Comparative study between backstepping and backstepping sliding mode controller for suspension of vehicle with a magneto-rheological damper

Received Oct 17, 2021 Revised Apr 1, 2022 Accepted Apr 16, 2022 The Suspensions are among the important components of vehicles, providing comfort for passengers and protecting the chassis and freight. They are normally provided with dampers that mitigate these harmful and uncomfortable vibrations. This work aims at investigating the potential of the use of the Magnetorheological Damper as a semi-active suspension for passenger vehicles to improve their ride dynamics and to, ensure their manipulability, and reduce unwanted vibrations levels. To achieve desired performances, a hybrid controller based on a backstepping-sliding mode control strategy is derived to increase the dynamic performance of the automotive suspension component and providing comfortable passenger traveling. Numerical simulation is used to test the efficiency of the proposed controller. The obtained results show that the proposed backstepping sliding mode controller is more efficient and robust against road profile excitation and external disturbances in comparison to a classical controller based on backstepping and sliding mode control.

. The basic configuration of the MR damper The mathematic equations of the damper can be used to describe the applied voltage or current or to the MR circuit with the produced damping force. Several scholars have constructed statistical models to describe the hysteretic behaviour of MR damper in the research. Figure 2 depicts the well-known enhanced Bouc-Wen model, which describes the characteristics of MR dampers [29]- [31]. Spencer et al. [29] created a phenomenological framework wherein the force of the MR damper is described as a function of the spring stiffness, accumulation stiffness, viscous damping at a low velocity, and damping or stiffness at high speeds. The following is an expression for the forces generated by the MR dampers using the generalized Bouc-Wen model: [12], [14], [29]- [31].
Where x, and indicate the damper's displacement, speed, supply voltage, and produced force, respectfully, z is an evolving variable that whih characterizes the hysteresis behavior of the MR damper force versus the damper's movement and velocity. The hysteresis loop's form and size are described by the parameters, and A. The device's extra parameters may be represented as a function of applied voltage as follows [29]- [31]: where u is a variable that defines the dynamic of the system given by: where v is the control voltage supply at terminals of the driver and denotes the first-order filter time constant.
Parameters of the mangnetorheological damper model adopted are given as: A sinusoidal signals displacement with the amplitude of 1.25cm and 1.5 Hz frequency is performed for varied supplied input voltages to validate the behavior and hysteresis features of the proposed MR damper type (0, 0.5, 0.75, 1, and 1.25 V). Figure 3 depict the produced force's temporal response, the pressure against MR damping displacement, as well as force-velocity hysteresis. The various plotted charts of the produced force clearly show the hysteresis behavior of the MR damper in between time Figure 3(a) and velocity x and the damper force as shown in Figure 3(b), and the non-linearity characteristics of according to x variation is remarkable (see Figure 3(c)).

DYNAMIC MODEL OF MR QUARTER-VEHICLE SUSPENSION SYSTEM
Despite the fact that the genuine suspension may be treated in a way more sophistical, the suggested model simplifies the complicated geometry of the suspension structure [32]. To investigate the robustness of the suspension control strategy we suppose a quarter vehicle system as shown in Figure 4 [13], [20], [30]. It is a basic structural model since it consists primarily of two subsystems: the wheel mass which represents tyre subsystem and also the body mass which represents the subsystem of suspension. These two subsystems are connected between them via elastic and dissipative elements and are subjected to external disturbances, which are represented by the road profile. From the mechanical model depicted in the Figure 4, the mathematic equations of the 1/4-vehicle model are written as follows [13], [20], [30]: with , and denote the vertical displacement of the sprung mass, the unsprung mass and the external disturbance due to the road profile, respectively. the force produced by the MR damper; and respectively are the car body and the wheel assembly masses. and represent the stiffness of the suspension and the stiffness of tire. is the total damping constant of the suspension. We define the state variables , the input force vector u and the road profile disturbance as follows:

CONTROL ALGORITHMS OF VEHICLE SUSPENSION DYNAMICS
This section proposes semi-active controls of the vehicle suspension dynamics depending on multiple nonlinear control schemes. There are three forms of nonlinear control formed: backstepping control, slide mode control, as well as a hybridation backstepping-sliding mode control scheme. All of these kinds of controls are dependent on the Lyapunov stability criteria, which will be shown in the next subsections.

Vehicle suspension control using backstepping strategy
Backstepping seems to be a systematic and recursive nonlinear feedback control design approach ideal for the wide category of feedback nonlinear systems. This technique ensures global pursuit for such nonlinear systems category that can be transformed into parametric-strict survey form [13], [14], [33]- [35]. The control law's backstepping technique relies on a recursive selection of some relevant functions using system states as virtual-control inputs for the under dimension subsystems. Each step of the backstepping startegy generates a virtual-control signal, which is expressed in relation of the previous design phases virtual-control designs. When technique steps are completed, a feedback disposition for the actual control input signal is produced, which meets the desired design target by summing the Lyapunov activities connected with each individual design phase [13], [14], [33]- [35].
Step 1: if we opt to reduce the suspension travel, the oscillating subsystem problem will remain unaddressed. Because we know that proper controlled variables can help to avoid unpredictable zero dynamic features, the regulated output is set to be [36]: where 1 x is the displacement of car body, 3 x is a filtered version of the wheel displacement 3 x : where  is a positive constant.
Because of that the error 1 e must be equal to zero, state variable 2 x is taken as the virtual control to make the first subsystem stable. In order to compute the stabilizing function, we defined the following Lyapunov function as: We differentiate to get: where, 1 K is constant gain. The error is achieved if we define stabilization function as: with 1 = 2 and 1 > 0 a positive design constant. By substituting (14) and (11) into (13) yields Step 2: now, in the next step we try to make the virtual control 2 as desired. So, we define the following error signal 2 = 2 − 1 (16) and the error equation is: To determine the stabilizing function, the augmented Lyapunov function is defined as: So, its derivative is defined as follow: The control tracking is achieved if one defines the following stabilizing function: where denotes the desired active force which must be produced by the MR damper (i.e. = ).
2 is a positive constant, by substituting (21) into (20) yields: thus, the system is asymptotically stable. The system is simulated in the Simulink as shown in Figure 5.

Vehicle suspension control using sliding mode control
Many research studies have used the variable structure with the sliding mode, or sliding mode control, in the previous decades for non -linear control processes [7], [13], [17], [20]- [22], [37]- [39]. The sliding mode controller (SMC) seems to be a non-linear control mechanism that modifies the dynamics of a non-linear system, causing it to slide. The SMC is widely renowned for its resilience, high stability, and straightforward implementation [40]- [42]. The characteristic of the sliding mode relies on directing a system's state trajectory toward this sliding surface and directing it to switch across it to the equilibrium point using proper switching logic, resulting in the sliding phenomena. The benefits of SMC control are numerous and considerable, including improved accuracy, a very rapid reaction time, and, most importantly, resilience.
[43], [44]. In general, the sliding mode control design procedure can divide in two steps. The first step of SMC design involves the selection of the sliding surface (sometimes referred to the switching surface or switching manifold ( ) = 0) in the state space where the control undergoes discontinuities [13], [20], [21]. This sliding surface is chosen to simulate the required closed loop performances.
Insecond phase, the control rule should be devised such that such system's state trajectory are driven toward and remain on the sliding surface. The reaching phase refers to the system state trajectory before to reaching the sliding surface. The generic expression of Slotine and Li's sliding surface is calculated by: where ( ) = − denotes the tracking error, is a positive constant, is the variable state. The sliding-mode control law u is consisting of two components. One is the equivalent control ueq and the other is switching control input us, i.e. [7], [20]. where is called equivalent control which dictates the motion of the state trajectory along the sliding surface and is known as switching control part which is given in the basic form as: where is a positive constant. The control input should be designed such that the following sliding condition is satisfied: using a sign function often causes chattering in practice. Among the solution of this problem is to introduce a boundary layer around the switching surface, and the switching component is become as: where sat(x) is the saturation function defined as: where denotes the boundary layer thickness around the sliding surface. The conventional SMC uses a control law with large control gains and can produce some drawbacks associated with large control chattering. The control-input chattering is undesired and it may excite high-frequency response and lead to instability.

Sliding mode speed controller design
The goal of the control technique is to create a nonlinear control system based on sliding -mode that achieves high accuracy and precision tracking under uncertainties and disturbances. The following stages are included in the design of the proposed sliding -mode for the semi-active suspensions control. The following is the definition of error tracking: then, the general sliding surface is given by: a sliding condition must be established in order for the state to travel toward and achieve the sliding surface. The suggested sliding condition is: [20], [21].
where is a positive constant. The damping force is calculated by deriving (29) since it forces trajectory to point toward this sliding surface. The systems trajectories, in particularly, adhere to the surface then on surface, i.e. = 0 From (28) and (29): The best approximation for the desired damping force , a control law should achieve ̇= 0 thus; In order to satisfy sliding condition, a term discontinuous across the surface is added to the equivalent control , then the desired control force a F can be expressed as: with > 0. The system is simulated in the Simulink as shown in Figure 6

The hybrid controller backstepping-sliding mode design
The control goal is to construct a backstepping sliding-mode control [45]- [49] system to monitor the reference trajectory for the outputs of the vehicle suspension illustrated in (9). The suggested backstepping sliding-mode control system is intended to meet the goal of automobile body displacement tracking. This controller's output is really the reference force which must be delivered by the MR damper, and it is discussed in: Step 1: the tracking error is chosen as shown in section 4.1: the derivative of 1 e is computed as: the first Lyapunov function is chosen as: define the first virtual control variable, then, the derivative of 1 V is: with 1 is a positive design gain, Step 2: we define a sliding surface So, it is easy to find the following expression: The second Lyapunov function is chosen as: where S is the sliding surface. So, the derivative of 2 is expressed as follow therefore, 1 and 2 will converge to zero. The stability of the proposed backstepping sliding mode control system can be guaranteed. The system is simulated in the Simulink as shown in Figure 7.

Semi-active control system
The semi-active control system consists of two controllers. The first one is the system controller which produces the appropriate active damping force ( ) corresponding to the suspension responses due to the road profile excitation, while the role of the second controller is the adaptation of the input voltage level which keeps the MR damper force on the desired active damping force ( ). In this paper, the semi-active controller is essentially composed of backstepping controller, sliding mode control or backstepping sliding mode control (system controller) and a function unit which used for converting the desired control force to an input control voltage (damper controller). The usually adopted algorithm for the conversion of the force to a adjusted voltage for the conversion is a clipping algorithm [29], [32]: where is the maximum applied voltage, {⋅} is a Heaviside step function, is a measured MR damper force, and denotes the desired active control force produced by the system controller (backstepping control/SMC or Backstepping-sliding mode control (B-SMC)).
The structure of the closed-loop semi-active controller using the proposed nonlinear controllers for vehicle suspension with an MR damper is depicted in Figure 8. In the first step, the actions specified control force is computed using one of the nonlinear controllers based on vehicle suspension system outputs. Because the control input directly represents the damping force , the Heaviside step function is employed to adjust the input voltage control according to the desired damping function and the real force realized by the MR damper.

SIMULATION RESULTS
The quarter vehicles suspension structure is subjected to two types of road stimulation in order to assess the performance of the semi-active control with MR damper utilising the suggested nonlinear controllers. The bump in Figure 9(a) is the very first stimulation signal studied, and it is typically used to highlight dynamic response properties. The bump excitation is described by: where = 2 denotes the frequency [rad/sec], represents the bump height. The second road excitation tested is a step excitation in Figure 9(b) with an amplitude of 0.2 . These types of road excitations, illustrated in Figures 9(a) and 9(b) are applied to the vehicle suspension system as shown in Figure 8.
The following are the different parameters of the suggested adaptive sliding mode control that are regarded to produce high transient control stability and performance which are used for control loop:   Figures 10(a)-(c) show the vertical displacement of an automobile body, acceleration of the sprung mass, and vertical displacement of the unsprung mass utilizing nonlinear controllers (backstepping control, sliding mode control, and backstepping-sliding control) for the bump road excitation. In contrast to the passive suspension, the outcomes shown in Figure 10 clearly demonstrate that using the suggested semiactive suspension system can obtain excellent performances in which the undesired vibrations due to road excitation have really been clearly diminished (lower amplitude for the vehicle acceleration, vehicle displacement, and wheel displacement). It definitely demonstrates the efficacy and durability of the recommended semi-active vehicle suspension control with MR dampers for induced vibrations mitigation. Figures 11 (a)-(c) depicts the acquired results for vertically vehicle displacement, vertical accelerating of sprung mass, and wheels vertical displacement for step road excitation. According to simulation results, the systems displacement and accelerating responses are significantly decreased, and the steady-state duration of controller system responds is significantly quicker than the passive system. We see that the displacements as well as accelerations have been reduced significantly, confirming that the performance have been enhanced. Thus, it can be stated that high passenger comfort may be gained by minimizing body movement as much as feasible, as demonstrated by the nonlinear semi-active control. Based on the dynamic response characteristics, we can conclude that the suggested semi-active control with the MR damper based on backstepping-sliding mode approach outperforms the other investigated nonlinear controls (i.e. Sliding mode control and backstepping control). Figure 12 depicts the applied voltage level to the semi-active suspension's MR damper. We can see that the Voltage level is constantly switching between zero and the maximum voltage based on differences between produced forces and clipped-control. Figure 13 depicts a contrast of the real damping force, desired damping force, and active force. Figure 13 means that the actual damping forces generated by the MR dampers can following the reference damping force with precision and decrease tracking error, demonstrating the usefulness of the clipper optimum control utilised in damping force control.    Figure 12. depicts the input voltage provided to the MR damping driver of semi-active control using a back stepping-sliding mode technique for a bumpy road excitation testing Figure 13. A comparison of the real damp force and also the target damping force for a bump roadway excitation test using a semi-active power based on a backstepping-sliding mode technique

CONCLUSION
To reduce undesirable vibrations caused by road irregularity, a semi-active control based on a clipper optimum control method for MR damper is suggested in this study for a vehicle suspension system. To improve the performance of the vehicle MR damper suspension system, the suggested control strategy includes the creation of a classic control scheme, a back stepping control approach, and a hybrid back stepping-sliding phase controller. The nonlinear control techniques and the clipper control algorithm are used to generate the active damping power and the MR damper driver's modified input voltage. In the first stage, a generalised model of the MR damper based on modified Bouc-Wen model is provided. Next, for a semiactive controlling, 3 types of nonlinear control methods, backstepping, slide mode, and backstepping-sliding authority, are created and used to the quarter-vehicle rear suspension.
Numerical methods utilizing Matlab/Simulink programs are used to evaluate the effectiveness of the suggested control techniques for various forms of road excitations. It has been quantitatively demonstrated that the semi-active controller performs well and can significantly minimise or suppress the produced vibrations. Moreover, we observe that the hybrid backstepping-sliding mode controlling is more durable and effective than the other nonlinear systems. Thus, the obtained findings indicate that proposed control mechanisms connected with the chosen MR suspension system may greatly increase passenger comfort.