[american institute of aeronautics and astronautics aiaa guidance, navigation, and control...

Hammerstein Model-Based Correlation UIO Method

for Fault Detection of

Nonlinear Flight Control Systems

Kai-Yew Lum∗ and Jun Xu†

Temasek Laboratories, National University of Singapore, Singapore 117411

Ai-Poh Loh‡

Dept. of Electrical & Computer Engineering, National University of Singapore, Singapore 117576

In this paper, we propose an alternative approach – the correlation UIO method –

for fault detection of nonlinear control systems. The approach exploits a property of

the Hammerstein model with separable input process, in which case the cross-correlation

function between the input process and the residual of a suitably designed linear UIO is

decoupled from the nonlinearity. In an application to nonlinear flight-control systems, a

Hammerstein model of the closed-loop system is obtained by a novel approach for system

identification, in which the input-output correlation functions are used as data. To apply

the correlation UIO method, a very weak (compared to disturbances) separable signal

(sinusoid) is injected to the closed-loop system as a diagnostic signal. Simulation results

based on an F-16 model show that the scheme is able to detect actuator lock-in-place fault

even at trim deflection and in straight-level flight, which is the most difficult situation for

flight-control fault detection. Moreover, the detection threshold is independent of the fault

and control signals; under some assumptions, it can be arbitrarily increased by increasing

the amplitude of the diagnostic signal. The simplicity and benefits of the proposed method

are demonstrated through comparison with the standard linear UIO design.

I. Introduction

High demand for reliability and safety of complex control systems, such as flight-control systems, has

drawn extensive attention to the problem of fault detection and isolation (FDI). Among the different ap-

proaches that have emerged over the years, observer-based FDI has become an important class of techniques.

Particularly, the classical unknown input observer theory has been applied to several linear settings. Ide-

ally, the main feature of UIO is to guarantee a complete decoupling between the fault-estimation errors

(or, residuals) and the unknown disturbance inputs. Important treatises on UIO and other observer-based

FDI techniques can be found in Chen & Patton,1 and Frank & Ding.2 Kalman filter estimators have also

been proposed as FDI observers. Saberi et al.3 gave an interesting analysis of the connection between UIO

and Kalman filtering. Meanwhile, some recent development have focused on robustness UIO and threshold

design,4,5 multiple-model observers approach and adaptive estimation.6–9 Meanwhile, real-world systems,

∗Principal Research Scientist, AIAA Member, kaiyew [email protected]†Research Scientist‡Associate Professor

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American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation, and Control Conference2 - 5 August 2010, Toronto, Ontario Canada

AIAA 2010-8155

Copyright © 2010 by Kai-Yew Lum. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

especially flight-control systems, are inherently nonlinear except when operating at close proximity of a trim

condition. Previous linear methods have been applied to the linearized flight dynamics via extended Kalman

filtering6,7 or gain-varying UIO.10 An early extension to nonlinear UIO approach was given in Seliger &

Frank11 based on Taylor series expansion of the system model and partial-differential UIO conditions. More

recently, Ref.12–14 proposed various nonlinear UIO designs for the special case where nonlinearities in the

state equations are Lipschitz. However, this latter assumption is hard to verify for a flight-control system.

One of the reasons for robust FDI of general nonlinear system to remain difficult is that it is practically

impossible for the residuals to be completely decoupled from the unknown disturbances as well as unmodeled

nonlinear dynamics.2 Even in the case of exact model, there is always a trade-off between disturbance

rejection and sensitivity to faults.15 Another reason is system complexity.16 Observer design for a high-order

nonlinear system is generally difficult; even for a linear system, UIO design requires an observability condition

which may be too restrictive for high-order systems. For example, model decomposition or reduction may

be necessary to ensure observability of the model (not necessarily that of the plant).17,18 A third reason

lies in the effect of the control signals on the residuals. Indeed, most observer-based approaches rely on

the plant’s input and output signals to produce an estimate of the outputs, which are compared with the

measured outputs to construct the residuals. In system identification, one is familiar with the idea of

persistent excitation. However, for a control system in operation, the input-output pair may not provide

sufficient excitation for the FDI observer. In the case of flight-control systems, an actuator may experience

a lock-in-place fault at or near its trim (i.e. neutral) deflection while the aircraft is flying straight and level,

in zero or small gust disturbances. In such a scenario, the effect of the fault is very small and hard to detect.

For example, Ducard19 proposed using a small excitation signal to improve detectability.

This paper attempts to address the above difficulties for the actuator fault detection problem using

a rather simple idea. Basically, we return to the linear UIO approach, but instead of a linearization of

the nonlinear system model, we seek an alternative linear model based on which to construct the UIO. In

fact, we consider the entire closed-loop flight-control system as the plant on which to conduct UIO design.

An additive input signal, which we call diagnostic signal, will be injected to the closed loop and serve as

the input signal for fault detection. First, model reduction is carried out by approximating the closed-loop

system with a Hammerstein model. Although it is simple, the Hammerstein model (and other block-oriented

models) has been widely employed as an approximation for general nonlinear systems.20–23 Next, a linear

UIO is constructed for the Hammerstein model. Now, by the classical UIO approach, the estimation error is

certainly not decoupled from the nonlinearity of the model. However, we show by a result of Nuttall24 that,

if the input signal (in this case, the diagnostic signal) is a separable process, the input and output correlation

functions of the Hammerstein model are simply related by a linear transfer function h(τ) corresponding to

the Hammerstein model’s linear component multiplied by a constant, the latter being a function of the input

signal’s statistics. Using this property, we are able to achieve two things: first, h(τ) is obtained by system

identification, but using input-output correlation functions as data; second, designing the standard UIO for

h(τ), we show that the correlation function between the estimation error and the diagnostic input signal

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is decoupled from the model’s nonlinearity. Thus, the error-input correlation is now used as the residual,

instead of the estimation error.

This idea is somewhat analogous to frequency-domain methods in FDI, where a pre-filter is applied to

the input and output signals.2,25 Here, the pre-filter is basically characterized by the power spectrum of the

diagnostic signal. Intuitively, one may conjecture that such a pre-filter retains mainly linear input-output

relations (by the principle of harmonic response), whereas nonlinear responses are likely out-of-band and

filtered off. For this conjecture to work, the input signal needs to be separable. Conveniently, sinusoids

and phase-modulated sinusoids are separable. The advantage of this method is that, for a given level of

exogenous disturbance, the detection threshold can be made arbitrarily large by increasing the amplitude of

the diagnostic signal, provide this is tolerable by the system. In fact, we shall show in a simulation example,

based on a nonlinear F-16 model, that a very weak diagnostic signal is sufficient for achieving an acceptable

threshold. The example also shows that the proposed method is able to detect a lock-in-place fault at trim

deflection in straight-level flight, whereas a standard UIO scheme fails in this setting.

II. Main Approach

A. Hammerstein Model with Separable Input Process

1. Scalar Case

We consider the Hammerstein model20,26 composed of a static input nonlinearity F [·] followed by a linear

block represented by its impulse response function h(τ) (Figure 1), where v, u and y are scalars. We shall

denote such a Hammerstein model by the duplet {F [·], h(τ)}. We assume hereafter that h(τ) is strictly

rational, i.e. without feedthrough.

F [·] h(τ)u

yv

Figure 1. Hammerstein model

In general, there is no simple formulation for the cross-correlation between the input and output signals.

However, the result of Nuttall24 states that the cross-correlation function φuv for a wide class of static

nonlinear functions F [·] is related to the auto-correlation function φuu of the input in the following manner:

φuv(τ) = CF (v)φvv(τ), CF (v) =1

φvv(0)

∫

vF [v] · p(v)dv (1)

provided that the input is separable. In (1), p(v) is the probability distribution function of the input process

v, and CF (v) is a constant depending only on F [·] and the statistics of v.27 Separability in the sense of

Nuttall means that the conditional expectation of u satisfies

E{v(t− τ)|v(t)} = c(τ)v(t) (2)

with c(τ) = φuu(τ)/φuu(0). Nuttall24 showed that Gaussian, sine-wave, and phase-modulated sine-wave

processes are separable, among others.

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With (1), one can also establish a general relationship between the input and output correlation functions

of the Hammerstein model. Indeed, since h(τ) is linear, one has

φyw(τ) = h(τ) ∗ φuw(τ)

for any signal w(t), where ∗ denotes the convolution operator. In particular, letting w = v and using (1),

one obtains the following lemma.

Lemma 1 Provided the input is a separable process, the cross-correlation function of the input v and output

y of the scalar Hammerstein model is proportional to the input auto-correlation function passed through h(τ):

φyv(τ) = CF (v)h(τ) ∗ φvv(τ). (3)

2. Multivariable Case

The above result can be extended to the multivariable case where F [·] is a diagonal matrix function. Indeed,

let v ∈ Rm, u ∈ R

m, y ∈ Rm, h(τ) = {hji(τ)}j,i be a p-by-m impulse response matrix, and

F [·](v) =

F1[·] 0 . . . 0

0. . .

......

. . ....

0 . . . . . . Fm[·]

v1...

vm

=

F1[v1]...

Fm[vm]

. (4)

For simplicity, assume that the vi’s are mutually independent, and that each process vi is separable in the

sense of Nuttall. Then, for each i, j ∈ (1, ...,m),

φuivj(τ) = 0 if i 6= j, (5a)

φuivi(τ) = CFi

(vi)φvivi(τ), CFi

(vi) =1

φvivi(0)

∫

viFi[vi] · p(vi)dvi. (5b)

Applying the argument as in the scalar case for each pair of input and output components, one obtains

φyjvi(τ) =

∑

k

hjk(τ) ∗ φukvi(τ) = hji(τ) ∗ CFi

(vi)φvivi(τ) = hji(τ)CFi

(vi) ∗ φvivi(τ),

for i ∈ (1, ...,m), j ∈ (1, ..., p). The above can be stated in matrix form as in the following lemma:

Lemma 2 Provided the input processes are mutually independent and individually separable, the cross-

correlation matrix of the input and output of the multivariable Hammerstein model with diagonal F [·] as

given in (4), is equal to the input auto-correlation matrix passed through h(τ)CF (v):

φyv(τ) = h(τ)CF (v) ∗ φvv(τ), (6a)

CF (v) =

CF1(v1) 0 . . . 0

0. . .

......

. . ....

0 . . . . . . CFm(vm)

. (6b)

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3. System Identification by Correlation Method

Calculation of the constant CF (v) for a given process v is generally cumbersome. However, an immediate

consequence of (3) and (6) is that by treating the pair of signals (φvv, φyv) as input and output, then they

are related by a linear system whose ‘effective’ impulse response is h(τ)∆= h(τ)CF (v). Hence, determination

of h can be performed using standard system identification techniques for linear systems. This observation

is summarized in the following lemma and will be useful in a later application to fault detection of nonlinear

flight-control system.

Lemma 3 (Two-step identification of Hammerstein model22) Given a separable process v, the Ham-

merstein model {F [·], h(τ)} can be identified using the following two-step algorithm:

Step 1: Identify h∆= (τ)CF h(τ) in matrix autoregressive moving-average (ARMA) representation {Ai, b′

j}i;j

by least-squares estimation, using the correlation functions (φvv(τ), φyv(τ)) as “input-output” data.

Step 2: Assume a p-order polynomial form for F [v] = γ1v + ... + γpvp, and estimate the γj’s according to

the equation

[I + A(q)]y(k) = b′(q) [γ1v(k) + ...+ γpvp(k)] + e(k).

Stack the data points in rows to yield a matrix equation Z = Φθ+E where θ = (γ1, . . . , γp)T. Com-

pute a least-squares estimate θ. Finally, CF (v) = 1/γ1, and obtain a matrix ARMA representation

of h(τ) by bj = γ1b′j.

Remark 1 Two-step identification methods have been commonly employed for models consisting of static

nonlinearity and linear time-invariant (LTI) systems in cascade. Lemma 3 bears some resemblance to an

early work of Billings & Fakhouri,28 where Gaussian input and correlation functions were employed for two

LTI systems sandwiching a static nonlinear element. More recently, D’Amato et al.29 also employed a

two step method for Wiener system identification; here, the static nonlinearity was first approximated by a

nonparametric model, whereas standard LTI system identification was performed in the second step.

B. Unknown Input Observer (UIO) for Fault Detection

1. Standard Linear UIO

Here, we recall a standard result of UIO applied to fault detection of linear control systems. Consider a

discrete-time nth-order, m-input-p-output linear control system with state x ∈ Rn, control input v ∈ R

m

and output y ∈ Rp, and subject to unknown disturbance w ∈ R

q:

xk+1 = Axk +Buk + Ewk, A ∈ Rn×n, B ∈ R

n×m, (7a)

yk = Cxk, C ∈ Rp×n, E ∈ R

n×q. (7b)

Suppose the system (7) is susceptible to an input fault represented by two possible values of the input

distribution matrix: B ∈ {B0, Bf}, where B0 is the nominal value, whereas Bf is the faulty value. The fault

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occurs by unknown switching of B from B0 to Bf . A UIO for fault detection can be constructed with the

following nth-order observer:

zk+1 = Lzk + TBuk +Kyk (8a)

xk = zk +Hyk (8b)

yk = Cxk (8c)

where H ∈ Rn×p, L ∈ R

n×n, and K ∈ Rn×p are chosen to satisfy the following UIO conditions:

(I −HC)E = 0, T∆= I −HC, L = TA−K1C, K2 = LH, K = K1 +K2. (9)

The UIO (8) provides an observation of the output. Defining the error xk∆= xk−xk and residual ek

∆= yk−yk,

it can be easily found that the residual evolves with the following dynamics:

xk+1 = Lxk + TBuk, ek = Cxk, where B = B − B. (10)

Now, if we set B = B0, then when the system is in its nominal mode, B = 0, and the residual dynamics are

autonomous and decoupled from the unknown disturbance w; by choosing K1 such that L is asymptotically

stable (e.g. by pole placement), the residual will decay to zero from its initial condition. On the other hand,

when the system is faulty, B = Bf −B0 6= 0, and the residual dynamics are driven by Buk, which provides

a means to detect the presence of the fault. Note that for this detection method to work, uk must not be

uniformly zero. In addition, it is necessary to compute a fault-detection index which usually takes the form

of a cumulative sum of ‖ek‖ over a finite horizon, with or without some forgetting factor. Moreover, in the

presence of noise and model errors, the residual is not exactly zero in the nominal mode, which necessitates

the setting of some detection threshold.

2. UIO Fault Detection of Hammerstein Model using Correlation Function

Consider now the multivariable Hammerstein model of the previous section, in which we shall replace the

input v by (v, w), where w is again an unknown disturbance. We assume the nonlinear block to be diagonal

as before, i.e. F [(v, w)] = (Fv[v], Fw[w]), and CF (v, w) = diag(CFv(v), CFw

(w)).

Now, let h(τ) ∼ {A, [BE], C, 0} be a realization of the linear block. Then, a state-space representation

of the Hammerstein model is given by

xk+1 = Axk +BFv[vk] + EFw[wk], (11a)

yk = Cxk. (11b)

We consider again the problem where the system (11) is susceptible to the fault model B ∈ {B, Bf}. Instead

of nonlinear UIO, we simply adopt the standard linear UIO given by (8) and (9), but now formulated for

the linear system h(τ) ∼ {A, [BE], C, 0}. This yields the following residual dynamics:

xk+1 = Lxk + T{BF [vk] − Bvk}, ek = Cxk. (12)

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The signal uk = F [vk], which captures the effects of nonlinearity, is an internal variable of the Hammerstein

model and not physically measured. Thus, the above residual equation does not provide a ready means to

detect the fault. However, if (v, w) are mutually independent and individually separable, then the result of

Lemma 2 provides a simply solution via correlation functions.

Indeed, denoting by L(τ) the impulse response of {L, In×n, C, 0}, (12) can be written as:

e(τ) = L(τ) ∗ T{Bu(τ) − Bv(τ)}. (13)

Convolving with v(τ) and applying (5b) gives:

φev(τ) = L(τ) ∗ T{Bφuv(τ) − Bφvv(τ)} = L(τ) ∗ T{BCFv(v) − B}φvv(τ). (14)

By setting B = BCFv(v) and treating φev as the residual instead of e, we effectively have a UIO fault-

detection scheme for the Hammerstein model (Figure 2): in the nominal mode, the residual φev is zero at

steady state, whereas in the faulty mode, it is driven by the auto-correlation of the input v. Meanwhile, it

can be easily shown that

h(τ) ∼ {A, [BE], C, 0} ⇐⇒ h(τ) = h(τ)CF (v, w) ∼ {A, [BCFv(v) ECFw

(w)], C, 0}. (15)

As stated in Lemma 3, h(τ) can be identified in Step 1 using input-output correlation functions, whose state-

space realization directly yields B = BCFv(v) as the input distribution matrix, rather than B and CFv

(v)

separately; in fact, for our purpose it is not necessary to do Step 2. The above result can be summarized as

follows:

Proposition 4 (UIO by correlation method) Consider a Hammerstein model

{(Fv[·], Fw[·]), h(τ)}

with input v and unknown disturbance w. Assume that (v, w) are independent and individually separable,

and let h(τ) = h(τ)CF (v, w). Then a standard linear UIO synthesized for the linear model h(τ) provides a

fault detection scheme for the Hammerstein model by using the error-input correlation function φev, whose

response is given in (14), as residual function.

*UIO+

_

h(τ)0Fv[·]

0 Fw[·]

uy

w

v

yφev

e

Figure 2. UIO Fault Detection of Hammerstein Model by Correlation Method

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Remark 2 The above scheme requires input separability. As Nuttall has shown, the white noise is separable,

thus the unknown disturbance w can be considered separable. However, the control input generally depends

on the closed-loop trajectory and cannot be guaranteed to be separable. As we shall see in a flight-control

application, v will be a ’diagnostic signal’ chosen to be a separable process (e.g. sinusoidal) and applied as

an additive input to the closed-loop system.

3. Error Analysis: Hammerstein Model as an Approximation of a Nonlinear System

Physical nonlinear systems such as flight-control systems do not naturally take on a Hammerstein form.

Nevertheless, the Hammerstein model may provide a reasonable approximation of the original system. To

apply the proposed UIO fault-detection scheme, one needs to determine the effect of model-approximation

errors on the residual.

For simplicity of argument, suppose the Hammerstein model in state-space representation (11) is an

approximation of the following nonlinear system of the same order:

xk+1 = f(xk) +BFv[vk] + E′Fw[wk], B ∈ {B′0, Bf}; (16a)

yk = Cxk. (16b)

Comparing (16a) with (11a) it can be seen that, in addition to the modeling errors due to f(xk) and E′,

in the nominal mode there also exists a model error on the input distribution matrix: B′0 −B0. The above

results in the following residual dynamics:

e(τ) = L(τ) ∗[

T{Bu(τ) − Bv(τ)} + f(x(τ)) + TEw(τ)]

,

where f(x) = f(x) −Ax, E = E′ − E and w = Fw[w]. Then, the residual correlation function is given by

φev(τ) = L(τ) ∗ T{BCFv(v) − B}φvv(τ) + L(τ) ∗ φf(x)v(τ) + L(τ) ∗ TEφwv(τ). (17)

With B = B0CFv(v) as before, the residual correlation function takes the following forms depending on

whether the system is in the nominal or faulty mode:

φnominalev (τ) = L(τ) ∗ T{B′

0 −B0}CFv(v)φvv(τ) + L(τ) ∗ φnominal

f(x)v(τ) + L(τ) ∗ TEφwv(τ); (18a)

φfaultyev (τ) = L(τ) ∗ T{Bf −B0}CFv

(v)φvv(τ) + L(τ) ∗ φfaulty

f(x)v(τ) + L(τ) ∗ TEφwv(τ). (18b)

The two functions φnominalf(x)v

(τ) and φfaulty

f(x)vsignify that the system state x behaves differently in either modes.

Taking the difference of the two, which is essentially the fault-detection threshold, one finds

∆φ(τ) = L(τ) ∗ T{Bf −B′0}CFv

(v)φvv(τ) + L(τ) ∗(

φfaulty

f(x)v(τ) − φnominal

f(x)v(τ))

. (19)

Roughly speaking, if the input signal v is weak compared to the disturbance w, f(x) is predominantly driven

by the disturbance, and the second term is linear in v. On the other hand, the first term is always quadratic

in v. In other words, one can approximate the above expression by ‖∆φ‖ ≈ α‖v‖2 ± βw‖v‖, where βw

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depends on the disturbance level. By choosing v with a sufficiently large magnitude such that ‖v‖ > βw/α,

one can guarantee detection.

The above argument certainly requires further and more rigorous analysis, which we shall leave for future

work; later in this paper, we shall simply demonstrate it in an application example. Meanwhile, in practice

one may also consider obtaining an empirical cumulative distribution function for φnominalev using real-time

data,10 with which to adaptively tune the diagnostic signal level.

III. Application: Actuator-Fault Detection for Flight Control System

A. Model Description

In this section, we shall apply the proposed correlation UIO scheme to the problem of actuator-fault detection

for a flight-control system. The system dynamics are represented by a full 6-degrees-of-freedom (6DOF) flight

mechanical model with aerodynamics data based on the F-16.30,31 Without going into the details, this model

can be written as a 14-state nonlinear system, non-affine in the control given by (20a):

x = f(x) + g(x, δ, T, w), (20a)

δ = Asδ + u, (20b)

y = Cx, (20c)

with

x = (V, α, β, p, q, r, h, β, p, r, ϕ, θ, ψ, ηe), w = (αg, βg)

δ = (δel, δer, δal, δar, δr), u = (uel, uer, ual, uar, ur),

y = (α, β, θ, β, p, r, ϕ), g(x, δ, T, w) = (X,Y,Z, L,M,N)

where (X,Y,Z, L,M,N) are the aerodynamic forces and moments expressed in their general form as:

X =1

2ρ(h)V 2S CX(α+ αg, β + βg, p, q, r, δ),

Y =1

2ρ(h)V 2S CY (α+ αg, β + βg, p, q, r, δ),

X =1

2ρ(h)V 2S CZ(α+ αg, β + βg, p, q, r, δ),

L =1

2ρ(h)V 2Sb CL(α+ αg, β + βg, p, q, r, δ),

M =1

2ρ(h)V 2Sc CM (α+ αg, β + βg, p, q, r, δ),

N =1

2ρ(h)V 2Sb CN (α+ αg, β + βg, p, q, r, δ).

Additionally, (20b) represents the actuator dynamics which in this study are taken to be a first-order system

for each actuator. See Table 1 for a list of definitions.

The aerodynamic coefficients CX , CY , CZ , CL, CM , CN are obtained from lookup tables based on Ref.31;

in this example, the tables have been artificially altered to allow the elevators and ailerons to be independent

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(as opposed to symmetric elevators and anti-symmetric ailerons), so as to simulate a lock-in-place fault in

one of them.

To complete the model, a fault-tolerant control (FTC) consisting of state feedback, flight-path and turn-

rate tracking is implemented.32 Effectively, the F-16 in closed loop maintains straight and level flight at

an altitude of 3 km and speed of 200 m/s (Mach 0.6). Including the 2 internal states of the controller, the

total number of state variables for the closed-loop system is 21. See the lower portion of Figure 3 for a

block-diagram representation of the flight-control closed loop.

UIO+

_

F16

FTC

+

+

+

__

*y

φeve

C

w

yxu

(γc = 0, ψc = 0)uFTC

(v, 0, 0, 0, 0)

Figure 3. UIO Fault Detection of Flight Control System by Correlation Method

The purpose of this example is to demonstrate detection of a lock-in-place fault in the left elevator. For

this, a scalar signal v is injected into the closed loop via the left-elevator command, i.e.

uel = uFTCel + v.

Next a Hammerstein model is obtained by system identification, and a UIO is then constructed; notice that

only a subset y of the flight-dynamics state x is needed by the UIO.

B. Identification of a Hammerstein Model for the F-16 Closed Loop

To apply the method proposed in the previous section, an approximate Hammerstein model for the F-16

closed loop is first obtained by system identification using simulated data. For this, the signal v is chosen to

be a band-limited chirp (basically, a sinusoid with phase modulated by another sinusoid) within the frequency

range of 2∼4 Hz, with a magnitude of 0.1 degrees, while w = (αg, βg) is a pair of similar band-limited chirp

corresponding to perturbations of 0.02 degrees in magnitude in the angles of attack and sideslip.

The identification method consists of two steps: first, following Lemma 3, a matrix ARMA model of

h(τ) = h(τ)CF (v, w) is obtained using the correlation functions of the inputs (v, w) and output y. In this

example, the ARMA model obtained has 3 inputs and 7 outputs, with a regression order of 2, moving-average

orders of 3 in v, and 2 in αg and βg, respectively. By a slight abuse of time- and z-domain notations, the

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V : Total airspeed

α : Angle of attack

β : Sideslip angle

p : Roll angular rate

q : Pitch angular rate

r : Yaw angular rate

ϕ : Roll angle

θ : Pitch angle

ψ : Yaw angle

h : Altitude

ηe : Engine internal state

δel : Left elevator position

δer : Right elevator position

δal : Left aileron position

δar : Right aileron position

δr : Rudder position

u : Commands to actuators

T : Throttle position

X : Force component in the x(nose)-axis

Y : Force component in the y(starboard)-axis

Z : Force component in the z(belly)-axis

L : Roll moment

M : Pitch moment

N : Yaw moment

αg : Gust disturbance to the angle of attack

βg : Gust disturbance to the sideslip angle

S : Wing area

c : Mean chord length

b : Wing span

ρ : Air density (function of altitude)

Table 1. Definition of variables for F-16 model

ARMA model can be expressed as:

(

I7×7 + A1z−1 + A2z

−2)

y(t)

=(

bv1z

−1 + bv2z

−2 + bv3z

−3)

v(t) +(

bαg

1 z−1 + bαg

2 z−2)

αg(t) +(

bβg

1 z−1 + bβg

2 z−2)

βg(t) (21)

where Ai ∈ R7×7 and b∗

i ∈ R7.

Next, a minimal (and balanced) realization of the ARMA model is obtained via the Hankel-matrix method

(also known as the eigensystem realization algorithm – ERA).33,34 See also Ref.35 for a detailed description

of the method. Order determination of the state-space realization is based on two considerations: the rank

of the associated Hankel matrix, and existence of a UIO solution in the next step. Hence, this step may need

to be iterated with the next. In the end, a 12th-order model is selected.

Figures 5–6 show partial comparisons of the Hammerstein model output with the measured (i.e. simu-

lated) F-16 model output. It can be seen that, while the Hammerstein model has a reduced order compared

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to the true model, the fit is generally good.

C. UIO Design

Once the Hammerstein model is obtained, a UIO is designed according to the method of Section II.B.2. The

UIO conditions (9) need to be solved; in particular, pole placement is performed to determine the UIO state

matrix L. As mentioned above, it may be necessary to iterate this step with the previous, especially when

the order of the Hammerstein model is too high for the UIO solution to exist.

For fault-detection, the diagnostic signal v is chosen to be a 3 Hz sinusoidal signal with magnitude 0.1 or

0.2 degrees (see later). The 12th-order discrete-time UIO is implemented with a time step of 0.01 seconds, and

the correlation functions φev(τ) (7 functions for 7 residuals) is computed at each time step over a preceding

window of 3 seconds (300 time steps); in other words, φev(τ) is computed for τ ∈ {−299, ..., 299}. At the

same time, the root-sum-squares of the correlation functions is computed and serves as the fault-detection

index:

Φev∆=

√

√

√

√

7∑

i=1

(

299∑

τ=−299

|φeiv(τ)|2

)

, e = (α, β, θ, β, p, r, ϕ). (22)

D. Fault Detection Results

In the following fault-detection simulation, the 6DOF F-16 closed-loop model with UIO is subjected to gust

disturbance consisting of 0∼1 Hz, 1-degree peak-to-peak random perturbations in the angles of attack (αg)

and sideslip (βg). Figures 7–8 show the aircraft’s responses in the angles of attack and sideslip, altitude,

heading and roll angle. It can be seen that these responses are predominantly driven by the lower-frequency

gust disturbance, and the effect of the 3 Hz diagnostic signal is hardly noticeable.

1. Performance of Correlation UIO Method

The next result shows that even with a weak diagnostic signal, the correlation UIO method is effective.

For this, a lock-in-place fault is simulated in the left elevator at mid-point (t = 50 sec.) of the simulation

(Figure 9). The locked position of the elevator is in fact its trim deflection of 0.6 degrees, hence the effect

of the fault on closed-loop performance is very small; with the fault-tolerant control the aircraft is able to

continue tracking its flight path. Observing the portions of Figures 7, 8 and 9 after t = 50 sec., it is hard to

notice the fault in the signals except δel.

Figure 10 shows the time history of the fault index Φev for the true F-16 model with and without

disturbance, and the case where the same UIO is applied to the linear model h(s) (recall Proposition 4),

with a diagnostic signal level of |v| = 0.1 degrees . From Figure 10 it can be seen that, in the nominal mode

and without disturbance, Φev obtained with the F-16 model is only slight different from that of the linear

model, whereas in the faulty mode the two match almost perfectly. This shows that the Hammerstein model

is indeed a good approximation of the nonlinear closed-loop dynamics. Moreover, the step change in Φev

12 of 23


provides an indication of fault as expecteda. For the F-16 model with gust disturbance Φev is perturbed,

with perturbations corresponding to the terms

L(τ) ∗ φ{mode}

f(x)v(τ) + L(τ) ∗ TEφwv(τ)

in (18). As a result, the distinction between faulty and nominal values of Φev is less obvious, although the

level Φev = 0.58 × 10−5 obtained from the noiseless case may still be an effective fault-detection threshold.

To improve the threshold, the diagnostic signal level is next doubled: |v| = 0.2 degrees. As shown in

Figure 11, the threshold is quadrupled to 2.40 × 10−5 due to the quadratic term

L(τ) ∗ T{Bf −B′0}CFv

(v)φvv(τ)

in (19). This shows that the correlation method can be made arbitrarily sensitive to the fault for a given

disturbance environment.

As further evidence that the diagnostic signal is weak enough to not interfere with control performance,

Figure 12 shows the power spectral densities (PSD) of the actuators and state variables for the higher signal

level of |v| = 0.2 degrees. It can bee seen that the left elevator’s PSD contains a peak at 3 Hz corresponding

to the diagnostic signal v added to it. However, this frequency component is at least 25dB below those at

lower frequencies. Meanwhile, PSD of the other actuators hardly contain any component of 3 Hz, but show

dominant components around 0.5∼1 Hz which are due to the gust disturbance. From the PSD of the angles

of attack and sideslip, and the angular rates, it is clear that the diagnostic signal is weak and practically

rolled-off by the bandwidth of the closed-loop system.

Finally, it can be noticed in the lower plots of Figures 10–11 that the residual itself, although showing

somewhat higher peaks in the faulty mode, is not steady and cannot readily provide a means of fault

detection.

2. Performance under aircraft maneuver

The Hammerstein model has been identified using data obtained near a straight-level flight condition. While

the nonlinear function F [·] captures the nonlinear input-output relationship around this trim condition, the

model cannot be expected to be valid over a wide envelope of operation. To examine the performance of

the correlation UIO outside of the design conditions, a simulation is conducted with the aircraft executing

a turning flight at 20 degrees bank angle. Figure 13 shows the histories of Φev in banking and straight-level

flights. Ignoring the transient between t=0∼10 seconds, it can be seen that the effect of disturbance is more

pronounced in the nominal mode during banking flight. Interestingly, in the faulty mode, the correlation UIO

is little affected by the change in flight condition. Besides further increasing the amplitude of the diagnostic

signal, it will be worthwhile to examine more closely how the Hammerstein model varies over different flight

conditions, and investigate the possibility of parametrization in order to cover a wider envelope.

aThe step is in fact a steep ramp with an interval of 3 seconds equal to the window length for computation of Φev .

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3. Comparison with Standard UIO Method

It should be clear now that although the proposed correlation UIO method using the Hammerstein model as

an intermediary, the basic design is still a linear UIO. One may question its benefit, if any, over the standard

UIO method. For comparison, a standard UIO as described in Section II.B.1 is designed for the linearized

equation of the 19th-order plant, i.e. equation (20). The UIO fault detection setup is as shown in Figure 4.

As before, we simulate a left elevator lock-in-place fault at its trim deflection during straight-level flight. The

top-left plot in Figure 14 shows the residuals without gust disturbance. It can be seen that the fault cannot

be detected at all. Indeed, as explained in Section II.B.1, detection relies on the residual being driven by the

control signal (see equation (10)), which is zero in this case as shown in the right-hand plots of Figure 14.

The bottom-left plots shows that, even with gust disturbances, the perturbation in the control signal does

not provide sufficient excitation for the residual.

+

_

F16

FTC+

__

UIOy e

C

w

yxu

(γc = 0, ψc = 0)uFTC

Figure 4. Fault Detection of Flight Control System by Standard UIO

Figure 15 compares the correlation UIO with the standard UIO in the case the left elevator locks in-place

at a different position than its trim deflection. In this case, the larger fault introduces a biased input in the

system dynamics, which is reflected the residual of the standard UIO. This example shows that the detection

ability of the standard UIO is dependent on the fault and control signals, as is well known.19

In contrast, the behavior of Φev is not different from the previous fault scenario. Indeed, it is obvious

from (14) that the detection ability of the correlation UIO relies on the auto-correlation of the diagnostic

signal, and not the control signal. Clearly, the correlation UIO presents the advantage of being able to detect

arbitrary fault deflections, including the most difficult case of fault at trim deflection.

4. Comparison with Linear Model Obtained by Standard Identification

Since the correlation UIO method is basically linear UIO design for a particular linear model, it may appear

that one can also apply it to a linear model obtained by standard system identification, i.e. using input

and output data (u, y), instead of the correlation-based identification of Lemma 3. Figure 16 compares

the performance of the fault index Φev for the two models, where discrepancies can be noticed. In effect,

implementing the correlation UIO method for the linear model obtained through standard identification

results in greater sensitivity to disturbances. This is because the correlation identification method is better

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able to extract the linear relationship between input and output. Thus, it is a necessary step in setting up

the correlation UIO method.

IV. Concluding Remarks

In this paper, we have proposed the correlation UIO method for fault detection of nonlinear systems.

As has been shown, while the method employs a Hammerstein model as an intermediary, it is essentially a

linear UIO design, but with the novelty that correlation functions are used for both system identification and

residual construction. Besides using correlation functions, the other key differences with existing approaches

are: first, the UIO design procedure remains the same as in the standard method, but a Hammerstein model

of the closed loop must first be obtained; second, a diagnostic signal of a special type (i.e. separable) is

injected for fault detection. In the F-16 example, a very weak sinusoidal signal has proved to be sufficient for

effective detection of lock-in-place actuator faults even in trim condition. Meanwhile, a few remarks about

the method are in order:

a. In the example, fault detection for one actuator was demonstrated. Similarly, the method can certainly

be employed for all the actuators via a multiple-model implementation. However, simultaneous operation

of the correlation UIOs will require each model (corresponding to a particular actuator) to represent the

diagnostic inputs in the other actuators as unknown disturbances, which will result in a high-order model.

Moreover, there can be cross-coupling among the different diagnostic signals. Instead, the multiple UIOs

can simply be operated sequentially by applying the diagnostic signal to one actuator at a time, i.e.

sequential polling.

b. The method does require some effort in obtaining a suitable Hammerstein model, as was discussed in

Sections III.B-C. In particular, it was necessary to model the effect of disturbance during system iden-

tification, which was possible using simulated data. However, disturbance data is usually not available

experimentally. This may be a shortcoming of the proposed method.

c. Additionally, the frequency of the diagnostic signal has to be chosen appropriately so as to be independent

from the disturbance, and also not to affect the closed-loop behavior. A rule of thumb is to place this

frequency beyond the disturbance bandwidth and roll-off bandwidth of the closed-loop. However, overly

high diagnostic frequency may not be suitable due to mechanical and other practical considerations. In

this study, we have not modeled sensor noise, especially high-frequency noise, which should also be taken

into consideration.

d. Theoretically, the correlation function should be calculated over an infinite horizon. In practice, how-

ever, it must be approximated over a limited sliding window at each time step. The window should be

long enough in comparison with the diagnostic frequency. However, a longer window will incur heavier

computation cost, as well as delay in detection.

e. As mentioned in Section III.D, although the Hammerstein model may capture nonlinearities around a trim

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condition, it is not sufficient to model an entire flight envelope. Some kind of parametric Hammerstein

model may be considered in future work.

f. A final and most important question is that of threshold selection. As we have already alluded to in

Section II.B.3, under some assumptions, the threshold can be arbitrarily increased by increasing the

diagnostic signal’s amplitude. This was also apparent in the simulation example. Moreover, the simulate

results show that in the faulty mode, the values of Φev with disturbance stay close to the case without

disturbance. These properties require formal analysis as part of future work. Similarly, whether they can

be exploited for the estimation of partial faults remains to be studied.

References

1Chen, J. and Patton, R. J., Robust model-based fault diagnosis for dynamic systems, Kluwer Academic Publishers,

Boston, 1999.

2Frank, P. M. and Ding, X., “Survey of robust residual generation and evaluation methods in observer-based fault detection

systems,” Journal of Process Control , Vol. 7, No. 6, 1997, pp. 403–424.

3Saberi, A., Stoorvogel, A. A., and Sannuti, P., “Connections between H2 optimal filters and unknown input observers -

Performance limitations of H2 optimal filtering,” International Journal of Control , Vol. 79, No. 2, 2006, pp. 171–183.

4Amato, F. and Mattei, M., “Design of full-order unknown input observers with H∞ performance,” Proc. Conference on

Control Applications, Vol. 1, Glasgow, UK, Sep. 2002, pp. 74–75.

5Lum, K.-Y., Xu, J., Xie, L., and Loh, A.-P., “A Novel UIO-based Approach for Fault Detection and Isolation in Finite

Frequency Domain,” Proceedings of the 7th International Conference on Control Applications, Christchurch, NZ, 9-11 December

2009.

6Fisher, K. A. and Maybeck, P. S., “Multiple model adaptive estimation with filter spawning,” IEEE Trans. on Aerospace

and Electronic Systems, Vol. 38, No. 3, 2002, pp. 755–768.

7Rupp, D., Ducard, G., Shafai, E., and Geering, H. P., “Extended multiple model adaptive estimation for the detection

of sensor and actuator faults,” Proc. Conference on Decision and Control , Seville, Spain, Dec. 2005, pp. 3079–3084.

8Wang, D. and Lum, K.-Y., “Adaptive unknown input observer approach for aircraft actuator fault detection and isola-

tion,” International Journal of Adaptive Control and Signal Processing, Vol. 21, No. 1, 2007, pp. 31 – 48.

9Zhang, Y. and Jiang, J., “Integrated active fault-tolerant control using IMM approach,” IEEE Trans. on Aerospace and

Electronic Systems, Vol. 37, No. 4, 2001, pp. 1221–1235.

10Xu, J., Lum, K.-Y., and Loh, A.-P., “A Gain-varying UIO Approach with Adaptive Threshold for FDI of Nonlinear F16

Systems,” Journal of Control Theory and Applications, Vol. 8, No. 3, 2010.

11Seliger, R. and Frank, P. M., “Robust component fault detection and isolation in nonlinear dynamic systems using

nonlinear unknown input observers,” IFAC Symposia /IMACS Symposium on Fault Detection, Supervision and Safety for

Technical Processes, Baden-Baden, Germany, 1992, pp. 277–282.

12Chen, W. and Saif, M., “Fault detection and isolation based on novel unknown input observer design,” Proc. American

Control Conference, Mineapolis, MN, USA, Jun. 2006, pp. 5129–5134.

13Mondal, S., Chakraborty, G., and Bhattacharyya, K., “Robust unknown input observer for nonlinear systems and its

application to fault detection and isolation,” Journal of Dynamic Systems, Measurement and Control, Trans. of the ASME ,

Vol. 130, No. 4, 2008, pp. 445031–445035.

14Pertew, A. M., Marquez, H. J., and Zhao, Q., “Design of unknown input observers for Lipschitz nonlinear systems,”

Proc. American Control Conference, Portland, OR, USA, Jun. 2005, pp. 4198–4203.

16 of 23


15Wang, W., Bo, Y., Zhou, K., and Ren, Z., “Fault Detection and Isolation for Nonlinear Systems with Full State Infor-

mation,” Proc. 17th IFAC World Congress, Seoul, Korea, Aug. 2008, pp. 8908–8912.

16Shields, D. N., “Models, residual design and limits to fault detection for a complex multi-tank hydraulic control system,”

International Journal of Control , Vol. 76, No. 8, 2003, pp. 781–793.

17Chen, W. and Saif, M., “Adaptive Fault Detection for a Class of Nonlinear Systems Based on Output Estimator Design,”

Proc. 17th IFAC World Congress, Seoul, Korea, Aug. 2008, pp. 10118–10123.

18Shumsky, A., “Algebraic Approach to the Problem of Fault Accommodation in Nonlinear Systems,” Proc. 17th IFAC

World Congress, Seoul, Korea, Aug. 2008, pp. 1884–1889.

19Ducard, G. and Geering, H. P., “Efficient nonlinear actuator fault detection and isolation system for unmanned aerial

vehicles,” Journal of Guidance, Control and Dynamics, Vol. 31, No. 1, 2008, pp. 225–237.

20Billings, S. A. and Fakhouri, S. Y., “Nonlinear system identification using the Hammerstein model,” International Journal

of System Science, Vol. 10, No. 5, 1979, pp. 567–578.

21Crama, P. and Schoukens, J., “Hammerstein-Wiener system estimator initialization,” Automatica, Vol. 40, No. 9, 2004,

pp. 1543–1550.

22Lum, K.-Y. and Lai, K. L., “Identification of a Hammerstein Model for Wing Flutter Analysis Using CFD Data and

Correlation Method,” Proceedings of the American Control Conference, Baltimore, Maryland, USA, 30 June – 2 July 2010.

23Zeng, J., Baldelli, D. H., and Brenner, M., “Novel Nonlinear Hammerstein Model Identification: Application to Nonlinear

Aeroelastic/Aeroservoelastic System,” Journal of Guidance, Control, and Dynamics, Vol. 31, No. 6, 2008, pp. 1677–1686.

24Nuttall, A. H., Theory and application of the separable class of random processes, Ph.D. thesis, Research Laboratory of

Electronics, MIT, 1958.

25Fong, K. F., Loh, A. P., and Tan, W. W., “A frequency domain approach for fault detection,” International Journal of

Control , Vol. 81, No. 2, 2008, pp. 264–276.

26Baldelli, D. H., Lind, R., and Brenner, M., “Nonlinear Aeroelastic/Aeroservoelastic Modeling by Block-Oriented Identi-

fication,” Journal of Guidance, Control, and Dynamics, Vol. 28, No. 5, 2005, pp. 1056–1064.

27Billings, S. A. and Fakhouri, S. Y., “Theory of separable processes with applications to the identification of nonlinear

systems,” Proceedings of the IEE , Vol. 125, No. 10, 1978, pp. 1051–1058.

28Billings, S. A. and Fakhouri, S. Y., “Identification of Systems Containing Linear Dynamic and Static Nonlinear Elements,”

Automatica, Vol. 18, No. 1, 1982, pp. 15–26.

29D’Amato, A. M., Teixeira, B. O., and Bernstein, D. S., “Semiparametric Identification of Wiener Systems Using a Single

Harmonic Input and Retrospective Cost Optimization,” Proceedings of the American Control Conference, 2010.

30Stevens, B. L. and Lewis, F. L., Aircraft Control and Simulation, Wiley, 2003.

31Sonneveldt, L., Nonlinear F-16 Fighter Model , Delft University of Technology, The Netherlands, 2009.

32Yang, G.-H. and Lum, K.-Y., “Fault-Tolerant Flight Tracking Control with Stuck Faults,” Proceedings of the American

Control Conference, Vol. 1, Denver, CO, United States, 2003, pp. 521 – 526.

33Chen, C. T., Linear System Theory and Design, HRW, New York, 1984.

34Juang, J. N., Applied System Identification, Prentice-Hall, Englewood Cliffs, NJ, 1994.

35Lai, K. and Lum, K.-Y., “Reduced-order based flutter analysis for complex aeroelastic systems,” Proceedings of the 26th

AIAA Applied Aerodynamics Conference, Honolulu, HI, United States, 2008.

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10 15 20 25 30 35 40 45 501.38

1.4

1.42

1.44

1.46Measured and Hammerstein−model Angles of Attack α (deg)

10 15 20 25 30 35 40 45 50−0.04

−0.02

0

0.02

0.04Measured and Hammerstein−model Angles of Sideslip β (deg)

Time (sec)

Figure 5. Comparison of angles of attack (α) and sideslip angles (β) obtained from the 12th-order Hammersteinmodel and the 21st-order nonlinear F-16 model.

10 15 20 25 30 35 40 45 50−0.2

0

0.2Measured and Hammerstein−model Rates (deg/sec)

Pitch Rate (q)

10 15 20 25 30 35 40 45 50−0.5

0

0.5 Roll Rate (p)

10 15 20 25 30 35 40 45 50−0.2

0

0.2 Yaw Rate (r)

Time (sec)

Figure 6. Comparison of the pitch, roll and yaw angular rates obtained from the 12th-order Hammersteinmodel and the 21st-order nonlinear F-16 model.

18 of 23


0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0

0.5

1

1.5

2

2.5

Time (s)

α

β

Angles of attack (α) and sideslip (β) (deg)

Figure 7. Responses of the angles of attack and sideslip for the nonlinear F-16 model subjected to 0∼1 Hz,1-degree peak-to-peak random gust disturbance. Left elevator lock-in-place fault at t = 50 sec.

0 20 40 60 80 100−5

0

5 h−3000

Height h (km), Roll Angle φ (deg), Heading ψ (deg), Angular rates (deg/s): pitch (q), roll (p), yaw (r)

0 20 40 60 80 100

−2

0

2 φ

0 20 40 60 80 100−1

0

1 ψ

Time (sec)

0 20 40 60 80 100

−1

0

1 q

0 20 40 60 80 100−10

0

10 p

0 20 40 60 80 100

−2

0

2 r

Time (sec)

Figure 8. Responses of the height, attitude and angular rates for the nonlinear F-16 model subjected to0∼1 Hz, 1-degree peak-to-peak random gust disturbance. Left elevator lock-in-place fault at t = 50 sec.

19 of 23


0 10 20 30 40 50 60 70 80 90 100−1

−0.5

0 δel

, δer

Actuator Positions (deg)

0 10 20 30 40 50 60 70 80 90 100−1

0

1 δal

, δar

0 10 20 30 40 50 60 70 80 90 100−2

0

2 δr

Time (sec)

Figure 9. Responses of the actuator deflections for the nonlinear F-16 model subjected to 0∼1 Hz, 1-degreepeak-to-peak random gust disturbance. Left elevator locks in-place at trim deflection, at t = 50 sec.

0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1x 10

−5 Correlation (Φev

)

0 10 20 30 40 50 60 70 80 90 1000

5x 10

−8 Residual (|e|2)

Time (sec)

nominal mode faulty mode

F16 model, with disturbance

F16 model, noiseless

Linear Model h(s)

0.58 × 10−5

Figure 10. Correlation UIO method: History of fault index Φev for the nonlinear F-16 model, with diagnosticsignal level |v| = 0.1 degrees. Left elevator locks in-place at trim deflection, at t = 50 sec.

20 of 23


0 10 20 30 40 50 60 70 80 90 1000

1

2

3x 10


)

0 10 20 30 40 50 60 70 80 90 1000

5x 10

−8 Residual (|e|2)

Time (sec)

nominal mode faulty mode

F16 model, with disturbance

F16 model, noiseless

Linear Model h(s)

2.40 × 10−5

Figure 11. Correlation UIO method: History of fault index Φev for the nonlinear F-16 model, with diagnosticsignal level |v| = 0.2 degrees. Left elevator locks in-place at trim deflection, at t = 50 sec.

0 5 10−200

−150

−100

−50

0

50PSD of state variables: α, β, p, q, r

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

0 5 10−200

−150

−100

−50

0

50PSD of actuator deflections

Frequency (Hz)

Pow

er/fr

eque

ncy

(dB

/Hz)

Pα, Pβ

Pv

Pp, Pq, Pr

Pδel

PSD of other actuators

Pv

Figure 12. Power spectral densities of actuator deflections (left) and state variables (right), subjected to0∼1 Hz, 1-degree peak-to-peak random gust disturbance and a diagnostic signal |v| = 0.2 degrees in the leftelevator.

21 of 23


0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4x 10


) −− 20 deg. banking flight

0 10 20 30 40 50 60 70 80 90 1000

2

x 10−6 Residual (|e|2) −− 20 deg. banking flight

Time (sec)

Straight−level flight

20 deg. banking flight

Transcient

Figure 13. Correlation UIO method: Comparison of Φev between straight-level and 20 deg. banking flights.Diagnostic signal |v| = 0.2. Left elevator locks in-place at trim deflection, at t = 50 sec.

0 50 100−1

−0.5

0 δel

, δer

δel

lock−in−place at trim (noiseless)

0 50 100−0.2

0

0.2 δal

, δar

0 50 100−0.2

0

0.2 δr

Time (sec)

0 20 40 60 80 1000

0.5

1x 10

−4 Standard UIO (|e|2) −− Noiseless case

0 20 40 60 80 1000

1

2x 10

−3 Standard UIO (|e|2) −− Gust disturbance

Time (sec)

Figure 14. Standard UIO method. Left: residuals for left elevator lock-in-place fault at trim deflection.Right: when without gust disturbance, fault at trim deflection does not cause any visible perturbation in theactuators.

22 of 23


0 20 40 60 80 1000

0.02

0.04Standard UIO Residual (|e|2)

0 20 40 60 80 1000

1

2

3x 10

−5 Correlation UIO (Φev

)

Time (sec)

0 20 40 60 80 100−2

−1

0 δel

, δer

δel

lock−in−place at 0 deg.

0 20 40 60 80 100−1

0

1 δal

, δar

0 20 40 60 80 100−2

0

2 δr

Time (sec)

Figure 15. Comparison of correlation-based and standard UIOs for left elevator lock-in-place fault at 0 deg.,i.e away from trim. Performance of correlation UIO (bottom-left) is similar to Figure 11.

0 10 20 30 40 50 60 70 80 90 1000

1

2

x 10−5 Correlation (Φ

ev) −− Noiseless

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4x 10


) −− Gust disturbance

Time (sec)

Correlation identification

Standard identificationCorrelation identification

Standard identification

Figure 16. Comparison of correlation-based and standard identification methods: to implement the correlationUIO method, h(s) should be obtained through correlation-based identification.

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