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Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System



J. Zhou and X. Huang
 
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ABSTRACT

In this study, the nonlinear fault detection and isolation (FDI) scheme is successfully applied to the aerocraft’s nonlinear closed-loop control system, which is established by using the dynamic inversion theory. The nonlinear FDI scheme consists of a bank of nonlinear adaptive estimators. One of them is the fault detection and approximation estimator, whereas the others are used for fault isolation (each associated with a specific type of fault). A type of fault that has occurred can be isolated if the residual associated with the matched isolation estimator remains below its corresponding adaptive threshold and at least one of the components of the residuals associated with all the other estimators exceeds its threshold at some finite time. The simulation results show the effectiveness of the application.

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  How to cite this article:

J. Zhou and X. Huang, 2008. Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. Information Technology Journal, 7: 1163-1169.

DOI: 10.3923/itj.2008.1163.1169

URL: https://scialert.net/abstract/?doi=itj.2008.1163.1169
 

INTRODUCTION

The failures which occur in the system usually cause enormous loss of productivity, expensive equipment and human lives ultimately, so the design and analysis of fault detection scheme has attached great importance and has received considerable attention. During the last two decades, various approaches to fault detection (FD) using model-based analytical redundancy have been represented. The studies based on these methods mainly focused on the robustness with respect to unknown factors (the disturbance, noise) and the sensitivity to early fault. Several state estimation-based FD methods are proposed, such as Luenberger robust observer (Ibaraki et al., 2005), linear matrix inequalities (LMI) (Wang et al., 2007), H-infinite estimator (Collins and Song, 2000{Collins, 2000 #2}) and multiple-model estimation method (Hofbaur and Williams, 2004).

Unlike the fault detection problem, which has been extensively investigated in the literature, the fault isolation (FI) problem has received less attention, especially in the case of nonlinear uncertain systems. In this study, a key design issue of the fault isolation scheme is the adaptive residual threshold associated with each isolation estimator. That is, the residual of each fault isolation estimator is associated with an adaptive threshold, which can be implemented online by using linear filtering methods (Basin et al., 2005). The occurrence of a particular fault is excluded if at least one of the residual components of the corresponding isolation estimator exceeds its threshold in a finite time. Fault isolation is achieved when all faults but one is excluded.

Meanwhile, the application of FD (without the consideration of FI) scheme aims at LTI and open-loop system. Such as the H estimation algorithms to faulty aircraft (Melody et al., 2001). The problem of fault detection was addressed through H robust estimator using Popov-Tsypkin multipliers in (Collins and Song, 2000). Their analysis were based on open-loop linear time-invariant (LTI) system response where the filters were designed for a nominal condition and tested for a second off-nominal LTI plant for robustness analysis. In (Felicio et al., 2002), the H-FD Ricatti-based approach is used to design FD filters for an inverted pendulum. But the application was not fully carried out in this case (i.e., no simulations). In fact, most practical conditions of the system are nonlinear in nature and most failures are more accurately modeled as nonlinear functions about the state/output and input variables.

So, in this study, a FDI scheme using some nonlinear estimators to detect and isolate the incipient faults in an aerocraft’s nonlinear uncertain closed-loop system is significative. In the presence of a failure, the function exported by the on-line approximator which is the important component of the nonlinear estimator can be used as an estimate of the possible nonlinear fault function. Once the fault is detected, the isolation estimators are activated for the purpose of fault isolation. A type of fault that has occurred can be isolated if the residual associated with the matched isolation estimator remains below its corresponding adaptive threshold and at least one of the components of the residuals associated with all the other estimators exceeds its threshold at some finite time. The simulation results show the effectiveness of the application.

PROBLEM FORMULATION

Here, the non-linear system in presence of the fault is formulated as below:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(1)

The differential equation Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systempresents the nominal model, η = [η,...ηn]T describe the model uncertainty, f = [f1,...fn] presents the fault occurs in the system. Many fault diagnosis schemes simplify the fault function to the time function, actually, most fault is the nonlinear function of the state variables x and input vector u.

Each time component βi(i = 1,...n) can be described as the form below (Trunov and Polycarpou, 2000):

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(2)

where, γi is an unknown constant and γi>0, it presents the rate at which the fault changes in state variable x evolves.

Assumption 1:The state variable x(t) is available for measurement

Assumption 2:Each modeling uncertainty function ηi is bounded by a known constant Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(3)

NONLINEAR FAULT DETECTION ISOLATION SCHEME

A class of N+1 nonlinear adaptive estimators are used in the proposed FDI scheme, where, N is the number of nonlinear faults. One of the nonlinear adaptive estimators is the fault detection and approximation estimator used to detect faults. The remaining ones are fault isolation estimators that are used for isolation purposes only after a fault has been detected. The structure of the fault detection estimator and the fault isolation estimators are given below, respectively.

Fault detection and approximation estimator: Based on (1), a nonlinear FD estimator of the form (4) is considered:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(4)

where, Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis the estimated state vector, Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systempresents an online approximation model, Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis an adjustable weight vector, Gd = (gdl,...,gdn) is a positive value. The initial value Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis chosed such that Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, which corresponds to the case where the system is normal (no fault).

A key component of the nonlinear adaptive estimator described by (4) is the online approximator, denoted by Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, whose ith component has the structure as (5):

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(5)

The next step in the construction of the fault detection estimator is the design of the learning algorithm for updating the weights Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. Let Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systembe the state estimation error. Using techniques from adaptive control (Lyapunov synthesis method) (Trunov and Polycarpou, 2000), the learning algorithm of the online approximator is chosen as follows:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(6)

The projection operator P restricts the weights estimate Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemto a predefined region Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis a symmetric positive definite learning rate matrix and Z denotes the gradient matrix of the online approximator with respect to its adjustable weights, i.e., Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. The dead-zone operator Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis defined as follows:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(7)

The dead-zone operator Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemwhich acts on the variable εdi such that the output of the operator is zero while no fault in system, that is Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. The detection of the fault occurs if at least one state εdi of the state estimation error exceeds its dead-zone boundary Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. More precisely, the fault detection time is defined as the first instant of time such that Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, that is:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(8)

By Eq. 1 and 4 it can be easily verified that each component of the state estimation error satisfies:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(9)

Because of Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, then using the Eq. (3), the FD threshold Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemcan be get as follow:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(10)

Next, the sensitivity of the fault detection scheme will be considered. The faults occur at some unknown time Tz and develop with the rate γi.

Theory 1: Consider the nonlinear fault diagnosis scheme described by Eq. 4 and 6. If there exists an interval of time [Tz+t1, Tz+t2], such that at least one component fi(x(t), u(t)) of the fault vector f(x(t), u(t)) satisfies:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(11)

then the fault can be detected, that is Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System(the proof can be found in Frank and Seliger (1991).

Fault isolation estimators: Once a fault has been detected, the isolation scheme is activated. Then construct the following nonlinear adaptive estimators as isolation estimators:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(12)

where, Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemfor i = 1,...,n, l...,N is the estimate of the fault parameter vector in the ith state variable. For notational simplicity and without loss of generality, assume that gli = gi, for all l = 1,...,N.

The design of fault isolation estimators is similar to the design of the fault detection estimator. Each isolation estimator corresponds to one of the possible types of nonlinear faults. SpecificallyImage for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, if let be the ith component of the state estimation error vector of the l th estimator, then the learning algorithm is chosen as (Zhang et al., 2002):

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(13)

where, Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis a symmetric positive definite learning rate matrix.

The fault-isolation decision scheme is based on the generalized observer scheme (GOS) principle: if the lth fault occurs at some time Tz and is detected at time Td, then a set of adaptive thresholds λli (t), 1,...,n can be designed such that the ith component of the state estimation error associated with the lth estimator satisfies Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, for all t>=Td. If for each rε{1,...,N}\{l}, there exist some finite time tr>=Td and some iε{1,...,n} such that Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, then the possibility that the fault l may have occurred can be deduced. The absolute fault isolation time is defined as Tls = tr.

Theorem 1: If the incipient fault l occurs, then for all t>=Td and for all iε{1,...,n}, the ith component of the state estimation error of the lth isolation estimator satisfies the following inequality:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(14)

Where:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(15)

represents the fault function estimation error in the case of a matched fault.

Proof: On the basis of (1) and (12), in the presence of the fault l, the ith component of the error dynamics of the lth isolation estimator for is given by:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

Then the solution to the previous differential equation is

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

where, ψli(t) is defined as (15). By taking norms, the Eq. 14 can be get

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

So, the proof is complete.

Because in Eq. 14 the fault approximation error ψli(t), the fault evolution rate γi and the fault occurrence time Tz are unknown, it cannot be directly used as a threshold function for fault isolation. However, as the estimateImage for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System belongs to the known compact parameter set Θli, letting the parameter set Θli be a hypersphere with center oli and radius Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemcan be get. Then

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(16)

Meanwhile, for the incipient fault time profile given by Eq. 2, assume that the unknown fault evolution rate satisfiesImage for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System , where, Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemdenotes a known lower bound on the unknown fault evolution rate γi . So, the Eq. 7 can be get:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(17)

Hence, based on (14), (16) and (17), the following threshold functions for fault isolation are chosen as below

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(18)

Next, the fault isolability condition of the proposed FDI scheme will be analyzed. Intuitively, faults are easier to isolate if they are sufficiently mutually different in terms of a suitable measure. In the following analysis, a fault mismatch function in the form is introduced:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(19)

which can be interpreted as the difference between the actual lth fault function in the ith state equation, represented by Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System, u(t)) and the estimated fault function Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemassociated with any other isolation estimator whose structure does not match the actual fault. The following theorem characterizes the class of incipient nonlinear faults that are isolable by the proposed FDI scheme.

Theorem 2: Consider the fault isolation scheme described by Eq. 12, (13) and (18). The incipient fault is isolable if for each rε{1,...,N}\{l} there exist some time tr> Td and some iε{1,...,n} such that the ith component mlri(t) of the fault mismatch function satisfies the following inequality:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(20)

Proof: Based on (1) and (14), in the presence of the fault, the ith component of the error dynamics associated with the estimator r is given by:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

the solution of the above differential equation is

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

By using the triangle inequality, Eq. 21 is obtained

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(21)

So, the threshold for the state estimation error of the rth estimator is:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

Therefore, if (20) is fulfilled, the occurrence of the fault r is excluded at time tr, i.e., Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. If this is satisfied for each rε{1,...,N}\{l}, then the ith fault can be isolated, thus concluding the proof.

AEROCRAFT’S NONLINEAR CLOSE-LOOP SYSTEM

Aerocraft’s nonlinear model: Consider the tail-controlled pitch-axis missile airframe (Kim et al., 2004) depicted in (Fig. 1).

The aircraft nonlinear model for the longitudinal axis are presented.

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
Fig. 1: Aerocraft airframe

Table 1: Model parameters
Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(22)

where, α denotes the angle of attack, q is the pitch rotational rate, δ is the tail fin deflection, Cn and Cm are the aerodynamic coefficients, the expressions of which depend on α, δ and the Mach number M (1.5~3). The parameters can be find in Table 1.

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(23)

The output of the system is the normal acceleration.

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(24)

The aerodynamic surface actuator and reaction jet actuator dynamics are modeled as:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(25)

Nonlinear control law design: Since the present missile configuration is tail controlled, it has a strong non-minimum phase behavior with respect to the normal acceleration. Consequently, a straightforward application of the feedback linearization technique will not be successful, because the unstable zerodynamics would not assure the internal stability of the system (Devaud et al., 2000).

Two approaches have been proposed to overcome this difficulty. First of these is the time-scale separation of the system dynamics into slow and fast modes. The second approach is to redefine the system outputs to suppress the zerodynamics. The first method will be adopt in the following sections. The first step in the two time-scale design process is to split the system dynamics into time scales based upon the notion of slow and fast dynamic modes. Note that even if a clear separation between the modes are not present in the open-loop dynamics, mode separation can be enforced during control system design. For the present case, the actuator dynamics, together with the pitch rate dynamics are included in the fast time-scale. The normal acceleration dynamics is considered to be the slow time scale mode.

Control of slow subsystem: The state is attack angle α and the output is the pitch angle rate q. The subsystem is described as follow:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(26)

The desired closed-loop aerodynamic of the slow subsystem can be described by first order module as follow:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(27)

Then the desired input of the slow subsystem can be computed by Eq. 26 and Eq. 27

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(28)

Control of fast subsystem: The state and output variants are identical q and the input is the rudder deflect angle δ. The system is described as follow:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(29)

The desired aerodynamic of the fast subsystem can be denoted by a second order modul as follow:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(30)

Proceeding as before, the nonlinear control law for the aerodynamic surface actuator in the fast time-scale is given by:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
(31)

SIMULATION RESULTS

Here, the proposed FDI scheme is applied to detect and isolate incipient faults in a nonlinear aerocraft’s control system described as Eq. 22, 24, 25 and 31.

The faults which occur in the aerocraft’s system are described as (two faults):

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

Where, θ1εΘ1 = [-1.5, 1.5] and θ2εΘ2 = [-1.5, 1.5]. Assume that the unknown incipient-fault evolution rate γ defined in (2) satisfies:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

Fault detection: Based on Eq. 22, the nonlinear FD estimator is constructed as follow:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

The online approximator Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemis implemented as a continuous radial basis function (RBF) neural network (Trunov and Polycarpou, 2000), which are described by:

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

where, the weight Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop Systemcan be tuned by Eq. 6 and choose a uniform width σ = 0.6 for the basis functions and 11 fixed centers cj (i.e., m = 11), which are evenly distributed in the interval [-2,2]. The stability and fault-detectability properties of the fault detection estimator have been investigated in (Trunov and Polycarpou, 2000).

Fault isolation: By using the methodology described earlier, a bank of two isolation estimators is designed

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System

The control input is set as Eq. 31. The modeling uncertainty is assumed to arise out of a 5% inaccuracy in the value of ai, bi, ci, (i = m, n). The bounding functionImage for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System. Moreover, set gi = 5(i = 1, 2).

The experiment results: Figure 2 shows the simulation results when an incipient fault of type 1, with θ1 = 0.75 and the fault evolution rate γ = 0.2, occurs at t = 10 sec.

Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System
Fig. 2:
(a) Time-behaviors of the fault function (solid line) and the RBF neural-network output (dash-dotted line) associated with the fault detection estimator (b) Time behaviors of the state estimation error (solid line) associated with the FDAE and the dead-zone threshold (dash-dotted line) (the fault detection time instant is shown by an arrow) (c) and (d) Time-behaviors of the state estimation errors (solid lines) and the thresholds (dash-dotted lines) associated with the two isolation estimators (the fault isolation time instant is shown by an arrow)

The evolution of the actual fault function f (solid line) and the output of the neural network approximator Image for - Application of a Nonlinear Fault Detection and Isolation Scheme to an Aerocraft’s Nonlinear Closed-Loop System(dash-dotted line) associated with the fault detection estimator are shown in Fig. 2a. The state estimation error (solid line) of the fault detection estimator and its corresponding dead-zone threshold (dash-dotted line) are shown in Fig. 2b. As shown, the fault is detected at approximately Td = 11.2 sec. Moreover, in Fig. 2c and d, the residuals εl(t) (solid lines) and their corresponding thresholds λl(t) (dash-dotted lines), associated with each isolation estimator, are shown. It can be seen that the residual of estimator 1 always remains below its threshold, whereas the residual of estimator 2 exceeds its threshold at approximately Tls = 15.5 sec, thus allowing the isolation of fault 1.

CONCLUSION

In this study, a fault detection and isolation scheme using some nonlinear estimators to detect and isolate the incipient faults is presented and an aerocraft’s nonlinear closed-loop system using dynamic inversion method is constructed. Then the FDI scheme is applied in the aerocraft’s faulty system constructed before. The simulation results show the effectiveness of the application.

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