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Adaptive Neuro-Fuzzy Control of Active Vehicle Suspension Based on ${\mathit{H}}_{\mathbf{2}}$ and ${\mathit{H}}_{\infty}$ Synthesis

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## Abstract

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## 1. Introduction

- Dynamic Controller Transition and Data Collection: The proposed methodology introduces dynamic switching between controllers based on road characteristics, a novel approach for improving suspension system performance. An emphasis on meticulous manual controller transitions and comprehensive data collection across various road profiles distinguishes this approach, paving the way for data-driven suspension optimization. The combination of fuzzy logic and neural network techniques allows the ANFIS controller to provide continuous and smooth control actions as the input conditions change. This smooth transition helps avoid sudden changes in control output, which can contribute to system stability and improved performance.
- Pioneering Utilization of ANFIS Controller as a Hybrid Controller Training Approach: One of the key innovations in this research is the training of the ANFIS controller using input–output data from both the ${H}_{2}$ and ${H}_{\infty}$ controllers. This hybrid training approach leverages the knowledge and performance characteristics of both controller types to enhance the capabilities of the ANFIS controller, resulting in a unique and powerful control system for suspension optimization.

## 2. Problem Formulation and Design Method

#### 2.1. Quarter-Car Model

#### 2.2. Design Objectives

- Ride comfort: Ride comfort is commonly assessed by quantifying the RMS value of acceleration experienced by passengers [7]. To attain optimal ride comfort, it is imperative to minimize this value to the greatest extent possible within the frequency range of passenger body sensitivity, typically falling between 1 Hz and 8 Hz [31]. When road roughness is predominantly modeled as white noise or impulse input (as is frequently the case), enhancing ride comfort involves the minimization of the ${H}_{2}$ norm of the transfer function from road displacement to chassis acceleration (${\ddot{z}}_{s}$), with appropriate weighting applied to the relevant frequencies. However, if the road disturbances include more complex waveforms than white noise (such as deterministic patterns), achieving optimal ride comfort may require the use of ${H}_{\infty}$ performance.
- Road-holding: To achieve good road-holding, it is necessary to maintain continuous contact between the tires and the road surface. In the context of a given road profile, this objective can be accomplished by minimizing the ${H}_{2}$ or ${H}_{\infty}$ norms of the transfer function from the road disturbance to tire displacement (${z}_{ur}={z}_{u}-{z}_{r}$). It is essential to emphasize that maintaining rigid contact between the tires and the road necessitates that the dynamic tire load does not surpass the static load [14], i.e.,$${k}_{u}\left({z}_{u}-{z}_{r}\right)<\left({m}_{s}+{m}_{u}\right)g,\forall t\ge 0$$
- Suspension stroke limits: Suspension deflection (${z}_{def}$) plays a crucial role in achieving the required road-holding specifications, and it is imperative to maintain the deflection limits to ensure optimal ride comfort and prevent any structural damage to the system. As a result, it becomes imperative to confine the transfer function ${z}_{def}/{z}_{r}$ within the established upper and lower bounds to achieve the desired outcomes. Deviating from these limits can result in compromised ride comfort and have an adverse impact on the overall performance of the vehicle. To avoid excessive suspension bottoming, it has to consider the limitations of suspension deflection and incorporate appropriate measures to maintain optimal performance, as follows:$$\left|{z}_{def}\right|=\left|{z}_{s}-{z}_{u}\right|\le {{z}_{def}}_{max}$$
- Control signal: The control signal is produced by a hydraulic actuator and is constrained due to its saturation. It is hypothesized that the normalized control signal is bounded, as expressed by the inequality:$$\left|u\right|<{u}_{max}$$

#### 2.3. Review of ${H}_{2}$ and H_{∞} Design Frameworks

#### 2.3.1. ${H}_{2}$ Synthesis

**Theorem**

**1.**

#### 2.3.2. H_{∞} Synthesis

## 3. Adaptive Neuro-Fuzzy Inference Systems (ANFIS)

- Layer 1: Every node in this layer is depicted as a square shape, each associated with a specific membership function:$${O}_{i}^{1}={\mu}_{{A}_{i}}\left(x\right)$$$${O}_{i}^{1}={\mu}_{{A}_{i}}\left(x\right)=\frac{1}{1+{\left[{\left(\frac{x-{c}_{i}}{{a}_{i}}\right)}^{2}\right]}^{{b}_{i}}}$$
- Layer 2: Each node in this layer is represented by a circular shape denoted by the symbol П, indicating that the incoming signals are multiplied. The output of the node is calculated as the T-norm (logical AND) multiplication of the input signals, as follows:$${O}_{i}^{2}={w}_{i}={\mu}_{{A}_{i}}\left(x\right)\times {\mu}_{{B}_{i}}\left(y\right),i=1,2$$
- Layer 3: Every node within this layer is a fixed node, symbolized by a circular shape marked as N. This layer executes a normalization procedure involving summation and arithmetic division. Specifically, the i-th node computes the ratio of the i-th rule’s firing strength to the total sum of all rules’ firing strengths, as follows:$${O}_{i}^{3}={\overline{w}}_{i}=\frac{{w}_{i}}{{w}_{1}+{w}_{2}},i=1,2$$
- Layer 4: Each node in this layer is represented by a square shape and performs the multiplication of the normalized output of layer 3, ${\overline{w}}_{i}$, with the “then” part of the fuzzy rule, denoted as ${f}_{i}$, as follows:$${O}_{i}^{4}={\overline{w}}_{i}{f}_{i}={\overline{w}}_{i}({p}_{i}x+{q}_{i}y+{r}_{i})$$
- Layer 5: The presented neural network architecture comprises a single node characterized by a circular shape and identified as Σ. The output of this layer is calculated through the algebraic summation of the input signals, represented as follows:$${O}_{1}^{5}={\sum}_{i=1}^{2}{\overline{w}}_{i}{f}_{i}=\frac{{\sum}_{i=1}^{2}{\overline{w}}_{i}{f}_{i}}{{\sum}_{i=1}^{2}{w}_{i}}$$

#### ANFIS Design

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**.**Ultimately, the control variables will be computed based on the following equations:

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**Figure 7.**Comparison of body acceleration of ${H}_{2}$ and ${H}_{\infty}$ controllers: (

**a**) bump input, (

**b**) real-world asphalt road.

**Figure 8.**Comparison of the proposed ANFIS controller with ${H}_{2}$ and ${H}_{\infty}$ controllers: (

**a**) weighted body acceleration, (

**b**) tire deflection, (

**c**) input force.

**Figure 9.**Comparison of ANFIS-based control performance with other controllers over passive suspension.

Symbol | Description | Value |
---|---|---|

${m}_{s}$ | Sprung mass | 315 kg |

${m}_{u}$ | Unsprung mass | 37.5 kg |

${k}_{s}$ | Suspension stiffness | 29,500 N/m |

${k}_{u}$ | Tire stiffness | 210,000 N/m |

${b}_{s}$ | Suspension damping coefficient | 1500 N·s/m |

$\left[{\underset{\_}{z}}_{def},{\overline{z}}_{def}\right]$ | Suspension deflection limits | [−8, 6] cm |

System | ${\ddot{\mathit{z}}}_{\mathit{s}}\mathbf{(}\mathbf{m}\mathbf{/}{\mathbf{s}}^{2}\mathbf{)}$ | ${\mathit{z}}_{\mathit{u}\mathit{r}}\mathbf{\left(}\mathbf{m}\mathbf{\right)}$ | ${\mathit{z}}_{\mathit{d}\mathit{e}\mathit{f}}\mathbf{\left(}\mathbf{m}\mathbf{\right)}$ | Force (kN) |
---|---|---|---|---|

Passive | 0.203 | $8.1\times {10}^{-4}$ | 0.103 | 0 |

${H}_{2}$ controller | 0.097 | $5.33\times {10}^{-4}$ | 0.091 | 1.51 |

${H}_{\infty}$ controller | 0.138 | $4\times {10}^{-4}$ | 0.073 | 1.20 |

ANFIS | 0.077 | $3.5\times {10}^{-4}$ | 0.97 | 1.40 |

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**MDPI and ACS Style**

Esmaeili, J.S.; Akbari, A.; Farnam, A.; Azad, N.L.; Crevecoeur, G.
Adaptive Neuro-Fuzzy Control of Active Vehicle Suspension Based on *Machines* **2023**, *11*, 1022.
https://doi.org/10.3390/machines11111022

**AMA Style**

Esmaeili JS, Akbari A, Farnam A, Azad NL, Crevecoeur G.
Adaptive Neuro-Fuzzy Control of Active Vehicle Suspension Based on *Machines*. 2023; 11(11):1022.
https://doi.org/10.3390/machines11111022

**Chicago/Turabian Style**

Esmaeili, Jaffar Seyyed, Ahmad Akbari, Arash Farnam, Nasser Lashgarian Azad, and Guillaume Crevecoeur.
2023. "Adaptive Neuro-Fuzzy Control of Active Vehicle Suspension Based on *Machines* 11, no. 11: 1022.
https://doi.org/10.3390/machines11111022