w}\right)\left(1\right),\phantom{\rule{0.277778em}{0ex}}b={T}_{o}$ | 1.0000 | 0.9943 | 0.9798 | 0.9675 |

${J}_{1/10}\left({u}_{w}\right)/{J}_{1/10}\left({u}_{w}\right)\left(1\right),\phantom{\rule{0.277778em}{0ex}}b=m{T}_{o}$ | 1.0000 | 0.9764 | 0.9531 | 0.9357 |

${J}_{1/10}\left({u}_{d}\right)/{J}_{1/10}\left({u}_{d}\right)\left(1\right)$ | 1.0000 | 0.9942 | 0.9880 | 0.9849 |

**Table 6.**PID control corresponding to the quasi-polynomial (28) with a double real pole ${p}_{o}$ and a pole pair ${p}_{o}/m$; ${K}_{d}={K}_{c}{T}_{D};{K}_{i}={K}_{c}/{T}_{i}$ with a measurement noise amplitude $\left|\delta \right|<0.01$; ${T}_{d}=1$; ${K}_{s}=1$; ${T}_{e}=0.1;n=N=1$; ${T}_{s}=0.001$.

- | $\mathit{m}=1$ | $\mathit{m}=2$ | $\mathit{m}=3$ | $\mathit{m}=4$ |
---|---|---|---|---|

$IA{E}_{w},b={T}_{o}$ | 8.9518 | 10.7147 | 12.8197 | 14.9720 |

$T{V}_{2}\left({u}_{w}\right),b={T}_{o}$ | 1529.2 | 1493.4 | 1445.4 | 1405.2 |

$IA{E}_{w},b=m{T}_{o}$ | 8.9518 | 8.9399 | 9.7348 | 10.7284 |

$T{V}_{2}\left({u}_{w}\right),b=m{T}_{o}$ | 1529.2 | 1493.3 | 1445.3 | 1405.2 |

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**MDPI and ACS Style**

Huba, M.; Vrancic, D.
Tuning of PID Control for the Double Integrator Plus Dead Time Model by Modified Real Dominant Pole and Performance Portrait Methods. *Mathematics* **2022**, *10*, 971.
https://doi.org/10.3390/math10060971

**AMA Style**

Huba M, Vrancic D.
Tuning of PID Control for the Double Integrator Plus Dead Time Model by Modified Real Dominant Pole and Performance Portrait Methods. *Mathematics*. 2022; 10(6):971.
https://doi.org/10.3390/math10060971

**Chicago/Turabian Style**

Huba, Mikulas, and Damir Vrancic.
2022. "Tuning of PID Control for the Double Integrator Plus Dead Time Model by Modified Real Dominant Pole and Performance Portrait Methods" *Mathematics* 10, no. 6: 971.
https://doi.org/10.3390/math10060971