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Parametrization and Optimal Tuning of Constrained Series PIDA Controller for IPDT Models^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. MRDP-Based PIDA Controller Design

#### 2.1. Parallel PIDA and PIDA Controllers with Automatic Reset

**Remark 1**

**.**The PIDA transfer function (5) describes the effect of the positive feedback loop from the output of the PDA controller in the proportional band of the control when the total output of the controller does not exceed the prescribed limits, i.e., when

#### 2.2. Process Approximation by IPDT Model

#### 2.3. Speed- and Shape-Related Performance Measures

**Remark 2**

**.**When the pure time delay ${e}^{-{T}_{d}s}$ is added to the loop with a single integrator and replaced by a finite number of terms of the Taylor expansion, the degree of the corresponding process transfer function increases. As a consequence, the dimensions of the state vector and the corresponding state controller also increase [35]. PIDA controllers are particularly suitable for the third-order model. The question arises whether the output of the controller should be evaluated according to the ideal 1P shape (given by the delay-free process) or according to the ideal 3P shape (with four monotonic intervals) by the $T{V}_{3}\left(u\right)$ measure [45] according to Feldbaum’s theorem [47].

#### 2.4. MRDP-PIDA Controllers for IPDT Models

#### 2.5. Design of Controller Filters

**Remark 3**

**.**Most of the existing works dealing with the design of PID, PDA and PIDA controllers either do not solve this problem at all or solve it inadequately. Namely, the filters must already be taken into account when approximating the process with the IPDT or FOTD model, as in [35,39]), or the filtering problem is solved separately for the derivative and for the acceleration component of the controller [14,16,17,48]. This unnecessarily increases the number of filter time constants, and the inclusion of different delays in specific controller channels complicates the analytical design. Therefore, to reduce the number of parameters and simplify the solution, it is much easier to filter all controller terms with a single binomial filter ${Q}_{n}\left(s\right)$

**Remark 4**

**.**The closed-loop system with transport delay generally has an infinite number of complex conjugate poles that can affect the stability of the closed-loop system. The stability of the dominant poles (14) is guaranteed by their negative sign. The stability and performance of the entire closed loop can be most easily checked by evaluating the shape of the Nyquist curve (see Figure 2). When the angular velocity ω (parameter of the Nyquist curve) increases for all considered n, the curve passes through the critical point $(-1,0j)$ on the left side with a sufficient margin. As can be seen from the Nyquist curves, the neglected complex poles of the circuit represented by the cycles around the origin, with the crossing points with the negative real axis closer to the critical point, obviously do not reduce the stability. Thus, although there are infinitely many stable conjugate complex poles, they do not significantly affect the linear step responses. These remain smooth with a minimum number of monotonic intervals for setpoint and disturbance step responses (see Figure 3). However, if the calculated ${T}_{e}$ is further reduced, the influence of neglected poles could already lead to a shift of the intersection points of the Nyquist curve closer to the critical point. In the time domain, this would contribute to a deformation of the ideal waveforms and could even lead to an instability of the loop.

**Proposition 1**

**.**In order to ensure stable closed-loop responses, the equivalent filter delay ${T}_{e}$ used in controller tuning according to (20) cannot be reduced arbitrarily. In terms of stability and monotonic responses, it is recommended to use the value ${T}_{e}\ge {T}_{dp}/5$, even in a situation with relatively low measurement noise and process uncertainties. For higher measurement noise amplitudes, ${T}_{e}$ should be increased accordingly.

#### 2.6. Basic Constrained Series MRDP-PIDA Modifications

#### 2.7. Absolute Stability Test

**Definition 1**

**.**Absolute stability (introduced by [57]) means that we can find $c>0$ and $\delta >0$ such that any closed loop solution of the system $x\left(t\right)$ satisfies the following relation of a monotonic decrease

#### 2.8. Design of Prefilter

## 3. Problem Formulation

#### 3.1. Effects of the Tuning Parameters on Absolute Stability

#### 3.2. Quantitative Evaluation of the Effects of the Tuning Parameters

**Remark 5**

**.**An optimal PIDA controller should guarantee the minimum of the chosen cost function J that is achievable under the given constraints on the shape-related deviations of the input and output variables (38).

#### 3.3. One-Dimensional Performance Test

#### 3.4. Evaluation of the Controller Tuning

**Proposition 2**

**.**“MRDP-optimal” values for the setpoint step responses are ${\tau}_{2}=0.232$ (23), or ${\tau}_{2}=0.22$ (24)–(28). The values increased by 50% (${\tau}_{2}=0.348$ and ${\tau}_{2}=0.33$, respectively) can be recommended as“near optimal” for the disturbance step responses.

**Proposition 3**

**.**Settings (27) and (28) are denoted by the symbols PIDA${}_{w0}$, or PIDA${}_{wm}$, respectively. They ensure near-minimum values of the $IA{E}_{s}$ for the setpoint step responses, with allowable deviations from the ideal shapes of the input and output process signals. The setting corresponding to the numerically simplest value ${\tau}_{2}=0.2$ can be calculated as follows:

## 4. SIMC Controller Design

#### 4.1. Simplified Process Modeling

#### 4.2. Design of the SIMC Controller

**Remark 6**

**.**Note that the model-based design (58), which is based on the cancellation of the process transfer function $F\left(s\right)$, is applicable only to stable models. Therefore, the SIMC method [31] has been significantly improved in practice by an ad hoc requirement of a double real dominant pole of a delay-free loop. In this way, the applicability of the method could be extended at least for integrating process models. Such models can also be interpreted as a limiting case of systems with a long dominant time constant, if, e.g.,

#### 4.3. SIMC Design for Higher-Order Models

- Once the desired closed-loop transfer function (57) has been chosen, the filter time constant ${T}_{f}$, which is required to implement the controller, should be systematically added to the process model using the aforementioned half-rule. Instead, the implementation of a series PID controller with an additional first order filter ${T}_{f}={T}_{D}/100$ was proposed in [31]$$U\left(s\right)={K}_{c}\frac{{T}_{i}s+1}{{T}_{i}s}\left(\right)open="("\; close=")">W\left(s\right)-\frac{{T}_{D}s+1}{{T}_{f}s+1}Y\left(s\right)$$The author admitted that in practice (especially for noisy processes) larger values of ${T}_{f}\in [{T}_{D}/10,{T}_{D}/5]$ have to be used. However, it is not clear why the filter was not systematically included in the design using the half rule. Indeed, the design of the controller without considering ${T}_{f}$ is not accurate, especially if you use a higher-order filter to reduce the noise level.
- The controller design is based on the setpoint response requirements, so the performance of disturbance responses can only be considered indirectly.
- The SIMC design does not address possible control signal (or state) constraints.

#### 4.4. Prefilter in SIMC Design

## 5. IPDT System Control

#### 5.1. Setpoint and Disturbance Responses without Measurement Noise

#### 5.2. Measurement Noise Generated by the Uniform Random Number Noise Generator

## 6. Stable Process Control

#### 6.1. Step Response Based Approximation of Stable Process by IPDT Model

#### 6.2. Setpoint and Disturbance Responses—No Noise

## 7. Discussion of the Results Obtained

**Remark 7**

**.**While the proposed methods for setpoint and disturbance rejection responses allow faster modes for controlling real processes with higher orders, the SIMC method does not have such an option. Moreover, refinement of the dead-time element approximation by higher-order controllers in SIMC leads to excessive controller effort due to the mismatch between the actual and desired first-order dynamics.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

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

3P | Three-Pulse, response with 4 monotonic segments (3 extreme points) |

ARC | Automatic-Reset Controller |

FOTD | First-Order Time-Delayed |

$IAE$ | Integral of Absolute Error |

IMC | Internal Model Control |

IPDT | Integrator Plus Dead-Time |

MRDP | Multiple Real Dominant Pole |

PDA | Proportional-Derivative-Accelerative |

PI | Proportional-Integral |

PIDA | Proportional-Integral-Derivative-Accelerative |

SIMC | SIMple Control |

SOTD | Second-Order Time-Delayed |

TOTD | Third-Order Time-Delayed |

TV | Total Variation |

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

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

TV${}_{3}$ | Deviation from 3P Shape |

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**Figure 1.**Automatic-reset-based controller with the positive feedback filter ${F}_{R}\left(s\right)$, prefilter ${F}_{P}\left(s\right)$, input disturbance ${d}_{i}$ and the measurement noise $\delta $.

**Figure 2.**Nyquist curves of the loop with IPDT process and PIDA-controllers tuned according to (15) with ${T}_{e}=1$ and ${T}_{e}=0.2$; ${K}_{s}=1$; ${T}_{dp}=1$; filter ${Q}_{n}\left(s\right),n\in [2,5]$ with ${T}_{f}={T}_{e}/n$.

**Figure 3.**Unit setpoint and disturbance step responses of the IPDT system (8) (${T}_{dp}=1$, ${K}_{sp}=1$) with PIDA controllers, $n\in [2,5]$, ${T}_{e}=1$ (

**full curve**) and ${T}_{e}=0.2$ (

**dotted**), ${T}_{f}={T}_{e}/n$, ${T}_{s}=0.001$, no noise.

**Figure 4.**Nonlinear standard form of the loop with the linear part ${L}_{s}\left(s\right)$ (31) and the saturation nonlinearity from the sector $[\u03f5,1],\phantom{\rule{0.277778em}{0ex}}\u03f5>0$.

**Figure 6.**Verification of absolute stability of PIDA-controllers tuned according to (25) with ${\tau}_{2}={T}_{2}/{T}_{d}\in [0.13,0.38]$, (20) for different orders of the filter ${Q}_{n}\left(s\right)$ (19), ${T}_{f}={T}_{e}/n,n\in [2,5]$ and ${T}_{e}=1$ (

**above**) and ${T}_{e}=0.2$ (

**below**); ${K}_{s}=1$; ${T}_{dp}=1$.

**Figure 7.**Performance measures of constrained series PIDA-controller tuned according to (25) with ${\tau}_{2}={T}_{2}/{T}_{d}\in [0.13,0.44],\Delta {\tau}_{2}=0.01$, (20) and ${T}_{e}=1$ (

**above**) and ${T}_{e}=0.2$ (

**below**); ${K}_{s}=1$; ${T}_{dp}=1$; filter ${Q}_{n}\left(s\right),{T}_{f}={T}_{e}/n,n\in [2,5]$, ${T}_{s}=0.001$ and prefilter (33); ${U}_{max}=0.1,{U}_{min}=-0.1$ for setpoint responses and ${U}_{max}=0.1,{U}_{min}=-1.1$ for disturbance step responses with the minimal $IAE$ values determined for admissible shape deviations (40) and (41) denoted by o and x.

**Figure 8.**Combined cost function (37) of constrained series PIDA-controllers tuned according to (25) with ${\tau}_{2}={T}_{2}/{T}_{d}\in [0.13,0.44],\Delta {\tau}_{2}=0.01$, (20) and ${T}_{e}=1$ (

**right**) and ${T}_{e}=0.2$ (

**left**); ${K}_{s}=1$; ${T}_{dp}=1$; filter ${Q}_{n}\left(s\right),{T}_{f}={T}_{e}/n,n\in [2,5]$, ${T}_{s}=0.001$ and prefilter (33); ${U}_{max}=0.1,{U}_{min}=-0.1$ for setpoint responses and ${U}_{max}=0.1,{U}_{min}=-1.1$ for disturbance step responses; $IA{E}_{s,min}$ and $IA{E}_{d,min}$ correspond to (40) (

**above**) and (41) (

**below**).

**Figure 9.**Unit setpoint and disturbance step responses of the IPDT system (8) (${T}_{dp}=1$, ${K}_{sp}=1$) with the constrained PIDA${}_{w0}$ and PIDA${}_{d0}$ controllers with $n=4$, ${T}_{e}=0.5,{T}_{f}={T}_{e}/n$ and SIMC-PI, PID and PIDA controllers (${}^{1}{R}^{1}$, ${}^{1}{R}^{2}$ and ${}^{1}{R}^{3}$) with ${T}_{cl}={T}_{dp}$ (

**dotted**) and ${T}_{cl}={T}_{dp}/2$ (

**dashed**), $u\in [-0.1,0.1]$ for setpoint responses and $u\in [-1.1,0.1]$ for disturbance responses, ${T}_{s}=0.001$, no noise.

**Figure 10.**Unit setpoint and disturbance step responses of the stable process (51) with the constrained PIDA${}_{w0}$ and PIDA${}_{d0}$ controllers, $n=4$, ${T}_{dp}=0.12$, ${K}_{sp}=1.01$, ${T}_{e}={T}_{dp},n=4,{T}_{f}={T}_{e}/n$, and SIMC-PI, PID and PIDA controllers based on the first-order models (52) with ${T}_{cl}=0.75{T}_{dp},{T}_{dp}=0.15$ (

**dotted**) and the second-order models (54) with ${T}_{cl}=0.5{T}_{dp},{T}_{dp}=0.05+{T}_{f},{T}_{f}=0.12$ (

**dashed**), $u\in [-1.1,1.1]$, ${T}_{s}=0.001$, no noise.

**Table 1.**IPDT system: Performance measures of unit setpoint step responses of PIDA${}_{w0}$, PIDA${}_{d0}$, ${}^{1}{R}_{a}^{1}$, ${}^{1}{R}_{b}^{1}$, ${}^{1}{R}_{a}^{2}$, ${}^{1}{R}_{b}^{2}$, ${}^{1}{R}_{a}^{3}$ and ${}^{1}{R}_{b}^{3}$ controllers introduced in Figure 9 left in the case of measurement noise with amplitude of the Uniform Random Number noise generator ${\Delta}_{n}=0.01$; ${T}_{e}=1$. Minimum (best) values are marked in bold and maximum (worst) in red.

Contr. | PIDA${}_{\mathit{w}0}$ | PIDA${}_{\mathit{d}0}$ | ${}^{1}{\mathit{R}}_{\mathit{a}}^{1}$ | ${}^{1}{\mathit{R}}_{\mathit{b}}^{1}$ | ${}^{1}{\mathit{R}}_{\mathit{a}}^{2}$ | ${}^{1}{\mathit{R}}_{\mathit{b}}^{2}$ | ${}^{1}{\mathit{R}}_{\mathit{a}}^{3}$ | ${}^{1}{\mathit{R}}_{\mathit{b}}^{3}$ |
---|---|---|---|---|---|---|---|---|

$IA{E}_{s}$ | 7.5825 | 7.7300 | 8.0912 | 6.7992 | 8.9700 | 7.4049 | 8.9746 | 7.4864 |

$T{V}_{0y}$ | 0.0258 | 0.0620 | 0.0048 | 0.0297 | 0.0191 | 0.0630 | 0.0202 | 0.0661 |

$T{V}_{1u}$ | 64.9307 | 148.6229 | 84.7714 | 97.4245 | 193.2384 | 206.3019 | 190.3595 | 203.8954 |

${J}_{1}.{10}^{-3}$ | 0.4923 | 1.1489 | 0.6859 | 0.6624 | 1.7333 | 1.5276 | 1.7084 | 1.5264 |

${J}_{10}.{10}^{-11}$ | 0.4079 | 1.1321 | 1.0194 | 0.2057 | 6.5164 | 1.0226 | 6.4522 | 1.1275 |

**Table 2.**IPDT system: Performance measures of unit input disturbance step responses of PIDA${}_{w0}$, PIDA${}_{d0}$, ${}^{1}{R}_{a}^{1}$, ${}^{1}{R}_{b}^{1}$, ${}^{1}{R}_{a}^{2}$, ${}^{1}{R}_{b}^{2}$, ${}^{1}{R}_{a}^{3}$ and ${}^{1}{R}_{b}^{3}$ controllers introduced in Figure 9 right in the case of measurement noise with amplitude of the Uniform Random Number noise generator ${\Delta}_{n}=0.01$; ${T}_{e}=1$. Minimum (best) values are marked in bold and maximum (worst) in red.

Contr. | PIDA${}_{\mathit{w}0}$ | PIDA${}_{\mathit{d}0}$ | ${}^{1}{\mathit{R}}_{\mathit{a}}^{1}$ | ${}^{1}{\mathit{R}}_{\mathit{b}}^{1}$ | ${}^{1}{\mathit{R}}_{\mathit{a}}^{2}$ | ${}^{1}{\mathit{R}}_{\mathit{b}}^{2}$ | ${}^{1}{\mathit{R}}_{\mathit{a}}^{3}$ | ${}^{1}{\mathit{R}}_{\mathit{b}}^{3}$ |
---|---|---|---|---|---|---|---|---|

$max\left(\right|e\left|\right)$ | 1.5701 | 1.3409 | 1.9613 | 1.7008 | 1.6964 | 1.4748 | 1.6583 | 1.4253 |

$IA{E}_{d}$ | 14.7998 | 10.9181 | 23.6137 | 17.5308 | 18.3922 | 12.9293 | 18.6143 | 12.0474 |

$T{V}_{1y}$ | 0.0431 | 0.1380 | 0.0473 | 0.1763 | 0.0098 | 0.0793 | 0.0114 | 0.0746 |

$T{V}_{1u}$ | 137.269 | 358.8607 | 36.4205 | 55.4752 | 104.4270 | 140.6965 | 106.0107 | 146.4343 |

${J}_{1}.{10}^{-3}$ | 2.0316 | 3.9181 | 0.8600 | 0.9725 | 1.9206 | 1.8191 | 1.9733 | 1.7641 |

${J}_{10}.{10}^{-15}$ | 0.0692 | 0.0086 | 1.9633 | 0.1521 | 0.4625 | 0.0184 | 0.5294 | 0.0094 |

**Table 3.**System (51): Performance measures of unit setpoint step responses of PIDA${}_{w0}$, PIDA${}_{d0}$, ${}^{1}{R}^{1}$, ${}^{1}{R}^{2}$, ${}^{1}{R}^{3}$, ${}^{2}{R}_{a}^{2}$, ${}^{2}{R}_{b}^{2}$ and ${}^{2}{R}^{3}$ controllers introduced in Figure 10 left in the case of measurement noise with amplitude of the Uniform Random Number noise generator ${\Delta}_{n}=0.01$ and measured for $t\in [0,6]$. Minimum (best) values are marked in bold and maximum (worst) in red.

Contr. | PIDA${}_{\mathit{w}0}$ | PIDA${}_{\mathit{d}0}$ | ${}^{1}{\mathit{R}}^{1}$ | ${}^{1}{\mathit{R}}^{2}$ | ${}^{1}{\mathit{R}}^{3}$ | ${}^{2}{\mathit{R}}_{\mathit{a}}^{2}$ | ${}^{2}{\mathit{R}}_{\mathit{b}}^{2}$ | ${}^{2}{\mathit{R}}^{3}$ |
---|---|---|---|---|---|---|---|---|

$IA{E}_{s}$ | 0.9143 | 0.8853 | 1.2911 | 1.3661 | 1.3661 | 1.3852 | 1.3852 | 1.5053 |

$T{V}_{0y}$ | 0.0072 | 0.0045 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

$T{V}_{1u}$ | 0.7706 | 1.4956 | 104.4903 | 247.3481 | 245.3201 | 128.4475 | 128.4475 | 390.3932 |

${J}_{1}$ | 0.7045 | 1.3241 | 134.9062 | 337.8936 | 335.1256 | 177.9287 | 177.9287 | 587.6513 |

${J}_{10}.{10}^{-4}$ | 0.0000 | 0.0000 | 0.1345 | 0.5598 | 0.5552 | 0.3341 | 0.3341 | 2.3317 |

**Table 4.**System (51): Performance measures of unit input disturbance step responses of PIDA${}_{w0}$, PIDA${}_{d0}$, ${}^{1}{R}^{1}$, ${}^{1}{R}^{2}$, ${}^{1}{R}^{3}$, ${}^{2}{R}_{a}^{2}$, ${}^{2}{R}_{b}^{2}$ and ${}^{1}{R}^{3}$ controllers introduced in Figure 10 right in the case of measurement noise with amplitude of the Uniform Random Number noise generator ${\Delta}_{n}=0.01$, measured for $t\in [0,6]$. Minimum (best) values are marked in bold and maximum (worst) in red.

Contr. | PIDA${}_{\mathit{w}0}$ | PIDA${}_{\mathit{d}0}$ | ${}^{1}{\mathit{R}}^{1}$ | ${}^{1}{\mathit{R}}^{2}$ | ${}^{1}{\mathit{R}}^{3}$ | ${}^{2}{\mathit{R}}_{\mathit{a}}^{2}$ | ${}^{2}{\mathit{R}}_{\mathit{b}}^{2}$ | ${}^{2}{\mathit{R}}^{3}$ |
---|---|---|---|---|---|---|---|---|

$max\left(\right|e\left|\right)$ | 0.2639 | 0.2267 | 0.3192 | 0.2922 | 0.2915 | 0.3054 | 0.3054 | 0.2598 |

$IA{E}_{d}$ | 0.2523 | 0.2101 | 0.4002 | 0.3940 | 0.3939 | 0.3851 | 0.392 | 0.3827 |

$T{V}_{1y}$ | 0.0057 | 0.0052 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 |

$T{V}_{1u}$ | 0.6915 | 1.5711 | 97.3952 | 244.2080 | 242.4337 | 123.0300 | 124.4798 | 389.8184 |

${J}_{1}$ | 0.1744 | 0.3301 | 38.9787 | 96.2102 | 95.4964 | 47.3735 | 48.8860 | 149.1859 |

${J}_{10}$ | 0.0000 | 0.0000 | 0.0103 | 0.0220 | 0.0218 | 0.0088 | 0.0109 | 0.0263 |

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**MDPI and ACS Style**

Huba, M.; Bistak, P.; Vrancic, D.
Parametrization and Optimal Tuning of Constrained Series PIDA Controller for IPDT Models. *Mathematics* **2023**, *11*, 4229.
https://doi.org/10.3390/math11204229

**AMA Style**

Huba M, Bistak P, Vrancic D.
Parametrization and Optimal Tuning of Constrained Series PIDA Controller for IPDT Models. *Mathematics*. 2023; 11(20):4229.
https://doi.org/10.3390/math11204229

**Chicago/Turabian Style**

Huba, Mikulas, Pavol Bistak, and Damir Vrancic.
2023. "Parametrization and Optimal Tuning of Constrained Series PIDA Controller for IPDT Models" *Mathematics* 11, no. 20: 4229.
https://doi.org/10.3390/math11204229