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EXTENDED MINIMAL RESOURCE ALLOCATING NEURAL NETWORKS FOR AIRCRAFT SFDIA

GIAMPIERO CAMPA Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV 26506 MARCELLO NAPOLITANO Department of Mechanical and Aerospace Engineering West Virginia University, Morgantown, WV 26506

MARIO LUCA FRAVOLINI (+) Department of Electronic and Information Engineering, Perugia University, 06100 Perugia, Italy

ABSTRACT This paper presents an Adaptive Neural Network (ANN) based tool for the modeling, simulation and analysis of aircraft Sensor Failure, Detection, Identification and Accommodation (SFDIA) problem. The tool is based on a SFDIA scheme in which learning NNs are used as on-line non-linear approximators of the analytically redundant portion of the system dynamics. This can provide validation capability to measurement devices, allowing sensors failures to be detected, identified and accommodated. In the context of online learning the issues of critical importance are learning speed, number of parameters to be updated, and stability of the learning algorithm. To address these problems, a library comprising different online learning Adaptive Neural Network is presented, and an Extended Minimal Resource Allocating Network (EMRAN) featuring a fully tuned Radial Basis Functions (RBF) is eventually selected between all the candidate architectures. The study has been performed on a detailed nonlinear 6DOF aircraft model of an aircraft.

INTRODUCTION

The traditional approach to provide fault tolerance following sensor failure is physical redundancy. However, there are special purpose aircraft (e.g. Unmanned Aerial Vehicles) and spacecraft where reduced complexity, lower costs, and weight optimization are major design specifications. For these classes of aircraft an alternative approach can take advantage of the analytical redundancy (Patton et al, 1989) (i.e. the functional relationship existing between the system’s outputs, states and inputs) existing in the system. Research on fault tolerance based on analytical redundancy has produced a quite mature framework especially for linear systems (Baruh and Choe, 1987, Kerr, 1982); currently, the research challenge is in the extension of the previous schemes to the case of nonlinear systems. In this context, a very promising approach is to employ Neural Networks as the main nonlinear approximators of an SFDIA scheme (Ha et al, 1992, Napolitano et al, 2000). Although the benefits of employing NNs for fault tolerance purposes within a non-linear flight control system are clear, the synthesis of these schemes requires dedicated

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simulation software in which it is possible to simulate the interaction between the closed loop dynamics of the Aircraft and the dynamics of the SDFIA system. In this work, a general SFDIA software has been designed in the Simulink environment. The tool allows evaluating either the open loop or the closed loop performance of the SFDIA scheme that employs different kinds of NN approximators and learning algorithms. A library comprising several different adaptive (i.e. online learning) NNs is presented. Finally, the results of a comparative study of different NN approximators applied to the SFDIA problem on a detailed nonlinear aircraft model.

ANALYTICAL REDUNDANCY BASED SFDIA

The used scheme is summarized in figure 1 (Napolitano et al, 2000).

AIRCRAFT

r (k )

z (k ) sensor

SFDIA

y (k )

sensor

u (k )

y a (k )

NN

y s (k )

Fig. 1 – General SFDIA scheme Analytical redundancy implies that some of the system variables are functionally related (Patton et al, 1989); namely, a variable y(k) can be expressed as function of a suitable set of other variables z(k) and inputs commands u(k). The residual signal r(k) is the difference between the sensor output y(k) and its estimation ys(k) provided by a proper estimator (in this work the estimator is a NN). When the square of this (filtered) residual exceeds a predefined threshold, the state of the corresponding sensor is declared suspect and a suitable procedure is called to decide on the health status of this sensor. If the state of the sensor is then declared faulty, a procedure is enabled, and an accommodated variable ya(k) is provided as output. In this work the accommodation procedure simply substitutes the faulty measure with the estimation given by the NN (ya(k)=ys(k)). Several options can be added to this basic scheme to increase robustness in presence of noisy measurements and/or intermittent sensor failures (Napolitano et al, 2000).

THE SIMULATION ENVIRONMENT

The Neural Network based SFDIA modeling and simulation toolbox was built under the Simulink® environment (by The Mathworks Inc). In particular the freely available Flight Dynamics and Control (FDC) toolbox for Matlab (Rauw, 1993) provides powerful tools for flight simulation, flight dynamic analysis, and flight control system design. In figure 2 the graphical interface of the proposed aircraft SFDIA tool is shown. The main blocks of the scheme are the following:

3 Aircraft Dynamics Pilot

CAS

SFDIA

Failures

Fig. 2 - Modeling and simulation environment • • • • •

Aircraft Dynamics: The tool has been built around a generic nonlinear aircraft model, which is configured for the DeHavilland DHC-2 Beaver aircraft, but can be adapted for many different airplanes. Control Augmentation System (CAS): the block is a Stochastic Optimal Feedforward and Feedback controller as in (Halyo et al, 1992). Pilot commands: the block emulates typical pilot commands. Hard and soft sensor failures: the block injects (adds) arbitrary soft and hard sensor failures to the desired measured signals SFDIA Group: It is the core of the tool and performs the main SFDIA procedures. It is constituted by two main sub-blocks: o Approximators: The block contains the Neural Network based function estimators. Different kinds of NN estimator blocks can be selected (some of the available NNs and learning algorithms will be described in the next section). o SFDIA LOGIC: The block performs the main threshold based sensor failure detection identification and accommodation operations. For more details about the SFDIA scheme see (Fravolini et al, 2001).

AN ADAPTIVE NEURAL NETWORKS LIBRARY FOR ON-LINE LEARNING

In order to select a suitable optimal SFDIA scheme it is necessary to test and to compare the performance achieved by different kinds of online Neural Network estimators, to this purpose, a Simulink Library containing several NN architectures has been built. The NNs in the library exploit linear, sigmoidal and radial basis as activation functions and employ different learning algorithms: Multi-Layer Perceptron with Extended Back-Propagation. (MLPEBP): It is a three layer NN, based on sigmoidal neurons with activation function:

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f ( net , U , L, T ) =

U −L

+L − net 1+ e T In the MLP-EBP the back-propagation algorithm it used not only to update the weights of the input and output matrices (W(k), V(k)), but also to update the parameters U,L,T, that define the shape of each neuron. The application of MLPEBP to SFDIA has been treated in (Napolitano et al, 2000). Radial Basis Function (RBF) (Standard): In the standard RBF Network the estimations ys ∈ ℜ m are expressed as a linear combination of M Gaussian Basis functions:

ys ( x) = We

−

( x − µ )T ( x − µ ) 2σ 2

where x ∈ ℜ n is the input vector, the parameters µj and σj are the basis center and width respectively. In the standard implementation the hidden layer neurons are a priori statically allocated on a uniform grid that covers the whole input space and only the weight wij are updated. This approach requires an exponentially increasing number of basis functions versus the dimension of the input space. Fully Tuned Extended Minimal Resource Allocation Network RBF (EMRAN-RBF): In order to avoid the dimensionality problems generated by standard RBF, (Platt 1991) proposed a sequential learning technique for RBFNs. The resulting architecture was called the Resource Allocating Network (RAN) and has proven to be suitable for online modeling of non-stationary processes with only an incremental growth in model complexity. The RAN learning algorithm proceeds as follows: At each sampling instant, if the following 3 criteria are met some units are added: Current estimation error criteria, error must be bigger than a threshold:

e(k ) = y(k ) − yˆ (k ) ≥ E1

Novelty criteria, the nearest center distance must be bigger than a threshold: M

inf x( k ) − µ j (k ) ≥ E2 j =1 Windowed mean error criteria, windowed mead error must be bigger than a threshold:

1

T

T∑ i =0

[ y ( k − T + i ) − yˆ ( k − T + i )] ≥ E3

If one (or more) of the above criteria are not met, the existing network parameters (the centers µj, the weights wij and the variances σj) are adjusted using a suitable online learning algorithm. To avoid an excessive increase of the Network size a pruning strategy can also be applied. When this happens the network is called Minimal RAN (MRAN) (Lu et al, 2000). The adaptation algorithm is called Extended MRAN (EMRAN) when the parameters are updated following a “winner takes it all” strategy, i.e. only the parameters of the most activated neuron are updated, while all the other are unchanged. This strategy implies a significant reduction of the number of parameters to be updated online with just a small performance degradation with respect to the MRAN (Li et al, 2000).

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SIMULATION EXAMPLE

In this paragraph the comparative results of SFDIA capabilities provided by different NN estimators are shown. The SFDIA procedure is applied to the estimation of the gyro rate q(k). The failure of the sensor occurring at exactly Tf=1098 s. Two different NN architectures were tested: EMRAN and MLP-EBP (of comparable complexity). To evaluate the goodness of the accommodation system, the parameters MAEE(k) and VAEE(k) that represent respectively the mean and variance of the absolute estimation error, before and after the fault are considered. To evaluate the goodness of the detection/identification system, the detection ratio S2/S1 between the main peak of the filtered residual signal during the failure transient (1096 < t < 1120) and the peek of the filtered residual before failure (0 < t < 1090) is considered. This ratio quantifies the detectability provided by the scheme. Furthermore, the time percentage in which LE=0 before the true-fault detection, is reported to quantify the false detection (false alarm) rate, and the time percentage in which AE=0 before the true-fault declaration, is reported to quantify the false accommodation rate. All these results are reported in table I. As expected the sixth failure is the most difficult do detect by any NNs. The best detectability performance, despite the simplicity of the learning algorithm, was given by the EMRAN, which exhibit the best approximation capability. On the contrary, the MLP-EBP after the accommodation causes the instability of the closed loop system, testifying, in this case, the non accurate online mapping achieved by this architecture. CONCLUSIONS

In this paper an aircraft SFDIA analysis tool was discussed. The tool allows the investigation of the interactions between the closed loop aircraft dynamics and the dynamics of the SFDIA system. A clear understanding of this interaction is of great importance since an incorrect choice of the internal SFDIA approximation architecture could result in the instability of the feedback control system. This aspect has been clearly pointed out by means of a simulation example in which instability occurs in the accommodation phase even if the sensor failure is correctly and promptly detected and isolated. In this respect, the main feature of the tool is the possibility of testing and comparing in closed loop the capabilities of SFDIA schemes exploiting different kinds of Neural Networks as nonlinear approximators. This capability is a consequence of the extensive modularity of the whole simulation tool, and allows an easy change of aircraft dynamics and feedback control law as well as NN estimators and SFDIA scheme.

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Subscript “1”: before failure; subscript “2”: after . AmplitudRise Tdetec MAEE VAEE MAEE

VAEE

S1/S

EMRAN

2.5 1 2.5 1 2.5 1

0.05 0.05 0.3 0.3 4 4

998 1002.35 998.1 1002.45 1000.1 1003.7

0.363857 0.364221 0.363857 0.364221 0.364804 0.362527

0.157222 0.15791 0.157222 0.15791 0.158085 0.157196

2.5 1 2.5 1 2.5 1

0.05 0.05 0.3 0.3 4 4

996.25 997.85 996.4 997.95 998.05 999.65

0.397254 0.398244 0.397254 0.398244 0.398244 0.397947

0.123787 0.123971 0.123787 0.123971 0.123971 0.124038

0.362104 0.359562 0.362086 0.359376 0.360973 0.357567

0.175498 0.173116 0.175737 0.173011 0.175841 0.172815

False

False

Simtim

6.441479 3.284092 6.425145 3.270282 6.482528 2.99019

0.275745 0.391507 0.273231 0.391507 0.283286 0.323527

0 0 0 0 0 0

405.41 404.63 407.82 405.96 407.88 408.65

3.286115 1.784715 3.298431 1.778442 3.244961 1.81221

7.031626 7.146301 7.03449 7.146301 7.066003 7.146301

1.571744 1.613026 1.571744 1.613026 1.571744 1.613026

452.53 450.77 450.5 450.99 453.91 452.86

MLP-EBPA 0.420127 0.422461 0.420459 0.42241 0.420935 0.421829

0.192858 0.191947 0.192938 0.192146 0.192985 0.192255

Table 1: SFDIA Performance Parameters

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