# PID Tuning Method Based on IMC for Inverse-Response Second-Order Plus Dead Time Processes

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## Abstract

**:**

## 1. Introduction

## 2. Inverse Response Second Order Plus Dead Time System Model

## 3. Design of Tuning Rules

#### 3.1. Internal Model Control

#### 3.2. Optimization Objective Function

#### 3.3. Scaling and Nondimensionalization

#### 3.4. Tuning Parameter Computation

^{®}/Simulink

^{®}. This computational tool allows one to find the value of the objective function for each data set of parameters of the process’ transfer function and $\gamma $. Since we used a central composite experimental design for rotatability, the optimization tool correspond to the steepest-descent gradient method [58].

## 4. Application Methodology

- Determine the transfer function of the inverse response second order plus dead time system and write it in the form of (5).
- Compute ${\widehat{\tau}}_{cult}$ by using (25).
- Select the value of $\gamma $, depending on the user’s needs. If $\gamma =0$, $OF=IAE$.
- Compute ${\widehat{\tau}}_{cOF}$ by using (35).
- Compute ${{\tau}_{c}}_{OF}$ by using (36).
- Check if the controller meets the desired performance.

## 5. Discussion on Performance Analysis

- $IAE={\int}_{0}^{\infty}\left|e\left(t\right)\right|dt$, $[=]$ %TO·UT;
- $ISE={\int}_{0}^{\infty}{e}^{2}\left(t\right)dt$, $[=]$(%TO)${}^{2}$·UT;
- $IMV={\int}_{0}^{\infty}\left|{\displaystyle \frac{dMV\left(t\right)}{dt}}\right|dt$, $[=]$ %CO;
- $MP$ (maximum peak), $[=]$ %TO;

## 6. Discussion on Robustness Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Experimental Design Results

${\widehat{\mathit{\tau}}}_{2}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cult}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|0}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|1}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|2}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|3}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|4}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|5}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|6}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|7}$ | ${\widehat{\mathit{\tau}}}_{\mathit{cFO}|8}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.26 | 0.89 | 0.21 | 0.49 | 1.10 | 1.70 | 1.94 | 2.10 | 2.10 | 2.10 | 2.57 | 2.57 | 3.44 |

0.26 | 0.89 | 0.80 | 0.55 | 1.66 | 2.09 | 2.39 | 2.63 | 2.63 | 2.75 | 2.88 | 2.88 | 2.88 |

0.26 | 3.21 | 0.21 | 1.50 | 2.96 | 4.25 | 4.69 | 5.33 | 5.87 | 6.59 | 7.22 | 8.05 | 8.05 |

0.26 | 3.21 | 0.80 | 2.11 | 4.07 | 4.64 | 5.35 | 5.70 | 5.92 | 6.04 | 6.90 | 6.90 | 7.41 |

0.74 | 0.89 | 0.21 | 0.89 | 1.56 | 2.28 | 2.65 | 3.22 | 3.22 | 3.41 | 3.41 | 3.55 | 3.55 |

0.74 | 0.89 | 0.80 | 0.93 | 2.17 | 2.54 | 3.01 | 3.38 | 3.55 | 3.87 | 4.19 | 4.19 | 4.19 |

0.74 | 3.21 | 0.21 | 2.38 | 4.22 | 5.32 | 5.94 | 6.46 | 6.88 | 6.88 | 7.81 | 7.81 | 7.81 |

0.74 | 3.21 | 0.80 | 2.87 | 5.20 | 5.57 | 6.40 | 7.02 | 7.34 | 7.75 | 8.13 | 8.39 | 8.70 |

0.10 | 2.05 | 0.50 | 0.97 | 2.09 | 2.74 | 2.99 | 3.54 | 3.78 | 4.40 | 4.89 | 5.36 | 5.82 |

0.90 | 2.05 | 0.50 | 2.04 | 3.51 | 4.09 | 4.75 | 5.27 | 5.75 | 5.99 | 6.13 | 6.13 | 6.35 |

0.50 | 0.10 | 0.50 | 0.19 | 0.71 | 1.23 | 1.62 | 1.85 | 1.96 | 2.00 | 2.00 | 2.00 | 2.10 |

0.50 | 4.00 | 0.50 | 2.80 | 5.21 | 5.94 | 7.08 | 7.32 | 7.60 | 7.96 | 8.24 | 9.60 | 8.68 |

0.50 | 2.05 | 0.01 | 1.28 | 2.16 | 3.10 | 5.39 | 5.80 | 5.80 | 5.86 | 5.86 | 5.80 | 5.86 |

0.50 | 2.05 | 0.01 | 1.71 | 3.50 | 3.10 | 4.48 | 4.88 | 5.25 | 5.59 | 5.59 | 5.59 | 5.59 |

0.50 | 2.05 | 0.50 | 1.54 | 3.08 | 3.71 | 4.23 | 4.63 | 4.97 | 4.97 | 4.97 | 5.35 | 5.35 |

## Appendix B. Performance Results for Remaining Sets of Parameters

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 16.7200 | 13.6500 | 2.3560 | 2.0500 |

WN | 17.2200 | 11.9800 | 2.9170 | 1.8622 |

ZN | 30.9500 | 16.1800 | 8.2430 | 3.9959 |

CCCC | 22.4200 | 11.6200 | 2.9820 | 2.5230 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 6.9980 | 8.1680 | 0.7902 | 1.0007 |

WN | 6.7140 | 6.4190 | 7.2390 | 1.0000 |

ZN | 8.7490 | 8.5640 | 4.1430 | 1.3683 |

CCCC | 7.3370 | 5.2440 | 3.6010 | 1.0039 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 6.3040 | 8.2120 | 1.9840 | 1.3950 |

WN | 5.6930 | 6.4080 | 2.8620 | 1.2494 |

ZN | 6.9180 | 7.0620 | 4.3250 | 1.7070 |

CCCC | 8.5510 | 5.9770 | 4.0270 | 2.0238 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 4.5670 | 5.9050 | 0.7466 | 1.0010 |

WN | 4.1240 | 4.1000 | 12.8100 | 1.0001 |

ZN | 4.8090 | 4.1760 | 1.9400 | 1.0000 |

CCCC | 5.2480 | 3.0810 | 0.8966 | 1.0822 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 7.5120 | 10.2800 | 1.6450 | 1.3635 |

WN | 8.4030 | 8.8060 | 3.8160 | 1.5593 |

ZN | 10.8800 | 13.1600 | 38.0600 | 2.0353 |

CCCC | 7.2170 | 8.1150 | 1.6610 | 1.2914 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 6.0110 | 8.4260 | 0.7220 | 1.0031 |

WN | 5.8830 | 6.5500 | 21.0900 | 1.3184 |

ZN | 8.0590 | 12.2700 | 58.0600 | 1.3625 |

CCCC | 5.6740 | 6.2400 | 1.2470 | 1.0119 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 13.1300 | 16.7700 | 1.6610 | 1.5205 |

WN | 14.2600 | 11.4000 | 3.2050 | 1.8102 |

ZN | 20.9300 | 17.6900 | 7.8383 | 2.5812 |

CCCC | 12.5000 | 13.5400 | 1.4470 | 1.5975 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 8.9170 | 13.0100 | 0.7630 | 0.9951 |

WN | 7.6430 | 6.9760 | 19.3000 | 1.0624 |

ZN | 9.8820 | 11.7200 | 7.1650 | 1.1496 |

CCCC | 7.6740 | 7.6310 | 2.5230 | 1.0080 |

## Appendix C. Plots for the Remaining Sets of Parameters

**Figure A1.**Time response disturbance rejection for the set ${P}_{1}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A2.**Time response set-point tracking for the set ${P}_{1}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A3.**Time response disturbance rejection for the set ${P}_{2}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A4.**Time response set-point tracking for the set ${P}_{2}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A5.**Time response disturbance rejection for the set ${P}_{3}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A6.**Time response set-point tracking for the set ${P}_{3}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A7.**Time response disturbance rejection for the set ${P}_{5}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A8.**Time response set-point tracking for the set ${P}_{5}$. %TO: percentage of transmitter output, UT: units of time.

## Appendix D. Additional Simulation in the Limits of the Experimental Region

Parameters | ${\widehat{\mathit{\tau}}}_{2}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ |
---|---|---|---|

${P}_{6}$ | 0.900 | 0.100 | 0.01 |

Set | CCV | WN | ZN | CCCC | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\widehat{\mathit{T}}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\widehat{\mathit{T}}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\widehat{\mathit{T}}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\mathit{T}}_{\mathit{d}}$ | |

${P}_{6}$ | 1.194 | 1.901 | 0.474 | 9.500 | 1.900 | 0.473 | 9.847 | 0.710 | 0.177 | 1.768 | 1.000 | 0.900 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 0.349 | 1.647 | 1.160 | 0.379 |

WN | 0.010 | 0.201 | 2.287 | 0.077 |

ZN | 0.007 | 0.124 | 5.181 | 0.104 |

CCCC | 0.084 | 0.567 | 1.167 | 0.224 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 0.870 | 1.647 | 1.538 | 1.001 |

WN | 0.227 | 0.206 | 569.1 | 1.026 |

ZN | 0.686 | 1.370 | 93.13 | 1.769 |

CCCC | 0.462 | 0.630 | 9.152 | 1.043 |

**Figure A9.**Time response disturbance rejection for the set ${P}_{6}$. %TO: percentage of transmitter output, UT: units of time.

**Figure A10.**Time response set-point tracking for the set ${P}_{6}$. %TO: percentage of transmitter output, UT: units of time.

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**Figure 2.**Behavior of ${\widehat{\tau}}_{cOF}$ for different values of ${\widehat{\tau}}_{cult}$, using different values of $\gamma $. The range of ${\widehat{\tau}}_{cult}$ was selected arbitrarily for illustration purposes.

**Figure 3.**Behavior of ${m}_{OF}$ with respect to variations in $\gamma $. %TO: percentage of transmitter output, %CO: percentage of controller output, UT: units of time.

**Figure 5.**Time response disturbance rejection for the set ${P}_{4}$. %TO: percentage of transmitter output, UT: units of time.

**Figure 6.**Time response set-point tracking for the set ${P}_{4}$. %TO: percentage of transmitter output, UT: units of time.

${\mathit{OF}}_{\mathit{\gamma}|\mathit{i}}$ | R (Fraction) | ${\mathit{R}}^{2}$ (%) | Durbin-Watson (Nondimensional) |
---|---|---|---|

$O{F}_{\gamma |0}$ | 0.9965 | 99.31 | 2.16 |

$O{F}_{\gamma |1}$ | 0.9918 | 98.36 | 2.02 |

$O{F}_{\gamma |2}$ | 0.9872 | 97.46 | 2.19 |

$O{F}_{\gamma |3}$ | 0.9857 | 97.15 | 2.42 |

$O{F}_{\gamma |4}$ | 0.9865 | 97.33 | 2.47 |

R (Fraction) | ${\mathit{R}}^{2}$ (%) | |
---|---|---|

${m}_{OF}\left(\gamma \right)$ | 0.9994 | 99.89 |

Parameters | ${\widehat{\mathit{\tau}}}_{2}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ |
---|---|---|---|

${P}_{1}$ | 0.2622 | 3.2095 | 0.2107 |

${P}_{2}$ | 0.5000 | 2.0500 | 0.0100 |

${P}_{3}$ | 0.5000 | 2.0500 | 1.0000 |

${P}_{4}$ | 0.5000 | 4.0000 | 0.5050 |

${P}_{5}$ | 0.7378 | 3.2095 | 0.7993 |

Set | CCV | WN | ZN | CCCC | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\widehat{\mathit{T}}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\widehat{\mathit{T}}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\widehat{\mathit{T}}}_{\mathit{d}}$ | ${\mathit{K}}_{\mathit{c}}\mathit{k}$ | ${\widehat{\mathit{T}}}_{\mathit{i}}$ | ${\mathit{T}}_{\mathit{d}}$ | |

${P}_{1}$ | 0.166 | 1.345 | 0.278 | 0.197 | 1.262 | 0.208 | 0.230 | 1.956 | 0.489 | 0.191 | 1.000 | 0.262 |

${P}_{2}$ | 0.256 | 1.503 | 0.336 | 0.366 | 1.500 | 0.333 | 0.417 | 1.739 | 0.435 | 0.348 | 1.000 | 0.500 |

${P}_{3}$ | 0.210 | 1.747 | 0.533 | 0.366 | 1.500 | 0.333 | 0.410 | 3.500 | 0.874 | 0.163 | 1.000 | 0.500 |

${P}_{4}$ | 0.128 | 1.656 | 0.458 | 0.188 | 1.500 | 0.333 | 0.225 | 2.973 | 0.743 | 0.131 | 1.000 | 0.500 |

${P}_{5}$ | 0.150 | 1.936 | 0.580 | 0.271 | 1.738 | 0.425 | 0.315 | 3.695 | 0.924 | 0.133 | 1.000 | 0.738 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 19.9700 | 19.3400 | 1.9780 | 1.9049 |

WN | 22.4500 | 15.0100 | 3.0450 | 2.0593 |

ZN | 38.9600 | 23.1000 | 7.8490 | 3.6781 |

CCCC | 23.4000 | 14.7000 | 2.3570 | 2.1652 |

Method | $\mathit{ISE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IAE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IMV}\phantom{\rule{4pt}{0ex}}$ | $\mathit{MP}\phantom{\rule{4pt}{0ex}}$ | Total |
---|---|---|---|---|---|

CCV | 4 | 2 | 4 | 4 | 14 |

WN | 3 | 3 | 2 | 3 | 11 |

ZN | 1 | 1 | 1 | 1 | 4 |

CCCC | 2 | 4 | 3 | 2 | 11 |

Method | $\mathit{ISE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{a}}$ | $\mathit{IAE}{\phantom{\rule{4pt}{0ex}}}^{\mathit{b}}$ | $\mathit{IMV}{\phantom{\rule{4pt}{0ex}}}^{\mathit{c}}$ | $\mathit{MP}{\phantom{\rule{4pt}{0ex}}}^{\mathit{d}}$ |
---|---|---|---|---|

CCV | 9.6500 | 13.0000 | 0.8140 | 0.9957 |

WN | 8,7230 | 8.0000 | 10.7500 | 1.0000 |

ZN | 11,4600 | 13.0800 | 4.6960 | 1.0581 |

CCCC | 8.9600 | 7.6490 | 1.2030 | 1.0048 |

Method | $\mathit{ISE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IAE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IMV}\phantom{\rule{4pt}{0ex}}$ | $\mathit{MP}\phantom{\rule{4pt}{0ex}}$ | Total |
---|---|---|---|---|---|

CCV | 2 | 2 | 4 | 4 | 12 |

WN | 4 | 3 | 1 | 1 | 9 |

ZN | 1 | 1 | 2 | 2 | 6 |

CCCC | 3 | 4 | 3 | 3 | 13 |

Method | $\mathit{ISE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IAE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IMV}\phantom{\rule{4pt}{0ex}}$ | $\mathit{MP}\phantom{\rule{4pt}{0ex}}$ | Total |
---|---|---|---|---|---|

CCV | 17 | 9 | 19 | 17 | 62 |

WN | 14 | 16 | 12 | 15 | 57 |

ZN | 6 | 6 | 5 | 6 | 23 |

CCCC | 13 | 19 | 14 | 12 | 58 |

Method | $\mathit{ISE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IAE}\phantom{\rule{4pt}{0ex}}$ | $\mathit{IMV}\phantom{\rule{4pt}{0ex}}$ | $\mathit{MP}\phantom{\rule{4pt}{0ex}}$ | Total |
---|---|---|---|---|---|

CCV | 12 | 8 | 20 | 17 | 57 |

WN | 19 | 16 | 6 | 14 | 55 |

ZN | 6 | 7 | 9 | 8 | 30 |

CCCC | 13 | 19 | 15 | 11 | 58 |

Method | Set ${\mathit{P}}_{1}$ | Set ${\mathit{P}}_{2}$ | Set ${\mathit{P}}_{3}$ | Set ${\mathit{P}}_{4}$ | Set ${\mathit{P}}_{5}$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\eta}}$ | $\widehat{\mathit{\theta}}$ | |

CCV | 5.9710 | 9.0310 | 3.8950 | 6.8300 | 5.3140 | 11.6650 | 8.7890 | 16.6050 | 8.2950 | 18.0600 |

WN | 6.3060 | 5.8040 | 4.0000 | 3.6900 | 3.5860 | 3.6900 | 7.7520 | 7.2350 | 6.0180 | 5.8100 |

ZN | 3.8050 | 0.7400 | 2.7560 | 3.8000 | 2.0930 | 1.0460 | 4.8640 | 1.1530 | 4.0550 | 1.7000 |

CCCC | 5.0950 | 3.6720 | 5.5650 | 7.0000 | 5.5590 | 7.0000 | 7.3040 | 6.4480 | 7.0650 | 7.5700 |

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## Share and Cite

**MDPI and ACS Style**

Castellanos-Cárdenas, D.; Castrillón, F.; Vásquez, R.E.; Smith, C.
PID Tuning Method Based on IMC for Inverse-Response Second-Order Plus Dead Time Processes. *Processes* **2020**, *8*, 1183.
https://doi.org/10.3390/pr8091183

**AMA Style**

Castellanos-Cárdenas D, Castrillón F, Vásquez RE, Smith C.
PID Tuning Method Based on IMC for Inverse-Response Second-Order Plus Dead Time Processes. *Processes*. 2020; 8(9):1183.
https://doi.org/10.3390/pr8091183

**Chicago/Turabian Style**

Castellanos-Cárdenas, Duby, Fabio Castrillón, Rafael E. Vásquez, and Carlos Smith.
2020. "PID Tuning Method Based on IMC for Inverse-Response Second-Order Plus Dead Time Processes" *Processes* 8, no. 9: 1183.
https://doi.org/10.3390/pr8091183