Research Article | Open Access

Jie Lan, Tongyu Xu, "Adaptive Finite-Time Fault-Tolerant Control for Half-Vehicle Active Suspension Systems with Output Constraints and Random Actuator Failures", *Complexity*, vol. 2021, Article ID 5964034, 16 pages, 2021. https://doi.org/10.1155/2021/5964034

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

**Academic Editor:**Rui Wang

#### 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.

#### 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 Section 2, 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

Figure 1 shows a half-vehicle suspension model. represents the mass of the vehicle body. is the mass moment of inertia. and are the defined masses of front and rear wheels, respectively. stands for the vertical displacement of the vehicle body. represents the pitch angle. and stand for the displacements of the front and rear vehicle body, respectively. and are the displacements of the front and rear wheels, respectively. and represent the road inputs of corresponding wheels. , , , and are defined as the forces produced by the related stiffness. , , , and are defined as the forces produced by the related dampers. and represent the control forces of the front and rear ASSs.

For half ASSs, the state space equations of vertical motion and pitch motion are shown in the following equation:where and .

The stochastic actuator failures are considered and described as follows:where and are the inputs of the front actuator and rear actuator, respectively. and are the independent irreducible right continuous homogeneous Markovian processes on the probability space , taking values in a finite set with generator matrix , where is the transition rate from mode to mode if , and .

In addition, and are the stochastic functions which represent the failure scaling factors for two different actuators and take values on the interval , and , .

*Remark 1. *The values and meet as follows,(1)When or , the front actuator or the rear actuator is healthy(2)When or , there is partial failure of the corresponding actuatorEach 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:Then, it obtainswhere , , , and .

The mass and the mass moment of inertia are uncertain due to the uncertainties of passengers and the load quality of cargoes. Therefore, and 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 controllers and are subjected to random Markovian jumping failures in (2). The vertical displacement and pitch angle motion are considered and limited in safety ranges. and 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.,where , and .

Last, suspension constraints must be guaranteed because of the confined mechanical space, that is, suspension deflections should not exceed their maximum values.where , and .

There are some assumptions, definitions, and lemmas presented in order to facilitate the design and analysis of adaptive control scheme.

*Assumption 1. *For given constants , , , and , the following results on , , , and hold, i.e.,where , , , and , for positive parameters , , , , , and are selected and positive parameters.

*Assumption 2. *In general, the inputs of the road and their time derivative of and are limited. Therefore, existing positive parameters make sure the following inequality:where the parameters , , , and are positive.

*Definition 1 (see [35]). *Consider the following nonlinear systems:where and represent the state and input vectors, respectively. If any initial condition satisfies , for , the system (10) is semiglobal practical finite-time stable (SGPFS). In addition, the state vector satisfies , for , where is a positive parameter and a deposit time satisfies .

Lemma 1 (see [36]). *For any , , , , the following inequality can be constructed, namely,*

Lemma 2 (see [37]). *Consider the Lyapunov function with stochastic terms, and the following inequality holds for a nonlinear system (10), namely,where is the differential operator, the constants and are positive, and . and . Then, stochastic trajectory of (10) is SGPFS.*

Lemma 3 (see [40]). *For any and are real variables, the following inequality can be constructed, namely,where the parameters , , and are positive.*

Lemma 4 (see [23]). *For all , the following inequality can be constructed as*

#### 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.,where and are the virtual control signals. for is the error variable. and 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. *Select as a Lyapunov function candidate in the following form:where , , , , and . The parameters and are positive, and and represent the errors. is the estimator of . is the estimator of . Then, , , . It would make an assumption: , , , and .

*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 aswhere . The parameter is a design parameter, . and are the constraint bounded of corresponding variables, respectively. Moreover,Then, according to the infinitesimal generator of , it getswhere , , and is the transition rate, .

*Remark 3. * and are the additional terms due to the involvement of Markovian variables and , 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, , there areCombining with (19) and (20) can obtainFurthermore, taking the derivative of , it yieldsSubstituting (17) and (22) into (21), it can attainwhere , is a positive design parameter, and .

Based Lemma 3, let , , , , , and obtainTherefore, it getsConstructed in the same way, it obtainsSimilarly, it yieldsAccording to Lemma 4, for any constant , , the following inequality holds:Next, substituting (26)–(28) into (23), is expressed aswhere .

Taking the derivative of and , they yieldwhere , . has been designed in (17). Then, getswhere the unknown functions and are denoted asUsing the powerful approximating ability of NNs in [42, 43] for uncertain nonlinear systems, and can employwhere , , and is the approximation error and the designed positive constant that satisfies . More remarkable, and . Furthermore, we can draw a conclusion and , where and are the corresponding numbers of NNs nodes.

Then, applying Young’s inequality, it can get the following:where and the design parameter is positive.

Substituting (33)–(35) into (31), it obtainsSubstituting (36) into (29), it can obtainwhere .

Design the control input and that are subjects to random actuator faults aswhere the parameters and are positive.

The adaptive law is established aswhere the designed parameter is positive.

Based on (1), (2), and (39), and are shown asThe updating laws of estimated parameters are established:where represents the projection operator in order to avoid singular values in the denominator.

*Remark 4. *The adaptive estimated parameters and 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), can be expressed asBy Young’s inequality , they can obtain can be rewritten aswhere .

Based on Lemma 3, let , , , , and , one hasFurthermore, we can obtainApplying the inequality (44) and (46) can be expressed aswhere .

In addition, the number of passengers and the load conditions of the vehicle are limited. The mass is bounded which means is bounded, namely,Based on (48),From (16), (49) can be rewritten aswhere .

The following will apply the discrete expectation to address the random terms.There would be a , no matter how big it is; once more than , it would exceed . From (51) in , . (50) is bounded in .

Then, according to property of discrete expectations, it yieldswhere , , , and . The stability distribution law of Markovian variable is . , and . . Various parameters associated with the selection need to be satisfied, .where .

For and , there is a unique solution in the closed-loop systems.From the Lemma 2, 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 Lemma 2 in [37] and inequality (54), we can acquire that the trajectories of the all signals that satisfy , for any , in finite time, where is defined as with . In other words, all signals of the studied closed system are SGPFS.

Furthermore, the following result can be obtained, . , where , In addition, the tracking error remains in a small neighborhood of the origin by finite-time control.

When and , we can get . Thus, when and , it can obtain . By using , it yields . In short, is constrained in its limit.

##### 3.2. Zero Dynamics and Performance Analysis

Through the analysis in Section 3.1, the signals and are bounded. Then, we are going to prove the zero dynamics that consist of the other four states , , , and .

Therefore, substituting (38)–(41) into (5), let . It yields the following form:where