# Development of a Universal Adaptive Control Algorithm for an Unknown MIMO System Using Recursive Least Squares and Parameter Self-Tuning

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The proposed algorithm estimates the RLS-based error dynamics coefficients and does not require information regarding the system. Therefore, it can be used as a universal-purpose controller in various unknown systems.
- (2)
- In this study, a virtual test drive simulator, CarMaker, and an actual DC motor platform were used to evaluate the reasonable performance of the proposed universal controller. In the case of the CarMaker-based evaluation, this study attempted to verify the performance of the proposed algorithm in various systems using front-wheel steering vehicles and front-and-rear-wheel steering vehicles.

## 2. Adaptive Control Algorithm Using RLS and Parameter Self-Tuning

#### 2.1. MIMO System Error Dynamics

- (A1)
- All the control errors have a complex influence on each other.
- (A2)
- At this stage, $n=p$ and the number of control errors and control inputs are the same.

#### 2.2. RLS-Based Coefficient Estimation

#### 2.3. Derivation of Control Input Based on the Lyapunov Direct Method

## 3. Performance Evaluation

#### 3.1. Performance Evaluation of DC Motor-Based Adaptive Speed Control

#### 3.2. Performance Evaluation of CarMaker-Based Adaptive Path Tracking Control

#### 3.2.1. Front-Wheel Steering Vehicle-Based Adaptive Path Tracking Control

#### 3.2.2. Front-and-Rear-Wheel Steering Vehicle-Based Adaptive Path Tracking Control

**Proof.**

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Velocity tracking result: (

**a**) Estimated coefficient ${C}_{11}$; (

**b**) Estimated coefficient ${D}_{1}$.

**Figure 6.**Velocity tracking result: (

**a**) Residual; (

**b**) Magnitude of injection term; (

**c**) Voltage; (

**d**) Current.

**Figure 10.**Path tracking result: (

**a**) Yaw rate; (

**b**) Lateral velocity; (

**c**) Longitudinal acceleration; (

**d**) Lateral acceleration.

**Figure 11.**Path tracking result: (

**a**) Lateral preview error; (

**b**) Yaw angle error; (

**c**) Integrated error; (

**d**) Integrated error differential.

**Figure 13.**Path tracking result: (

**a**) Estimated coefficient; (

**b**) Residual; (

**c**) Injection; (

**d**) Control input.

**Figure 15.**Path tracking result: (

**a**) Yaw rate; (

**b**) Lateral velocity; (

**c**) Longitudinal acceleration; (

**d**) Lateral acceleration.

**Figure 16.**Path tracking result: (

**a**) Lateral preview error; (

**b**) Yaw angle error; (

**c**) Integrated error; (

**d**) Control input.

**Figure 19.**Path tracking result: (

**a**) Yaw rate; (

**b**) Lateral velocity; (

**c**) Longitudinal acceleration; (

**d**) Lateral acceleration.

**Figure 22.**Path tracking result: (

**a**) Estimated coefficient; (

**b**) Residual; (

**c**) Injection; (

**d**) Control input.

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

Resistance | $\mathsf{\Omega}$ | $8.4$ |

Torque constant | $\mathrm{N}\mathrm{m}/\mathrm{A}$ | $0.042$ |

Motor back-EMF constant | $\mathrm{V}/(\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s})$ | $0.042$ |

Rotor inductance | $\mathrm{m}\mathrm{H}$ | 1.16 |

Inertia | $\mathrm{K}\mathrm{g}{\mathrm{m}}^{2}$ | 4.6 $\times {10}^{-6}$ |

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

Decay rate of the Lyapunov function ($\alpha $) | - | 0.0001 |

Reachability factor (η) | - | 0.0001 |

Initial value of estimated states (${\widehat{C}}_{11},{\widehat{D}}_{1}$) | - | $\left(\begin{array}{cc}0.1,& -2.73\end{array}\right)$ |

Initial value of covariance (${P}_{11},{P}_{12}$) | - | $\left(\begin{array}{cc}0.00008,& 0.00009\end{array}\right)$ |

Forgetting factor $({\lambda}_{11},{\lambda}_{12})$ | - | $\left(\begin{array}{cc}0.99995,& 0.99993\end{array}\right)$ |

Lower scale factor threshold (${\dot{e}}_{low,th})$ | - | 0.01 |

Upper scale factor threshold (${\dot{e}}_{up,th})$ | - | 0.08 |

Normalized threshold $\left({s}_{max}\right)$ | - | 1.001 |

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

Mass ($m$) | $\mathrm{k}\mathrm{g}$ | 2108 |

Wheelbase ($l$) | $\mathrm{m}$ | 2.97 |

Distance between CG * and front/rear axle (${l}_{f},{l}_{r}$) | $\mathrm{m}$ | $\begin{array}{cc}1.47,& 1.5\end{array}$ |

Z-axis rotational inertia (${I}_{z}$) | $\mathrm{k}\mathrm{g}\xb7{\mathrm{m}}^{2}$ | 3170.6 |

Estimated cornering stiffness of front/rear wheels (${C}_{f},{C}_{r}$) | $\mathrm{N}/\mathrm{r}\mathrm{a}\mathrm{d}$ | $\begin{array}{cc}\mathrm{118,270},& \mathrm{117,990}\end{array}$ |

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

Decay rate of the Lyapunov function ($\alpha $) | - | 1 |

Reachability factor (η) | - | 0.01 |

Initial value of estimated states (${\widehat{C}}_{11},{\widehat{D}}_{1}$) | - | $\begin{array}{cc}0,& 0\end{array}$ |

Initial value of covariance (${P}_{11},{P}_{12}$) | - | $\begin{array}{cc}0.001,& 0.001\end{array}$ |

Forgetting factor $({\lambda}_{11},{\lambda}_{12})$ | - | $\begin{array}{cc}0.9999,& 0.9999\end{array}$ |

Lower scale factor threshold (${\dot{e}}_{low,th})$ | - | 0.03 |

Upper scale factor threshold (${\dot{e}}_{up,th})$ | - | 0.1 |

Normalized threshold $\left({s}_{max}\right)$ | - | 1.008 |

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

Decay rate of the Lyapunov function ($\alpha $) | - | 1.2 |

Reachability factor (η) | - | 0.01 |

Initial value of estimated states (${\widehat{C}}_{11},{\widehat{D}}_{1}$) | - | $\left(\begin{array}{cc}0,& 0\end{array}\right)$ |

Initial value of estimated states (${\widehat{C}}_{21},{\widehat{D}}_{2}$) | - | $\left(\begin{array}{cc}0,& 0\end{array}\right)$ |

Initial value of covariance (${P}_{11},{P}_{12}$) | $\left(\begin{array}{cc}0.001,& 0.001\end{array}\right)$ | |

Initial value of covariance (${P}_{21},{P}_{22}$) | $\left(\begin{array}{cc}0.00001,& 0.00001\end{array}\right)$ | |

Forgetting factor $({\lambda}_{11},{\lambda}_{12})$ | - | $\left(\begin{array}{cc}0.9999,& 0.9999\end{array}\right)$ |

Forgetting factor $({\lambda}_{21},{\lambda}_{22})$ | - | $\left(\begin{array}{cc}0.99999,& 0.99999\end{array}\right)$ |

Lower scale factor threshold (${\dot{e}}_{low,th})$ | - | 0.03 |

Upper scale factor threshold (${\dot{e}}_{up,th})$ | - | 0.1 |

Normalized threshold $\left({s}_{max}\right)$ | - | 1.008 |

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**MDPI and ACS Style**

La, H.; Oh, K.
Development of a Universal Adaptive Control Algorithm for an Unknown MIMO System Using Recursive Least Squares and Parameter Self-Tuning. *Actuators* **2024**, *13*, 167.
https://doi.org/10.3390/act13050167

**AMA Style**

La H, Oh K.
Development of a Universal Adaptive Control Algorithm for an Unknown MIMO System Using Recursive Least Squares and Parameter Self-Tuning. *Actuators*. 2024; 13(5):167.
https://doi.org/10.3390/act13050167

**Chicago/Turabian Style**

La, Hanbyeol, and Kwangseok Oh.
2024. "Development of a Universal Adaptive Control Algorithm for an Unknown MIMO System Using Recursive Least Squares and Parameter Self-Tuning" *Actuators* 13, no. 5: 167.
https://doi.org/10.3390/act13050167