Procedia Engineering 15 (2011) 979 – 983

1877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2011.08.181

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Procedia Engineering 00 (2011) 000–000

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Advanced in Control Engineering and Information Science

Robust H ∞ Fault–tolerant Control against Sensor and Actuator Failures for Uncertain Descriptor Systems

Bing Liang a, Tian-qing Chang, Guo-sheng Wang a*Department of Control Engineering, Academy of Armored Force Engineering, Beijing, 100072, China

Abstract

HBased on ∞ theory in descriptor systems, a method of designing output-feedback dynamical compensators for uncertain descriptor systems with sensor and actuator failures is presented. The dynamical compensator designed by the proposed method can guarantee that the resultant closed-loop uncertain descriptor system is regular, impulse-free, stable and keeps certain H∞

H∞

norm performance in the normal condition as well as in the event of sensor and actuator failures. A numerical example shows the effect of the proposed method.

© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [CEIS 2011]

Keywords: Fault-tolerant control, control, uncertain descriptor systems, sensor failures, actuator failures

1. Introduction

Descriptor systems capture the dynamic behaviour of many natural phenomena, and have wide applications in many fields, such as network theory and robotics, and have attracted much attention in recent years. In practical applications, parameter uncertainties contained in the dynamical behaviour of many physical processes are unavoidable and failures of control components often occur. Therefore to design robust fault-tolerant controllers for uncertain descriptor systems is necessary. There are some

* Corresponding author. E-mail address: gswang@126.com.

980 Bing Liang et al. / Procedia Engineering 15 (2011) 979 – 9832 Bing Liang ,et al/ Procedia Engineering 00 (2011) 000–000

[ ]⎪⎪⎩

⎪⎪⎨

⎧

=

+=++Δ+=

TTT uHxz

wCxyGwBuxAAxE

)(

)]([

1

0σ&

articles to discuss the problem of state-feedback robust fault-tolerant control for descriptor systems, see for instance [1-4]. For state-feedback control approaches, all system state variables should be available. However, this is not the case for most practical systems. For certain design problems, the usual static feedback controllers cannot meet the required design criteria. Therefore dynamical compensators are considered. In the other hand, these papers only consider sensor failures or actuator failures. However few articles so far appear to discuss fault-tolerant control against simultaneous failures of sensor and actuator by dynamical compensators for uncertain descriptor systems.

2. Problem Formulation

Consider the following descriptor linear system

, (1)

where , , and nx R∈ mu R∈ py R∈ qz R∈ are the state vector, the input vector, the measured output

vector, and the controlled output vector, respectively; and are the disturbance input

vectors; and G are known matrices of appropriate dimensions with ;

sw R∈0pw R∈1

,nn×R B,, AE ∈nn

HC, nrE ≤=rankA ×∈Δ R is a uncertain parameter matrix and is assumed to be of the following form:

NMLA )(σ=Δ , (2a) where σ is an uncertain parameter vector, )(σL satisfies

, (2b) ILL T 2)()( ρσσ ≤and M , are known real constant matrices with appropriate dimensions. For convenience, N AΔ is said to be admissible uncertainty if (2) holds.

We consider an output-feedback dynamical compensator in the following form:

. (3) ⎪⎩

⎪⎨⎧

=+=

ξξξ

KuyBAE ˆˆ&

Applying (3) to system (1), gives the following closed-loop descriptor system:

. (4)

[ ]⎪⎪⎩

⎪⎪⎨

⎧

=

++=

++Δ+=

TTT KHxz

wBCxBAE

GwBKxAAxE

)()(

ˆˆˆ)(

1

0

ξ

ξξ

ξ&

&

Let

ξ−= xe , , .⎥⎦

⎤⎢⎣

⎡=

MM

M 0 [ ]00 NN =

Then system (4) can be rewritten as

, (5) ⎩⎨⎧

=+Δ+=

cc

ccccccc

xHzwGxAAxE )(&

where

⎥⎦

⎤⎢⎣

⎡=

ex

xc , , , , ,⎥⎦

⎤⎢⎣

⎡=

ww

wc0

⎥⎦

⎤⎢⎣

⎡=

EE

Ec 00

⎥⎦

⎤⎢⎣

⎡−

=BG

GGc ˆ

0⎥⎦

⎤⎢⎣

⎡−

=KK

HH c

0

981Bing Liang et al. / Procedia Engineering 15 (2011) 979 – 983Author name / Procedia Engineering 00 (2011) 000–000 3

⎥⎦

⎤⎢⎣

⎡−−−+

−+=

BKACBABKABKBKA

Ac ˆˆˆ , .00 LNMAc =Δ

Now, concepts of sensor and actuator failures are discussed. The system (1) has sensors and m actuators, denoted by and , respectively. Define a set of sensor failures as

of system (1) , and a set of actuator failures as

p},2,1{ pL=Ω },2,1{ mL=Σ

Σ⊂Λ Ω⊂F , Introduce the following decomposition FF BBB += , ΛΛ += CCC , (6)

where is formed from B by zeroing out columns corresponding to FB F , formed from by zeroing out rows corresponding to Λ ,

ΛC C

FF BBB −= and ΛΛ −= CCC . Similar to (6), we have the following decomposition

FF KKK += , ΛΛ += BBB ˆˆˆ , (7)

where is formed from FK K by zeroing out rows corresponding to F , and formed from by zeroing out columns corresponding to .

ΛB BΛ

In the case of sensor and actuator failures according to Γ and F respectively, system (1) and (5) may be rewritten as

(8)

[ ]⎪⎪⎩

⎪⎪⎨

⎧

=

+=++=

ΛTTT

F

uHxz

wxCyGwuBAxxE

)(

0&

and

, (9) ⎩⎨⎧

=+Δ+=

ccF

ccFcccFcc

xHzwGxAAxE )(&

where

⎥⎦

⎤⎢⎣

⎡−−−+

−+=

ΛΛ FFFF

FFFFcF KBACBAKBA

KBKBAA ˆˆˆ , , .⎥

⎦

⎤⎢⎣

⎡−

=ΛBG

GGcF ˆ

0⎥⎦

⎤⎢⎣

⎡−

=FF

cF KKH

H0

For convenience, we call system (1) the normal descriptor linear system and system (5) the normal closed-loop descriptor system. Correspondingly, (8) is called the fault descriptor system and (9) is called the fault closed-loop descriptor system.

With the above description, the problem about fault-tolerant control for uncertain descriptor systems to be studied in this paper can be described as follows.

∞H

Problem RFT (Robust Fault-Tolerant): Given the descriptor system (1), the system (8) and a constant , determine real matrices 0>a nmK ×∈ R , nnA ×∈ Rˆ and pnB ×∈ Rˆ in the output-feedback dynamical

compensator (3) such that the normal closed-loop system (5) and the fault closed-loop system (9) are stable, impulse-free, regular for all admissible uncertainties and the transfer functions

cccccc GAAsEHT )( Δ−−=and

cFccFcccF GAAsEHT )( Δ−−=

satisfy aTc < and aTcF < , respectively.

982 Bing Liang et al. / Procedia Engineering 15 (2011) 979 – 9834 Bing Liang ,et al/ Procedia Engineering 00 (2011) 000–000

3. Robust ∞H Fault-Tolerant Controller Design

First, let us introduce some concepts that are fundamental in our development. Definition 1：(i) The descriptor system (1) or the matrix pair is regular if is not identically zero.(ii) The descriptor system (1) or the matrix pair is impulse-free if . (iii)The descriptor system (1) or the matrix pair ( is admissible, if it is regular, impulse-free and stable.

),( AE )det( AsE −),( AE

), AEEAsE rank)det(deg =−

In order to solve Problem RFT, we present the following results, which will be used in the proof of our main result. Lemma 1[5]： Given the descriptor system

⎩⎨⎧

=+=

CxyBuAxxE&

,

then the system is admissible and γ<∞|||| G if and only if there exists nnX ×∈ R such that

0≥= EXXE TT , 012 <+++ XBBXCCAXXA TTTTT

γ,

where andBAsECG 1)( −−= 0>γ is a constant.

Lemma 2[6] Given matrices X, Y, F of appropriate dimensions, if IFFT ≤ , then for arbitrary constant 0>λ , the following inequality holds：

YYXXXFYXFY TTTTT

λλ 1+≤+ .

Based on Lemmas 1 and 2, we can give a sufficient condition for the existence of solutions to Problem RFT. Theorem 1：Given the descriptor system (1) with the uncertain parameter matrix satisfying (2), the system (8) with sensor failures and a constant .a1) Then Problem RFT has a solution if there exists a matrix nnP ×∈ R and an inverse matrix

satisfying the following conditions: nnQ ×∈ R

, (12) 0≥= EPPE TT

2 22

1( ) ( )T T T T T T T T T T TF F F FA P P A P B B B B P a C C P GG a MM P H H N N

aρΛ Λ+ − − + + + + + <2 0 , (13)

, (14) 0≥= EQQE TT

2 22

1( 2 )T T T T T T T T T TF FA Q Q A Q GG a MM a B B Q P BB P C C

a+ + + + + − <% % 0 , (15)

where

PMMPGGa

AA TT ++= 21~ . (16)

2) When the conditions (14)-(17) are met, one such dynamical compensator of the form of (3) is given by

CCQPBBMMGGa

AA TTTTT −−−++= )1(ˆ2 , (17)

(18) TT CQB −=ˆ

andPBK T−= . (19)

983Bing Liang et al. / Procedia Engineering 15 (2011) 979 – 983Author name / Procedia Engineering 00 (2011) 000–000 5

4. A numerical example

Consider a descriptor system in the form of (1) with the following parameters

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

000010001

E , , , , , , ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

100001011

A⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

011202

B⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

1.0001.0

G ⎥⎦

⎤⎢⎣

⎡=

172056

C [ ]010=H

0.1 0sin 0 0.1 0.05 0.05

( ) ( ) 0 0.050 cos 0 0 0.1

0 0.1A ML N

σσ σ

σ

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥Δ = = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

.

It is easy to verify that . When the second actuator and the first sensor of the system fails, denoted by and , then the matrices and in form of (6) are

ILL T ≤)()( σσ}1{=Λ}2{=F FB ΛC

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

010202

FB , .⎥⎦

⎤⎢⎣

⎡=Λ 172

000C

We choose

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

4/100020001

P , .⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−=

16000140417

Q

It is easy to prove that the matrices satisfy (14)-(17) ( /2). By Theorem 1, we can obtain

matrices

QP, 1=a

K , and B A :

, , .⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−−−

×=0013.00044.00021.01275.04440.19110.00305.03470.02210.0

10ˆ 3A⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

0625.001271093026

B ⎥⎦

⎤⎢⎣

⎡−

−−−=

02025.042

K

References [1] [Duan G R, Liang B. Reconfiguring descriptor linear systems by state feedback eigenstructure assignment: a

parametric approach. Dynamics of Continuous Discrete and Impulsive Systems-Series A-Mathematical Analysis; 2006, vol. 13. p. 1085-1092

[2] Hu G, Ren J C, Xie S. Robust fault-tolerant control for a class of nonlinear singular systems with uncertainty. International Conference Physics and Control, Saint Petersburg, Russia; 2003, p. 1377-1381

[3] Yao B, Zhang Q L, Chen Y P. Robust quadratic stability and stabilization with integrity for uncertain discrete descriptor systems. Proceedings of the 4th World Congress on Intelligent Control and Automation, Shanghai, P. R. China; 2002, p. 1706-1710.

[4] Hu G, Xie X S. Roust fault-tolerant control for singular systems with uncertain parameters. Journal of Shanghai Maritime University; 2001, vol. 22, no. 3, p. 146-148.

[5] Masubuchi I, Kamitanc Y, Ohara A. ∞H control for descriptor systems: a matrix inequalities approach. Automatica; 1997, vol. 33, no. 4, p. 669-673.

[6] Zhao M Y, Liu H P. Fault Tolerant Control for Networked Control Systems with Packet Loss and Time Delay . International Journal of Automation and Computing; 2011, vol. 8, no. 2, p. 244-253.