Novel neural observer based fault estimation,reconstruction and fault-tolerant control scheme for nonlinear systems

Abstract

This research is concerned with the adaptive neural network observer based fault approximation and fault-tolerant control of time-varying nonlinear systems. A new strategy for adaptively updating the weights of neural network parameters is proposed to enhance fault detection accuracy. Lyapunov function theory (LFT) is applied for adaptively updating the learning parameters weights of multi-layer neural network (MLNN). The purpose of using adaptive learning rates to update the weight parameters of MLNN is to obtain the global minima for highly nonlinear functions without increasing the computational complexities and costs and increase the efficacy of fault detection. Results of the proposed adaptive MLNN observer are compared with conventional MLNN observer and high gain observer. The effects of various faults or failures are studied in detail. The proposed strategy shows more robustness to disturbances, uncertainties, and unmodelled system dynamics compared to the conventional neural network, high gain observer and other existing techniques in literature. Fault tolerant control (FTC) schemes are also proposed to account for the presence of various faults and failures. Separate sliding mode control (SMC) based FTC schemes are designed for each observer to ensure stability of the faulty system. The suggested strategy is validated on Boeing 747 100/200 aircraft. Results demonstrate the effectiveness of both the proposed adaptive MLNN observer and the FTC based on the proposed adaptive MLNN compared to the conventional MLNN, high gain observer and other existing schemes in literature. Comparison of the performance of all the strategies validates the superiority of the proposed strategy and shows that the FTC based on proposed adaptive MLNN strategy provides better robustness to various situations such as disturbances and uncertainties. It is concluded that the proposed strategy can be integrated into the aircraft for the purpose of fault diagnosis, fault isolation and FTC scheme for increasing the performance of the system.

Keywords

Sensors fault detection fault-tolerant control neural networks sliding mode control observer

1 Introduction

Over the past few decades, safety and reliability of the dynamic systems have become critical factors in industry. Requirements for fault tolerant systems are increasing beyond the safety-critical systems like high-rise buildings, engines, nuclear reactors and aircrafts to new and innovative systems such as autonomous systems [1 –4] and human systems [5, 6]. For all the mentioned systems, Fault Diagnosis (FD) or Fault Detection is one of the important methods that can elude faults and failures and avoid breakdowns and disasters resulting in potential human mortalities. Diverse schemes and approaches have been utilized for Fault Diagnoses such as hybrid scheme [7], model-based scheme and computing-intelligence based scheme [8]. For enhancing the performance of aircraft, sensor fault diagnosis and isolation is essential and important. The main advantage of using model-based schemes over hardware redundancy for sensor fault detection and isolation is the weight reduction of the aircraft which enhances the aircraft performance and leads to both noise reduction and fuel economy [9]. Over the past few decades, many schemes were suggested for sensor fault detection and isolation by making use of the model information [10]. Sensor faults and failures were also considered in projects related to fault detection and isolation in aerospace engineering, such as reconfiguration [11] and advance fault diagnosis [9]. Diverse model-based methods have been suggested for aircraft sensor faults detection and isolation. Residual generation approach based on the H_∞ algorithm is proposed by utilizing the dynamics of the NASA Generic Transport Model (GTM) aircraft in combination with pitot-static fault model, in their approach, they have designed two H_∞ filers for faults detection, one filter is dependent on the measurements of the aircraft whereas the other filter is additionally incorporated for control commands [12]. In [13], an online approach is proposed for the fault detection of chillers, which is the subsystem of heating ventilation air conditioning (HVAC), in first step they have proposed recursive one-class support vector machine (ROSVM), in order to improve the efficiency and effectiveness of their approach, the ROSVM is combined with the Extended Kalman Filter (EKF). Their approach is for the systems which fall into faulty status gradually, how the proposed approach used in this research is suggested for abrupt faults occurrence. In [14], the distributed filtering algorithm is suggested for the soft fault detection of the wireless sensor networks (WSNs) of the nonlinear stochastic systems, they have proposed IT2 T-S fuzzy based distributed fault detection filter in order to avoid the coupling between the wireless sensors as wireless sensors can receive the information as well as from other neighboring sensors. In [15], the cross Long-short term Memory (cross-LSTM) and sliding window Long-short term Memory (SW-LSTM) neural networks are used for the fault detection of electro-mechanical actuators of the aircraft, they have suggested these algorithms based on known time series and known faults type whereas the proposed algorithm used in this research article is for any random and unknown faults detection and isolation. The aircraft sensors and actuators fault detection, isolation and accommodation strategy is suggested in [16], they have used unknown input observers for this purpose, they have assumed redundant sensors existence in the system, therefore, the sensor accommodation is done through switching technique, while the actuator accommodation is done by using the gain scheduling, if a fault occurs in one actuator, the remaining (n-1) actuators are considered. In [17], active fault tolerant control approach is developed for the aircraft, their approach is comprised of fault detection and diagnosis module generates the approximation of the faults which is exploited by additional feedback loop in order to accommodate the fault. In [18], sliding mode observer is designed for the approximation of the pitch and roll attitude angles based on which the fault detection and isolation strategy is developed for gyroscope and accelerometer sensors bias fault, in their work, two detection estimators are suggested, allowing the simultaneous estimation of accelerometer and gyroscope sensors bias fault as well as additional adaptive nonlinear observer is used for the robust approximation of the unknown measurements of the sensors bias. Control based fault detection and isolation which utilize the information of control was also applied by [19]. The multiple-model adaptive technique was applied for detection and isolation of sensor faults [20]. The performance and effectiveness of model-based techniques are dependent on the model of the system used. Because of the complexities, the real-time systems are normally simplified with restricted precision. For complex systems such as aircraft, accurate aerodynamic moments and forces are not always possible to obtain, as a result uncertainties exist in the system model. Often systems are affected by external turbulences that affect the performance and effectiveness of the model-based schemes. During flight, disturbances like wind and time-varying wind are inevitable. The existence of disturbances such as turbulence and wind are also the main factors for aircraft fault detection and isolation. Wind and turbulence can exist in diverse conditions during the flight.

Inertial measurement unit (IMU) is comprised of accelerometers which calculate the specific forces (A_x, A_y and A_z) and gyros which calculate the angular rates (p, q and r). Yaw rate (r) sensor faults are investigated in “Advanced Fault Diagnosis for Sustainable Flight Guidance and Control (ADDSAFE)” project [9] utilizing the parameter varying sliding mode method for the reduction of the effect of linearization errors. Linearization is not required in the airspeed based kinematic model (AS-KM) [21]. Another benefit of using the airspeed based kinematic model (AS-KM) is that the aerodynamic forces and moments calculations are not required. As the accurate calculation of aerodynamic forces and moments is not possible, the kinematic model reduces the uncertainties for fault detection and isolation. Diverse schemes are suggested for the sensor fault detection and isolation (FDI) utilizing the AS-KM, such as double model adaptive approximation [22, 23], sliding mode technique [24] and adaptive Kalman filter [22, 25]. Sensors fault were also approximated by utilizing the same kinematic model [22, 24]. The suggested approach was validated by simulation which shows the effectiveness of innovative adaptive learning rate approach.

Fault detection and diagnosis are categorized into two classes such as linear algorithms [26, 27] and nonlinear algorithms [21 , 28–34]. Both can be used for detection of faults in actuators and sensors. As most of the systems are nonlinear in nature so the linearized techniques may not provide detailed information about the faults in the system. Beside this fact, most of the researchers applied linear methods for fault detection due to their simplicity and low computational cost [26, 27].

In literature, most of the existing model-based fault detection and isolation (FDI) techniques are designed by using the linear time-invariant (LTI) models, these linear FDI schemes do not have the capabilities to detect and isolate faults with high preciseness in case of time-varying nonlinear systems [22]. Therefore, in order to enhance the accuracy of fault detection and isolation (FDI) schemes for nonlinear systems, numerous different nonlinear techniques such as Kalman Filter, sliding mode observer (SMO) [21, 30], neural networks [13 , 32–37]s and fuzzy logic [34, 38] are proposed for the approximation of faults as well as parameter uncertainties of nonlinear systems.

Amongst these algorithms, neural networks (NN) are considered to be the most effective and efficient method, because of learning properties as well as approximation property of the nonlinear functions. Therefore, neural networks (NN) is the best mathematical technique for fault detection (FD). The use of adaptive neural networks has been increased significantly in last few decades [32 –34]. One of the drawbacks of using conventional neural network algorithms is that the weight parameter are updated slowly, due to which they are not more effective in situation like abrupt faults. Therefore, numerous techniques are used to update the weight parameters of NN adaptively. The technique of updating the neural networks weights by utilizing Extended Kalman Filter (EKF) was presented by [39]. They implemented a strategy for satellite actuators fault detection utilizing neural network observer in which weights were updated by utilizing EKF. The designed algorithm lacks generality and is applicable only to a perticular category of nonlinear systems. Sensor faults which have great influence on system performance were also not included. Besides this, the proposed approach was not applied to nonlinear systems such as aircrafts which require quick fault detection system.

In this research article, the model-based novel nonlinear algorithm is designed for sensors fault detection and isolation of an aircraft. For establishing the effectiveness of the adaptive multi-layer neural network observer, Sliding Mode based fault-tolerant control is also designed based on the proposed observer. The learning parameters are updated by an adaptive learning rate technique in the designed algorithm. The key contribution and benefits of this research are summarized as;

The adaptive multi-layer neural network (MLNN) observer based on a novel online adaptive technique called Lyapunov function theory (LFT) is proposed which shows more robustness to faults, disturbances and uncertainties compared previously used techniques such as [32, 40].

The purposed scheme provides high accuracy in the detection of various kinds of faults, in terms of magnitude as well as shape, without increasing the computational complexities and cost.

In order to ensure the stability of the faulty system, fault-tolerant control (FTC) scheme based on Sliding Mode Control (SMC) is also designed for proposed adaptive MLNN observer, whereas in [41], they have proposed tracking control scheme instead of FTC. Fault-tolerant control (FTC) is an additional feature which was missing in the previous works [32, 40].

Compared to existing methods in literature, the proposed strategy is more effective in ensuring the fault detection and stability of the faulty system.

Remarks 1. The Lyapunov Function Theory (LFT) based multi-layer neural network observer is used in this research for fault detection and fault-tolerant control schemes. The main purpose of the Lyapunov Function Theory (LFT) is to update the weight updating parameters adaptively in order to enhance the accuracy of the fault detection and isolation in terms of magnitude, shape and time occurrence, as in conventional NN the weighting parameters are updated based on fixed or constant learning due to which the response of NN is slow in detecting the faults. By observing the results, it can be seen that the proposed strategy is far better than the conventional NN and other techniques existing in literature [32, 40]. Adaptive observer based fault detection strategy was used previously [32, 40] for fault detection and isolation. By comparing the fault detection and isolation results and root mean square error (RMSE) values to the approaches used in [32, 40], it can be observed that the proposed method suggested in this article is more effective, accurate and efficient in fault detection and isolation. Inaccuracy in fault detection can lead to serious damage or system failure. In this study, the proposed method is compared with conventional NN and high gain observer. The main advantage of LFT-NN over [32, 40] is its ability to deal with different flight conditions with high accuracy and robustness in the presence of disturbances.

An adaptive fault-tolerant control scheme is also designed based on each observer. In previous work [32, 40], adaptive strategy was used without incorporating any fault-tolerant control scheme. However, in [41] tracking control algorithm is proposed without consideration of faults, while the strategy proposed in this article is specifically suggested for ensuring the stability of the system in the presence of the unknown faults.

The remainder of this paper is organized as follows. In Section 2, the system dynamical model is presented. In Section 3, fault detection strategy is discussed. In Section 4, sensors fault reconstruction are discussed. States approximation and faults reconstruction results under various faults are given in section 5, fault-tolerant control scheme and results are given in section 6, section 7 explains the effectiveness of the proposed strategies.

2 System’s dynamical model

For precise and effective fault detection (FD) and fault-tolerant control (FTC), accurate dynamical model of the system is required. In this section, the aircraft nonlinear dynamics is presented. Aileron (δ_A), elevator (δ_E), and rudder (δ_R) deflectionsare the control inputs of the aircraft. The aircraft’s 6-DoF (Degree of freedom) nonlinear dynamical equations of motion are: $\begin{matrix} \dot{p} = \frac{I_{z} l_{aero} + I_{xz} n_{aero}}{I_{x} I_{z} - I_{xz}^{2}} r + \frac{I_{xz} (I_{x} - I_{y} + I_{z}) pq}{I_{x} I_{z} - I_{xz}^{2}} \\ + \frac{I [I_{z} (I_{y} - I_{z}) - I_{xz}^{2}] qr}{I_{x} I_{z} - I_{xz}^{2}} \end{matrix}$ (1) $\dot{q} = \frac{1}{I_{y}} [m_{aero} + pr (I_{z} - I_{x}) + I_{xz} (r^{2} - p^{2})]$ (2) $\begin{matrix} \dot{r} = [\frac{(I_{x} - I_{y} + I_{z}) I_{xz}}{I_{x} I_{z} - I_{xz}^{2}} r + \frac{I_{x} (I_{x} - I_{z}) + I_{xz}^{2}}{I_{x} I_{z} - I_{xz}^{2}} p] q \\ + \frac{I_{xz} l_{aero} + I_{xx} n_{aero}}{I_{x} I_{z} - I_{xz}^{2}} \end{matrix}$ (3) $\begin{matrix} \dot{β} = p sin α - r cos α + \frac{1}{mv} [mg sin γ sin μ] \\ + \frac{1}{mv} [Y cos β - T sin β cos α] \end{matrix}$ (4) $\begin{matrix} \dot{α} = q - (p cos α + r sin α) tan β + \frac{1}{mv cos β} * \\ [- L + Mg cos γ cos μ] + \frac{1}{mv cos β} [- T sin α] \end{matrix}$ (5) $\begin{matrix} \dot{μ} = \frac{1}{cos β} (p cos α + r sin α) - \frac{g}{v} tan β cos μ cos γ \\ + \frac{L + T sin α}{mv} [tan γ sin μ + tan β] \\ + \frac{Y}{mv} tan γ cos μ cos β \end{matrix}$ (6) $\begin{matrix} \dot{γ} = \frac{1}{mv} [L cos μ - mg cos γ - Y sin μ cos β] \\ + \frac{T}{mv} [sin μ sin β cos α + cos μ sin α] \end{matrix}$ (7) $\dot{v} = \frac{1}{m} [- D + Y sin β - mg sin γ + T cos β cos α]$ (8)

3 Fault detection strategy

In this part, the suggested strategy for sensors fault detection is illustrated and different observers used for fault detection are explained.

3.1 Problem description

A general mathematical expression for a nonlinear system in the presence of faults and disturbances is given as; $\begin{matrix} \dot{x} (t) = a (x (t)) + b (x (t)) u (t) + D (x, t) + f_{A} (x, t) \\ y (t) = c (x (t)) + f_{s} (x, t) \end{matrix}$ (9)

Where, the symbols x (t) ∈ Rⁿ and a : Rⁿ → Rⁿ are the state vector and state function of the system, b : Rⁿ → R^n×m is the input function and u (t) ∈ R^m is the input vector, y (t) ∈ R^r is the output function, D ∈ Rⁿ are the uncertainties and disturbances existing in the system, f_s (x, t) and f_A (x, t) represent sensor and actuator faults respectively. The fact of the existence of some disturbances, deficiencies and uncertainties in nonlinear systems, necessitates some sort of system to detect faults or failures in order to avoid any kind of damage to the system and its performance degradation. For this reason, an adaptive multi-layer neural network observer is proposed to detect any kind of fault or failure that may occur in the system’s sensors. For ensuring the stability of the faulty system, some assumptions are made which are given below [42].

Assumption 1: It is assumed that all of the system state variables, such as [pqrβαuγvφθψ], are computable and the availability of the required variables for the fault detection (FD) strategy such as gyro rates variables p, q, r is ensured, as the main purpose of this article is to detect and isolate faults of gyroscope sensors and p, q, r are representing the roll rate sensor, pitch rate sensor and yaw rate senor of gyroscope respectively.

Assumption 2: Every system has some certain capability, in which the system can operate effectively, therefore, limitations on the maximum and minimum values of the sensor faults f_s (x, t) = (f_p, f_q, f_r) are imposed in order to avoid unbounded or uncontrolled faults and anomalies in control signal.

Assumption 3: Maximum values of the plant uncertainties are also limited to a specified magnitude (∥ D (x, t) ∥ ⩽ X_max) in order to ensure the stability of the system.

3.2 Sensor faults and measurements

The system dynamical model considered in this research is of Boeing 747 100/200 aircraft. To ensure more realistic and effective validation, noise is also added in the measurements. Thus validation of the performance of the proposed approaches is accomplished with noisy data. The standard deviations of the noises added in the sensors are given in Table 1. For proving the efficacious, preciseness and accuracy of the suggested strategy, therefore, noises are also considered in the system dynamics as shown in Table 1. The symbol σ (·) depicts the standard deviations of added noises.

Table 1
Addition of Noises in Sensors

Symbols Values Units

σ _V 0.3 [m/s]

σ_α, σ_β 0.4π/180 [m/s]

σ_∅, σ_θ, σ_φ 0.03π/180 rad

σ_p, σ_q, σ_r 0.03π/180 rad/s

Symbols	Values	Units
σ _V	0.3	[m/s]
σ_α, σ_β	0.4π/180	[m/s]
σ_∅, σ_θ, σ_φ	0.03π/180	rad
σ_p, σ_q, σ_r	0.03π/180	rad/s

Instead of considering the fault occurrence at a specific period or at specific time, in this study the occurrence of faults is considered randomly in order to validate the performance of the suggested method. The fault f _i, given in Table 2, occurred during time 7s < t < 25s, such that f_i = 0 when t ⩽ 7s and t ⩾ 30s. In addition to faults like bias or oscillatory, various other faults are also considered, which are illustrated in the following section. In next section, the complete sensors fault reconstruction strategy is explained.

Table 2

Fault Statistics

Time Period	Symbols	Fault Category	Fault Value	Units
	f _p	bias	(2.5 * π)/180	rad/s
7s < t < 25s	f _q	oscillatory	(π * sin (t - 10))/180	rad/s
	f _r	bias	(2.5 * π)/180	rad/s

4 Sensors fault reconstruction

This part of research concerns with the problem of fault reconstruction. Various types of faults are considered such as abrupt, intermittent, incipient and simultaneous faults in order to avoid any kind of failure or damage such as aircraft crashes.

Detection of simultaneous faults is a difficult task for a fault detection system. Occurrences of multiple faults are also considered in this research work, as multiple sensors can have faults simultaneously.

Intermittent faults occur in irregular intervals and are very common. Detection of these faults is very difficult due to their irregular occurrence. These faults can take place due to various reasons such as the inappropriate connection of wires to the sensors. The probability of occurrences of these faults is enhanced with the complexity of the system.

Detection of incipient faults is difficult for any fault detection system due to their small influence on residuals. Incipient faults usually occur due to some deficiency in sensors or due to their partial failure. The mathematical representation of incipient faults is given as; $f_{i} (t - T_{0}) = {\begin{matrix} 0 if t < T_{0} \\ 1 - e^{- ω_{i} (t - T_{0})} if t ⩾ T_{0} \end{matrix}$ (10)

Where, ω depicts the fault evolution rate and for incipient faults the value of ω is small. By utilizing the larger value of ωthe fault is changed into step form which is defined as abrupt fault or failure.

Abrupt faults occur due to sudden changes in parametric values, the response of these faults is more rapid compared to the nominal dynamic process. Tracking of such rapid changes and variations is a challenging task for detection systems. In this research, three different types of abrupt faults are taken into account, as shown in Fig. 1, which can take place due to several reasons such as short circuit, metal flakes separation and intense vibrations. Such faults are considered for the validation and testing of the suggested strategy.

Fig. 1

Types of Abrupt Faults.

4.1 Multi-layer neural network

In this research, the multi-layer neural network (MLNN) observer [43] is used for the fault detection and isolation of the aircraft sensors. Therefore, the general mathematical model of the MLNN observer is presented in this section.

In above Fig. 2, the states x₁, x₁, …. x_n depict the input states to neural network and the states y₁, y₁, …. y_n depict the output states of the neural network. In this specific problem, the Gyro rate sensor faults are detected and isolated, therefore, p, q and r are the input states to neural network obtained from the system given in Equations (1–8) and $\hat{p}, \hat{q} and \hat{r}$ are the observed values obtained from the neural network observer.

Fig. 2

Multi-layer Neural Network Structure.

The output of the jth neuron of the network is given as; $v_{j} = Φ (h_{j}) = \frac{1 - e^{- h_{j}}}{1 + e^{h_{j}}}$ (11)

The term v_j is the general for representing the outputs of the first hidden layer of MLNN.

Where $h_{j} = \sum_{k = 1}^{n} w_{jk} x_{k}$ , $x_{k} = {[\begin{matrix} p & q & r \end{matrix}]}^{T}$ , Φ (.) is the sigmoid activation nonlinear function used for the training of the NN. The final output or the final response of the neural network is given as; $y_{i} = Φ (g_{i}) = \frac{1 - e^{- g_{i}}}{1 + e^{- g_{i}}}$ (12)

The term y_i is the general for representing the outputs of the second hidden layer of MLNN. Where $g_{i} = \sum_{j = 1}^{m} w_{ij} v_{j},$

The generalized form for the weight update law for the output layer is given as [43]; $\begin{matrix} w_{ij} (t + 1) = w_{ij} (t) + η (y_{i}^{d} - y_{i}) y_{i} (1 - y_{i}) v_{j} \\ = w_{ij} (t) + η δ_{i} v_{j} \end{matrix}$ (13)

Where $δ_{i} = (y_{i}^{d} - y_{i}) y_{i} (1 - y_{i})$ is the error propagated back from the output layer. Similarly, the weight update law from the input layer to the hidden layer is given as: $w_{jk} (t + 1) = w_{jk} (t) + η δ_{j} x_{k}$ (14)

Where $δ_{j} = v_{j} (1 - v_{j}) \sum_{i = 1}^{n} δ_{i} w_{ij}$ , $\sum_{i = 1}^{n} δ_{i} w_{ij}$ is the total error from the output layer, which is back propagated between the input layer and the hidden layer. The generalized delta rule for weight update law is given as: $W_{i_{l} i_{l - 1}} (t + 1) = W_{i_{l} i_{l - 1}} (t) + η δ_{i_{L}} v_{i_{l - 1}}$ (15)

v_{i
_l-1} is the output of i^th neuron of layer l - 1. $δ_{i_{L} =} (y_{i_{L}}^{d} - y_{i_{L}}) y_{i_{L}} (1 - y_{i_{L}})$ for output layer L. $δ_{i_{l}} = v_{i_{l}} (1 - v_{i_{l}}) \sum_{i = 1}^{n} δ_{i + 1} W_{i_{l + 1}}$

The common activation function that is used in multilayer perceptron is sigmoidal nonlinearity and the general mathematical expression of which is given as; $φ (v (n)) = \frac{1}{1 + exp (- av (n))}, a ≻ 0$ (16)

By applying the nonlinearity rule, the range of output is 0 < y < 1. Differentiating Equation (16) yields; $φ^{'} (v (n)) = \frac{a exp (- av (n))}{[1 + exp (- av (n))]^{2}}$ (17)

The Equations (16–17) representing the nonlinear activation functions which are used in Equations (11–12) for obtaining the output of the hidden layers of MLNN. Activation functions given in Equations (11–12) and (16–17) are different forms of activation functions, the main purpose of these nonlinear activation functions is to train the MLNN. Initial conditions used for the training of the MLNN can be observed in Table 3 Appendix B.

Table 3

Root Mean Square Error (RMSE) Comparison for Sensors

Fault 1 (rectangular)			Fault 2 (triangular)			Fault 3 (sigmoid)
Adaptive MLNN	Conventional MLNN	High Gain Observer	Adaptive MLNN	Conventional MLNN	High Gain Observer	Adaptive MLNN	Conventional MLNN	High Gain Observer
0.0162	0.1191	0.1710	0.0032	1.673	1.521	0.0035	1.787	1.995
0.0210	0.6110	0.6620	0.0025	0.1541	0.1711	0.0047	0.2195	0.3215
0.0261	0.1015	0.2252	0.0045	0.1163	0.1832	0.0039	0.0712	0.1234

4.2 Lyapunov function theory (LFT) based adaptive weight update law

For rapid faults detection, neural network weights need to be updated. In this section, an adaptive weight updating algorithm based on Lyapunov Stability Theory is presented. The main purpose of the algorithm is to update the neural network weight parameters online with a rapid convergence learning rate. The main purpose of the adaptive learning rate is to obtain global convergence for highly nonlinear functions without enhancing the computational complexities and costs.

In gradient-decent, the learning rate is kept fixed or constant, for the purpose of avoiding local minima and for ensuring global minima, it is needed to have a large learning rate for a situation far away from the global minimum and a small learning rate for a situation close to the local minima. For such conditions and situations, adaptive learning rate plays an important role.

The Lyapunov stability theory is widely used in autonomous control systems. If a Lyapunov function candidate V ((x, t) , t) is chosen such that:

Condition 1: V ((x, t) , t) is positive definite

Condition 2: $\dot{V} ((x, t), t)$ is negative definite

Then, such a system is called asymptotically stable system.

Hypothesis: Assume that X^* is the equilibrium point of the system $\dot{X} = F (x)$ with the function F is continuously differentiable.

Definition 1. A real-valued function L is a Lyapunov function if it is continuously differentiable from an open neighborhood $O ∋ X^{*}$ and when it has the following two properties:

L > 0 on $O ∖ X^{*}$ and L (X^*) = 0.

$\nabla_{\times} L (X) \cdot F (X) ⩽ 0 on O$ .

It is a strictly Lyapunov function when in addition.

$\nabla_{\times} L (X) \cdot F (X) < 0 on O ∖ X^{*}$ .

The term F(X) is the general form representing the dynamics of the system like the aircraft dynamics given in Equations (1–8).

As the neural network is comprised of weights, inputs, and outputs, the mathematical representation is: ${\hat{y}}^{p} = f (W, x^{p}) p = 1, 2, \dots N$ For training the neural network the quadratic cost function is defined as: $E = \frac{1}{2} \sum_{p = 1}^{N} {(y^{p} - {\hat{y}}^{p})}^{2}$ (18)

y^p is the desired output and ${\hat{y}}^{p}$ is the predicted output by the neural network. Suppose the Lyapunov candidate function for the system is defined as $\begin{matrix} V = \frac{1}{2} ({\tilde{y}}^{T} \tilde{y} + λ {\dot{W}}^{T} \dot{W}) \\ = \frac{1}{2} {\tilde{y}}^{T} \tilde{y} + \frac{λ}{2} {\dot{W}}^{T} \dot{W} \end{matrix}$ (19) $∴ \tilde{y} = [y^{1} - {\hat{y}}^{1} \dots \dots y^{p} - {\hat{y}}^{p} \dots . . y^{N} - {\hat{y}}^{N}]$ (19a)

Where ${\hat{y}}^{p} = f (W, x^{p}) p = 1, 2, . . . . . N$ (19b) In Equation (19), the Lyapunov candidate function is defined for proving the stability of the system based on the new derived weight updating parameter given in Equation (27) with adaptive learning rate and acceleration rate given in Equations (28–29).

The time derivation of Equation (19) and by applying the product rule such that $\frac{dV}{dt} = \frac{dV}{dW} \cdot \frac{dW}{dt}$ Therefore, by using the Equations (19 a, 19 b) we get $\begin{matrix} \dot{V} = - {\tilde{y}}^{T} \frac{\partial y}{\partial W} \dot{W} + λ {\ddot{W}}^{T} \dot{W} \\ = - {\tilde{y}}^{T} (\frac{\partial y}{\partial W} \dot{W} - λ \frac{1}{{∥ \tilde{y} ∥}^{2}} \tilde{y} {\ddot{W}}^{T} \dot{W}) \\ = - {\tilde{y}}^{T} (\frac{\partial y}{\partial W} - λ \frac{1}{{∥ \tilde{y} ∥}^{2}} \tilde{y} {\ddot{W}}^{T}) \dot{W} \\ = - {\tilde{y}}^{T} (J - D) \dot{W} \end{matrix}$ (20)

$Where, J = \frac{\partial y}{\partial W}, J ɛ R^{N \times M}$ is the Jacobian matrix and $D = λ \frac{1}{{∥ \tilde{y} ∥}^{2}} \tilde{y} {\ddot{W}}^{T} \in R^{N \times m}$

Now to find $\dot{W}$ such that $\dot{V} < 0$ by using the following theorem.

Theorem 1. If the update law for the weight vector W follows the dynamics given by following nonlinear function $\dot{W} = σ (W) J^{T} \tilde{y} - σ (W) λ {\ddot{W}}^{T}$ (21)

Where $σ (W) = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ}$ is a scalar function of weight vector W and ɛ is a small positive constant then $\tilde{y}$ converges to zero under the condition that $(J - D)^{T} \tilde{y}$ is non-zero along the convergence trajectory.

Proof. $\dot{W} = σ (W) J^{T} \tilde{y} - σ (W) λ {\ddot{W}}^{T}$ is defined in the following form; λ is constant. $\begin{matrix} \dot{W} = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} J^{T} \tilde{y} - \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} λ {\ddot{W}}^{T} \\ = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} (J^{T} \tilde{y} - λ {\ddot{W}}^{T}) \\ = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} (J^{T} - λ {(\frac{1}{{∥ \tilde{y} ∥}^{2}} \tilde{y})}^{T} {\ddot{W}}^{T}) \tilde{y} \\ = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} (J^{T} - D^{T}) \tilde{y} \\ = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} {(J - D)}^{T} \tilde{y} \end{matrix}$

Therefore, $\dot{W} = \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} (J - D)^{T} \tilde{y}$ (22)

By substituting $\dot{W}$ in Equation (20) we get $\dot{V} = - \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} {∥ (J - D)^{T} \tilde{y} ∥}^{2} ⩽ 0$ (23)

Since $(J - D)^{T} \tilde{y}$ is non-zero, $\dot{V} ≺ 0$ for all $\tilde{y} \neq 0$ and $\dot{V} = 0 iff \tilde{y} = 0$ . If $\dot{V}$ is uniformly continuous and bounded, then according to Barbalat’s Lemma $t \to 0, \dot{V} \to 0 and \tilde{y} \to 0 .$

Lemma 1. Suppose f (t) : R → R is differentiable and has a finite limit as t ⟶∞. If $\dot{f} (t)$ is uniformly continuous then $\dot{f} (t) \to 0$ as t ⟶∞.

The instantaneous weight update law using Lyapunov Function algorithm can be finally expressed in the difference equation model as follows; $\begin{matrix} W (t + 1) = W (t) + (μ \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ}) (J_{p} - D)^{T} \tilde{y} \\ = W (t) + (μ \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ}) J_{p}^{T} \tilde{y} . . . \\ - μ_{1} \frac{\ddot{W} (t)}{{∥ J^{T} \tilde{y} ∥}^{2} + ɛ} \end{matrix}$ (24)

Where p represents each pattern and μ₁ = μλ and the acceleration $\ddot{W} (t)$ is defined as; $\ddot{W} (t) = \frac{1}{(Δ t)^{2}} [W (t) - 2 W (t - 1) + W (t - 2)]$ (25)

∴Δt is taken to be one time unit for simulation.

By applying the Gradient Descent rule to $V = \frac{1}{2} {\tilde{y}}^{T} \tilde{y} + \frac{λ}{2} {\dot{W}}^{T} \dot{W}$ $\begin{matrix} Δ W = - η {(\frac{\partial V}{\partial W})}^{T} \\ = - η {(\frac{\partial V}{\partial W})}^{T} - η {[\frac{d}{dW} (\frac{λ}{2} {\dot{W}}^{T} \dot{W})]}^{T} \\ - (\frac{\partial y}{\partial W}) \tilde{y} - η λ \ddot{W} \end{matrix}$ (26)

Thus, the weight update equation for Gradient Descent can be written as; $W (t + 1) = W (t) + η^{'} J_{p}^{T} \tilde{y} - μ^{'} \ddot{W}$ (27)

Where $μ^{'} \ddot{W}$ is the acceleration term. So the adaptive learning rate and adaptive acceleration rate defined as; $η_{a}^{'} = μ \frac{{∥ \tilde{y} ∥}^{2}}{{∥ J_{P}^{T} \tilde{y} ∥}^{2} + \in}$ (28) $μ_{a}^{'} = \frac{λ}{∥ J_{p}^{T} \tilde{y} ∥ + \in}$ (29)

The global minimum for V is defined as; $\tilde{y} = 0, \dot{W} = 0 (\tilde{y}) \in R^{n}, W \in R^{m}$ . The global minimum is guaranteed with the condition that $\dot{W}$ exists along the trajectory.

The local minimum is analyzed through the following conditions and $\dot{W}$ does not exist or vanishes in the following cases:

Condition 1: J = D (J, D ∈ R^n×m), in neural networks the probability that J will be equal to D is very small, so this condition is ruled out for multi-layer neural networks.

Condition 2: ${(J - D)}^{T} \tilde{y} = 0$ , for avoiding local minima it is assumed that J ≠ D, RANKρ (J - D) = n assures the global convergence.

Condition 3: $J^{T} \tilde{y} = D^{T} \tilde{y}$ , the solution for this condition exists for every vector $\ddot{W} R^{m}$ whenever RANKρ (J) = m.

Remarks 2. For neural networks, there are m-n vectors $\ddot{W} R^{m}$ for which no solution exists and hence local minima is avoided. Increasing number of hidden layers or increasing number of hidden neurons i.e increasing m, the occurrence of local minima is reduced. Increasing number of both m and n as well n/m, the local minima can be avoided.

Rewriting the weight update law as; $\begin{matrix} W (t + 1) = W (t) + Δ W (t + 1) \\ = W (t) - η^{'} \frac{\partial V}{\partial W} (t) - μ^{'} \ddot{W} (t) \end{matrix}$

By considering the point B as shown in Fig. 3, at time interval of (t –1), the weight update computation for the time interval [t - 1, t].

Fig. 3

Global minima and local minima illustration.

$\begin{matrix} Δ W (t) = Δ W_{1} (t - 1) + Δ W_{2} (t - 1) \\ Δ W_{1} (t - 1) = - η \frac{\partial V}{\partial W} (t - 1) > 0 \\ Δ W_{2} (t - 1) = - μ \ddot{W} (t - 1) \\ = - μ (Δ W (t - 1)) - Δ W (t - 2) > 0 \end{matrix}$

It can be observed that ΔW (t - 1) < ΔW (t - 2) because of the decrease in the velocity towards local minima and ΔW (t) >0 speed increases and the local minimum is avoided.

By considering the point A at a time interval (t), the weight increments are defined as;

\begin{matrix} Δ W_{1} (t) = - η \frac{\partial V}{\partial W} (t - 1) = 0 \\ Δ W_{2} (t) = - μ \ddot{W} (t) \\ = - μ (Δ W (t)) - Δ W (t - 1) > 0 \end{matrix}

\begin{matrix} Δ W (t) < Δ W (t - 1) \Rightarrow Δ W_{2} (t) > 0 and \\ Δ W (t + 1) = Δ W_{1} (t) + Δ W_{2} (t) > 0 \end{matrix}

Hence, the local minima is avoided

By considering the point D at time interval.

\begin{matrix} Δ W_{1} (t + 1) = - η \frac{\partial V}{\partial W} (t + 1) < 0 \\ Δ W_{2} (t + 1) = - μ \ddot{W} (t + 1) \\ = - μ (Δ W (t + 1)) - Δ W (t) > 0 \end{matrix}

Because of the back-propagation term approaches to zero as the slope $\frac{\partial V}{\partial W} > 0$ so $\begin{matrix} Δ W (t + 1) < Δ W (t) \\ Δ W (t + 2) = Δ W_{1} (t + 1) + Δ W_{2} (t + 1) > 0 if \\ Δ W_{2} (t + 1) > Δ W_{1} (t + 1) \end{matrix}$

Hence, the local minima is avoided.

4.3 High gain observer design

In this part, the high gain observer design is described. The general mathematical form of the high gain observer is defined as; $\begin{matrix} {\dot{\hat{x}}}_{1} = {\hat{x}}_{2} - \frac{a_{n}}{ɛ} ({\hat{x}}_{1} - v (t)) \\ {\dot{\hat{x}}}_{2} = {\hat{x}}_{3} - \frac{a_{n - 1}}{ɛ} ({\hat{x}}_{1} - v (t)) \\ ⋮ \\ {\dot{\hat{x}}}_{n - 1} = {\hat{x}}_{n} - \frac{a_{2}}{ɛ^{n - 1}} ({\hat{x}}_{1} - v (t)) \\ {\dot{\hat{x}}}_{n} = - \frac{a_{1}}{ɛ^{n}} ({\hat{x}}_{1} - v (t)) \end{matrix}}$ (30)

Where the terms [x₁x₂x₃ ….. x_n] depict the system state vector, $[{\hat{x}}_{1} {\hat{x}}_{2} {\hat{x}}_{3} \dots . . {\hat{x}}_{n}]$ depict the estimated system state vector, v(t) is the measured variable of the system states, for the purpose of the faults detection and isolation through high-gain observer faults are injected in system states v(t), the variables a_i (i = 1, 2, 3, …, n) are chosen such that the polynomial sⁿ + a_ns^n-1+ ... ..+a₂s + a₁ = 0 is Hurwitz. By utilizing the high gain observer, states of the system are approximated such that the following condition is fulfilled. $lim_{ɛ \to 0} x_{i} = v^{(i - 1)} (t), i = 1, 2, . . ., n$ (31)

4.3.1 Stability analysis

For the nonlinear system, if p, q and rare the roll, pitch and yaw measurement variables of gyro rate sensors then the high gain observer is designed as; as it is required to detect faults in Gyro rate sensors then by using Equations 1–8, the roll, pitch and yaw rate sensor equations can be written as;

Roll rate sensor: $\begin{matrix} \dot{p} = \frac{I_{z} l_{aero} + I_{xz} n_{aero}}{I_{x} I_{z} - I_{xz}^{2}} r + \frac{I_{xz} (I_{x} - I_{y} + I_{z}) pq}{I_{x} I_{z} - I_{xz}^{2}} \\ + \frac{I [I_{z} (I_{y} - I_{z}) - I_{xz}^{2}] qr}{I_{x} I_{z} - I_{xz}^{2}} \\ \dot{φ} = p \end{matrix}}$ (32)

Pitch rate sensor: $\begin{matrix} \dot{q} = \frac{1}{I_{y}} [m_{aero} + pr (I_{z} - I_{x}) + I_{xz} (r^{2} - p^{2})] \\ \dot{θ} = q \end{matrix}}$ (33)

Yaw rate sensor: $\begin{matrix} \dot{r} = [\frac{(I_{x} - I_{y} + I_{z}) I_{xz}}{I_{x} I_{z} - I_{xz}^{2}} r + \frac{I_{x} (I_{x} - I_{z}) + I_{xz}^{2}}{I_{x} I_{z} - I_{xz}^{2}} p] q \\ + \frac{I_{xz} l_{aero} + I_{xx} n_{aero}}{I_{x} I_{z} - I_{xz}^{2}} \\ \dot{ψ} = r \end{matrix}}$ (34)

In case of pitch rate sensor, if p is the measurement variable then by using the Equation (30), the high gain observer for pitch rate sensor is defined as; $\begin{matrix} \dot{\hat{p}} = θ - \frac{a_{2}}{ɛ} (\hat{p} - p) \\ \dot{\hat{θ}} = - \frac{a_{1}}{ɛ^{2}} (\hat{p} - p) \end{matrix}}$ (35)

Where the term p is the measured value obtained from the system given in Equation (1) and $\hat{p}$ is the observed value.

The role of variables a₁, and a₂is to make the system Hurwitz. $s^{2} + a_{1} s + a_{2} = 0$ (36)

The observed errors are defined as $η = {[η_{1} η_{2}]}^{T}$ (37)

Where, $η_{1} = \frac{p - \hat{p}}{ɛ^{2}}, η_{2} = \frac{θ - \hat{θ}}{ɛ}$ (38)

Because of $\begin{matrix} ɛ {\dot{η}}_{1} = \frac{\dot{p} - \dot{\hat{p}}}{ɛ^{2}} = \frac{1}{ɛ} (θ - (θ - \frac{a_{1}}{ɛ} (\hat{p} - p))) \\ = \frac{1}{ɛ} (θ - \hat{θ} - \frac{a_{1}}{ɛ} (p - \hat{p})) \\ = - \frac{a_{1}}{ɛ^{2}} (p - \hat{p}) + \frac{1}{ɛ} (θ - \hat{θ}) \\ = - a_{1} η_{1} + η_{2} \end{matrix}$ (39) $\begin{matrix} ɛ {\dot{η}}_{3} = ɛ \frac{θ - \dot{\hat{θ}}}{ɛ} = (bu - (bu + \frac{a_{2}}{ɛ} (p - \hat{p}))) \\ = - \frac{a_{2}}{ɛ} (p - \hat{p}) \\ = - a_{2} η_{1} \end{matrix}$ (40)

So the equation of observer error is defined as $ɛ \dot{η} = \bar{A} η$ (41) where, $\bar{A} = [\begin{matrix} - a_{1} & 1 \\ - a_{2} & 0 \end{matrix}]$ , the characteristic equation of $\bar{A}$ is defined as; $| λ I - \bar{A} | = | \begin{matrix} λ + a_{1} & - 1 \\ a_{2} & λ \end{matrix} |$ (42)

Hence, λ² + a₁λ + a₂ = 0.

The design of a_i (i = 1, 2), for the requirement of expression λ² + a₁λ + a₂ = 0 to be negative. As for the stability of the system and for the design of observer, the real parts of the system equation must be negative; therefore, the matrix $\bar{A}$ is Hurwitz. ${\bar{A}}^{T} P + P \bar{A} + Q = 0$ (43)

By the utilization of the stability theory, for any random positive-definite matrix Q ∈ R³, there exists a positive definite symmetrical solution P given in the following Lyapunov equation.

For the system observing error, the Lyapunov equation is utilized given below: $V_{\circ} = ɛ η^{T} P η$ (44)

Therefore, $\begin{matrix} {\dot{V}}_{\circ} = ɛ {\dot{η}}^{T} P η + ɛ η^{T} P \dot{η} = (\bar{A} η)^{T} P η + η^{T} P (\bar{A} η) \\ = η^{T} {\bar{A}}^{T} P η + η^{T} P \bar{A} η \\ = η^{T} ({\bar{A}}^{T} P + P \bar{A}) η ⩽ - η^{T} Q η \end{matrix}$ (45)

Therefore, ${\dot{V}}_{\circ} ⩽ - λ_{min} (Q) {∥ η ∥}^{2}$ (46)

Where, the variable λ_min depicts the minimum value $∥ η ∥ ⩽ \frac{2 ɛ L ∥ P \bar{B} ∥}{λ_{min} (Q)}$ (47)

From the above analysis, it is concluded that the convergence velocity of the observing error η is dependent on the parameter ɛ. When the parameter ɛ has a very small value, then the error dynamics is a rapid variant sub-system. Smaller the ɛ, the more abrupt the convergent velocity of the error dynamic η is ∥η ∥ is O (ɛ), similarly, with decreasing the value of ɛ, the observing error approaches to zero.

The above stability analysis is proved for the case of roll rate sensor, similar procedure is followed for the pitch and yaw rate sensors.

5 Faults reconstruction and states approximation

In this section, faults reconstruction and states approximation results based on three different strategies namely adaptive multi-layer neural network observer, conventional multi-layer neural network observer and high gain observer are presented. The model of aircraft used for this purpose is Boeing 747 100/200. The different categories of faults given in section 4 and the disturbances shown in Table 1 are supposed to be present. Figure 4 shows the strategy adopted for the fault detection and isolation based on the new proposed strategy in this research. The main purpose of this strategy is to approximate and reconstruct the faults in terms of the magnitude, shape and time occurrences. In Fig. 4, the outputs $\hat{p}, \hat{q}, \hat{r}$ depicts the approximated and reconstructed faults based on the proposed strategy in order to isolate it from the system, therefore, these outputs are not used as an input to the controller.

Fig. 4

Fault Reconstruction Strategy Illustration.

The results of faults reconstruction are shown in the following figures. The estimated faults $({\hat{f}}_{p}, {\hat{f}}_{q}, {\hat{f}}_{r})$ and their difference $(Δ {\hat{f}}_{p}, Δ {\hat{f}}_{q}, Δ {\hat{f}}_{r})$ are computed. In Fig. 5, the three fault reconstruction techniques adaptive MLNN, conventional MLNN and high gain observer are compared in the presence of rectangular and oscillatory biases, the faults are assumed to be unknown to fault reconstruction techniques. The estimated faults and their differences are presented in Fig. 5(a) and 5(b). It can be observed that noticeable differences occur at the beginning or at the end of fault occurrence time period ( ${\hat{f}}_{p}$ , ${\hat{f}}_{r}$ ) while there is only a difference at the end of ${\hat{f}}_{q}$ . In all three fault cases, the faults occurrence time period is 7s < t < 25s.

Fig. 5

Comparison of Three Strategies based on Rectangular and Oscillatory Biases.

After reconstruction of faults $({\hat{f}}_{p}, {\hat{f}}_{q}, {\hat{f}}_{r})$ utilizing the adaptive MLNN, conventional MLNN and high gain observer, it can be seen that the faults approximation using the adaptive MLNN can better track the true faults compared to the other two strategies. The performance of the conventional MLNN and high gain observer for faults reconstruction is not satisfactory. Approximation of ${\hat{f}}_{p}$ , using high gain observer considerably deviates from the actual fault at time around 15s < t < 19s and also at the start and end of fault time period. The fault reconstruction performance of conventional MLNN is also not satisfactory especially at the start and end of fault occurrence.

In approximation of ${\hat{f}}_{r}$ , the high gain observer’s approximation again deviates from the true fault at the time around 12s < t < 16s and also at the start and end of fault time duration. In a similar way, the performance of conventional MLNN is also not satisfactory.

In the case of oscillatory fault reconstruction in ${\hat{f}}_{q}$ , the adaptive MLNN can track the true fault well as compared to conventional MLNN and high gain observer.

In Fig. 6(a) and 6(b), the approximated $({\hat{f}}_{p}, {\hat{f}}_{q}, {\hat{f}}_{r})$ and the difference of approximated values from true value, $(Δ {\hat{f}}_{p}, Δ {\hat{f}}_{q}, Δ {\hat{f}}_{r})$ are presented based on sigmoidal fault. The proposed adaptive MLNN observer is compared to conventional MLNN and high gain observer. It can be seen from Fig. 6(a) and 6(b) that the proposed adaptive MLNN can track the true fault well and with high accuracy as compared to the other two observers. Again it is assumed that the fault is unknown to the design of fault reconstruction techniques. In approximation of ${\hat{f}}_{p}$ , both the conventional MLNN and high gain observer deviates from true fault at the time period around 2.5s < t < 3.5s ( ${\hat{f}}_{p}$ , $Δ {\hat{f}}_{p}$ ). Similarly, in approximation ${\hat{f}}_{q}$ , the adaptive MLNN follow the true fault well while the conventional MLLN and high gain observer deviate from true fault at time duration around 2.5s < t < 3.9s. In case of fault reconstruction of ${\hat{f}}_{q}$ , the conventional MLNN and high observer deviate at two different time durations around 0.5s < t < 1.4s and 2.9s < t < 3.9s which is obvious in ( ${\hat{f}}_{q}$ , $Δ {\hat{f}}_{q}$ ), the adaptive MLNN shows satisfactory performance.

Fig. 6

Comparison of Three Strategies based on Sigmoidal Fault.

In Fig. 7(a) and 7(b), the incipient fault based results are presented and the performance of the three observers is compared. It can be seen that the performance of adaptive MLNN is much better as compared to other two observers. For incipient fault based reconstruction ( ${\hat{f}}_{p}$ ), the high gain observer shows some overshoot at the beginning of fault time (t > 10) and conventional MLNN also deviates from the true fault trajectory, the ${\hat{f}}_{p}$ also shows deviation at time period around 19.8 < t < 22.2 which is also obvious from observing ${\hat{f}}_{p} and Δ {\hat{f}}_{p}$ . At time duration around 9.8s < t < 10.5s and 11.9s < t < 13.5s in fault reconstruction of ${\hat{f}}_{r}$ , both the conventional MLNN and high gain observer show deviations from the real fault path as compared to adaptive MLNN. Similarly, by observing the results of ${\hat{f}}_{q}$ and $, Δ {\hat{f}}_{q}$ , deviations of convention MLNN and the high gain observer can be seen at time duration around 10s < t < 10.5s and 14s < t < 16.2s.

Fig. 7

Comparison of Three Strategies based on Incipient Fault.

Figures 8(a) and 8(b) show the results based on intermittent faults reconstruction, by comparing the results of approximated faults $({\hat{f}}_{p}, {\hat{f}}_{q}, {\hat{f}}_{r})$ and their difference $(Δ {\hat{f}}_{p}, Δ {\hat{f}}_{q}, Δ {\hat{f}}_{r})$ , it is concluded that the tracking capability of adaptive MLNN is better compared to conventional MLNN and high gain observer which have greater tracking error in following the true fault trajectory.

Fig. 8

Comparison of Three Strategies based on Intermittent Fault.

In the Fig. 9, results based on simultaneous faults are presented. Two kinds of faults are injected simultaneously, the rectangular fault in pitch rate p and triangular fault in yaw rate r. Different deviations during different time durations such as 7s < t < 7.3s, 10s < t < 18s and 25s < t < 25.4s in ${\hat{f}}_{p}$ can be observed. During these time periods conventional MLNN and high gain observer show deviations from true fault, similarly in time period around 10.8s < t < 13.3s of ${\hat{f}}_{r}$ deviations can be seen while adaptive MLNN can track the real fault path well compared to conventional MLNN and high gain observer.

Fig. 9

Comparison of Three Strategies based on Simultaneous Fault.

In order to evaluate the effectiveness of our proposed algorithm, MLNN based on adaptive learning rate, over the pure MLNN with fixed learning rate, the root mean square error (RMSE) of both algorithms is also calculated and compared as shown in Table 3.

The RMSE is mathematically represented as: $RMSE = \sqrt{\frac{\sum_{i = 1}^{N} (y_{D_{i}} - y_{i})^{2}}{N}}$

Where, y_{D
_i} is the detected fault and y_i is the actual fault and N is the total number of simulation samples. From Table 3, it is clear that the adaptive Multi-Layer neural network has less RMSE compared to conventional Multi-Layer neural and high gain observer. It has also been proved that the neural network with adaptive MLNN shows much better response compared to conventional MLNN and high gain observer. The sudden and sharp changes have always importance and significance for aircrafts and they have a great impact on the performance of aircrafts, even small faults can lead to a big failure, therefore, it is important for aircrafts to have such algorithms which can detect faults much faster with much better accuracy and preciseness to avoid any kind of failure. The presented strategy adaptive MLNN is capable of, to detect faults more rapidly with high accuracy and preciseness and has the ability to enhance the safety as well.

6 Fault-tolerant control (FTC) scheme based on sliding mode controller (SMC)

In this section, a fault-tolerant control scheme based on sliding mode control (SMC) is designed. The main purpose of this scheme is to maintain the stability of the system and to achieve the desired trajectory in presence of faults in Gyro rate sensors such as roll rate p, yaw rate r and pitch rate q. The SMC based FTC scheme is designed for all three observers separately for the purpose of comparing the effectiveness and performance of FTC based on different observer strategies.

The basic sliding mode control (SMC) design process is accomplished in two steps. In the first step, the sliding surface is designed according to trajectory tracking error, and in the second step a suitable Lyapunov function is designed that satisfies the sliding surface conditions. Because of such sliding regime, the closed-loop control system becomes insensitive and robust to any kind of faults/failures, disturbances, modeling and parameter uncertainties. The main purpose of the FTC scheme is to following or tracking of desired trajectories that represent the translation and rotational motions in the presence of Gyro rate sensors fault.

Suppose the nonlinear system is represented by the following mathematical expression; $\ddot{x} = bu (t) + f (t)$ (48)

Where x is the positional signal and $\dot{x}$ is the speed signal of the system, u(t) is the control input and f(t) represents the disturbances, faults or uncertainties which is bounded |f (t) | ⩽ l_f, where l_fis a positive constant.

In this section, the control laws are derived based on neural network observers and high gain observers. The observed values such as $\hat{p}, \hat{q} and \hat{r}$ are based on neural network observer or high gain observer.

Suppose that r is the desired trajectory then the errors are computed as; $\begin{matrix} e_{1} = θ - r \\ e_{2} = p - \dot{r} \end{matrix}}$ (49)

Where the terms pand θ are the measured values obtained from the system given in Equations (1–8).

Therefore, we get $\begin{matrix} {\dot{e}}_{1} = e_{2} \\ {\dot{e}}_{2} = f (t) - \ddot{r} + bu (t) \end{matrix}}$ (50)

The sliding mode surface is defined as; $s = e_{2} + {ce}_{1}$ (51)

Where variable c is a positive constant. The observing sliding mode surface is defined as; $\hat{s} = {\hat{e}}_{2} + c {\hat{e}}_{1}$ (52)

and the observing errors are defined as; $\begin{matrix} {\hat{e}}_{1} = \hat{θ} - r \\ {\hat{e}}_{2} = \hat{p} - \dot{r} \end{matrix}}$ (53)

Therefore, we get $\dot{s} = {\dot{e}}_{2} + c {\dot{e}}_{1} = f (t) - \ddot{r} + bu (t) + {ce}_{2}$ (54)

The Lyapunov function is defined as; $V = \frac{1}{2} s^{2}$ (55)

By differentiation of Equation (55) we get $\dot{V} = s \dot{s} = s (f (t) - \ddot{r} + bu (t) + {ce}_{2})$ (56)

The selected control law is defined as; $u = \frac{1}{b} (\ddot{r} - c {\hat{e}}_{2} - l sgn (\hat{s}))$ (57)

Where, l > l_f > 0, therefore $\begin{matrix} \dot{V} = s (f (t) - \ddot{r} + b \frac{1}{b} (\ddot{r} - c {\hat{e}}_{2} - l sgn (\hat{s})) + {ce}_{2}) \\ = s (- l sgn (\hat{s}) + f (t) + c (e_{2} - {\hat{e}}_{2})) \\ = - (\hat{s} + s - \hat{s}) l sgn (\hat{s}) + (\hat{s} + s - \hat{s}) f (t) \\ + (\hat{s} + s - \hat{s}) c (e_{2} - {\hat{e}}_{2}) \\ ⩽ - \hat{sl} sgn (\hat{s}) + l | s - \hat{s} | + (| f (t) | + c | e_{2} - {\hat{e}}_{2} |) | \hat{s} | \\ + | s - \hat{s} | (| f (t) | + c | e_{2} - {\hat{e}}_{2} |) \\ = - l | \hat{s} | + l | s - \hat{s} | + (| f (t) | + c | p - \hat{p} |) \\ + | s - \hat{s} | (| f (t) | + c | p - \hat{p} |) \\ ⩽ - l | \hat{s} | + (l_{f} + c | p - \hat{p} |) | \hat{s} | + | s - \hat{s} | (l + l_{f} \\ + c | p - \hat{p} |) \end{matrix}$ (58)

Because of the convergence property of the observer the terms $| x_{2} - {\hat{x}}_{2} |$ , $| s - \hat{s} |$ are bounded and have sufficiently small values, therefore $| s - \hat{s} | (l + l_{f} + c | x_{2} - {\hat{x}}_{2} |)$ is sufficiently small, and $\dot{V} < 0 .$

The above control law is derived for the roll rate sensor, similarly control procedure is followed for the pitch and yaw control laws.

6.1 SMC (Fault-tolerant Control) scheme based Results

In this section, the performance of the proposed FTC scheme is tested through various kinds of sensor faults. The FTC based on three different observers adaptive MLNN, conventional MLNN, and the high gain observer is proposed and their performance is compared. The three controllers in the presence of different faults are compared, it can be clearly seen that the FTC scheme performance based on adaptive MLNN is better than the other two strategies. The Fig. 10 shows the adopted strategy for fault tolerance control (FTC) based on the strategy proposed in this research.

Fig. 10

Fault-tolerant Control (FTC) Scheme Illustration.

6.1.1 Results in the presence of a fault in roll rate p

In this part, various fault-tolerant control schemes are presented under different kinds of faults in roll rate p. Three different types of fault-tolerant control schemes such as adaptive MLNN based SMC, conventional MLNN based SMC and high gain observer based SMC are compared by utilizing the different lateral dynamics such roll rate p, yaw rate r, sideslip angle β and roll angle φ.

Figure 11 shows the effect of intermittent fault in p at 7 s, the results show that the performance of roll rate p and sideslip angle β is not degraded in case of adaptive MLNN based SMC and conventional MLNN based SMC but the results based on high gain observer based SMC are much degraded having more oscillations and overshoot from the desired path. In case of states, yaw rate r and roll angle φ, both the conventional MLNN and high gain observer based SMC show degradation, therefore, adaptive MLNN based SMC is more effective in case of an intermittent fault in achieving the desired trajectory.

Fig. 11

FTC performance due to intermittent fault in roll rate p.

Figure 12 shows the performance of fault-tolerant control (FTC) scheme in case of triangular fault in roll rate p at 5 s. It can be clearly seen that the conventional MLNN based SMC and high gain observer based SMC results are much degraded after the occurrence of a triangular fault in p. Therefore, the adaptive MLNN based SMC scheme shows greater robustness in maintaining the stability of the system.

Fig. 12

FTC performance due to triangular fault in roll rate p.

In Fig. 13, sigmoidal fault based results are compared, it can be observed that performance of conventional and adaptive MLNN based results are comparable but high gain observer based control scheme is showing greater oscillations along the desired path. Therefore, adaptive MLNN based control scheme is much better in achieving the desired path.

Fig. 13

FTC performance due to Sigmoidal fault in roll rate p.

6.1.2 Results in the presence of the fault in yaw rate r

In this section, results of all the three FTC schemes in the pressence of different faults in yaw rate r are presented. In Fig. 14 intermittent faults based results are shown, it can be seen that results of all three FTC schemes are comparable but adaptive MLNN based FTC still shows better performance in maintaining the stability of the system and has lesser deviations, oscillations and overshoot as compared to other two schemes. In Fig. 15, performance of different FTC schemes are investigated based on triangular fault in yaw rate r. It can be clearly observed that the adaptive MLNN based FTC scheme is much better in acquiring the stability of the system and has lesser overshoot after the occurrence of the fault. Thus, the proposed scheme responds more effectively to the fault as compared to the conventional and high gain observer based FTC. In case of sigmoidal fault, as shown in Fig. 16, it is proved that the proposed adaptive MLNN based FTC scheme is much better, effective and robust in achieving the desired path as compared to other two schemes. Conventional MLNN FTC and high gain observer FTC have larger deviations especially the high gain observer FTC has much overshoot and deviations in achieving the desired path. Therefore, adaptive MLNN based FTC is better and robust in achieving the path.

Fig. 14

FTC performance due to intermittent fault in yaw rate r.

Fig. 15

FTC performance due to triangular fault in yaw rate r.

Fig. 16

FTC performance due to sigmoidal fault in yaw rate r.

6.1.3 Results in the presence of fault in pitch rate q

In this section, performance of SMC based FTC schemes in the presence of various faults in pitch rate q is compared. All the three FTC schemes are compared by considering the longitudinal dynamics states such as pitch rate q, pitch angle θ and angle of attack α. Intermittent fault based results are shown in Fig. 17, it can observed that after the occurrence of fault at 5 s, in state pitch rate q, the conventional MLNN and high gain observer based FTC shows some oscillations and deviations while the adaptive MLNN based FTC has smooth performance in achieving the desired path as well as showing good performance in other states θ and α. In Figs. 18 and 19, results are based on triangular fault and sigmoidal fault in pitch rate q, by comparing and observing the performances of all fault-tolerant control (FTC) schemes, it is proved that the adaptive MLNN based FTC scheme is much better and robust in maintaining stability and achieving the desired path.

Fig. 17

FTC performance due to intermittent fault in pitch rate q.

Fig. 18

FTC performance due to triangular fault in pitch rate q.

Fig. 19

FTC performance due to sigmoidal fault in pitch rate q.

7 Conclusion

A new strategy for aircraft sensor fault reconstruction using the Lyapunov function theory for adaptively updating the weights of neural networks is proposed. Adaptive multi-layer neural network (MLNN) approach is developed for the approximation and reconstruction of faults and states. MLNN parameter weights for faults approximation are adaptively updated by the proposed adaptive learning rate strategy. The faults and states approximation performance of the proposed strategy is compared with the conventional MLNN, high gain observer and another approach used in literature [32, 40]. Numerous fault types and conditions are considered to authenticate the efficiency of the proposed strategy. Desired trajectories were acquired in the presence of faults or failures of sensors and uncertain measurements were approximated effectively by using the proposed strategy. The performance of the proposed adaptive multi-layer neural network (MLNN) strategy outweighs other approaches in the approximation and reconstruction of faults and states.

A fault-tolerant control (FTC) strategy is also proposed. FTC is designed separately for each observer for ensuring the stability of the system in the presence of faults. After thorough evaluation of performance of all the strategies, it is established that the FTC based on proposed adaptive MLNN shows better robustness to situations such as disturbances and uncertainties compared to [41].

It is concluded that the proposed strategy can be integrated into aircraft system for the purpose of both faults diagnosis and isolation and FTC scheme for increasing the performance of the system.

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Footnotes

Acknowledgments

I am grateful to all the reviewers for reviewing my paper. This research is supported by, National Natural Science Foundation of China (61773245, 61806113, 61873048, 91848206, 61973200, 62073199), the Natural Science Foundation of Shandong Province (ZR2018ZC0436), and Taishan Scholarship Construction Engineering.

Appendix A: Nomenclature

Parameter	Definition
L	Lift, [N]
D	Drag, [N]
g	Gravitational acceleration, [m/s²]
T	Thrust, [N]
v	velocity, [m/s]
ρ	Air density, kg/m³
p, q, r	Aircraft angular velocity about x, y, z axes, [rad/s]
δ_a, δ_e, δ_r	Aileron, elevator, rudder deflection, [rad]
S	Wing area, [m²]
m	Mass, [kg]
β	Sideslip angle, [rad]
α	Angle of attack, [rad]
I_x, I_y, I_z	Moment of inertia, [kg. m²]
φ (.)	Neural Network activation function
w	NN weight parameter
η	NN learning rate parameter
C_L, C_D	Coefficients of lift and drag,
C_X, C_Y, C_Z	Coefficients of aerodynamic force
C_l, C_m, C_n	Coefficients of aerodynamic moment

Appendix B

Table 4

RBFNN Training Parameters

Parameter	Value	Definition
X	03	Number of inputs
Y	03	Number of outputs
N	02	Number of hidden layers
w _jk	0.01	The initial value of the weight updating parameter of the first hidden layer
w _ij	0.01	The initial value of the weight updating parameter of the second hidden layer
η _jk	0.99	Learning rate from input to hidden layer
η _ij	0.99	Learning rate from the hidden layer to the output layer

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Novel neural observer based fault estimation,reconstruction and fault-tolerant control scheme for nonlinear systems

Abstract

Keywords

1 Introduction

2 System’s dynamical model

3.1 Problem description

Table 1 Addition of Noises in Sensors Symbols Values Units σ V 0.3 [m/s] σ α , σ β 0.4π/180 [m/s] σ∅, σ θ , σ φ 0.03π/180 rad σ p , σ q , σ r 0.03π/180 rad/s

Declaration of conflicting interest

Footnotes

Acknowledgments

Appendix A: Nomenclature

Appendix B

References

Table 1
Addition of Noises in Sensors

Symbols Values Units

σ _V 0.3 [m/s]

σ_α, σ_β 0.4π/180 [m/s]

σ_∅, σ_θ, σ_φ 0.03π/180 rad

σ_p, σ_q, σ_r 0.03π/180 rad/s