Adaptive Neural Command Filtered Control for Pneumatic Active Suspension With Prescribed Performance and Input Saturation

In this paper, an adaptive neural command filtered backstepping scheme is proposed for the pneumatic active suspension with the vertical displacement constraint of sprung mass and actuator saturation. A quarter car model with a pneumatic spring is first fabricated on the basis of thermodynamic theory to describe the dynamic characteristics. To overcome the lumped unknown nonlinearities and enhance the requirement of modeling precision, the radial basis function neural networks (RBFNNs) are proposed to approximate unknown continuous functions caused by the uncertain body mass and other factors of pneumatic spring. To solve the explosion of complexity problem in the traditional backstepping designs, a proposed command filter control is applied by using the Levant differentiators which approach the derivative of the virtual control signals. Nussbaum gain technique is then incorporated into the controller to avoid the problem of the completely unknown control gain and control directions of a pneumatic actuator. In addition, the prescribed performance function (PPF) is suggested to guarantee that the tracking error of the sprung mass displacement does not violate the constraint boundaries. Based on the command filtered backstepping control with PPF, the Lyapunov theorem is then applied to indicate the system stability analysis. Finally, the comparative simulation examples for the pneumatic suspension are given to verify the effectiveness and reliability of the proposed control.


I. INTRODUCTION
The pneumatic suspension has been widely used in the automotive industry to improve passenger comfort and vehicle handling stability [1], [2]. Compared with the various actuators such as hydraulic [3] or electromagnetic [4], pneumatic actuators are low cost, clean, and high power-to-weight ratio characteristics [5], [6]. Pneumatic springs can provide flexible stiffness and generate the control forces according to various uncertain masses of passengers by controlling the internal pressure [7]. However, the high nonlinearity is one of the drawbacks of pneumatic systems, which will make it The associate editor coordinating the review of this manuscript and approving it for publication was Juntao Fei . difficult and complicated to design the suspension model and control scheme [8]. Besides, it not easy to maintain chassis stability under various loads of passengers due to the presence of unknown parameters in the pneumatic servo system [9].
To overcome the above limitations, many control strategies have been widely used to improve vehicle performance such as optimal control [10], [11], sliding mode control [2], [12], and model predictive control [13], [14]. Although these controllers can improve the suspension performance, they may be sensitive to external disturbances because of the fixed control parameters. To solve the problem of height tracking of pneumatic suspension, the backstepping control was proposed and addressed the parametric uncertainties and unmodeled dynamics [9]. However, a common disadvantage in the traditional backstepping design process is the explosion of complexity caused by its virtual controller derivatives, which increases computational complexity [15], [16]. Recently, a dynamic surface control (DSC) method has been developed to address this problem by including a first-order filter in each control design step to approximate these derivatives [17], [18]. But the DSC technique does not consider the errors arising from the first-order filters, which can reduce system control efficiency [19]. To solve the same inherent problem of traditional backstepping design, Farrell et al. [20] proposed a command filtered control technique. By using a command filter to approximate the differential coefficient of the virtual control signal at every step of the control design, CFC can obtain better system tracking performance [21]- [23]. Furthermore, the compensating signals are proposed to reject the errors caused by the command filters which can solve the limitation of the dynamics surface approach [24], [25].
Generally, it is well known that the function approximation based intelligent design technique has been shown to be a powerful method for dealing with unknown nonlinearities and uncertainties [26], [27]. In particular, neural networks can provide an effective tool to approximate the unknown functions or parametric uncertainties in the pneumatic systems [28], [29]. Bao et al. [30] designed a fuzzy adaptive sliding mode control to enhance passenger comfort and vehicle controllability of the pneumatic active suspension. However, the authors did not consider the robust control system in the presence of unmodeled dynamics, and the transient tracking performance cannot be quantitatively guaranteed in previous designs [31]. Although the neural networks can provide a good approximation ability for unknown continuous functions, the traditional backstepping requires that the repeated differentiation of virtual input must be resolved at each step of the design process [32]. Consequently, the drawback of complexity explosion cannot be handled, and this also limits the applications of traditional backstepping. In this paper, adaptive neural networks-based command filtered backstepping technique has been proposed to improve the performance of the pneumatic active suspension.
Significantly, most of the ASSs did not consider the input saturation problem during the control design process; however, the control performance of pneumatic active suspension can be seriously restricted. In order to improve the control efficiency, the effect of actuator saturation needs to be properly regarded in the control design procedure [33]. Although the input saturation of nonlinear systems can be addressed with an adaptive NNs controller [34], it will be more difficult when control gains are unknown time-varying nonlinearities because of the singularity problems [35]. Besides, the control directions are very important in nonlinear system design because they are not easily detected from the prior knowledge of the signs of the parameters, which makes the control design more complicated and challenging. The previous controllers of ASSs are not designed to accommodate unknown control directions for pneumatic systems. As we have known, the Nussbaum gain technique has been incorporated in the controller to handle the problem of unspecified control coefficient [36], [37]. The main point of Nussbaum's approach is that using the switching functions which can obtain the signs of the control directions [38], [39]. This is suitable to overcome the disadvantage of pneumatic active suspension which involves unknown nonlinear functions and depends on many physical parameters.
Recently, a new control scheme with output constraints called prescribed performance control (PPC) was introduced by Bechlioulis to guarantee the convergence of system outputs, maximum overshoot, and steady-state error into an arbitrarily small predefined region [40]. PPC has been used in many control engineering applications requiring output constraints [41], [42]. Zhang et al. [43] proposed a novel proportional-integral approximation-free control by using PPFs for nonlinear robotic systems without employing any function approximation. To stabilize the vertical and pitch displacements of active suspensions with parametric uncertainties, an adaptive control with PPF constraints was proposed by Jing Na in [44]. Liu et al. [45] designed an adaptive control scheme to ensure the convergence of the tracking error and maximum overshoot of the suspension system with the PPF constraints and actuator failure. However, these results assume that all the system states are available or directly measurable which is rarely satisfied in practical applications [46], and those external disturbances have a negative impact on control performance [47]. Besides, Shi et al. [48] proposed the output-feedback control for time-delay systems with PPF constraints. In response to the explosion of the complex problem, a dynamic surface control technique has been proposed by Zhai et al. [49]. Nonetheless, the PPF constraints with CFC are not considered for pneumatic active suspension.
Based on the aforementioned discussion, we propose a new active suspension system using a pneumatic spring in this research. Although pneumatic actuators can meet the requirements of flexible suspension, controlling the stability of the sprung mass within a small predefined boundary remains a challenge because of their parametric uncertainties and external disturbances. As we know that PPF constraints can guarantee the vertical displacement for the active suspension, but the unknown parameters may cause the problem of singularity and instability. Besides, actuator saturation usually leads to the performance degradation of the pneumatic actuator, so the system design becomes more difficult without knowledge of control directions. In this study, a novel control is established for ASSs with PPF constraint while the result did not require prior knowledge of control gains. By using neural networks, the unknown parameters are compensated to guarantee the performance of the pneumatic suspension. Furthermore, a command filtered control has been studied to solve the explosion of complexity in traditional backstepping controllers. Command filtered control combined with PPF is proposed not only to handle the explosion of complexity problem in the traditional backstepping techniques but also to guarantee the tracking error of sprung mass displacement does not violate the constraint boundaries. The Levant differentiators are introduced in the CFC control scheme to compute the derivative approximation of the virtual control signals and the compensating signals are then designed to eliminate the errors caused by the command filters. To design the CFC technique with input saturation, the Gaussian error function is employed to express the saturation nonlinearity as a continuous differentiable form. In addition, the Nussbaum gain function is applied to solve the difficulty of the control design due to the uncertain control directions of active suspension. Based on the Lyapunov stability analysis, the control scheme can ensure that all the signals are semi-global uniformly ultimately bounded. The main contributions of this paper can be summarized as follows 1. Adaptive neural command filtered backstepping control is proposed for the pneumatic active suspension which considers the problem of actuator saturation and unknown control direction.
2. RBFNNs are developed to approximate the parametric uncertainties of the pneumatic spring and the unknown various loads of passengers in the nonlinear ASSs.
The rest of this paper is organized as follows. System description of the pneumatic quarter car model is given in Section II. Adaptive neural command filtered control and the system stability are presented in Section III. Besides, to demonstrate the effectiveness of the proposed control, the comparative simulation results are provided in section IV. Finally, Section V gives some conclusions.

A. NONLINEAR QUARTER CAR MODEL
The quarter car model with pneumatic spring is designed as shown in Fig. 1. In this design, the chassis is affected by external disturbances that cause continuous excitations to the passengers. The suspension system is designed to dissipate this vibration for the passenger comfort. The dynamic equations of active suspension are demonstrated as where m s and m u denote sprung mass and unsprung mass, respectively. The unsprung mass represents an assembly of the vehicle axis and wheel while the sprung mass is the total weight of the chassis and passengers. These above forces are created by the stiffness of pneumatic spring, mechanical springs, damper, and tire, which can be expressed as F s (z s , z u , t) = (k s + k a ) (z s − z u ) , where z s and z u determine the position of the sprung mass and unsprung mass, z r presents the road profile; k s , c a are the stiffness coefficient and damping coefficient of active suspension; and k t , c t represent the stiffness and the damping coefficient of the tire. The tire force which depends on the road holding condition is expressed by the equation (2), where g denotes the gravitational acceleration [50].
Define the systems state variables: x 1 = z s , x 2 =ż s , x 3 = z u , x 4 =ż u , the state space form of active suspension can be written as followṡ To provide the active force for the active suspension system, an air below is installed between the sprung and unsprung masses. The active fore of air bellow F u is calculated by the following formula F u = A as P as (4) where A as represents the working area and P as is the internal pressure of the air bellow. Because the aerodynamic properties of the air bellow are caused by the action of the twisted-wire rubber material under the effect of external forces, there is a challenge to describe the absolute mathematical model. Therefore, the nonlinear dynamic model can be investigated by [2] P as = γ RT v as a 0 q as − P as A as ( where R is the ideal gas constant, γ represents the polytropic index, T denotes the air temperature, q as is the areanormalized mass flow rate, and a 0 denotes the orifice open area of the solenoid valve.

VOLUME 9, 2021
The air bellow volume v as depends on the relative motion between the sprung mass and unsprung mass where z as0 denotes the initial altitude of the air bellow. Assumption 1: The spool position is proportional to the signal applied to the control valve. Hence, the valve dynamics can be ignored in the system model, and the orifice open area a 0 of the servo valve can be described by where σ sv is the coefficient factor of the servo valve and u is the control signal of supply voltage.
Using (6) and (7), we can write the dynamic model of air bellow (5) as followṡ Define the new state variable x 5 = (A as /m s )P as , we obtaiṅ The nonlinear pneumatic stiffness k a exists in the air bellow, depending on the internal pressure and the axial displacements. Some previous studies have examined this stiffness based on the thermodynamic theory [51] but it cannot be applied to the control design process because of different working conditions. In this study, the air spring stiffness is considered as an uncertain parameter and then compensated by NNs. Therefore, we can define the unknown continuous function as follows Besides, the pressure of air bellow is a high nonlinearity model as it is affected by external disturbances, payload variations, and unmodeled dynamics. Thus, the dynamic model (9) should consider the parameter deviations which are lumped to the unmodeled terms. The state-space form of the quarter car ASSs must be extended using the pneumatic stiffness and unmodeled parameters of air spring as followṡ where p (t) is the time-varying modeling error of air bellow pressure. Assumption 2: d (t) and p (t) are the unknown bounded time-varying disturbances. Therefore, there are two constants d (t) andp (t) satisfying |d (t)| ≤d (t) and |p (t)| ≤p (t).
Assumption 3: Due to the limitations of the mechanical structure and physical performance, the mass of the vehicle body is limited by m s min < m s < m s max , where m s min and m s max are the lower and upper limits.
To ensure the ride comfort, the air bellow is used to create the active force that isolates the external vibrations in the suspension design. However, because of the limitations of the pneumatic actuator, the dissipation of vibration will be considered during the control design process. Thus, the problem of input saturation is solved for pneumatic active suspension in this study. The control signal u that is the output of the saturation actuator can be assumed by where v is the actual input signal and u B is the known bound of u.
It can be seen that the relationship between u and v in equation (12) is a saturation nonlinearity which has sharp corners as |v| = u B . Thus, it cannot be directly applied to the backstepping control. To overcome this limitation, a Gaussian error function is employed to express the saturation nonlinearity that can be used for the control design.
Definition 1 [34]: Gauss error function erf (x) is described by a nonelementary function of sigmoid shape Remark 1: Error function erf (x) is a continuously differentiable function, it has real value and has no singularity (except that at infinity).
To facilitate the control design later, the output signal u is defined by [52] where the smooth function h v is used to approximate the saturation nonlinearity and d (v) is bounded by |d(v)| ≤ . Control objectives: The control scheme is designed to meet three requirements of active suspension 1. Ride comfort: The controller is proposed to dissipate continuous excitations and guarantee the vertical displacement of sprung mass within the bounded constraints. 2. Handling stability: The oscillation space cannot exceed the limited range of suspension displacement.
To meet this requirement, the relative suspension deflection (RSD) must be ensured to be less than 1.
where z R which is called rattle space is defined as the distance between the tire and chassis at rest position. 3. Road holding: The dynamic tire load must be limited to ensure that the tire is always kept in contact with the road profile. It means that the relative tire fore (RTF) is kept smaller than 1.
Remark 2: Although some advance controllers have been proposed for the active suspension to provide passenger comfort by limiting the sprung mass displacement, the objective of handling stability cannot be guaranteed simultaneously because they conflict with each other. The proposed control in this study can improve all three objectives of active suspension and ensure that the tracking error of sprung mass displacement does not violate the PPF constraints.

C. NOTATIONS
In this study, a norm of vector x is defined by x = √ x T x. The estimate of generic constant quantity θ is indicated byθ . Moreover, the estimation error and its time derivative areθ = θ −θ andθ = −θ , respectively. Some symbols and their descriptions are given in Table 1.

III. ADAPTIVE NEURAL COMMAND FILTERED CONTROL WITH PRESCRIBED PERFORMANCE A. SOME DEFINITIONS AND LEMMAS
To guarantee the vertical displacement of sprung mass within boundary constraints, the proposed control design will focus on the dynamic equations of the sprung mass as followṡ By setting }σ sv q as , we can rewrite (17) as followsẋ In the above equation (18), it can be seen that f 2 and f 3 are unknown smooth functions because the system state x 4 is not considered in the control design. In real suspension, the sprung mass m s is an unknown parameter, depending on different passenger masses. Besides, the damping properties of air bellow cannot be accurately described and are often ignored in the active suspension.
Remark 3: The nonlinear smooth function g 3 contains the unknown parameter m s and depends on the relative values of x 1 and x 3 . Thus, the control directions are specified as the sign of variable control gain g 3 .
Assumption 4: The unknown function g 3 is bounded and there is a known positive constant satisfying |g 3 | ≤ υ.
Lemma 1 [26]: The RBFNNs can approximate any unknown continuous function f (X ) where W = [w 1 , w 2 , . . . , w n ] T ∈ R n denote the weight vector, S (X ) = [s 1 (X ) , s 2 (X ) , . . . , s n (X )] T is the Gaussian function vector, η (X ) illustrates the approximation error, n > 1 is the node number of RBFNNs, and X represents the input vector.
The Gaussian functions are described by where ξ i and σ i are the center and width of the Gaussian functions.
Lemma 3 [54]: Let V and ζ be smooth functions defined on [0, t f ) with V (t) ≥ 0. If the following inequality holds on t ∈ [0, t f ) where c 0 , c 1 > 0 are the constant parameters, then V (t), ζ (t), and t 0 g (τ ) N (ζ )ζ e c 1 τ dτ must be bounded on [0, t f ). Lemma 4 [55]: The command filters of CFC are defined based on Levant differentiator as followṡ where x i,c = L i1 andẋ i,c =L i1 are the output of each filter used to define tracking error e i = x i − x ic . Each command filter is designed to compute x ic andẋ ic without differentiation. The design parameters of differentiators are chosen by R 1 and R 2 while α i−1 are the virtual control at each step. If the input signal α i−1 and their derivatives are bounded and satisfied |α i−1 | ≤ µ 1 and |α i−1 | ≤ µ 2 in a finite time with µ 1 > 0, µ 2 > 0, the following inequality holds where µ > 0 is a positive constant. The finite-time error convergence characteristics of the Levant's differentiators were proved in [55] and [56]. Remark 4: The Levant differentiators can improve the limitations of traditional CFC by providing a precise filter of the input signals to obtain the differential signals and guarantee the convergence of the filter in a finite-time.
Lemma 5 [20]: It can be seen that the tracking errors caused by command filters can lead to an increase in system errors. To eliminate the effect of the errors caused by command filters, the compensating signals ω i for command filtered control are selected bẏ where ω i (0) = 0 for t ∈ [0, T 1 ], and k i are designed positive constants. According to [21], the compensating signals are bounded by where To guarantee the stability of the chassis does not violate the boundaries in vertical displacement, a PPF constraint is introduced into the control design. First, let the tracking error of the system state x 1 be defined as where x d is the desired position trajectory. Definition 2 [40]: The PPF constraint is chosen by a positive smooth function as follows where ϕ > 0 denotes the convergence rate, ρ 0 is the initial value, and ρ ∞ indicates the allowable steady-state error, which must be chosen to satisfy the initial conditions lim t→0 ρ (t) = ρ 0 > 0, lim t→∞ ρ (t) = ρ ∞ > 0, and ρ 0 > ρ ∞ .
From (28), the tracking error of sprung mass displacement can be guaranteed by the following inequality −κρ (t) < e 1 <κρ (t) , t > 0 (29) where κ,κ > 0 are the positive parameters chosen by the designers. Remark 5: Based on (28) and (29), the lower bound of the undershoot is defined by −κρ (0) whileκρ (0) serves as the upper bound of the maximum overshoot. By choosing the appropriate design parameters κ,κ, ρ 0 , ρ ∞ , ϕ, the steadystate performance of the system can be guaranteed.
To design the control scheme with PPF constraint, an output transformation is used to construct the prescribed performance boundary into an equality form. For this purpose, a smooth and strictly increasing function S (z 1 ) is introduced as follows [40].
Furthermore, the function S (z 1 ) satisfies Then, the performance condition (29) can be transferred as follows Because S (z 1 ) is strictly monotonically increasing and PPF constraint was chosen to satisfy ρ (t) > ρ ∞ > 0, the inverse transfer function z 1 can be expressed by Set β = e 1 /ρ (t), we can write the transform function of z 1 as follows Lemma 6 [57]: Based on the above analysis, the system state (17) is transformed by the smooth function S (z 1 ) of equation (30) and the stability of the signal e 1 can guarantee the regulation of x 1 according to the prescribed performance constraint (29).

C. ADAPTIVE NEURAL COMMAND FILTERED BACKSTEPPING CONTROL
In this section, an adaptive neural command filtered scheme is designed based on a modified backstepping algorithm. The unknown parameters are approximated by the NNs while the PPF is introduced to ensure the tracking error of sprung mass displacement within the boundary constraints. Besides, the stability of the proposed control scheme is demonstrated by the Lyapunov theorem. The controller diagram can be described in Fig. 2. Step 1: Identify the inverse transfer function z 1 with PPF in section III.B Step 2: Design the virtual control α 1 The derivative of z 1 is given by using (33) as followṡ where ς = 1 2ρ 1 β+κ − 1 β−κ A candidate Lyapunov function is chosen as V 1 = (1/2) z 2 1 . Then, the time derivative of V 1 is obtained bẏ In this step, the control algorithm is based on the basis of the backstepping algorithm, we havė The virtual control α 1 is selected as follows Substituting (37) into (36) leads tȯ Step 3: Design the virtual control α 2 The tracking error of x 2 is designed based on command filtered backstepping theory where x 2c is the output signal of the command filter while the virtual controller α 1 goes through the filter. In this step, the compensated tracking error is redefined based on the theory of command filtered backstepping as follows The error compensation is proposed based on (25) Choose the candidate Lyapunov function V 2 Taking the derivative of V 2 , we havė Using (39), (40), and (41), we can write (43) as followṡ Choose the virtual control α 2 where r 2 is a positive constant,θ denotes adaptive law which is determined later. Then, the equation (45) will be written aṡ According to (47), the virtual control signal is difficult to be designed because of the unknown function f 2 . In this step, f 2 will be approximated by NNs according to Lemma 1.
It can be seen from (17), the unknown function f (X 2 ) is related to these variables x 1 and x 2 , so we choose X 2 = [x 1 , x 2 ] T , and η 2 is the approximation error satisfying |η 2 | < λ 2 . Basing on Young's inequality theorem and Assumption 2, we have Rewrite equation (47) using (49) and (50), we havė Step 4: Design the control signal u In this step, the tracking error of x 5 is considered by where x 3c is the output signal of the command filter. The compensated tracking error is defined as follows The error compensation is suggested by (25) Choose the Lyapunov function V 3 as follows Then, the derivative of V 3 can be transformed intȯ Similar to step 3, the unknown function f 3 (X 3 ) = (f 3 + g 2 z 2 −ẋ 3c ) can be approximated by NNs.
where X 3 = [x 1 , x 2 , x 5 ] T . Let η 3 be the approximation error satisfying |η 3 | < λ 3 , and apply Young's inequality theorem and Assumption 2, we obtain where r 3 is the positive control parameter. Using the form of real control law (14) and substituting equations (58) and (59) into (56), we can write the derivative of V 3 as followṡ Because the control coefficient is unknown, the Nussbaum function is introduced to handle the problem of unknown control directions. Hence, the actual input signal v can be designed by where N (ζ ) is Nussbaum-type function and the smooth function ζ is chosen aṡ Apply Young's inequality theorem, we can write Then, (60) can be rewritten using (61), (62), and (63) as followṡ Consider the Lyapunov function V where m is the positive parameter andθ = θ −θ is the estimation error. By choosing θ = max W 2 2 , W 3 2 for the time derivative of V , we havė The adaptive law is designed as followṡ where q is the design parameter. Theorem: Consider the pneumatic active suspension system (17) satisfying Assumptions 1 -4, the virtual controllers (37), (46), actual control (61), and adaptation law (67) are designed. The proposed control can ensure that all system signals are semi-globally uniformly ultimately bounded. This leads to the estimation errors converging to a small set around zero asymptotically. It means that the sprung mass displacement is guaranteed within the PPF constraint.
Then according to (65), the following conditions are satisfied Moreover, the tracking errors e 1 , e 2 , e 3 are also bounded because z 1 , z 2 , z 3 and ω i are bounded. Then, the system states x i , i = 1, 2, 3 are also bounded by the selection of design parameters. Thus, the proposed control can guarantee the dynamic behavior of the system under the influence of the unknown nonlinear dynamics and various loads of passengers.
Remark 7: To obtain the objectives of active suspension, the control parameters should be chosen to meet the balance between the demand of output performance and the actual operating conditions of the system. The PPF constraint is designed to satisfy the initial condition −κρ (t) < x 1 < κρ (t) by selecting the boundary parameters κ,κ, ρ 0 . A good control performance can be achieved with large ϕ and small ρ ∞ but lead to large control actions. Besides, the positive parameters R 1 and R 2 of Levant differentiators are designed to obtain the differential signals and guarantee the convergence of the filter in a finite-time. The large control gains k i could enhance the convergence rate of the system but they require more control actions and lead to high chattering.

D. HANDLING STABILITY AND ROAD HOLDING ANALYSIS
In the previous analysis, the proposed control scheme has been designed to demonstrate the stability of the suspension system. By introducing a PPF constraint, the sprung mass displacement is guaranteed within the prescribed performance boundaries, which means the first objective of ride comfort is achieved. Other suspension requirements for handling stability and road holding are also analyzed in this section by choosing the appropriate control parameters.
Firstly, we consider the dynamic equations of the unsprung mass of the system (11). Based on the above results, the tracking errors e i , i = 1, 2, 3 are proved to be bounded. Besides, the RBFNNs are used to approximate the unknown continuous function f 2 (X 2 ). Then, substituting (48) into (11), we obtainẊ Because the tracking errors e i are bounded, ϒ is also bounded and there is a constantῩ such that ϒ ≤Ῡ.
Choose a Lyapunov function as follows where P is a positive definite symmetric matrix. Then, the time derivative of V z is written aṡ Substituting (75) into (77), we obtaiṅ There is a positive definite symmetric matrix Q > 0 so that the Lyapunov equation M T P+PM = −Q is satisfied. Applying Young's inequality theorem, we can rewrite 2X T PNY and 2X T PY 0 as follows where γ 1 > 0 and γ 2 > 0 are the positive constants. According to (79), we can write (78) as followṡ where λ max and λ min denote the maximal and minimal eigenvalues of the matrix Choose the positive constants γ 1 , γ 2 , and appropriate matrix P, Q, we obtain From (81), there are two constants χ and ψ satisfying Substituting (82) and (83) into (80), the time derivative of V z is described byV Then, integrating both sides of equation (84), we obtain Based on the above results, the system x 3 and x 4 are satisfied From (15), the handling stability condition can be written as follows Therefore, the inequality (15) is satisfied by selecting the PPF parametersκ, κ, ρ (0) and the positive parameters γ 1 , γ 2 , P such that |z s − z u | ≤ z R .
Similarly, the tire forces F st and F dt are computed by Substituting (88) and (89) into (2), we can get the relative tire force condition (16) as From (90), the RTF constraints can be obtained by selecting the positive parameters γ 1 , γ 2 , and matrix P to ensure Based on the above analysis, the requirements of handling stability and road holding are satisfied by the selection of initial conditions and control parameters.
Remark 8: The proposed control can ensure not only the transient response of vertical displacement of the sprung mass but also the mechanical structure and safety condition of the pneumatic suspension. Furthermore, by choosing the appropriate PPF constraints and control design parameters, the proposed control can improve the performance requirements of pneumatic active suspension.

A. SIMULATION DESCRIPTION
In this section, the numerical simulation examples for pneumatic active suspension are provided to demonstrate the effectiveness of the proposed method compared with passive suspension, traditional backstepping, CFC, and PPF controllers. To evaluate the ride comfort of active suspension, in addition to reducing the sprung mass displacement, the human body's sensitivity to acceleration should be considered during the control design process. According to ISO 2361 criteria, humans are sensitive to vertical vibration in the frequency range of 4 -8 Hz, and the active suspension systems must be guaranteed to a minimum in this domain. Therefore, the root mean square (RMS) values of sprung mass acceleration are examined with a filter is proposed in [58]. Besides, the objectives of handling stability and road holding are also investigated in this study by considering two parameters RSD and RTF. The main parameters of pneumatic active suspension are listed in Table 2. The simulation examples are demonstrated by the sin road profile with amplitude 0.02 m and frequency 1 Hz as z r = 0.02 sin (2π t). The initial values of the system states are set by x 1 (0) = 0.05 (m), x 2 (0) = x 3 (0) = x 4 (0) = 0 (m), x 5 (0) = 0.5 × 10 5 (Pa). The PPF constraint is defined by ρ 0 = 0.058, ρ ∞ = 0.0029, ϕ = 1.5 and design parameters κ = 0.98,κ = 0.98. To investigate the comparative results, the control parameters are given in Table 3.

B. SIMULATION RESULTS
The comparative simulation results of sprung mass acceleration and displacement, relative suspension deflection, relative tire force, and control signals of passive, traditional backstepping, CFC, PPF, and proposed control with sin road profile are provided in Figs. 3-7. The passenger comfort, driving safety, and handling stability are strongly improved   with the proposed control because the responses of acceleration and deflection, RSD, and RTF are all guaranteed. The proposed control can enhance the ride comfort compared with the other methods because the sprung mass displacement is ensured inside the boundary constraints as shown in Fig. 3. In particular, the proposed control can obtain better regulation performance and ensure the convergence of the tracking error does not violate the maximum overshoot as shown in Fig. 4. By introducing the PPF constraint, the time of zero convergence for the error signal e 1 can be achieved faster and around t = 1.3 (s). Although CFC can reduce the sprung mass displacement compared with passive suspension, there  are some peak values due to external disturbance that can affect the passenger comfort. Besides, the PPF control can guarantee the tracking error of vertical displacement within the boundaries but it cannot converge to zero because of parametric uncertainties. As can be seen in Fig. 4, the traditional backstepping cannot fulfill the convergence time requirement of the prescribed performance constraint. In addition, the proposed control can provide the magnitude of the RSD smaller than other methods as shown in Fig. 6, and this RSD is also guaranteed to be less than 1. Due to the influence of uncertain parameters in the dynamic system, the RSD of passive suspension is the biggest value, and there are some maximum peak points. Furthermore, the traditional backstepping cannot provide a good performance of RSD because it is affected by unknown parameters. By keeping the sprung mass vibration under the PPF boundaries, the proposed control scheme can not only provide passenger comfort but also guarantee the magnitude of RSD and RTF within the limit values. From  Fig. 7, the dynamic stroke constraints are guaranteed within the limits to ensure the stability of the chassis. Moreover, the RTF value of the proposed control is also smaller than CFC, PPF, and traditional backstepping designs.
The comparative simulation results of control signals are shown in Fig. 8. The objectives of suspension can be guaranteed in the presence of pneumatic actuator saturation with the proposed control. The control signals of PPF and traditional backstepping controllers are larger and more chattering  than the proposed control because of external disturbances. The CFC scheme can avoid the chattering problem of traditional backstepping control. Compared with the basic control designs, the proposed control scheme with NNs can overcome the unknown parameters to provide good performance for pneumatic active suspension. Besides, the simulation results of Nussbaum function signals ζ and N (ζ ) are shown in Fig. 9. It can be seen clearly that the Nussbaum gain N (ζ ) moves in to correct direction to ensure the tracking performance.

V. CONCLUSION
This paper presents an adaptive neural command filtered control with prescribed performance and input saturation for the pneumatic active suspension system. The proposed control scheme can not only retain sprung mass vertical displacement within the prescribed performance boundaries to get the ride comfort but also guarantee handling stability and road holding. To approximate the unknown continuous functions of the pneumatic suspension system with an air spring actuator, the RBFNNs are developed in the control design. In addition, the Gaussian error function is employed to characterize the saturation nonlinearity as a continuous differentiable form, and the Nussbaum gain function is then applied to scope with unknown control directions of pneumatic active suspension. A command filtered backstepping control has been employed to solve the explosion of complexity in traditional backstepping designs. Finally, the effectiveness of the proposed control is verified by the simulation examples which indicate that the controller design can be more efficient than other adaptive controllers. Therefore, this approach can be a promising method for the automotive industry.