# A NARX Model Reference Adaptive Control Scheme: Improved Disturbance Rejection Fractional-Order PID Control of an Experimental Magnetic Levitation System

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## Abstract

**:**

## 1. Introduction

- i.
- Linearization of a nonlinear ML system makes the reference model valid at the modeling conditions such as operating temperature, sphere motion ranges etc. This limitation decreases the accuracy of the reference model.
- ii.
- Identification of the inner loop is an initial process and this static reference model is not adaptive for changes in condition and behavior of the real systems.

## 2. Theoretical Background and Preliminaries

#### 2.1. Fractional Calculus and Fractional-Order Systems

#### 2.2. Multi-Loop Mrac

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

## 3. Mathematical Model of the Ml System: From a First Principles Model to a Narx Black Box Approximation

#### 3.1. Nn-Narx Modeling of the Ml System

#### 3.2. Off-Line Results

#### 3.3. Real-Life Results

## 4. Multi-Loop Mrac-Fopid Control with Narx Reference Model for Ml System Control Application

- i.
- The closed-loop retuning FOPID control system was implemented and FOPID controller was optimally tuned by using FOMCON toolbox [47].
- ii.
- The control signal and sphere position data from the designed closed-loop control system described above were collected and these data were used to train the NARX model in the virtual closed-loop control system. Thus, the virtual closed-loop PID control loop with the NARX model is used as reference model to represent closed-loop retuning FOPID control system.
- iii.
- The outer loop is connected to inner loop according to MIT rule as shown in Figure 10.

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Implementation of the retuning fractional-order proportional integral derivative (FOPID) system by using the PID control loop [9].

**Figure 2.**Block diagram of multi-loop Model Reference Adaptive Control (MRAC)-FOPID system [9].

**Figure 4.**Representation of neural network (NN)-nonlinear autoregressive neural network with external inputs (NARX) structure. The first layer activation function ${f}_{1}$ is generally a nonlinear activation function to model nonlinear relations in the data. The second layer activation function ${f}_{2}$ is a linear activation function that rescales NARX output to original output data. Thus, it can yield satisfactory predictions of a nonlinear system dynamics.

**Figure 11.**Complete experimental configuration for evaluating the multi-loop control structure. There are three control loops in total: the original PID control loop with reference input ${r}^{\prime \prime}\left(t\right)$, the retuning loop with reference input ${r}^{\prime}\left(t\right)$ that replaces the dynamics of the original loop with those of the optimally tuned FOPID controller, and the MRAC loop to which the original reference input $r\left(t\right)$ is connected. By bypassing reference inputs in various ways, it is possible to achieve different simulation scenarios with the MRAC loop and retuning control loop enabled or disabled independently. This schematic diagram serves as the basis for both pure software simulations and real-time experiments in MATLAB/Simulink software.

**Figure 12.**Disturbance responses of the multi-loop MRAC-FOPID control with NARX reference model and the FOPID control loop (When MRAC is disabled).

**Figure 13.**Step disturbance responses of the multi-loop MRAC-FOPID control with NARX reference model and the FOPID control loop (When MRAC is disabled).

**Figure 14.**Sinusoidal disturbance responses of the multi-loop MRAC-FOPID control with NARX reference model and the FOPID control loop (When MRAC is disabled).

Parameter | Physical Description | Unit |
---|---|---|

$m=5.7100\times {10}^{-2}$ | mass of ball | [kg] |

$g=9.81$ | gravity constant | [$m/{s}^{2}$] |

${F}_{em}=f({x}_{1},{x}_{3})$ | [N] | |

${F}_{emP1}=1.7521\times {10}^{-2}$ | electromagnetic force | [H] |

${F}_{emP2}=5.8231\times {10}^{-3}$ | electromagnetic force | [m] |

${f}_{i}\left({x}_{1}\right)$ | [1/s] | |

${f}_{iP1}=1.4142\times {10}^{-4}$ | [ms] | |

${f}_{iP2}=4.5626\times {10}^{-3}$ | [m] | |

${c}_{i}=2.4300\times {10}^{-2}$ | actuator value | [A] |

${k}_{i}=2.5165$ | actuator value | [A] |

${x}_{3MIN}=3.8840\times {10}^{-2}$ | limitation for current | [A] |

${u}_{MIN}=4.9800\times {10}^{-3}$ | limitation for voltage |

Parameter | Value |
---|---|

Sample rate | $0.001$ s |

Simulation time | 30 s |

Hidden layer | 12 neurons |

X, input | Output of controller |

Delay (external input for control current) | 1 |

T, target | Output of system (ball position, velocity and coil current) |

Iteration | 1000 |

Reference signals | Pulse, Sine, Chirp (1 Hz to 6 Hz) |

Compare Type | MSE |
---|---|

ANN Model off-line (YEXP-YANN) | $2.32\times {10}^{-7}$ |

Real-Time Simulation (YEXP-UEXP) | $2.12\times {10}^{-7}$ |

Mathematical Model (YMATH-UEXP) | $8.02\times {10}^{-8}$ |

ANN Model (YANN-UEXP) | $2.83\times {10}^{-7}$ |

**Table 4.**Comparison of experimental performances of control systems with the MRAC loop disabled and enabled.

Control Structure | Peak Values of $\left|\mathit{y}\right(\mathit{t})-\mathit{r}(\mathit{t}\left)\right|$ in Disturbance Responses [m] | Settling Time after Step Disturbance [s] | Cumulative Absolute Control Error (Additive Disturbance) | Cumulative Absolute Control Error (Harmonic Disturbance) |
---|---|---|---|---|

FOPID control loop (MRAC disabled) | $3.603\times {10}^{-3}$ | $1.23$ | $8.6403$ | $28.6592$ |

Multi-loop MRAC-FOPID control with NARX reference model | $2.021\times {10}^{-3}$ | $0.43$ | $3.6240$ | $8.2300$ |

Multi-loop original PID control with NARX reference model | $5.552\times {10}^{-3}$ | $1.08$ | $16.3261$ | $18.4087$ |

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**MDPI and ACS Style**

Alimohammadi, H.; Alagoz, B.B.; Tepljakov, A.; Vassiljeva, K.; Petlenkov, E. A NARX Model Reference Adaptive Control Scheme: Improved Disturbance Rejection Fractional-Order PID Control of an Experimental Magnetic Levitation System. *Algorithms* **2020**, *13*, 201.
https://doi.org/10.3390/a13080201

**AMA Style**

Alimohammadi H, Alagoz BB, Tepljakov A, Vassiljeva K, Petlenkov E. A NARX Model Reference Adaptive Control Scheme: Improved Disturbance Rejection Fractional-Order PID Control of an Experimental Magnetic Levitation System. *Algorithms*. 2020; 13(8):201.
https://doi.org/10.3390/a13080201

**Chicago/Turabian Style**

Alimohammadi, Hossein, Baris Baykant Alagoz, Aleksei Tepljakov, Kristina Vassiljeva, and Eduard Petlenkov. 2020. "A NARX Model Reference Adaptive Control Scheme: Improved Disturbance Rejection Fractional-Order PID Control of an Experimental Magnetic Levitation System" *Algorithms* 13, no. 8: 201.
https://doi.org/10.3390/a13080201