Research Article | Open Access
Actuator Fault Estimation for Vehicle Active Suspensions Based on Adaptive Observer and Genetic Algorithm
This paper is concerned with the problem of actuator fault estimation (FE) for vehicle active suspension systems. First, the fast FE approach, which combines the output error term with its derivative term in the FE algorithm, is extended to the active suspension system with actuator fault and external disturbance input. Then, considering three typical kinds of actuator faults, i.e., constant gain change fault, drift fault, and stuck fault, genetic algorithm (GA) is employed to optimize the adjustable parameters in the FE algorithm, which are usually determined by trials. Finally, simulation results of FE and fault-tolerant control (FTC) are presented to illustrate the effectiveness and applicability of the proposed FE method.
Vehicle suspensions play a significant role in ride comfort, safety, and the overall performance of modern vehicles. According to different control forms, vehicle suspensions are commonly classified into three types: passive, semiactive, and active suspensions. The passive suspension composed of a parallel spring and damper which has been acknowledged for its simple structure and low cost . The semiactive suspension [2, 3], which consists of a spring and tunable damper, can make desirable improvements over the passive one by effectively varying the rate of the energy dissipation. Compared with passive and semiactive suspensions, active suspensions are not widely used due to large energy demand and complex structure. However, they can provide high control performance in a wide frequency range and have the best potential to overcome conflicts between ride comfort and vehicle safety. As the key part for active suspensions, advanced control strategies have attracted considerable interest, such as control , nonlinear control [5, 6], predictive control , adaptive control [8, 9], fuzzy control [10, 11], and finite-time control . It should be pointed out that all the abovementioned active suspension control results are under a full-functional and fault-free assumption of all system components. However, various faults, especially actuator and sensor faults, are likely to be encountered in active suspension systems. Ignorance of these faults may result in poor system performance or even instability. In this case, it is reasonable to resolve fault diagnosis and fault-tolerant control (FTC) problems, which could increase system security and reliability.
Over the past few decades, fault diagnosis and FTC have been extensively investigated and many results have been obtained, see [13–18] and the references therein. Yet, for vehicle suspension systems, it seems that the corresponding results are relatively few. In , the fault diagnosis problem was investigated for a semiactive vehicle suspension system by using the particle filter-based approach. In [20–22], passive fault-tolerant control (PFTC) strategies were studied for active suspensions. Some active fault-tolerant control (AFTC) methods were proposed in [23–25] for full-car, quarter-car, and half-car active suspensions. Different from PFTC, AFTC guarantees the acceptable system performance through fault accommodation (FA) or system reconfiguration. As an important part in both fault diagnosis and FA, fault estimation (FE) has a strong ability to obtain the fault information, such as the size and shape [26–28]. Therefore, FE has received considerable attention, and a great number of theoretic results have been reported, e.g., based on adaptive observer [26, 29, 30], disturbance observer , extended state observer (ESO) , unknown input observer (UIO) [28, 32], and sliding mode observer (SMO) . In , the FE-based AFTC problem was discussed for a full-car semiactive suspension system. Note that only the measurement output error was used in the adaptive observer design. In , the AFTC method based on FE was developed for active suspension systems in finite-frequency domain. It could be easily seen that the output error was utilized in the FE signal, which implied that only the derivative term of the output error was included in the FE algorithm. In [26, 35], it was concluded that if both the output error and the derivative of it were used in the observer design, better estimation performance can be achieved compared with the traditional proportional-integral (PI) observer. However, some parameters or matrices in the FE algorithm are often chosen by trials, which may limit the improvement of FE performance.
In this paper, motivated by the above analysis, the FE problem is studied for an active suspension system based on the parameter optimization technique. The main contributions of this paper can be summarized as follows: (1) the fast FE approach, which combines the output error term with its derivative term in the FE algorithm , is extended to the active suspension system with the actuator faults and external disturbance input; (2) in order to improve the FE performance, genetic algorithm (GA) is used to optimize the adjustable parameters in the FE algorithm, which are usually determined by trials.
The remainder of this paper is organized as follows: a quarter-vehicle active suspension model and three kinds of actuator faults are described in Section 2. The actuator FE method based on an adaptive observer and GA is developed in Section 3. Simulation results and analysis are given in Section 4 to show the validity of the proposed method. Finally, the paper is concluded in Section 5.
Notation. refers to the Euclidean norm of a vector or the induced norm of a matrix. The notation (respectively, ), where X and Y are real symmetric matrices, means that the matrix is positive definite (respectively, negative definite). I and 0 denote the identity and zero matrices with appropriate dimensions, respectively. denotes the minimum of eigenvalues of a real symmetric matrix P. The superscript T denotes the transpose for vectors or matrices. The symbol denotes the term that is induced by symmetry. denotes the space of vector functions that are square integrable over . For , . Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.
2. Quarter-Car Active Suspension with Actuator Fault
2.1. Quarter-Car Active Suspension Model
A simple quarter-car active suspension model is considered in this paper, as shown in Figure 1. In this model, is the sprung mass, representing the chassis; is the unsprung mass, representing the mass of the wheel assembly; represents the active control force; and denote the displacement of the sprung mass and the unsprung mass, respectively; denotes the road displacement; and denote damping and stiffness of the uncontrolled suspension, respectively; the tire is simplified as a spring and denotes its stiffness.
The dynamical equations of the suspension system in Figure 1 can be written as
Define the following state variables: which is the suspension stroke, which is the sprung mass speed, which is the tire deflection, and which is the unsprung mass speed. Choose as the disturbance input caused by road roughness. Choose the sprung mass speed and suspension stroke as measurement outputs. Then, the dynamical equation (1) can be rewritten as the following state-space form:where is the state vector. is the output vector.
2.2. Actuator Fault Description
In this paper, three typical kinds of actuator faults are considered, i.e., constant gain change, drift, and stuck [36, 37]. Then, the lumped form of the actuator output can be given aswhere δ represents the actuator loss in efficiency, represents the value of drift or stuck failure, and represents the control command to the actuator. It is easily known that and correspond to the healthy actuator, and correspond to the actuator with constant gain change fault, and correspond to the actuator with drift fault, and and correspond to the actuator with stuck fault.
In order to separate the fault, the actuator output is rewritten by
Then, the actuator fault is obtained as
3. Actuator Fault Estimation
In this section, the fast FE idea  is introduced to the active suspension system with actuator faults and external disturbance input. Then, in order to improve the FE accuracy, GA is employed to optimize the parameters in the FE algorithm.
3.1. Adaptive Fault Estimation Design
For system (7), consider the following adaptive observer:where and are the state vector and the output vector of the observer, respectively, is the estimation of the fault, and L is the gain matrix to be designed. Denote
We can obtain the following error dynamics:
The adaptive FE algorithm is presented aswhere is the given learning rate, N is a suitable dimensional matrix, and is the weight scalar of the integral item.
Theorem 1. For system (10), given matrix , scalars and , if there exist matrices , M, and N, such that the following conditions hold:where . Then, and are uniformly ultimately bounded with . Moreover, under the zero initial condition,holds for all nonzero .
Proof. Choose the following Lyapunov function:From (10), (11), and (13), we can getWhen , we havewhere andIt follows from (12) and (13) that holds. Letting , we obtainFrom (19), it is seen that when . Therefore, with , and are uniformly ultimately bounded.
Moreover, defineAccording to (13) and , under the zero initial condition, we obtainwhere .
From (12), it is known that . Then, for nonzero , we haveHence, we obtainwhich is equivalent to (14). The proof is completed.
Remark 1. Note that the conditions in Theorem 1 are not LMIs (linear matrix inequalities) due to the existence of (13). Fortunately, (13) can be transformed into the following approximate condition :where η is a sufficiently small positive constant. By solving LMIs (12) and (24), we can easily obtain the gain matrix L in the adaptive observer (8): .
3.2. Parameter optimization
For the FE algorithm (11), the adaptive learning rate and the integral weight σ are undetermined parameters which are usually selected by trials. It is known that these two parameters may influence the accuracy and rapidity of the FE. For example, if a small is selected, the tracking speed of the observer will be slow. Otherwise, if a large is selected, a large overshoot on the FE may be generated. Therefore, in order to improve FE performance, and σ can be viewed as parameters that need to be optimized.
GA  is a random, global searching, and population-based optimization method. After decades of research and development, GA is known as a kind of effective and robust optimization method which can be employed to solve a variety of optimization problems [39–41], including problems in which the objective function is discontinuous, nondifferentiable, stochastic, or highly nonlinear. In this study, GA is used to optimize and σ in the FE algorithm. For the quarter-car suspension system (7), it is easily seen that the learning rate is a scalar. Hence, the optimization vector is defined as . Three common kinds of actuator faults, which are, respectively, constant gain change, drift, and stuck, are all considered in the construction of the objective function for the sake of comprehensiveness. The initial expression of the objective function is described aswhere and , respectively, represent the estimated fault and actual fault at the kth sampling point. , , and , respectively, represent the constant gain change, drift, and stuck faults. represents the number of corresponding data points. In (25), the mean absolute percentage error (MAPE) is chosen as the error indicator. However, when the actual fault value is equal or very close to zero, the MAPE value will be too large, which is not suitable for the parameter optimization. In order to avoid this situation, the objective function is modified as
From (26), it is clearly seen that the relative error is employed to construct the objective function if the absolute value of the actual fault is far greater than 1. In contrast, the absolute error is employed if that of the actual fault is far less than 1.
Based on GA, the parameter optimization process is described as follows.
Step 1. Set main parameters for GA, such as population size, maximum generation, crossover fraction, and so on.
Step 2. Randomly generate the initial population: .
Step 3. Obtain observer (8) by solving the following convex optimization problem:
Step 4. Calculate the objective function (26) with the obtained observer.
Step 5. If the stopping criteria are satisfied (maximum number of iterations is reached or objective function is achieved with certain accuracy), exit with the optimized vector . Otherwise, go to Step 6.
Step 6. Form a new generation of α by “selection,” “crossover,” and “mutation” operations, and go to Step 3.
4. Results and Analysis
4.1. Fault Estimation Analysis
In this section, we will employ the above FE method to a quarter-car suspension model described in Section 2. In addition, the disturbance input is represented as a white-noise signal whose power spectral density (PSD) is given by , where is the road roughness coefficient, is the reference spatial frequency, and is the vehicle forward velocity. Some parameters of the quarter-car suspension system are listed in Table 1.
For the fault-free situation, the control method  is employed to improve the ride comfort as well as guarantee hard constraints. These hard constraints usually include good road holding ability, allowable suspension stroke, and maximal active force . Considering the above control objective, the quarter-car active suspension system can be described aswhere , , A, B, and D have been defined in (2). is the constrained output which reflects the hard constraints mentioned above. is the control output which reflects the ride comfort.where and are upper bounds for the suspension stroke and active control force, respectively.
Choosing and and applying Theorem 4.3.1 in  with , the following state feedback controller can be obtained:
For the optimization of and σ in the FE algorithm (11), three types of actuator faults are supposed to be generated to evaluate the actual fault in the object function (26). In this simulation, every type of fault lasts 2.5 seconds. δ and in (6) are listed in Table 2. For convenience, the fault in Table 2 is called the reference actuator fault.
Then, GA is used to optimize the vector . Main parameters of GA are given in Table 3. Letting and utilizing the optimization procedure in Section 3.2 (the procedure is run 5 times), we can get the optimized results shown in Table 4. It is easily seen that the best and the average one is . To verify the availability of the proposed FE method, several groups of actuator faults, which are different from the reference one, are selected and corresponding results are given in Table 5. In Table 5, and is defined aswhere , , j, and have the same meaning as in (26). From Table 5, it can be seen that the calculated are very small under different fault situation, which indicates that the estimated fault can accurately track the actual fault.
Since the vehicle velocity varies frequently during practical driving, the optimized observer is applied under different vehicle velocity conditions. Estimation results are presented in Table 6. These results prove that the proposed FE method has good robustness against uncertain vehicle velocity, which is applicable to practical applications.
To further demonstrate the effectiveness of the proposed FE method, fault tracking effects are compared for three types of actuator faults. Three FE methods are employed: the proposed one, the one without optimization of and σ, the one in which only the output error term is included in the FE algorithm. For the second method, and σ are fixed as 10 and 1 . For the third method, solve the LMIs (24) in  and obtain the corresponding observer. Simulation results obtained by using the above three methods are denoted as “optimized observer,” “nonoptimized observer,” and “derivative-free observer,” respectively. Fault tracking effects with constant gain change fault, drift fault, and stuck fault are, respectively, depicted in Figures 2∼4. According to these simulation results, we can see that the proposed FE method is well suited for different types of actuator faults. Compared with other two methods, the proposed one can effectively improve the FE accuracy and rapidity.
Similarly, another type of disturbance input (usually called shock road disturbance) caused by an isolated bump is also considered in this paper. The ground displacement can be referred to (25) in , where the height and length of the bump are chosen as 0.1 m and 5 m. The vehicle forward velocity is chosen as 20 m/s. Fault tracking effects with constant gain change fault, drift fault, and stuck fault are, respectively, depicted in Figures 5∼7. These results show good robustness of the proposed FE method against different types of disturbance input.
4.2. Fault-Tolerant Control Based on Fault Estimation
Based on the proposed FE method, a simple FTC strategy is adopted for the active suspension system (7). Fault tolerance is realized by adjusting the actuator output with the estimated fault.
According to (6), let and , we have
The following linear functions are given to describe the relationship between the estimated fault value and the normal output of the actuator:where and are approximations of a and b, respectively.
Since and are linearly correlated, we can obtain a and b by linearly fitting the sampling values of and over a period of time. Based on the fitting results of a and b, we can qualitatively identify the actuator fault type. Detailed criteria are shown in Table 7, where ε and are given small positive constants (also called the thresholds of and in fault identifying).
Based on the FE results in Figures 2∼4, the fault type can be obtained with the aid of the least square fitting approach. Detailed fitting results under different types of faults are given in Table 8. Then, the FTC strategy can be adopted to compensate the corresponding fault.
Taking the gain change fault as an example, according to (4), the ideal fault-tolerant controller for the gain change fault can be expressed as
Then, based on the fitting results, the fault-tolerant controller to be designed can be expressed as
In simulation, the FTC is applied for the gain change fault. Simulation time interval is set from 0 to 3 s (second). The gain change fault starts from 0.5 s. The FTC is employed from 1.5 s. Response of the active suspension system is depicted in Figure 8. It is easily seen that, from 0.5 s to 1.5 s, during which the fault exists while the FTC is not employed, the body acceleration attenuation effect of the suspension system with fault gets worse than that of the fault-free system. At about 1.7 s (0.2 s after the FTC is employed), the body acceleration response is similar to that of the fault-free system. Moreover, it is known that the suspension stroke and actuator output force are guaranteed to be within the required constraints when the FTC is adopted. The relative dynamic tire load (RDTL) may exceed the upper bound 1 occasionally and briefly, which is acceptable in some driving situations .
Comparison of root-mean-square (RMS) values of body acceleration from 2 s to 3 s is given in Table 9. For the active suspension system, RMS values of with FTC and without FTC are denoted as “Active (FTC)” and “Active (WFTC),” respectively. From Table 9, it is easily seen that, for the active suspension with the gain change fault, the RMS value of body acceleration obtained with FTC is reduced than that obtained from the passive system or without FTC. In addition, by applying the FTC strategy, the RMS value of body acceleration is almost equal to that of the fault-free system. The FTC results also demonstrate the applicability and effectiveness of the developed FE method.
The FE problem is addressed in this paper for the quarter-car active suspension system with actuator faults. In the design process of the adaptive observer, the output error term and its derivative term are included in the FE algorithm. Then, GA is used to optimize some parameters in the FE algorithm, which are commonly selected by trials. Three typical kinds of actuator faults are taken into consideration for the GA optimization. Simulation results show that, for every kind of actuator fault, the proposed method can improve the rapidity and accuracy of FE. Furthermore, the proposed FE method is successfully utilized in the FTC of the active suspension system to improve the ride comfort and guarantee hard constraints. The experimental tests of the proposed FE method, which are important for practical applications, will be studied in the future.
The MATLAB data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
The authors would like to appreciate the support provided by the National Natural Science Foundation of China (grant nos. 51705206 and 61873114), China Postdoctoral Science Foundation (grant no. 2018T110457), and Project Foundation for Priority Academic Program Development of Jiangsu Higher Education Institutions.
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