Robust Static Output Feedback Control of a Semi-Active Vehicle Suspension Based on Magnetorheological Dampers

Abstract

This paper proposes a novel design method for a magnetorheological (MR) damper-based semi-active suspension system. An improved MR damper model that accurately describes the hysteretic nature and effect of the applied current is presented. Given the unfeasibility of installing sensors for all vehicle states, an MR damper current controller that only considers the suspension deflection and deflection rate is proposed. A linear matrix inequality problem is formulated to design the current controller, with the objective of enhancing ride safety and comfort while guaranteeing vehicle stability and robustness against any road disturbance. A series of experiments demonstrates the enhanced performance of the proposed MR damper model, which exhibits greater accuracy than other state-of-the-art damper models, such as Bingham or bi-viscous. An evaluation of the vehicle behavior under two simulated road scenarios has been conducted to demonstrate the performance of the proposed output feedback MR damper-based semi-active suspension system.

1. Introduction

The suspension system plays an important role in the overall performance of a vehicle, influencing aspects such as ride comfort, handling stability, and safety [1,2,3,4,5,6]. Vehicle suspension systems can be classified into three main categories: passive, semi-active, and active. Passive suspension systems typically comprise springs and dampers, which serve to absorb impacts and control spring motions, respectively. It is important to be aware that passive suspension systems have inherent limitations in their ability to provide an optimal balance between road holding and ride comfort [7]. To remedy this problem, an active or semi-active suspension can be employed. An active suspension system includes various actuators which pull down or push up the vehicle body to suppress the vibrations produced by road irregularities [8,9]. The principal disadvantage of active suspensions is their complexity, the elevated cost of manufacture and maintenance, and the associated customer service expenses [10,11]. Semi-active suspensions do not add energy to the suspension system but can change suspension parameters such as the damping coefficient of the shock absorber in real time, depending on the road profile and the vehicle dynamic state [12,13]. In light of the aforementioned considerations, it can be inferred that the most adequate solution can be a semi-active suspension system, which displays superior performance in comparison to a classical (passive) system. Furthermore, it offers a more straightforward and cost-effective alternative to an active system.

With regard to semi-active suspensions, magnetorheological (MR) dampers have been the subject of considerable interest over several decades [14]. An MR damper is a semi-active device that employs a MR fluid with adjustable viscosity, which is controlled by an external magnetic field. The fluid within MR dampers comprises micron-sized particles that are magnetisable and suspended in an oil-based medium. When exposed to magnetic fields, the particles align with the field, resulting in a notable increase in the fluid’s viscosity and enhanced damping capabilities [15]. MR dampers possess the following beneficial characteristics: variable damping force, rapid response time, fault safety, and low power consumption [16,17].

One of the primary challenges associated with integrating MR dampers for semi-active suspensions is their highly non-linear behaviour, which is inherently difficult to model [18]. For that purpose, a variety of models have been proposed to characterize the dynamic damper behavior. These include the Bingham model [19], the bi-viscous hysteresis model [20], and the phenomenological Bouc–Wen model [21]. While all of these models are capable of capturing the post-yield behavior of MR dampers, the modeling of nonlinear hysteretic damper behavior presents a significant challenge. For that purpose, the Bingham and Bouc–Wen models can be considered as two extremes. While Bingham is the simplest model, only accounting the magnitude and sign of the deflection rate of the damper [22], Bouc–Wen uses differential equations that depend on 14 experimental parameters [23]. The bi-viscous model represents a middle ground, utilizing a set of six equations based on piston velocity to estimate damper force [24]. Despite the fact that a higher-order model, such as the Bouc–Wen, has the potential to more accurately represent the actual behavior of a damper, several factors act as obstacles to its acceptance as a viable alternative. These include adjusting a large set of experimental parameters, which implies a complex and time-consuming identification process, as well as the difficulty in achieving a real-time implementation of such a complex model [25].

In order to adapt the behavior of semi-active or active suspensions according to the road conditions, it is often assumed that all vehicles are equipped with sufficient sensors to accurately measure their respective states, which is not a realistic assumption, as it is impractical to expect mass-produced vehicles to have such advanced features [26]. In [27], Lyapunov function–control barrier function–quadratic programming is used for active suspension control. In [28], a novel optimal control strategy for regenerative active suspension system to enhance energy harvesting is described. These works consider that some states, such as the vertical displacement and velocity of the sprung and unsprung mass, as well as tire deflection, to be measurements that can be readily obtained. In order to address the issue of incomplete state information, it is possible to incorporate an observer to estimate those vehicle states that cannot be directly measured. In [29], an active suspension system uses an unknown input observer to estimate the unmeasured vehicle states. In [30], an observer that estimates the effect of the road disturbance is presented. Nevertheless, the incorporation of supplementary observation stages leads to an increase in overall system complexity. In order to provide a more straightforward approach, it is possible to design controllers that rely solely on the data obtained from the available sensors, at the cost of a more complex design stage [31]. In this particular case, the majority of research findings are related to active suspension systems. This is due to the fact that, from a theoretical standpoint, they have fewer limitations and complexity than semi-active suspensions. In [32,33,34,35], different methodologies for active suspensions are presented, where the actuator force can be generated as a function of the vertical acceleration of the sprung mass and/or suspension deflection. In general terms, there is a notable lack of research on semi-active suspensions that considers the inherent incompleteness of vehicle state information. In [36], a hybrid damping control of magnetorheological semi-active suspension is used; however, a filter is used to estimate the unmeasured vehicle states. In [37], an adaptive backstepping control design for semi-active suspension with MR dampers is presented; however, the totality of the vehicle states is assumed to be measured. In [38], a MR damper-based semi-active suspension system that assumes all vehicle states available is designed using an LQR controller incorporating NAGA-II algorithm.

In consideration of the aforementioned reasons, this paper presents a methodology for the design of semi-active suspensions utilizing MR dampers, with due consideration of the limited availability of vehicle sensors. The main contributions are summarized as follows:

• An improved model for the MR damper behavior is presented and validated from experiments. The model accurately describes the hysteretic nature and the effect of the applied current on the MR damper, demonstrating superior performance compared to other state-of-the-art damper models, such as Bingham or bi-viscous models.

• Convex conditions are obtained to design a robust static output feedback MR damper current controller, only utilizing suspension deflection information, which can be practically measured by a Linear variable differential transformer (LVDT). Stability and robustness of the proposed semi-active suspension system are guaranteed under Lyapunov and ${\mathcal{H}}_{\infty }$ criteria, respectively. An evaluation of the vehicle behavior under two road scenarios demonstrates that the proposed MR damper-based semi-active suspension system exhibits enhanced performance compared to a passive suspension system.

The remainder of this article is structured as follows. In Section 2, the problem to be addressed is formulated, with the proposed MR damper model presented in Section 2.1, the semi-active suspension model shown in Section 2.2, and the MR current controller design described in Section 2.3. The validity of this work is presented in Section 3, where the experimental tests conducted to validate the MR damper are detailed in Section 3.1 and an evaluation of the MR damper-based semi-active suspension is provided in Section 3.2. This paper is concluded in Section 4.

Notation. The set of non-negative integers is denoted by ${\mathbb{Z}}_{+}$. For a matrix X, $X^{\top }$ denotes its transpose. The function $\mathrm{He}\left(X\right)$ returns $\mathrm{He}\left(X\right)=X+X^{\top }$. If Y is a square matrix, $Y>0$ denotes that Y is positive definite. In a symmetric matrix, the symbol * indicates the transpose of the symmetric term. The function $diag\left\{\begin{array}{c}{X}_1,X_2\end{array}\right\}$ retrieves a block-diagonal matrix composed by $X_1$ and $X_2$. If not stated, matrices are supposed to have compatible dimensions. Arguments are omitted when their meaning is clear.

2. Problem Formulation

2.1. Magnetorheological Damper Model

A schematic illustration of an MR damper is shown in Figure 1. It is essential to count on an appropriate dynamic model to calculate the command current (or voltage) for the MR damper in order to ensure optimal suspension performance in line with the vehicle status.

MR dampers exhibit significant hysteretic behavior when the velocity range of the damper is limited. However, for larger velocities, the force demonstrates a nearly linear variation with velocity. The hyperbolic tangent function is frequently used to model hysteretic behavior, as it effectively captures the nonlinear, memory-dependent characteristics typical of such systems. This function’s smooth, S-shaped curve allows for accurate representation of gradual transitions, making it a valuable tool for simulating the complex, cyclic responses often observed in materials and control applications with hysteresis. Inspired by [39], the following equation is proposed for modeling the MR damper:

(1)$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{md}\left(t\right)=& (a_0+a_{1}i\left(t\right))\tanh \left(a_2\left({\dot{z}}_{def}\left(t\right)+{\displaystyle \frac{V_0}{X_0}}{z}_{def}\left(t\right)\right)\right)\hfill \\ \hfill & +a_3\left({\dot{z}}_{def}\left(t\right)+{\displaystyle \frac{V_0}{X_0}}{z}_{def}\left(t\right)\right)\hfill \end{array}\end{array}$

where $F_{md}$ is the MR damper force. $a_i$, $i\in 0,\dots ,3$, are experimental parameters to be adjusted. The damper deflection is denoted by $z_{def}$. The ratio $V_0/X_0$ depends on the absolute values of the hysteretical critical velocity $V_0$ and hysteretical critical displacement $X_0$, respectively (see Figure 2). These denote the piston velocity and displacement when $F_{md}=0$. The stiffness ($k_{mr}$) and damping ($c_{mr}$) parameters of the MR damper are defined by

(2)$\begin{array}{c}\hfill \begin{array}{cc}\hfill k_{mr}& =a_3{\displaystyle \frac{V_0}{X_0}}\hfill \\ \hfill c_{mr}& =a_3\hfill \end{array}\end{array}$

The employed MR model (1) is an extension of the Bingham model [22], which models the damper force as $F_{md}=c_{1}sgn\left({\dot{z}}_{def}\right)+c_2{\dot{z}}_{def}$. It is crucial to acknowledge that the Bingham model does not incorporate the elastic characteristics of the damper. Additionally, the employed model utilizes the tanh function, which has a more gradual behavior than the sgn function for analyzing the damper hysteresis. It is worth noting that the Bingham model is a specific instance of the proposed model, when the hysteresis, pre-yield, and current effects on the damper are neglected.

In contrast to the bi-viscous hysteresis model [20], which employs a six-section piecewise function incorporating three parameters to model the MR damper behavior, the proposed method offers a straightforward solution for analyzing the dynamic behavior of MR dampers, as only one function is required.

In this work, the parameters of the model (1) were obtained through experimentation. The experimental setup and tests are presented in Section 3.1.

2.2. Semi-Active Suspension Model

A quarter-car suspension model, as depicted in Figure 3, is considered for this work, wherein the vehicle body is assumed to be rigid and to possess freedom of movement in the vertical direction.

The governing equations of motion for the quarter-car model, which account for the semi-active suspension behavior resulting from the MR damper (1), are as follows:

(3)$\begin{array}{c}\hfill \begin{array}{cc}\hfill & m_s{\ddot{z}}_s\left(t\right)+k_s\left(z_s\left(t\right)-z_u\left(t\right)\right)+F_{md}\left(t\right)=0\hfill \\ \hfill & m_u{\ddot{z}}_u\left(t\right)+k_s\left(z_u\left(t\right)-z_s\left(t\right)\right)-F_{md}\left(t\right)\hfill \\ \hfill & +k_t\left(z_u\left(t\right)-z_r\left(t\right)\right)+c_t\left({\dot{z}}_u\left(t\right)-{\dot{z}}_r\left(t\right)\right)=0\hfill \end{array}\end{array}$

where $z_s$, $z_u$, and $z_r$ denote the displacements of the sprung mass, unsprung mass, and road profile, respectively. The quarter-car model parameters are presented in Table 1.

Furthermore, the following performance aspects must be taken into account with regard to the vehicle suspension [40]:

1. In order to enhance ride comfort, the vertical acceleration ${\ddot{z}}_s$ has to be minimized

   (4)$\min {\ddot{z}}_s\left(t\right)=-{\displaystyle \frac{k_s\left(z_s\left(t\right)-z_u\left(t\right)\right)+F_{md}\left(t\right)}{m_s}}$

2. The suspension deflection is limited by the mechanical structure and cannot exceed a maximum value

   (5)$\left|\begin{array}{c}{z}_s\left(t\right)-z_u\left(t\right)\end{array}\right|\le z_{max}$

3. In order to guarantee a firm and uninterrupted contact of the wheels on the road, it is necessary that the dynamic tyre load be less than the static tyre load

   (6)$k_t\left(z_u\left(t\right)-z_r\left(t\right)\right)<(m_s+m_u)g$

Following (1), (2) and (3)–(6), the state-space representation of the quarter-car semi-active suspension model is written as

(7)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{x}\left(t\right)& =\mathcal{A}x\left(t\right)+\mathcal{B}u\left(t\right)+{\mathcal{B}}_{\omega }\omega \left(t\right)\hfill \\ \hfill y\left(t\right)& =C_{y}x\left(t\right)\hfill \\ \hfill z\left(t\right)& =C_{z}x\left(t\right)+D_{z}u\left(t\right)\hfill \end{array}\end{array}$

where x is the state vector, u is the control input, and $\omega$ is the road disturbance. y represents the measured output from the suspension system; in this case, it is assumed that the suspension deflection and its rate can be measured by a linear variable differential transformer (LVDT) [41]. The LVDT serves to convert a linear displacement into a proportional electrical signal. LVDT sensors can be situated in close proximity to the suspension shock-absorbers, on the same screws as the dampers. They are capable of measuring the relative displacement of the suspension system. Due to the high precision of the LVDT signals, they can be differentiated for the estimation of the relative velocity of the suspension elements. The performance vector z is a quantitative indicator utilized to assess whether the suspension deflection, vertical acceleration, and dynamic tire load remain within an acceptable range.

(8)$\begin{array}{c}\hfill \begin{array}{cc}\hfill & x\left(t\right)=\left[\begin{array}{c}{z}_s\left(t\right)-z_u\left(t\right)\\ z_u\left(t\right)-z_r\left(t\right)\\ {\dot{z}}_s\left(t\right)\\ {\dot{z}}_u\left(t\right)\end{array}\right],y\left(t\right)=\left[\begin{array}{c}{z}_s\left(t\right)-z_u\left(t\right)\\ {\dot{z}}_s\left(t\right)-{\dot{z}}_u\left(t\right)\end{array}\right],\omega \left(t\right)={\dot{z}}_r\left(t\right),\hfill \\ \hfill & u\left(t\right)=(a_0+a_{1}i\left(t\right))\tanh \left(a_2\left({\dot{z}}_{def}\left(t\right)+{\displaystyle \frac{V_0}{X_0}}{z}_{def}\left(t\right)\right)\right),\hfill \\ \hfill & z_{def}\left(t\right)=z_s\left(t\right)-z_u\left(t\right)\hfill \end{array}\end{array}$

The state space matrices of system (7) are

$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \mathcal{A}=\left[\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 1\\ -{\displaystyle \frac{k_s+k_{mr}}{m_s}}& 0& -{\displaystyle \frac{c_{mr}}{m_s}}& {\displaystyle \frac{c_{mr}}{m_s}}\\ {\displaystyle \frac{k_s+k_{mr}}{m_u}}& {\displaystyle \frac{k_t}{m_u}}& {\displaystyle \frac{c_{mr}}{m_u}}& -{\displaystyle \frac{c_{mr}+c_t}{m_u}}\end{array}\right],\mathcal{B}=\left[\begin{array}{c}0\\ 0\\ -{\displaystyle \frac{1}{m_s}}\\ {\displaystyle \frac{1}{m_u}}\end{array}\right],{\mathcal{B}}_{\omega }=\left[\begin{array}{c}0\\ -1\\ 0\\ {\displaystyle \frac{c_t}{m_u}}\end{array}\right],\hfill \\ \hfill & C_y=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& -1\end{array}\right],C_z=\left[\begin{array}{cccc}-{\displaystyle \frac{k_s+k_{mr}}{m_s}}& 0& -{\displaystyle \frac{c_{mr}}{m_s}}& {\displaystyle \frac{c_{mr}}{m_s}}\\ {\displaystyle \frac{1}{z_{max}}}& 0& 0& 0\\ 0& {\displaystyle \frac{k_t}{(m_s+m_u)g}}& 0& 0\end{array}\right],D_z=\left[\begin{array}{c}-{\displaystyle \frac{1}{m_s}}\\ 0\\ 0\end{array}\right]\hfill \end{array}\end{array}$

2.3. MR Damper Current Control

This section presents a static output feedback controller designed to vary the current in the MR damper, enabling the vehicle to adapt to varying road conditions. This work emphasizes the constraints associated with the utilization of sensors in mass-produced vehicles. Given that not all vehicle states can be directly measured, the control signal must be generated in accordance with the available information. Furthermore, given that external disturbances (in this instance, the unknown road profile variation) affect the vehicle, it is essential that the designed controller is robust against such influences, thereby ensuring an adequate response in any scenario [42].

First, in order to facilitate the real-time implementation, the system (7) is transformed into its discrete counterpart by applying Euler’s discretization [43]

(9)$\dot{x}\left(t\right)\simeq {\displaystyle \frac{x(k+1)-x\left(k\right)}{T_d}},\forall k\in {\mathbb{Z}}_{+}$

leading to the discrete state-space model

(10)$\begin{array}{c}\hfill \begin{array}{cc}\hfill x(k+1)& =Ax\left(k\right)+Bu\left(k\right)+B_{\omega }\omega \left(k\right)\hfill \\ \hfill y\left(k\right)& =C_{y}x\left(k\right)\hfill \\ \hfill z\left(k\right)& =C_{z}x\left(k\right)+D_{z}u\left(k\right)\hfill \end{array}\end{array}$

with the matrices

(11)$A=I+T_d\mathcal{A},B=T_d\mathcal{B},B_{\omega }=T_d{\mathcal{B}}_{\omega }$

The control signal u is generated by

(12)$u\left(k\right)=F{G}^{-1}y\left(k\right)$

where $F\in {\mathbb{R}}^{1\times 2}$ and $G\in {\mathbb{R}}^{2\times 2}$ are matrices to be designed.

At a given instant, the MR damper current i is obtained by solving (8), for known $z_{def}$, ${\dot{z}}_{def}$ and control signal u

(13)$i\left(k\right)={\displaystyle \frac{u\left(k\right)}{a_1\tanh \left(a_2\left({\dot{z}}_{def}\left(k\right)+\frac{V_0}{X_0}{z}_{def}\left(k\right)\right)\right)}}-{\displaystyle \frac{a_0}{a_1}}$

The following theorem presents a set of sufficient conditions for the design of an MR damper current controller for the semi-active suspension (3), taking into account the available sensors on the suspension system.

Theorem 1.

The semi-active suspension system (10) is stable and robust with an ${\mathcal{H}}_{\infty }$ performance index $\gamma >0$ if there exists a symmetric matrix $Q>0$, matrices F and G and scalars ${\epsilon }_1$ and ${\epsilon }_2$, if the following linear matrix inequality (LMI) convex optimization problem returns a feasible solution:

$\min \gamma \mathrm{s}.\mathrm{t}.\left(14\right)$

where

(14) $\left[\begin{array}{cccccc}-Q& ★& ★& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★& ★& ★\\ AQ+BF{C}_y& B_{\omega }& -Q& ★& ★& ★\\ C_{y}Q-G{C}_y& 0& {\epsilon }_1{F}^{\top }{B}^{\top }& -{\epsilon }_1\mathrm{He}\left(G\right)& ★& ★\\ C_{z}Q+D_{z}F{C}_y& 0& 0& 0& -I& ★\\ C_{y}Q-G{C}_y& 0& 0& 0& {\epsilon }_2{F}^{\top }{D}_z^{\top }& -{\epsilon }_2\mathrm{He}\left(G\right)\end{array}\right]<0$

Proof. 

First define

(15)$\left[\begin{array}{cccccc}I& 0& 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0\\ 0& 0& 0& I& 0& 0\\ 0& 0& 0& 0& I& D_{z}F{G}^{-1}\end{array}\right]$

now perform a congruent transformation by pre and post multiplying the LMI in (14) by (15) and its transpose to return

(16)$\left[\begin{array}{ccccc}-Q& ★& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★& ★\\ AQ+BF{C}_y& B_{\omega }& -Q& ★& ★\\ C_{y}Q-G{C}_y& 0& {\epsilon }_1{F}^{\top }{B}^{\top }& -{\epsilon }_1\mathrm{He}\left(G\right)& ★\\ C_{z}Q+D_{z}F{G}^{-1}{C}_{y}Q& 0& 0& 0& -I\end{array}\right]<0$

now perform the Schur complement to (16) to obtain

(17)$\left[\begin{array}{cccc}-Q+Q{(C_z+D_{z}F{G}^{-1}{C}_y)}^{\top }(C_z+D_{z}F{G}^{-1}{C}_y)Q& ★& ★& ★\\ 0& -{\gamma }^{2}I& ★& ★\\ AQ+BF{C}_y& B_{\omega }& -Q& ★\\ C_{y}Q-G{C}_y& 0& {\epsilon }_1{F}^{\top }{B}^{\top }& -{\epsilon }_1\mathrm{He}\left(G\right)\end{array}\right]<0$

Now define

(18)$\left[\begin{array}{cccc}I& 0& 0& 0\\ 0& I& 0& 0\\ 0& 0& I& BF{G}^{-1}\end{array}\right]$

perform a congruent transformation by pre and post multiplying the LMI (17) by (18) and its transpose to obtain

(19)$\left[\begin{array}{ccc}-Q+Q{(C_z+D_{z}F{G}^{-1}{C}_y)}^{\top }(C_z+D_{z}F{G}^{-1}{C}_y)Q& ★& ★\\ 0& -{\gamma }^{2}I& ★\\ AQ+BF{G}^{-1}{C}_{y}Q& B_{\omega }& -Q\end{array}\right]<0$

now apply the change of variable $Q=P^{-1}$ and pre and post multiply (19) by $\mathrm{diag}\{P^{-1},I,I\}$

(20)$\left[\begin{array}{ccc}-P+{(C_z+D_{z}F{G}^{-1}{C}_y)}^{\top }(C_z+D_{z}F{G}^{-1}{C}_y)& ★& ★\\ 0& -{\gamma }^{2}I& ★\\ A+BF{G}^{-1}{C}_y& B_{\omega }& -P^{-1}\end{array}\right]<0$

By means of the Schur complement again, (20) equals

(21)${\mathcal{W}}^{\top }P\mathcal{W}+\left[\begin{array}{cc}-P+{(C_z+D_{z}F{G}^{-1}{C}_y)}^{\top }(C_z+D_{z}F{G}^{-1}{C}_y)& ★\\ 0& -{\gamma }^{2}I\end{array}\right]$

where

(22)$\mathcal{W}=\left[\begin{array}{cc}A+BF{G}^{-1}{C}_y& B_{\omega }\end{array}\right]$

Multiplying (21) on the left by $\left[\begin{array}{cc}x{\left(k\right)}^{\top }& \omega {\left(k\right)}^{\top }\end{array}\right]$ and its transpose on the right, the following expression is obtained:

(23)$\Delta \mathcal{V}\left(k\right)+{\mathcal{J}}_{{\mathcal{H}}_{\infty }}\left(k\right)<0$

where $\Delta \mathcal{V}\left(k\right)$ is the variation of the Lyapunov function candidate $\mathcal{V}\left(k\right)=x^{\top }\left(k\right)Px\left(k\right)$ used to analyze the closed-loop stability of system (10), and ${\mathcal{J}}_{{\mathcal{H}}_{\infty }}$ is the ${\mathcal{H}}_{\infty }$ robustness criteria used to design a controller that minimizes the effect of the disturbance $\omega$ over system (10), such that

(24)$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Delta \mathcal{V}\left(k\right)& =x^{\top }(k+1)Px(k+1)-x^{\top }\left(k\right)Px\left(k\right)<0\hfill \\ \hfill {\mathcal{J}}_{{\mathcal{H}}_{\infty }}\left(k\right)& =z^{\top }\left(k\right)z\left(k\right)-{\gamma }^2{\omega }^{\top }\left(k\right)\omega \left(k\right)<0\hfill \end{array}\end{array}$

It follows from (24) that system (10) is robustly stable with an ${\mathcal{H}}_{\infty }$ performance index $\gamma$ if the LMI (14) returns a feasible solution. This completes the proof. □

3. Results

This section begins with an overview of experimental tests conducted to model the behavior of the MR damper. This is followed by a presentation of the tests carried out to validate the effectiveness of the proposed current controller for the MR damper in a simulated semi-active suspension system.

The advantage of using LMI-based controllers lies in the fact that the control gain matrix is designed offline, eliminating the need for recalculation with each control signal generation. This reduces the control signal computation to simple algebraic operations that can be executed in real time with low computational burden.

3.1. Experimental Validation of the Proposed MR Damper Model

In order to validate the MR model (1), a series of experimental data were obtained for a commercial MR damper. The most common method for determining the behavior of an MR damper is to excite it with a sinusoidal displacement in a series of test cases with varying frequencies of excitation, piston displacements, and applied currents. The experimental data were obtained from our laboratory using the damper test machine and setup depicted in Figure 4.

In the test machine, a hydraulic actuator was used to drive the MR damper through sinusoidal displacement cycles with amplitudes of 10, 15, and 20 mm, within a frequency range of 2–6 Hz, and under the application of magnetizing currents ranging from 0 to 1.8 A.

The damper stroke was positioned at its center before the test was initiated in order to avoid extreme damper stroke positions. The test machine is equipped with a displacement sensor to measure the displacement $z_{def}$ and velocity ${\dot{z}}_{def}$ of the MR damper piston, as well as a load cell to measure the output force $F_{md}$. The signals for displacement and force are sampled at a rate of 0.5 kHz.

After processing the experimental data, the MR damper model parameters introduced in (1) were adjusted and are presented in Table 2. The Microsoft Excel Solver tool was employed in this context.

Figure 5 and Figure 6 illustrate the comparison of the response curves for the experimental and estimated MR damper behavior corresponding to piston displacement amplitudes of 15 and 20 mm, with excitation frequencies of 3 Hz and 6 Hz, respectively. As observed, within the range of small velocities, the force variation exhibits significant hysteretic behavior, whereas at higher velocities, the force varies almost linearly with velocity. With an increase in current, the force required to yield the fluid also increases, leading to behavior characteristic of a plastic material in parallel with a viscous damper. The model’s efficacy in estimating the MR damper force is evidenced by two key metrics: the consistently low estimation error, which always remains below 0.1 kN, and the high correlation coefficient, which exceeds 0.995 in every case.

Figure 7 shows a comparison of the proposed damper model to existing literature alternatives, such as the Bingham and bi-viscous approaches. In the presented example, an experiment with an amplitude of the piston displacement of 20 mm and a current of 1.8 A at a frequency of excitation of 6 Hz, the maximum estimation errors for the Bingham and bi-viscous models are 1.67 and 0.25 kN, respectively. In contrast, the estimation error for the proposed model remains below 0.1 kN in all cases, which serves to illustrate its superiority.

3.2. Vehicle Suspension Simulation

Several simulations were conducted in Matlab 2024a to demonstrate the functionality of the proposed MR damper-based semi-active suspension in a vehicle. The quarter-car vehicle model (10) is simulated following the parameters in Table 1. The MR damper (1) is simulated according to the experimental parameters introduced in Table 2. Figure 8 illustrates the architectural framework of the simulation. The measured output $y={[z_s-z_u,{\dot{z}}_s-{\dot{z}}_u]}^{\top }$ is transmitted to the controller, which adjusts the current supplied to the MR damper in order to modify the vertical force generated by it. The current applied to the damper is saturated, in order to be in the range of 0–1.8 A.

By setting ${\epsilon }_1={\epsilon }_2=200$, a feasible solution for the controller was found by solving the proposed Theorem 1, with ${\gamma }_{min}=6.82$. The Matlab robust control toolbox was used for solving the LMIs. In this particular instance, the specific value of the controller gain is

(25)$K=\left[\begin{array}{cc}-2303.97& 955.70\end{array}\right]$

In order to evaluate the system performance, two road disturbance profiles were subjected to analysis:

1. A road bump with a height of $H=6$ cm and a length of L = 5 m , at a vehicle speed of $V=30$ km/h. The following formula represents the road bump profile over time [40]:

   (26)$z_r\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{H}{2}}\left(1-\cos \left({\displaystyle \frac{2\pi V}{L}}t\right)\right),& 0\le t\le {\displaystyle \frac{L}{V}}\\ 0,& t>{\displaystyle \frac{L}{V}}\end{array}\right.$

   The bump test is necessary for evaluating the vehicle’s suspension performance in mitigating the impact of high-energy disturbances.

2. A Class A random road profile, according to ISO 8608 [44], is a vehicle speed of $V=80$ km/h. The objective of evaluating the vehicle suspension under this random road profile is to guarantee that the performance is not compromised under typical highway conditions, i.e., that the semi-active suspension does not affect the vehicle when it is not required.

The aforementioned road profiles are illustrated in Figure 9.

In this work, three vehicle suspensions are compared:

1. Passive suspension. A traditional suspension system with springs and dampers that cannot adjust to varying road conditions. The objective of this analysis is to assess the performance of a conventional vehicle.

2. Semi-active suspension with MR damper. The suspension system employs a magnetorheological damper that is capable of modifying its behavior in real time in order to adapt to the road profile. The MR damper current (13) is generated according to the required suspension control input (12). The objective of this analysis is to evaluate the potential for enhancement of vehicle behavior through the implementation of the proposed system.

3. Active suspension. Actuators are integrated into the suspension system with the objective of counteracting the impact of the varying road profile on the vehicle. The actuator force is generated according to the required suspension control input (12). The objective of this analysis is to evaluate the performance of the vehicle with the optimal suspension system installed, which would be costly and challenging to implement in practice.

The bump road simulation results are presented in Figure 10. The proposed controller can achieve a lower value of body acceleration over time for the vehicle with semi-active suspension system than for the vehicle with passive one, which guarantees better ride comfort. The RMS values for the vertical acceleration are 1.101, 0.974, and 0.697 m/s² for the passive, semi-active, and active vehicle suspensions, respectively. Comparing the frequency components of the acceleration using the power spectral density (PSD), the maximum frequency component has been reduced from 2.42 (m/s²)²/Hz in the passive suspension to 1.60 (m/s²)²/Hz for the semi-active suspension. Furthermore, it is evident that the maximum suspension deflection does not exceed $z_{max}=0.05$ m under any circumstances, thereby substantiating the viability of implementing the proposed methodology without compromising the integrity of the vehicle suspension system. In every instant, the results demonstrate that the suspension deflection is observed to be lower over time for the semi-active suspension in comparison with the passive suspension. The RMS values for the suspension deflection are 0.014, 0.012, and 0.011 m for the passive, semi-active, and active vehicle suspensions, respectively. It is evident that the normalized dynamic tire load (NDTL) never surpasses the value of 1, which serves to confirm that all vehicles are consistently maintaining contact with the road.

It has been demonstrated that the semi-active suspension system has the capacity to enhance the vehicle’s performance in terms of comfort when compared to the passive suspension system. Moreover, the system does not compromise driving safety, as it ensures that the tires maintain consistent contact with the road surface and that the vehicle suspension does not deflect beyond its mechanical limits.

Although active suspensions are naturally superior in terms of performance, due to the fact that the MR damper force is limited by the sign of the term ${\dot{z}}_{def}+{\displaystyle \frac{V_0}{X_0}}{z}_{def}$ in (1) and the maximum allowable MR damper current, it has been demonstrated that the proposed semi-active suspension system is an effective mean of enhancing vehicle performance with its inherent benefits when compared to active suspensions, which are that they do not add energy to the vehicle system, and they have a lower energy cost.

Additional simulation tests for the road bump under various vehicle speeds are detailed in Table 3.

The Class A road simulation results are presented in Figure 11. Once again, it is demonstrated that the semi-active suspension can enhance the vehicle behavior compared to the passive suspension. The lower power spectral density (PSD) of the vertical acceleration for the semi-active suspension indicates that it has a more smooth and comfortable drive experience rather than the passive suspension. With regard to the suspension deflection, a notable enhancement is observable over time, with RMS values of 0.0025 m, 0.0015 m, and 0.0013 m for the passive, semi-active, and active suspensions, respectively.

The results of this test demonstrate that safety is not compromised during typical highway driving conditions. Additionally, the vertical acceleration of the vehicle is reduced, which enhances ride comfort.

Additional simulation tests for the Class A road under various vehicle speeds are detailed in Table 4.

To demonstrate the interest of the proposed methodology, Table 5 provides a comparison with alternative approaches.

4. Conclusions

This paper presents a design method for a robust static output feedback control for a MR damper-based semi-active suspension system, which adapts the MR damper current based solely on the information received from suspension deflection sensors.

In consideration of the existing literature, an enhanced Bingham mathematical model has been formulated to characterize the mechanical response of the MR damper in relation to the magnitude of the current applied to it. A series of experimental tests were conducted to validate the model utilized in this study. A high overall correlation between experimental and predicted forces of $R^2=0.995$ was found.

Considering that not every vehicle state can be measured, convex LMI conditions were obtained to design practical robust MR damper controllers that depend only on the information received from suspension deflection sensors. The Lyapunov stability theory is employed to ensure the stability of the vehicle suspension system. A practical implementation of the proposed methodology in real time is guaranteed, as no supplementary estimation stages are required for those states that cannot be measured.

The performance of the proposed system was evaluated under two distinct conditions: a road bump and a Class A road profile, in accordance with the ISO 8608 standard. It has been demonstrated that the incorporation of the proposed semi-active suspension system, as a substitute for a traditional passive suspension, can enhance ride comfort and safety in vehicles. In this case, vehicle vertical acceleration and suspension deflection are reduced by $50\%$ and $73\%$, respectively, in the most unfavorable scenarios.

Given the inherent theoretical mechanical advantage of active suspensions over semi-active suspensions, as they have less limitations on the force they can provide, at the expense of high energy consumption and cost, a viability analysis will be conducted as part of future works. This study will assist in identifying optimal application scenarios, balancing performance, energy efficiency, and cost-effectiveness, thereby guiding decision-making on the appropriate suspension type for various operational contexts.

Author Contributions

Conceptualization, F.V.-M., M.M.-U., M.J.-S., and B.L.B.; methodology, F.V.-M., M.M.-U., M.J.-S., and B.L.B.; software, F.V.-M.; validation, F.V.-M., M.M.-U., and M.J.-S.; formal analysis, F.V.-M.; investigation, F.V.-M., M.M.-U., and M.J.-S.; resources, F.V.-M., M.M.-U., M.J.-S., and B.L.B.; data curation, F.V.-M.; writing—original draft preparation, F.V.-M., M.M.-U., M.J.-S., and B.L.B.; writing—review and editing, F.V.-M., M.M.-U., M.J.-S., and B.L.B.; visualization, F.V.-M.; supervision, B.L.B.; project administration, B.L.B.; funding acquisition, F.V.-M. and B.L.B.All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

LMI	Linear Matrix Inequality
LVDT	Linear Variable Differential Transformer
MR	Magnetorheological
NDTL	Normalized Dynamic Tire Load
PSD	Power Spectral Density
RMS	Root Mean Square

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Common schematic of an MR damper.

Figure 2. A typical curve of the proposed model (a) Suspension deflection vs. Damper force, (b) Suspension deflection rate vs. Damper force.

Figure 3. Semi-active quarter-car suspension model.

Figure 4. Experimental MR damper setup.

Figure 5. Experimental and estimated MR damper curves for an amplitude of the piston displacement of 15 mm and different levels of applied current at a frequency of excitation of 3 Hz: (a) Deflection vs. Force, (b) Deflection Rate vs. Force, (c) Estimation Error, (d) Correlation.

Figure 6. Experimental and estimated MR damper curves for an amplitude of the piston displacement of 20 mm and different levels of applied current at a frequency of excitation of 6 Hz: (a) Deflection vs. Force, (b) Deflection Rate vs. Force, (c) Estimation Error, (d) Correlation.

Figure 7. Experimental and estimated MR damper curves for an amplitude of the piston displacement of 20 mm and a current of 1.8 A at a frequency of excitation of 6 Hz: (a) Deflection vs. Force, (b) Deflection Rate vs. Force, (c) Estimation Error, (d) Correlation.

Figure 8. The architectural framework of the simulation.

Figure 9. Simulated road profiles: (a) Road bump, (b) Class A road according to ISO 8608.

Figure 10. Simulation results under a road bump: (a) Vertical acceleration, (b) Vertical acceleration PSD, (c) Suspension deflection, (d) Normalized dynamic tire load, (e) Input force, (f) MR damper current.

Figure 11. Simulation results under a Grade A road: (a) Vertical acceleration, (b) Vertical acceleration PSD, (c) Suspension deflection, (d) Normalized dynamic tire load, (e) Input force, (f) MR damper current.

Table 1

Quarter-car model parameters.

Parameter	Name	Value
$m_s$	Vehicle sprung mass	243.2 kg
$m_u$	Vehicle unsprung mass	28.5 kg
$k_s$	Spring stiffness	10,680 N/m
$k_t$	Tire stiffness	25,278 N/m
$c_t$	Tire damping	3.65 Ns/m
$z_{max}$	Maximum suspension deflection	0.05 m

Table 2

Experimental MR damper parameters.

Parameter	Value
$a_0$	236.3 N
$a_1$	655.8 N/A
$a_2$	4.67 s/m
$a_3$	343.1 N s/m
$V_0/X_0$	6.05 s−1

Table 3

RMS results for the vertical acceleration of the vehicle under different bump simulations.

Vehicle Speed	Passive	Semi-Active	Active
15 km/h	0.789 m/s2	0.722 m/s2	0.510 m/s2
20 km/h	1.079 m/s2	0.959 m/s2	0.635 m/s2
25 km/h	1.133 m/s2	0.998 m/s2	0.683 m/s2

Table 4

RMS results for the suspension deflection of the vehicle under different Class A road simulations.

Vehicle Speed	Passive	Semi-Active	Active
90 km/h	0.0026 m	0.0015 m	0.0014 m
100 km/h	0.0027 m	0.0016 m	0.0015 m
110 km/h	0.0029 m	0.0017 m	0.0016 m

Table 5

Comparison with other methodologies.

Method	Output Feedback	MR Damper Model	ExperimentalModel Validation
[13]	✓	Improved Bingham	✗
[36]	✗	Polynomial	✓
[37]	✗	Bouc–Wen	✗
Proposed	✓	Current adaptiveimproved Bingham	✓

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