# Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models

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## Abstract

**:**

## 1. Introduction

- In the instance of a small or zero time delay $\tau =0$, we propose a variant in which the Maximum Time Delay Error (MTDE) $d{\tau}_{max}>0$ is the tuning parameter (see Section 3.2).
- Two optimal settings for the MP parameter are presented in Section 4. These are optimal in the sense that they minimise a Pareto performance objective (i.e., integrated absolute error for combined step changes in output and input disturbances) on two different aspects. One additional MP parameter is deduced from approximating the time delay with a (2, 1) Pade approximation in Section 3.3.
- Additional MP parameter settings are suggested for minimising a variety of given indices.
- The presented method (including variants of this) is demonstrated and compared to the Pareto-Optimal (PO) and SIMC (when possible) tuned PI controllers on various motivated (possible) higher order process model examples in Section 5.

## 2. Preliminary Theory

#### 2.1. Definitions

- ${\mathrm{IAE}}_{vu}$ evaluates the performance in case of a step input disturbance (${H}_{v}\left(s\right)={H}_{p}\left(s\right)$), $v=1$ (default), with the reference, $r=0$.
- ${\mathrm{IAE}}_{vy}$ evaluates the performance in case of a step output disturbance (${H}_{v}\left(s\right)=1$), $v=1$ (default), with the reference, $r=0$.
- ${\mathrm{IAE}}_{r}$ evaluates the performance in case of a reference unit step, $r=1$, with the disturbance, $v=0$.

#### 2.2. Lag-Dominant Systems

#### 2.3. SIMC Tuning Rules

## 3. Tuning for Maximum Time Delay Error

#### 3.1. Integrator Plus Time Delay Process

**Algorithm**

**1**

**Example**

**1**

#### 3.2. Pure Integrating Process

#### 3.3. Using a ($2,1$) Pade Approximation

## 4. Optimal Performance Settings

**length(${M}_{s}$)**.

^{o}(${\delta}_{i}$) is pre-calculated as follows

## 5. Simulation Examples

**Example**

**2**

**Example**

**3**

**Example**

**4**

**Example**

**5**

**Example**

**6**

## 6. Discussion

#### Remarks to Section 3

## 7. Concluding Remarks

- Two optimal settings for the MP parameter are presented in Section 4. These are optimal in the sense that they minimise the main performance objective ${V}_{M}$ on two different aspects. Interestingly, one of the MP parameters may (arguably) be deduced from approximating the time delay with a (2, 1) Pade approximation in Section 3.3.
- In the case of a small or zero time delay $\tau =0$, we propose a variant in which the MTDE $d{\tau}_{max}$ is the tuning parameter.
- Note that for an IPTD model, the SIMC tuned PI controllers are seen far from optimal, i.e., PO (or (almost) equivalently, Algorithm 1 with the MP parameter setting as $\overline{c}=2.5$). See Section 4.
- The presented method (and variants of this) is successfully demonstrated and compared to the SIMC and PO PI controllers on numerous motivated process model examples in Section 5.
- Note that, for the higher order process models in Examples 4 and 5, we use the PRC model reduction technique, which is generally easier to apply than the half-rule technique proposed in [12]. The half-rule technique is not compatible with handling complex poles.
- Some surprisingly optimal results are documented for Example 6, where a tuning method based on varying the gain velocity, $k=\zeta {R}_{1}$, (${R}_{1}$, is the ZN unit reaction rate), i.e., the tuning parameter is $\zeta $. Note that setting the RTDE $\delta =\overline{c}$ (i.e., an ad hoc choice) equal a constant is advisable.
- Note that the results in Section 5 are based on the original (possible) higher order models. The approximated IPTD models are only used for the PI controller design.

## Author Contributions

## Conflicts of Interest

## Abbreviations

PI | Proportional Integrating |

IPTD | Integrator Plus Time Delay |

FOPTD | First Order Plus Time Delay |

ZN | Ziegler–Nichols |

IAE | Integrated Absolute Error |

ITAE | Integrated Time-weighted Absolute Error |

ISE | Integrated Square Error |

ITSE | Integrated Time-weighted Square Error |

TV | Total input Value |

MP | Method Product |

IMC | Internal Model Control |

SIMC | Simple/Skogestad Internal Model Control |

GM | Gain Margin |

PM | Phase Margin |

DM | Delay Margin |

MTDE | Maximum Time Delay Error |

PO | Pareto-Optimal |

RTDE | Relative Time Delay Error |

## References

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**Figure 1.**Consider PI control of the FOPTD process model, ${H}_{p}\left(s\right)=\frac{{e}^{-s}}{s}$. The figure shows the robustness ${M}_{s}$ as a function of the MP parameter $\overline{c}$, given constant robustness, for the interval $1.5\le \overline{c}\le 4.0$.

**Figure 2.**Consider a control feedback system where the plant model is described by the process model, ${H}_{p}\left(s\right)$, PI controller, ${H}_{c}\left(s\right)={K}_{p}\frac{1+{T}_{i}s}{{T}_{i}s}$, and the disturbance model, ${H}_{v}\left(s\right)$, where disturbance v at the input when, ${H}_{v}\left(s\right)={H}_{p}\left(s\right)$, and at the output when, ${H}_{v}\left(s\right)=1$. Input u, output y and reference r.

**Figure 3.**Reference example (Example 2). Consider PI control of the IPTD model, ${H}_{p}\left(s\right)=\frac{{e}^{-s}}{s}$. The figure illustrates the trade-off between the Pareto performance objective J (Equation (41)) and robustness ${M}_{s}$ (Equation (10)). It illustrates the MP parameters $\overline{c}=2.5$ and $\overline{c}=2.7$ for Algorithm 1 proposed in Section 4. SIMC is added for comparison.

**Figure 4.**Consider PI control of an IPTD process, ${H}_{p}\left(s\right)=k\frac{{e}^{-\tau s}}{s}$ with process parameters $k=1$ and $\tau =1$. PI controller ${H}_{c}\left(s\right)={K}_{p}\frac{1+{T}_{i}s}{{T}_{i}s}$ with settings as in Algorithm 1. The figure shows the indices ${M}_{s}$, ${\mathrm{ITAE}}_{vu}$, ${\mathrm{IAE}}_{r}$, ${\mathrm{ITAE}}_{r}$, ${\mathrm{IAE}}_{r}$, $\mathrm{TV}$, $\mathrm{ISE}$, ${\mathrm{ITAE}}_{vu}$ and $\mathrm{IAE}$ as a function of varying the MP parameter $\overline{c}\in [1.5,4.0]$ and with prescribed RTDE $\delta =1.6$.

**Figure 5.**Example 2 (Reference example). Consider PI control of an IPTD process model, ${H}_{p}\left(s\right)=\frac{{e}^{-s}}{s}$. The figure illustrates the time-domain responses, given a prescribed robustness ${M}_{s}=1.59$, of the following methods: the PO PI, SIMC with prescribed closed loop time constant ${T}_{c}=1.24\phantom{\rule{0.166667em}{0ex}}\tau $ and Algorithm 1 where the MP parameter $\overline{c}=2.5$ (proposed in Section 4) and RTDE $\delta =1.79$. An output disturbance unit step is presented at time $t=0$ and an input disturbance unit step at time $t=50$.

**Figure 6.**Example 3. Consider PI control of the FOPTD process model, ${H}_{p}\left(s\right)=K\frac{{e}^{-\tau s}}{Ts+1}$, where $K=5.7$, $\tau =4$ and $T=60$. The figure shows the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness ${M}_{s}$ (Equation (10)). It illustrates the MP parameters $\overline{c}=2.5$ and $\overline{c}=2.7$ for Algorithm 1 (proposed in Section 4). SIMC with set-point time constant ${T}_{c}$ is added for comparison.

**Figure 7.**Example 3. Consider PI control of the FOPTD process model, ${H}_{p}\left(s\right)=K\frac{{e}^{-\tau s}}{Ts+1}$, where $K=5.7$, $\tau =4$ and $T=60$. The figure illustrates the time-domain responses, given a prescribed robustness ${M}_{s}=1.59$, of the following methods: the PO PI, SIMC with prescribed closed loop time constant ${T}_{c}=1.10\phantom{\rule{0.166667em}{0ex}}\tau $, and Algorithm 1 where the MP parameter $\overline{c}=2.5$ (proposed in Section 4) and RTDE $\delta =1.56$. An output disturbance unit step is presented at time $t=0$ and an input disturbance unit step at time $t=140$.

**Figure 8.**Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness ${M}_{s}$ (Equation (10)). It shows the MP parameter settings $\overline{c}=2.5$ and $\overline{c}=2.7$ for Algorithm 1 proposed in Section 4. SIMC is added for comparison.

**Figure 9.**Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the time-domain responses, given a prescribed robustness ${M}_{s}=1.59$, of the following methods: the PO PI controller vs. SIMC with closed loop time constant ${T}_{c}=1.33\phantom{\rule{0.166667em}{0ex}}\tau $, and PRC + Algorithm 1 where the MP parameter setting $\overline{c}=2.7$ (proposed in Section 4) and RTDE $\delta =1.63$. An output disturbance unit step is presented at time $t=0$ and an input disturbance unit step at time $t=35$.

**Figure 10.**Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure shows the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness ${M}_{s}$ (Equation (10)). It illustrates the PO PI controllers and PRC + Algorithm 1 variants with MP parameter settings $\overline{c}=4.0$, $\overline{c}=2.5$ and $\overline{c}=2.7$.

**Figure 11.**Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness ${M}_{s}=1.59$, of following methods: the PO PI and the PRC + Algorithm 1 where the MP parameter setting $\overline{c}=2.5$ (proposed in Section 4) and RTDE $\delta =2.20$. An output disturbance unit step is presented at time $t=0$ and an input disturbance unit step at time $t=80$.

**Figure 12.**Example 6. PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the trade-off curves with the Pareto performance objective J (Equation (41)) and robustness ${M}_{s}$ (Equation (10)). It shows the PO PI controllers with robustness ${M}_{s}$ and the ζ-PRC + Algorithm 1 variant where the RTDE $\delta =\overline{c}=2.7$ is fixed and the main tuning parameter is ζ.

**Figure 13.**Example 6. PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness ${M}_{s}=1.59$, for the following methods: the PO PI and the ζ-PRC + Algorithm 1 variant with MP parameter and MTDE settings $\overline{c}=\delta =2.7$, and tuning parameter $\zeta =0.74$. An output disturbance unit step is presented at time $t=0$ and an input disturbance unit step at time $t=80$.

**Table 1.**The table shows the recommended MP parameters $\overline{c}$ if one wants to minimise the main performance objective ${V}_{M}$ (Equation (42)) for different servo-regulator parameters ${s}_{r}$ in Equation (41). The optimal $\overline{c}$ values as indicated are almost constant in the interval $\delta \in [1.1,3.4]$ ([2]).

${\mathit{s}}_{\mathit{r}}$ | 0 | 0.1 | 0.25 | 0.5 | 0.75 | 1 |
---|---|---|---|---|---|---|

$\overline{c}$ | 2.4 | 2.5 | 2.6 | 2.7 | 3.7 | ∞ |

**Table 2.**Reference example (Example 2), i.e., an IPTD model, ${H}_{p}\left(s\right)=\frac{{e}^{-s}}{s}$. Comparing the different settings for the MP parameters for Algorithm 1 and SIMC using the main performance objective ${V}_{M}$ (Equation (42)).

Method | $\overline{\mathit{c}}$ = 2.5 | $\overline{\mathit{c}}$ = 2.7 | SIMC |
---|---|---|---|

${V}_{M}$/e-4 | 0.02 | 1.56 | 592.75 |

**Table 3.**Summary: The table shows the recommended settings for the MP parameter $\overline{c}$ for minimizing the objectives in the first row.

${\mathit{M}}_{\mathit{s}}$ | ${\mathbf{IAE}}_{\mathit{vu}}$ | ${\mathbf{ITAE}}_{\mathit{vu}}$ | ${\mathbf{ITAE}}_{\mathit{r}}$ | ${\mathbf{IAE}}_{\mathit{r}}$ | ${\mathit{V}}_{\mathit{M}}(\mathit{\delta}\mathit{O})$ | ${\mathit{V}}_{\mathit{M}}\left(\mathit{t}\right)$ | |
---|---|---|---|---|---|---|---|

$\overline{c}$ | 2.0 | 2.4 | 2.4 | 2.6 | 4.0 | 2.7 | 2.5 |

**Table 4.**Consider PI controller settings for an IPTD system, ${H}_{p}\left(s\right)=k\frac{{e}^{-\tau s}}{s}$, with varying gain velocity, k, and time delay $\tau \ge 0$. The table illustrates the ${\overline{c}}_{\mathrm{min}}={\mathrm{arg}\mathrm{min}}_{\overline{c}}{M}_{s}$, i.e., the minimum of the ${M}_{s}$, ${\mathrm{IAE}}_{vu}$, $\mathrm{ITAE}$ and ${\mathrm{ITAE}}_{vu}$ indices as a function of $\overline{c}$, with PI controller settings from Algorithm 1.

k | $\mathit{\tau}$ | ${\mathit{M}}_{\mathit{s}}$ | ${\mathbf{IAE}}_{\mathit{vu}}$ | $\mathbf{ITAE}$ | ${\mathbf{ITAE}}_{\mathit{vu}}$ |
---|---|---|---|---|---|

1 | 0.1 | 2.0 | 2.45 | 2.45 | 2.45 |

1 | 0.3 | 2.0 | 2.4 | 2.45 | 2.4 |

1 | 0.5 | 2.0 | 2.4 | 2.45 | 2.4 |

1 | 1 | 2.0 | 2.4 | 2.4 | 2.4 |

1 | 2 | 2.0 | 2.4 | 2.4 | 2.4 |

1 | 4 | 2.0 | 2.4 | 2.4 | 2.4 |

0.1 | 1 | 2.0 | 2.4 | 2.5 | 2.4 |

0.1 | 2 | 2.0 | 2.4 | 2.45 | 2.4 |

0.1 | 4 | 2.0 | 2.4 | 2.45 | 2.4 |

**Table 5.**Consider PI controller settings for an IPTD system, ${H}_{p}\left(s\right)=k\frac{{e}^{-\tau s}}{s}$, with varying gain velocity, k, and time delay $\tau \ge 0$. The table illustrates ${\overline{c}}_{\mathrm{min}}={\mathrm{arg}\mathrm{min}in}_{\overline{c}}{M}_{s}$, i.e., the minimum of ${\mathrm{IAE}}_{r}$, $\mathrm{Tv}$, $\mathrm{ISE}$, ${\mathrm{ITAE}}_{r}$ and $\mathrm{ITSE}$ indices as a function of $\overline{c}$, with PI controller settings from Algorithm 1.

k | $\mathit{\tau}$ | ${\mathbf{ITAE}}_{\mathit{r}}$ | $\mathbf{ITSE}$ | $\mathbf{ISE}$ | $\mathbf{TV}$ | ${\mathbf{IAE}}_{\mathit{r}}$ |
---|---|---|---|---|---|---|

1.0 | 0.1 | 2.7 | 3.4 | 4.0 | 4.0 | 4.0 |

1.0 | 0.3 | 2.7 | 3.2 | 4.0 | 4.0 | 4.0 |

1.0 | 0.5 | 2.7 | 3.1 | 3.5 | 4.0 | 4.0 |

1.0 | 1.0 | 2.6 | 3.1 | 3.2 | 4.0 | 4.0 |

1.0 | 2.0 | 2.6 | 3.0 | 3.1 | 4.0 | 4.0 |

1.0 | 4.0 | 2.6 | 3.0 | 3.1 | 4.0 | 4.0 |

0.1 | 1.0 | 2.6 | 3.9 | 4.0 | 4.0 | 4.0 |

0.1 | 2.0 | 2.6 | 3.2 | 4.0 | 4.0 | 4.0 |

0.1 | 4.0 | 2.6 | 3.1 | 3.6 | 4.0 | 4.0 |

**Table 6.**Example 2. Consider PI control of the IPTD process model, ${H}_{p}\left(s\right)=\frac{{e}^{-s}}{s}$. The table shows the controller parameter, indices and margins are given for prescribed robustness ${M}_{s}=1.59$ for the following methods: Alg. 1 ($\overline{c}=2.5,\delta =1.79$), SIMC (${T}_{c}=1.24\phantom{\rule{0.166667em}{0ex}}\tau $) and PO PI (${M}_{s}=1.59$).

Alg. 1 | SIMC | PO PI | |
---|---|---|---|

${K}_{p}$ | 0.41 | 0.45 | 0.41 |

${T}_{i}$ | 6.14 | 8.96 | 6.28 |

${\mathrm{IAE}}_{vy}$ | 4.39 | 4.24 | 4.37 |

${\mathrm{IAE}}_{vu}$ | 15.26 | 20.06 | 15.39 |

J | 1.52 | 1.64 | 1.52 |

TV | 3.33 | 3.12 | 3.31 |

GM | 3.56 | 3.34 | 3.54 |

PM | 44.57 | 50.02 | 44.94 |

DM | 1.79 | 1.90 | 1.80 |

${M}_{s}$ | 1.59 | 1.59 | 1.59 |

**Table 7.**Example 3. The table shows the comparison of the settings for Algorithm 1 and SIMC using the main performance objective ${V}_{M}$ (Equation (42)).

Method | $\overline{\mathit{c}}$ = 2.5 | $\overline{\mathit{c}}$ = 2.7 | SIMC |
---|---|---|---|

${V}_{M}$/e-2 | 0.57 | 1.08 | 6.96 |

**Table 8.**Example 3. The table shows the PI controller parameter, indices and margins are given for prescribed robustness ${M}_{s}=1.59$.

Alg. 1 | SIMC | PO PI | |
---|---|---|---|

${K}_{p}$ | 1.17 | 1.25 | 1.12 |

${T}_{i}$ | 22.55 | 33.60 | 19.47 |

${\mathrm{IAE}}_{vy}$ | 15.13 | 13.53 | 15.70 |

${\mathrm{IAE}}_{vu}$ | 17.73 | 25.16 | 15.89 |

J | 1.39 | 1.52 | 1.37 |

TV | 3.94 | 3.70 | 4.05 |

GM | 3.36 | 3.22 | 3.46 |

PM | 50.49 | 56.21 | 47.83 |

DM | 7.51 | 8.08 | 7.26 |

${M}_{s}$ | 1.59 | 1.59 | 1.59 |

**Table 9.**Example 4. The table shows the comparison of the different settings for Algorithm 1 and SIMC using the main performance ${V}_{M}$ (Equation (42)).

Method | $\overline{\mathit{c}}$ = 2.5 | $\overline{\mathit{c}}$ = 2.7 | SIMC |
---|---|---|---|

${V}_{M}$/e-3 | 0.7 | 0.3 | 6.2 |

**Table 10.**Example 4. The table shows the PI controller parameters, indices and margins are given for prescribed robustness ${M}_{s}=1.59$.

Alg. 1 | SIMC | PO PI | |
---|---|---|---|

${K}_{p}$ | 0.78 | 0.91 | 0.85 |

${T}_{i}$ | 5.35 | 7.04 | 6.06 |

${\mathrm{IAE}}_{vy}$ | 3.62 | 3.35 | 3.48 |

${\mathrm{IAE}}_{vu}$ | 6.83 | 7.74 | 7.14 |

J | 1.41 | 1.41 | 1.39 |

TV | 3.77 | 3.77 | 3.77 |

GM | 6.74 | 6.13 | 6.39 |

PM | 43.63 | 46.74 | 45.19 |

DM | 1.54 | 1.49 | 1.51 |

${M}_{s}$ | 1.59 | 1.59 | 1.59 |

**Table 11.**Example 5. The table shows the different MP parameter settings for the PRC + Algorithm 1 variant with corresponding main performance ${V}_{M}$ (Equation (42)).

$\overline{\mathit{c}}$ | 2.5 | 2.7 | 4 |
---|---|---|---|

${V}_{M}$/e-2 | 0.84 | 1.05 | 6.13 |

**Table 12.**Example 5. The corresponding controller parameter, indices and margins are given for prescribed robustness ${M}_{s}=1.59$.

PRC + Alg. 1 | PO PI | |
---|---|---|

${K}_{p}$ | −1.42 | −1.70 |

${T}_{i}$ | 12.18 | 14.90 |

${\mathrm{IAE}}_{vy}$ | 6.41 | 5.88 |

${\mathrm{IAE}}_{vu}$ | 8.59 | 8.72 |

J | 1.04 | 1.00 |

TV | 4.62 | 5.06 |

GM | 13.80 | 11.84 |

PM | 43.90 | 44.20 |

DM | 3.03 | 2.74 |

${M}_{s}$ | 1.59 | 1.59 |

**Table 13.**Example 6. Comparing the following variants, ζ-PRC + Algorithm 1 with MP parameter and MTDE settings $\overline{c}=\delta =2.7$ and $\zeta =0.74$, and the PRC + Algorithm 1 with MP parameter setting $\overline{c}=2.5$, using the main performance ${V}_{M}$ defined in Equation (42).

Variant | PRC | $\mathit{\zeta}$-PRC |
---|---|---|

${V}_{M}$/e-4 | 83.6 | 0.02 |

**Table 14.**Example 6. The corresponding controller parameter, indices and margins are given for prescribed robustness ${M}_{s}=1.59$. ζ-PRC + Algorithm 1.

$\mathit{\zeta}$-Alg. 1 | PO PI | |
---|---|---|

${K}_{p}$ | −1.70 | −1.70 |

${T}_{i}$ | 14.82 | 14.90 |

${\mathrm{IAE}}_{vy}$ | 5.88 | 5.88 |

${\mathrm{IAE}}_{vu}$ | 8.69 | 8.72 |

J | 1.00 | 1.00 |

TV | 5.06 | 5.06 |

GM | 11.85 | 11.84 |

PM | 44.15 | 44.20 |

DM | 2.74 | 2.74 |

${M}_{s}$ | 1.59 | 1.59 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Dalen, C.; Di Ruscio, D.
Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models. *Algorithms* **2018**, *11*, 86.
https://doi.org/10.3390/a11060086

**AMA Style**

Dalen C, Di Ruscio D.
Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models. *Algorithms*. 2018; 11(6):86.
https://doi.org/10.3390/a11060086

**Chicago/Turabian Style**

Dalen, Christer, and David Di Ruscio.
2018. "Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models" *Algorithms* 11, no. 6: 86.
https://doi.org/10.3390/a11060086