Fault-tolerant control of a two-degree of freedom

helicopter using LPV techniques

Saul Montes de Oca*, Vicenc Puig*, Marcin Witczak** and Joseba Quevedo*

Abstract— In this paper, a new active Fault Tolerant Control(FTC) strategy for Linear Parameter Varying (LPV) systemsis proposed. The key contribution of the proposed approach isan integrated FTC design procedure of the fault identificationand fault-tolerant control schemes using LPV techniques. Faultidentification is based on the use of an observer. Once the faulthave been identified, the FTC controller is implemented as astate feedback controller. This controller is designed such that itcan stabilize the faulty plant using LPV techniques and LinearMatrix Inequalities (LMIs). To assess the performance of theproposed approach a two degree of freedom helicopter is used.

I. INTRODUCTION

Fault Tolerant Control (FTC) is a new idea recently intro-

duced in the research literature [1] which allows to maintain

current performances closed to desirable performances and

preserve stability conditions in the presence of component

and/or instrument faults. Accommodation capability of a

control system depends on many factors such as severity

of the failure, the robustness of the nominal system and

mechanisms that introduce redundancy in sensors and/or

actuators. From the point of view of the control strategies,

the literature considers two main groups: the active and

the passive techniques. The passive techniques are control

laws that take into account the fault appearance as a system

perturbation. Thus, within certain margins, the control law

has inherent fault tolerant capabilities, allowing the system

to cope with the fault presence. In [2], [3], [4], [5] and [6],

among many others, complete descriptions of passive FTC

techniques can be found. On the other hand, the active fault

tolerant control techniques consist on adapting the control

law using the information given by the Fault Detection

and Isolation (FDI) block [7]. With this information, some

automatic adjustments are done trying to reach the control

objectives. See [8] for a recent review of active FTC. The

whole active FTC scheme can be expressed using the three-

layer architecture for FTC systems proposed by [7] where

the first layer corresponds to the control loop, the second

layer corresponds to the fault diagnosis and accommodation

modules while the third layer is the supervisor system.

Active FTC strategies make it possible to consider more

faults than passive ones do: some research works deal with it

and underline the problem of closed-loop system stability in

This work is supported by the Research Commission of the Generalitatof Catalunya (Grup SAC ref. 2005SGR00537)

* S. Montes de Oca and V. Puig are with Advanced Control SystemsGroup, Technical University of Catalonia, Pau Gargallo, 5, 08028 Barcelona,Spain. Saul.Montes.de.Oca,Vicenc.Puig@upc.edu

** M. Witczak is with Institute of Control and Computation Engineering,University of Zielona Gora, 50 Podgorna, 65-246 Zielona Gora, Poland.M.Witczak@issi.uz.zgora.pl

the presence of multiple actuator failures [9], [10], [11], [12],

[13]. Active FTC is characterized by on-line FDI scheme

[14] and an automatic control reconfiguration mechanism.

Moreover, active FTC is often dedicated to linear systems or

the linearization of non-linear systems, but rarely to Linear

Parameter Varying (LPV) systems [15]. The main motivation

for LPV systems comes from the analysis and control of

nonlinear systems. For nonlinear systems, the design of Fault

Torerant controller is rather complicated. Nonlinear systems

based on multiple linear models, represents an attractive

solutions to deal with the control of nonlinear systems [16],

[17] or where nonlinear dynamic systems are described by

a number of locally linearized models based on the idea of

Tagaki-Sugeno fuzzy models [18].

Various system modeling techniques in the fault-free case

are presented in [19], [20], [21]. In [15] develop a solu-

tion to handle FTC and polytopic LPV systems with an

Static Output Feedback design. On the other hand, [22]

proposed an integrated FTC design procedure of the fault

identification and fault-tolerant control for Takagi-Sugeno

fuzzy systems using Lyapunov theory and Linear Matrix

Inequalities (LMIs).

In this paper, a new active FTC strategy for LPV systems is

proposed, which is based on the general principle presented

in [22]. The key contribution of the proposed approach is an

integrated FTC design procedure of the fault identification

and fault-tolerant control schemes. Fault identification is

based on the use of an observer. Once the fault have been

identified, the FTC controller is implemented as a state

feedback controller. This controller is designed such that it

can stabilize the faulty plant using an LMI pole placement

technique [23], [11].

The paper is organized as follows: Section II presents the

details regarding the proposed FTC Strategy. In Section III

an integrated design procedure of an observer and a state

feedback controller is developed. Section IV describes the

two-degree of freedom helicopter being a subject of an

illustrative example, which exhibits the performance of the

proposed approach.

II. FAULT TOLERANT CONTROL STRATEGY

Consider the following discrete LPV representation in the

fault-free case:

xk+1 = A(θ)xk + B(θ)uk, (1)

yk = C(θ)xk + D(θ)uk. (2)

where xk ∈ Rn represents the reference state vector,

yk ∈ Rm is the reference output vector and uk ∈ R

r de-

16th Mediterranean Conference on Control and AutomationCongress Centre, Ajaccio, FranceJune 25-27, 2008

978-1-4244-2505-1/08/$20.00 ©2008 IEEE 1204

notes the nominal control input. The system (1)-(2) describes

a set of equilibrium points parameterized by a scheduling

variable denoted by θ. In this paper, the kind of LPV system

considered are those whose parameters vary affinely in a

polytope Θ. In particular the state-space matrices range in a

polytope of matrices defined as the convex hull of a finite

number of matrices N . Each polytope vertex corresponds to

a particular value of scheduling variable θ. In other words,(

A(θ) B(θ)

C(θ) D(θ)

)∈ Co

(Aj Bj

Cj Dj

), j = 1, . . . , N

: =N∑

j=1

αj(θ)

(Aj Bj

Cj Dj

), (3)

with αj(θ) ≥ 0 and∑N

j=1αj(θ) = 1. Because of this pro-

perty, this type of LPV systems is refered as polytopic [24].

In the following, we consider only strictly proper systems

such that D = 0. Consequently, the LPV system (1)-(2) can

be expressed as follows:

xk+1 =

N∑

j=1

αjk(θ) [Ajxk + Bjuk] , (4)

yk =

N∑

j=1

αjk(θ) [Cjxk] . (5)

Let us also consider an additive actuator fault representa-

tion, the system (4)-(5) can be described by the following

polytopic representation:

xf,k+1 =N∑

j=1

αjk(θ) [Ajxf,k + Bjuf,k + Ljfk] , (6)

yf,k =

N∑

j=1

αjk(θ) [Cjxf,k] , (7)

where xf,k ∈ Rn represent the system state vector,

yf,k ∈ Rm is the system output, uf,k ∈ R

r denotes the

system input, fk ∈ Rs, (s ≤ m) is the fault vector and

αjk(θ) = α(θj(k), k) and θj(k) is the value of θj at the

sample k, see [25], [26] for more details about LPV polytopic

representation. Here Aj ∈ Rn×n, Bj ∈ R

n×r, Cj ∈

Rm×nand Lj ∈ R

n×s are time-invariant matrices defined

for j-th model. The polytopic system is scheduled through

functions designed as follows: αjk(θ), ∀j ∈ [1, . . . , N ] lie in

a convex set:

Ω =αj

k(θ) ∈ RN , αk(θ) =

[α1

k(θ), . . . , αNk (θ)

]T,

αjk(θ) ≥ 0, ∀j,

N∑

j=1

αjk(θ) = 1

. (8)

The polytopic parameters are calculated by fast algorithms

with the scheduling variable θ where each linear model Mi,

i = 1, . . . , N has follow time-varying parameter:

αjk (θ) =

l∏

v=1

ϑjv (θm) with

ϑjv (θm) =

0 if θ /∈

[θj

v, θj+1v

]

ϑjv if θ ∈

[θj

v, θj+1v

] (9)

where ϑjv =

(θj+1v −θv)

(θj+1v −θ

jv)

and v is the number of scheduling

variables.

The main objective of this paper is to propose a novel

FTC control strategy which can be used for determining the

system input uf,k such that:

• the control loop for the system (6)-(7) is stable,

• xf,k+1 converges asymptotically to xk+1 independently

of the presence of the fault fk.

The subsequent part of this section shows the development

details of the scheme that is able to settle such a challenging

problem.

The crucial idea is to use the following control strategy

proposed by [22]:

uf,k = −Sfk + K1(xk − xf,k) + uk, (10)

where fk is the fault estimate. The purpose of first factor

(Sfk) is estimate the fault, the purpose of the term K1(xk −

xf,k) is eliminate the estimation error and uk is the control

input without fault. Since it is assumed that xk and xf,k are

not available, the estimators xk and xf,k can be used instead.

Thus, the following problems arise:

• to determine fk,

• to design K1 in such a way that the control loop is

stable, i.e. the stabilization problem.

A. Fault Identification

Additive actuator fault in (6) can be identified considering

that it is an unknown input. This enable to use existing results

on unknown input observer (UIO) theory [27],[28] to identify

a fault. The applications of these results require the following

rank condition condition is satisfied:

rank(CL) = rank(L) = s. (11)

This implies that it is possible to calculate H = (CL)+ =[(CL)T CL

]−1

(CL)T . As it is suggested by [28] for the LTI

case, multiplying (7) by H and then substituting (6) in each

polytope vertex (3), the fault estimate can be expressed as:

fk =

N∑

j=1

αjkHj(yf,k+1 − CjAjxf,k − CjBjuf,k). (12)

for the LPV case.

Thus, if xf,k is used instead of xf,k then the fault estimate

is given as follows:

fk =

N∑

j=1

αjkHj(yf,k+1 − CjAj xf,k − CjBjuf,k), (13)

and the associated fault estimation error is:

fk − fk = −

N∑

j=1

αjkHjCjAj(xf,k − xf,k). (14)

Unfortunately, the crucial problem with practical imple-

mentation of (13) is that it requires yf,k+1 and uf,k to

1205

calculate fk and hence it cannot be directly used to obtain

(10). To settle this problem, it is assumed that there exists

a diagonal matrix βk such that fk = βkfk−1 and hence the

practical form of (10) boils down to:

uf,k =

N∑

j=1

αjk

[−Sjβkfk−1 + K1,j(xk − xf,k)

]+uk. (15)

B. Stabilization Problem

By substituting (10) into (6) it can be shown that:

xf,k+1 =

N∑

j=1

αjk

[Ajxf,k − BjSfk

+BjK1,jek + Bjuk + Ljfk] , (16)

where ek = xk − xf,k stands for the tracking error. Let us

assume that S satisfies the following equality BS = L, e.g.

S = I for actuator faults. Thus:

xf,k+1 =

N∑

j=1

αjk

[Ajxf,k + Lj(fk − fk)

+BjK1,jek + Bjuk] . (17)

Finally, by substituting (14) into (17) and then applying

the result into ek+1 = xk+1 − xf,k+1 yields:

ek+1 =N∑

j=1

αjk [(Aj − BjK1,j)ek

+LjHjCjAjef,k] . (18)

where ef,k = xf,k−xf,k stands for the state estimation error.

C. Observer design

As was already mentioned, the fault estimate (13) is

obtained based on the state estimate xf,k . This raises the

necessity for an observer design. Consequently, by substitut-

ing (12) into (6) it is possible to show that

xf,k+1 =N∑

j=1

αjk

[Ajxf,k + Bjuf,k + Ljyf,k+1

], (19)

where Aj = (I − LjHjCj)Aj , Bj = (I − LjHjCj)Bj and

Lj = LjHj .

Thus, the observer structure, which can be perceived as

an unknown input observer (see, e.g. [27], [28]), is given by

xf,k+1 =

N∑

j=1

αjk

[Aj xf,k + Bjuf,k + Ljyf,k+1+

+K2,j(yf,k − Cj xf,k)] . (20)

Finally, the state estimation error can be written as follows:

ef,k+1 =

N∑

j=1

αjk(Aj − K2,jCj)ef,k. (21)

III. FAULT TOLERANT CONTROL DESIGN FOR LPV

SYSTEMS

The main objective of this section is to summarize the

presented results within an integrated framework for the

development of fault identification and fault-tolerant control

scheme. First, let us start with the following two crucial

assumptions that are required to apply existing results on

LPV gain scheduling control (see [24]):

• the pair ( ¯A(θ), C(θ)) is detectable,

• the pair (A(θ), B(θ)) is stabilizable.

for all θ ∈ Θ.Moreover, note that the first assumption is typical for

unknown input observers (see [29] and references therein).

Under these assumptions, it is possible to design the matrices

K1,j and K2,j in such a way that the extended error:

ek =

[ek

ef,k

], (22)

described by

ek+1 =N∑

j=1

αjk

[Aj − BjK1,j LjHjCjAj

0 Aj − K2,jCj

]ek

=

N∑

j=1

αjkA0,j ek, (23)

converges asymptotically to zero.

The detectability and stability of the closed-loop system

is design the problem using an efficient LMI pole placement

technique. For many problems, an exact pole assignment may

not be necessary and it suffices to locate the poles of the

closed-loop system in a subregion of the unit circle [23],

[11].

Consequently, define a disk region LMI called D included

in the unit circle with an affix (−q, 0) and a radius r such that

(q+r) < 1. These two scalars q and r are used to determine

a specific region included in the unit circle so as to place

closed-loop system eigenvalues. The pole placement of the

closed-loop system (6)-(7) for all models j ∈ [1, . . . , N ] in

the LMI region can be expressed as follows:(

−rXj qXj + (A0,jXj)T

qXj + A0,jXj −rXj

)< 0, (24)

A0,j is stable if and only if there exists a symmetric matrix

such that Xj = XTj > 0.

Assuming that Xj has the block diagonal form Xj =diag(X1,j, X2,j) and defining A0,j of (23) as:

A0,j =

(Aj − BjK1,j LjHjCjAj

0 ATj − CT

j KT2,j

). (25)

It can be observed from the structure of A0,j in (25)

that the eigenvalues of the matrix A0,j are the union of

Aj −BjK1,j and ATj −CT

j KT2,j . This clearly indicates that

the design of the state feedback and the observer can be

carried out independently (separation principle). Thus, the

inequalities are:

1206

»

−rX1,j qX1,j + XT1,j

(ATj− KT

1,jBT

j)

(q + Aj − BjK1,j)X1,j −rX1,j

–

< 0,

(26)"

−rX2,j qX2,j + XT2,j(Aj − K2,jCj)

(q + ATj − CT

j KT2,j)X2,j −rX2,j

#

< 0.

(27)

We should note that expression (26) and (27) are Bilinear

Matrix Inequalities (BMIs) which cannot be solved with

classical tools, but substituting W1,j = K1,jX1,j and W2,j =KT

2,jX2,j it is possible to transform into:»

−rX1,j qX1,j + XT1,j

ATj− W T

1,jBT

j

(q + Aj)X1,j − BjW1,j −rX1,j

–

< 0,

(28)"

−rX2,j qX2,j + XT2,jAj − W T

2,jCj

(q + ATj )X2,j − CT

j W2,j −rX2,j

#

< 0.

(29)

Finally, the design procedure boils down to solving

the LMIs (28) and (28), and then determining K1,j =

W1,j(X1,j)−1 and K2,j =

(W2,j(X2,j)

−1)T

.

IV. ILLUSTRATIVE EXAMPLE

The proposed active FTC for LPV systems presented in

this paper was implemented and tested on a two-degree of

freedom helicopter, which is called a twin-rotor multiple

input multiple output (MIMO) system (TRMS).

The main objective of this section is to extend the ap-

proach proposed in section (II) and (III) using an simulation

of TRMS. In particular, the proposed design procedure was

applied to obtain K1,j and K2,j and then (15) (with βk = I)

was applied as a control strategy.

A. Description of Twin-Rotor Multiple Input Multiple Output

System

The application of this research is based on a two degree

of freedom helicopter equipment (twin-rotor multiple input

multiple output (MIMO) system (TRMS) is a laboratory

set-up developed by Feedback Instruments Limited [30])

available at the laboratories of the Advanced Control Systems

Research in the Automatic Control Department (ESAII) of

Technical University of Catalonia (UPC).

The modeling of UAVs and helicopters, which resemble

the characteristics of a TRMS in certain aspects, has also

been reported comprehensively in the literature. For instance,

dynamic general modeling and feedback control of a side-

by-side rotor tandem helicopter have been proposed in [31].

The mathematical model of the TRMS becomes a sets

of four non-linear differential equations with two linear

differential equations and four non-linear functions [30].

Some of the parameters of this model can be obtained from

[30], while some others should be obtained by experiments

such as: magnitudes of physical propeller, length, mass,

inertia, coefficients of friction, and impulse force. The system

can be defined by the input vector U = [uh, uv]T where uh is

the input voltage of the tail motor and uv is the input voltage

of the main motor. Also can be defined by the output vector

Fig. 1. Aero-dynamical model of the Twin Rotor MIMO System

Y = [Ωh, αh, ωt, Ωv, αv, ωm]T where Ωh is the angular

velocity around the vertical axis, αh is the azimuth angle

of beam, ωt is the rotational velocity of the tail rotor, Ωv

is the angular velocity around the horizontal axis, αv is the

pitch angle of beam and ωm is the rotational velocity of the

main rotor.

B. LPV Model

An alternative way to obtain an LPV model is to linearize

the non-linear system around of different operating points

(uh = 0, u1v = 0, u2

v = 0.05, u3v = 0.1 and u4

v = 0.15).

Consider a system described by N = 4 models, these four

models should be adapted from an LPV model. The discrete

state space representation (4)-(5) and (6)-(7) consists in the

following matrices:

A1 =

2

6

6

6

6

6

4

0.9812 −0.0105 0.1847 0 0 0

0 0.9657 0 0 0 0

0 0 0.878 0 0 0

0 0.0152 −0.0254 0.9908 −0.1718 0

0 0.0004 0.1367 0.0498 0.9957 0

0.0495 0.0276 0.0047 0 0 1

3

7

7

7

7

7

5

,

A2 =

2

6

6

6

6

6

4

0.9814 −0.0103 0.1841 0 0.0004 0

0 0.9657 0 0 0 0

0 0 0.878 0 0 0

0 0.02 −0.0254 0.9908 −0.1718 0

0 0.0005 0.1367 0.0498 0.9957 0

0.0495 0.0274 0.0046 0 −0.001 1

3

7

7

7

7

7

5

,

A3 =

2

6

6

6

6

6

4

0.9818 −0.0098 0.183 0 0.0007 0

0 0.9657 0 0 0 0

0 0 0.878 0 0 0

0 0.0405 −0.0254 0.9908 −0.1718 0

0 0.001 0.1367 0.0498 0.9957 0

0.0495 0.0268 0.0045 −0.0001 −0.002 1

3

7

7

7

7

7

5

,

A4 =

2

6

6

6

6

6

4

0.9826 −0.009 0.1809 0 0.001 0

0 0.9657 0 0 0 0

0 0 0.878 0 0 0

0 0.0734 −0.0254 0.9908 −0.1717 0

0 0.0018 0.1367 0.0498 0.9957 0

0.0496 0.0256 0.0044 −0.0001 −0.003 1

3

7

7

7

7

7

5

,

B1 =

2

6

6

6

6

6

4

47.2 −2.60.0 491.4

468.8 0.0−5.4 5.0

35.0 0.10.8 6.9

3

7

7

7

7

7

5

× 10−4, B2 =

2

6

6

6

6

6

4

46.9 −2.50.0 491.4

468.8 0.0−5.4 10.2

35.0 0.20.8 6.8

3

7

7

7

7

7

5

× 10−4,

B3 =

2

6

6

6

6

6

4

46.3 −2.30.0 491.4

468.8 0.0−5.4 18.5

35.0 0.30.7 6.5

3

7

7

7

7

7

5

× 10−4, B4 =

2

6

6

6

6

6

4

45.4 −2.00.0 491.4

468.8 0.0−5.4 28.4

35.0 0.50.7 6.0

3

7

7

7

7

7

5

× 10−4,

1207

C1 =

2

4

0 0 0 0 0 1

0 0 0 0 1 0

0 896.24 0 0 0 0

3

5 ,

C2 =

2

4

0 0 0 0 0 1

0 0 0 0 1 0

0 894.14 0 0 0 0

3

5 ,

C3 =

2

4

0 0 0 0 0 1

0 0 0 0 1 0

0 879 0 0 0 0

3

5 ,

C4 =

2

4

0 0 0 0 0 1

0 0 0 0 1 0

0 851.01 0 0 0 0

3

5 ,

L1 = B1, L2 = B2, L3 = B3, L4 = B4.

where the scheduling variable is the azimuth angle of beam,

which can be determined from (9).

C. Fault scenarios

The proposed design procedure was applied to obtain K1,j

with the parameters q = 0 and r = 0.8 and K2,j with the

parameters q = 0 and r = 0.2. The fault scenarios and the

results are presented as follows:

1) Input voltage fault of the tail motor:

fh = fk,1 =

0, for k < 3040−0.005, for 3040 ≤ k < 3800

0, for k ≥ 3800. (30)

where k = 0, . . . , 4000 with T = 0.05s.

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T

−5

0

5

x 10−3

Time

Co

ntr

ol In

pu

t

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T−5

0

5

10x 10

−3

Time

Fau

lt

Fault Fault estimate

Fig. 2. (Top) Control input voltage of the tail motor. (Bottom) Input voltagefault of the tail motor and its estimate

As a result, the Fig. 2 shows the result of input voltage

fault of the tail motor and its estimate, the estimator presents

deviation of nominal value due to changes in reference. The

fault is presented in the response of the second change of

reference. Estimation achieve the real value in approximately

250 samples after the fault presence.

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T

−0.6

−0.4

−0.2

0

0.2

Time

Azim

uth

an

gle

of

beam

With FTC

Without FTC

Reference

Fig. 3. Azimuth angle of beam (horizontal position) with Fault TolerantControl, without Fault Tolerant Control and its Reference

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T

−0.93

−0.925

−0.92

−0.915

−0.91

Time

Pit

ch

an

gle

of

beam

With FTC

Without FTC

Reference

Fig. 4. Pitch angle of beam (vertical position) with Fault Tolerant Control,without Fault Tolerant Control and its Reference

The Fig. 3 represents the performance of control strategy

proposed and the response of the azimuth angle of beam

without control. The system is stabilized with the FTC in

spite of actuator fault. Fig. 4 show the pitch angle of beam

and its trajectory was not change significantly after the fault.

2) Input voltage fault of the main motor:

fv = fk,2 =

0, for k < 14000.025, for 1400 ≤ k < 2200

0, for k ≥ 2200. (31)

where k = 0, . . . , 4000 with T = 0.05s.

The Fig. 5 illustrates that the input voltage fault of the

main motor can be estimated with very high accuracy.

Figs. 6-7 represent the azimuth and the pitch angle of

the beam, respectively. The evolution was not change with

control strategy in fault presence.

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T0

0.05

0.1

Time

Co

ntr

ol In

pu

t

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T0

0.01

0.02

0.03

Time

Fau

lt

Fault

Fault estimate

Fig. 5. (Top) Control input voltage of the main motor. (Bottom) Inputvoltage fault of the tail motor and its estimate

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T0

0.05

0.1

0.15

0.2

Time

Azim

uth

an

gle

of

beam

With FTC

Without FTC

Reference

Fig. 6. Azimuth angle of beam (horizontal position) with Fault TolerantControl, without Fault Tolerant Control and its Reference

V. CONCLUSIONS

An active FTC strategy for LPV system has been presented

in this paper. First, this new approach has been developed in

the context of linear systems and emphasizes the importance

1208

0 500T 1000T 1500T 2000T 2500T 3000T 3500T 4000T

−0.93

−0.925

−0.92

−0.915

−0.91

Time

Pit

ch

an

gle

of

beam

With FTC

Without FTC

Reference

Fig. 7. Pitch angle of beam (vertical position) with Fault Tolerant Control,without Fault Tolerant Control and its Reference

of non-linear system based on LPV representation. Con-

trollers are designed for each separate linear model through

LMIs pole placement. This method is suitable for partial ac-

tuator faults. The proposed approach is an integrated FTC de-

sign procedure of the fault identification and control scheme.

Fault identification is based on the use of an observer.

Once the fault has been identified, the FTC controller is

implemented as state feedback controllers. These controllers

are designed for each separate linear model through LMIs

pole placement. The proposed design of controllers place the

eigenvalues of the closed-loop system in a predetermined

LMI region inside the unit circle. The main contribution

is the development a more practical active FTC for non-

linear system based on a polytopic LPV representation.

The illustrative example of a polytopic LPV system was

presented to show the effectiveness of the proposed active

FTC approach.

VI. ACKNOWLEDGMENTS

This work has been funded by the grant CICYT DPI2005-

04722 of Spanish Ministry of Education and by Research

Comission of the Generalitat de Catalunya (group SAC ref.

2005SGR00537). The work was also financed as a research

project with the science funds for years 2007-2010.

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