Adaptive fuzzy fault-tolerant control for active seat suspension systems with full-state constraints

In this paper, an adaptive fuzzy fault-tolerant control (FTC) scheme is proposed for active seat suspension systems. The addressed system contains electromagnetic actuator faults. In addition, the constraints of the displacements of the car seat, active suspension, and wheel, their vertical vibration speeds and current intensity are included. In the control design, since the systems have dynamic characteristics of complexities and spring non-linearities, the fuzzy logic systems are utilized to approximate the unknown nonlinear dynamics. By the aid of Barrier Lyapunov functions and based on the adaptive backstepping recursive design algorithm, a novel adaptive fuzzy FTC scheme is formulated. When the electromagnetic actuator fails, the developed control scheme can guarantee that all the vertical vibration states are stable. Eventually, the effectiveness of the presented control method is offered to illustrate by a random road surface.


Introduction
With the increasing demand for automobiles in society, comprehensive performance improvement has become one of the hot issues in current automotive research.The suspension systems, as a necessary part of the carchassis, which can effectively improve the overall safety and stability of the vehicle.Therefore, it is really important to promote the quality of the suspension system.There are some considerable works that have been published (see Hrovata, 1997;Lin & Lian, 2011).According to different control methods, vehicle suspension systems can be divided into three types: passive suspension (Tamboli & Joshi, 1999), semi-active suspension (Morato et al., 2019;Pang et al., 2018) and active suspension (Li et al., 2018;Zhang et al., 2021).Although the passive suspension system and the semi-active suspension system are simple and cannot require external energy input, the vibration performance of the shock absorbers is poor, it is difficult to adapt to the complex random road surfaces.Subsequently, scholars proposed the concept of an active suspension system (Cao et al., 2008;Hu et al., 2016).The active suspension system can collect vehicle information from relevant sensors and respond autonomously to random roads through its energy, which can reduce vibration and improve vehicle safety and stability (Li et al., 2018;Li et al., 2021;Zhang et al., 2021).Therefore, compared with the passive suspension system and the semi-active CONTACT Yongming Li l_y_m_2004@163.comsuspension system, the control effect of the active suspension system is optimal.
Compared with the traditional suspension system, since the active seat suspension systems have better versatility and integrity, which can also more truly reflect the vertical performance of the vehicle and more directly reflect the comfort of passengers.Thus, the research of active seat suspension systems has attracted considerable attention of many scholars (see Sun et al., 2011;Wei et al., 2020;Zhao et al., 2010).The authors in Zhao et al. (2010) designed a novel state-feedback controller for vehicle active seat suspension systems to handle actuator saturation and time-varying input delay.The authors in Wei et al. (2020) proposed a H∞ robust control strategy for an automotive active seat suspension system with sampled measurements.Subsequently, the authors in Sun et al. (2011) presented a H∞ control method for an active seat suspension system in the limited frequency domain to match the human body characteristics.
Although many studies have been proposed to improve the performance of active seat suspension systems, to the best of our knowledge, there are no results on adaptive fuzzy fault-tolerant control (FTC) for active seat suspension systems with full-state constraints.In general, there are actuator failures for many practical physical systems, which will degrade the performance and affect the stability of the controlled systems, or even lead to catastrophic accidents (Deng & Yang, 2019;Li, 2019;Liang et al., 2020;Zhang et al., 2019).The authors in Deng and Yang (2019) addressed the problem of cooperative output regulation for linear multi-agent systems with actuator failures.The authors in Li (2019) solved the finite-time control problem for uncertain nonlinear systems with unknown actuator failures, where the total number of allowable failures is infinite.The authors in Liang et al. (2020) investigated the cooperative FTC problem for stochastic nonlinear systems with actuator failure and input saturation.For a class of switched nonlinear systems with actuator faults, the authors in Zhang et al. (2019) focused on the static output feedback control problem and designed a reliable SOF controller to protect against actuator faults.Therefore, to compensate for the failure of the active seat suspension systems, it is of great significance to study the failure of the electromagnetic actuator.In addition, the states of many physical systems need to be constrained to a certain range during operation, which comes from the operational specifications and safety considerations.In recent years, using BLF methods to solve the full-state constraint problems has attracted extensive attention (see Edalati et al., 2018;He et al., 2017;Min et al., 2020).The authors in He et al. (2017) studied the control problem of an uncertain robotic manipulator with n-degrees freedom, which is constrained by time-varying outputs.The authors in Edalati et al. (2018) investigated the adaptive fuzzy dynamic surface control strategy for uncertain strict-feedback nonlinear systems with asymmetric time-varying state constraints.The authors in Min et al. (2020) investigated an adaptive fuzzy inverse optimal control problem for a class of vehicle active suspension systems with the constraints of the displacements and their vertical vibration speeds of the sprung and unsprung masses.Therefore, as an inevitable existence in the active seat suspension systems with full-state constraints, fault has always been a problem to be overcome to ensure the safety and comfort of passengers.
Inspired by the above considerations, this paper proposes an adaptive fuzzy FTC method for active seat suspension systems with full-state constraints.Different from the existing results, the main contributions of this paper are as follows: (1) This paper proposes an adaptive fuzzy FTC method for active seat suspension systems with full-state constraints.Although the previous control schemes (Min et al., 2020) also addressed the control problem for active suspension systems with full-state constraints, the actuator faults are not solved.(2) When designing a fault-tolerant controller, this paper takes the seat suspension system as an integral structure and the electromagnetic actuator is selected.Note that (Zhang et al., 2021) also investigated active seat suspension systems, but they mainly studied the stability of car seats.In addition, although Zhang et al. (2021) and Liu et al. (2016) addressed the FTC problem, but they all ignored the electromagnetic actuator, and the internal characteristics of the actuator are not considered.

Electromagnetic actuator
The simplified circuit diagram of the electromagnetic actuator is shown in Figure 1.where U S is the voltage, i is the current flowing into the electromagnetic actuator, F U D is the back electromotive force, which is produced by U D .L and R are the self-inductance and resistance of the electromagnetic actuator, respectively.The relations from Figure 1 can be expressed as where T m = k T i is the output torque of the permanent magnet motor, k T is the equivalent torque.P h is the guide in the electromagnetic actuator.Since k T and P h are the known constants, then A = 2π k T /P h is a known constant and F U D = Ai.

Quarter active seat suspension system model
The quarter active seat suspension system model is shown in Figure 2. According to Newton's second law, the dynamics differential equations of the active seat suspension system can be given as follows where M DS , M CS and M US represent the masses of the car seat, the active suspension, and the wheel.D P , D S and D W denote the displacements of the car seat, the active suspension, and the wheel.F 1 , F 2 and F 3 are elastic forces generated by the stiffness coefficients of the car seat, active suspension and wheel, respectively.D 1 and D 2 are the damping forces, which are generated by the damping coefficients of the car seat and active suspension.The dynamic output forces produced by the springs, dampers and wheel, which can be expressed as where k d and k t are the stiffness coefficients of the car seat and wheel.k a1 and k a2 are the linear and cubic terms of the active suspension stiffness coefficient.c d is the damping coefficient of the car seat, c a1 and c a2 are the linear and cubic terms of the active suspension damping coefficient.r is the road excitation.
Define the following state variables as x 1 = D P , x 2 = ḊP , x 3 = D S , x 4 = ḊS , x 5 = D W , x 6 = ḊW and x 7 = i.The quarter active seat suspension system is converted into the following form: Remark 2.1: In the actual situation, the complex road surfaces can cause the electromagnetic actuator to fail in the active seat suspension systems.The addressed actuator faults in the study are as follows: where β ∈ (0, 1] is the loss of effectiveness factor, u e is the actual control input, -- λ(t) is the bias fault, which is assumed that | -- λ(t)| ≤ − λ.Note that (5) implies the following two situations: (1) If β = 1 and -- λ(t) = 0, then u(t) = u e (t), there is faultfree. (2 , there is a partial loss of effectiveness fault.
Our control objective is to propose an adaptive fuzzy FTC scheme, which can ensure the displacements of car seat, active suspension, and wheel, their vertical vibration speeds and current intensity in the active seat suspension systems are stable.At the same time, the vertical vibration states are required to be bounded regardless of electromagnetic actuator faults.

Fuzzy logic systems
In this study, the unknown nonlinear is contained in the active seat suspension systems, we employ the FLSs to approximate the unknown nonlinear.
Define the optimal parameter vector θ * as follows: Then one has: where ε is the fuzzy minimum approximation error.

Adaptive fuzzy FTC design
In this section, an adaptive fuzzy FTC strategy for an active seat suspension system will be proposed by utilizing the backstepping control design technique and basing on barrier Lyapunov functions.Define the change of coordinates as follows: where α t−1 is the virtual control function of the active seat suspension system.
Step 1: Consider the following BLF candidate as where k bi , i = 1, 2, 3, 4, 5, 6, 7 is a positive constant.Under the backstepping structure, the virtual controller α 1 is designed as where c 1 > 0 is a positive design parameter.
Step 2: From ( 4) and ( 12), the time derivative of z 2 can be given as Consider the following BLF candidate as where γ 2 > 0 is the design positive parameter.
is the estimation error, and ˆ i is the estimation of * i .In the same way, θ2 = θ * 2 − θ2 is the estimation error, θ2 is the estimation of θ * 2 .
The time derivative of V 2 is where Since the approximation capability of the FLSs, the unknown function H 2 (x 1 , x 2 , x 4 ) can be approximated as where Based on the backstepping recursive design algorithm, the virtual controller α 2 and the parameter adaptation law ˙ 2 are designed as where c 2 > 0, σ 2 > 0 and τ are the positive design parameters.s2 = [x 1 , x 2 ] T .
Step 3: From (12), we can know z 3 = x 3 − α 2 .Construct the following BLF candidate as Similar to the virtual controller design in Step 1, α 3 is given by where c 3 > 0 is a positive design parameter.
Step 4: According to (4) and ( 12), differentiating z Construct the following BLF candidate as The time derivative of V 4 along with ( 23) is where Based on the approximation capability of the FLSs, we can assume that where In view of the virtual controller design in Step 2, the virtual controller α 4 and the parameter adaptation law ˙ 4 are given by where c 4 > 0 and σ 4 > 0 are the positive design parameters s4 = [x 1 , x 2 , x 3 , x 4 ] T .
Consider the following BLF candidate as Then, the virtual control controller α 5 is designed as where c 5 > 0 is a positive design parameter.
Step 6: According to Step 2 and Step 4, differentiating Consider the following BLF candidate as The time derivative of V 6 along with ( 31) is where The unknown function H 6 (x) can be approximated as where |ε 6 (x)| ≤ ε * 6 , ε * 6 is a positive constant.The virtual controller α 6 and the parameter adaptation law ˙ 6 are designed as where c 6 > 0 and σ 6 > 0 are the positive design parameters.s6 = [x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ] T .
Step 7: From ( 4) and ( 12), the derivative of z 7 = x 7 − α 6 with respect to time is given by Inspired by (Li (2019), there are the following definitions as s = inf  The time derivative of V 7 along with ( 37) is where H 7 (x) = R 4 (x 7 ) − α6 , χ is an intermediate control law, which is used to get the actual control input.
The intermediate control law can be designed as where c 7 > 0 and τ 1 > 0 are known constants.Similar to the above steps, the unknown function H 7 (x) can be approximated as where |ε 7 (x 7 )| ≤ ε * 7 , ε * 7 is a positive constant.Under the backstepping structure, the actual controller u e , the parameter adaptation laws ˙ 7 , ℘ and ζ are designed as where σ 7 > 0, ς 1 > 0 and are the positive design parameters.

Stability analysis
In this part, the stability analysis and proof of the adaptive fuzzy FTC method for active seat suspension systems will be given.

Proof:
From (11), the time derivative of z 1 can be given as The time derivative of V 1 along with ( 46) is Considering the virtual control controller ( 14), the time derivative of the Lyapunov function ( 47) satisfies Substituting ( 18) into ( 17), one has According to Lemma 2.2, one gets: where τ is the design parameter, s2 = [x 1 , x 2 ] T .
Based on ( 50) and ( 51), the time derivative of the Lyapunov function ( 49) holds By utilizing the virtual controller ( 19) and the parameter adaptation law (20), we can obtain From Lemma 2.2, one has Substituting ( 54) into ( 53) results in where The time derivative of V 3 along with ( 56) is Considering the virtual control controller ( 22), the time derivative of the Lyapunov function ( 54) satisfies According to Step 2, by utilizing the virtual controller (27) and the parameter adaptation law (28), we can obtain where The time derivative of V 5 along with ( 60) is Considering the virtual control controller (30), the time derivative of Lyapunov function (61) satisfies According to Step 2 and Step 4, Substituting the virtual controller ( 35) and the parameter adaptation law (36) into (33) yields where Substituting ( 40) and ( 64) into (39), one has By utilizing the actual controller (42), parameter adaptive laws ( 43), ( 44), ( 45) and the intermediate control law (40) into (65), the following inequality can be derived: )) According to Lemma 2.1, we have Inspired by Li (2019), the inequality 0 ≤ |ξ |−ξ tanh( ξ τ ) ≤ 0.2785τ is used to handle term tanh(•), thus the following inequality holds Substituting ( 67) and ( 68) into (66) yields By applying the Young's inequality, it has From ( 70) and ( 71), ( 69) can be rewritten as According to Ren et al. (2010) and Liu et al. (2018) , the equality log holds.Thus, we have Choosing C = min 2c 1 , 2k Then, the following inequality holds From ( 76), one can obtain By considering physical factors of the active seat suspension system, k ci i = 1, 2, 3, 4, 5, 6, 7 is the maximum state variable to ensure the stability in the control process.From (77), we can know |z i | ≤ k bi , it is obvious that z i ≤ k bi .Since z 1 = x 1 , then|x 1 | ≤ k b1 .Moreover, from ( 14), there must exist a constant A 1 , which is satisfying Likewise, according to the feasible condition in (Li et al., 2019;Liu et al., 2021), we can get |x i | < k ci (i = 3, 4, 5, 6, 7).Thus, we can obtain the displacements of car seat, active suspension and wheel, their vertical vibration speeds, and current intensity of the active seat suspension systems are bounded.

Simulation
In the simulation part, the results are given by the following example to verify the effectiveness of the proposed control method.
The selection of system parameters comes from the data acquired from the testing rig or real vehicles, which accords with basic mechanism of conforming suspension system.Consider the active seat suspension system (4), according to Liu et al. ( 2021 This paper considers the electromagnetic actuator failure of an active seat suspension system.The faults model is selected as where δ = 0.5 and − λ = 0.1, and the failure time is t h = 40s. In this study, to validate the performance of the proposed control scheme, the random road displacement input is r(t) = 0.025 sin(10π t). Figure 3 shows the trajectory of the random road displacement input.To demonstrate the advantages of the presented control approach, we consider the electromagnetic actuator faults at 40s.Among them, Figures 4-6 display the vertical displacements of the car seat, active suspension and the wheel with FTC.Figures 7-9 show their vertical vibration speeds with FTC, Figure 10 exhibits the electric current intensity of the electromagnetic actuator with FTC, Figure 11 expresses the trajectory of control input for the active seat suspension with FTC.In Figures 12 and 13, the spaces of the car seat and active suspension are displayed.It can be obviously seen that the proposed control method can ensure the vertical vibration states are all stabilized.In addition, when the electromagnetic actuator fails, the vertical displacements of car seat, active suspension and wheel, their vertical vibration speeds and current intensity are all constrained.
To further illustrate the advantages of the developed adaptive fuzzy FTC method in this paper, we provide           It can be seen from Figures 14-21 that when the electromagnetic actuator fails at 40s, the vertical displacements of car seat, active suspension and wheel, their vertical vibration speeds and current intensity are all unstable.The comparative simulation results between the proposed FTC method and without FTC method can further illustrate that the fact that the presented FTC method can perform desirably and ensure the stability of the active suspension with full-state constraints in the presence of actuator electromagnetic failures.

Conclusion
This paper has investigated an adaptive fuzzy FTC problem for active seat suspension systems with electromagnetic actuator, in which the states are fully constrained.The systems had dynamic characteristics of complexities and spring non-linearities.Besides, the electromagnetic actuator is subject to the loss and bias faults.To control the active seat suspension systems, FLSs had been exploited to approximate the unknown nonlinear dynamics.By constructing BLFs, a novel adaptive fuzzy FTC scheme has been proposed, which can guarantee the displacements of car seat, active suspension and wheel, their vertical vibration speeds, and current intensity in the active seat suspension systems to be stable, and the states are bounded regardless of electromagnetic actuator faults.Then, the safety and comfort of passengers are improved.Future research directions will focus on adaptive output feedback fuzzy FTC methods for semi-vehicle and full-vehicle active seat suspension system models with electromagnetic actuator faults and time-varying constraints.

Figure 1 .
Figure 1. Circuit diagram of the electromagnetic actuator.

Figure 2 .
Figure 2. Quarter active seat suspension system model.

Figure 3 .
Figure 3.The random road displacement input.

Figure 4 .
Figure 4.The vertical displacement of the car seat with FTC.

Figure 5 .
Figure 5.The vertical displacement of the active suspension with FTC.

Figure 6 .
Figure 6.The vertical displacement of the wheel with FTC.

Figure 7 .
Figure 7.The vertical speed of the car seat with FTC.

Figure 8 .
Figure 8.The vertical speed of the active suspension with FTC.

Figure 9 .
Figure 9.The vertical speed of the wheel with FTC.

Figure 10 .
Figure 10.The electric current intensity with FTC.

Figure 11 .
Figure 11.The trajectory of control input u with FTC.

Figure 12 .
Figure 12.The space of the car seat with FTC.

Figure 13 .
Figure 13.The space of the active suspension with FTC.

Figure 14 .
Figure 14.The vertical displacement of the car seat without FTC.

Figure 15 .
Figure 15.The vertical speed of the car seat without FTC.

Figure 16 .
Figure 16.The vertical displacement of the active suspension without FTC.

Figure 17 .
Figure 17.The vertical speed of the active suspension without FTC.

Figure 18 .
Figure 18.The vertical displacement of the wheel without FTC.

Figure 19 .
Figure 19.The vertical speed of the wheel without FTC.

Figure 20 .
Figure 20.The electric current intensity without FTC.

Figure 21 .
Figure 21.The trajectory of control input u without FTC.