A new adaptive sliding mode control for Macpherson strut suspension system with magneto-rheological damper

Abstract

This work proposes a new adaptive sliding mode controller to enhance ride comfort and steering stability of automobile associated with a semi-active magneto-rheological damper. In this study, a Macpherson strut type suspension system which is widely used in light vehicles is considered. The dynamic model of the Macpherson strut with magneto-rheological damper is obtained and the governing equations are then formulated using kinematic properties of the suspension system following Lagrange’s formulation. In the formulation of the model, both the rotation of the wheel assembly and the lateral stiffness of the tire are considered to represent the nonlinear characteristic of Macpherson type suspension system. Subsequently, in order to effectively reduce unwanted vibrations, a new adaptive sliding mode controller is designed by adopting moving sliding surface instead of conventional fixed sliding surface. In order to demonstrate the effectiveness of the proposed controller, a cylindrical magneto-rheological damper is designed and manufactured on the basis of practical application conditions such as required damping force. Then, ride comfort, suspension travel, and road handling are evaluated and some benefits of the proposed controller such as enhanced ride comfort are evaluated.

Keywords

vehicle suspension Macpherson strut equivalent linearization magneto-rheological damper sliding mode control fuzzy adaptive control moving sliding surface

Introduction

Attenuation of vibration level in an automotive vehicle is done through suspension systems. The suspension systems are generally of three types: passive, active, and semi-active suspension systems. The passive systems are designed for specific operating conditions. It does not perform well when the operating condition changes. To remove this drawback, active systems came into application where a continuous supply of energy helps the suspension to perform better. The high demand of energy for active systems gives rise to the concept of semi-active systems where energy is added to the system only when required. It provides a good balance between passive and active systems. Most of the suspension system–related studies considered the system to be a quarter car system where only the vertical movements of the masses are considered. Less focus has been given on the kinematic or dynamic analysis of the suspension system. For proper analysis, a detailed kinematic and dynamic model of suspension system is needed. One of the commonly used suspension system is Macpherson strut type of suspension system. It was first developed by Earl Macpherson for Ford Motor Company. Some research studied about the kinematic and dynamic analysis of Macpherson strut are found in literatures (Fallah et al., 2008, 2009; Hurel et al., 2012). Some literature (Hong et al., 1999; Sohn et al., 2000) considered the sprung and unsprung masses as point elements where rigid body movements of the masses are ignored. Cronin (1981) and Attia (2003) studied the kinematics of Macpherson strut suspension systems. Anderson (2007) developed a dynamic model of Macpherson strut system using Lagrange multiplier for the constrained equations and presented a system of algebraic equations. The equations are solved using Hilber–Hughes–Taylor integrator.

Electro-rheological (ER) and magneto-rheological (MR) dampers are widely used as semi-active element in suspension systems. Choi et al. (2000) proposed cylindrical ER damper suspension systems for small-sized vehicles. Sims et al. (1999) showed that the MR fluids are more stable for a wide range of temperature and high yield stress and thus preferred to ER fluids. Carlson et al. (1996) proposed a commercially available MR damper which is applicable to the vehicle suspension systems. Spencer et al. (1997) proposed a new phenomenological model for predicting the force–displacement relationship of the MR damper. The response of a quarter car suspension system with MR damper for step and sinusoidal bump had been studied by Butz and Von Stryk (2002). Most of the researches are carried out on the semi-active control of quarter car suspension systems. Karnopp et al. (1974) first developed the concept of semi-active control strategies. Since then, several control strategies have been developed which provide comparable performances of suspension systems. Yao et al. (2002) studied semi-active sky-hook control of the quarter car MR suspension. Seong et al. (2011) studied different semi-active control algorithms for suspension systems with MR damper in which sliding mode controls are found out to be effective. Many studies had been carried out on different semi-active control strategies for quarter car systems (Ahmadian and Pare, 2000; Choi et al., 2002; Dong et al., 2010; Du et al., 2005). Semi-active suspension systems using MR damper as semi-active element have not been studied much. Some researchers (Hong et al., 1999; Sohn et al., 2000) proposed one Macpherson suspension system with MR damper assuming unsprung mass to be a point mass. Thus, the important parameters of Macpherson system like camber angle and track alteration have been neglected in this work.

As far as the authors’ best knowledge, a study on the design of a new controller for Macpherson strut suspension integrated with the semi-active MR damper is considerably rare. Consequently, the technical main contributions of this work are summarized as follows: (1) the derivation of an equivalent linear model of Macpherson strut type suspension system with MR damper considering both the dynamic and kinematic relationships, (2) the design of a new fuzzy logic–based adaptive sliding mode controller using a moving sliding surface, and (3) the suspension performance evaluation by tracking the variation of the sliding surface gradients. In the formulation of the suspension model, both the rotation of the wheel assembly and the lateral stiffness of the tire are considered to represent the nonlinear characteristic of Macpherson type suspension system. In order to demonstrate the effectiveness of the proposed controller, a cylindrical MR damper is manufactured on the basis of practical applicability and its field-dependent damping characteristics are tested. These properties are used for designing a new adaptive sliding mode controller whose switching gain is changed based on adaptive fuzzy logic. The suspension characteristics like ride comfort, suspension travel, and road handling are evaluated in time and frequency domains.

MR damper

Since the invention of MR fluids in 1940s by Jacob Rabinow, these fluids are used in various applications including suspension systems, shock absorbers, haptic devices, and engine mounts due to its property of changing viscosity with changing magnetic field. MR fluids are used in dampers to provide controllable dampers whose property varies with the applied magnetic field. The cylindrical MR damper used in this study is shown in Figure 1. The damper is tested for its field-dependent force performances. The damper consists of accumulator, cylinder, and piston. The magnetic choke is placed within the piston head. A floating piston is placed between the accumulator and the chamber containing MR fluid. The accumulator provides stiffness to the damper and helps prevent cavitation in the fluid during off state. The MR fluid is allowed to pass from one chamber to another chamber through the annulus gap between the outer wall of the piston head and the inner wall of the cylinder. The magnetic field is applied in the coils placed within the piston head. During off-state, the MR fluid behaves like a viscous fluid and provides damping force only due to the viscous resistance of the fluid. When the magnetic field is applied, an additional damping force due to the yield stress of the MR fluid arises. Due to this behavior, the damping force of the MR damper can be easily controlled using an external magnetic field.

In this study, a commercially available fluid, MRF 132-LD, is used in the MR damper. Various researchers obtained different mathematical modeling of MR damper to find out force–displacement or force–velocity relationships of MR damper. This study considers the damper model adopted by Choi et al. (1998) as this model is found out to be suitable. The damping force of the MR damper is given by

F_{D} = K_{e} Δ L + C_{e} \overset{\cdot}{Δ} L + F_{MR} sign (\overset{\cdot}{Δ} L)

(1)

where K_e and C_e are the equivalent stiffness and damping coefficients of the MR damper which are considered to be equal to K_s and C_s (stiffness and damping coefficients of passive damper), ΔL is the deflection of the MR damper, and the force due to MR effect is given by

F_{MR} = (A_{p} - A_{r}) \frac{2 L_{p}}{h_{g}} α_{1} H^{α_{2}}

(2)

The coefficients L_p, h_g, A_p, A_r, α₁, and α₂ are length of the pole, annular gap, area of piston, area of piston rod and coefficients, respectively. H is the magnetic field obtained as $H = NI / 2 H_{g}$ , where N is the number of coil turns and I is the current.

A cylindrical type MR damper is designed and manufactured. The design parameters are listed in Table 1. The field-dependent damping force for different piston velocities at different magnetic fields is plotted in Figure 2(a). The plot is obtained by calculating the maximum damping force for a given piston velocity. These types of plots are used to characterize a damper. The piston velocity is changed by varying the excitation frequency from 0.5 to 3 Hz and the excitation amplitude is taken as 0.02 m. It is clear from the figure that the damping force increases by a substantial amount (from 580 to 2270 N) as the input current varies (from 0 to 2 A) at a constant piston velocity of 0.377 m/s. The measured damping force is plotted in Figure 2(a) which shows the change in the damping force with a change in the applied current. The time constant is 23 ms which introduces a delay in attaining the higher damping value. The time required to reach 70% of the final steady-state value is considered as time constant. In Figure 2(b), time required to reach 788 N which is 70% of the steady-state force 1125 N is 0.023 s. For this field-dependent characteristic, the MR damper is used in vehicle suspension in this study.

Table 1.

Parameters of magneto-rheological damper.

A_p	A_r	L_p	N	h_g	α₁	α₂
7.07 × 10⁻⁴ m²	2.54 × 10⁻⁴ m²	1 × 10⁻² m	150	1 × 10⁻³ m	0.083	1.25

Mathematical model

Kinematics and dynamics

The schematic diagram of Macpherson suspension system is shown in Figure 3. The important points of the system is marked as A, B, C, D, E, P, and O. The MR damper is connected between C and E, AB represents the control arm, and P is the center of the wheel. The following assumptions are made to mathematically model the suspension system: (1) the chassis, that is, the sprung mass undergoes vertical motion only, (2) all the suspension systems except tire are rigid, (3) the control arm and the strut have negligible masses, (4) the wheel assembly is subjected to rotational and translational motion, (5) all the joints are ideal, and (6) damper and springs have linear behavior. The camber angle (ϕ) and the control arm rotation (θ) are shown in Figure 4. During equilibrium, the point A was at A₀ making an angle θ₀ with Y-axis. At equilibrium position, the origin of the coordinate system O coincides with B. The lateral displacement of the tire (d_tl) is shown in Figure 4. Consider (y_P₀, z_P₀), (y_C₀, z_C₀), (y_E₀, z_E₀), and (y_A₀, z_A₀) are the positions of points P, C, E, and A at the equilibrium. The control arm rotates at angle θ which is measured counterclockwise and the sprung mass moves by an amount z_s. The suspension kinematics is analyzed using displacement matrix method. The displacement of the points on the spindle-wheel assembly can be formulated as follows

[\begin{matrix} y_{C} & y_{E} & y_{P} \\ z_{C} & z_{E} & z_{P} \\ 1 & 1 & 1 \end{matrix}] = D_{w} [\begin{matrix} y_{C 0} & y_{E 0} & y_{P 0} \\ z_{C 0} & z_{E 0} & z_{P 0} \\ 1 & 1 & 1 \end{matrix}]

(3)

where D_w is the displacement matrix of the wheel-spindle assembly and given by

D_{w} = [\begin{matrix} a_{11} & a_{12} & y_{A} - (a_{11} y_{A 0} + a_{12} z_{A 0}) \\ a_{21} & a_{22} & z_{A} - (a_{21} y_{A 0} + a_{22} z_{A 0}) \\ 0 & 0 & 1 \end{matrix}]

(4)

The coefficients a₁₁ = a₂₂ = cos ϕ and a₁₂ = −a₂₁ = sin ϕ, where ϕ is the wheel rotation about x-axis, that is, the camber angle. In practical situations, the angle ϕ possesses small values, thus the above set of equations can be reduced by considering a₁₁ = a₂₂ = 1 and a₁₂ = −a₂₁ = ϕ as follows

y_{c} = y_{c 0} + φ z_{c 0} + y_{A} - (y_{A 0} + φ z_{A 0}) = a_{C} + b_{C} φ + y_{A}

(5)

y_{E} = a_{E} + b_{E} φ + y_{A}

(6)

y_{P} = a_{P} + b_{P} φ + y_{A}

(7)

z_{C} = - φ a_{C} + b_{C} + z_{A}

(8)

z_{E} = - φ a_{E} + b_{E} + z_{A}

(9)

z_{P} = - φ a_{P} + b_{P} + z_{A}

(10)

where the coefficients are obtained from the following equilibrium coordinates

a_{C} = y_{C 0} - y_{A 0}, a_{E} = y_{E 0} - y_{A 0}, a_{P} = y_{P 0} - y_{A 0}

and

b_{C} = z_{C 0} - z_{A 0}, b_{E} = z_{E 0} - z_{A 0}, b_{P} = z_{P 0} - z_{A 0}

The constraints on the Macpherson strut can be found as follows

\tan α = \frac{y_{C} - y_{E}}{z_{C} - z_{E}}

(11)

Again

\tan α = \frac{y_{A} - y_{D}}{z_{A} - z_{D}}

(12)

The coordinate (y_A, z_A) can be obtained from the rotation of the control arm and the constrained equations of the control arm are given by

y_{A} = L_{2} \cos (θ_{0} + θ)

(13)

and

z_{A} = L_{2} \sin (θ_{0} + θ) + z_{S}

(14)

where L₂ is the length of the control arm and $θ_{0}$ is the initial angle of the control arm. The above equations can be linearized considering the angle $θ$ small as follows

y_{A} = L_{2} \cos θ_{0} \cos θ - L_{2} \sin θ_{0} \sin θ = L_{C} - L_{S} θ

(15)

and

z_{A} = L_{2} \sin θ_{0} \cos θ + L_{2} \cos θ_{0} \sin θ + z_{S} = L_{S} - L_{C} θ + z_{S}

(16)

Substituting the values in equations (11) and (12) and then equating these two equations yields the following equation

\frac{a_{C} + b_{C} φ - a_{E} - b_{E} φ}{- φ a_{C} + b_{C} + φ a_{E} - b_{E}} = \frac{L_{C} - L_{S} θ - y_{D 0}}{L_{S} - L_{C} θ - z_{D 0}}

(17)

Neglecting the higher order terms of θ, the above equation leads to the follow equation

φ = P_{1} + P_{2} θ + P_{3} θ^{2}

(18)

where

\begin{matrix} P_{1} = \frac{Q_{1}}{Q_{3}}, P_{2} = \frac{Q_{1}}{Q_{3}} (\frac{Q_{2}}{Q_{1}} - \frac{Q_{4}}{Q_{3}}), \\ and P_{3} = - \frac{Q_{1} Q_{4}}{Q_{3}^{2}} (\frac{Q_{2}}{Q_{1}} - \frac{Q_{4}}{Q_{3}}) \end{matrix}

(19)

\begin{matrix} Q_{1} = b_{CE} (L_{C} - y_{D 0}) - a_{CE} (L_{S} - z_{D 0}), \\ Q_{2} = - (a_{CE} L_{C} + b_{CE} L_{S}) \\ Q_{3} = b_{CE} (L_{S} - z_{D 0}) - a_{CE} (L_{C} - y_{D 0}), \\ Q_{4} = b_{CE} L_{C} - a_{CE} L_{S} \\ a_{CE} = a_{C} - a_{E} and b_{CE} = b_{C} - b_{E} \end{matrix}

Now, from equations (7) and (10), the unknowns (y_P, z_P) can be calculated without solving the other four equations. The equations become as follows

\begin{matrix} y_{P} = a_{P} + b_{P} (P_{1} + P_{2} θ + P_{3} θ^{2}) \\ + L_{C} - L_{S} θ = P_{4} + P_{5} θ + P_{6} θ^{2} \end{matrix}

(20)

\begin{matrix} z_{P} = - a_{P} (P_{1} + P_{2} θ + P_{3} θ^{2}) + b_{P} + L_{S} - L_{C} θ \\ + z_{S} = P_{7} + P_{8} θ + P_{9} θ^{2} + z_{S} \end{matrix}

(21)

where

\begin{matrix} P_{4} = a_{P} + L_{C} + b_{P} P_{1}, P_{5} = b_{P} P_{2} - L_{S} \\ P_{6} = b_{P} P_{3}, P_{7} = - a_{P} P_{1} + L_{S} + b_{P} \\ P_{8} = - a_{P} P_{2} - L_{C} and P_{9} = - a_{P} P_{3} \end{matrix}

The deflection of the spring, ΔL is given by

Δ L = L_{3} - L_{3}'

(22)

where L₃ is the length of the spring at equilibrium position and $L_{3}'$ is the instantaneous length of spring.

From geometry, we can develop the following equations

\begin{matrix} L_{3}^{2} = L_{1}^{2} + L_{2}^{2} - 2 L_{1} L_{2} \cos γ \\ L_{3}^{' 2} = L_{1}^{2} + L_{2}^{2} - 2 L_{1} L_{2} \cos (γ - θ) \end{matrix}

Neglecting the highest order terms of θ, the deflection of the spring equation is obtained as follows

Δ L = - \frac{K_{D}}{2 L_{3}} θ, where K_{D} = 2 L_{1} L_{2} \sin γ

(23)

In Figure 4, the angle γ is shown as the angle between the control arm and the line joining points B and D. Tire lateral deflection is computed as follows

d_{tl} = y_{u} - φ R = R_{1} + R_{2} θ + R_{3} θ^{2}

(24)

where R is tire effective radius and given by the relationship

R_{1} = P_{4} - P_{1} R, R_{2} = P_{5} - P_{2} R, and R_{3} = P_{6} - P_{3} R

Derivation of equation of motion

The equation of motion of the Macpherson strut with MR damper can be obtained using Lagrange’s method. The kinetic energy (T), potential energy (V), and the dissipation energy (D) of the system can be obtained as follows

\begin{matrix} T = \frac{1}{2} m_{S} {\overset{\cdot}{z}}_{S}^{2} + \frac{1}{2} m_{u} ({\overset{\cdot}{z}}_{u}^{2} + {\overset{\cdot}{y}}_{u}^{2}) + \frac{1}{2} I_{u} {\overset{\cdot}{φ}}^{2} + \frac{1}{2} I_{C} {\overset{\cdot}{θ}}^{2} \\ V = \frac{1}{2} K_{S} {(Δ L)}^{2} + \frac{1}{2} K_{t} {(z_{u} - z_{r})}^{2} + \frac{1}{2} K_{tl} d_{tl}^{2} \\ D = \frac{1}{2} C_{S} {(\overset{\cdot}{Δ} L)}^{2} + \frac{1}{2} C_{t} {({\overset{\cdot}{z}}_{u} - {\overset{\cdot}{z}}_{r})}^{2} \end{matrix}

(25)

where z_u = z_P − z_P₀ and y_u = y_P − y_P₀ are the displacements of the unsprung mass in z- and y-axes.

The Lagrangian of the system is given by L = T − V. The equations of motion of the system can be written as follows

\frac{d}{dt} (\frac{\partial L}{\partial {\overset{\cdot}{q}}_{i}}) - \frac{\partial L}{\partial q_{i}} + \frac{\partial D}{\partial {\overset{\cdot}{q}}_{i}} = Q_{i}

(26)

where q_i is (z_s, θ) and Q_i is the generalized force, that is, $Q_{i} = F_{MR} sign (\overset{\cdot}{Δ} L) (\partial Δ L / \partial q_{i})$ .

The dynamics of the MR damper can be expressed as the following equation

{\overset{\cdot}{F}}_{M} = - \frac{1}{τ} F_{M} + \frac{1}{τ} u

(27)

The equation of motion can be reduced to the following form

\begin{matrix} {\overset{\cdot\cdot}{z}}_{s} = F_{1} (z_{s}, {\overset{\cdot}{z}}_{s}, θ, \overset{\cdot}{θ}, z_{r}, {\overset{\cdot}{z}}_{r}, F_{MR}, u) \\ \overset{\cdot\cdot}{θ} = F_{2} (z_{s}, {\overset{\cdot}{z}}_{s}, θ, \overset{\cdot}{θ}, z_{r}, {\overset{\cdot}{z}}_{r}, F_{MR}, u) \\ {\overset{\cdot}{F}}_{M} = F_{3} (z_{s}, {\overset{\cdot}{z}}_{s}, θ, \overset{\cdot}{θ}, z_{r}, {\overset{\cdot}{z}}_{r}, F_{MR}, u) \end{matrix}

(28)

Now, we consider the state variables as $[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}]^{T} = [z_{s}, {\overset{\cdot}{z}}_{s}, θ, \overset{\cdot}{θ}, F_{M}]^{T}$ .

Equivalent linearization

The above equations of motion are highly nonlinear. For further work and to incorporate control strategies for the system, the system of equations should be linearized. Thus, the above nonlinear equations are linearized at the equilibrium position, $[x_{1 e}, x_{2 e}, x_{3 e}, x_{4 e}, x_{5 e}]^{T} = [0, 0, 0, 0, 0]$ . The linearized equation becomes as follows

\overset{\cdot}{x} = Ax + Dw + Bu, x_{0} = 0

(29)

where

A = [\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ \frac{\partial F_{1}}{\partial x_{1}} & \frac{\partial F_{1}}{\partial x_{2}} & \frac{\partial F_{1}}{\partial x_{3}} & \frac{\partial F_{1}}{\partial x_{4}} & \frac{\partial F_{1}}{\partial x_{5}} \\ 0 & 0 & 0 & 1 & 0 \\ \frac{\partial F_{1}}{\partial x_{1}} & \frac{\partial F_{2}}{\partial x_{2}} & \frac{\partial F_{2}}{\partial x_{3}} & \frac{\partial F_{2}}{\partial x_{4}} & \frac{\partial F_{2}}{\partial x_{5}} \\ 0 & 0 & 0 & 0 & - \frac{1}{τ} \end{matrix}] = \frac{1}{A_{D}} [\begin{matrix} 0 & D & 0 & 0 & 0 \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} \\ 0 & 0 & 0 & D & 0 \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55} \end{matrix}]

A_{D} = m_{u}^{2} M_{2}^{2} - M_{1} (m_{s} + m_{u})

a_{21} = - K_{t} m_{u} M_{2}^{2} + K_{t} M_{1}

a_{22} = - C_{t} m_{u} M_{2}^{2} + C_{t} M_{1}

\begin{matrix} a_{23} = M_{1} (P_{8} + P_{9} θ) K_{t} \\ - M_{2} m_{u} (R_{4} K_{tl} + M_{2}^{2} K_{t} + \frac{K_{s} K_{D}^{2}}{4 L^{2}}) \end{matrix}

\begin{matrix} a_{24} = M_{1} M_{2} C_{t} - M_{2} m_{u} (M_{2}^{2} C_{t} + \frac{C_{s} K_{D}^{2}}{4 L^{2}}), \\ a_{25} = \frac{M_{2} K_{D} m_{u}}{2 L_{S}} \end{matrix}

\begin{matrix} a_{41} = K_{t} M_{2} (m_{s} + m_{u}) - K_{t} M_{2} m_{u}, \\ a_{42} = C_{t} M_{2} (m_{s} + m_{u}) - C_{t} M_{2} m_{u} \end{matrix}

\begin{matrix} a_{43} = - M_{2} (P_{8} + P_{9} θ) K_{t} m_{u} \\ + (R_{4} K_{tl} + M_{2}^{2} K_{t} + \frac{K_{s} K_{D}^{2}}{4 L^{2}}) (m_{s} + m_{u}) \end{matrix}

a_{44} = C_{t} M_{3}^{2} m_{u} - (m_{s} + m_{u}) (M_{2}^{2} C_{t} + \frac{C_{s} K_{D}^{2}}{4 L^{2}})

a_{45} = \frac{K_{D} (m_{u} + m_{s})}{2 L_{S}} and a_{55} = - \frac{A_{D}}{τ}

\begin{matrix} D = [\begin{matrix} 0 & 0 \\ \frac{\partial F_{1}}{\partial z_{r}} & \frac{\partial F_{1}}{\partial {\overset{\cdot}{z}}_{r}} \\ 0 & 0 \\ \frac{\partial F_{2}}{\partial z_{r}} & \frac{\partial F_{2}}{\partial {\overset{\cdot}{z}}_{r}} \\ \frac{\partial F_{3}}{\partial z_{r}} & \frac{\partial F_{3}}{\partial {\overset{\cdot}{z}}_{r}} \end{matrix}] \\ = \frac{1}{A_{D}} [\begin{matrix} 0 & 0 \\ K_{t} m_{u} M_{3}^{2} - M_{1} K_{t} & C_{t} m_{u} M_{3}^{2} - M_{1} C_{t} \\ 0 & 0 \\ K_{t} {m_{u} M_{3} - (m_{s} + m_{u}) M_{3}} & C_{t} {m_{u} M_{3} - (m_{s} + m_{u}) M_{3}} \\ 0 & 0 \end{matrix}] \end{matrix}

and

{[w_{1}, w_{2}]}^{T} = {[z_{r}, {\overset{\cdot}{z}}_{r}]}^{T}

B_{2} = [\begin{matrix} 0 \\ \frac{\partial F_{1}}{\partial F_{MR}} \\ 0 \\ \frac{\partial F_{2}}{\partial F_{MR}} \\ \frac{\partial F_{3}}{\partial F_{MR}} \end{matrix}] = \frac{1}{2 L A_{D}} [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ \frac{1}{τ} \end{matrix}]

where $M_{1} = m_{u} P_{5}^{2} + m_{u} P_{8}^{2} + I_{u} P_{2}^{2} + I_{C}, M_{2} = P_{8}, and R_{4} = R_{2}^{2} + 2 R_{1} R_{3}$ .

The sprung mass varies as the number of passenger and the weight of the passenger varies. Thus, uncertainty in the sprung mass is taken into account by considering variation in the sprung mass as

m_{s} = m_{s 0} + Δ m_{s}, | Δ m_{s} | < 0.5 m_{s 0}

(30)

The restriction in the uncertainty is obtained from the knowledge of commercial vehicle with passenger restrictions. Now the equations (29) and (30) can be combined to obtain the state-space equation of motion considering the parameter uncertainty as

\overset{\cdot}{x} = (A_{0} + Δ A) x + Dw + (B_{0} + Δ B) u

(31)

where $A_{0}$ and $Δ A$ are the nominal and uncertain parts of the system matrix and $B_{0}$ and $Δ B$ are the nominal and uncertain parts of the control input matrix. It is assumed that there exists a matrix function, say P such that $Δ B = B_{0} P$ and $| P | < φ_{n} < 1$ .

Control strategy

Sliding mode control

To evaluate the performance of the suspension system, first a sliding mode controller (SMC) has been developed in this study. After the modification, the available SMC have been studied. As the first step, a sliding surface is defined as

σ = Cx = [\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} & C_{5} \end{matrix}] x

(32)

where C, $(C \in R^{1 \times 5})$ is the surface gradient matrix and has to be determined such a way that the sliding surface should be itself asymptotically stable. To determine the gradient matrix C, the eigenvector matrix $W \in R^{5 \times 4}$ associated with the desired eigenvalues should be obtained (Hui and Żak, 1992) as

A_{0} W - WJ = B_{0} L

(33)

where $J \in R^{4 \times 4}$ is the desired eigenvalue matrix in Jordan Canonical form and L is an arbitrary matrix chosen to provide linear combinations of the column B₀. Now, the surface gradient matrix is given as

C = B_{0}^{g}

(34)

Here, $B_{0}^{g}$ is the generalized inverse of $B_{0}$ . The surface gradient matrix is obtained by inverse of expansion matrix $[W ⋮ B_{0}]$ . The inverse of $[W ⋮ B_{0}]$ is $[\begin{matrix} W^{g} \\ B_{0}^{g} \end{matrix}]$ . Thus, C forms five rows from last row. In this work, the eigenvalues of −3 is chosen to obtain the surface gradient matrix C, as $C = [\begin{matrix} - 3.594 e 3 & 0.055 e 3 & - 0.848 e 3 & 8.63 & 0.023 \end{matrix}]$ .

The main objective of formulating the sliding mode controller is to guarantee stability and high performance for uncertain system parameters. In this study, we consider that each uncertain element of ΔA is to be bounded as $| δ a_{ij} | \leq {\bar{a}}_{ij} < \infty$ (Choi et al., 1994). A matrix $\bar{A}$ is defined such that the uncertain element has the upper bound of ${\bar{a}}_{ij}$ . Now, the sliding mode controller is defined as

\begin{matrix} u = - \frac{{(C B_{0})}^{- 1}}{1 - φ_{n}} \\ {[| CAx | + | C \bar{A} x |] sgn (σ (x)) + K sgn (σ (x))} \end{matrix}

(35)

where K is a discontinuous positive gain which is calculated as $| GDn |$ , $n = {[\begin{matrix} v_{1} & v_{2} \end{matrix}]}^{T}$ when $v_{i} > | w_{i} |$ , for i = 1,2. Now, the uncertain system with the control law provides a stable system as

\begin{array}{l} σ \dot{σ} = σ C \dot{x} = σ [C {(A_{0} + Δ A) x + D w + (B_{0} + Δ B) u}] \\ = σ [C (A_{0} + Δ A) x + C D w + C B_{0} (I + P) u] \\ = σ [{C A_{0} x - ψ | C A x | sgn (σ (x))} + {C Δ A x - ψ | C \bar{A} x | sgn (σ (x))} + {C D w - ψ K sgn (σ (x))}] < 0 \end{array}

(36)

where $ψ = (I + P) / (1 - φ_{n}) > 1$ with $I \in R^{5 \times 5}$ is the identity matrix. The discrete gain can be changed by introducing saturation function sat(s) in place of sgn(σ) as follows

sat (σ) = {\begin{matrix} \frac{σ}{ε}, for | \frac{σ}{ε} | \leq 1 \\ sgn (σ), otherwise \end{matrix}

(37)

Adaptive fuzzy moving sliding mode controller

It is very much required to reduce the reaching phase of the sliding mode controller to make it more effective. By changing the surface gradient, the reaching time can be reduced. Figure 5 shows two methods of changing the surface gradient, rotating and shifting. Thus, the sliding surface for this moving sliding mode controller can be determined as (Choi et al., 1993, 2008)

\begin{matrix} σ = c_{1} (x_{1}, x_{2}) x_{1} + c_{2} x_{2} + c_{3} x_{3} \\ + c_{4} x_{4} + c_{5} x_{5} + α_{m} (x_{1}, x_{2}) \end{matrix}

(38)

c_{1} (x_{1}, x_{2}) = \frac{(Δ_{rot} - x_{2})}{x_{1}} and α_{m} (x_{1}, x_{2}) = c_{10} x_{1} + x_{2} - Δ_{sf}

(39)

where $c_{10}$ is the initial value of $c_{1}$ and $Δ_{rot}$ and $Δ_{sf}$ are the coefficients for rotating and shifting surfaces, respectively. Differentiating equations (39) and (38), we get the following equation

c_{1} {\overset{\cdot}{x}}_{1} + {\overset{\cdot}{c}}_{1} x_{1} = - {\overset{\cdot}{x}}_{2} and {\overset{\cdot}{α}}_{m} = c_{10} {\overset{\cdot}{x}}_{1} + {\overset{\cdot}{x}}_{2}

(40)

\begin{matrix} \overset{\cdot}{σ} = c_{1} {\overset{\cdot}{x}}_{1} + {\overset{\cdot}{c}}_{1} x_{1} + c_{2} x_{2} + c_{3} x_{3} + c_{4} x_{4} + c_{5} x_{5} + {\overset{\cdot}{α}}_{m} \\ = c_{10} {\overset{\cdot}{x}}_{1} + c_{2} x_{2} + c_{3} x_{3} + c_{4} x_{4} + c_{5} x_{5} = C \overset{\cdot}{x} \end{matrix}

(41)

Now, we can consider the same stability analysis as equation (36) by substituting $\overset{\cdot}{σ}$ from equation (41) to show the stability condition: $\overset{\cdot}{σ} σ < 0$ .

The moving sliding mode controller is defined as follows

\begin{matrix} u = - \frac{{(C B_{0})}^{- 1}}{1 - φ_{n}} \\ {[| CAx | + | C \bar{A} x |] sgn (σ (x)) + K sgn (σ (x)) + α_{m} (x_{1}, x_{2})} \end{matrix}

(42)

The major three performance characteristics of vehicle suspension are ride comfort (sprung mass acceleration), suspension travel (relative displacement between sprung and unsprung mass), and road handling (relative displacement between unsprung mass and the road). Thus, an adaptive technique is proposed so that these performances improve and the settling time for bump input reduces by a significant amount. To form the adaptation rule, a fuzzy logic–based controller is used which has three input variables, sprung mass acceleration (a_s), relative displacements between masses (x_rel), and relative displacement between unsprung mass and road (x₂₀) and a single output $(\tilde{θ})$ . The fuzzy control system consists of three stages, fuzzification, fuzzy interface, and defuzzification. In the fuzzification stage, the input values are converted into fuzzy values. In the second stage, the fuzzy controller calculates the controller output using the inputs and rule base and the output is converted to the desired control force in defuzzification stage. The rule base, which is presented in Table 3, is prepared. The abbreviations used are listed below:

For input: N, negative; Z, zero; P, positive.

For output: ON, one; SP, small positive.

The three rules are written as

IF (a_s = P) OR (x_rel = N) OR (x₂₀ = P) THEN (u = ON).

IF (a_s = N) OR (x_rel = P) OR (x₂₀ = N) THEN (u = ON).

IF (a_s = Z) OR (x_rel = Z) OR (x₂₀ = Z) THEN (u = SP).

The control force is given by

\begin{matrix} u = - \frac{{(C B_{0})}^{- 1}}{1 - φ_{n}} \\ {[| CAx | + | C \bar{A} x |] sgn (σ (x)) + α_{m} (x_{1}, x_{2}) + \tilde{θ} K sgn (σ (x))} \end{matrix}

(43)

It is noted here that the switching gain in the controller (43) makes the control algorithm run faster and reduces the unwanted chattering of the system. The adaptive switching gain helps the system to settle faster than the ordinary sliding mode controller. The stability of the controller does not affect as $\tilde{θ} > 1$ is always positive. The control input u is designed in the above sections in active manner. But the MR dampers can be operated in semi-active manner by applying the control input force according to the following semi-active condition

u = {\begin{matrix} u, if u ({\overset{\cdot}{z}}_{s} - {\overset{\cdot}{z}}_{u}) > 0 \\ 0, otherwise \end{matrix}

(44)

This semi-active condition implies that the controller only assures the increment of the energy dissipation of the system and it increases the stability of the system. The current required to obtain the control input is calculated from equation (2). The maximum current supplied to the damper is limited to 2 A. The control parameters for the proposed control strategies are chosen as $φ_{n} = 0.01, K = 300, and ε = 0.001$ .

Results and discussion

To find out the response of the Macpherson strut, the parameters are listed in Table 2. The coordinate values of Macpherson strut are shown in Table 3. The proposed control system is compared with the uncontrolled and other control strategies when the strut is subjected to two types of inputs, namely, bump input and random input.

Table 2.

Parameters of the Macpherson strut.

m_s	453 kg	m_u	71 kg
K_s	27,358 N/m	C_s	570 Ns/m
K_t	183,887 N/m	C_t	50 Ns/m
I_u	0.021 kg/m²	I_c	0.01 kg/m²
K_tl	190,000 N/m	R	0.3 m

Table 3.

Coordinates of important points of Macpherson strut.

	Y-coordinate (mm)	Z-coordinate (mm)
A	249	−60.8
B	0	0
C	134.5	374.8
D	107.4	582.5
E	152.6	236.2
P	292.1	27.5

Responses for bump input

Control characteristics of the suspension system is measured for bump type transient road input which is given as

z_{r} = z_{b} [1 - \cos (ω_{r} t)]

(45)

where $z_{b}$ is the half bump height (0.035 m), $ω_{r} = 2 π V / D_{r}$ , D is the width of the bump (0.8 m), and V is the wheel velocity which is considered as 0.8 m/s (2.88 km/h) in this study.

Figure 6 shows the responses of the suspension system, when there is no uncertainty, after applying proposed control algorithm. The sprung mass displacement is plotted in Figure 6(a). It is seen that the proposed control strategy attenuates the vibration level well and the settling time for the response after applying the control is improved. For the proposed adaptive fuzzy moving sliding mode controller (AMSMC), the settling time is faster than that of the conventional sliding mode controller (SMC) which can be seen from Figure 6(b). Settling time has been reduced to 1.991 s in AMSMC from 3.48 s for uncontrolled system. The settling time for SMC was 2.245 s. The settling time is calculated as the time required for the system to reach a steady state. In this study, time required to reach the system response within 20% of the excitation amplitude is considered. For the present case, the maximum input amplitude was 0.07 m. Thus, time required for the system to settle within 0.014 m is considered as settling time of the system which is shown in Figure 6(b). The settling points for each case are marked in Figure 6(b). It is noted here that in this work MR damper is designed to have same damping force as the passive damper for vehicle suspension. Thus, the uncontrolled case of the results in Figure 6 can be considered as a conventional passive damper to compare the result.

The control input current for SMC and AMSMC cases is plotted in Figure 7. Figure 7 shows that there is not much difference in the current plots. It is calculated that the root mean square (RMS) value of input current for AMSMC case is 0.3607 A whereas it is 0.3382 A for SMC case. But, after the settling point, input current for AMSMC is low as compared to that in case of SMC. The other three major performance criteria of the suspension system are sprung mass acceleration (a_s), relative displacement between masses (z_rel), and relative displacement between unsprung mass and road (z₂₀) as shown in Figure 8. The performances are also shown to be improved using the proposed control logic. It is seen that the adaptive sliding mode controller (AMSMC) makes the system settle faster than the ordinary sliding mode controller (SMC). In the acceleration response, chattering is observed in the controlled responses. To check the robustness of the controller an additional mass, $Δ m_{s} = 50 kg$ is attached to the sprung mass. The system matrices changes according to equation (31). The responses of the system are shown in Figure 9. It is seen that the sprung mass displacement, acceleration, and suspension travel are reduced well by adopting AMSMC even in the presence of uncertain parameter.

The kinematic parameters like camber angle (ϕ), king-pin angle, and track alteration are obtained for different control strategies. Camber angle, shown in Figure 10(a), shows that the angle is reduced by considerable amount. The king-pin angle (α) is given by the angle between the line through the points A and D and the vertical axis in case of Macpherson strut

α = \tan^{- 1} \frac{y_{D} - y_{A}}{z_{D} - z_{A}}

The king-pin angle, as shown in Figure 10(b), is also reduced by a good amount for AMSMC and the settling time reduces by a good amount. The track alteration is the change in the distance between the centers of the two wheels. It is given by the expression, $d_{yl} = y_{u} - φ R$ . The track alteration is also improved for AMSMC as shown in Figure 10(c).

Response for random excitation

The proposed control strategy is checked for random road condition. The power spectral density (psd) of sprung mass displacement and acceleration are plotted in Figure 11(a) and (b), respectively. It is well shown in the figure that the proposed control performs well near the body resonance. However, since the proposed sliding mode control only deals with the displacement and velocities, the acceleration response does not improve much within the frequency range between body resonance and wheel resonance. It is seen from Figure 11(c) and 11(d) that the suspension travel and tire deflection (relative displacement between unsprung mass and road, $(z_{2} - z_{0}) or z_{20}$ ) are reduced by a significant amount in both the resonance frequencies. In case of tire deflection, the proposed control logic almost eliminates the first peak of the responses. The same phenomenon is obtained for system with uncertainty also which is shown in Figure 12. The amount of uncertainty is same as that of the previous case.

Conclusion

A new semi-active control strategy of Macpherson strut type vehicle suspension system has been studied in this study. The nonlinear equations of motion are first obtained using Lagrange’s principle. The equivalent linear equations were constructed at the equilibrium position to make it suitable to use in state-space form. The different parameters of the Macpherson strut, like camber angle, track width, and king-pin angle, have been calculated. The MR damper which is employed in the suspension system has been controlled using semi-active control logic. A new fuzzy-based adaptive sliding mode controller was used to provide better result of the system. In the formulation of the controller, a moving sliding surface has been used instead of conventional fixed sliding surface. The proposed control strategy provides good vibration control result when parameter uncertainty is present in the system. It is also seen that the proposed controller can produce better vibration control results for both the bump type of input and random road input than conventional sliding mode controller. In addition, it is observed that the kinematic parameters such as camber angle can also be controlled well using the proposed controller. From the frequency response of the system when subjected to random road input, it is clearly seen that the proposed control logic performs excellent vibration control near the resonance neighbors. It is remarked that the integration of steering model with Macpherson strut type suspension will be done to study the lateral stability and experimental validation work will be undertaken in the future.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Inha University Research Grant. This financial support is gratefully appreciated.

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