# Adaptive Tracking PID and FOPID Speed Control of an Elastically Attached Load Driven by a DC Motor at Almost Step Disturbance of Loading Torque and Parametric Excitation

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## Abstract

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## 1. Introduction

#### 1.1. Scope of the Selected Achievements

- Knowledge of real-time estimation of disturbances and system parameters are unnecessary in Reference [3] to suppress time-varying disturbances of a higher order.
- The results found in Reference [7] confirm that the PSO-based controller is very promising.
- A fractional-order adaptive law presented in Reference [8] shows the path that we follow at a final stage of this work.
- FOPID controller is proposed in Reference [9] for BLDC motor to achieve effective control of torque and speed.
- An adaptive FLC considered in Reference [10] in the DC motor speed control improves disturbance rejection of the motor response due to the load.

#### 1.2. The Improved Control Strategy Demonstrated in This Work

## 2. The Physical Model

## 3. The Trajectories of Disturbances

## 4. Control Strategies

#### 4.1. Block Diagram of the Object of Control

#### 4.2. The First Strategy

#### 4.3. The Second Strategy

#### 4.4. The Third Strategy

## 5. Results

- classical PID—see Section 4.2,
- FOPID (PID${}^{\alpha}$)—see Section 4.3,
- adaptive PID with PI action at the second stage of error compensation—see Section 4.4.

## 6. Effectiveness of the Proposed Solutions

#### 6.1. Estimating Performance of the Control System

#### 6.2. Verifying Robustness of the Control System

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Network |

BAT | Bat Algorithm |

BLDC | Brushless DC |

DC | Direct Current |

FFA | Firefly Algorithm |

FLC | Fuzzy Logic Controller |

FOPID | Fractional-Order PID |

ITAE | Integral Time-Weighted Absolute Error |

LQR | Liner Quadratic Regulator |

PID | Proportional Integral Derivative |

PSO | Particle Swarm Optimization |

WSA | Wolf Search Algorithm |

## References

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**Figure 1.**Physical model of the Direct Current (DC) permanent magnet motor driving a softly attached mechanical load.

**Figure 2.**Block diagram of the open-loop model reference control system with the additional time-dependent loading disturbance ${T}_{la}\left(t\right)$, while variation of the additional damping coefficient ${b}_{la}\left(t\right)$ is turned off, ${G}_{2}$ is the second-order dynamical operator given by Equation (9).

**Figure 3.**Block diagram of the closed-loop model reference Proportional Integral Derivative (PID) control with the additional time-dependent disturbances of the damping coefficient ${b}_{la}\left(t\right)$ (case A) and the loading torque ${T}_{la}\left(t\right)$ (case B).

**Figure 4.**Block diagram of the closed-loop model reference adaptive PID (fractional PID, as well) control system with the additional time-dependent disturbances of the damping coefficient ${b}_{la}\left(t\right)$, and a PI compensation of the resulting second-stage error dynamics.

**Figure 5.**The open-loop reference model control system response at the presence of the additional time-dependent disturbances of: (

**a**) damping of load ${b}_{la}\left(t\right)$—see on the right-hand vertical axis; (

**b**) load ${T}_{la}\left(t\right)$ with adaptation of PID parameters and PI compensation of error dynamics. Green line represents the desired rotational velocity trajectory, while the red line the actual velocity of the load (low-frequency case).

**Figure 6.**The closed-loop reference model control system response at the presence of: (

**a**) ${b}_{la}\left(t\right)$; (

**b**) ${T}_{la}\left(t\right)$ disturbance and a PID regulator (low frequency)—see case A in the block diagram in Figure 3.

**Figure 7.**The closed-loop reference model control system response at the presence of: (

**a**) ${b}_{la}\left(t\right)$; (

**b**) ${T}_{la}\left(t\right)$ disturbance and a Fractional-Order PID (FOPID) regulator (low frequency)—see case B in the block diagram in Figure 3.

**Figure 8.**The closed-loop reference model control system response at the presence of the disturbed ${b}_{la}\left(t\right)$ (

**a**) and ${T}_{la}\left(t\right)$ (

**b**) and an adaptive PID regulator with PI compensation of error dynamics (low frequency)—see case B in the block diagram in Figure 4.

**Figure 9.**The open-loop reference model control system response at presence of the additional time-dependent disturbances of damping (

**a**) ${b}_{la}\left(t\right)$ and loading (

**b**) ${T}_{la}\left(t\right)$ (high frequency).

**Figure 10.**The closed-loop reference model control system response at the presence of ${b}_{la}\left(t\right)$ (

**a**) and ${T}_{la}\left(t\right)$ (

**b**) disturbance and a PID regulator (high frequency)—see case A in the block diagram in Figure 3.

**Figure 11.**The closed-loop reference model control system response at the presence of the disturbed ${b}_{la}\left(t\right)$ (

**a**) and ${T}_{la}\left(t\right)$ (

**b**) and a FOPID regulator (high frequency)—see case B in the block diagram in Figure 3.

**Figure 12.**The closed-loop reference model control system response at the presence of ${b}_{la}\left(t\right)$ (

**a**) and ${T}_{la}\left(t\right)$ (

**b**) and an adaptive PID + PI (high frequency)—see case B in the block diagram in Figure 4.

**Figure 13.**The closed-loop reference model FOPID control system response at the presence of low-frequency ${T}_{la}\left(t\right)$ and the disturbances of electric model parameters: R (see $\tilde{R}\left(t\right)$) and L (not shown here as $\tilde{L}\left(t\right)$ has an opposite and scaled form). This solution is comparable with Figure 7b.

Parameter | Notation | Value | Unit |
---|---|---|---|

rotor inertia | ${J}_{m}$ | $2.3\xb7{10}^{-3}$ | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

load inertia | ${J}_{l}$ | $10\xb7{J}_{m}$ | $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ |

motor damping | ${b}_{m}$ | $1.2\xb7{10}^{-4}$ | $\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}/\mathrm{rad}$ |

load damping | ${b}_{l}$ | $0.02$ | $\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}/\mathrm{rad}$ |

torque constant | ${K}_{t}$ | $0.5$ | $\mathrm{N}\xb7\mathrm{m}/\mathrm{A}$ |

inductance | L | $1.4\xb7{10}^{-4}$ | H |

resistance | R | $0.9$ | $\mathsf{\Omega}$ |

back-EMF | ${K}_{b}$ | $0.5$ | $\mathrm{V}\xb7\mathrm{s}/\mathrm{rad}$ |

stiffness | ${k}_{rl}$ | 400 | $\mathrm{N}\xb7\mathrm{m}/\mathrm{rad}$ |

damping | ${b}_{rl}$ | $0.01$ | $\mathrm{N}\xb7\mathrm{m}\xb7\mathrm{s}/\mathrm{rad}$ |

reference control | $\beta $ | 1 | $\mathrm{V}/\mathrm{rad}$ |

**Table 2.**The Integral Time-Weighted Absolute Error (ITAE) performance index for various numerical experiments.

ITAE | Controller | Control Loop | Disturbed Parameter | Disturbance Frequency | Figure |
---|---|---|---|---|---|

$0.0368$ | PID | closed | ${T}_{l}$ | low | Figure 6b |

$0.0551$ | Adaptive PID | closed | ${T}_{l}$ | low | Figure 8b |

$0.0839$ | Adaptive PID | closed | ${b}_{l}$ | low | Figure 8a |

$0.0962$ | FOPID | closed | ${T}_{l}$ | low | Figure 7b |

$0.1104$ | PID | closed | ${T}_{l}$ | high | Figure 10b |

$0.1313$ | Adaptive PID | closed | ${T}_{l}$ | high | Figure 12b |

$0.2210$ | PID | closed | ${b}_{l}$ | high | Figure 10a |

$0.2263$ | Adaptive PID | closed | ${b}_{l}$ | high | Figure 12a |

$0.2383$ | FOPID | closed | ${T}_{l}$ | high | Figure 11b |

$0.4780$ | FOPID | closed | ${b}_{l}$ | low | Figure 7a |

$0.7453$ | – | open | ${T}_{l}$ | high | Figure 9b |

$1.2115$ | FOPID | closed | ${b}_{l}$ | high | Figure 11a |

$1.4949$ | – | open | ${T}_{l}$ | low | Figure 5b |

$1.5168$ | PID | closed | ${b}_{l}$ | low | Figure 6a |

$1.9720$ | – | open | ${b}_{l}$ | high | Figure 9a |

$2.4874$ | – | open | ${b}_{l}$ | low | Figure 5a |

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**MDPI and ACS Style**

Olejnik, P.; Adamski, P.; Batory, D.; Awrejcewicz, J.
Adaptive Tracking PID and FOPID Speed Control of an Elastically Attached Load Driven by a DC Motor at Almost Step Disturbance of Loading Torque and Parametric Excitation. *Appl. Sci.* **2021**, *11*, 679.
https://doi.org/10.3390/app11020679

**AMA Style**

Olejnik P, Adamski P, Batory D, Awrejcewicz J.
Adaptive Tracking PID and FOPID Speed Control of an Elastically Attached Load Driven by a DC Motor at Almost Step Disturbance of Loading Torque and Parametric Excitation. *Applied Sciences*. 2021; 11(2):679.
https://doi.org/10.3390/app11020679

**Chicago/Turabian Style**

Olejnik, Paweł, Paweł Adamski, Damian Batory, and Jan Awrejcewicz.
2021. "Adaptive Tracking PID and FOPID Speed Control of an Elastically Attached Load Driven by a DC Motor at Almost Step Disturbance of Loading Torque and Parametric Excitation" *Applied Sciences* 11, no. 2: 679.
https://doi.org/10.3390/app11020679