A Nonlinear Adaptive Observer-Based Differential Evolution Algorithm to Multiparameter Fault Diagnosis

In this paper, a novel adaptive diagnosis scheme is proposed for multiparametric faults of nonlinear systems by using the model and intelligent optimization-based approaches. ,e key idea of the proposed method is to analyze the correlation of the output signals between the real system and the fault identification system instead of residual. A new adaptive scheme is built based on an adaptive observer and differential evolution algorithm. Meanwhile, the conditions of detectability and identifiability of faults are analyzed.,e isolation and estimation of the multiparametric fault are formulated as the solution of an optimization problem that is solved by using a differential evolutionary algorithm (DE). ,e fitness function of DE is constructed by the correlation coefficient equations in which the faulty components are contained. ,e application on a coupled three water tank model attests the feasibility and validity of the suggested approach. Simulation and experimental results show that the developed method is applicable to diagnose either single or multiparameter faults on-line.


Introduction
Nonlinearity is considered to be ubiquitous in industrial systems. Due to complicated structures of the systems and harsh working conditions, multifault may coexist in practical applications [1]. In order to improve the safety and reliability of practical systems, prompt detection, robust isolation, and accurate estimation of multifault have become active research areas [2]. Generally, the difficulties of multifault diagnosis mainly originate from two aspects: (1) under severe noise interference, the defect signals generated by different faults will overlap and counteract in the time domain; (2) different faults may excite common or different resonance [1]. Furthermore, the uncertainty and nonlinear characteristics of modern industrial systems have made multifault diagnosis to become a challenging task.
Faults can be classified into catastrophic faults and parametric faults. Catastrophic faults cause the complete failure of the systems. By parametric faults, they mean any deviation in the nominal value implying unwanted changes in the behavior of one or more components of the system. Parameterization fault is mainly caused by component aging, wear, manufacturing tolerances, and environmental conditions [3]. Degradation of system performance is closely related to the respective parameters. Scientists are prone to believe that efficient diagnosis of small bursts or early failure can prevent rapid degradation of system performance and larger failure [4]. If the parametric faults are not detected and estimated immediately, the system is likely to suffer serious catastrophic faults [5]. e problem of fault diagnosis (FD) thus amounts to the detection and isolation of parameter changes in a model. Multiparametric FD is crucial for the evaluation of fault severity, fault cause analysis, residual life prediction, and fault tolerant control [6]. For these reasons, the focus of this paper considers the multiparametric fault diagnosis based on continuous process monitoring.
Aiming to detect and isolate multiple parametric faults of nonlinear systems, different diagnostic techniques have been investigated. Diagnosis methods may be classified as two types: model-based approach and computationally intelligent-based fault diagnosis methodologies [7]. In modelbased diagnosis, a residual is an indicator of the deviation from its expected/normal behavior of a system. A conventional model-based approach is typically relied on a set of residual generators and residual evaluation logic [7]. In an ideal case, a residual generator should be sensitive to a fault. erefore, the number of possible residual generator candidates grows exponentially with the degree of redundancy of the model [8]. By definition of a residual, a fault-free system means that the ideal output of a residual is zero [9]. However, under the influence of model uncertainty and measurement noise, the residual output of the fault-free system may not be zero, which causes the fault false alarm. To address this problem, much literature has devoted to the thresholding of the residual [10] and the optimization of threshold research [11][12][13]. In a fault diagnosis system, the objective is not only to develop an algorithm that detects faults as quickly as possible, but also to minimize the false alarm. erefore, the threshold should be sufficiently small so that low severity faults can be detected in their early stages; on the contrary, thresholds have to be set to sufficiently large values to avoid excessive false alarms due to the presence of noise and uncertainties [7]. In terms of the multiparametric fault estimation for nonlinear systems, optimal selection of multiple threshold bands and designing of multiple observers are still very difficult. For a conventional adaptive observer, the parameter estimation algorithm is formed according to the gradient change of the residual. Due to the coupling of effects of the multiple parameter faults to residual, multiparameter estimation is easy to fall into local optimization.
As the close relative approaches based on intelligent computing have developed rapidly in recent years, evolutionary algorithm, support vector machine algorithm, neural network/deep learning algorithm, and others have been introduced into fault diagnosis methods [4,7]. In order to improve the ability of fault parameter estimation in the case of multiparameter faults and nonlinear system, intelligent computing algorithms could be used in the residual analysis process.
In the above types of approach, each has its own advantages and drawbacks. No single method is able to satisfy various diagnostic requirements, especially in the case of nonlinearity, uncertainty, and multiple faults. In recent years, there has been a growing interest in integrating different methods to each other. e development of hybrid approaches could improve the fault diagnosis performances and overcome the limitations of individual methods used separately [14]. Indeed, model-based methods are better for discovery causal relationships, while the methods which are computationally intelligent based are better at finding correlations between faults and symptoms. erefore, the combination of two methods may potentially derive benefits from each approach and overcome their eventual limitations.
In this paper, a hybrid fault diagnosis scheme is employed. Considering the sensitivity of a residual to a fault, the residual still acts as the basis of fault detection to achieve prompt detection of fault in this paper. To improve the performance of fault isolation, a new adaptive observer based on correlation analysis of the output is built. e system parameters being identified are formulated as the solution of an optimization problem that is solved by using a differential evolutionary (DE) algorithm. e latest identification values of parameters will be sent back to the observer to overcome the uncertainty of the model. It is necessary to mention that different from the usual intelligent computingbased fault diagnosis methods, in this paper, the differential evolution algorithm is not used for the construction of the main framework for fault diagnosis, but for the correlation analysis between the actual system output and the model system output. e fault diagnostic system structure is in principle model based. e main contributions of this paper are as follows. First, using output correlation coefficient analysis by a differential evolution algorithm, a novel multiparameter fault diagnosis framework is built which is a new construction on an adaptive observer. Secondly, a new scheme is proposed which fuses the advantage of the model-based method in structured analysis and the ability in parameter identification of DE, and the efficiency of fault isolation and identification are improved significantly.
is paper is organized as follows. e considered fault diagnosis problem is formulated in Section 2. e proposed adaptive fault diagnosis scheme-based DE is presented in Section 3. e correlation analysis method and the DE algorithm tool are investigated in Section 4. Validation studies on a benchmark model are addressed in Section 5. Concluding remarks are made in Section 6.

Nonlinear System
Model. Without loss of generality, a parameterized nonlinear system is considered: where x ∈ R l , θ ∈ R p , u ∈ R m , and y ∈ R n denote the system state, parameter, input vector, and output vector, respectively, θ is an unknown time-varying parameter, and the term φ(x, t) is known as nonlinear functions with appropriate dimensions.

Assumption 1.
e uncertain time-varying parameter θ(t) satisfies where θ 0 is known as parameter nominal values and Γ is known as a constant vector. Here, the assumed nonlinear system M is linear on parameters. If the process parameter nominal value θ o is known, the nominal model of the system is e output error r(t) in the fault detection system is as follows: Assumption 2. e fault identification system M ′ has the same model structure with M: where y ′ is the output of the fault identification system.

Parametric Fault in Dynamic
System. In a system model, a parametric fault is usually related to parameter changes. erefore, it is natural to monitor the parameter changes to achieve FD. e nominal parameter value θ 0 characterizing the fault-free mode of the system is known, and a parametric fault is characterized by a deviation from θ 0 .
To determine whether the actual system is faulty, the difference between the parameter value θ of the actual system model M and the parameter value θ 0 of the nominal model M o can also be used. When the difference between θ and θ 0 exceeds a certain threshold, it means the system is faulty. Generally, system parameters cannot be measured directly. If y(t) and y o (t) are used to represent the behavior of the actual system and the nominal system, respectively, then the residual r(t) represents the difference between the behavior of the actual system and the one of the nominal system. If the difference r(t) caused by parameter changes exceeds the acceptable level in the view of engineering, it means that there is a fault in the system. erefore, we can judge whether the system is faulty according to whether the residual r(t) exceeds a certain threshold. e size of the threshold depends on the specific engineering requirements.

Diagnosability of Multiple Faults of Linear Parametric
Variation Model (LPV). Fault diagnosability includes fault detectability, isolability, and identifiability. For multiparameter fault diagnosis, the following definition of fault diagnosability is used to formulate the fault indicator and fitness function selection problem.
Definition 3 (fault identifiability). A fault f i identifiability refers to whether a fault can be uniquely estimated when the input and output data of the model are known.
In this paper, a residual acts as fault test quantity and the output vectors correlation coefficient is used for fault isolation instead of the residual. e correlation coefficient ρ yy′ of the two output variables can reflect the directivity and waveform change of the two variables and the similarity degree of every unit change. Our approach to fault diagnosis is under the following assumptions: (1) the considered system is component based: a fault f i corresponds to a component. (2) Faults do not change the model structure.
(3) e system is observable with its normal working excitation. (4) e parameters are changed at most once in a time window T.
e parameters θ of a linear parametric variation model can be estimated consistently through the output correlation function if the following necessary conditions are satisfied: e input signal u(t) is exactly measurable or known; When t ∈ [t 0 , t 0 + T], the output correlation coefficient n i�1 ρ yy′ � n (n is the dimension of the output vector y or y ′ ) between the fault detection system and the fault identification system; e systems' initial state satisfies e systems' state satisfies Proof. If n i�1 ρ yy′ � n, then the output y ′ is linearly dependent on y. Obviously, y ′ can be expressed as where K and D are the constant matrix. When t � t 0 , from (3), (5), and (6), there exist en, Since e proof is complete. e above proof means that the precise estimation of the parameters can be obtained if the output curves of two systems overlap, respectively, within a time window. Accordingly, the coordinates of the parameters in the parameter space and the fault mode (single fault or multiple faults) can be determined. erefore, the fault identification problem can be solved as an optimization problem. e optimization problem is as follows: where n denotes the output dimensions which is known for a certain system model and the correlation coefficient ρ y,y′ is the optimised variable. is diagnosis method can be seen as a parameter identification problem in engineering since the input and the output are known, and it is used to estimate the parameters of the model.

Fault Detection Based on Luenberger-Like Observer.
Considering a system of the form (1), a Luenberger-like observer acts as an estimator of the state x(t) from the measurable u(t) and y(t). e state observer can ensure that the estimated state keeps track of the system state in time.
When parameter θ subjects to changes, the residual between the estimated output and the system output is clearly not equal to 0. us, the fault is detected. For fault detection, a parameter vector of the nominal system is used: where x is the estimate of x and K obv represents the gain of the fault detection observer.
Under certain assumptions about the system structure and input excitation, and if the gain vector K obv is chosen appropriately, for example, according to Lyapunov theorem, the estimated state vector can converge to its actual value.
us, the necessary condition (5) of eorem 1 can be ensured. ere are many ways to obtain the observer gain matrices, but we will not deal with this problem in this paper. e interested reader is referred to [13]. Note: when system (1) is deterministic, the estimation of x(t) amounts in principle to the estimation of the initial state x(t 0 ). However, due to modeling errors, it makes sense to estimate the trajectory x(t) instead of the initial state x(t 0 ).
Residual of output variables: r(t) � y − y is used for fault detection. If r(t) � 0, it means that the actual system is fault free. If the absolute value of r(t) is greater than a preset threshold, the system is determined faulty. Of course, in the case of considering system uncertainty, environmental noise, and other factors, the residual threshold will be more complex. Because the fault detection is not the main topic of this paper, interested readers please refer to the literature [13].

Fault Isolation and Identification Scheme.
In the parameter estimation-based fault diagnosis method, if the parameter vector value of the actual dynamic system is identified, the parameter component in the actual system parameter vector which is different relative to the one in the normal system parameter vector, i.e., the faulty parameter, is found, so the fault is isolated. At the same time, the magnitude of the deviation of the faulty parameter from the normal value is estimated, so the fault is identified.
As an initial motivation, a conventional Luenberger-like adaptive observer is designed to estimate the states and parameters. It is presented as follows: where θ represents the parameter estimates which are composed of parameters in which may occur faults according to prior knowledge and K θ denotes the gain of the parameter estimation observer. e state estimates the equation in the above observer and still uses an output variable residual r(t) for state correction. e objective of selecting the gain K obv in the state estimate equation is to ensure the sensitivity of the output residual of the parameter deviation θ − θ.
at is to say, if parameter deviation θ − θ ≠ 0, then the output residual r(t) ≠ 0. On the contrary, the selecting of the gain K obv should ensure that output residual r(t) and state error x(t) − x(t) have certain robustness to noise, uncertainty, initial condition difference, etc. at is to say, when the parameter deviation θ − θ � 0, the gain K obv can eliminate the effects of noise, uncertainty, and initial condition differences and make the output residual and the state error converge to zero.
In the above Luenberger-like adaptive observer, the parameter identification of the Luenberger-like adaptive observer chooses the search path according to the variation of the output variable output continuously. However, for a nonlinear dynamic system, especially in the case of multiparameters, in the continuous parameter identification search path, the case is often complex and difficult. ere may be local optimal points or flat road sections which cause the failure of parameter identification. To overcome the disadvantages of this kind of continuous path search method, the differential evolution (DE) algorithm-based relative analysis can be used to fulfill parameter estimation in the adaptive observer. Consequently, the following DEbased adaptive observer is obtained: where min θ DE * { } presents the optimization calculation using the differential evolution algorithm, ρ yy′ is a correlation coefficient which measures the degree of similarity between different signals, and the absolute value of ρ yy′ is larger when the degree of correlation between two time series is higher. e objective function of DE optimization calculation is chosen as min( n i�1 ρ yy′ − n). e choice of this objective function is based on the consideration that the correlation calculation only analyzes the morphological similarity between the variables and does not care about the difference in the amplitude of the variables.
To sum up, the fault diagnosis scheme based on an adaptive observer using DE is shown in Figure 1. Figure 1 shows the scheme of the fault diagnosis as a solution of an optimization problem. is article will use the above scheme of fault diagnosis based on parameter estimation in a nonlinear system. Such approach is presented in the next section.

Mathematical and Algorithm Tools
4.1. Correlation Analysis. As a statistical method, correlation analysis has been applied in evaluating the strength of the relationship between two quantitative variables with available statistical data. is technique is strictly connected to the linear regression analysis that is a statistical approach for modeling the association between a dependent variable and one or more explanatory or independent variables. In this paper, the Pearson correlation coefficient is chosen to extract the feature and apply it to multifault identification. is method has many advantages, such as high accuracy, high efficiency, and strong practicability.
e Pearson correlation coefficient presents a measure of how two random variables relate linearly in a sample. e Y and Y ′ vectors in the following formula define the residual correlation coefficient, which is generally denoted by the symbol ρ YY′ : where cov is the covariance, σ is the standard deviation, and E is the expectation. e Pearson correlation coefficient lies in the range −1 ⩽ ρ YY′ ⩽ 1. In order to obtain the correlation coefficient, obtaining σ Y′ , σ Y , and cov(Y, Y ′ ) is necessary. In practice, these parameters for the variables are either unknown or difficult to obtain. us, the alternative is using ρ YY′ which one can be obtained from a sample. e following formula expresses the sample correlation coefficient ρ YY′ : where in which T is the length of the data window used for fault identification; T s is the sampling step size; (y i ′ , y i ) is the i th pair observation value; and y ′ and y are sample means for y ′ and y, respectively. Pearson correlation is a straightforward approach to evaluate the relationship between two variables. e larger the |r| is, the more significant the relationship between Y and Y ′ is.
In the subject of signal and system, correlation analysis is an important method to describe signal characteristics in the time domain. Since the correlation concepts were introduced to study the statistical characteristics of random signals, they can be also theoretically applied to the study of the similarity between two deterministic signals (one of the reference waveform and one of the signal waveform collected).

Differential Evolution Algorithm (DE)
. Differential evolution (DE) is a simple and powerful intelligent optimization algorithm. As an advanced version of the genetic algorithm (GA), DE has been the subject of much attention due to its attractive characteristics of the need for lower parameters, simple structure, ease of use, and robustness [15,16]. DE has been extended for handling multiobjective, constrained, large-scale, dynamic, and uncertain optimization problems. Due to the advantages of DE, it was developed for many applications in engineering fields. In DE, a general description includes four operators of population initialization, mutation, crossover, and selection. An overview of the main steps of the DE algorithm is presented next.

Initialization.
Setting up the maximum generation number G max and generation index G � 1, the initial population within the range [θ min i , θ max i ] of parameters is randomly created.
where rand(0, 1) denotes a random number in the interval (0, 1), i � 1, 2, . . . , NP. e value of NP is the number of individuals of population that will affect computational time and fault identification performance. Increasing NP means high computational time but low risk to be trapped in local minimum.

Mutation.
Mutant individual is generated by using (21). e scale factor of difference perturbation F is chosen in the range of [0, 2].
where θ j , θ k , θ l (j ≠ k ≠ l) are randomly selected vectors, θ G is a mutant, and all of these vectors must be dissimilar so that the population contains at least four individuals.

Selection
Operation. Fitness computation f(θ) by (13) and comparison by (23) is used to update the population.
Mathematical Problems in Engineering 5 is step is repeated until the termination criterion is satisfied, and hence a population of solution is obtained.

Case Studies
e effectiveness of the proposed correlation-based robust FDI is demonstrated through typical nonlinear dynamic process systems with variable parameters.

Description of ree-Tank Model.
e three-tank system is widely used for comparison and demonstration purpose in control engineering as a benchmark process in laboratories for process control [17]. To illustrate the effectiveness of the proposed methodology, in this section, a simulation experiment on the three-tank system (TTS) is introduced. TTS is shown in Figure 2. e precise mathematical model can easily describe the multiply faults such as leakage/plugging/ sensor/actuator faults. Industrial systems consist of the plant (or system dynamics), sensors, and actuators [4]. e sensor and actuator faults are additive fault. e leakage and plugging faults are usually nonadditive. In the references, many efficient fault estimation methods of the sensors and the actor can be found [18][19][20][21]. However, from the FTC point of view, much more important is the problem of the plant (component) fault diagnosis.
ere have been few studies on nonadditive faults, especially for the nonlinear dynamic system. In this paper, two types of tank faults (leakage and plugging faults) are belonging to nonadditive faults.
Applying the incoming and outgoing mass flows with the Torricellies law, the dynamics of TTS is modeled by where A is the cross-section area of the tank; A n is the crosssection area of the pipe; Q 1 , Q 2 are the incoming mass flow (cm 3 /s); h i (t), i � 1, 2, 3, are the water level (cm) of each tank and measured; δ i , i � 1, 2, 3 is the coefficient of flow for pipes.
e state vector X, output vector Y, and input U will be e dynamic model fault free is as follows:

Real system
Residual evaluation r(t) > 0

Condition monitoring Input u(t)
Outside the threshold

Residual r(t)
Output y′(t) Adaptive observer Output y(t)

Fault isolation and identification
Parameters evaluation

Multifault Diagnosis Results.
In this paper, the process is affected by two types of faults: the leak faults and the plugging faults. e leaks in the three tanks can be modeled as additional outgoing mass flows of tanks [22]: where S 1 , S 2 , and S 3 are the cross-sectional area of leaks, which are unknown and depend on the size of the leaks, and a 1 , a 2 , and a 3 are the coefficients of flow for leaks and unknown. e plugging between two tanks and in the outlet pipe by tank 2 can be modeled by 1] are unknown. e model of plant faults is modeled by In general, the size of the leaks is less than the pipes, then the coefficient of flow for leaks is less than that of the pipes, thus let Tank 1  Tank 3  Tank 2 S sn Figure 2: ree-tank system by Ding [22].

Mathematical Problems in Engineering
where ρ i , k i are the scale factors and ρ i , k i ∈ [0, 1]. Note: as shown by equations (27) and (28), fault 2 and fault 6 are linearly correlated with respect to ���� 2gx 2 , which does not satisfy the first necessary condition of eorem 1. erefore, fault 2 and fault 6 cannot be accurately identified and isolated at the same time in this system. e defining Q * f2 � Q f6 − Q f2 is a trade-off in order to achieve a linear description on parameters.
e linear parameter-space representation then becomes where   (e) Figure 4: Estimation of the five parameters (SNR � 10). Mathematical Problems in Engineering e output residuals r 1 (t), r 2 (t), and r 3 (t) of the fault detection system are shown as follows: where y 1 , y 2 , and y 3 are the system outputs and y 1 ′ , y 2 ′ , and y 3 ′ are the Luenberger-like observer outputs. According to equation (13), the object function of the parameter optimization problem is

e Results in the Absence of Noise.
In order to verify the effectiveness of the proposed approach in the paper, the noise-free case is first examined with the aid of the threetank system. In the system, five parameters can constitute 31 fault modes. Considering that the essence of any fault mode is to estimate the coordinates of parameters in the parameter space, only five typical modes are selected for verification of the proposed approach. Five experiments were conducted for five fault modes. e process parameters of the system are given in Table 1. e diagnosis results with different fault dimension are shown in Table 2.

e Results in the Presence of
Noise. Now, the output signal is assumed to be corrupted by noise. If the noise is relatively high compared to the effect of fault on the output, then the correlation coefficient will decrease. us, performance of fault identification will degrade. Figure 3 shows the relationship between the correlation coefficient and signal to noise ratio (SNR). e values of the estimated parameters under different output SNR are shown in Table 3, which indicates that the higher the SNR, the better the estimation performance. From Figure 3, when SNR > 30, the correlation coefficient is close to 1 and changes slowly. When SNR < 30, the more the SNR, the larger the correlation coefficient. All the implementations were made in MATLAB R2016b. Figures 4  and 5 show the parameter identification results with different SNR. Comparing to Figure 5, Figure 4 shows the degradation of parameter estimation performance in the low SNR. Figure 6 shows the real output of the three tanks and the estimated output. From Figure 6, it can be seen that when the correlation coefficient is 3, the two curves almost overlap. Figure 7 shows the residual changes when the faults occur at 1000 and 2000 seconds.

Conclusion
In this paper, the diagnosis system follows a general modelbased architecture where the output signal correlation coefficient is used to detect inconsistencies between model predictions and sensor data. e fault detection system and fault identification system are designed. In the fault identification system, a variable parameter model is used to simulate the real system. e output correlation coefficient between the fault detection system and the fault identification system is formulated as an optimization problem. It is solved by evolutionary algorithms to estimate fault parameters. By integrated estimation of all parameters of the system, the coordinates of the fault point in the parameter space can be obtained and the faults severity and mode of the fault (single fault or multiple faults) can be determined. Comparing with common residual-based methods, e.g., filter and adaptive observer methods, the proposed method reduces the number and the complexity of the observer for the estimation of high-dimensional parameter. It overcomes the problem of combinatorial explosion of fault modes and simplifies the fault diagnosis structure. To our knowledge, there are few papers on the adaptive isolation and identification mechanism which combine the observer and intelligent optimization algorithms. Using the powerful global search ability of an evolutionary algorithm, the proposed method is more beneficial to overcome local optimality comparing with the traditional adaptive search mechanism based on parameter gradient change.

Data Availability
Since the experimental results are obtained by the simulation calculation of the mathematical model, the paper does not use explicit data. Models and parameters are present in the submission.

Conflicts of Interest
e authors declare that they have no conflicts of interest.