# Adaptive Finite-Time Fault-Tolerant Control for Half-Vehicle Active Suspension Systems with Output Constraints and Random Actuator Failures

Authors: Jie Lan; Tongyu Xu
Journal: Complexity
 (2021)

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## Abstract

The problem of adaptive finite-time fault-tolerant control (FTC) and output constraints for a class of uncertain nonlinear half-vehicle active suspension systems (ASSs) are investigated in this work. Markovian variables are used to denote in terms of different random actuators failures. In adaptive backstepping design procedure, barrier Lyapunov functions (BLFs) are adopted to constrain vertical motion and pitch motion to suppress the vibrations. Unknown functions and coefficients are approximated by the neural network (NN). Assisted by the stochastic practical finite-time theory and FTC theory, the proposed controller can ensure systems achieve stability in a finite time. Meanwhile, displacement and pitch angle in systems would not violate their maximum values, which imply both ride comfort and safety have been enhanced. In addition, all the signals in the closed-loop systems can be guaranteed to be semiglobal finite-time stable in probability (SGFSP). The simulation results illustrate the validity of the established scheme.


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## Body

## 1. Introduction

With the development of modern industrial automation, vehicles play a more and more important role in people’s production and life. Suspension as an important part, the damping effect to a great extent determines the comfort and safety of the automobile. Compared to the traditional passive and semiactive suspension systems, active suspension systems (ASSs) can provide better dynamic adjustment damping, potential road handling capacity, extreme ride comfort, and suspension deflection [1–3]. The design of complex mechanical engineering of ASSs had become a hot issue in the past two decades.ASSs are often simplified into the full-vehicle model, half-vehicle model, and quarter-vehicle model. The actuators are parallel inserted to the components that provide external forces to increase or dissipate the energy of the ASSs and manage the tradeoffs between conflicting performance indicators. Many remarkable results were reported in [4–8] with the vertical motion of the quarter ASSs as the research topic. However, pitch motion was ignored that also directly affected the ride comfort and safety. Research on half-vehicle ASSs mainly focused on the discussions of pitch motion and vertical motion [9–17]. For ASSs, some inevitable uncertainties were in the design of controllers in [9–13], such as body mass, mass moment of inertia, and modeling uncertainties. But, these control methods [9–11] did not achieve good performance in estimating real valves. In [12, 13], they proposed adaptive control schemes by adding new leakage items to the update rules. In [14], the damping coefficient and spring stiffness of the tires were considered in the suspension as random uncertain parameters. The uncertain actuator was discussed and eliminated the influence by continuous-time homogeneous [15]. However, many progresses have been made for uncertain nonlinear ASSs, and a few studies were on the constraint of half-vehicle models.The actual mechanical systems need to keep the output or states within certain ranges; otherwise, the system performance would degrade. The properties of prescribed performance control [17] and barrier Lyapunov functions (BLFs) provide effective methods to deal with constraints, and a large number of results have been obtained for output constraints of various nonlinear systems and practical systems [18–26]. The asymmetric BLFs were coped with the position constraint problem of the marine vessel [24]. Due to the limitation of physical factors in suspension fields, it is worth noting that [25, 26] reported the output and time-varying output constraint of vertical motion for the quarter ASSs. However, for the half-vehicle ASSs, considering above output constraints had been carried out.Theoretically, the aforementioned works can only guarantee the desired system performance as time approaches infinity. However, the actual mechanical control should achieve the expected transient system performance. The design of finite-time control for nonlinear systems has attracted considerable attention. The finite-time Lyapunov stability theorem was first proposed in [27]. Based on this theory, the continuous finite-time control for nonlinear systems was proposed in [28–34], robotic manipulators in [28], switched systems in [32–34], and Markovian jump systems in [35, 36]. The concept of semiglobal practical finite-time stability (SGPFS) was proposed in different forms in [31–36]. The adaptive fuzzy finite-time control scheme of general uncertain nonlinear systems is discussed in [36]. Furthermore, Cai and Xiang expanded SGPFS to nonstrict nonlinear systems in [37]. Sui et al. expanded SGPFS for nontriangular stochastic nonlinear systems in [38]. The finite-time results of nonlinear quarter ASSs had been made in [39, 40]. The finite-time results of nonlinear half ASSs had been made in [41], but there was no constraint study on the individual states. For nonlinear strict feedback systems, both output constraint and finite-time control design had been completed in [42, 43]. However, for uncertain nonlinear half ASSs, there are few results on how to implement finite-time control associated with output constraints.On the other hand, actuator failures are inevitable due to the influence of external environment, mechanical system failures, operation errors, and human factors in practical systems. These faults can seriously have an impact on system stability, degradation, and even catastrophic risks. Most of the above studies assumed that all actuators or sensors were in normal operation. Fault-tolerant control (FTC) strategies can compensate the faults and maintain acceptable system performance. Many methods to deal with actuator failures, such as the pseudoinverse method was in [44], model prediction method was in [45], and sliding control was in [46], by applying the adaptive backstepping techniques for linear systems in [47] and nonlinear strict feedback systems in [48–50]. Failures should be random. The actuator of states can switch between various modes in a random way. Given enough historical data, the states of the actuator can be modeled as Markov states in [51]. In the process, the failures of different actuators also meet the requirements of different Markov processes. Each actuator is independent and can fail at any sampling time. In [52], an adaptive fault compensation for a class of nonlinear uncertain systems with random actuator faults was studied and a random function to scale actuator faults by Markov correlation variables was proposed. In [53], random faults between different actuators in the half ASSs were considered for the first time. Motivated by the above observations and existing research results, this study proposes an adaptive NN finite-time FTC scheme for uncertain nonlinear half-vehicle ASSs with output constraints. The three main advantages of the proposed scheme can be listed as follows:(1)
Compared with existing adaptive FTC studies, the problem for half ASSs subject to infinite stochastic actuator failures and the states of multiactuators modeled by different Markovian processes has not received enough attention. Particularly, considering finite-time control, the additional correlation terms generated by the infinitesimal generator are handled by the stochastic finite-time control theorem.(2)
In comparison with existing constraints, it is asymmetry, which can restrain different outputs of displace and pitch angle more reasonable and reduce the vibration in uncertain nonlinear half ASSs. Moreover, the finite-time FTC control strategy can also enable the practical control systems to realize the transient system stability.(3)
In comparison with existing adaptive finite-time control, it considers a class of uncertain nonlinear half-ASSs subject to stochastic actuator failures. It should dispose random terms which makes the existing stability criteria in [53–55] are invalid. Concurrently, the asymmetric output constraints for different factors have been considered. The Lyapunov is proved SGPFS.This work is organized as follows. In Section2, the half active suspension systems and control objectives are shown. Section 3 presents the design procedures of the adaptive finite-time fault-tolerant controller designed based on stochastic Lyapunov function and zero dynamic. In Section 4, an example to show that the constructed method is effective. In Section 5, it demonstrates a conclusion about the results of this work and future work.


## 2. System Description and Preliminaries

### 2.1. Nonlinear Half-Vehicle Suspension Systems

Figure1 shows a half-vehicle suspension model. M represents the mass of the vehicle body. I is the mass moment of inertia. mf and mr are the defined masses of front and rear wheels, respectively. Dc stands for the vertical displacement of the vehicle body. φ represents the pitch angle. Df and Dr stand for the displacements of the front and rear vehicle body, respectively. D1 and D2 are the displacements of the front and rear wheels, respectively. D01 and D02 represent the road inputs of corresponding wheels. Fsr, Ftr, Fsf, and Ftf are defined as the forces produced by the related stiffness. Fdr, Fbr, Fdf, and Fbf are defined as the forces produced by the related dampers. uf and ur represent the control forces of the front and rear ASSs.Figure 1
The nonlinear half-vehicle suspension model.For half ASSs, the state space equations of vertical motion and pitch motion are shown in the following equation:(1)MD¨c+Fsf+Fdf+Fsr+Fdr−uz=0,Iφ¨+l2Fsf+Fdf−l1Fsr+Fdr−uφ=0,mfD¨1−Fsf−Fdf+Ftf+Fbf+uf=0,mrD¨2−Fsf−Fdr+Ftr+Fbr+ur=0,where uz=ur+uf and uφ=l2uf−l1ur.The stochastic actuator failures are considered and described as follows:(2)uft=g1r1tΨft,urt=g2r2tΨrt,where Ψft and Ψrt are the inputs of the front actuator and rear actuator, respectively. r1t and r2t are the independent irreducible right continuous homogeneous Markovian processes on the probability space Ω,E,P, taking values in a finite set S=1,2,…,N with generator matrix Π=ϒpqN×N, where ϒpq>0 is the transition rate from mode p to mode q if p≠q, and ϒpp=∑q=1,p≠qNϒpq.In addition,g1r1t and g2r2t are the stochastic functions which represent the failure scaling factors for two different actuators and take values on the interval g1min,1g2min,1, and g1min>0, g2min>0.Remark 1.
The valuesg1r1t and g2r2t meet as follows,(1)
Wheng1r1t=1 or g2r2t=1, the front actuator or the rear actuator is healthy(2)
Wheng1r1t∈g1min,1 or g2r2t∈g2min,1, there is partial failure of the corresponding actuator
Each actuator will switch randomly between above two states. The problem of FTC considered in the systems was assumed that once the actuator failed, it would keep the fault state for the rest operation. However, the failure may be intermittent, and the actuator may repetitiously fail with different failure modes. Meanwhile, the modes, times, and patterns of actuator failures are essentially random. Then, actuator failures in (2) are more complicated and practical.
In order to facilitate the design and analysis of the adaptive finite-time fault-tolerant control method, the state variables need to be defined as follows:(3)x1=Dc,x2=D˙c,x3=φ,x4=φ˙,x5=D1,x6=D˙1,x7=D2,x8=D˙2,x=x1x2x3x4x5x6x7x8T.
Then, it obtains(4)x˙1=x2,x˙2=F1+ρ1g1r1tΨft+g2r2tΨrt,x˙3=x4,x˙4=F2+ρ2l2g1r1tΨft−l1g2r2tΨrt,(5)x˙5=x6,x˙6=1mfFsf+Fdf−Ftf−Fbf−g1r1tΨft,x˙7=x8,x˙8=1mrFsr+Fdr−Ftr−Fbr−g2r2tΨrt,where ρ1=1/M, ρ2=1/I, F1=ρ1−Fsf−Fdf−Fsr−Fdr, and F2=ρ2−l2Fsf+Fdf+l1Fsr+Fdr.
The massM and the mass moment of inertia I are uncertain due to the uncertainties of passengers and the load quality of cargoes. Therefore, F1 and F2 are the uncertainty and unknown functions.


### 2.2. Control Objectives

For different types of vehicles, improving ride comfort and safety is one of the most important requirements. Due to the hardware limitations, the following three requirements must be considered in the design and control process of ASSs.First, the controllersuf and ur are subjected to random Markovian jumping failures in (2). The vertical displacement Dc and pitch angle motion φ are considered and limited in safety ranges. Dc and φ should be smaller which can largely enhance ride comfort.Second, considering the driving safety, the wheels should make uninterrupted contact with the ground. It means that the dynamic tire load must be less than the static load, i.e.,(6)Ftf+Fbf≤Ff,Ftr+Fbr≤Fr,where Ff+Fr=M+mf+mrg, and Ffl1+l2=mfgl1+l2+Ml1g.Last, suspension constraints must be guaranteed because of the confined mechanical space, that is, suspension deflections should not exceed their maximum values.(7)ΔDf≤ΔD¯f,ΔDr≤ΔD¯r,where ΔDf=Dc+l2sinφ−D1, and ΔDr=Dc−l1sinφ−D2.There are some assumptions, definitions, and lemmas presented in order to facilitate the design and analysis of adaptive control scheme.Assumption 1.
For given constantsk¯c1>0, k¯c1>0, k¯c2>0, and k¯c2>0, the following results on y1d, y3d, y˙1d, and y˙3d hold, i.e.,(8)−B¯11≤y1d≤B¯11,y˙1d≤B¯12,−B¯31≤y3d≤B¯31,y˙3d≤B¯32,where B¯11<k¯c1, B¯11≤k¯c1, B¯31<k¯c2, and B¯31≤k¯c2, for positive parameters B¯11, B¯11, B¯31, B¯31, B¯12, and B¯32 are selected and positive parameters.Assumption 2.
In general, the inputs of the road and their time derivative ofD01 and D02 are limited. Therefore, existing positive parameters make sure the following inequality:(9)D01t≤D¯11,D˙01t≤D¯12,D02t≤D¯21,D˙02t≤D¯22,where the parameters D¯11, D¯12, D¯21, and D¯22 are positive.Definition 1 (see [35]).
Consider the following nonlinear systems:(10)ζ˙=gζ,u,g0,0=0,ζ∈ℜn,u∈ℜn,where ζ and u represent the state and input vectors, respectively. If any initial condition ζt0 satisfies ζt0=ζ0, for t≥t0+T, the system (10) is semiglobal practical finite-time stable (SGPFS). In addition, the state vector ζt satisfies ζt≤ζ¯, for t≥t0+T, where ζ¯ is a positive parameter and a deposit time Tζ0,ζ¯ satisfies 0<Tζ0,ζ¯<∞.Lemma 1 (see [36]).
For anyγi∈R, i=1,2,…,n, 0<a<1, 0<b<2, the following inequality can be constructed, namely,(11)γ1+γ2+⋯+γna≤γ1a+γ2a+⋯+γna,γ12+γ22+⋯+γn2b≤γ1a+γ2a+⋯+γna2.Lemma 2 (see [37]).
Consider the Lyapunov functionVζ with stochastic terms, and the following inequality holds for a nonlinear system (10), namely,(12)LVζ≤−ℓVaζ+ℏwhere L is the differential operator, the constants ℓ and ℏ are positive, and 0<a<1. ζ∈Rn and t≥t0. Then, stochastic trajectory of (10) is SGPFS.Lemma 3 (see [40]).
For anyz and α are real variables, the following inequality can be constructed, namely,(13)zaαb≤aa+bμza+b+ba+bμ−a/bαa+b,where the parameters a, b, and μ are positive.Lemma 4 (see [23]).
For allγ1≤kb1, the following inequality can be constructed as(14)log1kb12−γ12≤γ12kb12−γ12.


## 2.1. Nonlinear Half-Vehicle Suspension Systems

Figure1 shows a half-vehicle suspension model. M represents the mass of the vehicle body. I is the mass moment of inertia. mf and mr are the defined masses of front and rear wheels, respectively. Dc stands for the vertical displacement of the vehicle body. φ represents the pitch angle. Df and Dr stand for the displacements of the front and rear vehicle body, respectively. D1 and D2 are the displacements of the front and rear wheels, respectively. D01 and D02 represent the road inputs of corresponding wheels. Fsr, Ftr, Fsf, and Ftf are defined as the forces produced by the related stiffness. Fdr, Fbr, Fdf, and Fbf are defined as the forces produced by the related dampers. uf and ur represent the control forces of the front and rear ASSs.Figure 1
The nonlinear half-vehicle suspension model.For half ASSs, the state space equations of vertical motion and pitch motion are shown in the following equation:(1)MD¨c+Fsf+Fdf+Fsr+Fdr−uz=0,Iφ¨+l2Fsf+Fdf−l1Fsr+Fdr−uφ=0,mfD¨1−Fsf−Fdf+Ftf+Fbf+uf=0,mrD¨2−Fsf−Fdr+Ftr+Fbr+ur=0,where uz=ur+uf and uφ=l2uf−l1ur.The stochastic actuator failures are considered and described as follows:(2)uft=g1r1tΨft,urt=g2r2tΨrt,where Ψft and Ψrt are the inputs of the front actuator and rear actuator, respectively. r1t and r2t are the independent irreducible right continuous homogeneous Markovian processes on the probability space Ω,E,P, taking values in a finite set S=1,2,…,N with generator matrix Π=ϒpqN×N, where ϒpq>0 is the transition rate from mode p to mode q if p≠q, and ϒpp=∑q=1,p≠qNϒpq.In addition,g1r1t and g2r2t are the stochastic functions which represent the failure scaling factors for two different actuators and take values on the interval g1min,1g2min,1, and g1min>0, g2min>0.Remark 1.
The valuesg1r1t and g2r2t meet as follows,(1)
Wheng1r1t=1 or g2r2t=1, the front actuator or the rear actuator is healthy(2)
Wheng1r1t∈g1min,1 or g2r2t∈g2min,1, there is partial failure of the corresponding actuator
Each actuator will switch randomly between above two states. The problem of FTC considered in the systems was assumed that once the actuator failed, it would keep the fault state for the rest operation. However, the failure may be intermittent, and the actuator may repetitiously fail with different failure modes. Meanwhile, the modes, times, and patterns of actuator failures are essentially random. Then, actuator failures in (2) are more complicated and practical.
In order to facilitate the design and analysis of the adaptive finite-time fault-tolerant control method, the state variables need to be defined as follows:(3)x1=Dc,x2=D˙c,x3=φ,x4=φ˙,x5=D1,x6=D˙1,x7=D2,x8=D˙2,x=x1x2x3x4x5x6x7x8T.
Then, it obtains(4)x˙1=x2,x˙2=F1+ρ1g1r1tΨft+g2r2tΨrt,x˙3=x4,x˙4=F2+ρ2l2g1r1tΨft−l1g2r2tΨrt,(5)x˙5=x6,x˙6=1mfFsf+Fdf−Ftf−Fbf−g1r1tΨft,x˙7=x8,x˙8=1mrFsr+Fdr−Ftr−Fbr−g2r2tΨrt,where ρ1=1/M, ρ2=1/I, F1=ρ1−Fsf−Fdf−Fsr−Fdr, and F2=ρ2−l2Fsf+Fdf+l1Fsr+Fdr.
The massM and the mass moment of inertia I are uncertain due to the uncertainties of passengers and the load quality of cargoes. Therefore, F1 and F2 are the uncertainty and unknown functions.


## 2.2. Control Objectives

For different types of vehicles, improving ride comfort and safety is one of the most important requirements. Due to the hardware limitations, the following three requirements must be considered in the design and control process of ASSs.First, the controllersuf and ur are subjected to random Markovian jumping failures in (2). The vertical displacement Dc and pitch angle motion φ are considered and limited in safety ranges. Dc and φ should be smaller which can largely enhance ride comfort.Second, considering the driving safety, the wheels should make uninterrupted contact with the ground. It means that the dynamic tire load must be less than the static load, i.e.,(6)Ftf+Fbf≤Ff,Ftr+Fbr≤Fr,where Ff+Fr=M+mf+mrg, and Ffl1+l2=mfgl1+l2+Ml1g.Last, suspension constraints must be guaranteed because of the confined mechanical space, that is, suspension deflections should not exceed their maximum values.(7)ΔDf≤ΔD¯f,ΔDr≤ΔD¯r,where ΔDf=Dc+l2sinφ−D1, and ΔDr=Dc−l1sinφ−D2.There are some assumptions, definitions, and lemmas presented in order to facilitate the design and analysis of adaptive control scheme.Assumption 1.
For given constantsk¯c1>0, k¯c1>0, k¯c2>0, and k¯c2>0, the following results on y1d, y3d, y˙1d, and y˙3d hold, i.e.,(8)−B¯11≤y1d≤B¯11,y˙1d≤B¯12,−B¯31≤y3d≤B¯31,y˙3d≤B¯32,where B¯11<k¯c1, B¯11≤k¯c1, B¯31<k¯c2, and B¯31≤k¯c2, for positive parameters B¯11, B¯11, B¯31, B¯31, B¯12, and B¯32 are selected and positive parameters.Assumption 2.
In general, the inputs of the road and their time derivative ofD01 and D02 are limited. Therefore, existing positive parameters make sure the following inequality:(9)D01t≤D¯11,D˙01t≤D¯12,D02t≤D¯21,D˙02t≤D¯22,where the parameters D¯11, D¯12, D¯21, and D¯22 are positive.Definition 1 (see [35]).
Consider the following nonlinear systems:(10)ζ˙=gζ,u,g0,0=0,ζ∈ℜn,u∈ℜn,where ζ and u represent the state and input vectors, respectively. If any initial condition ζt0 satisfies ζt0=ζ0, for t≥t0+T, the system (10) is semiglobal practical finite-time stable (SGPFS). In addition, the state vector ζt satisfies ζt≤ζ¯, for t≥t0+T, where ζ¯ is a positive parameter and a deposit time Tζ0,ζ¯ satisfies 0<Tζ0,ζ¯<∞.Lemma 1 (see [36]).
For anyγi∈R, i=1,2,…,n, 0<a<1, 0<b<2, the following inequality can be constructed, namely,(11)γ1+γ2+⋯+γna≤γ1a+γ2a+⋯+γna,γ12+γ22+⋯+γn2b≤γ1a+γ2a+⋯+γna2.Lemma 2 (see [37]).
Consider the Lyapunov functionVζ with stochastic terms, and the following inequality holds for a nonlinear system (10), namely,(12)LVζ≤−ℓVaζ+ℏwhere L is the differential operator, the constants ℓ and ℏ are positive, and 0<a<1. ζ∈Rn and t≥t0. Then, stochastic trajectory of (10) is SGPFS.Lemma 3 (see [40]).
For anyz and α are real variables, the following inequality can be constructed, namely,(13)zaαb≤aa+bμza+b+ba+bμ−a/bαa+b,where the parameters a, b, and μ are positive.Lemma 4 (see [23]).
For allγ1≤kb1, the following inequality can be constructed as(14)log1kb12−γ12≤γ12kb12−γ12.


## 3. Adaptive Finite-Time Fault-Tolerant Controller Design Based on Stochastic Lyapunov Function and Zero Dynamic

For a class of uncertain nonlinear half ASSs, the universal approximation property of NN solves the uncertainty, and then, the outputs of vertical motion and pitch angle motion are constrained by using asymmetrical log-BLFs. The adaptive backstepping technique is used to address the functions with random generality variables generated by the infinitesimal generator due to random actuator failures. The Lyapunov stability is proved by the SGPFS and zero dynamic theorem.In order to facilitate the design of controllers, there are some given coordinate transformations, i.e.,(15)γ1=x1−y1d,γ2=x2−β1,γ3=x3−y3d,γ4=x4−β3,where β1 and β3 are the virtual control signals. γi for 1≤i≤4 is the error variable. y1d and y3d are the desired vertical displacement and pitch angle, respectively.


### 3.1. Finite-Time Constraint Control Scheme Design for Vertical Motion and Pitch Angle Motion

Now, for the vertical motion of active suspension systems, more details on finite-time approach will be given in the next section.Step 1.
SelectV=Vxt,r1t,r2t as a Lyapunov function candidate in the following form:(16)V=∑i=1,31−qγi2logkai2kai2−γi2+qγi2logkbi2kbi2−γi2+∑i=1212ρiγ2i2+12ηiθ˜i2+12χig˜i2rit,where xt=x1t,x2t,x3t,x4t, γ1=x1,r1t,r2t, γ2=x1,x2,r1t,r2t, γ3=x3,r1t,r2t, and γ4=x3,x4,r1t,r2t. The parameters ηi and χi are positive, and θ˜i and g˜irit represent the errors. θ^i is the estimator of θi. g^irit is the estimator of girit. Then, θ˜i=θi−θ^i, g˜irit=girit−g^irit, i=1,2. It would make an assumption: ∀t>0, r1t=p1, r2t=p2, and p1,p2∈S.Remark 2.
For quarter ASSs, the study in [30] only showed the vehicle body’s displacement constraint method by the symmetric BLFs. In contrast, we further study the pitch angle constraint of nonlinear uncertain half ASSs by asymmetric BLFs in this study.
In addition, the virtual controller signal is designed as(17)βi=−ciγi21−qγikai2−γi2+qγikbi2−γi2+y˙id,where i=1,3. The parameter ci is a design parameter, ci>0. kai and kbi are the constraint bounded of corresponding variables, respectively. Moreover,(18)q⋅=1,⋅≥0,0,⋅<0.
Then, according to the infinitesimal generator ofV, it gets(19)LV=∑i=1,31−qγiγiγ˙ikai2−γi2+qγiγiγ˙ikbi2−γi2+∑i=121ρiγ2iγ˙2i+1ηiθ˜iθ˜˙i+1χig˜ipig˜˙ipi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2,where V1=xt,p2,q1, V2=xt,p1,q2, and ϒpiqi>0 is the transition rate, i=1,2.Remark 3.
∑q1=1Nϒp1q1V1 and ∑q2=1Nϒp2q2V2 are the additional terms due to the involvement of Markovian variables r1t and r2t, which do not exist in the determined situation. The extratransition rate-related terms appear in the infinitesimal generator of Lyapunov candidate function. These additions need further processing. It is a challenge that cannot be ignored in stability analysis.
From the previous definition,i=1,2, there are(20)g˜˙irit=−g^˙irit,θ˜˙i=−θ^˙i.
Combining with (19) and (20) can obtain(21)LV=∑i=1,3γi1−qγikai2−γi2+qγikbi2−γi2γ˙i+∑i=121ρiγ2iγ˙2i−∑i=121χig˜ipig^˙ipi+1ηiθ˜iθ^˙i+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2.
Furthermore, taking the derivative ofγi, it yields(22)γ˙i=γi+1+βi−y˙id,i=1,3.
Substituting (17) and (22) into (21), it can attain(23)γi1−qγikai2−γi2+qγikbi2−γi2γ˙i=λiγiγi+1−ciγi221−qγikai2−γi2+qγikbi2−γi2=λiγiγi+1−ci1−qγi2γi2kai2−γi2−ciqγi2γi2kbi2−γi2,where λi=1−qγi/kai2−γi2+qγi/kbi2−γi2, ci is a positive design parameter, and i=1,3.
Based Lemma3, let zi=γi2/2kai2−γi2, α=1, a=2n−1/2n+1, n∈ℕ, b=1−a, and μ=aa/1−a obtain(24)γi22kai2−γi2a≤1−aμ+γi22kai2−γi2.
Therefore, it gets(25)−γi22kai2−γi2≤1−aμ−γi22kai2−γi2a.
Constructed in the same way, it obtains(26)−1−qγiγi22kai2−γi2≤1−aμ−1−qγiaγi22kai2−γi2a.
Similarly, it yields(27)−γi22kbi2−γi2≤1−aμ−γi22kbi2−γi2a,−qγiγi22kbi2−γi2≤1−a−qγiaγi22kbi2−γi2a.
According to Lemma4, for any constant a, 0<a<1, the following inequality holds:(28)log1kai2−γi2a≤γi2kai2−γi2a,log1kbi2−γi2a≤γi2kbi2−γi2a.
Next, substituting (26)–(28) into (23), LV is expressed as(29)LV=∑i=1,3λiγiγi+1−ci1−qγi2log1kai2−γi2a−ciqγi2log1kbi2−γi2a+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ1+∑i=121ρiγ2iγ˙2i−1χig˜ipig^˙ipi−1ηiθ˜iθ^˙i,where ℏ1=∑i=1,32ci1−aμ.
Taking the derivative ofγ2 and γ4, they yield(30)γ˙2=F1+ρ1uz−β˙1,γ˙4=F2+ρ2uφ−β˙3,where β˙i=∂βi/∂xix˙i+∂βi/∂yidy˙id+∂βi/∂y˙idy¨id, i=1,3. βi has been designed in (17). Then, ∑i=121/ρiγ2iγ˙2i gets(31)1ρ1γ2γ˙2=γ2F¯1+uz,1ρ2γ4γ˙4=γ4F¯2+uφ,where the unknown functions F¯1 and F¯2 are denoted as(32)F¯1=ρ1−1F1−ρ1−1β˙1,F¯2=ρ2−1F2−ρ2−1β˙3.
Using the powerful approximating ability of NNs in [42, 43] for uncertain nonlinear systems, F¯1 and F¯2 can employ(33)F¯1=Θ1ΦZ1T+τ1,F¯2=Θ2ΦZ2T+τ2,where Z1=x1,x2,x3,x4,x5,x6,x7,x8,β1,β˙1,y1d,y˙1dT, Z2=x1,x2,x3,x4,x5,x6,x7,x8,β3,β˙3,y3d,y˙3dT, and τi is the approximation error and the designed positive constant τ¯i that satisfies τi≤τ¯i. More remarkable, 0<Θ1iΦZ1<1 and 0<Θ2iΦZ2<1. Furthermore, we can draw a conclusion 0<Φ1TZ1Φ1Z1≤N1 and 0<Φ2TZ2Φ2Z2≤N2, where N1 and N2 are the corresponding numbers of NNs nodes.
Then, applying Young’s inequality, it can get the following:(34)γ2iΘiΦZiT≤γ2i2Θi2ΦZiTΦZi2δi2+δi22=γ2i2θi2δi2+δi22,(35)γ2iτi≤12γ2i2+12τi2≤12γ2i2+12τ¯i2,where θi=NiΘi2 and the design parameter δi is positive.
Substituting (33)–(35) into (31), it obtains(36)∑i=121ρiγ2iγ˙2i≤γ2γ2θ12δ12+γ22+uz+γ4γ4θ22δ22+γ42+uφ+∑i=12δi22+τ¯i22.
Substituting (36) into (29), it can obtain(37)LV=∑i=1,3λiγiγi+1−ci1−qγi2log1kai2−γi2a−ciqγi2log1kbi2−γi2a+γ2γ2θ12δ12+γ22+uz+γ4γ4θ22δ22+γ42+uφ−∑i=12g˜ipig^˙ipiχi+θ˜iθ^˙iηi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ2,where ℏ2=ℏ1+∑i=12δi2/2+τ¯i2/2.
Design the control inputuz and uφ that are subjects to random actuator faults as(38)uz=−λ1γ1−γ22−γ2θ^12δ12−c2γ22a−1,uφ=−λ3γ3−γ42−γ3θ^22δ32−c4γ42a−1,where the parameters c2 and c4 are positive.
The adaptive law is established as(39)θ^˙i=ηi2δi2γ2i2−κiθ^i,i=1,2,where the designed parameter κi is positive.
Based on (1), (2), and (39), uf and ur are shown as(40)uf=l1uz+uφl1+l2,ur=l2uz−uφl1+l2,Ψft=1g^1l1+l2l1−λ1γ1−γ22−γ2θ^12δ12−c2γ22a−1−λ3γ3−γ42−γ3θ^22δ32−c4γ42a−1,Ψrt=1g^2l1+l2l2−λ1γ1−γ22−γ2θ^12δ12−c2γ22a−1+λ3γ3+γ42+γ3θ^22δ32+c4γ42a−1.
The updating laws of estimated parameters are established:(41)g^˙1p1=projg^1p1γ2+l1γ4χ1ψft−2cg1g^1p10,g^1p1=g1min,γ2+l1γ4χ1ψft−2cg1g^1p1,g^1p1,others,g^˙2p2=projg^2p2γ2−l2γ4χ2ψrt−2cg2g^2p20,g^2p2=g2min,γ2−l1γ4χ2ψrt−2cg2g^2p2,g^2p2,others,where proj represents the projection operator in order to avoid singular values in the denominator.Remark 4.
The adaptive estimated parametersg^˙1p1 and g^˙2p2 appear in the denominators in (41), which may cause the controllers to fail. Construct the projection form to avoid this situation.
Substituting (39)–(41) into (37), LV can be expressed as(42)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2iγ2i2a+∑i=12cgig˜ipig^ipiχi+κiθ˜iθ^iηi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ2.
By Young’s inequalityi=1,2, they can obtain(43)θ˜iθ^i≤12θ˜i2+12θi2,g˜ipig^ipi≤−12g˜i2pi+12g^i2pi.
LV can be rewritten as(44)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2iγ2i2a+cgig˜i2pi2χi+κiθ˜i22ηi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ3,where ℏ3=ℏ2+∑i=12κiθi2/2ηi+cgigi2pi/2χi.
Based on Lemma3, let z=θ˜i2/2ηia=2n−1/2n+1, n∈ℕ, b=1−a, α=1, and μ=aa/1−a, one has(45)θ˜i22ηia≤1−aμ+θ˜i22ηi.
Furthermore, we can obtain(46)−θ˜i22ηi≤1−aμ−θ˜i22ηia.
Applying the inequality (44) and (46) can be expressed as(47)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2iγ2i2a+cgig˜i2pi2χi+κiθ˜i22ηia+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ,where ℏ=ℏ3+1−aμ.
In addition, the number of passengers and the load conditions of the vehicle are limited. The massM is bounded which means ρi is bounded, namely,(48)ρ¯i≤ρi≤ρ¯i,i=1,2.
Based on (48),(49)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2i2ρ¯iaγ2i2a2ρ¯ia+cgig˜i2pi2χi+κiθ˜i22ηia+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ.
From (16), (49) can be rewritten as(50)LV≤−ℓVa+ℏ+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2,where ℓ=min−c22ρ¯1a,c42ρ¯2a,c1,c3,κ1,κ2,cg1,cg2.
The following will apply the discrete expectation to address the random terms.(51)limx⟶∞inft≥t0V=∞.
There would be aB, no matter how big it is; once more than t>tl, it would exceed B. From (51) in t0,tl, xt≤B. (50) is bounded in t0,tl.
Then, according to property of discrete expectations, it yields(52)ELV≤∑p1=1N∑p2=1NELVlπp1πp2≤−ℓ∑p1=1N∑p2=1NELVlaπp1πp2+ℏ+∑p1=1N∑p2=1NE∑q1=1Nϒp1q1V1l+∑q2=1Nϒp2q2V2lπp1πp2≤−ℓ∑p1=1N∑p2=1NELVla+ℏ+N∑k=12maxpk,qk=1NγpkqkπpkπqkEVla≤−ℓ−ℓlEVla+ℏ,where Vl=Vxtl,r1tl,r2tl, V1l=xtl,p2,q1, V2l=xtl,p1,q2, and Vla=Vaxtl,r1tl,r2tl. The stability distribution law of Markovian variable ritl is πpi=π1,π2,…,πN. ∑j=1Nπj=1, and πj>0. ℓl=N∑k=12maxpk,qk=1Nγpkqkπpk/πqk. Various parameters associated with the selection need to be satisfied, ℓ−ℓl>0.(53)EVla=Vt0a+E∫t0tleℓ−ℓlsLVsads≤Vt0a+ℏtl−t0−ℓ−ℓl∫t0tlEVsads≤Vt0a+ℏtl−t0,where Vt0a=Vaxt0,r1t0,r2t0.
For∀xt0 and r1t0,r2t0∈S, there is a unique solution xt in the closed-loop systems.(54)Eeℓ−ℓltlVla−Eeℓ−ℓlt0Vt0a=E∫t0tleℓ−ℓlsLVsads+ℓ−ℓlE∫t0tleℓ−ℓlsLVsads≤−ℓ−ℓlE∫t0tleℓ−ℓlsLVsads+∫t0tlℏeℓ−ℓlsds+ℓ−ℓlE∫t0tleℓ−ℓlsVsadsEeℓ−ℓlℓlVla≤eℓ−ℓlt0Vt0a+ℏℓ−ℓleℓ−ℓlt−eℓ−ℓlt0EVla≤e−ℓ−ℓlt−t0Vt0a+ℏℓ−ℓl1−e−ℓ−ℓlt−t0≤Vt0a+ℏℓ−ℓl.
From the Lemma2, it can be concluded that all the signals can be guaranteed SGPFS. Meanwhile, the constraints of two different output factors are bounded.Theorem 1.
The finite-time FTC control can be implemented by designing the virtual signal (17), the controller in (40), and the updating law (39) and (41). Meantime, by selecting appropriate design parameters, the tracking error of the systems can be arbitrarily small in a finite-time.Proof.
Through Lemma2 in [37] and inequality (54), we can acquire that the trajectories of the all signals that satisfy Vaγt,θ˜t≤ℏ/1−θ0ℓ12, for any t≥Treach, in finite time, where Treach is defined as Treach=1/1−θ0ℓ12V1−aγ0,θ10−ℏ/1−θ0ℓ121−a/a with γ0=γ10,γ20T. In other words, all signals of the studied closed system are SGPFS.
Furthermore, the following result can be obtained,i=1,3. xi−yid≤ki1−e−2ℏ/1−θ0ℓ121/a1/2, where ki=minkai,kbi, In addition, the tracking error γi remains in a small neighborhood of the origin by finite-time control.
Whenqγi=1 and γi≥0, we can get γi≤kbi1−e−2ℏ/1−θ0ℓ121/a1/2≤kbi. Thus, when qγi=0 and γi<0, it can obtain γi≥−kai1−e−2ℏ/1−θ0ℓ121/a1/2≥−kai. By using −B¯i1≤yid≤B¯i1, it yields −k¯ci≤−kai+yid≤xi≤kbi+yid≤k¯ci. In short, xi is constrained in its limit.


### 3.2. Zero Dynamics and Performance Analysis

Through the analysis in Section3.1, the signals xi and θ^ji=1,2,3,4,j=1,2 are bounded. Then, we are going to prove the zero dynamics that consist of the other four states x5, x6, x7, and x8.Therefore, substituting (38)–(41) into (5), let γ1=γ2=γ3=γ4=0. It yields the following form:(55)Z˙=AZ+B1D1+B0D0+W,where(56)Z=x5,x6,x7,x8T,D0=D01,D˙01,D02,D˙02T,D1=Df−D1,D˙f−D˙1,Dr−D2,D˙r−D˙2T,A=0100−kf2mf−bf2mf00001000−kr2mr−br2mr,B0=0000kf2mfbf2mf00000000kr2mrkr2mr,B1=0000g˜1kf2mfg^1g˜1bf2mfg^100000000g˜2kr2mrg^2g˜2br2mrg^2,w=0−l2w1−w2mfl1+l20−l2w1+w2mrl1+l2,w1=−λ1γ1−12γ2−γ22δ12θ^1−c2γ22l−1,w2=−λ3γ3−12γ4−γ42δ32θ^2−c4γ42l−1.Notable is, due to the boundedness ofxi and θ^ji=1,2,3,4,j=1,2, w1 and w2 are bounded, namely, wi≤ϖi for the positive parameter ϖii=1,2. Furthermore, it means that W satisfies W≤W¯, for the parameter W¯>0. Meanwhile, according to Assumptions 1 and 2, D satisfies D≤D¯, for the parameter D¯>0. Choosing the positive matrixes P and Q can ensure the following equality holds, namely,(57)ATP+PTA=−Q.Define(58)VZ=ZTPZ,where VZ=VZZ,p1,p2, and P is a positive matrix.According to the infinitesimal generator in [50], (58) can be calculated at the mode of r1t,r2t=p1,p2 as(59)LVZ=Z˙TPZ+ZTPZ˙+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2,where Z1=Z1Z,p2,q1, and Z2=Z2Z,p1,q2.(60)LVZ=AZ+B1D1+B0D0+WTPZ,ZTPAZ+B1D1+B0D0+W+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2=ZTATP+PTAZ+2ZTPB1D1+2ZTPB0D0+2ZTPW,+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2.Based on Young’s inequality, it obtains(61)2ZTPB0D0≤λmaxPB0TB0Pϑ0Z2+ϑ0D¯02,2ZTPB1D1≤λmaxPB1TB1Pϑ1Z2+ϑ1D¯12,2ZTPW≤λmaxPPϑ2Z2+ϑ2W¯2,where the parameters ϑ0, ϑ1, and ϑ2 are positive.Combining (57) and (61), (60) can be rewritten as(62)LVZ=−ZTQZ+λmaxPB0TB0Pϑ0Z2+λmaxPB1TB1Pϑ1Z2+λmaxPPϑ2Z2+ℏZ+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2≤−ZTQ−λmaxPB0TB0Pϑ0−λmaxPB1TB1Pϑ1−λmaxPPϑ2Z+ℏZ+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2LVZ≤−ℓZVZ+ℏZ+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2,where(63)ℏZ=ϑ0D¯02+ϑ1D¯12+ϑ2W¯2,ℓZ=λminP−1/2QP−1/2−λmaxPϑ2−λmaxP1/2B0TB0P1/2ϑ0−λmaxP1/2B1TB1P1/2ϑ1.Then, same as the above discrete expectations, (62) yields(64)EVZ≤e−ℓ0t−t0VZ0+ℏZℓ01−e−ℓ0t−t0,where ℓ0=ℓZ−ℓl, and VZ0Zt0,r1t0,r2t0.According to Lemma2, all the signals of the zero dynamics are bounded.Now, the control objectives will be proved. The dynamic tire loads are given as(65)Ftf+Fbf≤kf2+bf2VZ0+ℏ3/ℓZλminP1/2+kf2D¯11+bf2D¯12,Ftr+Fbr≤kr2+br2VZ0+ℏ3/ℓZλminP1/2+kr2D¯21+br2D¯22.Thus, selecting appropriate parametersϑ1 and ϑ2 can ensure kf2+bf2ϑ3+kf2D¯11+bf2D¯12≤Ff, kr2+br2ϑ3+kr2D¯21+br2D¯22≤Fr hold, that is, (6) holds.Finally, we are going to learn about the suspension deflection performances. According to the analysis in Section3.1 and 3.2, one has(66)ΔD¯f≤x1+l2sinx3+x5≤x1+l2x3+x5≤kc1+l2kc2+VZ0+ℏ3/ℓZλminP1/2,ΔD¯r≤x1+l2sinx3+x7≤x1+l2x3+x7≤kc1+l2kc2+VZ0+ℏ3/ℓZλminP1/2.Thus, letkc1+l2kc2+VZ0+ℏ3/ℓZ/λminP1/2≤ΔD¯f and kc1+l2kc2+VZ0+ℏ3/ℓZ/λminP1/2≤ΔD¯r, and we have ΔD¯f≤ΔD¯f and ΔD¯r≤ΔD¯r, which imply the suspension deflection performances can be achieved.Remark 5.
The works [38, 39] mainly focused on the finite-time control problem of quarter active suspension systems. Compared with the foregoing, this study not only addresses the finite-time control of half-vehicle but also further studies constraints of displacement and pitch angle of the car body. Particularly, the random faults between different actuators are studied.


## 3.1. Finite-Time Constraint Control Scheme Design for Vertical Motion and Pitch Angle Motion

Now, for the vertical motion of active suspension systems, more details on finite-time approach will be given in the next section.Step 1.
SelectV=Vxt,r1t,r2t as a Lyapunov function candidate in the following form:(16)V=∑i=1,31−qγi2logkai2kai2−γi2+qγi2logkbi2kbi2−γi2+∑i=1212ρiγ2i2+12ηiθ˜i2+12χig˜i2rit,where xt=x1t,x2t,x3t,x4t, γ1=x1,r1t,r2t, γ2=x1,x2,r1t,r2t, γ3=x3,r1t,r2t, and γ4=x3,x4,r1t,r2t. The parameters ηi and χi are positive, and θ˜i and g˜irit represent the errors. θ^i is the estimator of θi. g^irit is the estimator of girit. Then, θ˜i=θi−θ^i, g˜irit=girit−g^irit, i=1,2. It would make an assumption: ∀t>0, r1t=p1, r2t=p2, and p1,p2∈S.Remark 2.
For quarter ASSs, the study in [30] only showed the vehicle body’s displacement constraint method by the symmetric BLFs. In contrast, we further study the pitch angle constraint of nonlinear uncertain half ASSs by asymmetric BLFs in this study.
In addition, the virtual controller signal is designed as(17)βi=−ciγi21−qγikai2−γi2+qγikbi2−γi2+y˙id,where i=1,3. The parameter ci is a design parameter, ci>0. kai and kbi are the constraint bounded of corresponding variables, respectively. Moreover,(18)q⋅=1,⋅≥0,0,⋅<0.
Then, according to the infinitesimal generator ofV, it gets(19)LV=∑i=1,31−qγiγiγ˙ikai2−γi2+qγiγiγ˙ikbi2−γi2+∑i=121ρiγ2iγ˙2i+1ηiθ˜iθ˜˙i+1χig˜ipig˜˙ipi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2,where V1=xt,p2,q1, V2=xt,p1,q2, and ϒpiqi>0 is the transition rate, i=1,2.Remark 3.
∑q1=1Nϒp1q1V1 and ∑q2=1Nϒp2q2V2 are the additional terms due to the involvement of Markovian variables r1t and r2t, which do not exist in the determined situation. The extratransition rate-related terms appear in the infinitesimal generator of Lyapunov candidate function. These additions need further processing. It is a challenge that cannot be ignored in stability analysis.
From the previous definition,i=1,2, there are(20)g˜˙irit=−g^˙irit,θ˜˙i=−θ^˙i.
Combining with (19) and (20) can obtain(21)LV=∑i=1,3γi1−qγikai2−γi2+qγikbi2−γi2γ˙i+∑i=121ρiγ2iγ˙2i−∑i=121χig˜ipig^˙ipi+1ηiθ˜iθ^˙i+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2.
Furthermore, taking the derivative ofγi, it yields(22)γ˙i=γi+1+βi−y˙id,i=1,3.
Substituting (17) and (22) into (21), it can attain(23)γi1−qγikai2−γi2+qγikbi2−γi2γ˙i=λiγiγi+1−ciγi221−qγikai2−γi2+qγikbi2−γi2=λiγiγi+1−ci1−qγi2γi2kai2−γi2−ciqγi2γi2kbi2−γi2,where λi=1−qγi/kai2−γi2+qγi/kbi2−γi2, ci is a positive design parameter, and i=1,3.
Based Lemma3, let zi=γi2/2kai2−γi2, α=1, a=2n−1/2n+1, n∈ℕ, b=1−a, and μ=aa/1−a obtain(24)γi22kai2−γi2a≤1−aμ+γi22kai2−γi2.
Therefore, it gets(25)−γi22kai2−γi2≤1−aμ−γi22kai2−γi2a.
Constructed in the same way, it obtains(26)−1−qγiγi22kai2−γi2≤1−aμ−1−qγiaγi22kai2−γi2a.
Similarly, it yields(27)−γi22kbi2−γi2≤1−aμ−γi22kbi2−γi2a,−qγiγi22kbi2−γi2≤1−a−qγiaγi22kbi2−γi2a.
According to Lemma4, for any constant a, 0<a<1, the following inequality holds:(28)log1kai2−γi2a≤γi2kai2−γi2a,log1kbi2−γi2a≤γi2kbi2−γi2a.
Next, substituting (26)–(28) into (23), LV is expressed as(29)LV=∑i=1,3λiγiγi+1−ci1−qγi2log1kai2−γi2a−ciqγi2log1kbi2−γi2a+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ1+∑i=121ρiγ2iγ˙2i−1χig˜ipig^˙ipi−1ηiθ˜iθ^˙i,where ℏ1=∑i=1,32ci1−aμ.
Taking the derivative ofγ2 and γ4, they yield(30)γ˙2=F1+ρ1uz−β˙1,γ˙4=F2+ρ2uφ−β˙3,where β˙i=∂βi/∂xix˙i+∂βi/∂yidy˙id+∂βi/∂y˙idy¨id, i=1,3. βi has been designed in (17). Then, ∑i=121/ρiγ2iγ˙2i gets(31)1ρ1γ2γ˙2=γ2F¯1+uz,1ρ2γ4γ˙4=γ4F¯2+uφ,where the unknown functions F¯1 and F¯2 are denoted as(32)F¯1=ρ1−1F1−ρ1−1β˙1,F¯2=ρ2−1F2−ρ2−1β˙3.
Using the powerful approximating ability of NNs in [42, 43] for uncertain nonlinear systems, F¯1 and F¯2 can employ(33)F¯1=Θ1ΦZ1T+τ1,F¯2=Θ2ΦZ2T+τ2,where Z1=x1,x2,x3,x4,x5,x6,x7,x8,β1,β˙1,y1d,y˙1dT, Z2=x1,x2,x3,x4,x5,x6,x7,x8,β3,β˙3,y3d,y˙3dT, and τi is the approximation error and the designed positive constant τ¯i that satisfies τi≤τ¯i. More remarkable, 0<Θ1iΦZ1<1 and 0<Θ2iΦZ2<1. Furthermore, we can draw a conclusion 0<Φ1TZ1Φ1Z1≤N1 and 0<Φ2TZ2Φ2Z2≤N2, where N1 and N2 are the corresponding numbers of NNs nodes.
Then, applying Young’s inequality, it can get the following:(34)γ2iΘiΦZiT≤γ2i2Θi2ΦZiTΦZi2δi2+δi22=γ2i2θi2δi2+δi22,(35)γ2iτi≤12γ2i2+12τi2≤12γ2i2+12τ¯i2,where θi=NiΘi2 and the design parameter δi is positive.
Substituting (33)–(35) into (31), it obtains(36)∑i=121ρiγ2iγ˙2i≤γ2γ2θ12δ12+γ22+uz+γ4γ4θ22δ22+γ42+uφ+∑i=12δi22+τ¯i22.
Substituting (36) into (29), it can obtain(37)LV=∑i=1,3λiγiγi+1−ci1−qγi2log1kai2−γi2a−ciqγi2log1kbi2−γi2a+γ2γ2θ12δ12+γ22+uz+γ4γ4θ22δ22+γ42+uφ−∑i=12g˜ipig^˙ipiχi+θ˜iθ^˙iηi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ2,where ℏ2=ℏ1+∑i=12δi2/2+τ¯i2/2.
Design the control inputuz and uφ that are subjects to random actuator faults as(38)uz=−λ1γ1−γ22−γ2θ^12δ12−c2γ22a−1,uφ=−λ3γ3−γ42−γ3θ^22δ32−c4γ42a−1,where the parameters c2 and c4 are positive.
The adaptive law is established as(39)θ^˙i=ηi2δi2γ2i2−κiθ^i,i=1,2,where the designed parameter κi is positive.
Based on (1), (2), and (39), uf and ur are shown as(40)uf=l1uz+uφl1+l2,ur=l2uz−uφl1+l2,Ψft=1g^1l1+l2l1−λ1γ1−γ22−γ2θ^12δ12−c2γ22a−1−λ3γ3−γ42−γ3θ^22δ32−c4γ42a−1,Ψrt=1g^2l1+l2l2−λ1γ1−γ22−γ2θ^12δ12−c2γ22a−1+λ3γ3+γ42+γ3θ^22δ32+c4γ42a−1.
The updating laws of estimated parameters are established:(41)g^˙1p1=projg^1p1γ2+l1γ4χ1ψft−2cg1g^1p10,g^1p1=g1min,γ2+l1γ4χ1ψft−2cg1g^1p1,g^1p1,others,g^˙2p2=projg^2p2γ2−l2γ4χ2ψrt−2cg2g^2p20,g^2p2=g2min,γ2−l1γ4χ2ψrt−2cg2g^2p2,g^2p2,others,where proj represents the projection operator in order to avoid singular values in the denominator.Remark 4.
The adaptive estimated parametersg^˙1p1 and g^˙2p2 appear in the denominators in (41), which may cause the controllers to fail. Construct the projection form to avoid this situation.
Substituting (39)–(41) into (37), LV can be expressed as(42)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2iγ2i2a+∑i=12cgig˜ipig^ipiχi+κiθ˜iθ^iηi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ2.
By Young’s inequalityi=1,2, they can obtain(43)θ˜iθ^i≤12θ˜i2+12θi2,g˜ipig^ipi≤−12g˜i2pi+12g^i2pi.
LV can be rewritten as(44)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2iγ2i2a+cgig˜i2pi2χi+κiθ˜i22ηi+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ3,where ℏ3=ℏ2+∑i=12κiθi2/2ηi+cgigi2pi/2χi.
Based on Lemma3, let z=θ˜i2/2ηia=2n−1/2n+1, n∈ℕ, b=1−a, α=1, and μ=aa/1−a, one has(45)θ˜i22ηia≤1−aμ+θ˜i22ηi.
Furthermore, we can obtain(46)−θ˜i22ηi≤1−aμ−θ˜i22ηia.
Applying the inequality (44) and (46) can be expressed as(47)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2iγ2i2a+cgig˜i2pi2χi+κiθ˜i22ηia+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ,where ℏ=ℏ3+1−aμ.
In addition, the number of passengers and the load conditions of the vehicle are limited. The massM is bounded which means ρi is bounded, namely,(48)ρ¯i≤ρi≤ρ¯i,i=1,2.
Based on (48),(49)LV≤−∑i=1,3ci1−qγi2log1kai2−γi2a+qγi2log1kbi2−γi2a−∑i=12c2i2ρ¯iaγ2i2a2ρ¯ia+cgig˜i2pi2χi+κiθ˜i22ηia+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2+ℏ.
From (16), (49) can be rewritten as(50)LV≤−ℓVa+ℏ+∑q1=1Nϒp1q1V1+∑q2=1Nϒp2q2V2,where ℓ=min−c22ρ¯1a,c42ρ¯2a,c1,c3,κ1,κ2,cg1,cg2.
The following will apply the discrete expectation to address the random terms.(51)limx⟶∞inft≥t0V=∞.
There would be aB, no matter how big it is; once more than t>tl, it would exceed B. From (51) in t0,tl, xt≤B. (50) is bounded in t0,tl.
Then, according to property of discrete expectations, it yields(52)ELV≤∑p1=1N∑p2=1NELVlπp1πp2≤−ℓ∑p1=1N∑p2=1NELVlaπp1πp2+ℏ+∑p1=1N∑p2=1NE∑q1=1Nϒp1q1V1l+∑q2=1Nϒp2q2V2lπp1πp2≤−ℓ∑p1=1N∑p2=1NELVla+ℏ+N∑k=12maxpk,qk=1NγpkqkπpkπqkEVla≤−ℓ−ℓlEVla+ℏ,where Vl=Vxtl,r1tl,r2tl, V1l=xtl,p2,q1, V2l=xtl,p1,q2, and Vla=Vaxtl,r1tl,r2tl. The stability distribution law of Markovian variable ritl is πpi=π1,π2,…,πN. ∑j=1Nπj=1, and πj>0. ℓl=N∑k=12maxpk,qk=1Nγpkqkπpk/πqk. Various parameters associated with the selection need to be satisfied, ℓ−ℓl>0.(53)EVla=Vt0a+E∫t0tleℓ−ℓlsLVsads≤Vt0a+ℏtl−t0−ℓ−ℓl∫t0tlEVsads≤Vt0a+ℏtl−t0,where Vt0a=Vaxt0,r1t0,r2t0.
For∀xt0 and r1t0,r2t0∈S, there is a unique solution xt in the closed-loop systems.(54)Eeℓ−ℓltlVla−Eeℓ−ℓlt0Vt0a=E∫t0tleℓ−ℓlsLVsads+ℓ−ℓlE∫t0tleℓ−ℓlsLVsads≤−ℓ−ℓlE∫t0tleℓ−ℓlsLVsads+∫t0tlℏeℓ−ℓlsds+ℓ−ℓlE∫t0tleℓ−ℓlsVsadsEeℓ−ℓlℓlVla≤eℓ−ℓlt0Vt0a+ℏℓ−ℓleℓ−ℓlt−eℓ−ℓlt0EVla≤e−ℓ−ℓlt−t0Vt0a+ℏℓ−ℓl1−e−ℓ−ℓlt−t0≤Vt0a+ℏℓ−ℓl.
From the Lemma2, it can be concluded that all the signals can be guaranteed SGPFS. Meanwhile, the constraints of two different output factors are bounded.Theorem 1.
The finite-time FTC control can be implemented by designing the virtual signal (17), the controller in (40), and the updating law (39) and (41). Meantime, by selecting appropriate design parameters, the tracking error of the systems can be arbitrarily small in a finite-time.Proof.
Through Lemma2 in [37] and inequality (54), we can acquire that the trajectories of the all signals that satisfy Vaγt,θ˜t≤ℏ/1−θ0ℓ12, for any t≥Treach, in finite time, where Treach is defined as Treach=1/1−θ0ℓ12V1−aγ0,θ10−ℏ/1−θ0ℓ121−a/a with γ0=γ10,γ20T. In other words, all signals of the studied closed system are SGPFS.
Furthermore, the following result can be obtained,i=1,3. xi−yid≤ki1−e−2ℏ/1−θ0ℓ121/a1/2, where ki=minkai,kbi, In addition, the tracking error γi remains in a small neighborhood of the origin by finite-time control.
Whenqγi=1 and γi≥0, we can get γi≤kbi1−e−2ℏ/1−θ0ℓ121/a1/2≤kbi. Thus, when qγi=0 and γi<0, it can obtain γi≥−kai1−e−2ℏ/1−θ0ℓ121/a1/2≥−kai. By using −B¯i1≤yid≤B¯i1, it yields −k¯ci≤−kai+yid≤xi≤kbi+yid≤k¯ci. In short, xi is constrained in its limit.


## 3.2. Zero Dynamics and Performance Analysis

Through the analysis in Section3.1, the signals xi and θ^ji=1,2,3,4,j=1,2 are bounded. Then, we are going to prove the zero dynamics that consist of the other four states x5, x6, x7, and x8.Therefore, substituting (38)–(41) into (5), let γ1=γ2=γ3=γ4=0. It yields the following form:(55)Z˙=AZ+B1D1+B0D0+W,where(56)Z=x5,x6,x7,x8T,D0=D01,D˙01,D02,D˙02T,D1=Df−D1,D˙f−D˙1,Dr−D2,D˙r−D˙2T,A=0100−kf2mf−bf2mf00001000−kr2mr−br2mr,B0=0000kf2mfbf2mf00000000kr2mrkr2mr,B1=0000g˜1kf2mfg^1g˜1bf2mfg^100000000g˜2kr2mrg^2g˜2br2mrg^2,w=0−l2w1−w2mfl1+l20−l2w1+w2mrl1+l2,w1=−λ1γ1−12γ2−γ22δ12θ^1−c2γ22l−1,w2=−λ3γ3−12γ4−γ42δ32θ^2−c4γ42l−1.Notable is, due to the boundedness ofxi and θ^ji=1,2,3,4,j=1,2, w1 and w2 are bounded, namely, wi≤ϖi for the positive parameter ϖii=1,2. Furthermore, it means that W satisfies W≤W¯, for the parameter W¯>0. Meanwhile, according to Assumptions 1 and 2, D satisfies D≤D¯, for the parameter D¯>0. Choosing the positive matrixes P and Q can ensure the following equality holds, namely,(57)ATP+PTA=−Q.Define(58)VZ=ZTPZ,where VZ=VZZ,p1,p2, and P is a positive matrix.According to the infinitesimal generator in [50], (58) can be calculated at the mode of r1t,r2t=p1,p2 as(59)LVZ=Z˙TPZ+ZTPZ˙+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2,where Z1=Z1Z,p2,q1, and Z2=Z2Z,p1,q2.(60)LVZ=AZ+B1D1+B0D0+WTPZ,ZTPAZ+B1D1+B0D0+W+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2=ZTATP+PTAZ+2ZTPB1D1+2ZTPB0D0+2ZTPW,+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2.Based on Young’s inequality, it obtains(61)2ZTPB0D0≤λmaxPB0TB0Pϑ0Z2+ϑ0D¯02,2ZTPB1D1≤λmaxPB1TB1Pϑ1Z2+ϑ1D¯12,2ZTPW≤λmaxPPϑ2Z2+ϑ2W¯2,where the parameters ϑ0, ϑ1, and ϑ2 are positive.Combining (57) and (61), (60) can be rewritten as(62)LVZ=−ZTQZ+λmaxPB0TB0Pϑ0Z2+λmaxPB1TB1Pϑ1Z2+λmaxPPϑ2Z2+ℏZ+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2≤−ZTQ−λmaxPB0TB0Pϑ0−λmaxPB1TB1Pϑ1−λmaxPPϑ2Z+ℏZ+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2LVZ≤−ℓZVZ+ℏZ+∑q1=1Nϒp1q1Z1+∑q2=1Nϒp2q2Z2,where(63)ℏZ=ϑ0D¯02+ϑ1D¯12+ϑ2W¯2,ℓZ=λminP−1/2QP−1/2−λmaxPϑ2−λmaxP1/2B0TB0P1/2ϑ0−λmaxP1/2B1TB1P1/2ϑ1.Then, same as the above discrete expectations, (62) yields(64)EVZ≤e−ℓ0t−t0VZ0+ℏZℓ01−e−ℓ0t−t0,where ℓ0=ℓZ−ℓl, and VZ0Zt0,r1t0,r2t0.According to Lemma2, all the signals of the zero dynamics are bounded.Now, the control objectives will be proved. The dynamic tire loads are given as(65)Ftf+Fbf≤kf2+bf2VZ0+ℏ3/ℓZλminP1/2+kf2D¯11+bf2D¯12,Ftr+Fbr≤kr2+br2VZ0+ℏ3/ℓZλminP1/2+kr2D¯21+br2D¯22.Thus, selecting appropriate parametersϑ1 and ϑ2 can ensure kf2+bf2ϑ3+kf2D¯11+bf2D¯12≤Ff, kr2+br2ϑ3+kr2D¯21+br2D¯22≤Fr hold, that is, (6) holds.Finally, we are going to learn about the suspension deflection performances. According to the analysis in Section3.1 and 3.2, one has(66)ΔD¯f≤x1+l2sinx3+x5≤x1+l2x3+x5≤kc1+l2kc2+VZ0+ℏ3/ℓZλminP1/2,ΔD¯r≤x1+l2sinx3+x7≤x1+l2x3+x7≤kc1+l2kc2+VZ0+ℏ3/ℓZλminP1/2.Thus, letkc1+l2kc2+VZ0+ℏ3/ℓZ/λminP1/2≤ΔD¯f and kc1+l2kc2+VZ0+ℏ3/ℓZ/λminP1/2≤ΔD¯r, and we have ΔD¯f≤ΔD¯f and ΔD¯r≤ΔD¯r, which imply the suspension deflection performances can be achieved.Remark 5.
The works [38, 39] mainly focused on the finite-time control problem of quarter active suspension systems. Compared with the foregoing, this study not only addresses the finite-time control of half-vehicle but also further studies constraints of displacement and pitch angle of the car body. Particularly, the random faults between different actuators are studied.


## 4. Simulation Example

In this section, an example is given to demonstrate the established approach for half-vehicle ASSs. In order to facilitate the implement of the simulation, the nonlinear stiffness and dampers are given in [6](67)Fsf=kf01ΔDf+kf02ΔDf3,Fsr=kr01ΔDr+kr02ΔDr3,Ftf=kf2D1−D01,Ftr=kr2D2−D02,Fbf=bf2D˙1−D˙01,Fbr=br2D˙2−D˙02,Fdf=de1Δz˙f,dc1Δz˙f,Fdr=de2Δz˙r,dc2Δz˙r,where kf01 and kf02 are the stiffness coefficients of linear and nonlinear terms for the front active suspension systems; kr01 and kr02 are the stiffness coefficients of linear and nonlinear terms for the rear active suspension systems; kf2 and bf2 stand for the damping coefficients of front and rear wheels; kr2 and br2 denote the damping coefficients of front and rear wheels; de1 and dc1 are the segmented coefficients of front damper of active suspension systems; and de2 and dc2 are the segmented coefficients of front damper of active suspension systems.The road profile is an important aspect of affecting suspension performance. Thus, two different road inputs are carried out and given as follows:S1: the road input of the front wheel is described as(68)D01=0.5h1+sin2πvLt,t1≤t≤th,0,otherwise,whereh and L are the height and the length of the bump, and v is the velocity of the vehicle. The road input D02 for the rear wheel is implemented as D02t=D01t−τ. Suppose h=0.05m, L=2.5m, v=18km/h, t1=2, th=2.5, and time delay τ=0.5.S2: the road input of the front wheel is presented as(69)D01=0.5hb1+sin2πvbLt,t1≤t≤Lvb,0,t>Lvb,where hb=0.08m, L=2.5m, and vb=25km/h. The road input D02 is carried out the same as S1.The parameters in half-vehicle active suspension systems are briefly illustrated:M=1200kg, I=1000kgm2, mf=mr=100kg, kf01=kr01=15000N/m, kf02=kr02=1000N/m, kf2=200000N/m, l1=1.5m, kr2=150000N/m, bf2=1500Ns/m, br2=2000Ns/m, de1=de2=1500Ns/m, dc1=dc2=1200Ns/m, and l2=1.2m. The control parameters in this work are given as follows: η1=4, η2=6, δ1=3, δ2=2, κ1=30, κ2=40, c1=c3=50, c2=c4=100, a=299/300, ka1=0.006, kb1=0.01, ka2=0.012, kb2=0.011, k¯c1=0.012, k¯c1=0.009, k¯c2=0.013, k¯c2=0.012, and the rest ones are assumed as zeros. In simulation examples, the initial state values are given as x0=0,0,0,0,0,0,0,0T, S0,1. For r1t: γpq2×2=−443−3; r2t: γpq2×2=−334−4.For the casesS1 and S2, the curves of r1t and r2t are shown in Figure 2; the trajectories of x1 which illustrates the displacement of vertical motion are shown in Figure 3. It can be seen that the displacement x1 does not exceeds its constraint bounded.Figure 2
The curves ofr1t and r2t.
(a)(b)Figure 3
The vertical motion displacementx1.The pitch angleφ is constrained under its bounded and can stabilize quickly in a finite time, as shown in Figure 4. Figure 5 shows the accelerations of vertical motion and pitch motion. It is observed that the maximums of vertical motion and pitch motion are 4 m/s2 and 5 m/s2, respectively. In addition, the stabilization of these motions is achieved in 1 sec for cases S1 and S2. The purpose of Figure 6 is to show the trajectories of θ^1 and θ^2. It indicates that for cases S1 and S2, they can converge to zero in a very short period of time. Figure 7 shows the controllers uf and ur, and we can observe that the control forces are calculated about 300 N. It means that the designed controllers play a key role in controlling the variation of active suspension systems. Figure 8 shows that the front wheel and rear wheel of the suspension spaces are all within the bounds. Through the above analysis, the proposed control schemes could be used to help reduce the variation of active suspension systems and achieve a good performance in a finite time.Figure 4
The pitch angleφ.Figure 5
The accelerations of active suspension systems motion.
(a)(b)Figure 6
Trajectories ofθ^ii=1,2.
(a)(b)Figure 7
The controllers with fault tolerant.
(a)(b)Figure 8
The suspension spaces.
(a)(b)


## 5. Conclusions

In order to achieve good suspension performances, the finite-time control for half active suspension systems have been discussed in this study. The asymmetric BLFs have been employed to constrain vehicle body’s displacement and pitch angle in a safety range. Then, utilizing the finite-time control design theory, the adaptive controllers were proposed for vertical motion and pitch motion, respectively. Finally, the effectiveness of designed schemes is demonstrated with the help of results of simulation examples. On the one hand, in the aspect of automobile research, we would further study the battery problem combined with the distributed controller of the multiagent consistency algorithm and apply it to distributed generators in the Energy Internet such as [56, 57]. On the other hand, we apply this method in the pipeline network system such as [58] and optimal deployment of agents in the pipeline network system in [59]. The study of control problem in the pipeline network system will be an interesting research topic.



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*Source: 5964034-2021-07-10.xml*