Design of an Adaptive Fuzzy Observer-Based Fault Tolerant Controller for Pneumatic Active Suspension With Displacement Constraint

This paper proposes an adaptive fuzzy observer based fault tolerant controller for a pneumatic active suspension system considering unknown parameters, actuator failures, and displacement constraints. A pneumatic spring is used for a quarter car model to enhance the vibration attenuation performance. Since the pneumatic system contains uncertain nonlinearities, fuzzy logic systems are utilized to approximate unknown nonlinear functions of unmodeled dynamics and various masses of passengers. Besides, a nonlinear disturbance observer is proposed to estimate the effects of the actuator failures, approximation errors, and external disturbances. By utilizing the disturbance estimation and fuzzy approximation techniques, an adaptive fault tolerant control (FTC) is designed to enhance the output performance of the vehicle suspension. Meanwhile, the command filtered scheme is introduced to solve the explosion of complexity problem in the traditional backstepping approach by avoiding virtual controller derivatives. In contrast to previous results, the proposed control can handle the fault tolerant problem and ensure the tracking error of vertical displacement converges into a small-predefined boundary by introducing the prescribed performance function. Moreover, the stability of the closed-loop system is analyzed according to the Lyapunov theory. Finally, comparative simulation examples and experimental studies are performed on the active pneumatic suspension test bench to verify the feasibility of the proposed scheme.


I. INTRODUCTION
With the development of the automotive industry, the suspension system is the most important component of the vehicle chassis, which can improve passenger comfort and driving safety [1]. Therefore, the active suspension designs have attracted significant attention during the past decades by adopting a wide variety of actuators, involving electromagnetic [2] and hydraulic actuators [3]. To better vibration isolation, pneumatic actuators are extensively considered and applied both in automobile research and manufacturing due to high reliability and ease of installation features [4]. By regulating the inlet and exhaust airflow between the air bellow and the pressure source, the ride height and stiffness coefficient The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau . of the suspension system can be controlled simultaneously to enhance ride comfort [5]. However, the dynamic model of air spring contains lumped parametric uncertainties due to the thermodynamic characteristics while the pneumatic stiffness depends on the bellow rubber's behavior under external force [6]. Besides, there are still several challenges in designing the controller for pneumatic suspension as it requires controlling the air spring and proportional pressure valve simultaneously. In particular, it is very important to ensure the chassis stability under the presence of unknown masses of passengers and various road conditions.
To further enhance the suspension performance, many advanced controllers have been investigated to handle various problems, including adaptive control [7], optimal control [8], sliding mode control [9], [10], and backstepping control [11]. Nazemian and Masih-Tehrani [12] developed an optimized controller to maximize the dynamic performance and reduce the energy assumption of the pneumatic suspension. Some adaptive sliding mode controllers were proposed to address the unknown dynamics of the pneumatic suspension [13], [14]. Rui [15] designed an adaptive sliding mode control considering the nonlinear dynamical characteristic of the pneumatic system. To improve the ride height motion for pneumatic active suspension under the presence of parametric uncertainties and unmodeled dynamics, the backstepping technique has been applied in [16], [17]. Besides, the problem of hard constraints for ASSs was solved by the backstepping control schemes in [18]. Zhang et al. [19] designed the adaptive backstepping control to enhance the ride comfort by regulating the chassis displacement. To guarantee the tracking error and convergence rate of the system states under the maximum overshoot, a new control scheme combining PPF was introduced to improve the output constraints [20], [21]. However, in most previous works, the suspension systems were assumed that accurate mathematical models were established even though parametric uncertainties or external disturbances in physical systems always exist, which make adverse effects on the control objectives.
Since the neural networks control and fuzzy logic systems technique have proved the ability to approximate unknown functions [22], several intelligent controllers using these approximation methods have been developed for the ASSs [23], [24]. In particular, these intelligent control methods can be well combined with backstepping to effectively improve the control performance of suspension systems [25], [26]. Because the ASSs contain uncertain parameters, neural networks are employed to estimate these unknown functions [27]. Li et al. [28] proposed an adaptive event-triggered fuzzy controller which considered the nonlinear uncertainties for the ASSs. To reject the disturbances caused by actuator saturation, a disturbance observer based Takagi-Sugeno fuzzy control was designed for an active seat suspension to guarantee passenger comfort by Ning et al. [29]. Although these aforementioned approaches can satisfy the requirement of suspension, the explosion of complexity problem induced by repetitive virtual controller derivatives can limit the scope of the traditional backstepping application [30], [31]. Fortunately, dynamic surface control can solve this limitation by introducing a first-order filter to estimate these derivatives. However, the errors caused by the filters are not considered in this technique, which could reduce the control performance definitely [3]. As an alternative, a command filtered control can solve this problem [32]. By adopting error compensation mechanisms, CFC can diminish the errors of the command filters to improve the control efficiency [33]. Nonetheless, only a few studies applied CFC and FLS for the pneumatic ASSs.
On the other hand, most previous results assumed that the suspension systems were in fault-free operating conditions and that ideal actuator behavior was implemented for the control designs [34], [35]. However, various unpredictable actuator failures often occur in the control system, which may degrade the output suspension performance and even lead to controller instability [36], [37]. To enhance the control efficiency and guarantee system reliability, the issue of fault tolerant should be investigated for the suspension control designs [38]. Hence, some researchers have proposed many adaptive controllers to solve the problems of sensor and actuator failures for many years [39]- [41]. A fault tolerant scheme is an effective approach to solve the problem of actuator failures and it can ensure the desired performance by combing the proper controller with an approximation technique [42], [43]. Jing and Yang [44] investigated an adaptive fuzzy observer-based fault tolerant tracking control for uncertain nonlinear systems which considered the unmatched external disturbances and actuator failure problems. Wang et al. [45] suggested the output feedback fault tolerant control to improve the passenger comfort for the ASSs. Nonetheless, these studies investigated adaptive fault tolerant schemes based on traditional backstepping technology, which cannot accommodate the explosion of complexity problem [46], [47]. Furthermore, there are a few adaptive fault tolerant controllers based on CFC technique for pneumatic ASSs with non-ideal actuator and parametric uncertainties, which make the motivation for this research.
Basing on the above discussions, we investigate an adaptive fault tolerant tracking control for the pneumatic ASSs using the fuzzy nonlinear disturbance observer method in this paper. Because the active suspension with the pneumatic actuator contains unknown parameters and nonlinearities, it is not easy to establish an accurate system model. Moreover, the suspension system is affected by external disturbances due to various masses of passengers. Although some advanced controllers have been applied to handle these unknown parameters, the actuator failure problems of air spring still need more attention to enhance the control performance. Thus, the FLSs are applied to approximate these unknown nonlinear functions and improve the requirement of modeling precision. Then, a nonlinear disturbance observer is introduced by constructing FLSs to better deal with the actuator failures, approximation errors, and external disturbances. Due to the limitations of the mechanical design, the control objectives of ride comfort and suspension deflection for the ASSs conflict with each other [48]. Besides, some studies can guarantee the control objectives and keep the vertical displacement does not violate the output constraint, but they did not consider the actuator failures. Unlike some previous fault tolerant controllers, the proposed control not only guarantees the tracking performance by introducing the PPF constraint but also eliminates the explosion of complexity problem with the CFC technique. The tracking errors can be guaranteed to converge a small neighborhood of the origin, and the stability of the closed-loop system is proven according to the Lyapunov theory. The main contributions of this paper are summarized as follows 1. A quarter car model is designed with a pneumatic spring to investigate the behavior of the actual suspension system.
2. An adaptive fuzzy observer-based fault tolerant control is designed for the pneumatic suspension system which considers unknown parameters and actuator failures.
3. CFC combined with PPF can solve the explosion of complexity problem in the traditional backstepping technique and guarantee the tracking error of sprung mass displacement does not violate the constraint boundaries.
The rest of this article is arranged as follows. The problem formulation and fault description are displayed in Section II. The design of an adaptive fuzzy observer-based fault tolerant scheme and system stability analysis are shown in Section III. Furthermore, Section IV verifies the efficiency of the developed scheme by the simulation results while the experimental studies are executed by Section V. Finally, some conclusions were given in Section VI.

A. PNEUMATIC QUARTER CAR SUSPENSION MODEL
The active suspension using a pneumatic spring is shown in Fig. 1. The total weight of the passengers and chassis is denoted by the sprung mass m s while the assembly of the vehicle wheel is represented by the unsprung mass m us . The sprung mass and unsprung mass positions are performed by z s and z us , respectively. The external disturbance caused by the road profile z r will generate continuous excitations to the passengers. To get ride comfort, active suspension systems are employed to disperse this vibration. The mechanical equations of the quarter car model can be obtained in the form m szs +F sp (z s , z us , t)+F dp (ż s ,ż us , t)−F p = 0 m uszus −F sp (z s , z us , t)−F dp (ż s ,ż us , t) +F st (z us , z r , t)+F dt (ż us ,ż r , t)+F p = 0 (1) These forces are generated by the stiffness and damping of mechanical structure and pneumatic spring actuator, which can be calculated by: F sp (z s , z u , t) = (k s +k p )(z s − z us ), F dp (ż s ,ż us , t) = c d (ż s −ż us ), F st (z us , z r , t) = k st (z us − z r ), F dt (ż us ,ż r , t) = c dt (ż us −ż r ), in which k s , c d denote the stiffness and damping coefficient of active suspension; k st , c dt are the stiffness and damping coefficient of the tire, and k p represents the stiffness coefficient of pneumatic spring.
To describe the road holding condition, the expression for the tire force can be given by [49] where g is the acceleration of gravity.
Determine the system state variables as: x 1 = z s , x 2 =ż s , x 3 = z us , x 4 =ż us , we can obtain the state space form of the active suspension system as followṡ Compared with the other suspension systems, a pneumatic spring is used to achieve the suspension modulation and the active force can be obtained by the following equation F p = A as P as (4) where P as denotes the internal pressure and A as is the working area of the pneumatic spring.
To archive the suspension performance, the control design is proposed to regulate the volume of airflow through the pneumatic spring. Thus, the thermodynamic theory is applied to describe the dynamic model of pneumatic spring [13] where R denotes the perfect gas constant, κ is the polytropic index, T represents the air temperature, q as is the areanormalized mass flow rate, and s 0 is the orifice open area of the proportional pressure valve. The working volume of air bellow V as depends on the relative position between the unsprung mass and sprung mass where z as0 denotes the initial stroke of the pneumatic spring. Assumption 1: The dynamic characteristic of the high response proportional valve is assumed to be linear in this study. Therefore, the spool position is proportional to the applied signal, and the orifice open area s 0 of the pneumatic valve can be indicated by where π sv is the coefficient factor and u is the control signal of the supply voltage. Substituting (6) and (7) into (5), the dynamic equation of pneumatic spring can be illustrated aṡ Consider x 5 = (A as m s )P as , we can obtain (8) bẏ To enhance the tracking control, it is necessary to consider the full dynamic behavior of the pneumatic system, especially the nonlinear pneumatic stiffness k p of the air bellow. Although some simulation methods can determine this parameter based on the thermodynamic theory [50], its application to the controller design is not easy because it depends on various working conditions and external disturbances. Thus, this stiffness coefficient is considered as an uncertain parameter in this study and we can specify the unknown function from (3) as On the other hand, the air pressure of pneumatic spring is a high nonlinearity parameter which is affected by the unmodeled dynamics and the behavior of the twisted cord rubber material under the effect of external forces. Therefore, the dynamic model (9) should consider the unmodeled term of internal pressure when applying for the control design process.
Control Objectives: The active suspension system is designed to meet three objectives 1) Passenger comfort: The proposed control is applied to stabilize chassis displacement, thereby enhancing the comfort of the passenger. 2) Driving safety: This objective guarantees that the tires are always in contact with the road surface. To satisfy this condition, the relative tire fore (RTF) must be maintained to be less than 1.
1) Handling stability: The suspension space must be smaller than the limitations of the mechanical architecture. For this purpose, the relative suspension deflection (RSD) is defined by where z R is the maximum value of the suspension displacement. Remark 1: Some controllers are designed to satisfy the three above requirements, but these objectives conflict with each other because increasing the passenger comfort will require a larger oscillation. Thus, the proposed control in this study not only solves this problem by providing the PPF constraint but also considers the actuator failures of the pneumatic system.

B. FAULT PROBLEM MODEL AND PRELIMINARIES
It is well-known that actuator faults could limit the system performance of the pneumatic active suspension. To deal with this problem, the actuator failures are considered in this article, which can be modeled as [51]. Two types of lockin-place and loss of effectiveness model are examined in the controller.
1) Lock-in-place model: This case shows the fact that the actuator gets stuck at a time t and it can be described by where is a constant denotes the float fault of the actuator. 2) Loss of effectiveness model: where u f represents the actual input signal and λ ∈ (0, 1] denotes the ratio of the actuator that remains effective after losing some effectiveness. For example, -λ = 0.6 means that the actual coefficient actuator is 60% and the actuator losses 40%. When -λ = 1, the actuator operates with the normal condition (no faults occur). Basing on two cases of actuator failures, we can describe the general form of fault as follows Remark 2: Note with the lock-in-place model, it means that λ = 0 and = 0 while the loss of effectiveness model is represented by -λ ∈ (0, 1]. Generally, considering the dynamic model of pneumatic spring (9) with unmodeled parameters, external disturbance, unknown function (10), and actuator fault (15), we can write the full state space form of the pneumatic suspension model aṡ where p (t) is the time-varying modeling error of air bellow pressure.
To guarantee comfort for the passenger, the controller will be designed to dissipate the external excitation into the chassis. For this purpose, we focus on the dynamic equations of VOLUME 9, 2021 the sprung mass as followṡ where Based on the dynamic equation (17), we can see that f 2 and f 3 are unknown smooth functions because x 3 , x 4 are not considered in the control design. Besides, the damping characteristic of the pneumatic spring cannot be accurately simulated, and it is usually neglected in the suspension models. These problems will be solved by the control design based on the following preliminaries.
Assumption 2: Under the limitation of mechanical structure and physical performance, the sprung mass is assumed to be bounded by m s min < m s < m s max , where m s max and m s min denote the upper and lower bound, respectively.
Remark 3: The sprung mass m s is an unknown variable depending on the various masses of passengers. Thus, the unknown function g 3 contains the parametric uncertainties and it can be considered as time-varying control gains.
FLSs have been shown to be a good approximation technique for the unknown continuous functions. The FLSs are composed of a fuzzy rule base, a fuzzifier, a fuzzy inference engine, and a defuzzifier. The knowledge base constitutes a series of fuzzy If-Then rules as follows where X = [x 1 , x 2 , . . . , x n ] T and y are the FLSs input and output; H l i and T l are fuzzy sets associated with fuzzy membership functions θ H l i (x i ) and θ T l (y); and N is the number of fuzzy rules [52].
Through singleton fuzzifier, center average defuzzification, and product inference, the FLSs can be expressed as Then, the fuzzy basic function can be designed by Determine S(X ) = [s 1 (X ), s 2 (X ), . . . , s N (X )] T and W T = [ȳ 1 ,ȳ 2 , . . . ,ȳ N ] = [w 1 , w 2 , . . . , w N ], one can obtain the fuzzy logic (19) as follows Lemma 1 [53]: Define a continuous vector function f (X ) on a compact set , for any given positive constant η > 0, there exist fuzzy logic systems W T S(X ) that satisfy where η > 0 is an error value of the fuzzy system approximation to an unknown function, W is the ideal FLSs weight matrix, and S (X ) is bounded by S (X ) ≤ ι.
In the following, the FLSs are used to approximate the unknown functions f (X ) with X ∈ Lemma 2 [32]: The command filter is employed to eliminate the derivative of the virtual control of traditional backstepping as followṡ where ρ m and τ m are the control parameters and α i−1 are the virtual control signals. The output of each filter is chosen by x c i = θ i1 andẋ c i =θ i1 . Let the input signal α i−1 and their derivatives are bounded and satisfied |α i−1 | ≤ 1 and |α i−1 | ≤ 2 in a finite time, there exist τ m ∈ (0, 1] and ρ m > 0 so that the following inequality holds where i > 0, i = 1, 2, 3 are the positive constants. Remark 4: The command filter control (24) is designed to compute the intermediate control signal x ic andẋ ic without a differentiator. Hence, the explosion of complexity problem in the basic backstepping technique is solved.
Lemma 3 [32]: To decline the effect of the command filter errors (θ i1 −α i−1 ), the compensation mechanisms ϕ i are employed at each step of the control design process.
where the initial condition is selected ϕ i (0) = 0 for t ∈ [0, T 1 ], and k i are design parameters. The compensation mechanisms are bounded by invoking [54] where k 0 = 1 2 min (k i )

A. ADAPTIVE FUZZY OBSERVER COMMAND FILTERED CONTROL WITH PRESCRIBED PERFORMANCE
This section proposes an adaptive fuzzy observer fault tolerant control scheme based on the command filtered backstepping technique. The FLSs are employed to approximate the unknown functions while the nonlinear disturbance observer is used to handle the effects of the actuator failures, approximation errors, and external disturbances. The PPF constraint is used to ensure the tracking error of vertical displacement does not violate the boundary. Finally, the Lyapunov theorem is applied to analyze the stability of the proposed control scheme.
Step 1: In this step, the tracking error constraint of vertical displacement is guaranteed by the PPF. First, we define the tracking error of sprung mass displacement by where x d is the reference trajectory. Definition 2 [20]: The prescribed performance is defined by a positive smooth function δ (t) where δ 0 indicates the initial value, δ ∞ denotes the ultimate error, and φ > 0 is the convergence rate. The initial conditions are chosen to satisfy lim t→0 The PPF (29) is used to establish the predefined boundaries which retain the tracking error e 1 by the following inequality where ϑ,θ > 0 are the positive design parameters. Remark 5: It can be seen from (29) and (30), the lower bound of tracking error is determined by −ϑδ(0) whileθδ(0) stands for the upper bound. Thus, the steady-state performance of tracking error can be satisfied by choosing the appropriate control parameters δ 0 , δ ∞ , φ, ϑ,θ.
To combine the PPF into the proposed control, the constrained transformation technique is applied to convert the prescribed performance boundary into an equality form. Hence, one can use a smooth and strictly increasing function G(z 1 ) which is given by [20].
Furthermore, the function G(z 1 ) satisfies: −ϑ < G(z 1 ) < ϑ and lim the inequality condition (30) can be converted by the following form Then, one can obtain the inverse transfer function z 1 as follows Set χ = e 1 δ (t), the transform function z 1 can be written by Lemma 4 [55]: According to the above analysis, the transform function z 1 of the system state is transferred by the smooth function G (z 1 ) and the stability of the tracking error e 1 can be ensured within the predefined boundaries (30).
Step 2: Choose the virtual control α 1 We obtain the time derivative of z 1 from (34) as followṡ Choose a candidate Lyapunov function V 1 = 1 2 z 2 1 . Thus, the time derivative of V 1 is calculateḋ Based on the traditional backstepping technique in this step, we obtainV 1 as followṡ Select the virtual control α 1 by Using (38), we can write (37) as followṡ Step 3: Choose the virtual control α 2 In this step, the command filtered backstepping algorithm is employed to design the control scheme, and the tracking error of x 2 is defined by where x c 2 denotes the output signal of the command filter through the virtual controller α 1 .
We can define the compensated tracking error The error compensation is designed by (26) where x c 3 is the output signal of the command filter through the virtual controller α 2 , which is determined later. Therefore, we can express the time derivative of z 2 by using (17)ż Based on Lemma 1, the unknown function f 2 is approximated by FLSs, we have: f 2 = W T 2 S 2 (X 2 )+η 2 (X 2 ), where X 2 = [x 1 , x 2 ] T . Then, we can write (43) as followṡ Define the approximate error η 2 (X 2 ) and uncertain parameter d 2 (t) by using the unknown function D 2 (t) = η 2 (X 2 )+ d 2 (t), we can arrange (44) agaiṅ Substituting (42) into (45), we can obtaiṅ Choose the virtual control α 2 whereD 2 is the estimation of the disturbance D 2 (t) andŴ 2 is the estimation of the ideal weight vector W 2 . Besides, the adaptive law is proposed aṡ where β 2 > 0 and ς 2 > 0 are the design parameters. Substitute (47) into (46), we havė whereD 2 = D 2 −D 2 is the error of disturbance observer and W 2 = W 2 −Ŵ 2 is the error of weight estimation. From (49), we obtain To estimate the disturbance D 2 (t), the fuzzy nonlinear disturbance observer is designed bŷ Using (45) and (51), we obtaiṅ Then, the time derivate of disturbance error can be derived as followsḊ The candidate Lyapunov function V 2 is selected by Taking the time derivative of V 2 using (39), we havė Using (50) and (53), we can write (55) aṡ Substituting (48) into (56), we havė Step 4: Design the fault tolerant control signal u f Similar to step 3, the tracking error of x 5 can be defined by Then we can calculate the compensated tracking error From (26), the compensating signals is useḋ Using (17) and (58), we can obtain the time derivative of z 3 as followsż Therefore, we can choose the desired control u d Hence, we can write (62) as follows Then, the adaptive fault tolerant control with disturbance observer is proposed by whereD 3 the estimation of the disturbance D 3 (t). The adaptive law is designed aṡ where β 3 > 0 and ς 3 > 0 are the design parameters. Substituting (64) into (61), we obtaiṅ In this step, the fuzzy nonlinear disturbance observer is designed for FTC as followŝ where υ 3 is control parameter, µ 3 and 3 are the auxiliary variable and the intermedial variable, which are defined by where b 3 > 0 is the design parameter. The intermedial variable λ 3 is proposed bẏ And define the adaptive lawŴ Ḟ where β F > 0 and ς F > 0 are control parameters.
To design the disturbance observer for FTC, we set where υ 3 > 0 is the design parameter. By employing FLSs for (70) and (73), we havė Basing on Young's inequality theorem for nonnegative real numbers a and b, we have ab ≤ a 2 (2ρ)+(b 2 ρ) 2, where ρ > 0 is the adjustable parameter. Hence, we can obtain Similarity, we can obtain the time derivative ofD 3 from (67), (69), and (74) as followṡ Therefore, we can obtain the disturbance erroṙ Proof: Choose the candidate Lyapunov function V 3 and V We can derive the time derivative of V from (81) as followṡ Applying (57), (66), (76), and (79) into (82), one obtains the derivative of V as followṡ Applying Young's inequality and (65), (71), we have Using (65) and (71), we receive So, we can write (83) as followṡ We also apply Young's inequality to prove the following inequalities Thus, we can arrange (84) as followṡ By choosing the corresponding control parameters to satisfy the system conditions, we can determine the following elements as follows And we can simplify (85) as followṡ Multiplying (88) by e t on both sides and then integrating, we receive Therefore, we can conclude the following conditions according to (81), It can be seen that transformation errors z i , i = 1, 2, 3 and compensate signals ϕ i , i = 1, 2, 3 are bounded, hereby the tracking errors e 1 , e 2 , e 3 are also bounded. Therefore, by selecting the control parameters and PPF constraints, the stability of the closed-loop system is guaranteed.
Remark 7: The external disturbances, unmodeled dynamics, and unknown coefficient of actuator failures are compensated by fuzzy logic systems and nonlinear disturbance observers. Here, the propped control is designed by the combination of the command filter technique and prescribed performance constraint theory. It not only reduces the computational complexity but also enhances the suspension performance in the presence of unknown parameters and actuator fault.

B. HANDLING STABILITY AND DRIVING SAFETY ANALYSIS
Based on the above results, the first objective of passenger comfort has been satisfied by the proposed control. Besides, two objectives of handling stability and driving safety can be ensured by choosing suitable design parameters that will be analyzed in this section. For these purposes, we concentrate on analyzing the dynamic equations of unsprung mass (16). Besides, the tracking errors e i , i = 1, 2, 3 are assumed to be bounded according to (91). Applying the FLSs for f 2 = W T 2 S 2 (X 2 )+η 2 (X 2 ), we obtain (16) as followṡ According to (91), we can see that M is bounded because the tracking errors z 1 , z 2 , and z 3 are bounded. There exists a constantM so that M ≤M is hold.
The candidate Lyapunov function is chosen by where P is a positive definite symmetric matrix. Thus, we can obtain the time derivative of V Ḋ Rewrite (94) using (92), we havė There exists a positive definite symmetric matrix Q > 0 that satisfies the equation C T P+PC = −Q. Besides, we can write the form of 2X T PDY and 2X T PY 0 according to Young's inequality theorem as follows where ρ i > 0, i = 1, 2 are adjustable parameters. Substituting (96) into (95), we havė where λ max and λ min are the maximal and minimal eigenvalues of the matrix. We can select the appropriate matrix P, Q to meet the following inequalities Based on (98), we can determine two parameters ϒ and such that Hence, we can rewrite the inequality (97) as followṡ Multiplying (101) by e t on both sides and integrating, we can write According to (93), the system states (16) are bounded by Substituting (103) into the handing stability (12), we can get |z s −z us | ≤ |x 1 |+|x 3 | ≤θδ (0) From (104), the handling stability condition can be guaranteed by choosing control parameters ρ 1 , ρ 2 , P, and the approximate PPF constraintθ, ϑ, δ 0 so that |z s −z us | ≤ z R .
Besides, the tire forces F st and F dt can be expressed by Substituting (105) into (2), we can obtain relative tire force by According to (106), the relative tire force condition (11) can be ensured by selecting the design parameters ρ 1 , ρ 2 , P which satisfy the inequality |F st +F dt | ≤ (m s +m us ) g.
Remark 8: From the above analysis, the suspension objectives of handling stability and driving safety are guaranteed by setting initial conditions and approximate design parameters. Therefore, the driving condition of the pneumatic suspension system is ensured by the mechanical structure.

A. SIMULATION DEFINITION
To verify the effectiveness of the developed method, the comparative simulations are executed in comparison with passive suspension and traditional backstepping. The results of RSD and RTF parameters are compared to analyze the handling stability and driving safety of active suspension. The main parameters of pneumatic suspension are chosen as in Table 1.

B. SIMULATION RESULTS
The simulation results of sprung mass displacement and acceleration, relative suspension deflection, relative tire VOLUME 9, 2021 force, and control signals of passive, traditional backstepping, and proposed control are shown in Fig. 3 -6. The proposed scheme can improve passenger comfort by reducing the sprung mass displacement and acceleration as shown in Fig. 2 -3. With the PPF constraint, the tracking error of vertical displacement is kept inside the predefined boundary. Although traditional backstepping can reduce the amplitude of sprung mass displacement, the output performance cannot satisfy the requirement of output constraint. Furthermore, the objectives of analyzing the handling stability and driving safety are enhanced by the proposed control because the RSD and RTF parameters are guaranteed in Fig. 4 -5. The control signal of the proposed scheme with the CFC technique is shown in Fig. 6. Although the proposed method requires a large control signal at the beginning to guarantee the speed convergence of tracking error, its required signal will be smaller as soon as the constraint boundary is satisfied.

A. EXPERIMENTAL PLATFORM
The quarter car test bench for active suspension with a pneumatic spring is designed to verify the effectiveness of the proposed controller, which is displayed in Fig. 7. Due to the limitation of mechanical structure, the main parameters are chosen in accordance with the platform design. There are three separate plates that are used to represent the chassis, tire, and road profile, and they are connected by mechanical and pneumatic springs. The chassis is denoted by the top plate and it is supported by a pneumatic spring and two mechanical    springs. The middle plate is used to simulate the motion of tire and structure assembly. The road profile is simulated by the motion of the bottom plate, which is driven by the servo motor   to create various road conditions. The stiffness coefficient of a tire is represented by two mechanical springs. Besides, three cable sensors are used to measure the position of the three plates. A piezoelectric accelerometer is employed to determine the acceleration of the chassis. A PCI card is used to record and send the control signal between the computer and the proportional pneumatic valve.

B. EXPERIMENTAL RESULTS
The experiment results of the pneumatic active suspension carried on the test bench are displayed in Fig. 8 -12. We can   see that the experimental results also demonstrate the effectiveness of the proposed control in comparison with the backstepping approach. From Fig. 8, the vertical displacement of sprung mass can be restrained within the PPF constraint of the proposed scheme. Since the active suspension enhances the ride comfort resulting in a larger suspension stroke, the RSD value of the proposed control is greater than the passive suspension and traditional backstepping. Besides, the RSD and RTF parameters are also maintained to be less than 1 to achieve the goals of handling stability and driving safety of the pneumatic active suspension. Although there are different values between the simulation and experimental results, both are able to demonstrate the efficiency of the proposed controller. Similar to the simulation results, the control signal of the proposed controller is smaller than that of the traditional backstepping technique.

VI. CONCLUSION
This article investigates an adaptive fuzzy observer fault tolerant scheme for the active suspension which considers the uncertain parameters and the failure problem of pneumatic spring. The adaptive fuzzy observer based command filtered is designed to solve the problem of actuator failures where both of loss of effectiveness model and lock-in-place model are considered. The PPF constraint is employed to restrain the sprung mass movement to get the passenger comfort while the CFC technique is proposed to avoid the explosion of complexity problem of the traditional backstepping approach. By using the fuzzy logic systems method to estimate unknown nonlinear functions and then incorporate them into the nonlinear disturbance observer, the external disturbances and unknown actuator failure coefficients are estimated. Moreover, the effectiveness of the designed controller has been verified by both simulation and experiment, which considers control objectives of the passenger comfort, driving safety, and handling stability for the active suspension system.
Further research includes the implementation of the proposed fault tolerant control design to stabilize the full-car pneumatic active suspension.