December 2014 

NASA/TM–2014-218669 

Identifiability of Additive, Time-Varying Actuator 

and Sensor Faults by State Augumentation 

Jason M. Upchurch 

Langley Research Center, Hampton, Virginia 

Oscar R. Gonzάlez 

Old Dominion University, Norfolk, Virginia 

Suresh M. Joshi 

Langley Research Center, Hampton, Virginia 
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December 2014 

NASA/TM–2014-218669 

Identifiability of Additive, Time-Varying 

Actuator and Sensor Faults by State 

Augmentation 

Jason M. Upchurch 

Langley Research Center, Hampton, Virginia 

Oscar R. Gonzάlez 

Old Dominion University, Norfolk, Virginia 

Suresh M. Joshi 

Langley Research Center, Hampton, Virginia 
Available from: 

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NASA Langley Research Center 

Hampton, VA 23681-2199 

Fax: 757-864-6500 

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Space Administration. 
Abstract

Recent work has provided a set of necessary and sufficient conditions for

identifiability of additive step faults (e.g., lock-in-place actuator faults,

constant bias in the sensors) using state augmentation. This paper extends these results to an important class of faults which may affect linear,

time-invariant systems. In particular, the faults under consideration are

those which vary with time and affect the system dynamics additively.

Such faults may manifest themselves in aircraft as, for example, control

surface oscillations, control surface runaway, and sensor drift. The set of

necessary and sufficient conditions presented in this paper are general,

and apply when a class of time-varying faults affects arbitrary combinations of actuators and sensors.

The results in the main theorems are illustrated by two case studies, which provide some insight into how the conditions may be used

to check the theoretical identifiability of fault configurations of interest

for a given system. It is shown that while state augmentation can be

used to identify certain fault configurations, other fault configurations

are theoretically impossible to identify using state augmentation, giving

practitioners valuable insight into such situations. That is, the limitations of state augmentation for a given system and configuration of faults

are made explicit. Another limitation of model-based methods is that

there can be large numbers of fault configurations, thus making identification of all possible configurations impractical. However, the theoretical

identifiability of known, credible fault configurations can be tested using

the theorems presented in this paper, which can then assist the efforts

of fault identification practitioners.

i
Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 A Short Review of Key Concepts 2

2.1 A Review of Observability and Detectability . . . . . . . . 2

2.2 Deriving the Rosenbrock System Matrix . . . . . . . . . . 3

3 Time-Varying Fault Modeling 4

3.1 A Representation for Time-Varying Faults . . . . . . . . . 4

3.2 Actuator Fault Modeling . . . . . . . . . . . . . . . . . . . 5

3.3 Sensor Fault Modeling . . . . . . . . . . . . . . . . . . . . 6

3.4 Simultaneous Actuator and Sensor Fault Modeling . . . . 7

4 Conditions for Identifiability of Time-Varying Actuator

and Sensor Faults 9

4.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Time-Varying Actuator Fault Identifiability . . . . . . . . 10

4.3 Time-Varying Sensor Fault Identifiability . . . . . . . . . 14

4.4 Simultaneous Time-Varying Actuator and Sensor Fault

Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . 18

5 Conclusions and Future Research 23

ii
1 Introduction

1.1 Background

In this paper, the term fault denotes a state in a dynamical system,

which may result in a malfunction or failure of the system [1]. A failure

is either an intermittent or permanent interruption of a system’s ability

to fulfill a desired function [2]. In dynamical systems with actuators and

sensors, faults in these components may lead to failures characterized by,

for example, instability and loss of control.

In domains such as civil aviation and space operations, such actuator

and sensor faults can have particularly serious implications for safety and

reliability. For example, actuator faults such as rudder runaway have

been implicated in multiple aviation incidents (for example, see [3] and

[4]). Other actuator faults such as undesired control surface oscillations

(e.g., the oscillatory failure case) can increase the structural loads on

an aircraft and compromise its structural integrity in flight [5, 6]. As

a further example, sensor-bias faults have contributed to the failure of

missions such as NASA’s Demonstration of Autonomous Rendezvous

Technology (DART) [7]. Many more examples of aviation incidents and

accidents where actuator or sensor faults were implicated as contributing

factors can be found in [8].

In the aerospace industry, safety and reliability concerns associated

with actuators and sensors are primarily addressed through techniques

based on hardware redundancy [9, 10]. The counterpart to hardware

redundancy is generally referred to as analytical redundancy, a broad

class of techniques which make use of mathematical models of a system

to detect and identify actuator and sensor faults. Such model-based

fault detection and identification (FDI) methods have received significant

attention in the literature over the last several decades. For surveys on

a variety of model-based FDI techniques see [11–14].

The model-based technique investigated in this paper uses multiple

models, where each model corresponds to the nominal system state augmented by a set of fault configurations of interest. Typically, a bank of

detection filters is used to estimate the present state of the aircraft, and

multiple-hypothesis testing determines if a fault has occurred [15–17].

Several authors have proposed using state augmentation, e.g., [16–21].

However, a key requirement for state augmentation to be effective is that

each model, or fault configuration of interest, be identifiable [22].

Identification of constant bias-type faults was initially treated in [23].

The preliminary conditions for identifiability of these faults in arbitrary

combinations of actuators and sensors were presented in [17], and a subsequent, detailed analysis was given in [24]. A complete characterization

of a set of necessary and sufficient conditions, including numerical examples, can be found in [22]. This paper extends these results to the

broader class of additive, time-varying faults, which includes faults such

1
as the sinusoidal and ramp faults, while still remaining applicable to the

step fault.

This paper treats time-varying faults as outputs of exogenous, linear, time-invariant (LTI) systems which then additively affect a given

LTI system of interest. In particular, a set of necessary and sufficient

conditions for identifiability of additive, time-varying faults affecting arbitrary combinations of (1) actuators only, (2) sensors only, and (3)

combinations of actuators and sensors are presented. These conditions

fully characterize the identifiability of such faults using state augmentation, which provides designers a deeper understanding of the theoretical

reasons for non-identifiability and have practical implications on the limitations of using state augmentation alone, especially when the theory

shows a fault configuration of interest is not identifiable by state augmentation. The approach in this paper is the investigation of the causes

of non-identifiability beyond the level of (numerical or analytical) rank

tests, i.e., it seeks to provide exactly the reason(s) why such rank tests

will fail for a particular system and set of faults.

The remaining sections of this paper are organized as follows: Section

2 gives a short review of observability, detectability, and the Rosenbrock

System Matrix; Section 3 develops a state-space representation for timevarying actuator and sensor faults using state augmentation; Section 4

presents a set of necessary and sufficient conditions for identifiability

of additive, time-varying faults affecting arbitrary combinations of actuators, sensors, or both, where examples using a practical system are

included; Section 5 presents the conclusions of the research and suggestions for future directions.

2 A Short Review of Key Concepts

This section provides a short review of the concepts of observability,

detectability, and the Rosenbrock System Matrix (RSM).

2.1 A Review of Observability and Detectability

First, consider a system having state-space representation given by

x ̇(t) = Ax(t) + Bu(t), x(0) = x0, (1)

y(t) = Cx(t) + Du(t), (2)

where A ∈ R

n×n

, B ∈ R

n×m, C ∈ Rl×n

, and D ∈ R

l×m. Furthermore,

x(t) ∈ R

n

, u(t) ∈ R

m, and y(t) ∈ Rl are the state, input, and output

vectors, respectively. In many practical systems, certain states may not

be directly measurable, so state estimators (i.e., observers) are used to

infer the unknown state(s) by making use of the output vector y(t) and

the input vector u(t). For an observer to be effective, a system must

generally be observable, or at the very least detectable.

2
A system is said to be observable if there exists a time t1 > 0 such

that any initial state x0 can be uniquely determined from y(t), t ∈ [0, t1]

[25]. Two equivalent tests for observability given in [26] are the PopovBelevitch-Hautus (PBH) rank test and the PBH eigenvector test.

Test 1. PBH Rank Test. The system given by Equations (1) and (2) is

observable if and only if for every eigenvalue λi of A, that is for every

λi ∈ Λ(A), where Λ(A) denotes the set of eigenvalues of A,

rank 

λiI − A

C



= n

for i = 1, 2, . . . , n.

Test 2. PBH Eigenvector Test. The system given by Equations (1)

and (2) is observable if and only if there does not exist a nonzero γ ∈ C

n

such that



λiI − A

C



γ = 0

for i = 1, 2, . . . , n. That is, the pair (C, A) is observable if and only if

Cγ 6= 0 for every eigenvector γ of A.

If either Test 1 or 2 fails for any value λi, i = 1, . . . , n, then λiis an

eigenvalue corresponding to an unobservable mode of A. An eigenvalue

λ is an open left-half plane (OLHP) eigenvalue if and only if Re(λ) < 0.

Let Λu(A) denote the set of eigenvalues that lie on the closed right

half plane. The system given by Equations (1) and (2) is said to be detectable if and only if all of the eigenvalues associated with unobservable

modes of A lie in the OLHP [25]. Thus, a system is detectable if and only

if either Test 1 or 2 is satisfied for {λj : Re(λj ) ≥ 0}, j = 1, 2, . . . , k,

where k ≤ n [25].

Since observability requires that Test 1 be satisfied for all eigenvalues

of A, observability implies detectability. However, because detectability requires only those eigenvalues with non-negative real parts correspond to observable modes of A, an unobservable system may still be

detectable. Finally, a system may be detectable, but if it has eigenvalues

with negative real parts associated with unobservable modes of A, the

system is still not observable. Thus, observability implies detectability,

but detectability does not imply observability.

2.2 Deriving the Rosenbrock System Matrix

The Rosenbrock System Matrix (RSM) and some of its important properties are used in the proofs of conditions for time-varying actuator and

sensor fault identifiability. The derivation of the RSM follows.

Consider the system given by Equations (1) and (2). This system

can be represented in the frequency domain by its one-sided Laplace

3
transformation as

sxˆ(s) − x0 = Axˆ(s) + Buˆ(s), (3)

yˆ(s) = Cxˆ(s) + Duˆ(s), (4)

where ˆx(s), uˆ(s), and ˆy(s) are the Laplace transforms of the state, input,

and output vectors, respectively. Furthermore, x0 is the initial condition

at time t = 0, that is, x(0). Here, the one-sided Laplace transform, Υ( ˆ s),

of a function Υ(t) is defined as

Υ( ˆ s) = Z ∞

0−

Υ(t)e

−st dt. (5)

Now, the representation given by Equations (3) and (4) can be expressed

as



sI − A −B

C D   xˆ(s)

uˆ(s)



=



x0

yˆ(s)



. (6)

The coefficient matrix in (6) is referred to as the Rosenbrock System

Matrix of the system having realization {A, B, C, D}.

3 Time-Varying Fault Modeling

This following development gives a representation for time-varying faults

by treating them as outputs of an exogenous LTI system driven only by

initial conditions. It is assumed throughout that the fault of interest has

a one-sided Laplace transform as defined by Equation (5).

3.1 A Representation for Time-Varying Faults

Let f(t) be a vector of faults. Such faults may be modeled as the output

of an LTI system having state-space representation given by

x ̇ f (t) = Afxf (t), xf (0) = xf0, (7)

f(t) = Cfxf (t), (8)

where Af ∈ R

nf ×nf , Cf ∈ Rμ×nf , xf (t) ∈ Rnf , and f(t) ∈ Rμ

. It is

assumed that Cf has full row rank, that is, rank(Cf ) = μ. Taking the

Laplace transform of Equations (7) and (8), and solving for ˆxf (s) in

Equation (7) gives

xˆf (s) = (sI − Af )

−1xf0

, (9)

ˆf(s) = Cfxˆf (s). (10)

Now, substituting Equation (9) into Equation (10) gives

ˆf(s) = Cf (sI − Af )

−1xf0

.

Thus, the frequency domain representation of the fault vector is the

zero-input response of the system given by Cf (sI − Af )

−1xf0

.

4
3.2 Actuator Fault Modeling

Consider a system given by Equations (1) and (2). If mk of the m actuators are affected by additive, time-varying faults, the system dynamics

can be represented as

x ̇(t) = Ax(t) + X

j∈Fak

bjuj (t) + X

j /∈Fak

bjuj (t) + ωp(t), (11)

= Ax(t) + B

k

u

k

(t) + B

kuk

(t) + ωp(t), (12)

y(t) = Cx(t) + ωs(t), (13)

where B

k

∈ R

n×mk , Bk ∈ Rn×(m−mk)

, u

k

(t) ∈ R

mk , uk

(t) ∈ R

m−mk ,

ωp(t) ∈ R

n

, and ωs(t) ∈ R

l

. Furthermore, Fak in Equation (11) denotes

the set of indices corresponding to the failed actuators, uj (t) ∈ R denotes

a faulty input associated with a faulty actuator at time t, uj (t) ∈ R denotes a non-faulty input associated with a non-faulty actuator at time

t, and bj ∈ R

n denotes the particular column of B (see Equation (1))

affected by the corresponding faulty or non-faulty actuator. Thus, Equation (12) represents the system subject to a particular actuator fault

configuration.

Now consider the state-space representation for time-varying faults

given by Equations (7) and (8). Observe that such a representation may

model time-varying faults in the actuators, resulting in a system with

fault state vector, xa(t), and output u

k

(t), i.e.,

x ̇ a(t) = Aaxa(t) + ωpa(t), xa(0) = xa0, (14)

u

k

(t) = Caxa(t) + ωsa(t), (15)

where Aa ∈ R

na×na, Ca ∈ Rmk×na, and xa(t) ∈ Rna. Additionally,

terms for the exogenous actuator fault process and measurement noise,

i.e., ωpa(t) ∈ R

na and ωsa

(t) ∈ R

mk , respectively, have been added.

These terms allow for the modeling of faults which might not manifest

themselves as entirely deterministic representations in practice. Now,

Equations (12)-(15) can be combined to represent the interconnected

system. Thus, the state-augmented system can be represented as



x ̇(t)

x ̇ a(t)



=

"

A B

k

Ca

0 Aa

#

| {z }

Ak

ξ



x(t)

xa(t)



| {z }

ξ

k(t)

+



Bk

0



| {z }

Bk

ξ

u

k

(t)

+

"

B

k

ωsa(t) + ωp(t)

ωpa(t)

#

| {z }

ω

k

ξp

(t)

, (16)

y(t) = 

C 0



| {z }

Ck

ξ



x(t)

xa(t)



+ ωs(t), (17)

5
where, after making the appropriate substitutions indicated by the

braces, the system above can be expressed more compactly as

 ̇ξ

k

(t) = A

k

ξ

ξ

k

(t) + B

k

ξ u

k

(t) + ω

k

ξp

(t), (18)

y(t) = C

k

ξ

ξ

k

(t) + ωs(t). (19)

Equations (18) and (19) model the general case of arbitrary, additive,

time-varying actuator faults treated in this paper. In the case of step

faults, Aa = 0mk×mk, and Ca = Imk. Thus, Equation (16) reduces to



x ̇(t)

x ̇ a(t)



=

"

A B

k

0 Imk

# 

x(t)

xa(t)



+



Bk

0



u

k

(t)

+

"

B

k

ωsa(t) + ωp(t)

ωpa(t)

#

, (20)

which is identical to Equation (4) in [22], where the case of actuator step

faults was considered.

3.3 Sensor Fault Modeling

Consider again a system given by Equations (1) and (2). If q of the l

sensors for the given system are affected by additive, time-varying faults,

the system dynamics can be represented as

x ̇(t) = Ax(t) + Bu(t) + ωp(t), (21)

y(t) = 

y1(t)

y2(t)



, (22)

= Cx(t) + 

0

ys(t)



+ ωs(t), (23)

where A, B, C, x(t), u(t), y(t), ωp(t), and ωs(t) are as previously defined,

and y1(t) ∈ R

l−q and y2(t) ∈ Rq

represent the vectors containing the

fault-free sensor measurements and the faulty sensor measurements, respectively. Finally, ys(t) ∈ R

q

is the vector containing the additive timevarying sensor faults affecting the q faulty sensors.

Furthermore, consider the fault dynamics represented by Equations (7) and (8). Such a representation can be modified to address

the specific case of sensor faults as

x ̇ s(t) = Asxs(t) + ωps(t), xs(0) = xs0, (24)

ys(t) = Csxs(t) + ωss(t), (25)

where As ∈ R

ns×ns

, Cs ∈ R

q×ns

, xs(t) ∈ R

ns

, ys(t) ∈ R

q

, and ωps(t) ∈

R

ns and ωss

(t) ∈ R

q are exogenous sensor fault process and measurement

noise, respectively. Observe that Equations (24) and (25) model the

sensor faults present in Equation (23). Thus, Equations (21) and (23)

6
can be viewed as an LTI system with q of its outputs affected by timevarying sensor faults given by the output of the exogenous LTI system in

Equations (24) and (25). The interconnected system can be represented

in augmented state-space form by letting ωs(t) = 

ω

T

s,1

(t) ω

T

s,2

(t)

T

and by substituting Equation (25) into Equation (23) as follows



x ̇(t)

x ̇ s(t)



=



A 0n×ns

0ns×n As



| {z }

Aη



x(t)

xs(t)



| {z }

η(t)

+



B

0ns×m



| {z }

Bη

u(t) + 

ωp(t)

ωps(t)



| {z }

ωηp(t)

,

(26)

y(t) = 

C1 0(l−q)×n

C2 Cs



| {z }

Cη



x(t)

xs(t)



+



ωs1(t)

ωs2(t) + ωss(t)



| {z }

ωηs(t)

, (27)

where by making the appropriate substitutions indicated by the braces,

the system above can be expressed more compactly as

η ̇(t) = Aηη(t) + Bηu(t) + ωηp(t), (28)

y(t) = Cηη(t) + ωηs(t). (29)

Equations (28) and (29) model the general case of additive, time-varying

sensor faults treated in this paper.

In the case of sensor step faults, As = 0q×q, and Cs = Iq. Thus, when

all of the sensor faults are step faults, Equations (26) and (27) reduce to



x ̇(t)

x ̇ s(t)



=



A 0

0 0   x(t)xs(t)



+



B

0



u(t) + 

ωp(t)

ωps(t)



, (30)

y(t) = 

C1 0

C2 Iq

  x(t)

xs(t)



+



ωs1(t)

ωs2(t) + ωss(t)



. (31)

Equations (30) and (31) are identical to Equations (17) and (18) in [22],

where the case of sensor step faults was considered.

3.4 Simultaneous Actuator and Sensor Fault Modeling

Consider the case of time-varying actuator faults represented by Equations (14) and (15) and the case of time-varying sensor faults represented

by Equations (24) and (25). Now, in order to represent simultaneous,

additive, actuator and sensor faults, it is sufficient to augment the states

in the form 

x(t)

T xa(t)T xs(t)T

T

. Thus, when q of the l sensors

and mk of the m actuators are affected by time-varying faults, the aug7
mented system can be expressed as





x ̇(t)

x ̇ a(t)

x ̇ s(t)



 =





A B

k

Ca 0n×ns

0na×n Aa 0na×ns

0ns×n 0ns×na As





| {z }

Aφ





x(t)

xa(t)

xs(t)





| {z }

φ(t)

+





Bk

0

0





| {z }

Bφ

u

k

(t)

+





B

k

ωsa

(t) + ωp(t)

ωpa(t)

ωps(t)





| {z }

ωφp(t)

, (32)



y1(t)

y2(t)



=



C1 0(l−q)×na0(l−q)×ns

C2 0q×na Cs



| {z }

Cφ





x(t)

xa(t)

xs(t)





+



ωs,1(t)

ωs,2(t) + ωss(t)



| {z }

ωφs(t)

, (33)

where after making the appropriate substitutions indicated by the braces,

the system above can be expressed compactly as

φ ̇(t) = Aφφ(t) + Bφu

k

(t) + ωφp(t), (34)

y(t) = Cφφ(t) + ωφs(t). (35)

Equations (34) and (35) model the general case of simultaneous, additive,

time-varying actuator and sensor faults treated in this paper.

For the case when all actuator and sensor faults are step faults, Aa =

0mk×mk, As = 0q×q, Ca = Imk, and Cs = Iq. Then, Equations (32)

and (33) reduce to





x ̇(t)

x ̇ a(t)

x ̇ s(t)



 =





A B

k

0n×q

0mk×n 0mk×mk0mk×q

0q×n 0q×mk0q×q









x(t)

xa(t)

xs(t)



 +





Bk

0

0



 u

k

(t)

+





B

k

ωsa(t) + ωp(t)

ωpa(t)

ωps(t)



 , (36)



y1(t)

y2(t)



=



C1 0(l−q)×mk0(l−q)×q

C2 0q×mkIq







x(t)

xa(t)

xs(t)





+



ωs,1(t)

ωs,2(t) + ωss(t)



. (37)

Equations (36) and (37) are identical to Equations (31) and (33) presented in [22], where the case of simultaneous actuator and sensor step

faults was considered.

8
4 Conditions for Identifiability of Time-Varying

Actuator and Sensor Faults

The main results of this paper are developed in this section, that is, a set

of necessary and sufficient conditions for the identifiability of additive,

time-varying faults affecting combinations of

(1) actuators only,

(2) sensors only, and

(3) actuators and sensors, simultaneously.

The conditions are presented as three separate theorems accompanied

by a proof for each of the indicated fault configurations. It is further

shown through corollaries that, when all of the faults are step faults, a

set of necessary and sufficient conditions for items (1), (2), and (3) above

reduce to those presented in [22].

Mathematically, a fault is identifiable if the augmented system is

detectable. However, in some practical applications, observability is preferred [22]. This paper presents conditions for both detectability and

observability, where satisfying either property shall imply identifiability,

in the weak and the strong sense, respectively. Before the conditions

for identifiability and the associated proofs are given, several relevant

assumptions are addressed.

4.1 Assumptions

It is assumed in the proofs in this section that the following rank conditions are satisfied:

(A1) rank(B

k

) = mk, where B

k

corresponds to the columns of B associated with the mk faulty actuators,

(A2) rank(C) = l,

(A3) rank(Ca) = mk, where Ca represents the output matrix associated with an exogenous LTI system which generates the mk timevarying actuator fault signals,

(A4) rank(Cs) = q, where Cs represents the output matrix associated

with an exogenous LTI system which generates the q time-varying

sensor fault signals,

(A5) rank(C1) = l−q, where C1 corresponds to the rows of C associated

with the l − q non-faulty sensors, and

(A6) rank(C2) = q, where C2 corresponds to the rows of C associated

with the q faulty sensors.

9
Assumptions (A1) and (A2) are taken directly from [22], that is: (1) it

is assumed that inputs associated with linearly dependent columns of

B

k

have been aggregated and that B

k

is full column rank, and (2) it

is assumed that outputs associated with linearly dependent rows of C

have been aggregated and that C is full row rank. If (A1) or (A2) do not

hold, faults associated with linearly dependent columns of B

k

, or linearly

dependent rows of C will not be uniquely identifiable. Similarly, Assumptions (A3) and (A4) follow from the representation of time-varying

actuator and sensor faults shown in Sections 3.2 and 3.3, respectively.

Finally, Assumptions (A5) and (A6) follow from Assumption (A2).

4.2 Time-Varying Actuator Fault Identifiability

Before the main conditions are provided, a preliminary notation is developed, and two useful lemmas are provided. First, let Σ(s) denote the

RSM given by

"

sI − A −B

k

C 0

#

,

and let Γξ(s) ⊂ C

n+mk denote the right nullspace of Σ(s), where s ∈

C. Furthermore, let Γξ

mk

(s) ⊂ C

mk denote the subspace spanned by

the last mk components of a basis for Γξ(s). The elements of Γξ

mk

(s)

are characterized next in terms of (extended) invariant zeros and inputzero directions. For a recent treatments of these objects in the context

intended in this paper, see [27].

1. When s is an (extended) invariant zero but not an outputdecoupling zero of Σ(s), then Γξ

mk

(s) = G

ξ

mk

(s)∪{0}, where G

ξ

mk

(s)

is the set of all input-zero directions of Σ(s),

2. When s is an (extended) invariant zero and an output-decoupling

zero of Σ(s), then Γξ

mk

(s) = G

ξ

mk

(s) = {0} is the only input-zero

direction of Σ(s), and

3. When s is not an invariant zero of Σ(s), then Γξ

mk

(s) = G

ξ

mk

(s) is

the subspace spanned by all of non-input-zero directions of Σ(s).

Lemma 1. The pair (Ca, Aa) is detectable (observable) if and only if the

pair (B

k

Ca, Aa) is detectable (observable).

Proof. Observe that the pair (B

k

Ca, Aa) is detectable (observable) if and

only if

rank "

sI − Aa

B

k

Ca

#

= na for s ∈ Λu(Aa) (s ∈ Λ(Aa)),

rank "

In 0

0 B

k

# 

sI − Aa

Ca



= na for s ∈ Λu(Aa) (s ∈ Λ(Aa)).

10
By Sylvester’s inequality (see [28]) and Assumptions (A1) and (A3),

rank 

sI − Aa

Ca



≤ rank "

In 0

0 B

k

Ca

# 

sI − Aa

Ca



≤ rank 

sI − Aa

Ca



.

Therefore, rank "

sI − Aa

B

k

Ca

#

= rank 

sI − Aa

Ca



for s ∈ Λu(Aa) (s ∈

Λ(Aa)), so that the pair (Ca, Aa) is detectable (observable) if and only

if the pair (B

k

Ca, Aa) is detectable (observable).

Lemma 2. The pair (C

k

ξ

, Ak

ξ

) is detectable (observable) if and only if all

of the following conditions are satisfied:

(i) the pair (C, A) is detectable (observable),

(ii) (Ca, Aa) is detectable (observable), and

(iii) for (λa, v) an (eigenvalue, eigenvector) pair of Aa with λa ∈

Λu(Aa) (λa ∈ Λ(Aa)), Cav /∈ Γ

ξ

mk

(λa).

Proof. Applying the PBH eigenvector test, the pair (Cξ, Aξ) is detectable

(observable) if and only if





sI − A −B

k

Ca

0na×n sI − Aa

C 0l×na







ζ

v



= 0

is satisfied only by the trivial solution (that is, 

ζ

T vT

T = 0) for s ∈

Λu(A) ∪ Λu(Aa) (for s ∈ Λ(A) ∪ Λ(Aa)). The first n columns of the

PBH test matrix are independent for s ∈ Λu(A) (for s ∈ Λ(A)) if and

only if the pair (C, A) is detectable (observable). The last na columns

are independent for s ∈ Λu(Aa) (for s ∈ Λ(Aa)) if and only if the pair

(Ca, Aa) is detectable (observable) (see Lemma 1). For s /∈ Λ(Aa) the

last na columns are independent of the first n columns. For s = λa ∈

Λu(Aa) (s = λa ∈ Λ(Aa)) the last na columns are independent of the

first n columns if and only if

"

λaI − A −B

k

C 0

#

| {z }

Σ(λa)



ζ

Cav



= 0 (38)

does not have a nontrivial solution for (λa, v) an (eigenvalue, eigenvector)

pair of Aa, that is, if and only if Cav /∈ Γ

ξ

mk

(λa).

The following theorem provides a necessary and sufficient condition for

the identifiability of additive, time-varying actuator-only faults.

11
Theorem 1. The pair (C

k

ξ

, Ak

ξ

) is detectable (observable) if and only if

all of the following conditions are satisfied:

(i) for (λa, v) an (eigenvalue, eigenvector) pair of Aa with λa ∈

Λu(Aa) (λa ∈ Λ(Aa)), when l < mk and when λa is not an invariant zero of Σ(λa), then Cav /∈ G

ξ

mk

(λa),

(ii) the pair (C, A) is detectable (observable),

(iii) the pair (Ca, Aa) is detectable (observable), and

(iv) for (λa, v) an (eigenvalue, eigenvector) pair of Aa with λa ∈

Λu(Aa) (λa ∈ Λ(Aa)), when λa is an invariant zero of Σ(λa), then

Cav /∈ Gξ

mk

(λa).

Proof. The pair (C

k

ξ

, Ak

ξ

) is detectable (observable) if and only if Conditions (i)-(iii) of Lemma 2 are satisfied. Thus, Conditions (ii) and (iii)

of Theorem 1 are established. It remains only to be shown that Conditions (i) and (iv) of Theorem 1 now hold if and only if Condition (ii) of

Lemma 2 holds. Now, suppose that Conditions (i) and (iv) of Theorem 1

hold and that Condition (iii) of Lemma 2 does not. Then Equation (38)

has a nontrivial solution. The following two possible cases will be considered and shown to result in a contradiction:

1. rank{Σ(λa)} < rank{Σ(s)}, and

2. rank{Σ(λa)} = rank{Σ(s)} with

(a) rank{Σ(s)} = n + min{mk, l}, or

(b) rank{Σ(s)} < n + min{mk, l}.

Case 1 leads to a nontrivial solution if and only if (λa, Cav) is an (invariant zero, input-zero direction) pair of Σ(λa), a contradiction of Condition

(iv) of Theorem 1, that is, Cav ∈ Gξ

mk

(λa). Case 2(a) leads to a nontrivial solution if and only if l < mk and Cav ∈ G

ξ

mk

(λa), a contradiction of

Condition (i) of Theorem 1. Case 2(b) implies that Σ(s) is degenerate,

that is, every λa ∈ C is an invariant zero Σ(s). Thus, Case 2(b) leads

to a nontrivial solution if and only if Cav ∈ Gξ

mk

(λa), a contradiction of

Condition (iv) of Theorem 1. Thus, Conditions (i)-(iv) of Theorem 1 are

necessary and sufficient for identifiability of the pair (C

k

ξ

, Ak

ξ

).

The following corollary considers the case when the geometric multiplicity of λa is equal to na. One such instance is when Aa = 0na×na

, (a

step fault in na = mk actuators).

Corollary 1. Let λa ∈ Λu(Aa) (λa ∈ Λ(Aa)) have geometric multiplicity

γa. If γa = na then Condition (iii) in Lemma 2 becomes

(i) l ≥ mk, and

12
(ii) Σ(s) has no invariant zeros at s = λa.

Proof. Let g ∈ Γ

ξ

mk

(λa), and consider whether there exists v ∈ R

na

such that Cav = g, where v is an eigenvector of Aa in the associated

eigenspace W, with dim(W) = γa. Next, observe that Ca : R

na → Rmk

is a surjective linear transformation since Ca has full row rank, that is,

rank(Ca) = mk (see Assumption A3). Therefore, for any g ∈ R

mk , there

always exists a vector τ ∈ C

na such that Caτ = g. Now, when γa = na,

W = C

na, and any nonzero vector in C

na is an eigenvector. Thus, for

any nonzero solution τ , let v = τ , and it follows that Cav = g. Therefore,

when λa has geometric multiplicity γa = na, Σ(λa) must have full column

rank so that Γξ

mk

(λa) = {0}, implying that l ≥ mk and that λa is not an

invariant zero of Σ(s)

Corollary 2. For the special case when all of the faults are constant

biases, that is, a step fault in each of the mk faulty actuators, Conditions

(i)-(iv) of Theorem 1 reduce to:

(i) l ≥ mk,

(ii) the pair (C, A) is observable, and

(iii) Σ(s) has no invariant zeros at s = 0.

Proof. First, observe that for step faults, na = mk, Aa = 0na×na and

Ca = Ina×na. Thus, Aa has one distinct eigenvalue at zero with geometric

multiplicity na, that is, any nonzero vector v ∈ R

na is an eigenvector

of Aa, and by Corollary 1, Conditions (i) and (iii) of Corollary 2 are

established. Furthermore, since (Ca, Aa) is observable, Condition (iii) of

Theorem 1 is not needed. The detectability (observability) requirement

for the pair (C, A) is now strictly an observability requirement, since

s = 0 does not lie in the open, left-half plane, thus establishing Condition

(ii) of Corollary 2.

Remark 1. The conditions in Corollary 2 are identical to the conditions

for constant, bias-type actuator fault identifiability presented in Theorem

1 in [22].

Example 1. Consider the fourth-order, linearized longitudinal dynamics model used in [17] and fully examined for additive, bias-type fault

identifiability in [22]. The model represents a large, transport aircraft in

13
wings-level, cruise condition and is given by

A =









−0.6803 0.0115 −1.0490 0

−0.0026 −0.0062 −0.0815 −0.1709

1.0050 −0.0344 −0.5717 0

1 0 0 0









;

B =









−44.5192 0

−11.4027 0

0.88254 1.3287

−0.0401 0









. (39)

The state vector consists of the pitch rate, forward speed, angle of attack,

and pitch angle, that is, x(t) = 

q(t) v(t) α(t) θ(t)

T

. The control

vector consists of the elevator deflection and engine thrust, that is, u(t) =



ue(t) uT (t)

T

.

Suppose that one must identify sinusoidal faults having frequency

ω = 1 radian per second in the elevator actuator and bias-type faults

having any amplitude in the thrust actuator (i.e., a “stuck actuator”

fault). Observe that there are 45 possible fault configurations for the

particular faults of interest, where either or both actuators are faulty

(

P4

kl=1 

4

k



sensor configurations, where for each configuration there are

3 actuator fault configurations possible). Out of these, there are two

cases of non-identifiability, both due to violation of Condition (iv) of

Theorem 1. The discussion of these two cases follows.

Case 1: y(t) = q(t) and both actuators are affected by their respective fault type of interest, λa = 0 is both an eigenvalue of Aa and an

invariant zero of the triple (C, A, B

k

). Furthermore, it can be verified

that n

1 0 T

,



0 1 T

o

is a basis for G

ξ

mk

, therefore for any nonzero

vector Cav in R

2

it follows that Cav ∈ Gξ

mk

.

Case 2: y(t) = q(t) and the thrust actuator becomes stuck. Again

λa = 0 is both an eigenvalue of Aa and an invariant zero of the triple

(C, A, B

k

), and in this case it can further be verified that for any Cav ∈

R, Cav ∈ Gξ

mk

.

An interesting observation is that in three of the 45 possible fault

cases, both actuators are faulty and l < mk, yet the faults are still identifiable, that is, all of the conditions of Theorem 1 hold. This situation

is different from the case when all the faults are “stuck actuators” as

described in [22], where it is required that l ≥ mk.

4.3 Time-Varying Sensor Fault Identifiability

This section presents conditions for time-varying sensor fault identifiability. Consider the augmented system given by Equations (26) and (27),

the model of an arbitrary LTI system subject to time-varying sensor

faults. The identifiability of such a fault requires that the pair (Cη, Aη)

14
be detectable (observable). The following theorem gives a necessary and

sufficient condition for identifiability of time-varying sensor faults.

Theorem 2. The pair (Cη, Aη) is detectable (observable) if and only if

all of the following conditions are satisfied:

(i) the pair (C, A) is detectable (observable),

(ii) the pair (Cs, As) is detectable (observable), and

(iii) when (λs, ζ) and (λs, ψ) are eigenvalue, eigenvector pairs of A and

As, respectively, and λs is not a detectable (observable) eigenvalue

of the pair (C1, A), then C2ζ 6= αCsψ, where α ∈ C.

Proof. The pair (Cη, Aη) is detectable (observable) if and only if

rank













sI − A 0n×ns

0ns×n sI − As

C1 0(l−q)×ns

C2 Cs











 = n + ns

for s ∈ Λu(A) ∪ Λu(As) (for s ∈ Λ(A) ∪ Λ(As)). The first n columns

of the PBH test matrix are linearly independent for all s ∈ Λu(A) (for

all s ∈ Λ(A)) if and only if (C, A) is detectable (observable). The last

ns columns are linearly independent for s ∈ Λu(As) (for s ∈ Λ(As)) if

and only if (Cs, As) is detectable (observable). Furthermore, whenever

s /∈ Λ(A)∪Λ(As) the last ns columns are linearly independent of the first

n columns. Now, let s = λs ∈ Λu(As) ∪ Λu(A) (s = λs ∈ Λ(As) ∪ Λ(A)).

The last ns columns are linearly independent from the first n columns if

and only if









λsI − A 0n×ns

0ns×n λsI − As

C1 0(l−q)×ns

C2 Cs











ζ

ψ



= 0 (40)

only for 

ζ

T ψT

T = 0. Now, suppose there exists 

ζ

T ψT

T

6= 0

such that (40) is satisfied. Then

(λsI − A)ζ = 0 (41)

(λsI − As)ψ = 0 (42)

C1ζ = 0 (43)

C2ζ + Csψ = 0, (44)

must hold. Since the pair (C, A) is detectable (observable) ψ 6= 0, and

since the pair (Cs, As) is detectable (observable) ζ 6= 0. Observe that

Equation (42) is satisfied since ψ is nonzero and, thus, an eigenvector of

As associated with λs. The remaining equations may be expressed as



λsI − A

C1



ζ = 0, (45)

C2ζ + Csψ = 0. (46)

15
If λs is not a detectable (observable) eigenvalue of (C1, A) then Equation (45) is satisfied. Now, Equation (46) is satisfied if and only if

C2ζ 6= αCsψ, where α ∈ C. Thus, Conditions (i)-(iii) of Theorem 2

are necessary and sufficient for identifiability of the pair (Cη, Aη).

The following corollary considers the special case when λs has geometric multiplicity equal to ns.

Corollary 3. Let λs ∈ Λu(As) ∩ Λu(A) (s ∈ Λ(As) ∩ Λ(A)) have geometric multiplicity equal to γs, and let ψ be an eigenvector of As from

the associated eigenspace V corresponding to the eigenvalue λs. Observe

that dim(V ) = γs. If γs = ns then Condition (iii) in Theorem 2 becomes

(i) the pair (C1, A) is detectable (observable) with respect to s = λs.

Proof. Observe that Cs : R

ns → Rq

is a surjective linear transformation

since rank(Cs) = q (see Assumption (A4)). Therefore, there always

exists a vector, p ∈ C

ns

, such that Csp = −C2ζ. Now, when γs = ns,

V = C

ns

, and any vector in C

ns

is an eigenvector. Then, for any solution

p let ψ = p, and it follows that Csψ = −C2ζ. That is, whenever γs = ns,

Equation (46) is satisfied. Thus, it is required that λs be a detectable

(observable) eigenvalue of the pair (C1, A).

Corollary 4. For the special case when all of the faults are constant

biases, that is, step faults, Conditions (i)-(iii) in Theorem 2 reduce as

follows:

(i) the pair (C, A) is observable, and

(ii) s = 0 corresponds to an observable mode of the pair (C1, A).

Proof. First, observe that Condition (i) of Theorem 2 is identical to Condition (i) of Corollary 4. Now, As = 0 and Cs = I. Thus, Condition (ii)

of Theorem 2 is no longer necessary, since the pair (Cs, As) is observable.

Furthermore, note that the only eigenvalue of Aa is s = 0, having algebraic and geometric multiplicity q = ns, and by Corollary 3, Condition

(ii) of Corollary 4 is established, that is, if (C1, A) is not observable when

s = 0 then the pair is not detectable.

Note that Conditions (i) and (ii) of Corollary 4 are exactly those

presented in Theorem 2 of [22] for the sensor step fault case.

Example 2. Consider the 6th-order linearized longitudinal dynamics for

the Cranfield A3 Observer, a fixed-wing research UAV presented in [29].

The UAV is in cruise condition, and the airframe is in a gust-insensitive

16
configuration. The dynamics are given by

A =

















−0.146 −0.016 0.557 −9.809 0 0.001

−0.63 −4.487 34.57 0.161 0 0

0.001 0.039 −0.894 0 0 0

0 0 1 0 0 0

−0.016 −1 0 35.2 0 0

665.7 −6.89 0 0 0 −8.57

















;

B =

















0 −1.368

0 −19.96

0 −15.96

0 0

0 0

45910 0

















,

and the output is specified as

C =













1 −0.014 0.019 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0.984 0

0 0 0 0 0 1













.

The state vector consists of the forward speed, vertical speed,

pitch rate, pitch angle, altitude, and engine rpm, that is, x(t) =



v(t) w(t) q(t) θ(t) h(t) NE(t)

T

. The control vector consists of

engine thrust and elevator deflection, that is, u(t) = 

uT (t) ue(t)

T

.

The output vector consists of the measured speed error, pitch rate,

pitch angle, perturbed altitude, and engine rpm, that is, y(t) =



ve(t) q(t) θ(t) he(t) NE(t)

T

. It can be verified that Λ(A) =

{0, −4.8345, 1.9641 · 10−1, −4.0534 · 10−1 ± i2.0428 · 10−1, −8.6482} and

that the system realization {A, B, C, 0} is minimal, that is, the system

is both controllable and observable.

Suppose that it is of interest to identify drift faults in the sensors,

which may be modeled as ramp faults. In the absence of actuator faults,

there are 31 possible sensor fault configurations, excluding the faultfree case. Of these, there are 16 cases of non-identifiability, and all 16

cases of non-identifiability are associated with at least a drift fault in the

perturbed altitude measurement. Furthermore, any drift faults in any

combination of the other measurements are always identifiable, provided

that the altitude measurement is not faulty.

It can be verified that the pairs (C, A) and (Cs, As) are always observable. Furthermore, it can be verified that zero is a common eigenvalue

of A and any form of As for the fault configurations of interest, and that

the eigenvalue at zero corresponds to an unobservable mode of the pair

(C1, A) whenever y4(t) is faulty. The test of whether or not αCsψ = C2ζ

17
for α ∈ C is unnecessary, since the eigenspace of As associated with

λs = 0 is such that there always exists ψ which satisfies the equality.

Thus, for the 16 cases of non-identifiability, Condition (iii) of Theorem 2

fails.

4.4 Simultaneous Time-Varying Actuator and Sensor

Fault Identifiability

Before presenting the conditions for identifiability for simultaneous actuator and sensor faults, a preliminary notation is given. First, let Σ(s)

denote the RSM given by

"

sI − A −B

k

C1 0

#

.

Thus, the RSM associated with Σ(s) is to be understood in the context

of the fault configuration of interest, i.e., whether it is of interest to

identify faults in the actuators alone or in combination with sensors.

Furthermore, let Γφ(s) ⊂ C

n+mk denote the right nullspace of Σ(s),

where s ∈ C, and let Γφ

mk

(s) ⊂ C

mk denote the subspace spanned by the

last mk components of a basis for Γφ(s). The elements of Γφ

mk

(s) may be

characterized similarly to those of Γξ

mk

(s) in Section 4.2, where G

φ

mk

(s)

corresponds to the input-zero directions, and G

φ

mk

(s) corresponds to the

non-input-zero-directions.

Theorem 3. The pair (Cφ, Aφ) is detectable (observable) if and only if

all of the following are satisfied:

(i) The pair (C

k

ξ

, Ak

ξ

) is detectable (observable),

(ii) The pair (Cη, Aη) is detectable (observable),

(iii) for λa,s ∈ Λu(Aa)∩Λu(As) (λa,s ∈ Λ(Aa)∩Λ(As)) not an invariant

zero of Σ(s) with v and ψ eigenvectors of Aa and As associated with

λa,s, respectively, when l < mk + q then either Cav /∈ G

φ

mk

(λa,s) or

C2ζ 6= αCsψ, for α ∈ C,

(iv) for λa,s ∈ Λu(Aa) ∩ Λu(As) (λa,s ∈ Λ(Aa) ∩ Λ(As)) an invariant

zero of Σ(s), with v and ψ are eigenvectors of Aa and As associated

with λa,s, respectively, either Cav /∈ Gφ

mk

(λa,s) or C2ζ 6= αCsψ, for

α ∈ C.

Proof. The pair (Cφ, Aφ) is detectable (observable) if and only if the

equation













sI − A −B

k

Ca 0n×ns

0na×n sI − Aa 0na×ns

0ns×n 0ns×nasI − As

C1 0(l−q)×na0(l−q)×ns

C2 0q×na Cs

















ζ

v

ψ



 = 0, (47)

18
is satisfied only by the trivial solution for s ∈ Λu(A) ∪ Λu(Aa) ∪ Λu(As)

(for s ∈ Λ(A) ∪ Λ(Aa) ∪ Λ(As)). All of the possible solution cases to be

considered are enumerated in Table 1 and treated subsequently. Observe

Case ζ = 0 v = 0 ψ = 0

(a) T T T

(b) T T F

(c) T F T

(d) T F F

(e) F T T

(f) F T F

(g) F F T

(h) F F F

TABLE 1: General form of possible solutions to Equation (47).

that Case (a) represents the trivial solution, and so is always a solution

to Equation (47). Case (b) can be reduced to considering the equation



sI − As

Cs



ψ = 0. (48)

A nontrivial solution to Equation (47) exists for Case (b) if and only if

Equation (48) is also nontrivially satisfied. That is, if the pair (Cs, As)

is not detectable (observable), which corresponds to Condition (i) of

Theorem 2.

Case (c) can be reduced to considering the equation

"

sI − Aa

−B

k

Ca

#

v = 0. (49)

A nontrivial solution to Equation (47) exists for Case (c) if and only if

Equation (49) is also nontrivially satisfied. By Lemma 1, this is possible

only in the case when the pair (Ca, Aa) is not detectable (observable),

which corresponds to Condition (iii) of Theorem 1.

Case (d) can be reduced to considering the equation









−B ̄kCa 0

sI − Aa 0

0 sI − As

0 Cs











v

ψ



= 0. (50)

However, Equation (50) can be further represented as two separate equations, which must be simultaneously satisfied (i.e., Equations (48) and

(49)), thus reestablishing the conditions covered under Cases (b) and (c),

respectively.

19
Case (e) represents a nontrivial solution to Equation (47) if and only

if the equation



sI − A

C



ξ = 0, (51)

has a nontrivial solution. Such a solution exists for Case (e) if and

only if the pair (C, A) is not detectable (observable). This condition is

addressed in Conditions (ii) and (i) of Theorems 1 and 2, respectively.

Case (f) can be reduced to considering the following equation.









sI − A 0

0 sI − As

C1 0

C2 Cs











ζ

ψ



= 0. (52)

By inspection of Equation (52), a nontrivial solution exists if either or

both of the pairs (C, A) and (Cs, As) are not detectable (observable).

These cases are covered through of Condition (ii) of Theorem 1 and

Conditions (i) and (ii) of Theorem 2. Furthermore, a nontrivial solution

to Equation (52) (where both ζ and ψ are nonzero) still exists if: 1) s =

λs ∈ Λ(As), where ψ is a corresponding a eigenvector, 2) λs corresponds

to an undetectable (unobservable) eigenvalue of the pair (C1, A), where

ζ is a corresponding eigenvector, and 3) C2ζ ∈ αCsψ, where α ∈ C. The

situation of 1) through 3) for Case (f) is addressed in Condition (iii) of

Theorem 2.

Case (g) can be reduced to considering the following equation.





sI − A −B ̄kCa

0 sI − Aa

C 0







ζ

v



= 0. (53)

Inspection of Equation (53) reveals a nontrivial solution if either pair or

both pairs, (C, A) and (Ca, Aa), are not detectable (observable). Conditions (ii) and (iii) of Theorem 1 and Condition (i) of Theorem 2 together

address such a solution. Now, a nontrivial solution to Equation (53)

still exists if: 1) s = λa ∈ Λ(Aa) and 2) either Condition (i) or (iv) of

Theorem 1 are violated.

Cases (a)-(g) are addressed entirely by Theorems 1 and 2, where it

is furthermore the case that all conditions have been applied (i.e., no

conditions remain unused). Thus, Conditions (i) and (iii) of Theorem 3

are established.

The only remaining case to consider is the situation in Case (h),

where ζ, v, and ψ are all nonzero vectors. Considering a nontrivial solution to Equation (47), it is necessary that s = λa,s ∈ Λ(Aa) ∩ Λ(As),

where v and ψ are corresponding eigenvectors. Under these assumptions,

20
Equation (47) can be reduced and expressed as









λa,sI − A −B

k

0

C1 0 0

C2 0 Iq















ζ

Cav

Csψ







= 0. (54)

Equation (54) has a nontrivial solution if and only if either:

(h.1) for λa,s ∈ Λu(Aa)∩Λu(As) (λa,s ∈ Λ(Aa)∩Λ(As)) not an invariant

zero of Σ(C1,A,B

k

)

(s), when l < mk + q then both Cav ∈ G

φ

mk

(λa,s)

and C2ζ = αCsψ, where ζ ∈ C

n and α ∈ C, or

(h.2) for λa,s ∈ Λu(Aa) ∩ Λu(As) (λa,s ∈ Λ(Aa) ∩ Λ(As)) an invariant

zero of Σ(C1,A,B

k

)

(s), both Cav ∈ Gφ

mk

(λa,s) and C2ζ = αCsψ,

where ζ ∈ C

n and α ∈ C.

Thus, Conditions (iii) and (iv) of Theorem 3 are established, and Conditions (i)-(iv) are together a necessary and sufficient condition for detectability (observability) of the pair (C

φ, Aφ).

Corollary 5. For the special case when all of the simultaneous faults

are constant biases, i.e., step faults, Conditions (i)-(iv) of Theorem 3

reduce as follows.

(i) the pair (C, A) is observable,

(ii) l ≥ mk + q, and

(iii) Σ(s) has no invariant zeros at the origin.

Proof. Observe that when all the actuator and sensor faults are step

faults, Equation (47) can be expressed as













sI − A −B

k

0

0 sImk0

0 0 sIq

C1 0 0

C2 0 Iq

















ζ

v

ψ



 = 0. (55)

First, observe the requirement that the pair (C, A) be observable (since

s = 0 does not lie strictly in the open, left-half plane). This requirement

ensures that the first n columns are linearly independent. Letting s = 0,

Equation 55 can be expressed in three equations as

−Aζ − B

k

v = 0 (56)

C1ζ = 0 (57)

C2ζ + Iqψ = 0 (58)

21
By inspection, Equation 58 is satisfied for any ζ. Thus, it is required

that the remaining two equations be satisfied only by the trivial solution.

The remaining two equations can be expressed as

"

−A −B

k

C1 0

# 

ζ

v



= 0. (59)

Now, by appeal to Corollary 2 and the symmetry of the problem, the

conditions that l ≥ mk + q and Σ(s) has no invariant zeros at the origin

follows.

Remark 2. Conditions (i)-(iii) of Corollary 5 are identical to those presented in Theorem 3 of [22] for the simultaneous actuator and sensor

step fault case.

Example 3. Consider the 4th-order, linearized vertical-plane dynamics

of a vertical takeoff and landing (VTOL) aircraft, flying in the airspeed

range of 60-170 knots, given in [21] and [30] given by

A =









−0.0336 0.0271 0.0188 −0.4555

0.0482 −1.0100 0.0024 −4.0208

0.1002 0.2855 −0.7070 1.3229

0 0 1 0









;

B =









0.4422 0.1761

3.0447 −7.5922

−5.5200 4.9900

0 0









;

C =



1 0 0 0

0 1 0 0 

.

The state vector is comprised of the horizontal velocity, vertical velocity,

pitch rate, and pitch angle, that is, x(t) = 

v w q θ T

. The control

vector is comprised of the collective pitch angle and longitudinal cyclic

pitch angle. The collective pitch angle input controls the vertical motion, and the longitudinal cyclic pitch angle input controls the horizontal

velocity [21]. The output vector is comprised of the horizontal velocity

and vertical velocity, that is, y(t) = 

v w T

. It can be verified that

Λ(A) = {2.8174 · 10−1 ± i9.7701 · 10−2, −3.3318 · 10−1, −1.9809}, and

that {A, B, C, 0} is a minimal realization, that is, both controllable and

observable.

For the case when the actuators are subject to ramp faults, and the

sensors are subject to step faults, there are a total of nine possible fault

configurations having simultaneous faults. A check of the conditions in

Theorem 3 show that

1. Conditions (i) and (ii) are satisfied,

22
2. for λa,s ∈ Λ(Aa) ∩ Λ(As) = {0} (where λa,s is not an invariant

zero of (C1, A, B

k

)), there are five cases when l < mk + q, i.e., the

one case when all actuators and senors are faulty, the two cases

when both actuators are faulty and either of the sensors are faulty,

and the two cases when both sensors are faulty and either of the

actuators are faulty. In all five cases, Cav ∈ G

φ

mk

(0). For each of

these cases, since the geometric multiplicity of the zero eigenvalue

of As is ns = q, it is always the case that C2ζ = αCsψ, where

α ∈ C. Thus, Condition (iii) of Theorem 3 is not satisfied for these

five cases, and the faults are not identifiable,

3. l ≥ m, therefore Condition (iv) of Theorem 3 is satisfied, and

4. for the four cases when Σ(s) has invariant zeros, that is when only

one or the other measurement is biased in conjunction with only

one or the other input being faulty, none of the invariant zeros are

at the origin. Thus for all four such fault cases, Condition (iv) of

Theorem 3 is satisfied.

Thus, for the case when it is of interest to identify ramp faults in the

actuators together with constant bias in the sensors, there are five fault

configurations which are theoretically not possible to identify using state

augmentation. In such cases, other identification methods need to be

considered.

5 Conclusions and Future Research

The fundamental problem addressed in this paper was the determination

of a theoretical set of necessary and sufficient conditions for identifiability

of arbitrary combinations of a class of additive, time-varying actuator

and sensor faults using state augmentation. The provided theorems give

the conditions which must be satisfied in order for a practical stateaugmentation-based solution to exist. One recommendation for future

work on this topic is to investigate identifiability of multiplicative faults

in the actuators and sensors. Such faults may manifest themselves as a

loss-of-effectiveness in the actuators and sensors. One relevant example

of such faults is a reduction in thrust provided by a propulsion system.

23
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26
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Identifiability of Additive, Time-Varying Actuator and Sensor Faults by 

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Upchurch, Jason M.; Gonzalez, Oscar R.; Joshi, Suresh M.

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14. ABSTRACT

Recent work has provided a set of necessary and sufficient conditions for identifiability of additive step faults (e.g., lock-in-place actuator faults, constant bias 

in the sensors) using state augmentation. This paper extends these results to an important class of faults which may affect linear, time-invariant systems. In 

particular, the faults under consideration are those which vary with time and affect the system dynamics additively. Such faults may manifest themselves in 

aircraft as, for example, control surface oscilations, control surface runaway, and sensor drift. The set of necessary and sufficient conditions presented in this 

paper are general, and apply when a class of time-varying faults affects arbitrary combinations of actuators and sensors. The results in the main theorems are 

illustrated by two case studies, which provide some insight into how the conditions may be used to check the theoretical identifiability of fault configurations 

of interest for a given system. It is shown that while state augmentation can be used to identify certain fault configurations, other fault configurations are 

theoretically impossible to identify using state augmentation, giving practicioners valuable insight into such situations. That is, the limitations of state 

augmentation for a given system and configuration of faults are made explicit. Another limitation of model-based methods is that there can be large numbers of 

fault configurations, thus making identification of all possible configurations impractical. However, the theoretical identifiability of known, credible fault 

configurations can be tested using the theorems presented in this paper, which can then assist the efforts of fault identification practicioners. 

15. SUBJECT TERMS

Fault detection; Identifiability

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