Author’s Accepted Manuscript Adaptive sliding fault tolerant control for nonlinear uncertain active suspension systems Shubo Liu, Huanyin Zhou, Xianxi Luo, Jing Xiao
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S0016-0032(15)00410-X http://dx.doi.org/10.1016/j.jfranklin.2015.11.002 FI2476
To appear in: Journal of the Franklin Institute Received date: 28 January 2015 Revised date: 12 October 2015 Accepted date: 9 November 2015 Cite this article as: Shubo Liu, Huanyin Zhou, Xianxi Luo and Jing Xiao, Adaptive sliding fault tolerant control for nonlinear uncertain active suspension s y s t e m s , Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.11.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Adaptive sliding fault tolerant control for nonlinear uncertain active suspension systems
Shubo Liu*, Huanyin Zhou, Xianxi Luo, Jing Xiao Jiangxi Province Engineering Research Center of New Energy Technology and Equipment, East China Institute of Technology, Nanchang 330013, PR China *Corresponding author. Tel.: +86-791-83897628. Fax: +86-791-83898562 E-mail address:
[email protected]
Abstract: This paper is concerned with the fault tolerant control problem of nonlinear uncertain active suspension systems with constraint requirements. A novel adaptive sliding fault tolerant controller, which does depend on accurate models, is designed to stabilize the active suspension systems and thus to improve the ride comfort, without utilizing the bounds of actuator faults and parameter uncertainties. Furthermore, an H∞ optimization scheme based on differential evolution (DE) algorithm and linear matrix inequalities (LMIs) is introduced to design appropriate parameters of the sliding surface, which guarantees the constraint requirements of active suspension systems. Finally, simulation results are included to illustrate the effectiveness of the proposed strategy. Keywords: Nonlinear uncertain active suspension; Sliding fault tolerant control; Differential evolution (DE); Linear matrix inequalities (LMIs) 1 Introduction With the development of automotive industry, vehicle suspensions have attracted significant attention of academics as well as engineers, due to the potential to improve ride comfort, vehicle maneuverability and safety of passengers. Since the vibration isolation capabilities of traditional passive suspensions and semi-active suspensions are restricted, active suspensions, in which the passive components are augmented by actuators that supply additional forces, are developed to achieve promising performance under different riding conditions. Consequently, many diverse actuator control strategies, such as optimal control [1], preview method [2-3], robust control [4-7], fuzzy control [8-9], adaptive control [10-11] among many others, have been proposed during the past few decades. A backstepping control design in [12] is proposed for the control of a vehicle active suspension system. In [13], a functional approximation based adaptive sliding controller with fuzzy compensation is
developed and successfully applied to a quarter-car hydraulic active suspension system, whereas in [14] a road-adaptive nonlinear control in active suspensions is developed, which adjusts the control parameters according to different road profiles. In lots of practical applications, actuator faults may occur occasionally in active suspensions, and therefore actuators are not able to generate the desired forces, resulting in serious degradation of the system stability and performance. Since the adverse effects of actuator faults have not been fully appreciated by most researchers, almost all the studies related to active suspension control only focus on normal cases, and few of them incorporate the actuator faults in the control design of active suspensions. In [15-16], optimal robust fault tolerant control for active suspension systems is investigated based on state feedback. In [17-18], a reconfigurable fault tolerant control strategy is put forward, which further enhances the reliability and quality of active suspension systems. However, in the aforementioned strategies, the vehicle suspension systems are assumed to be with linear dynamics which could be precisely obtained. As a matter of fact, active suspensions are typical nonlinear uncertain systems suffering from various road disturbances. Suspension springs and dampers possess the nonlinear characteristics, whose upper bounds are difficult to obtain in practice. In this sense, the performance of the aforementioned schemes based on a linear model may not be achieved for practical suspension systems with nonlinear dynamics. In addition, suspension systems have some parameters that are inherently uncertain in real vehicles. For example, sprung mass depends on the total load of the vehicle; springs as well as dampers may lose efficiency gradually during their lifetime due to wear and tear. Therefore, it is more appropriate to design fault tolerant controllers for active suspension systems if all the issues such as nonlinearity, uncertainty and disturbance rejection are taken into account. Sliding mode control (SMC) has been recognized as one of the effective tools to design robust controllers for nonlinear systems working under various uncertainty conditions. The major advantage of SMC is the low sensitivity to system uncertainties and disturbances, which relaxes the necessity of exact modeling. Over the past few years, SMC has been successfully designed for active suspension systems [19-22]. Unfortunately, few researches of fault tolerant control utilizing SMC for active suspensions have been reported so far. Therefore, it is challenging to design a SMC-based controller such that the system stability and performance of the active suspension closed-loop system can be tolerated with actuator faults. The above considerations motivate the study in this paper. In the present work, an adaptive sliding fault tolerant strategy for nonlinear uncertain active suspension systems is proposed to stabilize the active suspension systems and improve the ride comfort. Moreover, the parameters of the sliding surface are optimized by solving a finite number of linear matrix equalities (LMIs) embedded in differential evolution (DE) algorithm in order to ensure the constraint performances. The primary contributions of the paper can be summarized as follows: The proposed strategy is applicable in engineering since it just utilizes the practically accessible state variables, such as the vertical displacement as
well as velocity of sprung mass, as feedback signals. Unlike the conventional SMC, the proposed control strategy does not require an accurate mathematical model, which avoids the dependence on the upper bounds of system uncertainties and disturbances. A new mathematical model describing the nonlinear spring and damper characteristics is exploited in the controller design, which may cover more practical application cases compared to other available results [12, 23-25] based on some kind of specific nonlinear dynamics. Finally, in order to ensure the constraint performances of active suspensions, parameters of the sliding surface are ultimately selected based on an effective H∞ optimization approach using LMIs and DE without the trial-and-error effort. The subsequent parts of this paper are organized as follows. Section 2 formulates the problem of controller design for active suspension systems. The adaptive fault tolerant control design is proposed in Section 3, and an H∞ optimization algorithm for designing the sliding surface parameters is investigated in Section 4. Section 5 presents the design results and performance evaluations. Finally, we conclude our findings in Section 6. Notation: For a matrix P, PT denotes its transpose; the notation P>0(<0) means that P is real symmetric and positive (negative) definite. Rn×m is the set of all n×m real matrices. Also, I is used to denote the identity matrix of appropriate dimensions. In symmetric block matrices, an asterisk * represents a term that is readily inferred by symmetry. For a transfer function G(s) H∞, its norm is denoted by ||G(s)||∞ .The norm
of a vector α is defined as || || T . 2. Problem statement 2.1. Fault model of active suspension Consider the quarter car model of suspension system shown in Fig.1. ms is the sprung mass, which may vary due to changes in passenger load and cargo; mu is the unsprung mass, which represents mass of the wheel assembly; Fd and Fs are the forces generated by the nonlinear springs and dampers, respectively; Ft and Fb are the elasticity force and damping force of tires; zs and zu stand for the displacements of sprung and unsprung masses, respectively; zr is the input of the road displacement, and ua is the actual control input produced by actuators which may fail during the system operation. Derived from Newton’s second law, the dynamic behavior of an active suspension can be expressed as ms zs Fd ( zs , zu ) Fs ( zs , zu ) ua mu zu Fd ( zs , zu ) Fs ( zs , zu ) Ft ( zu , zr ) Fb ( zu , zr ) ua
where żs and żu are the derivatives of zs and zu , respectively.
(1)
In Eq.(1), the forces Fd and Fs produced by the dampers and springs are unknown nonlinear functions whose expressions and upper bounds are unavailable to obtain. Furthermore, ms and mu are probably changeable in actual process, whose variation bounds are unknown. To facilitate the control design, let the state x=[x1 x2 x3 x4]T be defined as x1 zs , x2 zs , x3 zu , x4 zu
(2)
Then Eq.(1) can be rewritten in the following state variable form as x1 x2 x 1 ( F ( z , z ) F ( z , z ) u ) 2 s s u d s u a ms
(3)
x3 x4 x 1 (F ( z , z ) F ( z , z ) F (z , z ) F (z , z ) u ) d s u t u r b u r a 4 mu s s u
(4)
zs
ms
Vehicle
Fd
Fs
ua Controller
Suspension
Tire
zu
mu
Wheel
Ft
Fb zr
Fig.1. Structure of quarter-car active suspension. Remark 1 Model (1) reflects some important nonlinear uncertain characteristics in active suspension control, which is simple without showing the concrete expressions of Fd and Fs. Therefore, the fault control strategy to be investigated based on model (1) in Section 3 and 4, does not depend on the accuracy of mathematical model, which means that the precise mathematical model of active suspension systems is not required. For such an active suspension system, an ideal control input u designed by a control scheme is applied to the system through the actuator which generates the actual input ua. For normal system operation, ua=u. When actuator faults occur, the actual control input is not equal to the ideal input, that is, ua≠u. Therefore, ua can be modeled as ua=δu, (5)
where δ is a time-varying parameter, and subjects to 0≤δ≤1. δ=1 denotes that the actuator works normally; δ=0 means that the actuator is completely failed; 0<δ<1 shows that the actuator is partially failed, but it can still work all the time. In this paper, the assumption 0<δ≤1 is made. Remark 2 In this paper, since active suspensions without actuator forces reduce to traditional passive suspensions, the fault case in which the actuator fails to work (i.e., δ=0) is not taken into account. Simulation results on passive suspensions will be presented in Section 5 as comparison to better illustrate the effectiveness of the proposed strategy. Assumption 1 The external road disturbances are continuous bounded functions, that is, there exist unknown positive constants v1 and v2 such that ||zr||≤v1, ||żr||≤v2, respectively.
2.2 Problem statement The performance requirements of a vehicle suspension include the following aspects. Firstly, one of the main tasks is to improve ride comfort, which means to design a controller that is able to stabilize the vertical motion of the car body and isolate passengers from vibrations arising from road roughness. Secondly, in order to avoid damaging vehicle components and causing more discomfort, the suspension should not hit its travel limit. Hence, we need to guarantee the suspension deflection z s zu zmax
(6)
where zmax is the maximum suspension deflection. Finally, in order to ensure the vehicle safety and road holding ability, the firm uninterrupted contact of wheels to road should be guaranteed, which means that the relative dynamic tire load (Ft+Fb)/(ms+mu)g should be less than one, i.e.,
Ft +Fb 1 (ms mu ) g
(7)
where g denotes the gravitational constant. Based on the aforementioned statements, this paper tries to design a constrained nonlinear controller to solve the following problem: Problem 1: For nonlinear uncertain active suspensions subject to actuators faults and external disturbances, propose an adaptive sliding fault tolerant strategy to stabilize the vertical motion of the car body and satisfy the suspension constraint requirements shown in Eqs.(6) and (7). 3 Adaptive fault tolerant control design The goal of desirable active suspension control is to make passengers unaware of
vibrations when rough roads are encountered, which reflects the perfect performance of isolation, so the desired outputs of sprung mass displacement and velocity denoted as x1r and x2r are chosen to be zero, i.e., x1r=0, x2r=0. Let e1=x1-x1r and e2=x2-x2r be the tracking errors for x1 and x2, respectively. Then according to Eq.(3), the error dynamical system can be described by e1 e2 e u 1 ( F F ) s d 2 m ms s
(8)
Define the error vector e=[e1 e2]T, thus system (8) can be written into state-space form as e Ae Bu f ( zs , zs , zu , zu )
0 where A 0
(9)
0 0 1 . , B , f ( zs , zs , zu , zu )= 1 ( Fs Fd ) 0 ms ms
Throughout the paper, it is assumed that the sprung mass ms resides in an interval, which can be expressed as msmin≤ms≤ msmax, (10) where msmin and msmax denote the minimum and the maximum sprung mass, respectively. Supposing that ms is the nominal value of ms and defining
b
ms
1 , ms
(11)
then Eq.(9) can be rewritten as e=Ae ( B0 B)u f ( zs , zs , zu , zu )
(12)
T
1 T where B0 0 and B 0 b . ms
In view of Eqs. (10) and (11), it can be obtained that
1 1 1 b , ms ms min ms
(13)
From (10) and (13), we have 0
1 1 m ms min 1 = s ms min ms ms min ms ms min
(14)
In this section, the sliding surface for system (12) is conventionally defined as s=Ge (15) where G=[σ1 σ2] is the sliding surface vector; σ1>0, σ2>0 should be chosen to
guarantee that the error vector e on the sliding surface has satisfying performances. Based on Eq.(12), the derivative of s is given by s Ge GAe (GB0 GB)u +Gf ( zs , zs , zu , zu )
(16)
Assumption2 In Eq.(16), Gf(żs, żu, zs, zu) is assumed to be upper bounded and satisfies ||Gf(żs, żu, zs, zu)||
G B0
2 , G B= 2 b . ms
(18)
In terms of (13) and (14), we have
G B
Let F
2 ms min
2 ms min
(19)
, and it is obvious that the following inequalities hold: F GB0 0
(20)
F (GB0 ) 1 1
(21)
Theorem1. Consider the nonlinear uncertain system (3), if the adaptive fault tolerant control law is designed as follows: u (GB0 )1 GAe (GB0 )1 sgn(s) | GAe | sgn(s)
(22)
with the following adaptive laws s GAe s
(23)
where α and β are adaptive rates, then the motion on the sliding surface given in (15) will be asymptotically stable. Proof. Define and as
0 , 0
(24)
F (GB0 )1 L , = 0 1 F (GB0 ) 1 F GB0
(25)
where
0 =
Choose a Lyapunov function candidate as 1 1 F (GB0 )1 1 2 1 F GB0 2 V s2 2 2 2
(26)
From (20) and (21), we can conclude that V is a positive definite function. Taking the time derivative of the Lyapunov function candidate V gives
( F (GB0 )1 1) F GB0 ( 0 )( s ) ( 0 )( GAe s ) ss L s ( F (GB0 )1 1) s ( F (GB0 )1 GAe s ( F GB0 ) GAe s )
V ss
(27)
Substituting the control law (22) into the sliding surface (15) gives
ss s GB0 GAe s (GB)(GB0 )1 (GAes s ) (GB) GAe s sGf ( zs , zs , zu , zu )
(28)
L s ( F (GB0 )1 1) s F (GB0 ) 1 GAe s ( F GB0 ) GAe s Taking (28) into (27), one can obtain V L s ( F (GB0 )1 1) s ( F (GB0 )1 GAe s ( F GB0 ) GAe s ) L s ( F (GB0 )1 1) s F (GB0 ) 1 GAe s ( F GB0 ) GAe s 0
(29)
Hence, the proof is completed. Remark 3. From Eq.(22), when the sliding surface parameters σ1 and σ2 are selected, the proposed control law only involves the accessible error vector e, which is easy to implement for real applications. Remark 4. It is unknown whether the actuator is working normally, but the control law designed in Eq.(22) can still ensure the stability of the system without requiring any information about possible actuator failures and parameter uncertainties. 4. Sliding surface parameter optimization In the previous analysis, the stability of system (3) has been proved in terms of the proposed control strategy, i.e., the suspension objective (1) is achieved. However, before resorting to any toolbox for numerical simulations, the sliding surface parameters σ1 and σ2 have to be chosen. In this section, we are devoted to present one feasible approach of designing the sliding surface parameters to meet the constraint requirements in (6) and (7). In Eq.(1), the tire is modeled as a linear spring, thus Ft and Fb are described as Ft kt ( x3 zr ) Fb ct ( x4 zr )
(30)
where kt and ct are the stiffness and damping coefficients of the tire, respectively. Without loss of generality, the following uniform model of the nonlinear forces Fs and Fd is exploited:
Fs k1 ( x1 x3 ) Fsn Fd c1 ( x2 x4 ) Fdn
(31)
where k1 is the linear stiffness coefficient; c1 is the linear damping coefficient; Fsn and Fdn are the nonlinear error terms of Fs and Fd, respectively.
Remark 3 In the existing results, it should be noted that the controller designs are specific to some kind of given nonlinear dynamics concerned with Fs and Fd, which are limited for practical implementation. In [12] and [25], for example, Fd obeys a piecewise linear representation, whereas Fd in [22] is denoted as the sum of a linear term and a quadratic term. In order to avoid the design limitation and make the controller more applicable in practice, nonlinear functions Fs and Fd are decomposed into two parts, i.e., a familiar linear term and an unknown bounded nonlinear term. Hence, expressions considered in Eq.(31) are of the more general forms. Substitute Eqs.(30), (31) and (5) into system (3)-(4), we have x A1 x B2u w1
(32)
0 0 0 1 k c k1 c1 1 1 T ms ms ms ms 0 where x is defined as in (2), A1 = , B2 0 , 0 0 0 1 ms mu k1 c1 k1 kt c1 ct m m mu mu u u 0 Fsn Fdn ms w1 0 Fsn Fdn kt zr ct zr mu
is the lumped disturbance.
Following a similar procedure to Eq.(12), let A1=A1m+ΔA1, B2=B2m+ΔB2, where A1m and B2m are the nominal matrices of A1 and B2, respectively; ΔA1 and ΔB2 are uncertain bounded matrices. Thus, Eq.(32) can be rewritten as x A1m x B2 mu w2
(33)
where w2 w1 A1 x B2u R4 . Considering matrix A in (9), B0 in (12) and G in (15), we can simplify (22) as u 0 e N sgn(s) T
where
(34)
1 ms , ms is the nominal value of ms, N (GB0 )1 | GAe | . 2
Substituting the control law (34) into (33) leads to x Acl x w3
(35)
where Acl A1m B2 m C , w3 w2 B2 m N sgn( s ) , C 0 1 0 0 . In order to meet the constraint requirements in (6) and (7), the controlled output vector z contains the
signals of x1-x3 and x3-zr, and then the output equations can be described as z C1 x zr
1 0 1 where C1 0 0 1
(36)
0 T . Define the augmented disturbance vector w w3 zr , and 0
rewrite Eqs.(34) and (35), thus the resulting closed-loop system is described as
x Acl x Bcl w z C1 x D1w 1 0 where Bcl 0 0
0 1 0 0
0 0 1 0
0 0 0 1
(37)
0 0 0 0 0 0 0 , D1 . 0 0 0 0 0 1 0
From Eq.(37), it is obvious that the parameter σ defined in Eq.(34) has an impact on the system outputs. Note that parameter σ may not be appropriate if it is chosen with trial-and-error work, so an effective design method for σ is needed to satisfy the constraint performances in (6) and (7). In this paper, we choose the H∞ norm in system (37) as the performance measure, and it is required that the closed-loop transfer function from disturbance to controlled output Twz should be as small as possible, so our goal is to minimize ||Twz||∞ to obtain the optimal parameter σopt. Theorem 2. For a given closed-loop system (37), let λ>0 be a given scalar, and the H∞ performance ||Twz||∞<λ is achieved if and only if there exist P=PT>0 and σ<0 satisfying the following inequality ( A1m B2 m C )T P P( A1m B2 m C ) PBcl C1T * I D1T 0 . * * I
(38)
The proof for the theorem, which can be easily derived by referring to bounded real lemma and other works [26, 27], etc., is omitted here for brevity. As a consequence of Theorem 2, to minimize ||Twz||∞ is equivalent to solving the following optimization problem
min subject to LMI (38)
(39)
Since C is not an identity matrix, Eq.(38) is a bilinear matrix inequality belonging to an NP hard problem. Therefore, for the present non-convex optimization problem (39), it cannot be easily resolved using traditional convex optimization algorithms. However, based on a given σ, inequality (38) turns to LMI which can be efficiently resolved using Matlab LMI Toolbox. In this section, DE is introduced thereafter to find the feasible solutions for inequality (38) due to its efficient stochastic search capability. In recent years, there has been an ever-increasing interest in the area of DE
presented by Storn and Price [28-31]. The advantages of employing DE for solving global optimization problems are its global solution finding property, simple but powerful search capability, compact structure with a few control parameters, and high convergence characteristics. DE has already participated in the First International IEEE Competition on Evolutionary Optimization. Among conference entries, DE proved to be the fastest evolutionary algorithm. Moreover, DE outperforms all of the minimization approaches, such as adaptive simulated annealing, genetic algorithm and stochastic differential method, in terms of required number of function evaluations necessary to locate a global minimum of the test functions. Hence, an approach which combines the feasible solution of the LMIs and the random search of DE will be proposed to solve the optimization problem (39). DE is a global optimization algorithm and individuals in the population are encoded by real number. The ith individual at Gth generation can be represented as σi,G(i=1, 2,…, Np, where Np does not change during the optimization process). The whole flow chart of the algorithm is shown in Fig.2 and the procedures can be summarized as follows. Step 1: Population initialization. In this procedure, an initial population (at G=0) of Np individuals, is randomly generated within a reasonable search space. Since not all the generated individuals are valid for the subsequent evolution, feasibility analysis on every σi,G will be conducted by solving problem (38). If problem (38) is infeasible with σi,G, it will be replaced with another feasible individual that could be regenerated for one or more times. As a result, all individuals in the initial population are feasible for problem (38) so as to improve the search efficiency. Step 2: Objective function evaluation. Determine the minimal λi by solving problem (39), and take every λi as the objective value corresponding to σi,G. Step 3: Mutation operation. For each target individual σi,G, a mutant individual is generated according to vi,G+1=σj,G+F(σq,G-σr,G), where random indexes j, q and r belong to {1,2,…, Np}and mutually different. The chosen j, q and r are also different from the running index i. F (0, 2] means mutation control parameter. Step 4: Crossover operation. In order to increase the diversity of the perturbed parameter vectors, crossover is introduced using the following scheme:
vi ,G 1 , if (rand ( ) CR) ui ,G +1 i ,G , otherwise
(40)
In the above formula, rand( ) is a uniform random number generator with outcome [0, 1] and CR is a crossover probability [0, 1]. Step 5: Selection operation. The trial individual vi,G+1 is compared with the target individual σi,G to decide whether it should become a member of the population at the next generation using a greedy criterion. The selection operation in DE is implemented according to the following rule
u , if (ui ,G 1 ) ( i ,G ) i ,G +1 i ,G 1 . i ,G , otherwise
(41)
As a result, all selected individuals are as good as or better than their counterparts in
the current generation. Step 6: If a stopping criterion is met, then output the optimal output σopt as well as its objective value; otherwise go back to Step 2.
Fig.2. Flow chart of the proposed control strategy. 5. Simulation In this section, numerical simulations are provided to indicate the effectiveness of the novel strategy. The performances active suspension with the proposed control scheme are assessed in the presence of different types of road profiles, parameter uncertainties and actuator faults.In the simulations, the unknown nonlinear forces Fs and Fd produced by the nonlinear spring and damper are expressed as
Fs k1 ( x1 x3 ) k2 ( x1 x3 )3 2 Fd c1 ( x2 x4 ) c2 ( x2 x4 )
(42)
where k1and k2 are the linear and nonlinear stiffness coefficients, respectively; c1 and c2 are the linear and nonlinear damping coefficients, respectively.The nominal values of the quarter car parameters are listed in Table 1 [22]. The DE parameters used in this paper are given as population size Np=5; crossover probability CR=0.5; Mutation factor F=0.8; maximum of generations N=10. By using the proposed method in Section 4 to solve the optimization problem (39), we can obtain that the achieved H∞ norm of the closed-loop transfer function from w to z is λopt=1.0005 with σopt=-10711. Based on Eq.(34), the sliding surface vector is therefore chosen as G=[37,1]. Moreover, adaptive rates α=0.08 and β=1.3 are chosen.
Parameter ms k1 c1 kt zmax
Table 1. Parameters of active suspensions. Nominal Value Parameter 290kg mu 14.5kN/m k2 1385.4Ns/m c2 190kN/m ct 0.12m
Nominal Value 59 kg 160kN/m 524.28Ns/m 170Ns/m
5.1 Robustness analysis An isolated bump, with relatively short duration and high intensity, in an otherwise smooth road surface is used. The corresponding ground displacement for the wheel is represented as 2 v L Am (1 cos t) 0 t zr 2 L v 0 otherwise
(43)
where Am and L are the height and the length of the bump. Assume Am=0.1m, L=5m, and the vehicle forward velocity as v=45km/h. When no actuator fault occurs, i.e., δ=1, the bump responses of passive suspensions and active suspensions are presented to illustrate the effectiveness and robustness of the control strategy. Simulations on the following two cases are implemented: Case1: Normal system without perturbed parameters; Case 2: ms=(1-20%)×290, c1=(1+20%)×385.4, kt=(1-30%)×190, δ=1. Bump responses of the sprung mass displacement, body acceleration, suspension deflection, relative dynamic tire load and the corresponding control input are plotted in Fig.3. It can be clearly seen from Fig.3 that the magnitudes and the settling time of the sprung mass displacement as well as body acceleration are much less than those of the passive suspension systems no matter whether the system parameters vary or not, and thus good vibration isolation is achieved. Additionally, there is no significant
change in the magnitudes of the displacements and accelerations, although several system parameters fluctuate around their nominal values. Finally, suspension deflection and relative dynamic tire load are guaranteed to be less than their limits as those defined in Eqs.(6) and (7) in spite of the large bump energy. Therefore, the designed active suspension can realize satisfying suspension performances regardless of system parameter variations, which indicates the robustness of the proposed control strategy.
Fig.3. Bump response without actuator fault. 5.2 Simulation of actuator failure 5.2.1 Bump response of constant failure with measurement noise The bump road profile is described in Eq.(43). When actuator fault in the presence
of white noises with noise power 0.01 is considered, the bump responses of passive suspensions and the proposed active suspensions with perturbed parameters are compared in Fig.4 with regard to the following cases: Case1: Passive suspension system with ms=(1-20%)×290, c1=(1+20%)×385.4 and kt=(1-30%)×190. Case2: Active suspension system with ms=(1-20%)×290, c1=(1+20%)×385.4 and kt=(1-30%)×190, δ=0.8; Case3: Active suspension system with ms=(1-20%)×290, c1=(1+20%)×385.4 and kt=(1-30%)×190, δ=0.5. From Fig.4, one can find that the responses are all similar to those shown in Fig.3. In spite of the presence of actuator fault and measurement noise, the proposed active suspension systems can bring about a significant improvement on ride comfort, suggesting excellent fault tolerance and robustness of the proposed control strategy in this paper.
Fig.4. Bump response with actuator constant failure and measurement noise. 5.2.2 Continuous bump response of time-varying failure with uncertainty The continuous bump road profile is expressed as zr=0.08cos(2πt)sin(0.6πt). (44) The responses for nominal passive suspension systems and active suspension systems are presented, where white noises with noise power 0.01 are taken into account, and the fast time-varying parameter δ is assumed as 1, 0.7, (t ) 14 t 10 , 0.8,
0t 2 2t 5 5t 8
.
(45)
8 t 10
It can be clearly seen from Fig.5 that the designed active suspension systems outperform the passive ones, since the magnitudes for the body displacements and accelerations are dramatically reduced, which indicates the effectiveness of the controller in spite of such a fast time-varying actuator fault. The actuator fault can be finally seen like a matched perturbation, which is easily compensated due to the inherent robustness of sliding mode algorithm to matched perturbation. Additionally, simulation for the active suspensions subject to parameter uncertainties is also implemented. In this case, the system parameters are chosen as ms=(1+20%)×290, c1=(1-20%)×385.4, and kt=(1+30%)×190. The maximum values and root-mean-square (RMS) values of vertical displacement, acceleration, suspension deflection and relative dynamic tire load are listed in Table 2. It can be observed that the improvement in ride comfort and the satisfaction of constraints can be achieved with the adaptive fault tolerant controller, indicating that the proposed control strategy is capable of handling such fault and uncertainty successfully.
Fig.5. Response with fast time-varying actuator fault. Table 2. Performance for suspension systems: maximum and RMS values. Relative Suspension 2 Displacement/m Acceleration/m/s dynamic tire Type of deflection/m load suspension Passive Active
max
RMS
max
RMS
max
RMS
max
RMS
0.129 0.025
0.063 0.013
5.856 1.130
2.816 0.508
0.103 0.071
0.056 0.038
0.479 0.138
0.244 0.064
5.3 Comparison analysis In order to further validate the superiority of the proposed control scheme, the existing results in [16] are presented for the objective of comparison. Let us consider the active suspension model and its parameters described in the literature [16], in which the fault-tolerant H∞ control (FHC) for a class of quarter-car active suspension systems with actuator faults is investigated. For the sake of verifying the effectiveness of the proposed control strategy, G=[41,1], α=0.08 and β=1.3 are chosen in the simulation. As shown in Fig.6, the proposed adaptive sliding fault tolerant control can obtain better regulation performance of body acceleration, and simultaneously satisfy the constraint requirements no matter whether the actuator fails or not.
Fig.6. Response curves using the proposed control strategy and FHC method in [16]. 6. Conclusion In this paper, a novel adaptive sliding fault tolerant control for nonlinear uncertain active suspension systems has successfully been proposed based on Lyapunov approach. An additional H∞ optimization scheme based on DE and LMIs is utilized to design the appropriate sliding surface parameters to meet the constraint requirements of active suspension systems. The results of simulation for different road profiles convincingly demonstrate that the vehicle performances are improved with the proposed controllers in spite of various actuator faults and parameter uncertainties. Since adaptive rates have effects on system performances, it is a challenge for the proper selection of them when applying the proposed strategy. The research on the optimization of the adaptive rates is still in progress, which will be finished in the future work. Acknowledgements The work is financially supported by the doctoral research fund of East China of Technology (DHBK1015), the Natural Science Foundation of China (51409047, 61463003) and the Open Project Program of Jiangxi Engineering Research Center of Process and Equipment for New Energy, East China Institute of Technology (JXNE2015-10). The author would like to thank the editor and anonymous reviewers for their helpful comments and suggestions to improve the quality of this paper.
Appendix. The analysis of the proposed approach In [23], a constrained adaptive backstepping control strategy, based on barrier lyapunov functions (BLF), has been proposed for active suspension systems to achieve multi-objective control without considering actuator faults. The mathematical model of active suspension systems with some specific nonlinear dynamics, i.e., nonlinear spring and piece-wise linear damper, is utilized to design the control strategy. As a matter of fact, nonlinear features of active suspension systems could not be exactly obtained. Moreover, the sophisticated control strategy, i.e., BLF control, takes zs, żs, Fd and Fs as feedback signals, in which Fd and Fs would be difficult to measure. By contrast, a practical control scheme which is independent on acquiring the nonlinear dynamics of active suspension systems is proposed in this paper. The proposed control scheme is simple to implement, since it just utilizes the practically accessible variables zs and żs. In what follows, simulation will be carried out to validate the effectiveness of the proposed scheme. The quarter-car model parameters and the road profile in simulation are the same as those used in [23], and sliding surface vector G=[23,1] is selected in the simulation. Case1: Normal system without actuator faults; Case2: δ=0.6; feedback signals of zs and żs which are subject to white noise with noise power 0.05.
Fig.7. Response curves using the proposed strategy and BLF approach in [23]. Fig.7 shows a comparison of the active suspension systems with the proposed controller and BLF controller in [23], and the following conclusions can be obtained. (1) As it is shown in Fig.7 (a) and (b), actuator fault and sensor measurement noise can cause obvious performance degradation to active suspension systems with BLF controller in [23], whereas the proposed active suspension systems with fault tolerance ability almost shows no degradation in performance. (2) Fig.7(c) and (d) shows that suspension deflection and relative dynamic tire load are all guaranteed to be less than their hard limits by the two approaches in spite of the large bump energy. Therefore, it can be concluded that good vibration isolation is achieved by the proposed control scheme in the presence of actuator fault and measurement noise. (3) Actuator output forces are plotted in Fig.7 (e), from which we can see that the proposed approach needs smaller actuator force than BLF control in [23].
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