A Neuro-Augmented Observer for Robust Fault Detection in Nonlinear Systems

Abstract

A new fault detection method using neural-networks-augmented state observer for nonlinear systems is presented in this paper. The novelty of the approach is that instead of approximating the entire nonlinear system with neural network, we only approximate the unmodeled part that is left over after linearization, in which a radial basis function (RBF) neural network is adopted. Compared with conventional linear observer, the proposed observer structure provides more accurate estimation of the system state. The state estimation error is proved to asymptotically approach zero by the Lyapunov method. An aircraft system demonstrates the efficiency of the proposed fault detection scheme, simulation results of which show that the proposed RBF neural network-based observer scheme is effective and has a potential application in fault detection and identification (FDI) for nonlinear systems.

1. Introduction

One of the basic requirements for successful residual-based fault detection and identification (FDI) is that during fault-free operation, the residuals should be zero in the deterministic case or zero-mean and uncorrelated in stochastic case. While this condition is easy to meet in linear systems, it is a difficult task for nonlinear systems. Reference [1] presents a fault-tolerant control (FTC) scheme for a class of multi-input-multi-output stochastic systems with actuator faults, in which the actuator fault diagnosis is based on the state estimation and the nominal controller is designed to compensate for the loss of actuator effectiveness. However, for nonlinear systems, the common approach of linearizing the nonlinear system with Taylor series expansion about an operating point may work reasonably well for control design, but the errors from the unmodeled nonlinear terms affect the residual in a way that renders residual-based FDI techniques unusable. Reference [2] proposes a fault-tolerant tracking control method based on adaptive control technique for near-space-vehicle attitude dynamics, in which Takagi-Sugeno fuzzy model is used to describe the nonlinear system, and then an adaptive tracking control scheme is developed based on on-line actuator faults estimation.

One way of eliminating this problem is to develop state observers that estimate the actual state vector that includes the linear and nonlinear terms. State observers of nonlinear dynamic systems are, therefore, becoming a growing topic for fault detection and identification (FDI) [3–6]. Several observer design methods have been proposed in recent years for nonlinear dynamic systems [7–9]. However, effective and accurate observer design methods for generic nonlinear systems are still an open research issue. Neural networks (NN) provide one of the newer approaches to robust FDI for nonlinear systems [10–12]. The theory and application research of the neural networks is still a hot topic in intelligent control fields. Reference [13] uses three neural networks as parametric structures for facilitating the implementation of the iterative algorithm, to solve the near-optimal control problem of the nonlinear discrete-time systems with control constraints. Reference [14] develops a weighting-delay-based method for stability analysis of a class of recurrent neural networks with time-varying delay, and the optimization method is used to calculate the optimal weighting-delay parameters. Applications of neural networks in FDI can be grouped into two categories of tasks: (1) neural network observers for detecting the faults and (2) neural network classifiers for classifying fault patterns.

In this paper we focus on the first group of tasks: we present a new method for fault detection using a neural-networks-augmented state observer. The novelty of our approach is that instead of using the entire nonlinear system for designing the neural network-based observer, we only use the unmodeled part that is left over after a conventional linearization with Taylor series. Instead of the multilayered back-propagation neural networks, we choose radial basis function (RBF) neural network, which is faster to train and does not have the problem of local minima [15]. This allows us to build a state observer with an output tracking error that approaches zero asymptotically. The technique is applicable to systems that have nonlinearities, but their accurate nonlinear models are either not available or are difficult (or too time-consuming) to use. However, it assumes that the linearized models of these systems are available.

2. Problem Formulation

Consider the affine nonlinear dynamic system as follows: 𝑥̇(𝑡)(𝑥(𝑡(𝑥(𝑡(𝑡),𝑦(𝑡)(𝑡),=𝑓))+𝑔))𝑢+𝛿=𝐶𝑥 (2.1) where 𝑥∈𝑅^𝑛,  𝑢∈𝑅^𝑚,  𝑦∈𝑅^𝑝,  𝛿∈𝑅^𝑛 is a norm-bounded fault vector, ‖𝛿(𝑡)‖≤𝐷, 𝐶∈𝑅^𝑝×𝑛,  𝑓(𝑥)=[𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑛(𝑥)]^𝑇, 𝑔(𝑥)=[𝑔_𝑖𝑗(𝑥)]_𝑛×𝑚, and 𝑓(⋅),  𝑔(⋅) are continuously differentiable vector functions. Our goal is to design a nonlinear state observer that is a combination of a conventional observer and a neural network (the neuro-augmented observer).

System (2.1) is linearized at operating point (𝑥₀,𝑢₀), resulting in the state space model ̇()(𝑡),𝑦(𝑡)(𝑡),𝑥=𝐴𝑥+𝐵𝑢+𝜙𝑥,𝑢,𝑡+𝛿=𝐶𝑥 (2.2) where 𝐴=(𝜕𝑓/𝜕𝑥)_𝑥=𝑥0∈𝑅^𝑛×𝑛 and 𝐵=𝑔(𝑥₀)∈𝑅^𝑛×𝑚 are constant matrices. 𝑓(𝑥) is expanded in a first-order Taylor series approximation at 𝑥=𝑥₀, and 𝑓(𝑥)=𝑓(𝑥₀)+𝐴𝑥+𝑓(𝑥). Similarly, 𝑔(𝑥)=𝑔(𝑥₀)+𝑔(𝑥). Now we can get 𝜙(𝑥,𝑢)=𝑓(𝑥₀)+𝑓(𝑥)+𝑔(𝑥)𝑢.

The nonlinear function 𝜙(𝑥,𝑢) in system (2.2) includes an unmodeled nonlinear high-order term. If the system has no fault, that is,  𝛿(𝑡)=0, the state observer of system (2.2) is 𝑥̇=(𝐴−𝐿𝐶)𝑥+𝐿𝑦+𝐵𝑢+𝜙(𝑥,𝑢), (2.3) where 𝑥 is the state estimate, and 𝐿∈𝑅^𝑛×𝑝 is the observer gain matrix. Define the state estimation error 𝜀=𝑥−𝑥, and we get 𝜀̇=(𝐴−𝐿𝐶)𝜀+𝜙(𝑥,𝑢)−𝜙(𝑥,𝑢). (2.4)

The task of observer design is to select a suitable matrix 𝐿 and make the state estimation error tend to zero asymptotically.

Reference [16] provides a method to check the stability of estimation error. However, it does not indicate how to choose the observer gain matrix 𝐿 and how to approximate the nonlinear term. For the state estimation error equation (2.4), if 𝜙_𝑒=𝜙(𝑥,𝑢)−𝜙(𝑥,𝑢) then the state estimation error at time 𝑡 is 𝜀(𝑡)=𝑒^{(𝐴−𝐿𝐶)𝑡}𝜀(0)+^𝑡₀𝑒^{(𝐴−𝐿𝐶)(𝑡−𝜏)}𝜙_𝑒(𝜏)𝑑𝜏. (2.5) From (2.5) we can see that minimizing 𝜙_𝑒 and driving it to zero would make the estimation error also approach zero asymptotically. Thus, the success of the observer depends on the successful approximation of the nonlinear term that resulted from the unmodeled portion of the original system. In order to achieve that goal, we propose the following structure shown as Figure 1, which incorporates an RBF neural network into the state estimation loop.

Figure 1 

Robust fault detection based on neural network nonlinear observer.

The RBF neural network is used to approximate the nonlinear term 𝜙(𝑥,𝑢), thereby driving 𝜙_𝑒 to zero, and thus improving the accuracy of state estimation. With that, we can expect the residuals during fault-free (healthy) situations to remain small, which is essential for successful FDI design.

3. Design of Neuro-Augmented Observer Based on RBF Neural Network

An RBF neural network observer is introduced for system (2.2), expressed by 𝑥̇=(𝐴−𝐿𝐶)𝑥+𝐿𝑦+𝐵𝑢+𝜙_𝑛𝑛(𝑥,𝑢,), (3.1) where 𝜙_𝑛𝑛(𝑥,𝑢) is the neural network nonlinear estimator. It is realized by a three-layer RBF neural network. The first layer is direct input layer, and the hidden layer consists of several nodes. Each node contains a parameter vector called a center. The node calculates the Euclidean distance between the center and network input vector and then passes the result through a nonlinear function. The output layer is a set of linear combiners. The overall input-output response of the RBF neural network is a mapping 𝜙_𝑛𝑛∶𝑅^𝑛+𝑚→𝑅^𝑛, then we get 𝜙_𝑛𝑛(𝑥,𝑢)=𝜙_𝑛𝑛1(𝑥,𝑢),𝜙_𝑛𝑛2(𝑥,𝑢),…,𝜙_𝑛𝑛𝑛(,𝜙()=𝑥,𝑢)𝑥,𝑢^𝑠_𝑗=1𝑤_𝑖𝑗𝜎_𝑗𝑥−𝑐_𝑗,𝜌_𝑗,𝑖=1,2,…,𝑛, (3.2)(3.3) where 𝑥∈𝑅^𝑛+𝑚 is input vector; 𝑛+𝑚,𝑛 are the number of inputs and outputs; 𝑠 is the number of nodes in the hidden layer; {𝑤_𝑖𝑗,𝑖=1∼𝑛,𝑗=1∼𝑠} are the weights of the linear combiners of the output layer; {𝑐_𝑗,𝑗=1∼𝑠} are the centers of hidden nodes; {𝜌_𝑗,𝑗=1∼𝑠} are the widths of the Gaussian function. The nonlinear activation function 𝜎(⋅) is selected as Gaussian function, and then 𝜎(𝑥,𝜌)=𝑒^{−𝑥2/𝜌}.

Define 𝜎𝑥,𝜌=𝜎₁𝑥,𝜌,𝜎₂𝑥,𝜌,…,𝜎_𝑠𝑥,𝜌^𝑇,𝑊=𝑤_𝑖𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑠. (3.4) Then 𝜙(𝑥,𝑢)=𝑊𝜎(𝑥,𝑢). (3.5)

Assumption 3.1. The RBF neural network can approximate the nonlinear function 𝜙(𝑥,𝑢) arbitrarily closely, that is, there exists an ideal weight matrix  𝑊^∗, satisfying 𝜙(𝑥,𝑢)=𝑊^∗𝜎(𝑥,𝜌), where 𝑥 is the input vector.

Assumption 3.2. For the system (2.2) and observer (3.3), the nonlinear term 𝜙(𝑥,𝑢) satisfies the global Lipschitz condition, that is,  ‖𝜙(𝑥,𝑢)−𝜙(𝑥,𝑢)‖≤𝛾‖𝑥−𝑥‖.

Theorem 3.3. The state estimation error approaches zero asymptotically (lim_𝑡→∞𝜀=0), provided (i)𝛾<𝜆_min(𝑄)/2𝜆_max(𝑃), where 𝑃,𝑄 are positive definite symmetric matrices and  𝜆_min and 𝜆_max, respectively, denote the minimum eigenvalue and maximum eigenvalue of the matrices,(ii) the Lyapunov equation (𝐴−𝐿𝐶)^𝑇𝑃+𝑃(𝐴−𝐿𝐶)=−𝑄 holds, (iii)the weight tuning law of RBF neural network is adopted as follows: 𝑤̇_𝑖=𝜂𝜎_𝑖𝑥,𝜌𝑃𝜀, (3.6)where 𝑤_𝑖 is the i th column of 𝑊 and 𝜂 is the learning rate in range of [0,1].

Proof. Select the Lyapunov function 1𝑉=2𝜀^𝑇1𝑃𝜀+𝑊2𝜂tr^𝑇𝑊, (3.7) where 𝑊=𝑊^∗−𝑊. Since the state estimation error is 𝜀=𝑥−𝑥, see from (2.4), we get 𝜀̇=(𝐴−𝐿𝐶)𝜀+𝜙(𝑥,𝑢,𝑡)−𝜙_𝑛𝑛()=()𝑥,𝑢,𝑡𝐴−𝐿𝐶𝜀+𝑊^∗𝜎𝑥,𝜌−𝑊𝜎𝑥,𝜌. (3.8) Adding and subtracting 𝑊^∗𝜎(𝑥,𝜌) in (3.8), we get 𝜀̇=(𝐴−𝐿𝐶)𝜀+𝑊^∗𝜎𝑥,𝜌−𝑊𝜎𝑥,𝜌+𝑊^∗𝜎𝑥,𝜌−𝑊^∗𝜎𝑥,𝜌. (3.9) Considering 𝜎=𝜎(𝑥,𝜌)−𝜎(𝑥,𝜌), 𝑊=𝑊^∗−𝑊, and 𝑤(𝑡)=𝑊^∗𝜎, the above equation can be converted to 𝜀̇=(𝐴−𝐿𝐶)𝜀+𝑊𝜎𝑥,𝜌+𝑤(𝑡). (3.10) Then we get 𝑉̇=12𝜀^𝑇̇1𝑃𝜀+2𝜀^𝑇𝑃̇1𝜀+𝜂tr𝑊̇^𝑇𝑊1=−2𝜀^𝑇𝑄𝜀+𝜀^𝑇𝑃(𝑡+1𝑊𝜎𝑥,𝜌+𝑤)𝜂tr𝑊̇^𝑇𝑊1=−2𝜀^𝑇𝑄𝜀+𝜀^𝑇𝑃𝑤(𝑡)+^𝑠_𝑖=1𝜀^𝑇𝑃𝑤_𝑖𝜎_𝑖+1𝑥,𝜌𝜂^𝑠_𝑖=1𝑤̇^𝑇_𝑖𝑤_𝑖, (3.11) where 𝑤_𝑖 is the 𝑖th column of 𝑊. And then 𝑉̇=−12𝜀^𝑇𝑄𝜀+𝜀^𝑇(𝑡)+1𝑃𝑤𝜂^𝑠_𝑖=1𝜂𝜀^𝑇𝑃𝜎_𝑖𝑥,𝜌+𝑤̇^𝑇_𝑖𝑤_𝑖. (3.12) If the weight tuning law of the RBF neural network is 𝑤̇_𝑖=−𝜂𝜎_𝑖(𝑥,𝜌)𝑃𝜀 or 𝑤̇_𝑖=𝜂𝜎_𝑖(𝑥,𝜌)𝑃𝜀, then 𝑉̇=−12𝜀^𝑇𝑄𝜀+𝜀^𝑇(𝑡)1𝑃𝑤≤−2‖𝜀‖𝜆_min(𝑄)‖𝜀‖+‖𝜀‖𝜆_max(𝑃𝑤(𝑡1)‖)‖=−2‖𝜀‖𝜆_min(𝑄)‖𝜀‖+‖𝜀‖𝜆_max(𝑃)𝑊^∗𝜎𝑥,𝜌−𝑊^∗𝜎1𝑥,𝜌=−2‖𝜀‖𝜆_min(𝑄)‖𝜀‖+‖𝜀‖𝜆_max(𝑃𝜙()(1)𝑥,𝑢−𝜙𝑥,𝑢)≤−2‖𝜀‖𝜆_min(𝑄)‖𝜀‖+‖𝜀‖𝜆_max(𝑃)𝛾‖𝑥‖1𝑥−≤−2‖𝜀‖𝜆_min(𝑄)‖𝜀‖+‖𝜀‖𝜆_max(𝑃)𝜆_min(𝑄)2𝜆_max(𝑃)‖𝜀‖≤0, (3.13) then 𝑉̇≤0. Therefore, the RBF neural network observer is stable, and lim_𝑡→∞𝜀=0.

4. Numerical Simulation Study

A fourth-order longitudinal aircraft model [15] is employed to demonstrate the effectiveness of the proposed method. The longitudinal perturbation equation is linearized in horizontal flight at a speed 𝑣₀=300 m/sec and at a height of 12000 m. The system matrices 𝐴, 𝐵, and 𝐶 have the following forms: ,,𝐴=−0.0170.0260−9.81−0.0143−1.021002.06−1.1200010𝐵=00−0.032−0.032−5.78−5.7800,𝐶=1000010000100001 (4.1) where 𝑥(𝑡)=[𝑣,𝛼,𝑞,𝜃]^𝑇 is state vector and the state variables 𝑣,𝛼,𝜃,𝑞 are forward velocity, attack angle, pitch angle, and pitch angular velocity, respectively. 𝑢=[𝛿_el,𝛿_er]^𝑇 is control input vector, 𝛿_el is the left elevator, and 𝛿_er is the right elevator.

Nonlinear state observer (3.3) is designed by placing the poles of (𝐴−𝐿𝐶) at −2, −3, −4, and −5. The observer gain matrix 𝐿 is 𝐿=1.9830.0260−9.81−0.01431.981002.062.8800015. (4.2)

The nonlinear term 𝜙(𝑥,𝑢) is approximated by RBF neural network estimator 𝜙_𝑛𝑛(𝑥,𝑢). It is realized by a three-layer RBF neural network with a 6-8-4 structure. The input layer consists of 6 nodes, and the input vector is [𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡),𝑥₄(𝑡),𝑢₁(𝑡),𝑢₂(𝑡)]^𝑇. The hidden layer consists of 8 nodes. The output layer is the estimation of nonlinear term [𝜙_𝑛𝑛1(𝑡),𝜙_𝑛𝑛2(𝑡),𝜙_𝑛𝑛3(𝑡),𝜙_𝑛𝑛4(𝑡)]^𝑇. The RBF neural network can be off-line trained during fault-free operation. In practice, 𝜙(𝑥,𝑢) will be variable with the changes of flight conditions. Therefore the off-line training and on-line learning can be combined to approximate the nonlinear term 𝜙(𝑥,𝑢), in order to improve the adaptability of the neural network observer.

In the simulation, the left-elevator-stuck fault occurs from 𝑡=2 sec. The residuals are shown in Figure 2. In Figure 2, we get that the state residuals change significantly at 2 sec and the fault-free residuals before 2 sec are zeros. Faults can be detected by comparing the output residuals with preselected thresholds, which are typically derived from the magnitudes of the residuals during fault-free cases.

Figure 2 

Residuals: left-elevator-stuck fault.

5. Conclusions

A new nonlinear observer based on RBF neural network is designed in this paper. The novelty of this work is that instead of approximating the entire nonlinear system with neural network, only the unmodeled part of nonlinear system after linearization is approximated. Furthermore, an RBF neural network that has better performance than the traditional neural networks is employed to design the state observer. The proposed neuro-augmented observer can generate residuals that are essential for fast fault detection. Simulation results of an aircraft model demonstrate the effectiveness of the proposed neuro-augmented observer and exhibit the application prospect of the robust fault detection scheme.