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

Abstract

This paper addresses the issue of a road-type-adaptive control strategy aimed at enhancing suspension performance. $H_2$ synthesis is employed for modeling road irregularities as impulses or white noise, minimizing the root mean square (RMS) of performance outputs for these specific road types. It should be noted, however, that this approach may lead to suboptimal performance when applied to other road profiles. In contrast, the $H_{\infty }$ controller is employed to minimize the RMS of performance outputs under worst-case road irregularities, taking a conservative stance that ensures robustness across all road profiles. To leverage the advantages of both controllers and achieve overall improved suspension performance, automatic switching between these controllers is recommended based on the identified road type. To implement this adaptive switching mechanism, manual switching is performed, gathering input–output data from the controllers. These data are subsequently employed for training an Adaptive Neuro-Fuzzy Inference System (ANFIS) network. This elegant approach contributes significantly to the optimization of suspension performance. The simulation results employing this novel ANFIS-based controller demonstrate substantial performance enhancements compared to both the $H_2$ and $H_{\infty }$ controllers. Notably, the ANFIS-based controller exhibits a remarkable 62% improvement in vehicle body comfort and a significant 57% enhancement in ride safety compared to passive suspension, highlighting its potential for superior suspension performance across diverse road conditions.

1. Introduction

In the automotive industry, ensuring vehicle safety and passenger comfort are two primary design challenges. The vehicle suspension system plays a crucial role in achieving these objectives. It is responsible for isolating the vehicle body from vibrations and shocks caused by road surface irregularities to provide a smooth ride. Additionally, it should prevent the wheels from bouncing to maintain stable contact with the road for ride safety. However, these goals can conflict with one another. Improving ride comfort requires a larger suspension stroke, which means a soft suspension, while ride safety requires stiff damping. As a result, achieving the best possible passenger comfort without compromising road-holding requires adjusting these suspension parameters to suit different road profiles.

In passive suspension systems, the damper coefficient and spring stiffness are typically set during the design phase and cannot be adjusted to respond to different road profiles. As a result, the development of more intelligent suspension systems, such as semi-active and active suspension systems, has become increasingly popular within the automotive industry and academic communities. These advanced suspension systems provide the capability to adjust the suspension parameters in real time to adapt to changing road conditions, resulting in improved ride comfort and safety.

The active suspension system employs a singular linear electromagnetic or hydraulic actuator, paired with its respective amplifier, at each corner of the vehicle. This configuration has the capability to either replace or function concurrently with the conventional damper–spring setup. In the last decade, several control methodologies have emerged for the enhancement of active suspension systems, including robust control [1,2,3,4,5,6], preview-based active suspension [7,8,9], optimal control [9,10,11,12], and adaptive control [4,13,14,15,16]. Since neural networks and fuzzy controls are the key branches of intelligent control, numerous studies have been conducted on using adaptive control methods, neural networks, and fuzzy control theory to control active suspension systems [17,18,19,20,21]. These approaches have been shown to improve ride comfort and handling performance by adapting to changing road conditions and providing real-time control of suspension dynamics, either independently or in conjunction with complementary methodologies [16,22,23,24,25]. Two additional prominent methodologies that have been extensively studied in the existing literature include Model Predictive Control (MPC) and Sliding Mode Control (SMC). MPC excels in dynamic adaptation through predictive modeling, resulting in improved ride comfort and vehicle handling, while SMC showcases robustness to uncertainties and disturbances, enhancing stability and control precision in suspension applications. In both cases, substantial performance improvements have been consistently observed when compared to passive suspension systems, affirming the effectiveness of these control methodologies [26,27,28,29,30].

In the domain of vibration analysis, researchers often model road surface irregularities as a stochastic process. This involves characterizing them either as white noise ground speed or as deterministic disturbances, such as road bumps or puddles acting as impulse inputs [31]. The $H_2$ norm, serving as a quantitative measure of the root mean square (RMS) amplitude of the system’s output in reaction to white noise or impulse inputs, is a prevalent metric employed for the evaluation of ride comfort within the context of vehicular dynamics analysis [10]. Consequently, the $H_2$ norm is frequently regarded as the preferred option, as it is typically less conservative than the $H_{\infty }$ norm when quantifying ride comfort in scenarios where the system is subjected to white noise or impulse inputs. As a result, many researchers have explored the use of Linear Quadratic Gaussian (LQG) and $H_2$ norm minimization in the design of active vehicle suspension systems [2,31,32,33].

It is crucial to emphasize that the efficacy of $H_2$ synthesis can diminish when the suspension system encounters road profiles different from those it was initially designed for, especially in cases where the road inputs are deterministic, such as bumps or potholes. On the other hand, $H_{\infty }$ control is designed to minimize the RMS value of the system output for worst-case road irregularities, ensuring a conservative approach that is effective for all road profiles, including deterministic inputs. As a result, numerous studies on $H_{\infty }$ or $\mu$ synthesis have been conducted in the field of suspension design, including [1,7,8] and [34,35,36].

To enhance the performance of the suspension system in response to a wide range of road irregularities, our novel approach recommends automatic controller switching contingent upon the road type. This is achieved through meticulous manual controller transitions and the systematic accumulation of input–output datasets from diverse road profiles. The resulting dataset serves as the foundation for training an innovative Adaptive Neuro-Fuzzy Inference System (ANFIS) controller. The main novelties of this work can be summarized as follows:

• 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.

The structure of this study is outlined as follows: In Section 2, a detailed description of the quarter-car model used for the suspension system in this paper is presented. Additionally, this section elaborates on the active control problem, providing specific details regarding the control objectives aimed at enhancing ride safety and comfort. Section 3 introduces the linear matrix inequality (LMI)-based $H_2$ and $H_{\infty }$ control methods as the foundation for designing the ANFIS-based controller structure. This section also outlines the ANFIS neuro-fuzzy architecture. Section 4 contains simulation studies, which serve to illustrate the effectiveness and advantages of the proposed method in achieving the control objectives when compared to the two previously employed control approaches. Lastly, in Section 5, concluding remarks are provided.

2. Problem Formulation and Design Method

2.1. Quarter-Car Model

In this study, a 2-degree-of-freedom (2-DOF) quarter-car model is employed to depict the suspension system, as illustrated in Figure 1. This model comprises two main components: the chassis, denoted as the sprung mass ($m_s$), and the axle, referred to as the unsprung mass ($m_u$). A suspension system is mounted between these components to facilitate the analysis. This model is widely used in active vehicle suspension studies due to its ability to effectively capture the essential characteristics of a real suspension system.

Figure 1. Quarter-car model of the active suspension system.

The equations of motion that elucidate the behavior of the sprung and unsprung masses in the active suspension system are expressed as follows:

$$m_s{\ddot{z}}_s=k_s\left(z_u-z_s\right)+b_s\left({\dot{z}}_u-{\dot{z}}_s\right)+u$$

(1)

$$m_u{\ddot{z}}_u=k_s\left(z_s-z_u\right)+b_s\left({\dot{z}}_s-{\dot{z}}_u\right)+k_u\left(z_r-z_u\right)-u$$

(2)

where $z_s$ and $z_u$ represent the displacement of the sprung and unsprung masses, while $k_s$ and $k_u$ signify the spring constant of the suspension system and the tire stiffness, respectively; $b_s$ is indeed the damping coefficient of the suspension system;$z_r$ is the road disturbance input; and u signifies the control input.

The state variables are chosen as follows:

$$x_1\left(t\right)=z_s-z_u$$

$$x_2\left(t\right)=z_u-z_r$$

$$x_3\left(t\right)={\dot{z}}_s$$

$$x_4\left(t\right)={\dot{z}}_u$$

The state-space representation of the quarter-car model is provided below:

$$\dot{x}\left(t\right)=Ax\left(t\right)+B_{1}w\left(t\right)+B_{2}u\left(t\right)$$

with

$$A=\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ -\frac{k_s}{m_s}& 0& -\frac{c_s}{m_s}& \frac{c_s}{m_s}\\ \frac{k_s}{m_u}& -\frac{k_u}{m_u}& \frac{c_s}{m_u}& -\frac{c_s}{m_u}\end{array}\right] , B_1=\left[\begin{array}{c}\begin{array}{ccc}0& 0& 0\\ -1& 0& 0\end{array}\\ \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\end{array}\right] , B_2=\left[\begin{array}{c}0\\ 0\\ \frac{1}{m_s}\\ -\frac{1}{m_u}\end{array}\right]$$

The road roughness-induced disturbance input is denoted as $w={\dot{z}}_r$. The parameter values employed in this study are presented in Table 1 [37].

Table 1. Parameters of “Renault Mégane Coupe” car.

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

2.2. Design Objectives

The assessment of body acceleration serves as a prevalent metric in evaluating passenger comfort within vehicular contexts. In the pursuit of enhancing passenger comfort, it becomes imperative to minimize both the vertical displacement and the rate of change in velocity (acceleration) experienced by the vehicle’s body. This objective can be effectively pursued through the optimization of damper displacement. The role of dampers, also known as shock absorbers, in this endeavor, is pivotal. Dampers, as integral components of a vehicle’s suspension system, significantly contribute to mitigating the effects of road irregularities on passenger comfort. Notably, the damper force exerted depends on the vehicle’s speed, increasing in magnitude as the vehicle’s speed rises. This modulation of damper force with speed is essential to effectively control and dampen body movements, maintaining both ride comfort and road-holding capabilities.

However, it is paramount to underscore that safety represents an equally essential consideration alongside comfort. Ensuring safe driving conditions necessitates the maintenance of optimal tire-to-road contact. Consequently, the design of the suspension system must be executed with meticulous attention to balance the dual imperatives of comfort and safety under various driving scenarios, all while considering the dynamic fluctuations in system parameters.

In designing a controller for a suspension system, the following specifications must be taken into account:

• 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.,

  $$k_u\left(z_u-z_r\right)<\left(m_s+m_u\right)g, \forall t\ge 0$$

  This fact can be utilized to normalize the dynamic tire load.
• 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:

  $$\left|u\right|<u_{max}$$

In order to optimize ride comfort, it is essential to minimize the RMS value of body acceleration, all the while allowing suspension deflection, tire deflection, and the control signal to fluctuate within their pre-defined limits.

Due to the predominant modeling of road roughness as white noise [31], the $H_2$ norm appears to be less cautious than the $H_{\infty }$ norm when quantifying any of the aforementioned outputs. Moreover, in the suggested approach, the $H_{\infty }$ controller is employed to achieve robust performance goals and ride stability by minimizing the associated transfer functions for road disturbances that are more inclusive than just white noise and impulse. In light of these factors, it is evident that there is a requirement for a controller that encompasses all of the aforementioned features of both controllers.

2.3. Review of $H_2$ and H_∞ Design Frameworks

2.3.1. $H_2$ Synthesis

The control objectives outlined in the preceding section necessitate the incorporation of normalizing weights into both control outputs $[z_{def} z_{ur} {\ddot{z}}_s]$ and exogenous inputs $[w n_1 n_2]$. Figure 2 depicts a general block diagram of the system. As mentioned earlier, it is crucial to identify specific frequency conditions concerning suspension deflection (aiming to reduce gain at lower frequencies), dynamic tire load, and the control signal. Input weights denote the frequency characteristics of the inputs, while output weights signify the desired frequency components of the controlled outputs [32]. Additionally, it is crucial to acknowledge the presence of measurement noise, denoted as $n_1$ and $n_2$, which can affect the accuracy of the measured outputs $y=[y_1 y_2]$. The practical importance of these measured outputs, specifically suspension deflection and body acceleration, cannot be overstated. They can be readily acquired through the utilization of suitable sensors. Therefore, the inclusion of normalizing weights accounts for the frequency characteristics of both inputs and measurements, contributing to the effective control of the system while considering these exogenous noise inputs in the design procedure.

Figure 2. Generalized block diagram of the active suspension system.

The following is a description of the augmented plant model for the design:

$$\dot{x}\left(t\right)=Ax\left(t\right)+B_{1}w\left(t\right)+B_{2}u\left(t\right)$$

$$z\left(t\right)=C_{1}x\left(t\right)+D_{11}w\left(t\right)+D_{12}u\left(t\right)$$

(3)

$$y\left(t\right)=C_{2}x\left(t\right)+D_{21}w\left(t\right)+D_{22}u\left(t\right)$$

The core aim is to develop an $H_2$ controller that guarantees the stability of the closed-loop system while meeting specific control objectives. These objectives encompass reducing the $H_2$ norm between the disturbance input $\omega$ and two key variables: body acceleration (${\ddot{z}}_s$) and tire deflection ($z_{ur}$). This optimization aims to enhance ride comfort and ride safety, respectively. Simultaneously, the control signal (u) and suspension deflection ($z_{def}$) must be maintained within their allowable boundaries. The choice of weights is made to align with the design framework within the specified frequency range.

For the $H_2$ design, let us describe the controller as follows:

$$K:=\left(\genfrac{}{}{0pt}{}{{\dot{x}}_c}{u}\right)=\left(\begin{array}{cc}{A}_c& B_c\\ C_c& D_c\end{array}\right)\left(\genfrac{}{}{0pt}{}{x_c}{y}\right)$$

(4)

The state-space representation of the closed-loop system is obtained as follows:

$${\mathcal{A}}_{cl}:=\left(\begin{array}{cc}A+B_2{D}_c{C}_2& B_2{C}_c\\ B_c{C}_2& A_c\end{array}\right)$$

$${\mathcal{B}}_{cl}:=\left(\begin{array}{c}{B}_1+B_2{D}_c{D}_{21}\\ B_c{D}_{21}\end{array}\right)$$

(5)

$$\begin{gathered} {\mathcal{C}}_{cl}:=\left(\begin{array}{cc}{C}_1+D_{12}{D}_c{C}_2& D_{12}{C}_c\end{array}\right) \\ {\mathcal{D}}_{cl}:=D_{11}+D_{12}{D}_c{D}_{21} \end{gathered}$$

Theorem 1.

The closed-loop system will be stable, and the $H_2$ norm from exogenous inputs to controlled outputs will be less than γ if, and only if, there exist symmetric positive definite matrices, $\mathcal{X}$ and $\mathcal{Z}$, that satisfy the following inequalities [33]:

$$\left(\begin{array}{cc}{\mathcal{A}}_{cl}\mathcal{X}+\mathcal{X}{\mathcal{A}}_{cl}^T& {\mathcal{B}}_{cl}\\ \ast & -I\end{array}\right)\prec 0$$

$$\left(\begin{array}{cc}\mathcal{Z}& {\mathcal{C}}_{cl}\mathcal{X}\\ \ast & \mathcal{X}\end{array}\right)\succ 0$$

(6)

$$Trace\left(\mathcal{Z}\right)<{\gamma }^2,\hspace{1em}\hspace{1em}\hspace{1em}{\mathcal{D}}_{cl}=0$$

2.3.2. H_∞ Synthesis

The primary objective of $H_{\infty }$ controller design is to minimize the $H_{\infty }$ norm of a system while preserving internal stability [38]. It is worth noting that the $H_{\infty }$ norm possesses diverse interpretations, particularly concerning its application in both the time and frequency domains. In the frequency domain, the objective is to diminish the peak singular value of the transfer function concerning frequency. This goal equates to minimizing the system’s maximum gain across the entire frequency spectrum:

$${‖G\left(s\right)‖}_{\infty }\triangleq \underset{\omega }{\mathit{sup}}\overline{\sigma }\left(G\right(j\omega \left)\right)$$

(7)

In the single-input, single-output (SISO) scenario, the $H_{\infty }$ norm is synonymous with the maximum magnitude of the transfer function. The $H_{\infty }$ norm’s interpretation in the time domain is expressed as the induced 2-norm, as detailed below [39]:

$${‖G\left(s\right)‖}_{\infty }=\underset{w\left(t\right)\ne 0}{\mathit{sup}}\frac{{‖z\left(t\right)‖}_2}{{‖w\left(t\right)‖}_2}$$

(8)

where the induced 2-norm, indicated as ${‖z\left(t\right)‖}_2=\sqrt{{\int }_0^{\infty }{\sum }_i{\left|z_i\left(t\right)\right|}^{2}dt}$, signifies the energy content of the signal vector. In the context of the time domain interpretation of the $H_{\infty }$ norm, the primary objective is to mitigate the energy of the output signal when subjected to the most adverse input signal conditions.

In accordance with the principles of linear dissipative systems, the controller can be derived by satisfying the bounded realness condition, commonly referred to as the bilinear matrix inequality (BMI) [40]:

$$\mathit{Minimize} {\gamma }_{\infty } \mathit{subject}\hspace{0.17em}\mathit{to}$$

$$\mathcal{X}\succ 0$$

(9)

$$\left(\begin{array}{ccc}{\mathcal{A}}_{cl}^{T}K+K{\mathcal{A}}_{cl}& K{\mathcal{B}}_{cl}& {\mathcal{C}}_{cl}^T\\ \ast & -{\gamma }_{\infty }^{2}I& {\mathcal{D}}_{cl}^T\\ \ast & \ast & -I\end{array}\right)\succ 0$$

By employing a change in variables similar to the $H_2$ design and an appropriate congruence transformation, the corresponding LMI (linear matrix inequality) conditions for $H_{\infty }$ can be obtained. The Appendix A is devoted to this transformation. Also, readers are encouraged to refer to [41,42] for a more detailed understanding of this process.

3. Adaptive Neuro-Fuzzy Inference Systems (ANFIS)

For simplicity, a fuzzy system architecture, commonly known as the Type-3 ANFIS structure, is considered with a two-input, two-rule fuzzy system that accommodates five layers inside it [43]. Suppose that two if–then rules of a Sugeno-type fuzzy system are as given below:

$$\mathrm{If} x \mathrm{is} A1 \mathrm{and} y \mathrm{is} B1, \mathrm{then} f_1=p_{1}x+q_{1}y+r_1$$

$$\mathrm{If} x \mathrm{is} A2 \mathrm{and} y \mathrm{is} B2, \mathrm{then} f_2=p_{2}x+q_{2}y+r_2$$

The equivalent ANFIS structure corresponding to this controller design is illustrated in Figure 3. The node functions within a given layer are members of the same function family, and this family is described as follows:

Figure 3. Type-3 ANFIS architecture.

• Layer 1: Every node in this layer is depicted as a square shape, each associated with a specific membership function:

  $$O_i^1={\mu }_{A_i}\left(x\right)$$

  where the superscripts above and below denote the layer number and the node index within the layer, respectively. The input to the i-th node in this layer is denoted by x (or y), while $A_i$ represents the corresponding linguistic term associated with this node (e.g., “small”, “large”, etc.). In this specific case, the membership function, denoted as ${\mu }_{A_i}\left(x\right)$, is selected to have a bell-shaped form with a maximum value of 1 and a minimum value of 0, as illustrated below:

  $$O_i^1={\mu }_{A_i}\left(x\right)=\frac{1}{1+{\left[{\left(\frac{x-c_i}{a_i}\right)}^2\right]}^{b_i}}$$

  (10)

  The parameter set $\left\{a_i,b_i,c_i\right\}$ corresponds to the specific parameters of the bell-shaped membership function. Any variation in the values of these parameters will result in changes to the shape of the bell-shaped function accordingly. As a result, a wide range of membership functions can be induced for the linguistic value $A_i$ by adjusting the values of these parameters.
• 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:

  $$O_i^2=w_i={\mu }_{A_i}\left(x\right)\times {\mu }_{B_i}\left(y\right) , i=1, 2$$

  where ${\mu }_{A_i}\left(x\right)$ and ${\mu }_{B_i}\left(y\right)$ are the membership functions for the input variables x and y, respectively, associated with the i-th node, and $\times$ represents the logical AND operator. The resulting value $w_i$ represents the firing strength or degree of membership for the i-th rule. Each output node corresponds to the firing strength of a specific rule.
• 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:

  $$O_i^3={\overline{w}}_i=\frac{w_i}{w_1+w_2} , i=1, 2$$

  The computed values, denoted as ${\overline{w}}_i$, are termed normalized firing strengths. These values signify the extent of the contribution of each rule towards the ultimate output of the ANFIS model.
• 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:

  $$O_i^4={\overline{w}}_i{f}_i={\overline{w}}_i(p_{i}x+q_{i}y+r_i)$$

  The set of parameters $\left\{p_i,q_i,r_i\right\}$ are referred to as the consequent parameters, and they determine the shape and position of the output surface of the ANFIS model.
• 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:

  $$O_1^5={\sum }_{i=1}^2{\overline{w}}_i{f}_i=\frac{{\sum }_{i=1}^2{\overline{w}}_i{f}_i}{{\sum }_{i=1}^2{w}_i}$$

  (11)

  Here, $w_i$ and $f_i$ denote the weight and input signal of the i-th input node, respectively. The output $O_1^5$ represents the weighted sum of the input signals normalized by the sum of the weights.

In this way, a Type-3 ANFIS structure is established, demonstrating functional equivalence to a Type-3 fuzzy reasoning system.

ANFIS Design

The ANFIS algorithm utilizes input–output datasets obtained from the design procedure of the $H_2$ and $H_{\infty }$ controllers to construct a fuzzy inference system (FIS). To generate a training dataset for the ANFIS controller in an active suspension system, data were collected for both the $H_2$ and $H_{\infty }$ controllers by simulating the vehicle over various road conditions while recording the input parameters (e.g., road roughness) and the corresponding output control actions from each controller. We labeled the data to indicate which controller was active during data collection. Specifically, the training data used as input to the ANFIS system comprised the deflection and acceleration of the suspension system, while the output data were generated from two distinct sources: The first component of the output data is related to the command force data generated by the $H_{\infty }$ controller in response to the bump, which is designed to enhance the robustness of the system. The second component of the output data is linked to the force command of the $H_2$ controller in response to the white noise road profile. The data were collected over a period of 10 s for each controller, resulting in a total of 60,000 data points. To train the ANFIS system, the first 70% of the data were used as the training dataset, while the remaining 30% were reserved for validating the system. ANFIS automatically determines when to switch smoothly between controllers during inference by evaluating the input conditions and the learned knowledge from the dataset. This allows it to make dynamic controller transitions for optimal suspension performance, as it continuously adjusts its output based on the current road conditions and the knowledge it has acquired from the training data. In this study, we analyze the response of closed-loop systems to an asphalt road profile comprising white noise and bump disturbance. For the purpose of this study, we define bump disturbance as follows:

$$z_r\left(t\right)=\left\{\begin{array}{c}\frac{h}{2}\left(1-\mathit{cos}\left(\frac{2\pi vt}{l}\right)\right) , 0\le t\le \frac{l}{v} \\ 0 , t>\frac{l}{v} \end{array}\right.$$

(12)

Bump disturbance is described using the following parameters: h = 0.1 m, l = 2 m, and v = 10 m/s (equivalent to 36 km/h); these represent the height and length of the bump, and the velocity of the vehicle, respectively. Figure 4 and Figure 5 illustrate the training data and validation results for the command force of both types of road inputs, along with their respective errors. Specifically, Figure 4a and Figure 5a depict the predicted command force and its ability to track the target command force, while Figure 4b,c and Figure 5b,c present the corresponding error and its RMS value, respectively. These results demonstrate that the ANFIS system is capable of accurately predicting the command force, as evidenced by the small magnitude of the error and its RMS value.

Figure 4. (a) Training data, (b) training error, and (c) training RMSE.

Figure 5. (a) Test data, (b) test error, and (c) test RMSE.

4. Simulation Results

The design procedures for the $H_2$ and $H_{\infty }$ controllers were performed separately, following the steps outlined in [32]. The simulations were conducted in the 2020b MATLAB/Simulink environment. In order to assess the effectiveness of the $H_2$ and $H_{\infty }$ controllers, a comparative analysis was conducted by examining the frequency response of the desired objectives in contrast to that of a passive suspension system across a broad frequency spectrum, as depicted in Figure 6. The results indicate a significant improvement in body acceleration and tire deflection for both controllers. As mentioned before, the $H_2$ controller is designed to minimize the mean squared value ($L_2$ norm) of the system’s output, which equates to reducing energy or power within specific frequency bands. This design characteristic renders the $H_2$ controller especially adept at mitigating vibrations. In the realm of active suspension systems, where vibrations within the 1 to 20 Hz frequency range significantly influence ride comfort and road handling, the $H_2$ controller excels by effectively managing vibrations within this critical frequency range, thereby underpinning its superior performance in this context.

Figure 6. Frequency responses: (a) body acceleration, (b) tire deflection.

In Figure 7, a comparison is presented illustrating the weighted body acceleration responses for both $H_2$ and $H_{\infty }$ controllers when subjected to bump and asphalt road inputs. The findings reveal that the $H_2$ controller excels in minimizing body acceleration and tire deflection under asphalt road input, whereas the $H_{\infty }$ controller outperforms the $H_2$ controller in response to bump input. This observation underscores the distinct strengths and weaknesses inherent in each controller. Therefore, the selection of the appropriate controller hinges on the specific road conditions and the defined design objectives.

Figure 7. Comparison of body acceleration of $H_2$ and $H_{\infty }$ controllers: (a) bump input, (b) real-world asphalt road.

Based on the aforementioned details, the proposed ANFIS-based controller is anticipated to incorporate the desirable properties of both the $H_2$ and $H_{\infty }$ controllers in response to road inputs. To evaluate the effectiveness of the proposed ANFIS-based controller, a comparison was conducted with the two aforementioned controllers in terms of body acceleration and tire deflection. These parameters serve as key indicators in suspension design, significantly influencing both ride safety and comfort. These comparisons are presented in Figure 8, with the road profile consisting of a real-world asphalt road profile with two tandem bumps. As demonstrated in Figure 8, the proposed ANFIS-based controller outperforms both the $H_2$ and $H_{\infty }$ approaches in terms of meeting the design objectives. Specifically, the proposed method achieves the best performance in minimizing body acceleration and tire deflection, thereby improving ride safety and comfort. In other words, the ANFIS-based controller behaves like the $H_{\infty }$ controller when confronted with the bump input, while it behaves like the $H_2$ controller in response to the asphalt road input. This property of the ANFIS-based controller allows it to achieve optimal performance under both types of road inputs. It thus represents a significant improvement over either the $H_2$ or $H_{\infty }$ controller alone.

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.

The RMS values of the design objectives are listed in Table 2 to compare the three methods.

Table 2. RMS values of design objectives.

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

In this comparative study of active suspension systems, we evaluated performance based on key design objectives (body acceleration (${\ddot{z}}_s$), suspension deflection ($z_{def}$), and unsprung mass displacement ($z_{ur}$)) using RMS values. The ANFIS-based controller emerges as the standout performer, achieving a remarkable 62.07% reduction in body acceleration and a significant 56.8% decrease in unsprung mass displacement compared to the passive suspension. The ANFIS-based controller exhibits a slight decrease of 5.83% in suspension deflection compared to the passive system.

Comparatively, when assessed against the $H_2$ controller, ANFIS demonstrates a 20.6% improvement in body acceleration, a substantial 34.33% reduction in unsprung mass displacement, a modest 6.6% increase in suspension deflection, and a slight 7% decrease in command force.

Similarly, when compared to the $H_{\infty }$ controller, ANFIS excels with a 44.2% reduction in body acceleration and a 42.9% increase in suspension deflection. Additionally, ANFIS reduces unsprung mass displacement by 12.5% and increases command force by 16.7%. Importantly, it is noteworthy that the ANFIS-based controller, while exhibiting slightly increased suspension deflection compared to the $H_2$ and $H_{\infty }$ controllers, still operates well within the predefined structural limitations of the suspension system deflection. This controlled increase in suspension deflection ensures that the system maintains stability and structural integrity within acceptable structural constraints.

In summary, these outcomes affirm the effectiveness of the proposed ANFIS-based controller in meeting the design goals of enhancing ride comfort and safety. Additionally, it ensures that suspension stroke and command force remain within their predetermined limits. Notably, the ANFIS-based controller exhibits distinct advantages over the $H_2$ and $H_{\infty }$ controllers, particularly in the reduction of body acceleration and unsprung mass displacement. This makes ANFIS a compelling choice for improving active suspension system effectiveness while ensuring structural integrity within defined constraints. These improvement percentages are illustrated in Figure 9. It is important to acknowledge that in this comparison, the passive suspension system serves as the baseline. Consequently, the force section is omitted from the plotted data.

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

5. Conclusions

In this research endeavor, we introduced an Adaptive Neuro-Fuzzy Inference System (ANFIS)-based controller as a novel approach to the design of active vehicle suspension systems. This ANFIS controller was formulated to encompass the comprehensive attributes associated with both $H_2$ and $H_{\infty }$ controllers, specifically emphasizing optimizing ride comfort and ensuring safety.

The ANFIS methodology was carefully employed to construct a fuzzy inference system, utilizing input–output datasets derived from the distinct design methodologies employed in $H_2$ and $H_{\infty }$ controllers. The ensuing simulation results, employing a quarter-car model, were meticulously compared with those obtained using the $H_2$ and $H_{\infty }$ controllers. Our investigation has unequivocally demonstrated that the ANFIS-based controller offers substantial performance advantages, achieving optimal ride comfort while effectively constraining suspension deflection and control force within their prescribed limits, thus concurrently upholding ride safety standards. Ultimately, our findings underscore the potential efficacy and efficiency of the ANFIS-based controller as a compelling solution in the domain of active vehicle suspension system design. This research presents a significant stride toward enhancing vehicular ride quality and safety through advanced control methodologies.

In future work, optimizing the ANFIS-based controller, evaluating its robustness under uncertainties, and validating its performance through practical experiments are essential steps for assessing its application and real-time performance.