October 2006

NASA/CR-2006-213925 

NIA Report No. 2005-07 

Gain-Scheduled Fault Tolerance Control Under 

False Identification 

Jong-Yeob Shin 

National Institute of Aerospace, Hampton, Virginia 
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October 2006

NASA/CR-2006-213925 

NIA Report No. 2005-07 

Gain-Scheduled Fault Tolerance Control Under 

False Identification 

Jong-Yeob Shin 

National Institute of Aerospace, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 

7121 Standard Drive 5285 Port Royal Road 

Hanover, MD 21076-1320 Springfield, VA 22161-2171 

(301) 621-0390 (703) 605-6000 
GAIN-SCHEDULED FAULT TOLERANCE CONTROL UNDER

FALSE IDENTIFICATION∗

Jong-Yeob Shin †

ABSTRACT

An active fault tolerant control (FTC) law is generally sensitive to false identification

since the control gain is reconfigured for fault occurrence. In the conventional FTC law

design procedure, dynamic variations due to false identification are not considered. In

this paper, an FTC synthesis method is developed in order to consider possible variations of closed-loop dynamics caused by false identification. An active FTC synthesis

problem is formulated into a linear matrix inequality (LMI) optimization problem to

minimize the upper bound of the induced-L2 norm which can represent the worst-case

performance degradation due to false identification. The developed synthesis method

is applied to control of the longitudinal motion of FASER (Free-flying Airplane for

Subscale Experimental Research). The designed FTC law of the airplane is simulated

for pitch angle command tracking under a false identification case.

1 INTRODUCTION

In the past few decades, there has been interest in a fault tolerant control (FTC) system which has the ability to detect actuator/sensor faults automatically and to prevent

faults from developing into a total system failure. Especially in constructing a flight control

system, an active FTC system has been researched for achieving single aircraft accident

prevention [2, 3, 7, 16]. An active FTC system consists of an FTC law, a fault detection

and isolation (FDI) module and a supervisory system. An FTC law should react to actuator/sensor faults through reconfiguration and an FDI module should detect actuator/sensor

fault occurrences. Based on the information provided by an FDI module, a supervisory

system decides which actuator/sensor is faulty and sends a signal to an FTC law for its

reconfiguration.

In general, an active FTC law is designed, based on an open-loop system modeled as a

function of fault parameters under the assumption that they are immediately identified by an

FDI module. Recently, using linear matrix inequality (LMI) optimization solutions [3, 7, 16],

an active FTC law has been synthesized in the form of a linear parameter varying (LPV)

system whose dynamics vary as scheduling parameters change. Open-loop dynamics are

modeled as an LPV system in which some of the scheduling parameters are fault parameters that represent fault occurrences at actuators/sensors. The designed LPV-FTC law

can robustly stabilize the closed-loop system and achieve desired performance during a fault

occurrence under the assumption that fault parameters are measured in real-time.

∗This work was supported by the National Aeronautics and Space Administration under NASA Cooperative Agreement NCC1-02043.

†National Institute of Aerospace (N.I.A.), Hampton, VA 23666. Email:shinjy@nianet.org

1
Typically, there is always some level of time-delay to detect faults regardless of FDI algorithms such as an extended Kalman FDI filter [16] or an LPV-FDI filter [17]. During a

time-delay interval, an open-loop system is in a faulty condition but the information provided

by an FDI module implies that the system is not in a faulty condition. It is also possible

that an FDI module and a supervisory system can produce false identification on healthy

actuator/sensors, which may lead the system to be locally unstable temporarily.

The objectives of FTC synthesis are for the closed-loop system to be stable and to be able

to achieve the desired performance level for fault occurrences and to reduce performance

degradation due to false identification. One approach to achieve the objectives is designing a

passive FTC law which has constant gains for both healthy and faulty systems. In that case,

obviously the closed-loop system can be stable for a false identification case since the control

gains are not changed due to false identification. The designed passive FTC law may be,

however, conservative in performance since the controller is constructed from the common

solutions of the LMI optimization problem[3] for both healthy and faulty systems.

Another approach is designing an active FTC law whose gains are reconfigured when a

fault occurs. One of the synthesis methods to design an active FTC law is a conventional

LPV synthesis method[18] in which it is assumed that scheduling parameters of an LPV

system are exactly measured in real-time. In the conventional LPV synthesis method, false

identification can not be considered. It is needed to modify the LPV synthesis method in

order to consider possible closed-loop dynamic variations caused by false identification.

In this paper, an LPV control synthesis method is developed to design an active FTC law

in order to consider the dynamic variations of an FTC system due to false identification,

based on the brief unstable LPV system analysis method[13, 8] and the conventional LPV

synthesis method[18]. In Ref.[8], the performance of an LPV system with brief instability

was analyzed in terms of the upper bound of the L2 norm as a function of duration time and

exponential decay rate over parameter subspaces. In Ref.[13], the LPV analysis method[8] is

applied to analyze an FTC system under false identification. In Ref.[13], during a short time

interval of false identification, the closed-loop system can be locally unstable. The unstable

rate was calculated and then used to calculate the upper bound of the L2 performance level

over all parameter spaces. In this paper, an LPV control synthesis method under possible

false identification is developed and applied for designing an active FTC laws of the longitudinal motion of FASER.

This paper is organized as follows. In Section 2, the analysis problem including false identification case is described and in Section 3 a gain-scheduled FTC synthesis methodology is

presented. In Section 4, the longitudinal motion of FASER is described in terms of control

sensitivities of each control surfaces and the details in designing FTC law are presented.

Section 5 concludes with a brief summary.

2 LINEAR PARAMETER VARYING ANALYSIS METHOD

An FTC system consists of an FTC law, an FDI module and a supervisory system (logic).

When a fault occurs, the FDI module and supervisory system detect it and generate signals

for evaluating/reconfiguring the FTC law. The FDI model requires some time to detect a

fault. Hereafter, the required time is called detection time for the FDI module. During

2
detection time, open-loop dynamics are in a faulty condition but the designed control law

is not yet reconfigured for it. Suppose there are redundant control actuators to prevent a

system having zero controllability for fault occurrence. The FDI module can also identify a

healthy actuator as a faulty one for a short time interval when a fault occurs on another. It

is called false identification. In Refs.[13, 8, 15], an analysis framework of the FTC system

is developed for false identification and detection time-delay cases. In this section, the LPV

analysis method including false identification is summarized.

Consider an open-loop system as an LPV system:









x ̇

e

y









=









A(ρp, ρf ) B1(ρp, ρf ) B2(ρp, ρf )

C1(ρp, ρf ) D11(ρp, ρf ) D12(ρp, ρf )

C2(ρp, ρf ) D21(ρp, ρf ) 0

















x

d

u









(1)

where ρp ∈ Rnp are physical parameters measurable by sensors in real-time such as velocity

and angle of attack in flight dynamics and ρf ∈ Rnf are fault parameters estimated by the

FDI module that can indicate open-loop dynamic variations due to fault occurrence. The

vectors x ∈ Rnx, e ∈ Rne

, y ∈ Rny

, d ∈ Rnd, and u ∈ Rnu are states, errors, measurements,

disturbances and input signals, respectively. Note that the matrices are functions of physical

parameters and fault parameters.

A control law is defined as

"

x ̇ k

u

#

=

"

Ak(ρp, ρ ̄f ) Bk(ρp, ρ ̄f )

Ck(ρp, ρ ̄f ) 0 # "xky

#

:= K

"

xk

y

#

(2)

where the vector xk is a control state vector and  ̄ρf are estimated fault parameters. Note

that the control gain is a function of physical parameters and estimated fault parameters.

Using Eqs.(1) and (2), the closed-loop system is

"

x ̇ cl

e

#

=

"

Acl(ρp, ρf , ρ ̄f ) Bcl(ρp, ρf , ρ ̄f )

Ccl(ρp, ρf , ρ ̄f ) Dcl(ρp, ρf , ρ ̄f )

# "xcl

d

#

(3)

where x

T

cl = [x

T xT

k

] ∈ Rn and

Acl(ρ) = A ̃(ρp, ρf ) + Bˆ(ρp, ρf )K(ρp, ρ ̄f )Cˆ(ρp, ρf )

Bcl(ρ) = B ̃(ρp, ρf ) + Bˆ(ρp, ρf )K(ρp, ρ ̄f )Dˆ

21(ρp, ρf )

Ccl(ρ) = C ̃(ρp, ρf ) + Dˆ

12(ρp, ρf )K(ρp, ρ ̄f )Cˆ(ρp, ρf )

Dcl(ρ) = D11(ρp, ρf ) + Dˆ

12(ρp, ρf )K(ρp, ρ ̄f )Dˆ21(ρp, ρf )

(4)

where the parameter matrices are

A ̃ =

"

A 0

0 0#

, B ̃ =

"

B1

0

#

, C ̃ =

h

C1 0

i

, Cˆ =

"

0 I

C2 0

#

(5)

3
Bˆ =

"

0 B2

I 0

#

, Dˆ

12 =

h

0 D12i

, Dˆ

21 =

"

0

D21#

(6)

Hereafter, parameters (ρp, ρf , ρ ̄f ) are denoted by ρ. Note that when an FDI module identifies

faults immediately, the estimated fault parameters  ̄ρf are equal to ρf . For analyzing the

closed-loop system for false identification,  ̄ρf and ρf are treated as independent parameters

to capture the closed-loop dynamic variations due to false identification.

The trajectories of parameters ρ(t) stay in the compact set called the entire parameter

space such that

P := {ρ(t) | ρp(t) ∈ Fp, ρf (t) ∈ Fρf, ρ ̄f (t) ∈ Fρ ̄f} (7)

where Fp, Fρf, and Fρ ̄fare bounded compact sets in Rnp, Rnf, and Rnf, respectively. Note

that Fp is called flight envelope when ρp are velocity and angle of attack for longitudinal

motion of an airplane.

The entire parameter set is divided into two subspaces P1 and P2 such that

P := P1 ∪ P2,

∅ = P1 ∩ P2,

(8)

The subspaces are defined based on the closed-loop dynamic changes due to false identification. Here, when all parameters stay in P1, the FDI module estimates fault parameters

accurately to within a small error bound, δρ. Thus, the subspace P1 can be defined as:

P1 := {ρf (t)| ρf (t) ∈ S( ̄ρf , δρ), ρ ̄f (t) ∈ Fρ ̄f, ρp ∈ Fp} (9)

where

S( ̄ρf , δρ) := {ρf (t) | ||ρf (t) − ρ ̄f (t)|| ≤ δρ, 0 ≤ δρ} . (10)

For false identification, ρf (t) is not in the set S( ̄ρf , δρ) since ρf (t) indicates a healthy condition but  ̄ρf (t) indicates a faulty condition. Parameter subspace P2 is defined for the false

identification. Assume that the closed-loop system is locally unstable along the trajectories

over the parameter subspace P2.

To represent dynamic variations of the closed-system in Eq. (3) over each parameter subspace, a duration time over each subspace is defined as follows [8]:

Definition 1 Duration time Tpiover each parameter subspace :

Tpi

(to, t) = Toi +

Z t

to

σi(ρ(s))ds, ∀t > to > 0, (11)

σi(ρ(s)) = (

0, ρ(s) ∈ P/ i,

1, ρ(s) ∈ Pi, i ∈ [1 2].

(12)

A duration time is bounded as

Toi ≤ Tpi ≤ Toi + αi(t − to) (13)

4
where 0 ≤ αi ≤ 1 and 0 ≤ Toi

. The constant αi represents a ratio of the duration time in

the i-th subspace to the total time and Toiis the duration time for a system to stay in the

i-th subspace during the interval [0, to]. Note that P2

i=1 αi = 1 and P2i=1 Toi = to.

For the closed-loop system of Eq. (3), an induced-L2 norm is defined as:

sup

ρ∈P,d∈L2,||d||26=0

||e||2

||d||2

. (14)

The performance level of the closed-loop system with false identification is calculated in

terms of the induced-L2 norm as follows:

Theorem 1: Consider the closed-loop system in Eq.(3). Suppose there exists a positive

definite matrix P(ρ) ∈ Rnx×nx such that









AT

cl(ρ)P(ρ) + P(ρ)Acl(ρ) + P ̇(ρ) + λ1P(ρ) P(ρ)Bcl(ρ) C

T

cl(ρ)

BT

cl(ρ)P(ρ) −γI DTcl(ρ)

Ccl(ρ) Dcl(ρ) −γI









< 0, ρ ∈ P1, (15)









AT

cl(ρ)P(ρ) + P(ρ)Acl(ρ) + P ̇(ρ) − κP(ρ) P(ρ)Bcl(ρ) C

T

cl(ρ)

BT

cl(ρ)P(ρ) −γI DTcl(ρ)

Ccl(ρ) Dcl(ρ) −γI









< 0, ρ ∈ P2. (16)

P > 0, ρ ∈ P (17)

where λ1 and κ are positive scalar constants. The induced-L2 norm from d to e of the

closed-loop system with given To2is no larger than Mγ where

Mγ = γ

r

e

(λ1+κ)To2 λ1

λ

, (18)

λ = λ1 − (λ1 + κ)α2. (19)

Under the condition α2 < αt:= λ1

(λ1+κ)

, the constant λ is positive.

Proof: See Ref.[15].

The constant Mγ represents the upper bound of the worst-case performance level due to

false identification. Thus, a synthesis problem for a controller K, as defined in Eq.(2), can

be formulated into an optimization problem:

min

K,γ>0,P >0

Mγ, s.t. Eqs.(15)-(17). (20)

with pre-defined λ1 and κ values. The optimization problem of Eq.(20) is converted into an

LMI optimization problem in the next section.

5
3 GAIN-SCHEDULED FTC SYNTHESIS METHODOLOGY

The projection lemma in Ref. [5] plays a key role in converting the analysis problem of

Eqs.(15)-(17) into a control synthesis problem. Here, for completeness, the lemma [5] is

described as follows:

Lemma 1: Given a symmetric matrix Ψ ∈ Rn×n and two matrices Q and R of column

dimension n, consider the problem of finding some matrix Θ of compatible dimensions such

that

Ψ + R

TΘTQ + QTΘR < 0 (21)

Suppose the matrices NR and NQ are the null space of R and Q, respectively. Then Eq.(21)

is solvable for Θ if and only if

N

T

R ΨNR < 0, and N

T

Q ΨNQ < 0 (22)

Proof: See Ref.[5].

To synthesize a gain-scheduled FTC law under false identification, λ1 and κ values are

defined by a control designer. The solvability condition of synthesizing a gain-scheduled

control under false identification is described as follows:

Theorem 2 Given the open-loop system in Eq.(1), the entire parameter spaces are partitioned (see Eq.(8)). Suppose there exist positive matrices X(ρ) and Y (ρ) that satisfy the

following conditions

N

T

X









A(ρ)

T X(ρ) + X(ρ)A(ρ) + X ̇ + λ1X(ρ) X(ρ)B1(ρ) CT

1

(ρ)

BT

1

(ρ)X(ρ) −γI DT

11(ρ)

C1(ρ) D11(ρ) −γI









NX < 0, ρ ∈ P1 (23)

N

T

Y









Y (ρ)A(ρ) + Y (ρ)AT(ρ) − Y ̇ + λ1Y (ρ) Y (ρ)C

T

1

(ρ) B1(ρ)

C1(ρ)Y (ρ) −γI D11(ρ)

BT

1

(ρ) DT

11(ρ) −γI









NY < 0, ρ ∈ P1, (24)

N

T

X









A(ρ)

T X(ρ) + X(ρ)A(ρ) + X ̇ − κX(ρ) X(ρ)B1(ρ) CT

1

(ρ)

BT

1

(ρ)X(ρ) −γI DT

11(ρ)

C1(ρ) D11(ρ) −γI









NX < 0, ρ ∈ P2, (25)

N

T

Y









Y (ρ)A(ρ) + Y (ρ)AT(ρ) − Y ̇ − κY (ρ) Y (ρ)C

T

1

(ρ) B1(ρ)

C1(ρ)Y (ρ) −γI D11(ρ)

BT

1

(ρ) DT

11(ρ) −γI









NY < 0, ρ ∈ P2, (26)

"

X(ρ) I

I Y (ρ)

#

> 0, ρ ∈ P, (27)

6
with given constants λ1 and κ, where

NX = Null([B

T

2

(ρ) D

T

12(ρ) 0]) (28)

NY = Null([C2(ρ) D21(ρ) 0]). (29)

Then the induced-L2 norm of d to e of the closed-loop system of Eq.(3) is bounded by Mγ of

Eq.(18).

Proof. : Define matrices as follows:

R = [Cˆ(ρ) Dˆ

21(ρ) 0] (30)

Q = [BˆT(ρ)P(ρ) 0 Dˆ T

12(ρ)] (31)

Ψ =









A ̃T(ρ)P(ρ) + P(ρ)A ̃(ρ) + λ1P(ρ) + P P  ̇ (ρ)B ̃(ρ) C ̃T(ρ)

B ̃T(ρ)P(ρ) −γI DT

11(ρ)

C ̃(ρ) D11(ρ) −γI









(32)

Using Eqs. (30)-(32) and (2), the LMI condition of Eq.(15) is equal to

Ψ + R

T KTQ + QT KR < 0. (33)

Using Lemma 1, the LMI condition of Eq. (33) is decomposed into the following two LMI

conditions:

N

T

R ΨNR < 0, (34)

N

T

Q ΨNQ < 0. (35)

The positive matrix P is parameterized as follows:

P(ρ) = "

X(ρ) N(ρ)

NT(ρ) X1(ρ)

#

, P −1(ρ) = "

Y (ρ) M(ρ)

MT(ρ) Y1(ρ)

#

. (36)

From Eqs.(30)-(32) and (36), it is proved that the LMI condition of Eq.(34) is equal to

Eq.(23).

Eq.(31) is rewritten as

Q = [BˆT(ρ) 0 Dˆ T

12(ρ)]S(ρ) (37)

where

S(ρ) :=









P(ρ) 0 0

0 I 0

0 0 I









. (38)

Using Eq.(37), the nullspace NQ is rewritten as

NQ = S

−1N ([BˆT

(ρ) 0 Dˆ T

12(ρ)]. (39)

After some algebraic manipulation with Eqs.(30)-(32) and (39), the LMI condition of Eq.(35)

is equal to Eq.(24). Using the above method, it is proved that the LMI in Eq.(16) is converted

into the two LMI conditions of Eqs.(25) and (26).

7
The control synthesis problem can be solved using the LMI Toolbox [6] such that

min

X(ρ), Y (ρ)

γ. (40)

In order to solve the optimization problem, the basis functions of X(ρ) and Y (ρ) are required.

X(ρ) := Xns

i=1

fi(ρ)Xi, Y (ρ) := Xns

i=1

gi(ρ)Yi (41)

where functions fi(ρ) : Rns → R and gi(ρ) : Rns → R are bounded and the unknown

matrices Xi and Yi are symmetric, respectively. Note that it is still unknown how to choose

optimal basis functions for predicted performance level of the closed-loop system. Under

the assumption that D12 and D21 are full-column and full-row rank for the entire parameter

space, respectively, the controller is constructed from the solutions X(ρ) and Y (ρ) in Eq.(40)

as follows:[1]

Dk = 0 (42)

Bk = N

−1Bˆ

k (43)

Ck = Cˆ

kM−T

(44)

Ak = N

−1

{XY ̇ + NM ̇ T + Aˆ

k − XAY − BˆkC2Y − XB2Cˆk}M−T

where

Bˆ

k =

1

γ

C

T

2

(D21G1D

T

21)

−1 + (XB1 +

1

γ

C

T

1 D11)(D

T

21D21)

−1DT

21 (45)

Cˆ

k = −

1

γ

(D

T

12G2D12)

−1BT

2 − D

T

12(D12D

T

12)

−1

(C1Y −

1

γ

D11B

T

1

) (46)

Aˆ

k = A

T

h

XB1 + Bˆ

kD21 C

T

1

i

"

γI −DT

11

−DT

11 γI #−1 "

BT

1

C1Y + D12Cˆ

k

#

(47)

G1 = (γ

2

I − D

T

11D11)

−1

, G2 = (γ

2

I − D11D

T

11)

−1

. (48)

Note that the matrices in Eqs.(42)-(48) are functions of the scheduling parameters. To

complete the control construction, the following factorization is used

N(ρ) = X(ρ), M(ρ) = X(ρ)

−1 − Y (ρ) (49)

to satisfy the equality condition:

I − X(ρ)Y (ρ) = N(ρ)MT(ρ). (50)

4 EXAMPLE

In this section, it is demonstrated how to synthesize a gain-scheduled FTC law for the

airplane of FASER shown in Fig. 1, by using the method described in the previous section.

Two gain-scheduled LPV FTC laws are designed by the conventional LPV synthesis and the

suggested method, respectively. For comparison, the designed control laws will be simulated

under false identification.

8
4.1 Mathematical Model of FASER

4.1.1 Description of FASER

FASER (Free-flying Airplane for Sub-scale Experimental Research) shown in Fig. 1 has been

developed at the NASA Langley Research Center, which has a conventional high wing with

tail configuration with 7 ft wingspan and a propeller driven by an electric motor[11]. The

detailed shape and physical quantities are taken from Refs.[10, 11]. The control surfaces of

the airplane are an elevator, ailerons, flaps and a rudder driven by JRDS8611 Digital servos.

For an INS system of the airplane, the MIDG GPS/INS unit built by Microbotics[9] is used.

The positions are calculated based on GPS signals received by the MIDG. To measure angle

of attack and side-slip angles, small props with small potentiometers are attached at both

tips of the main wing. The roll, pitch and yaw angle rates are also measured by the MIDG.

The on-board flight computer of FASER is the PC104 with 133MHz CPU, the 10 channel

PWM board (Quartz-MM Diamond system) and the 16 channel analog I/O board (DM6420

Real-Time Devices).

4.1.2 Nonlinear Equations of Longitudinal Motion

The longitudinal dynamics have four states: velocity (V ) in f t/s, angle of attack (α) in

radians, pitch rate (q) in rad/s, and pitch angle (θ) in radians, and inputs: elevator deflection

(δe) in radians and thrust (T) in lbs. The airplane (FASER) has the right/left ailerons

(δra/δla) and the right/left flaps (δrf /δlf ) to be operated independently. The longitudinal

equations of the motion are written as:

V ̇ =

1

m

{(qSCX + T) cos α + qSCZ sin α} + g(− sin θ cos α + cos θ sin α), (51)

α ̇ =q +

qSCZ cos α

mV

−

qSCX + T

mV

+

g

V

(cos α cos θ + sin θ sin α), (52)

q ̇ =

1

Jyy

qScC ̄ m, (53)

 ̇θ = q, (54)

where aerodynamics coefficients CX, CZ, and Cm are written as

CX =CXo(α) + CXq(α)

cq ̄

2V

+ CXδe(α)δe + CXδa(α)δaall + CXδf(α)δfall ,

CZ =CZo(α) + CZq(α)

cq ̄

2V

+ CZδe(α)δe + CZδa(α)δaall + CZδf(α)δfall ,

Cm =Cmo(α) + Cmq(α)

cq ̄

2V

+ Cmδe(α)δe + Cmδa(α)δaall + Cmδf(α)δfall .

Note that the positive deflection of both right and left aileron control surfaces is defined

in the same way of the conventional definition of the positive elevator deflection. In the

longitudinal motion, assume that the left and right aileron surfaces can symmetrically move

9
to change the lift and drag forces. Thus, the control inputs of the airplane in the longitudinal

motion are thrust T(lb), elevator deflection δe(deg), aileron deflection δaall = 0.5(δra + δla),

and flap deflection δfall = 0.5(δrf +δlf ). The detailed aerodynamic coefficients are calculated

by using a CFD code in Refs.[4, 12].

4.1.3 Quasi-LPV Model of Longitudinal Motion

It is required to represent the nonlinear dynamics as a quasi-LPV model in order to design

an LPV control law using the method described in Section 3. Using the quasi-LPV representation method in Ref.[14], the nonlinear equation of longitudinal motion can be represented

as the quasi-LPV model:

x ̇ = A(V, α)x + B(V, α)u (55)

where x = [V α q θ]

T and u = [T δe δaall ]T

. The detailed matrices A and B can be found

in Refs.[14, 12]. Note that the flaps are not included in the quasi-LPV model on purpose

since the wake generated by flap movement may be able to coupled with elevator deflection

movement. The coupling effect may generate mathematically unpredictable aerodynamic

force and moment on the airplane. Hereafter, control variables are set as thrust T, elevator

deflection δe and same directional aileron deflection δaall .

The flight envelope is defined as follows:

60ft/sec ≤ V ≤ 120ft/sec

2

◦ ≤ α ≤ 7◦

, (56)

based on the trim region (the line of the plot shown in Fig.2) for a constant level flight

condition (  ̇x = 0, α = θ). Note that the states V and α are also considered as scheduling

parameters ρp which represent flight conditions. The control action limits under the nominal

condition are as follows:

0 ≤ T ≤ 6 lb, −20◦ ≤ δe ≤ 20◦, −20◦ ≤ δaall ≤ 20◦. (57)

4.1.4 Elevator Fault Model and Parameter Space

In this example, elevator failure is considered as loss of elevator controllability. Assume that

the elevator is fixed at trim position for a failure case. Floating control surfaces, partially

damaged control surfaces and control surfaces fixed at any other positions can be considered

as failures and can be modeled. However, those failures are not considered here for simplicity.

In this paper, the fault is simply modeled as linear change of the matrix B(ρ) such as

B(ρ)u = B(ρp)









1 0 0

0 τ1 0

0 0 τ2









u (58)

10
where the failure parameters τ1 and τ2 can vary from 0 (failure condition) to 1 (healthy

condition).

Consider that one of two actuators can be failed at a time in order to prevent zero controllability of the system. In order to represent the failure condition, the fault parameter ρf

is defined as follows:

ρf (t) =







τ1(t) if 0 ≤ τ1(t) < 1, τ2 = 1,

2 − τ2(t) if τ1 = 1, 0 ≤ τ2 < 1.

(59)

The estimated fault parameter can be calculated from the estimations of the failure parameters τ1 and τ2.

The false identification case is considered as the estimated fault parameter indicates the

aileron failure for the elevator failure for a short time interval. In the time interval, aileron

actuator is actually healthy but an FDI module indicates it as a faulty one. Also, the elevator actuator is failed but a FDI indicates it as a healthy one. In that case, estimated fault

parameter  ̄ρf is equal to 2(aileron failure) instead of 0 (elevator failure).

The entire parameter space P and subspaces shown in Fig. 3 are defined as

P = P1 ∪ P2 (60)

where

P1 := {ρ | ρp ∈ Fp, ρf ∈ S( ̄ρf , 0.1), ρ ̄f ∈ [0 2]}

P2 := {ρ | ρf = 0, ρ ̄f ∈ [1.9 2]}.

(61)

Here, the set S( ̄ρf , 0.1) is defined as

S( ̄ρf , 0.1) := {ρf (t) | ||ρf (t) − ρ ̄f (t)|| < 0.1}. (62)

The set Fp is defined as the flight envelope in Eq.(56). Note that the fault parameter ρf

can vary abruptly from 1 (healthy condition) to 0 (faulty condition) but the estimated fault

parameter can smoothly vary from 1 to 0, based on the FDI module algorithm. Generally,

the LPV-FTC law is designed for the subspace P1 in Refs.[16, 7] under the assumption that

the FDI can estimate the fault parameter fast and accurately. In this paper, we consider

cases where the closed-loop system stays in the parameter subspace P2 for a short time

interval.

4.2 Control Synthesis Framework

The control design objectives are to track the pitch angle command with less than 4% tracking error under a nominal condition and to reduce tracking error under a faulty condition.

In the control design procedure, the variations of the predicted closed-loop dynamics due

to false identification are also considered by including LMI conditions of Eqs. (25) and (26)

with the pre-defined parameter κ. Recall that the parameter κ represents possible instability

of the closed-loop system under false identification [13].

The control design problem is formulated as a model matching problem shown in Fig.4.

11
In the block diagram shown in Fig.4, the block “Act” implies the actuator models. In this

example, the mathematical models of the JRDS8611 servo motor and the electric motor

are approximated as the first order filters 20

s+20 and 5s+5 , respectively. The first order model

of the servo motor is approximated with constant one in the control design procedure to

reduce the order of the designed control law. The first order model is, however, used for

simulations of the closed-loop system. The block Wnoise represents sensor noise such as

diag([0.1ft/sec, 1

◦ 1◦

, 1 ]). The performance weighting function Wp is set as 25

s/0.1+1 to

represent 4% error allowance in a low frequency domain and the ideal transfer function Tiis

set as 1

s/0.9+1 for ideal pitch angle responses. The block Wact is set as diag([1/10, 1/40 1/40])

to constrain control action. Using the block diagram shown in Fig.4, the augmented openloop is constructed with a linearized model around a trim point (Vt

, α) = (70, 5.4◦).

For simplicity in designing an FTC law under the false identification, the scheduling parameter is defined as the fault parameter ρf ∈ [0, 2] at the given trim point. The basis

function fi(ρ) of X in Eq. (41) is defined as

f1(ρf ) = 1, f2(ρf ) = ρf , f3(ρf ) = 1/ρf , (63)

and the basis function gi(ρ) of Y is set as same as fi(ρ). The LMI constraints of Eqs. (23)-

(26) are evaluated at each grid point in the set [0.01, 0.2 0.4 · · · 1.8 2] of ρf . Note that

the grid point of 0.01 that represents a faulty condition is chosen in order to avoid singularity

of 1/ρf when the basis functions are evaluated at every grid point. For evaluating the time

derivatives of X and Y , the parameter rate bound is defined as 1 sec−1.

To solve the LMI optimization in Eqs.(23)-(26), the scalar parameters λ1 and κ are predefined as 0.01 and 0.1, respectively. To calculate the best values of λ1 and κ, it is required

to solve another optimization by minimizing Mγ of Eq. (18). In this paper, we choose them

based on a few iterations of different values of λ1 and κ. For example, when λ1 is 0.2, there

is no feasible solution for the LMI optimization in Eqs. (23)-(26). Thus, λ1 value should be

smaller than 0.2. When κ is set as large number, it implies that the closed-loop system can

be unstable with very fast rate. It also implies that the upper bound of the induced-L2 norm

of Eq. (18) could be large. Thus, the parameter κ can not be set as large number. When

the parameter is set as small number close to zero, it implies that stability of the closedloop system is required under the false identification with expense of over all performance

degradation.

In this paper, two LPV FTC controllers K1 and K2 are designed by the conventional LPV

method and the method suggested in Section 3, respectively. The conventional LPV method

can be re-constructed from Eqs. (23) and (24) by setting λ1 as zero. The designed two LPV

controllers are simulated in the next section.

4.3 Simulation Results

For all simulations, pitch angle commands are set as a square signal such as 1 degree up at

5 second and 1 degree down at 20 second. The label “Ti” in Figs. 5-9 denotes the ideal

pitch angle time responses. The label δe and δec denote elevator deflection angle and elevator

deflection angle command, respectively. For all simulations, state’s initial values are set as

the trim values. The simulation results under a healthy condition are shown in Fig. 5. Note

12
that the controllers K1 and K2 achieve the desired performance objective that pitch angle

command tracking is less than 4% at low frequency.

For an elevator failure case, the closed-loop time responses are shown in Fig. 6. In the

simulation, it is assumed that elevator fails after 5 second and an FDI module detects the

fault immediately. The pitch angle time responses with the controllers are very similar to

each other and imply that the two controllers achieve the performance objective while the

elevator fails. The two controllers generates aileron deflection angle commands to compensate

the elevator failure.

Three false identification cases are simulated. 1) While the elevator fails after 5 second,

an FDI detects the aileron failure between 5 and 10 seconds and the elevator failure after

10 second. 2) While the two actuators are healthy, an FDI detects elevator failure between

5 and 10 second. 3) While the two actuators are healthy, an FDI detects aileron failure

between 5 and 10 second. The simulation results for the case 1 are shown in Fig. 7. Note

that estimated fault parameter responses are generally simulated by an FDI module with

the closed-loop system but in this paper, they are pre-defined in order to represent false

identification. It is observed from Fig. 7 that the controller K2 can reduce the performance

degradation due to the false identification, but the controller K1 can not. The controller K1

generates elevator command signals due to the estimated fault parameter which indicates

an aileron fault between 5 sec and 10 sec. The generated elevator command signals can not

operate the elevator actuator since the elevator is actually failed after 5 second. Again, in the

design process for the controller K1, there is assumption that the estimated fault parameter

is equal to the true fault parameter. In the false identification, the assumption is violated.

Meanwhile, in the controller K2 design process, the possible false identification is considered.

That leads to the reduction of performance degradation due to false identification. When

the pre-defined parameters λ1 and κ are different values, the time responses of the closedloop system would be different. It would be of interest to further investigate the sensitivity

the closed-loop performance and actuator responses to λ1 and κ for a false identification

case. For the false identification cases 2 and 3, the closed-loop time responses are shown

in Figs. 8 and 9, respectively. Note that the two controllers K1 and K2 can achieve the

performance objective. Since the two actuators are healthy, one of compensated actuator

command signals can operate healthy actuators, respectively. The second and third false

identifications do not affect pitch angle tracking of the system, respectively.

5 CONCLUSION

In this paper, an active fault tolerant control synthesis method is developed under false

identification. In the control synthesis method, possible closed-loop dynamic variations due

to false identification are considered. In the control synthesis, the closed-loop system under

false identification can be unstable for a brief time interval with trade-off of performance

degradation. The performance level of the closed-loop system is described as a function of

duration time of the parameter subspace, the induced-L2 norm index γ and exponential decay

rate in each parameter subspace. With given possible exponential decay rate of the predicted

closed-loop system, the control synthesis problem is formulated into the LMI optimization

to minimize the upper bound of the induced-L2 norm. The suggested LPV-FTC synthesis

13
method is applied to design an FTC law for the longitudinal motion of the airplane FASER.

According to simulation results, the designed control law is insensitive for false identification

while it achieve pitch angle tracking error less than 4% in a faulty condition.

6 ACKNOWLEDGMENTS

The author thanks the technical monitor Dr. Irene Gregory at the NASA Langley Research

Center and also acknowledge the contributions of David Cox (NASA Langley Research Center), and Pawel Chwalowski (Analytic Mechanics Associates) for developing the nonlinear

mathematical model of the airplane.

REFERENCES

[1] Apkarian, P. and Adams, R. Advanced Gain-Scheduling Techniques for Uncertain Systems. IEEE Transactions on Control Systems Technology, 6(1):21–32, 1998.

[2] Belcastro, Christine and Belcastro, Celeste. Application of failure detection, identification, and accommodation methods for improved aircraft safety. In Proceedings of

American Control Conference, pages 2623–2624. American Automatic Control Council,

Evanston, IL, 2001.

[3] Chen, J., Patton, R.J., and Chen, Z. An LMI approach to Fault-Tolerant Control of

Uncertain System. In Proceeding of the 1998 International Symposium on Intelligent

Control, pages 175–180. Institute of Electrical and Electronics Engineers, Piscataway,

N.J., 1998.

[4] Chwalowski, P. Vehicle Configuration Studies and Small Aircraft Formation Flying

Study. Technical Report 04-33, Analytical Mechanics Associates, Inc., Nov. 2004.

[5] Gahinet, P. and Apkarian, P. A Linear Matrix Inequality Approach to H∞ Control.

International Journal of Robust and Nonlinear Control, pages 421–448, 1994.

[6] Gahinet, P., Nemirovski, A., Laub, A.J., and Chilai. M. LMI Control Toolbox User’s

Guide. The MATH Works, Natick, Mass, 1995.

[7] Ganguli, S., Marcos, A., and Balas, G. Reconfigurable LPV Control Design for Boeing

747-100/200 Longitudinal Axis. In Proceedings of American Control Conference, pages

3612–3617, Anchorage, AK, 2002. American Control Conference Council, Evanston, IL.

[8] Hespanha, J., Yakimenko, O., Kaminer, I., and Pascoal, A. LPV Systems with Brief

Instabilities:Application to Integrated Vision/IMU Navigation. In IEEE Proceeding of

the Conference on Decision and Control, 2001.

[9] Microbotics, Inc. http://www.microboticsinc.com/midg.html. Microbotics, Inc., 2005.

[10] Monzon, B.R. Development of a Nonlinear Simulation for a Research Model Airplane.

Master’s thesis, MSE Thesis, Geroge Washington University JIAFS,, NASA Langley

Research Center, September 2001.

14
[11] Morelli, E.A. and DeLoach, R. Wind Tunnel Database Developement Using Mordern

Experiment Design and Multivariate Orthogonal Functions. In 41st AIAA Aerospace

Sciences Meeting and Exhibit, AIAA 2003-0653, Jan. 6-9 2003.

[12] Shin, J.-Y. Quasi-Linear Parameter Varying Representation of General Aircraft Dynamics Over Non-Trim Region. Technical report, National Institute of Aerospace, 2005.

[13] Shin, J-Y. and Belcastro, C. Analysis of A Fault Tolerant Control System: False Fault

Detection Case. In 5th IFAC Symposium on Fault Detection, Supervision and Safety of

Technical Processes, June 2003.

[14] Shin, J-Y. and Belcastro, C. Quasi-Linear Parameter Varying Representation over NonTrim Region. In AIAA Guidance, Navigation, and Control Conference and Exhibit,

AIAA-2004-5423. American Institute of Aeronautics and Astronautics, 2004.

[15] Shin, J-Y., Wu, N.E., and Belcastro, C. Linear Parameter Varying Control Synthesis for

Actuator Failure, Based on Estimated Parameter. In AIAA Guidance, Navigation and

Control, number AIAA-2002-4546. American Institute of Aeronautics and Astronautics,

2002.

[16] Shin, J-Y., Wu, N.E., and Belcastro, C. Adaptive Linear Parameter Varying Control

Synthesis for Actuator Failure. Journal of Guidance, Control, and Dynamics, 27(5):787–

794, Sept. 2004.

[17] Szaszi, I., Marcos, A., Balas, G., and Bokor, J. LPV Detection Filter Design for Boeing

747-100/200. In AIAA Guidance, Navigation and Control, number AIAA 2002-4957.

American Institute of Aeronautics and Astronautics, 2002.

[18] Wu, F. Control of Linear Parameter Varying Systems. PhD thesis, Department of

Mechanical Engineering, University of California, Berkeley, 1995.

15
Figure 1: The FASER

16
2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

60

70

80

90

100

110

120

angle of attack (deg)

 Velocity (ft/sec)

Flight Envelope

Figure 2: Flight envelop with trim points

P

ρf

ρf

 























































1

false identification

0

P2

2

2

Figure 3: Parameter subspaces.

17
P( ρ)

T ideal

Wnoise

Wact

Wp

V α q

ep

eu

Act.

cmd K

+

+

+

_

θ

d{4}

u

Figure 4: Model matching synthesis block diagram.

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

θ (deg)

Ti

K1

K2

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

ρ

fault parm

estimated parm

0 10 20 30

−4

−2

0

2

4

Time(sec)

δ

e

c

 (deg)

K1

K2

0 10 20 30

−10

−5

0

5

10

Time(sec)

δ

a

c

 (deg)

K1

K2

0 10 20 30

−4

−2

0

2

4

Time(sec)

δ

e (deg)

K1

K2

0 10 20 30

−10

−5

0

5

10

Time(sec)

δ

a (deg)

K1

K2

Figure 5: Time responses of the closed-loop system under a healthy condition.

18
0 10 20 30

0

0.5

1

1.5

2

Time(sec)

θ (deg)

Ti

K1

K2

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

ρ

fault parm

estimated parm

0 10 20 30

−5

0

5

Time(sec)

δ

e

c

 (deg)

K1

K2

0 10 20 30

−20

−10

0

10

20

Time(sec)

δ

a

c

 (deg)

K1

K2

0 10 20 30

−1

−0.5

0

0.5

1

Time(sec)

δ

e (deg)

K1

K2

0 10 20 30

−20

−10

0

10

20

Time(sec)

δ

a (deg)

K1

K2

Figure 6: Time responses of the closed-loop system under elevator failure.

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

θ (deg)

Ti

K1

K2

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

ρ

fault parm

estimated parm

0 10 20 30

−30

−20

−10

0

10

Time(sec)

δ

e

c

 (deg)

K1

K2

0 10 20 30

−40

−20

0

20

Time(sec)

δ

a

c

 (deg)

K1

K2

0 10 20 30

−1

−0.5

0

0.5

1

Time(sec)

δ

e (deg)

K1

K2

0 10 20 30

−40

−20

0

20

Time(sec)

δ

a (deg)

K1

K2

Figure 7: Time responses of the closed-loop system under the false identification case 1.

19
0 10 20 30

0

0.5

1

1.5

2

Time(sec)

θ (deg)

Ti

K1

K2

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

ρ

fault parm

estimated parm

0 10 20 30

−4

−2

0

2

4

Time(sec)

δ

e

c

 (deg)

K1

K2

0 10 20 30

−20

−10

0

10

Time(sec)

δ

a

c

 (deg)

K1

K2

0 10 20 30

−4

−2

0

2

4

Time(sec)

δ

e (deg)

K1

K2

0 10 20 30

−20

−10

0

10

Time(sec)

δ

a (deg)

K1

K2

Figure 8: Time responses of the closed-loop system under the false identification case 2.

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

θ (deg)

Ti

K1

K2

0 10 20 30

0

0.5

1

1.5

2

Time(sec)

ρ

fault parm

estimated parm

0 10 20 30

−5

0

5

Time(sec)

δ

e

c

 (deg)

K1

K2

0 10 20 30

−10

−5

0

5

10

Time(sec)

δ

a

c

 (deg)

K1

K2

0 10 20 30

−5

0

5

Time(sec)

δ

e (deg)

K1

K2

0 10 20 30

−10

−5

0

5

10

Time(sec)

δ

a (deg)

K1

K2

Figure 9: Time responses of the closed-loop system under the false identification case 3.

20
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2. REPORT TYPE 

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 4. TITLE AND SUBTITLE

Gain-Scheduled Fault Tolerance Control Under False Identification

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Shin, Jong-Yeob

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center National Institute of Aerospace (NIA)

Hampton, VA 23681-2199 100 Exploration Way

 Hampton, VA 23666

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NIA Report No. 2005-07

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Langley Technical Monitor: Christine Belcastro

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14. ABSTRACT

An active fault tolerant control (FTC) law is generally sensitive to false identification since the control gain is reconfigured for

fault occurrence. In the conventional FTC law design procedure, dynamic variations due to false identification are not

considered. In this paper, an FTC synthesis method is developed in order to consider possible variations of closed-loop

dynamics under false identification into the control design procedure. An active FTC synthesis problem is formulated into an

LMI optimization problem to minimize the upper bound of the induced-L2 norm which can represent the worst-case

performance degradation due to false identification. The developed synthesis method is applied for control of the longitudinal

motions of FASER (Free-flying Airplane for Subscale Experimental Research). The designed FTC law of the airplane is

simulated for pitch angle command tracking under a false identification case.

15. SUBJECT TERMS

Fault Tolerant Control; LPV Control; Mathematical Model

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