# Multi-Loop Model Reference Proportional Integral Derivative Controls: Design and Performance Evaluations

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## Abstract

**:**

## 1. Introduction

## 2. Multi-Loop Model Reference PID-MIT (ML-MR PID-MIT) Control: Analytical Tuning and Nonlinearity

- (i)
- Control error: When the controller of the inner loop is tuned properly, the value of the control error, which is written as ${e}_{c}={u}_{r}-y$, moves to zero, and thus ensures the control system output settles to the reference input (the signal ${u}_{r}$ is the modified reference input controlled by the MIT rule and $y$ is the system output).
- (ii)
- Model error: Discrepancy between the reference model output and controlled system output is defined as the model error, which is written as ${e}_{m}=y-{y}_{m}$ (the signal ${y}_{m}$ is the output of reference model). The objective of the outer loop is to shape the reference input signal ($r$) such that the model error converges to zero. The convergence of the model error to zero implies that the inner loop resembles the response of the reference model ${T}_{m}(s)$. The reference model describes the desired response of the control system that forms the inner loop.

## 3. Multi-Loop Model Reference PID-MIT with Controller Gain Modification (ML-MR PID-MIT-CGM): Adaptive RDR Adjustment for Dynamic Disturbance Rejection and the Linear Adaptation Rule

## 4. Multi-Loop Model Reference PID Internal Model (ML-MR PID-IM) Control Structure: Internal Model Linear Adaptation Rule

## 5. Multi-Loop Model Reference MIT-PID with Plant Function Adaptation (ML-MR MIT-PID-PFA): Nonlinear Adaptation Rule for Time-Delayed Systems

## 6. Multi-Loop Model Reference PID-PID with Plant Function Adaptation (ML-MR PID-PID-PFA): Linear Adaptation Rule for Time-Delayed Systems

## 7. Discussions and Conclusions

- (i)
- The proposed ML-MR adaptive control structures can be applied to existing control loops without modifying any parameters of the control loops. These structures do not require online return of the control loop, which may result in instant performance degradation while altering controller coefficients. Therefore, ML-MR control structures provide more consistent control performance than conventional MRAC structures that are used to perform online return of the control loop.
- (ii)
- The proposed ML-MR adaptive control structures increase the disturbance rejection performance without deteriorating the set-point control quality. This is an important contribution to the solution of performance tradeoffs between disturbance rejection and set-point control.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Lemma**

**A1.**

**Proof**

**A1.**

**Theorem A1**

**(Stability of the ML-MR PID-MIT Control Structure).**

**Proof**

**A2.**

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**Figure 1.**Block diagram of multi-loop model reference PID-MIT (ML-MR PID-MIT) control structure with input shaping.

**Figure 2.**Block diagram of the ML-MR PID-MIT control structure and a solution for numerical realization of reference models.

**Figure 3.**Comparison of disturbance rejection performance of ML-MR PID-MIT control and classical PID control for ${G}_{1}(s)$ and ${G}_{2}(s)$ plants.

**Figure 4.**Comparison of disturbance rejection performances of ML-MR PID-MIT control and classical PID control for the cases of $r(t)=1$ and $r(t)=0$. (

**a**) Step response and (

**b**) change of model error.

**Figure 5.**RDR spectra of ${C}_{\theta}(s)$ controllers for several $\theta $ values. This figure reveals the adjustment of RDR performance according to several $\theta $ configurations.

**Figure 6.**Block diagram of the ML-MR PID-MIT with controller gain modification (ML-MR PID-MIT-CGM) control structure and numerical realization of the reference model.

**Figure 7.**ML-MR PID-MIT-CGM control structure with control gain shaping that is implemented for improvement of the disturbance rejection performance of an optimal closed loop PID control of an AVR model.

**Figure 8.**Improvement of the disturbance rejection performance of the classical optimal PID loop by the proposed ML-MR PID-MIT-CGM control structure for the AVR control system model. (

**a**) Step response and (

**b**) change of model error.

**Figure 9.**(

**a**) Evolution of the adaptation gain $\theta $ of the ML-MR PID-MIT-CGM control structure. (

**b**) Corresponding adaptive modification of the RDR spectrum of the ${C}_{\theta}(s)$ controller in order to reject disturbances.

**Figure 10.**Improvement of the disturbance rejection performance of the classical optimal PID loop by the proposed ML-MR PID-I control structure for the model. (

**a**) Step response and (

**b**) change of model error.

**Figure 11.**(

**a**) Evolution of adaptation gain $\theta $ of the ML-MR PID-I control structure. (

**b**) Adaptive modification of the RDR spectrum of the ${C}_{\theta}(s)$ controller for disturbance rejection.

**Figure 13.**Implementation of the ML-MR PID-IM control structure for disturbance rejection control of the liquid level of the reboiler model.

**Figure 14.**Improvement of the disturbance rejection performance of the classical optimal PID loop by the proposed ML-MR PID-IM control structure for liquid level control in the reboiler model. (

**a**) Step response and (

**b**) change of model error.

**Figure 17.**Control performance of various control systems for robust performance optimal control of a large time delay plant model [34].

**Figure 18.**Comparisons of disturbance rejection performances of classical PID control and ML-MR PID-PID-PFA control structures for (

**a**) step disturbance, (

**b**) sinusoidal disturbance with a frequency of 0.001 rad/s, (

**c**) sinusoidal disturbance with a frequency of 0.01 rad/s, and (

**d**) sinusoidal disturbance with a frequency of 0.1 rad/s.

**Figure 19.**Experimental twin-rotor multi-input multi-output system (TRMS) setup [39].

**Table 1.**Mean absolute error (MAE) of control error signals obtained from control simulations for step and sinusoidal waveform disturbances.

Angular Frequencies $\omega $ (rad/s) | Step Disturbance | 0.001 | 0.005 | 0.01 | 0.05 | 0.1 | 0.5 |

Classical PID Control (Tavakoli et al.) | 0.0470 | 0.0868 | 0.1748 | 0.1605 | 0.0560 | 0.0267 | 0.0145 |

ML-MR PID-PID-PFA | 0.0234 | 0.0216 | 0.1037 | 0.1465 | 0.0690 | 0.0250 | 0.0140 |

ML-MR Control Structures | Character | Dependence to Set-Point Level | Negative Effects on Set-Point Performance | Applicability for Time Delay Systems | Indications in Simulation Results |
---|---|---|---|---|---|

ML-MR PID-MIT | Nonlinear | Dependent | None | None | Figure 3 and Figure 4 |

ML-MR PID-MIT-CGM | Nonlinear | Dependent | None | None | Figure 8 |

ML-MR PID-I | Linear | Independent | None | None | Figure 10 |

ML-MR PID-IM | Linear | Independent | None | None | Figure 14 |

ML-MR MIT-PID-PFA | Nonlinear | Dependent | None | Applicable to some degree | Figure 17 |

ML-MR PID-PID-PFA | Linear | Independent | None | Applicable to some degree | Figure 17 |

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**MDPI and ACS Style**

Alagoz, B.B.; Tepljakov, A.; Petlenkov, E.; Yeroglu, C. Multi-Loop Model Reference Proportional Integral Derivative Controls: Design and Performance Evaluations. *Algorithms* **2020**, *13*, 38.
https://doi.org/10.3390/a13020038

**AMA Style**

Alagoz BB, Tepljakov A, Petlenkov E, Yeroglu C. Multi-Loop Model Reference Proportional Integral Derivative Controls: Design and Performance Evaluations. *Algorithms*. 2020; 13(2):38.
https://doi.org/10.3390/a13020038

**Chicago/Turabian Style**

Alagoz, Baris Baykant, Aleksei Tepljakov, Eduard Petlenkov, and Celaleddin Yeroglu. 2020. "Multi-Loop Model Reference Proportional Integral Derivative Controls: Design and Performance Evaluations" *Algorithms* 13, no. 2: 38.
https://doi.org/10.3390/a13020038