Performance Improvement of Active MacPherson Suspension Using a Pneumatic Muscle and an Intelligent Vibration Compensator

This paper presents an intelligent control strategy for an active MacPherson suspension driven by a pneumatic muscle (PM) actuated system and evaluates its improved control performance on a self-developed test-rig. To fulfill the fully-functional test and analysis of active suspension systems, this test-rig comprises three units: 1. an active MacPherson suspension unit, which consists of a PM mechanism and two MacPherson struts, is built to isolate vibration from road disturbances; 2. a road profile generator (RPG) unit, which can produce a vertical force to lift the car-body according to various road profiles; 3. a PC-based control (PBC) unit, which computes, sends and receives signals for both of the active MacPherson suspension and the PC-based control unit. The objective of this study is to improve the MacPherson suspension in terms of the capability of road vibration isolation by using the PM actuation that can actively provide extra compensatory force for MacPherson struts. Then, for motion control of the PM, this study employs an interval type-2 adaptive fuzzy controller to approximate the optimal control law and adopts a self-tuning and fuzzy sliding mode compensator to compensate unmodeled dynamics for the active MacPherson suspension system. Three experiments are conducted to compare the active MacPherson suspension system with the original MacPherson struts through various road profiles on the test-rig. The results show the significant improvement for the proposed active MacPherson suspension system in suppressing the displacement and acceleration of the car-body.


I. INTRODUCTION
Vehicle suspension systems (VSS) are designed to isolate vibrations from the internal engine and external road disturbances to maintain driving comfort. VSS's can be categorized into three general systems: passive, semi-active and active suspension. A passive suspension system has the simplest structure with common and basic suspension components, including springs and dampers. This has been proven to work smoothly in all types of road situations [1]. However, a passive suspension system is extremely sensitive to road surface bumps and irregularities. A semi-active suspension The associate editor coordinating the review of this manuscript and approving it for publication was Juntao Fei . system is an electronic shock absorber that is capable of adjusting damping coefficients in real-time for different road conditions [2], [3]. Up to now, the Magneto Rheological (MR) damper is one of the most devices successfully applied to semi-active suspension systems [4]. An active suspension system with a force actuator absorbs shocks and maintains the chassis at an even level [5]- [7]. This is realized by monitoring the feedback forces from the road and generating movements that are perpendicular to the vehicle's body. Most early designs for suspension systems used purely mechanical components or a simple combination of mechanical and hydraulic components. Current suspension systems usually involve a combination of mechanical, hydraulic, pneumatic and even electronic components. However, this combination complicates the physical construction. In general, an active suspension system (ASS) contains an ordinary passive suspension component and active actuators. A hydraulic actuator [8], [9] is the most commonly used because it is strong and has a high power-to-weight ratio. However, fluid leaks may cause environmental pollutions. A hydraulic actuator is also large so it is not easily accommodated within the ASS. In recent years, pneumatic-based actuators have been the subject of many studies [10] and may replace hydraulic suspension due to the advantages such as low cost, compact structures, lightweight, and high reliability in operation. Zapateiro et al. [11] presented semi-active suspension control for vehicles using an MR damper. The nonlinearity in the rapid response of the MR damper was ignored while using back-stepping techniques. Banerjee et al. [12] designed and implemented a single axis DC attraction type suspension system in which four cascade lead compensators individually control four electromagnetic actuators. This suspension prototype achieved stable levitation with the desired operating gap by ignoring mutual cross-coupling. In [9], a hydraulicdriven active suspension system with fuzzy sliding controllers is presented to compensate for nonlinearity and time-varying behavior. This study simulated a suspension system on a test rig using a 20mm high bump. An ASS with a linear switched reluctance actuator is presented in [7]. The system incorporates a tracking differentiator (TD) into a PD controller to stabilize the system, but the displacement and the derivative of the sprung mass are acquired for the TD which significantly amplifies noise. To provide superior performance to a passive suspension and a PID-based ASS, a pneumaticdriven ASS test rig with four control feedback loops was proposed to improve force tracking and to compensate for disturbance [6], [7]. However, the maximum displacement of the sprung mass resulted only within 0.5cm high that limits the application in vehicle suspension systems. In [10], a state feedback control law based on the linear model of the prototyped ASS was investigated and verified using a real-roadconditions generator. In 13,14,15, an interval type-2 fuzzy controller (IT2FC) was proposed for a nonlinear active suspension system that is subject to system modeling uncertainties, measurement noise and external disturbances. Such a controller minimizes the displacement and acceleration of the sprung mass that leads to more system robustness and reliability. However, it remains difficult to yield appropriate membership functions and fuzzy rules for the interval fuzzy controller to deal with system uncertainty and random vibrations. In 2016, we presented a prototype [16] of the PM-drive active suspension system and used an interval type-2 fuzzy controller with an adaptive fuzzy sliding compensator for it to compensate road disturbances and improve the ride comfort.
The Pneumatic Muscle (PM) was developed in the 1950s to actuate the artificial limbs [10], [17]- [19]. The retraction strength of a PM is dependent on the strength of the individual fibers in the woven shell. A PM has several advantages over a pneumatic actuator (PA): 1. if the cylinders have the same diameter, the weight of the PM is much lighter of that for a PA but it supplies much more initial force during contraction; 2. the physical model of the PM can be regarded as a spring that is under constant pressure or which has a constant volume (similar to a damping) and exhibits similar actions as the springs work; 3. a PM is light but powerful because it has a better ratio for power-to-weight and 4. a PM does not experience fluid leaks and has better performance in rough environments. The experimental results in [20] show that the FESTO PM performs well with the AVSS at frequencies of up to 8Hz and that the FESTO PM is applicable to suspension systems with a provided control law regardless of the complicated and highly nonlinear items in the system dynamics. However, a PM is very difficult to be compensated using conventional control approaches since it has a highly nonlinear mathematical and physical properties in nature [21], [22].
MacPherson Struts [23] have a similar physical layout to a real vehicle suspension. They are packaged with a spring and a damper and can suppress vibrations. However, it cannot actively provide actuator force against vibration while encountering road variations; besides, it contains nonlinearities and uncertainties in the mathematical model while analyzing its dynamic behavior. Therefore, it is critical to design an active actuator force for MacPherson Suspension to improve its performance on vibration suppression. This paper proposes an active MacPherson suspension system (AMSS) that utilizes a FESTO PM, named as AMSS-PM, as the force actuator for active suspension control. To generate real road profiles, the developed RPG unit consists of a wheel, rollers, an induction motor and a pneumatic cylinder. Because of the PM's complicated dynamics, the designed AMSS-PM is nonlinear and uncertain. Thus, an interval type-2 adaptive fuzzy controller combined with a self-tuning fuzzy sliding mode compensator (IT2AFC-STFSMC) is presented to compensate for the system uncertainties and attenuate disturbances and noise. This paper expands from [16] to enhance its experimental results and stability analysis for the overall system. There are three different road conditions conducted to verify the feasibility of AMSS-PM in this paper, and stability proof in detail for an ASS with unknown uncertainties and disturbances is also addressed here. The major contributions of this paper are depicted as follows: • The utilization of PM actuator achieves the compact design, low energy consumption, and cost-effective implementation of the AMSS that utilizes a FESTO PM for a quarter-car suspension platform.
• A fully-functional test-rig for a quarter car is developed to validate the feasibility and performance of the designed AMSS-PM under various road conditions as provided by the road profile generator.
• The proposed IT2AFC-STFSMC can dominate the unmodeled dynamics and parametric uncertainties caused by external disturbances for a general nth nonlinear uncertain suspension system while guaranteeing stability and robustness for the tracking control problem. VOLUME 8, 2020 • The adaptive law of the IT2AFC-STFSMC provides an instantly tuning scheme for the controller gains of the IT2AFC-STFSMC, so that the optimal parameters can be found according to changed road conditions and the system stability remains within the Lyapunov framework.

II. DESCRIPTION OF THE MATHEMATICAL MODEL
The PM is a long tube that is made of natural rubber with fibers wrapped inside and metal fittings attached at each end. Due to the reversible physical deformation, the PM can produce linear motion during contraction and expansion of the muscle. The next section presents the mathematical model for the PM.

A. MATHEMATICAL MODEL OF THE PM
The PM changes shape when the pressure of the air that flows into the interior of the rubber tube is changed, as shown in Fig. 1. Injecting compressed air into the rubber tube expands the PM and produces a force that drags the sprung mass downward. Otherwise, the sprung mass is restored to its original shape when air is released from the PM. The reversible physical deformation during contraction and expansion produces a linear motion: Firstly, if the PM is modeled using a common static physical model [24]- [26], the length and the diameter of the PM are formulated as: and where L represents the actual length of the PM and is a variable of the pressure inside the PM, L 0 represents the original length of the PM, l represents the length of the thread, n represents the number of turns in the thread, D represents the diameter, and θ represents the angle that is resulted by threads relative to the longitudinal axis. Consider the PM as the cylindrical shape, and its volume can be expressed as: Substituting Eq. (1) and Eq. (2) into Eq. (3) yields: For the PM, the effective area of the cylinder changes over time so it is considered to be a pneumatic cylinder with a variable volume. According to Principle of the Conservation of Energy, the simple geometric force F a that is exerted by the PM is calculated as the product of the pressure and the change in the volume with respect to length, which is [24], [25]: where P represents the pressure inside the PM and P e represents atmospheric pressure. The term dV/dL can be expressed as an equivalent effective area: Substituting (6) into (5) yields: Due to the complicated dynamic behavior of the proportional pressure regulator (PPR), which regulates airflow into the PM, a high-order nonlinear dynamic equation is required to establish an accurate mathematical model. Fortunately, since the bandwidth of the PPR is much wider than that of the PM, the PPR model can be formulated as a first-order linear differential equation for the AMSS-PM. After simple manipulation, the transfer function between the input voltage and the output pressure is: where K v is a constant gain of the PPR, T s is a time constant for the PPR andū is the input voltage. Substituting Eq. (8) into Eq. (7) yields:

B. MATHEMATICAL MODEL FOR A QUARTER CAR
The AMSS-PM uses two MacPherson struts and a PM to support the sprung mass. The PM also provides vertical movement to reduce vibration. Figure 2 illustrates a model of the AMSS-PM, where Z s is the displacement of the sprung mass, Z u is the displacement of the unsprung mass. M u is the mass of the unsprung mass and M s is the mass of the rigid body (sprung mass). If the tire is in full contact with the road surface, the dynamics of the tire can be modeled as a spring with a spring constant K t and an unsprung mass of M u . In this condition, the quart car is modeled as a dynamic system with 2-DOF and the dynamic response and the control performance can be measured easily. The frictional force of the piston rod in the shock absorber inhibits the smooth travel of a MacPherson strut suspension system, so the frictional force, denoted as F µ , must be included in calculation concerning the suspension system. The motion equations of the AMSS-PM are given as follows: where Z r is the variation in the position of the road surface, B s is the average damping coefficient of a MacPherson strut, F a is the force that is exerted by the PM and K s is the average stiffness coefficient of a MacPherson strut, which connected between the sprung and unsprung masses. The characteristics  (11) for states x 1 to x 5 yields a dynamic equation with the output aṡ

III. INTERVAL TYPE-2 ADAPTIVE FUZZY CONTROLLER WITH SELF-TUNING FUZZY SLIDING MODE COMPENSATION (IT2AFC-STFSMC) FOR THE AMSS-PM
The AMSS-PM has a composition of the PM and the MacPherson strut so that the parameter variation and unmodeled dynamics happen at these two parts. Firstly, the MacPherson strut contains frictional force F µ , which acts as nonlinear behavior due to the external load, and damper friction. The MacPherson strut also shows the spring buckling in the suspension systems. Secondly, the AMSS-PM uses a PM to provide an actuated force. The actuated force generated by the PM is a nonlinear function of the thread length (l), the number of turns in the thread (n), the time constant (T s ) as well as gain (K v ) of PPR and the air pressure P inside the PM. In practice, these mentioned parameters for the PM are VOLUME 8, 2020 usually not given, even we have the type of PM for the AMSS-PM. Besides, the PM shows the complexity and uncertainties in its dynamic behavior because of the air-compressibility and the hysteresis phenomenon. Also, for the AMSS-PM, the nonlinear effects in the passive spring force, damping force, and vertical tire force would be amplified in some situations [2]. Therefore, the AMSS-PM exhibits complex nonlinear and time-varying behavior due to the fact that it contains these uncertainties and unmodeled dynamics from the MacPherson struts, the PM and the tire bouncing. The traditional model-based controller is challenging to implement for the AMSS-PM. To overcome the difficulty referred to, this paper proposed IT2AFC+STFSMC to control the AMSS-PM and evaluated its control performance.

A. FEEDBACK LINEARIZATION OF THE INPUT-OUTPUT MAP
In practice, many nonlinear systems may not be represented in an equivalent linear system. The feedback linearization is a common approach used for transforming a nonlinear system into an equivalent linear system. A general nonlinear system is given by:ż where z = z 1 z 2 · · · z n T ∈ n is the state vector, v and o are the input and output of the system, respectively, andF andh are nonlinear functions.
Assumption 1: If the general nonlinear system has a relative degree r and the control signal v linear with respect to the o (r) , then an equivalent linear system can be found, as given by: wheref andḡ are unknown functions, andḡ = 0 for z in controllable region U c . In this paper, some assumptions are made before linearization: (1) the dynamic equation (12) satisfies the assumption 1; (2) the external loading and the friction that is generated during the motion of the piston rod in the shock absorber are refined as uncertain, which are neglected for the purposes of linearization. Clearly, Eq. (12) can be expressed a 5 th order general nonlinear system, as given by: where the state vector is defined as and the corresponding vector fields f (χ ,ū) and h(χ) are described as shown at the bottom of the next page: and in which f (χ ,ū) and h(χ) are unknown and smooth vector functions on the set ∈ R 5 . In order to derive the relative degree r with 0 ≤ r < 5 of the nonlinear system (15), the outputȳ is differentiated along the f until the outputū appears explicitly [27], such as ∂/∂ū(L i f h(χ)) = 0, i ∈ {0, 1, 2} , ∂/∂ū(L 3 f h(χ)) = 0. This relevels that the relative degree is r = 3 < n = 5, which means that the Eq. (15) can be transformed into a 3 th -order affine system with the system statex = [ȳẏÿ] T , as given by: where In Eq. (16), y is the displacement of the sprung mass,x is the state vector andū is the control input (voltage) from the PPR. It is worthy of note that both F(x) and G(x) are highly nonlinear and uncertain, so it is more difficult for the AMSS-PM to define the boundaries for the states and to determine accurate dynamic models. The interval type-2 FLC, which is a type of intelligent controller, gives satisfactory performance for systems that are uncertain and imprecise. In this paper, an additional compensator is necessary to attenuate the lump of disturbances. A self-tuning fuzzy sliding mode compensator is proposed in [8] that allows disturbance to be rejected. Hence, this study combines an intelligent interval type-2 adaptive fuzzy control (IT2AFC) and a self-tuning fuzzy sliding model compensation (STFSMC) deal with the nonlinearity and uncertainty that is inherent in the AMSS-PM.

B. DESCRIPTION OF THE IT2AFC-STFSMC
To extend the control problem to a general case, Eq. (16) can be represented in the form of Eq. (14), as given by: where d(x) is an external disturbance and the unmodeled friction force of the piston rod within the shock absorber, x = [ yẏ · · · y (n−1) ] T ∈ R n , u ∈ R, y ∈ R be the state vector, the control input and the system output, respectively. It is assumed that |d(x)| ≤ D for all states x, and F(x) and G(x) are partially unknown functions with uncertain timevarying parameters. The boundary conditions of G(x) are expressed as 0 functions. Without loss of generality, G(x) can be assumed to be strictly positive. The reference signals are defined as ] T , so the tracking error vector is expressed as: The sliding surface is: where c i is specified such that n i=1 c i λ i−1 is a Hurwitz polynomial and λ is a Laplace operator. Taking [28] as a reference, we have the ideal control law as: where η > 0 is a constant, s = [0, c 1 , c 2 , · · · , c n−2 ] T is the constant vector, and S (t) = S − sat(S/ ), for which ≥ 0 is the width of the boundary layer of the sliding surface S. The properties of the function S are described below [29]: The conditions on Property 1 and Property 2 allow the adaptation terminates as researching the boundary layer. Differentiating Eq. (19) gives: Substituting Eq. (20) into Eq. (21) gives: The convergence for e(t) = [e(t)ė(t) . . . e (n−1 )(t)] T in Eq. (22) can be achieved while η > 0. However, some variables in Eq. (16) maybe unknown or perturbed in AMSS-PM, and d(x) may not be measurable, so the implementation of the ideal control law u * is impossible for the AMSS-PM. To this regard, the proposed IT2AFC+STFSMC yields the control forceû fz to approximate the ideal control law and a compensator u comp (S) to compensate for the disturbance and the modeling error, which is:

C. DESIGN OF THE INTERVAL TYPE-2 ADAPTIVE FUZZY CONTROLLER (IT2AFC)
In this section, a single-input IT2AFC is designed to formulate the control lawû fz (S,α), as mentioned in Eq. (23). For the IT2AFC with M rules, the ith fuzzy rule is designed as: where S is the input variable,F i T 2 S is the interval type-2 fuzzy set, andα i T 2 fz is the interval type-2 singleton fuzzy set. The output of the IT2AFC calculated by the singleton fuzzification, the product inference and the center-average defuzzification gives: where y l and y r respectively represent the farthest left and the farthest right points of the interval type-2 set. In (25), the weight vectorα T = α 1 ,α 2 , · · · ,α M is used to estimate the optimal weight vector α * T = α * 1 , α * 2 , · · · , α * M and the parameterα * T is reasonably assumed to be bounded. The farthest left point of the interval type-2 set is defined as: represent the upper and lower degrees of the membership function, respectively.
The farthest right point of the interval type-2 set is defined as: whereα i r is farthest right point ofα i (s). The parameters L and R, respectively in (26) and (27), can be calculated by using type reduction [30]. The adaptive law for the IT2AFC is given as: where η 1 > 0 is the adaptive learning rate.

D. DESIGN OF THE SELF-TUNING FUZZY SLIDING MODE COMPENSATION (STFSMC)
The STFSMC uses the sliding surface S as the input and the sliding control law u fs as the output. Figure 3 shows The fuzzy rules are simply expressed as:

R l
: if S is f l s then u fs is f 8−l u , l = 1, . . . , 7. (30) The output of STFSMC is calculated by the singleton fuzzification, the max-min inference and the center-average defuzzification gives: where F(u fs ) = 7 l=1 f l s (S) ∧ f 8−l u (u fs ) . To avoid heavy computational cost in the general fuzzy control algorithm, the calculation of u fs can be simplified as two following cases: (C1) k = S/ < 0: (C2) k = S/ ≥ 0: where > 0 denotes the width of the boundary layer. Notice that u fs = −sgn(S) if |S| ≥ , and thereafter the STFSMC is designed as: where k c is a compensation gain that is: where M 0 (x) and M 2 (x) are specified variables;ρ is a compensation gain [31] that is represented as: in which η 2 > 0 is a learning rate that is greater than zero. Figure 4 shows the overall-system for the proposed IT2AFC-STFSMC. For the IT2AFC, the parameter g s is used to ensure that the sliding surface S is within the range of the fuzzy input and the gain factor g u is used to regulate the fuzzy outputû fz . Assumption 2: The internal dynamics of the nonlinear system are stable under the influence of the IT2AFC+STFSMC.
Theorem 1: Consider the AMSS-PM in the form of an affine system (17) that satisfies Assumptions 1 to 2, the control law is designed as: whereû fz (S,α) represents the IT2AFC that is shown in (25) and u comp (S) represents the STFSMC that is shown in (34),  with the adaptive laws (28) and (36), respectively. Then, (i) the system state x and the control law u are bounded and (ii) the tracking errors converge to 0 as t → ∞. The proof of Theorem 1 is given in Appendix A.

IV. TEST RIG DESIGN AND DEVELOPMENT
The designed test-rig for a quarter of vehicle model is shown in Fig. 5. This test-rig consists of three main units: the AMSS-PM unit, the RPG unit and the PBC unit. The following details the construction of these three units.

A. AMSS-PM UNIT
The main components of the AMSS-PM unit are constructed using two MacPherson struts and a PM. This unit also contains a sprung mass which includes a counterweight and an upper support frame to represent a car-body, as shown in Fig. 6. To measure the displacement and the acceleration of the sprung mass, a linear encoder and an accelerometer are installed on the upper support frame and a linear scale is installed in the plane of the under support frame to measure vertical displacement. The PM, in parallel with two  MacPherson struts, are installed between the vehicle body and the wheel-axle to generate control force to the suspension. A proportional pressure regulator (PPR) regulates air into the PM and the PM produces vertical movement to attenuate external vibration from irregular roads and to maintain the upper support frame in a stable position. With the voltage range from 0V to 10V of the PPR, the system initially sets 5V to maintain the PM in a half-stretched condition so that the PM can react to road variations immediately. Therefore, the zero variation for the displacement of the sprung mass is set to 5V the half-stretched condition.

B. PBC UNIT
The PPR is installed inside a control box in the PBC unit, as shown in Fig. 7. The PBC is implemented on a National Instruments (NI) cRIO-9074 integrated system with an industrial 400 MHz real-time processor, a 64GB memory, a NI-9263 D/A card, a NI-9215 A/D card and a NI-9411 encoder card. The NI-9215 A/D card receives data, including the displacement of the unsprung mass and under support frame, and the acceleration of the sprung mass, from the AMSS-PM unit and the RPG unit. The NI-9411 encoder card receives data for the displacement of the sprung mass and the NI-9263 D/A card sends control signals to the proportional directional control valve (PDCV), the inverter and the PPR. The PM, the pneumatic cylinder and the induction motor are driven according to these signals.

C. RPG UNIT
The RPG unit, as shown in Fig. 8, consists of a wheel, rollers, an induction motor and a pneumatic cylinder. The induction motor, which is controlled by a frequency converter, rotates the rollers to induce rotation of the wheel. The maximum speed that the wheel can reach is 35 km/hr. A PDCV regulates the flow of air into the cylinder to create vertical displacements to take account of road conditions. The maximum pressure that is provided by the air compressor is 6 bar.
To connect the base and the frame for the rollers, the bottom of the pneumatic cylinder is fixed to the base and the rollers bear firmly on the tire. A linear scale which is installed on the support frame is applied to measure the vertical variation in the road profile that is created by the pneumatic cylinder.

V. EXPERIMENTAL RESULTS
With the experimental setup as detailed in Section IV, the objective of the proposed AMSS-PM is to isolate the vibration generated by road irregularities and offer a ride comfort performance. Primarily, this paper aims to use the PM to provide extra compensatory force for the MacPherson strut and enhance its performances on vibration reduction. In this section, two phases are presented here to show that the proposed AMSS-PM has much improvement than the MacPherson strut in vibration reduction and ride comfort. First, we examine the performance of the vibration reduction regarding their displacement variation and acceleration variation in the time-domain on three different road profiles, which are a sine wave road profile, a rough concave-convex road profile, and a two-bump excitation road profile. Secondly, we evaluate the performance of the ride comfort due to their car-body acceleration in the frequency-domain. Please note that the sine-wave road profile (Experiment 1) has a constant 0.5Hz frequency so that its frequency analysis can be neglected. For these three experiments, the initial control parameters are chosen as c 1 = 5, c 2 = 1, = 6.5, g s = 0.4, g u = 0.3, η 1 = 0.05, η 2 = 0.01, and k c = 0.65. The membership functions for the controllers u fs andû fz are given in TABLE 2, where M (S) and M (u fs ) for u fs is the type-1 fuzzy sets, M (S) forû fz is the gaussian type-2 fuzzy set, and M (û fz ) forû fz is the singleton type-2 fuzzy set. For M (S), m 1 and m 2 respectively stand for the mean of the upper and lower membership functions, and δ is the variation of the membership functions.

Experiment 1: Vehicle Riding on a Sine Wave Road Profile
In the first experiment, a sinusoid with an amplitude of 15mm and a frequency of 0.5Hz is used to verify the performance on vibration reduction. The road profile in Experiment 1 is given by Z r3 (t) = 15 sin(πt) for t ∈ [0, 10) .
(38) VOLUME 8, 2020   As can be seen in Fig. 9, while traveling on the sinusoid road, the maximum displacement of the sprung mass with the AMSS-PM by using the IT2AFC-STFSMC strategy is less than 6mm (red dashed line), and it for the AMSS-PM with FC is less than 7mm, and it for the MacPherson struts is up to 18mm. The resulted RMS values for the displacement of the sprung mass are 2.57mm, 4.07mm and 11.45mm, respectively. Figure 10 shows the acceleration comparison of the sprung mass, and Figure 11 shows the actuating pressure comparison of the FC and IT2AFC-STFSMC when the quarter car moves along a sine wave road profile. The maximum acceleration for the sprung mass is bounded within 0.26g (g = 9.8m/s 2 ) for the AMSS-PM using the IT2AFC-STFSMC while the maximum acceleration is 0.64g for the MacPherson struts and it for the AMSS-PM using the FC is 0.21g. In this case, the IT2AFC-STFSMC provides the larger initial pressure than the FC for the AMSS-PM so that it has bigger maximum acceleration. The experimental results also show that the AMSS-PM with the IT2AFC-STFSMC strategy can effectively suppress the vibration of the sprung mass. The performance comparisons are shown in Table 5.

Experiment 2: Vehicle Riding on a Rough Concave-Convex Road Profile
This experiment verifies the stability and robustness of the AMSS-PM running on a rough concave-convex road profile. The road profile is composed of a bump and a hollow with a sinusoidal disturbance superimposed to simulate a rough road surface that can be represented by where d r1 (t) is the sinusoidal disturbance. To make the simulation more challenging, a disturbance signal d r1 (t) is assigned as d r1 (t) = 1.75 sin(2πt) + 0.7 sin(7.5πt) mm. Figure 12 shows the variation in the displacement (red dashed line) of the sprung mass when the quarter car moves along a rough concave-convex road. As can be seen, the maximum displacement of the sprung mass for the AMSS-PM with the IT2AFC-STFSMC and the FC is less than 4.5mm and 8mm. Using the same test-rig and conditions, the displacement of the sprung mass using the MacPherson struts follows the variation in the road profile. The maximum displacement is 24mm because the MacPherson struts cannot supply sufficient opposing force to suppress vibration. The root-mean-square (RMS) values for the displacement of   the sprung mass are 1.92mm, 2.85mm and 9.71mm for the AMSS-PM with and the FC, respectively. However, the RMS for the displacement of the sprung mass is 9.71mm for the MacPherson struts. Figure 13 shows the acceleration comparison of the sprung mass and Figure 14 shows the actuating pressure comparison of the FC and IT2AFC-STFSMC when the quarter car moves along a rough concave-convex road VOLUME 8, 2020  profile. The maximum value for the acceleration of the sprung mass is around 0.11g for the AMSS-PM with the IT2AFC-STFSMC, 0.24g for the AMSS-PM with the FC, and 0.43g for the MacPherson suspension. The performance comparison is shown in Table 6.
For the ride comfort evaluation, the frequency analysis regarding body acceleration is supplemented in accordance with this road profile. Firstly, the frequency response of sprung mass acceleration subject to a rough concave-convex road condition for three control approaches is illustrated in Fig. 15. Figures 15(a) and 15(c) respectively show the magnitude and the power spectral density (PSD) for the road vertical varation excited by the RPG. As shown in Figs. 15(b) and 15(d), vibration and ride comfort for both sprung mass natural frequency range (1-2Hz) and human sensitive frequency range (4-8Hz) can be improved by the proposed controller since the magnitudes are reduced in the frequency domain. Besides, the proposed controller also can exhibit better performance on the ride comfort than the original McPherson and the FC.

Experiment 3: Vehicle Riding on a Two-Bump Excitation Road Profile
The road profile in Experiment 3 consists of two bumps: a 30mm high bump and a 2cm high bump. The road profile is given by 34092 VOLUME 8, 2020 Figure 16 shows the displacement of the sprung mass when the quarter car travels on a twin-bump excitation road using the AMSS-PM and MacPherson struts. The maximum displacement of the sprung mass for the AMSS-PM using the IT2AFC-STFSMC strategy is less than 8mm, while the maximum displacement of the sprung mass for the MacPherson struts is 27mm and it for the AMSS-PM using FC is around 12mm. Their RMS values for the displacement of the sprung mass are 1.77mm, 3.37mm and 7.98mm, respectively. Figure 17 shows the acceleration comparison of the sprung mass and Figure 18 shows the actuating pressure comparison of the FC and IT2AFC-STFSMC when the quarter car moves along a twin-bump excitation road profile. The maximum acceleration for the sprung mass is bounded within 0.22g for the AMSS-PM using the IT2AFC-STFSMC while the maximum acceleration is 0.42g for the MacPherson struts and it for the AMSS-PM using FC is around 0.27g. The performance comparisons between the IT2AFC-STFSMC with Macpherson and FC are shown in Table 7.
For the ride comfort evaluation, the frequency response of sprung mass acceleration subject to a twin-bump road condition for different control approaches is illustrated in Fig. 19. Figure 19(a) and Figure 19(c) respectively show the magnitude and the PSD for the road vertical varation excited by the RPG. From the observation in Figs. 19(b) and 19(d), the frequency response and the PSD of the proposed controller can outperform the original MacPherson and the FC for both sprung mass natural frequency range (1-2Hz) and human sensitive frequency range (4-8Hz). With Fig. 16, these results clearly indicate that the improvement in ride comfort but with less vibtation can be achieved using the proposed controller.

VI. CONCLUSION
This study proposes a conceptual model of an ASS that is driven by a light and low-cost PM actuator with a high powerto-weight ratio, named as AMSS-PM. The AMSS-PM is implemented on a quarter car test rig and its feasibility in terms of the vibration isolation and improved ride comfort is demonstrated from the experiments. Since the PM is modeled as a pneumatic cylinder with a variable sectional area, this study firstly presents a mathematical model for a quarter car with the proposed AMSS-PM. To address the problem of highly nonlinearity and uncertainty of the suspension system dynamics, the IT2AFC-STFSMC is designed and its stability is proved with a Lyapunov stability analysis. The experimental results show that the proposed control strategy for AMSS-PM can significantly reduce the displacement and the acceleration of the sprung mass while facing different road surface variations. Concludingly, the developed AMSS-PM with the IT2AFC-STFSMC can substantially achieve improved performance of the MacPherson suspension with regard to advantageous vibration isolation and better ride comfort. The future work in this study will intend to explore further applications of this novel AMSS-PM mechanism on a half car or a full car suspension system.

APPENDIXES APPENDIX A
This Theorem can be proven by two steps. In Step 1, the system state x and the adaptive control u are bounded when the IT2AFC-STFSMC is used for the AMSS-PM. Next, Step 2 shows that the tracking error will converge to zero asymptotically.

APPENDIX B
According to [27], we can linearize a SISO system by differentiating its output. For convenience in analysis, we can initially neglect the frictional force F µ and bring back later as uncertainties. By applying the feedback linearization theory and usingẏ =ẋ 1 = x 2 , we can find the second derivative and the third derivative of x 1 , which arë So, we can findȳ where