Research ArticleReconfigurability Analysis Method forSpacecraft Autonomous Control

Dayi Wang and Chengrui Liu

Beijing Institute of Control Engineering Beijing 100190 China

Correspondence should be addressed to Chengrui Liu liuchengruigmailcom

Received 11 December 2013 Accepted 19 March 2014 Published 10 April 2014

Academic Editor Xiaojie Su

Copyright copy 2014 D Wang and C LiuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a critical requirement for spacecraft autonomous control reconfigurability should be considered in design stage of spacecraftsby involving effective reconfigurability analysis method in guiding system designs In this paper a novel reconfigurability analysismethod is proposed for spacecraft design First some basic definitions regarding spacecraft reconfigurability are given Thenbased on function tree theory a reconfigurability modeling approach is established to properly describe systemrsquos reconfigurabilitycharacteristics and corresponding analysis procedure based on minimal cut set and minimal path set is further presented Inaddition indexes of fault reconfigurable degree and system reconfigurable rate for evaluating reconfigurability are defined and themethodology for analyzing systemrsquos week links is also constructed Finally the method is verified by a spacecraft attitudemeasuringsystem and the results show that the presented method cannot only implement the quantitative reconfigurability evaluations butalso find the weak links and therefore provides significant improvements for spacecraft reconfigurability design

1 Introduction

Nowadays autonomous control has become a key technologyfor increasing spacecraft survival capabilityThe reason is thatautonomous control regarding fault detection identificationand reconfiguration will be automatically activated to reducethe fault effect when faults emerge in a spacecraft Thereforehow to increase the ability of fault processing has become akey issue for autonomous control of spacecraft However itcan be concluded bymany recent serious spacecraft incidentsthat certain deficiencies exist in their fault diagnosis andprocessing procedure Further analysis reveals that thesedeficiencies are caused by reconfigurability lack of spacecraftFrom this viewpoint excellent reconfigurability has beenbecoming more and more critical for autonomous controlto ensure the increasing requirements of spacecraft safetyand reliability In order to improve spacecraft autonomouscontrol ability of tolerating faults reconfigurability shouldbe considered in design stage of spacecrafts and effectivereconfigurability analysis methodmust be presented to guidethe system design

As far as the authors know regarding reconfigurabilitydesign mass research aiming at enhancing flexibility about

environment changes and function variations has beenconducted in computing and manufacturing fields [1 2]For spacecraft although extensive attention to reconfigura-bility design has been devoted to controller designs afterfaults [3ndash9] or to system function changes [10] to satisfyother mission requirements little improvement has beenachieved regarding function recovery of faulty spacecraftby reconfigurability design Meanwhile some scholars havestudied control reconfigurability from the intrinsic andperformance-based perspectives The intrinsic reconfigura-bility of LTI systems can be evaluated by the controllabilityand observability Gramians [11] or by the smallest second-order mode which is the smallest eigenvalue of the com-bination of controllability and observability Gramians [12]The performance-based control reconfigurability is regardedas the ability of the considered system to keeprecoversome admissible system performance when certain faultoccurs Staroswiecki discussed the reconfigurability underenergy limitation constraints in [13] However all the studiesmentioned above did not consider systemrsquos components andconfiguration and thus they cannot settle reconfigurabilityanalysis and design problems for complex systems such asspacecrafts Consequently the critical objective of this study

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 724235 8 pageshttpdxdoiorg1011552014724235

2 Mathematical Problems in Engineering

is to construct an effective reconfigurability analysis methodbased on the function tree theory which can synthesizecomponents and reconfiguration strategies of spacecraft andestimate quantitative evaluation indexes

The rest of this paper is organized as follows Section 2presents some basic definitions and Section 3 constructsa reconfigurability modeling and analyzing method InSections 4 and 5 reconfigurability evaluation indexes andweak link analysis procedure for reconfiguration design arediscussed respectively In Section 6 the proposed approachis illustrated by a practical application regarding spacecraftattitude measuring system Some conclusions and relevantremarks are given in Section 7

2 Basic Definitions

Siddiqi indicated that different definitions exist in differentfields in [14] By summing up a series of definitions hedefined reconfigurable system and reconfigurability as fol-lows Reconfigurable system is a system that can reversiblyachieve distinct configurations (or states) through alterationof system form or function in order to achieve a desiredoutcome within acceptable reconfiguration while recon-figurability is a system architectural property that definesthe ease and extent to which a system is reconfigurableConsidering spacecraft reconfiguration is the problem ofreplacing the faulty part of the systemby anonfaulty one so asto still achieve control objectives and reconfigurability is theability of recovering all the functions or achieving degradedobjectives by reconfiguration when faults appear

System configuration is one of the basic factors that affectreconfigurability Two relevant definitions reconfigurationunit (RU) and minimal reconfiguration unit (MRU) shouldbe explained here RU is a combination of spacecraft compo-nents to achieve the anticipant function by reconfigurationitself or by switching to other RUs when the current RUfails MRU is a combination of spacecraft components toachieve the anticipant function only by switching to otherRUs when the current RU fails It is the minimal unit in thereconfiguration analysis

A novel reconfigurability model is established based onthe function tree theory in this study Function tree is a treediagram whose vertex corresponds to the system functionand whose branches are subfunctions decomposed fromthe system function and its roots are the MRUs Higherlevel functions and lower level functions in a function treeare connected by AND gates or OR gates The relationshipbetween function and MRUs can be clearly explained bythe corresponding function tree A typical function tree isillustrated in Figure 1

In order to evaluate the reconfigurability quantitativelydefinitions including cut set (CS) minimal cut set (MCS)path set (PS) and minimal path set (MPS) of a function treeare involved A CS is a set of MRUs When all MRUs in a CSare healthy the system functions can be achieved MCS is aspecial CS and if and only if all MRUs in MCS are in goodcondition the system functions can be achieved APS is also aset of MRUs When all MRUs in a PS fail the system will lose

System function

Higher level Higher level

Lowest levelLowest level

subfunction 1

subfunction 1 subfunction n

MRU MRU MRU MRU1 2 k minus 1 k

Vertex

Branches

Roots

middot middot middot

middot middot middot subfunction m

Figure 1 Function tree schematic diagram

its function MPS is a special PS and if and only if failureappears in every MRU in MPS the system function shouldhave been lost Furthermore theMCS set orMPS set is calledMCS family or MPS family

3 Reconfigurability Modeling

For reconfigurability evaluating and designing one firstneeds to build an effective reconfigurability model andestablish relationships between reconfigurability and MRUsThen evaluation indexes and weak links of the spacecraftreconfigurability can be analyzed

We define a reconfigurability model from viewpoint offunction tree which is similar to theory of fault tree Themodeling processes are discussed as below

Step 1 According to the system function define the recon-figuration strategy based on the system observability andcontrollability

For example consider the LTI deterministic system

(119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(1)

We adopt the observability criterion and controllability crite-rion

rank [119862 119862119860 sdot sdot sdot 119862119860

119899minus1

]

1015840

= 119899

rank [119861 119861119860 sdot sdot sdot 119861119860

119899minus1

] = 119899

(2)

to confirm the reconfiguration strategy by changing 119861 or119862 inthe system model and then obtain the component set 119862comeach one of which can perform the system function

Step 2 If any redundancy is involved in a system componentdecompose it to the functional module According to theredundancy relationship between themodules determine theMRUs Furthermore according to the MRUs functions theMRUs function set 119865MRU can be obtained And the elementsin 119865MRU are the lowest level function in the function tree

Mathematical Problems in Engineering 3

Power supply 1

Power supply 2

Data processing IO Gyro

sensor

RedundantGyro

MRU1 MRU2 MRU3

Structure decomposition

DeterminingMRU

Figure 2 Structure decomposition of gyro

Angle velocity measure

Power supply Measure and data process

Power supply 1

Power supply 2

Dataprocessing IO Gyro

sensor

Functiondecomposition

CorrespondingMRU

(MRU1)(MRU3)

(MRU2)

Figure 3 Function decomposition of gyro

To get a better understanding a gyro system is utilized asan example to illustrate this procedure A gyro can be decom-posed to several modules such as power supply module dataprocessing module IO module and gyro sensor module Ifthe power supply module is redundant while others are notany single power supply module can be considered as MRUand the rest can be treated as MRU Consequently 119865MRUof a gyro is 119901119900119908119890119903 119904119906119901119901119897119910 119898119890119886119904119906119903119890 119886119899119889 119889119886119905119886 119901119903119900119888119890119904119904Figure 2 shows the decomposition structure

Step 3 From the system function decompose higher levelfunctions into lower level functions (or subfunctions) untilthe functions are contained in 119865MRU

Return to the example of gyro ldquoAngle velocity measurerdquois the function of a gyro It can be decomposed into twosubfunctions ldquopower supplyrdquo and ldquomeasure and data processrdquoThen the decomposition process can be terminated becauseldquopower supplyrdquo and ldquomeasure and data processrdquo belong to119865MRU The decomposition process is illustrated in Figure 3

Step 4 Build a function tree by AND gate and OR gateThe vertex of this function tree is the system functionthe branches are the subfunctions and the roots are theMRUs AND gate and OR gate connect the higher layers andthe lower layers according to the relationship between thesubfunctions

AND gate and OR gate in function trees are depictedin Figure 4 The AND gate in Figure 4(a) shows that theupper level function 119884 can only be achieved when all thesubfunctions 119909

119894have been realized 119894 = 1 2 119899 while for

OR gate in Figure 4(b) it can be concluded that the upperlevel function 119884 can be realized when any single or multipleor all subfunctions 119909

119894are achieved 119894 = 1 2 119899

Y

x1 x2 xnmiddot middot middot

(a) AND gate

Y

x1 x2 xnmiddot middot middot

(b) OR gate

Figure 4 AND gate and OR gate

Angle velocity measure

Power supply Measure and data process

Vertex

Branches

RootsMRU1 MRU2 MRU3

Figure 5 Function tree of gyro

According to the stepsmentioned above the function treeof a gyro can be formed which is shown in Figure 5

In order to analyze the reconfigurability quantitativelythe MCS andMPS of function tree should be obtained firstly

Let C119894(119909

119895) denote the ith MCS for the jth level

function 119909

119895 and let C(119884) denote the CS family for the upper

level function 119884 For AND gate

C (119884) = C119894(119909

1) cup C119895(119909

2) cup sdot sdot sdot cup C

119896(119909

119899)

119894 isin (1 2

1003816

1003816

1003816

1003816

C (119909

1)

1003816

1003816

1003816

1003816

)

119895 isin (1 2 sdot sdot sdot

1003816

1003816

1003816

1003816

C (119909

2)

1003816

1003816

1003816

1003816

)

119896 isin (1 2 sdot sdot sdot

1003816

1003816

1003816

1003816

C (119909

119899)

1003816

1003816

1003816

1003816

)

(3)

For OR gate

C (119884) = C (119909

1) cup C (119909

2) cup sdot sdot sdot cup C (119909

119899) (4)

where |C(119909

119894)| 119894 = 1 2 119899 is the cardinal number of C(119909

119894)

which indicates MCS number in the MCS family for thesubfunction 119909

119894

Let R119894(119909

119895) be the 119894th MPS for the 119895th level function 119909

119895

and let R(119884) be the PS family of the upper level function 119884For AND gate

R (119884) = R (119909

1) cupR (119909

2) cup sdot sdot sdot cupR (119909

119899) (5)

For OR gate

R (119884) = R119894(119909

1) cup R119895(119909

2) cup sdot sdot sdot cup R

119896(119909

119899)

119894 isin (1 2

1003816

1003816

1003816

1003816

R (119909

1)

1003816

1003816

1003816

1003816

)

119895 isin (1 2

1003816

1003816

1003816

1003816

R (119909

2)

1003816

1003816

1003816

1003816

)

119896 isin (1 2

1003816

1003816

1003816

1003816

R (119909

119899)

1003816

1003816

1003816

1003816

)

(6)

4 Mathematical Problems in Engineering

where|R(119909

119894)| 119894 = 1 2 119899 is the cardinal number of R(119909

119894)

which corresponds to theMPS number of theMPS family forthe subfunction 119909

119894

Although C(119884) or R(119884) derived by (3) to (6) may not beMCS family or MPS family the MCS and MPS are neededin the upper level function analysis according to (3) to (6)Consequently the MCS and MPS of function 119884 can becalculated by the following steps

Step 1 Initialize Cmin(119884) or Rmin(119884) to be a null set

Step 2 ChooseCmin(119884) orRmin(119884)with a minimum cardinalnumber in all sets in C(119884) or R(119884) and transform it intoCmin(119884) or Rmin(119884)

Step 3 Check all remaining sets in C(119884) orR(119884) If there is aset containing all the MRUs in Cmin(119884) or Rmin(119884) delete itfrom C(119884) or R(119884) and go back to Step 2 otherwise

Step 4 Execute Steps 2 and 3 repeatedly until C(119884) or R(119884)

turns to a null set Then elements C119894(119884) or R

119894(119884) in Cmin(119884)

or Rmin(119884) are the expected MCS or MPS

4 Reconfigurability Evaluation Indexes

Based on the reconfigurability model constructed in thepreceding section reconfigurability evaluation indexes forspacecrafts are given as follows

41 Fault Reconfigurable Degree (FRD) FRD describeswhether the system has available resources and methods forreconfigurations after certain faults as

120574 =

1 fault is reconfigurable0 fault is unreconfigurable

(7)

When certain faults emerge the MCS family shouldbe activated by deleting all the MCSs including the faultreconfigurable units Consider 120574 = 0 if the MCS family isempty consider 120574 = 1 otherwise

42 System Reconfigurable Rate (SRR) SRR indicates the rateof reconfigurable faults with respect to all faults in the system

119903 =

sum

119898

119894=1119908

119894120574

119894

sum

119898

119894=1119908

119894

(8)

where 120574

119894is the FRD of the 119894th fault 119891

119894 119898 is the number

of all the system fault modes and 119908

119894is the weight of fault

119891

119894according to its severity and occurrence probability The

major fault has a bigger weight than aminor one and the faultwith high occurrence probability has a bigger weight thanthe one with low occurrence probability If the fault severitycan be defined as four levels as listed in Table 1 and theoccurrence probability can be divided into five levels as listedin Table 2 then119908

119894can be determined from Table 3 119878 denotes

the fault severity level and 119875 indicates the fault occurrenceprobability in Table 3

Table 1 Fault severity level definition

Level DefinitionI System function is lost or service life is shortened seriously

II System function is degraded seriously or service life isreduced by 14 to 12

III System function is degraded partially or service life isreduced below 14

IV There is little affection in system function and service life

Table 2 Fault occurrence probability definition

Level DefinitionA MRU fault probability ge 20 times total fault probability

B 20 times total fault probability gtMRU fault probability ge

10 times total fault probability

C 10 times total fault probability gtMRU fault probability ge 1times total fault probability

D 1 times total fault probability gtMRU fault probability ge 01times total fault probability

E MRU fault probability lt 01 times total fault probability

Table 3 119908119894matrix

119875

119878

I II III IVA 1 13 17 113B 12 15 19 116C 14 16 111 118D 18 110 114 119E 112 115 117 120

5 Weak Link Analysis inReconfigurability Design

For better reconfigurability the reconfiguration weak linksshould be improved in the design phase of a spacecraft Basedon the established configurability model the following twoindexes are proposed to determine weak links in reconfigu-ration

51 Importance Degree of MRU (IDMRU) IDMRU denotesthe rate of the number of MCSs that includes the MRU withrespect to the number of all MCSs as

119868

119872=

119873

119872

119873

119879

(9)

where 119868

119872is the IDMRU of MRU 119872 119873

119872is the number of

MCSs that comprise the MRU and 119873

119879is the number of all

MCSsFor any system the MRU with maximal IDMRU con-

tributes most in system function realization Consequentlynecessary redundancy or special reliability design should beconsidered for this MRU

52 System Fault Tolerance Degree (SFTD) SFTD representsthe maximal number of failure MRUs that the system can

Mathematical Problems in Engineering 5

tolerate without loss of system functions SFTD reflects thesystem reconfigurability as

119879 = min (

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

) minus 1

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

isin R 119894 = 1 2 |R| (10)

where 119879 denotes SFTD R119894is the 119894th minimal path set of the

function tree |R119894| is the cardinal number of R

119894

In a system the path set with the minimum numberof MPSs is the weakest link And for this part necessaryredundancy or special reliability design should be consideredaccording to the subfunctions of MRUs in the MPS

The four indexes proposed above are closely connectedto each other Let 119891

119894be a fault whose corresponding recon-

figurable degree is equal to zero 120574119894= 0 namely the corre-

sponding MRU cannot be reconfigured then the importancedegree 119868

119872of the MRU will be equal to one and the system

fault tolerance degree 119879 will become zero Otherwise if allfault reconfigurable degrees are one namely all theMRU canbe reconfigured thenwe can conclude that all the importancedegrees will be less than one the system fault tolerance degreewill be not less than one and the system reconfigurable ratewill be equal to 100

6 Empirical Results

In this section we focus on the practical performance ofthe proposed method Our experiment is presented for thereconfigurability analysis of an attitude measuring system ina spacecraft The dynamic functions regarding momentumdevices are shown in (11)The spacecraft is considered as rigidbody systems and the body coordinate system coincides withthe principle axes of inertia as

119868

119909

119909minus (119868

119910minus 119868

119911) 120596

119910120596

119911minus ℎ

119910120596

119911+ ℎ

119911120596

119910= minus

ℎ

119909+ 119879

119909

119868

119910

119910minus (119868

119911minus 119868

119909) 120596

119911120596

119909minus ℎ

119911120596

119909+ ℎ

119909120596

119911= minus

ℎ

119910+ 119879

119910

119868

119911

119911minus (119868

119909minus 119868

119910) 120596

119909120596

119910minus ℎ

119909120596

119910+ ℎ

119910120596

119909= minus

ℎ

119911+ 119879

119911

(11)

where 119868

119909 119868119910and 119868

119911are moments of inertia along axes 119874119909

119874119910 and 119874119911 respectively 120596 = [120596

119909 120596

119910 120596

119911]

119879 is the angularvelocity vector h = [ℎ

119909 ℎ

119910 ℎ

119911]

119879 is the synthesizing angularmomentum vector of all the momentum devices T =

[119879

119909 119879

119910 119879

119911]

119879 is the control torque vector applied on thespacecraft except for the torque from themomentumdevicesTherefore the control torque vector T = [119879

119909 119879

119910 119879

119911]

119879 in(11) includes torques from thrusters other space torques anddisturbing torques

If all attitudes vary in a small scale the dynamic functionscan be simplified as

120596

119909= minus 120596

0120595

120596

119910=

120579 minus 120596

0

120596

119911=

120595 + 120596

0120593

(12)

where 120593 120579 and 120595 are Euler angles 120596

0denotes the orbit

angular velocity with which the spacecraft circles around thecenter body

Then the linearization form of the attitude dynamicfunction can be derived based on (11) and (12) as

119868

119909 + [(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910] 120593

+ [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

120595

= minus

ℎ

119909+ 120596

0ℎ

119911+ 119879

119909

119868

119910

120579 + ℎ

119909(

120595 + 120596

0120593) minus ℎ

119911( minus 120596

0120595) = minus

ℎ

119910+ 119879

119910

119868

119909

120595 + [(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910] 120595

minus [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

= minus

ℎ

119911minus 120596

0ℎ

119909+ 119879

119911

(13)

Accordingly the dynamic function of the spacecraft canbe expressed by a state space form as shown in (1) with thefollowing notations

119909 = [120593 120579

120579 120595

120595]

119879

119860 =

[

[

[

[

[

[

[

[

0 1 0 0 0 0

119872

210 0 0 0 119872

26

0 0 0 1 0 0

119872

41119872

420 0 119872

45119872

46

0 0 0 0 0 1

0 119872

620 0 119872

650

]

]

]

]

]

]

]

]

119872

21= 119868

minus1

119909[(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910]

119872

26= 119868

minus1

119909[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

41= 119868

minus1

119910ℎ

119909120596

0

119872

42= minus119868

minus1

119910ℎ

119911

119872

45= 119868

minus1

119910ℎ

119911120596

0

119872

46= 119868

minus1

119910ℎ

119909

119872

62= minus119868

minus1

119911[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

65= 119868

minus1

119911[(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910]

(14)

Matrixes 119861 and 119862 in (1) can be determined accordingto the detailed configuration of the system For example asystem with two infrared earth sensors three orthogonalgyros and one main backup thruster can be described as

119906 (119905) = [119879

1199091119879

1199092119879

1199101119879

1199102119879

1199111119879

1199112]

119879

119910 (119905) = [120593

ℎ1120579

ℎ1120593

ℎ2120579

ℎ2119892

119909119892

119910119892

119911]

119879

6 Mathematical Problems in Engineering

119861 =

[

[

[

[

[

[

[

[

0 0 0 0 0 0

119868

minus1

119909119868

minus1

1199090 0 0 0

0 0 0 0 0 0

0 0 119868

minus1

119910119868

minus1

1199100 0

0 0 0 0 0 0

0 0 0 0 119868

minus1

119911119868

minus1

119911

]

]

]

]

]

]

]

]

119862 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

]

]

]

]

(15)

Considering a spacecraft system described by (1) whenfaults appear the premise of achieving system reconfigura-bility is that the remaining of the system is observable andcontrollable The corresponding criterion is given by (2)According to engineering experience one can assume that119868

119909= 119868

119910= 119868

119911and 120596

0= 0 Consider the following

(1) Only one infrared earth sensor is employed forattitude determination as

119862

1= [

1 0 0 0 0 0

0 0 1 0 0 0

] rank[

[

[

[

[

119862

1

119862

1119860

119862

1119860

5

]

]

]

]

]

= 6 (16)

(2) Three gyros are employed for attitude determinationas

119862

2=

[

[

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

rank[

[

[

[

[

119862

2

119862

2119860

119862

2119860

5

]

]

]

]

]

= 5 (17)

(3) One infrared earth sensor and three gyros areemployed for attitude determination as

119862

3=

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

rank[

[

[

[

[

119862

3

119862

3119860

119862

3119860

5

]

]

]

]

]

= 6 (18)

From (16) to (18) the attitude can be measured in thefollowing two ways

M1 by infrared earth sensorsM2 by infrared earth sensors and gyros

In addition it is assumed that two infrared earth sensorsshare one power supply and three gyros share another powersupply then Table 4 lists the MRUs and their correspondingsubfunctions

Table 4 MRUs and their corresponding functions

MRU FunctionsInfrared earth sensor power(ESP)

Power supply for infrared earthsensor (PS for ES)

Infrared earth sensor 1 (ES1) 120593 and 120579measureInfrared earth sensor 2 (ES2) 120593 and 120579measure

Gyro power (GPower) Power supply for gyros(PS for gyro)

Gyro 119909(119866119909) measure 120596

119909

Gyro 119910 (119866119910) measure 120596

119910

Gyro 119911 (119866119911) measure 120596

119911

Table 5 Results of reconfigurability analysis

MRU 120574 IESPower 0 1

ES1 1 05ES2 1 05

GPower 1 0119866

1199091 0

119866

1199101 0

119866

1199111 0

Figure 6 illustrates the function tree constructed by thereconfigurability modeling process The MCS family and theMPS family could be derived by analyzing the function treein Figure 6 as

C = ESPES1 ESPES2

R = ESP ES1ES2 (19)

Thus reconfigurability indexes can be calculated by (7)to (10) Table 5 lists the FRD and IDMEU of all the MRUsFurthermore suppose that the severity and occurrence pos-sibility for all MRUs are the same then 119908

119894= 1 119903 = 67 and

119879 = 0According to the analysis results of IDMRU and SFTD

of all MRUs the weakest link of this system is the power ofinfrared earth sensors Consequently it is better to store abackup in this link

7 Conclusion

To involve reconfigurability in spacecraft design phase forpotential faults a novel reconfigurability analysis method isinvestigated in this paper First on the basis of observabilityand controllability the reconfigurability criterion is givenfor spacecraft that is considered as a rigid body systemThen the function tree is built formodeling reconfigurabilityand evaluation indexes are proposed After that accordingto minimal cut set and minimal path set of the functiontree a quantitative evaluation method for reconfigurabilityindexes and an approach for determining system weak links

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

is to construct an effective reconfigurability analysis methodbased on the function tree theory which can synthesizecomponents and reconfiguration strategies of spacecraft andestimate quantitative evaluation indexes

The rest of this paper is organized as follows Section 2presents some basic definitions and Section 3 constructsa reconfigurability modeling and analyzing method InSections 4 and 5 reconfigurability evaluation indexes andweak link analysis procedure for reconfiguration design arediscussed respectively In Section 6 the proposed approachis illustrated by a practical application regarding spacecraftattitude measuring system Some conclusions and relevantremarks are given in Section 7

2 Basic Definitions

Siddiqi indicated that different definitions exist in differentfields in [14] By summing up a series of definitions hedefined reconfigurable system and reconfigurability as fol-lows Reconfigurable system is a system that can reversiblyachieve distinct configurations (or states) through alterationof system form or function in order to achieve a desiredoutcome within acceptable reconfiguration while recon-figurability is a system architectural property that definesthe ease and extent to which a system is reconfigurableConsidering spacecraft reconfiguration is the problem ofreplacing the faulty part of the systemby anonfaulty one so asto still achieve control objectives and reconfigurability is theability of recovering all the functions or achieving degradedobjectives by reconfiguration when faults appear

System configuration is one of the basic factors that affectreconfigurability Two relevant definitions reconfigurationunit (RU) and minimal reconfiguration unit (MRU) shouldbe explained here RU is a combination of spacecraft compo-nents to achieve the anticipant function by reconfigurationitself or by switching to other RUs when the current RUfails MRU is a combination of spacecraft components toachieve the anticipant function only by switching to otherRUs when the current RU fails It is the minimal unit in thereconfiguration analysis

A novel reconfigurability model is established based onthe function tree theory in this study Function tree is a treediagram whose vertex corresponds to the system functionand whose branches are subfunctions decomposed fromthe system function and its roots are the MRUs Higherlevel functions and lower level functions in a function treeare connected by AND gates or OR gates The relationshipbetween function and MRUs can be clearly explained bythe corresponding function tree A typical function tree isillustrated in Figure 1

In order to evaluate the reconfigurability quantitativelydefinitions including cut set (CS) minimal cut set (MCS)path set (PS) and minimal path set (MPS) of a function treeare involved A CS is a set of MRUs When all MRUs in a CSare healthy the system functions can be achieved MCS is aspecial CS and if and only if all MRUs in MCS are in goodcondition the system functions can be achieved APS is also aset of MRUs When all MRUs in a PS fail the system will lose

System function

Higher level Higher level

Lowest levelLowest level

subfunction 1

subfunction 1 subfunction n

MRU MRU MRU MRU1 2 k minus 1 k

Vertex

Branches

Roots

middot middot middot

middot middot middot subfunction m

Figure 1 Function tree schematic diagram

its function MPS is a special PS and if and only if failureappears in every MRU in MPS the system function shouldhave been lost Furthermore theMCS set orMPS set is calledMCS family or MPS family

3 Reconfigurability Modeling

For reconfigurability evaluating and designing one firstneeds to build an effective reconfigurability model andestablish relationships between reconfigurability and MRUsThen evaluation indexes and weak links of the spacecraftreconfigurability can be analyzed

We define a reconfigurability model from viewpoint offunction tree which is similar to theory of fault tree Themodeling processes are discussed as below

Step 1 According to the system function define the recon-figuration strategy based on the system observability andcontrollability

For example consider the LTI deterministic system

(119905) = 119860119909 (119905) + 119861119906 (119905)

119910 (119905) = 119862119909 (119905)

(1)

We adopt the observability criterion and controllability crite-rion

rank [119862 119862119860 sdot sdot sdot 119862119860

119899minus1

]

1015840

= 119899

rank [119861 119861119860 sdot sdot sdot 119861119860

119899minus1

] = 119899

(2)

to confirm the reconfiguration strategy by changing 119861 or119862 inthe system model and then obtain the component set 119862comeach one of which can perform the system function

Step 2 If any redundancy is involved in a system componentdecompose it to the functional module According to theredundancy relationship between themodules determine theMRUs Furthermore according to the MRUs functions theMRUs function set 119865MRU can be obtained And the elementsin 119865MRU are the lowest level function in the function tree

Mathematical Problems in Engineering 3

Power supply 1

Power supply 2

Data processing IO Gyro

sensor

RedundantGyro

MRU1 MRU2 MRU3

Structure decomposition

DeterminingMRU

Figure 2 Structure decomposition of gyro

Angle velocity measure

Power supply Measure and data process

Power supply 1

Power supply 2

Dataprocessing IO Gyro

sensor

Functiondecomposition

CorrespondingMRU

(MRU1)(MRU3)

(MRU2)

Figure 3 Function decomposition of gyro

To get a better understanding a gyro system is utilized asan example to illustrate this procedure A gyro can be decom-posed to several modules such as power supply module dataprocessing module IO module and gyro sensor module Ifthe power supply module is redundant while others are notany single power supply module can be considered as MRUand the rest can be treated as MRU Consequently 119865MRUof a gyro is 119901119900119908119890119903 119904119906119901119901119897119910 119898119890119886119904119906119903119890 119886119899119889 119889119886119905119886 119901119903119900119888119890119904119904Figure 2 shows the decomposition structure

Step 3 From the system function decompose higher levelfunctions into lower level functions (or subfunctions) untilthe functions are contained in 119865MRU

Return to the example of gyro ldquoAngle velocity measurerdquois the function of a gyro It can be decomposed into twosubfunctions ldquopower supplyrdquo and ldquomeasure and data processrdquoThen the decomposition process can be terminated becauseldquopower supplyrdquo and ldquomeasure and data processrdquo belong to119865MRU The decomposition process is illustrated in Figure 3

Step 4 Build a function tree by AND gate and OR gateThe vertex of this function tree is the system functionthe branches are the subfunctions and the roots are theMRUs AND gate and OR gate connect the higher layers andthe lower layers according to the relationship between thesubfunctions

AND gate and OR gate in function trees are depictedin Figure 4 The AND gate in Figure 4(a) shows that theupper level function 119884 can only be achieved when all thesubfunctions 119909

119894have been realized 119894 = 1 2 119899 while for

OR gate in Figure 4(b) it can be concluded that the upperlevel function 119884 can be realized when any single or multipleor all subfunctions 119909

119894are achieved 119894 = 1 2 119899

Y

x1 x2 xnmiddot middot middot

(a) AND gate

Y

x1 x2 xnmiddot middot middot

(b) OR gate

Figure 4 AND gate and OR gate

Angle velocity measure

Power supply Measure and data process

Vertex

Branches

RootsMRU1 MRU2 MRU3

Figure 5 Function tree of gyro

According to the stepsmentioned above the function treeof a gyro can be formed which is shown in Figure 5

In order to analyze the reconfigurability quantitativelythe MCS andMPS of function tree should be obtained firstly

Let C119894(119909

119895) denote the ith MCS for the jth level

function 119909

119895 and let C(119884) denote the CS family for the upper

level function 119884 For AND gate

C (119884) = C119894(119909

1) cup C119895(119909

2) cup sdot sdot sdot cup C

119896(119909

119899)

119894 isin (1 2

1003816

1003816

1003816

1003816

C (119909

1)

1003816

1003816

1003816

1003816

)

119895 isin (1 2 sdot sdot sdot

1003816

1003816

1003816

1003816

C (119909

2)

1003816

1003816

1003816

1003816

)

119896 isin (1 2 sdot sdot sdot

1003816

1003816

1003816

1003816

C (119909

119899)

1003816

1003816

1003816

1003816

)

(3)

For OR gate

C (119884) = C (119909

1) cup C (119909

2) cup sdot sdot sdot cup C (119909

119899) (4)

where |C(119909

119894)| 119894 = 1 2 119899 is the cardinal number of C(119909

119894)

which indicates MCS number in the MCS family for thesubfunction 119909

119894

Let R119894(119909

119895) be the 119894th MPS for the 119895th level function 119909

119895

and let R(119884) be the PS family of the upper level function 119884For AND gate

R (119884) = R (119909

1) cupR (119909

2) cup sdot sdot sdot cupR (119909

119899) (5)

For OR gate

R (119884) = R119894(119909

1) cup R119895(119909

2) cup sdot sdot sdot cup R

119896(119909

119899)

119894 isin (1 2

1003816

1003816

1003816

1003816

R (119909

1)

1003816

1003816

1003816

1003816

)

119895 isin (1 2

1003816

1003816

1003816

1003816

R (119909

2)

1003816

1003816

1003816

1003816

)

119896 isin (1 2

1003816

1003816

1003816

1003816

R (119909

119899)

1003816

1003816

1003816

1003816

)

(6)

4 Mathematical Problems in Engineering

where|R(119909

119894)| 119894 = 1 2 119899 is the cardinal number of R(119909

119894)

which corresponds to theMPS number of theMPS family forthe subfunction 119909

119894

Although C(119884) or R(119884) derived by (3) to (6) may not beMCS family or MPS family the MCS and MPS are neededin the upper level function analysis according to (3) to (6)Consequently the MCS and MPS of function 119884 can becalculated by the following steps

Step 1 Initialize Cmin(119884) or Rmin(119884) to be a null set

Step 2 ChooseCmin(119884) orRmin(119884)with a minimum cardinalnumber in all sets in C(119884) or R(119884) and transform it intoCmin(119884) or Rmin(119884)

Step 3 Check all remaining sets in C(119884) orR(119884) If there is aset containing all the MRUs in Cmin(119884) or Rmin(119884) delete itfrom C(119884) or R(119884) and go back to Step 2 otherwise

Step 4 Execute Steps 2 and 3 repeatedly until C(119884) or R(119884)

turns to a null set Then elements C119894(119884) or R

119894(119884) in Cmin(119884)

or Rmin(119884) are the expected MCS or MPS

4 Reconfigurability Evaluation Indexes

Based on the reconfigurability model constructed in thepreceding section reconfigurability evaluation indexes forspacecrafts are given as follows

41 Fault Reconfigurable Degree (FRD) FRD describeswhether the system has available resources and methods forreconfigurations after certain faults as

120574 =

1 fault is reconfigurable0 fault is unreconfigurable

(7)

When certain faults emerge the MCS family shouldbe activated by deleting all the MCSs including the faultreconfigurable units Consider 120574 = 0 if the MCS family isempty consider 120574 = 1 otherwise

42 System Reconfigurable Rate (SRR) SRR indicates the rateof reconfigurable faults with respect to all faults in the system

119903 =

sum

119898

119894=1119908

119894120574

119894

sum

119898

119894=1119908

119894

(8)

where 120574

119894is the FRD of the 119894th fault 119891

119894 119898 is the number

of all the system fault modes and 119908

119894is the weight of fault

119891

119894according to its severity and occurrence probability The

major fault has a bigger weight than aminor one and the faultwith high occurrence probability has a bigger weight thanthe one with low occurrence probability If the fault severitycan be defined as four levels as listed in Table 1 and theoccurrence probability can be divided into five levels as listedin Table 2 then119908

119894can be determined from Table 3 119878 denotes

the fault severity level and 119875 indicates the fault occurrenceprobability in Table 3

Table 1 Fault severity level definition

Level DefinitionI System function is lost or service life is shortened seriously

II System function is degraded seriously or service life isreduced by 14 to 12

III System function is degraded partially or service life isreduced below 14

IV There is little affection in system function and service life

Table 2 Fault occurrence probability definition

Level DefinitionA MRU fault probability ge 20 times total fault probability

B 20 times total fault probability gtMRU fault probability ge

10 times total fault probability

C 10 times total fault probability gtMRU fault probability ge 1times total fault probability

D 1 times total fault probability gtMRU fault probability ge 01times total fault probability

E MRU fault probability lt 01 times total fault probability

Table 3 119908119894matrix

119875

119878

I II III IVA 1 13 17 113B 12 15 19 116C 14 16 111 118D 18 110 114 119E 112 115 117 120

5 Weak Link Analysis inReconfigurability Design

For better reconfigurability the reconfiguration weak linksshould be improved in the design phase of a spacecraft Basedon the established configurability model the following twoindexes are proposed to determine weak links in reconfigu-ration

51 Importance Degree of MRU (IDMRU) IDMRU denotesthe rate of the number of MCSs that includes the MRU withrespect to the number of all MCSs as

119868

119872=

119873

119872

119873

119879

(9)

where 119868

119872is the IDMRU of MRU 119872 119873

119872is the number of

MCSs that comprise the MRU and 119873

119879is the number of all

MCSsFor any system the MRU with maximal IDMRU con-

tributes most in system function realization Consequentlynecessary redundancy or special reliability design should beconsidered for this MRU

52 System Fault Tolerance Degree (SFTD) SFTD representsthe maximal number of failure MRUs that the system can

Mathematical Problems in Engineering 5

tolerate without loss of system functions SFTD reflects thesystem reconfigurability as

119879 = min (

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

) minus 1

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

isin R 119894 = 1 2 |R| (10)

where 119879 denotes SFTD R119894is the 119894th minimal path set of the

function tree |R119894| is the cardinal number of R

119894

In a system the path set with the minimum numberof MPSs is the weakest link And for this part necessaryredundancy or special reliability design should be consideredaccording to the subfunctions of MRUs in the MPS

The four indexes proposed above are closely connectedto each other Let 119891

119894be a fault whose corresponding recon-

figurable degree is equal to zero 120574119894= 0 namely the corre-

sponding MRU cannot be reconfigured then the importancedegree 119868

119872of the MRU will be equal to one and the system

fault tolerance degree 119879 will become zero Otherwise if allfault reconfigurable degrees are one namely all theMRU canbe reconfigured thenwe can conclude that all the importancedegrees will be less than one the system fault tolerance degreewill be not less than one and the system reconfigurable ratewill be equal to 100

6 Empirical Results

In this section we focus on the practical performance ofthe proposed method Our experiment is presented for thereconfigurability analysis of an attitude measuring system ina spacecraft The dynamic functions regarding momentumdevices are shown in (11)The spacecraft is considered as rigidbody systems and the body coordinate system coincides withthe principle axes of inertia as

119868

119909

119909minus (119868

119910minus 119868

119911) 120596

119910120596

119911minus ℎ

119910120596

119911+ ℎ

119911120596

119910= minus

ℎ

119909+ 119879

119909

119868

119910

119910minus (119868

119911minus 119868

119909) 120596

119911120596

119909minus ℎ

119911120596

119909+ ℎ

119909120596

119911= minus

ℎ

119910+ 119879

119910

119868

119911

119911minus (119868

119909minus 119868

119910) 120596

119909120596

119910minus ℎ

119909120596

119910+ ℎ

119910120596

119909= minus

ℎ

119911+ 119879

119911

(11)

where 119868

119909 119868119910and 119868

119911are moments of inertia along axes 119874119909

119874119910 and 119874119911 respectively 120596 = [120596

119909 120596

119910 120596

119911]

119879 is the angularvelocity vector h = [ℎ

119909 ℎ

119910 ℎ

119911]

119879 is the synthesizing angularmomentum vector of all the momentum devices T =

[119879

119909 119879

119910 119879

119911]

119879 is the control torque vector applied on thespacecraft except for the torque from themomentumdevicesTherefore the control torque vector T = [119879

119909 119879

119910 119879

119911]

119879 in(11) includes torques from thrusters other space torques anddisturbing torques

If all attitudes vary in a small scale the dynamic functionscan be simplified as

120596

119909= minus 120596

0120595

120596

119910=

120579 minus 120596

0

120596

119911=

120595 + 120596

0120593

(12)

where 120593 120579 and 120595 are Euler angles 120596

0denotes the orbit

angular velocity with which the spacecraft circles around thecenter body

Then the linearization form of the attitude dynamicfunction can be derived based on (11) and (12) as

119868

119909 + [(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910] 120593

+ [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

120595

= minus

ℎ

119909+ 120596

0ℎ

119911+ 119879

119909

119868

119910

120579 + ℎ

119909(

120595 + 120596

0120593) minus ℎ

119911( minus 120596

0120595) = minus

ℎ

119910+ 119879

119910

119868

119909

120595 + [(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910] 120595

minus [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

= minus

ℎ

119911minus 120596

0ℎ

119909+ 119879

119911

(13)

Accordingly the dynamic function of the spacecraft canbe expressed by a state space form as shown in (1) with thefollowing notations

119909 = [120593 120579

120579 120595

120595]

119879

119860 =

[

[

[

[

[

[

[

[

0 1 0 0 0 0

119872

210 0 0 0 119872

26

0 0 0 1 0 0

119872

41119872

420 0 119872

45119872

46

0 0 0 0 0 1

0 119872

620 0 119872

650

]

]

]

]

]

]

]

]

119872

21= 119868

minus1

119909[(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910]

119872

26= 119868

minus1

119909[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

41= 119868

minus1

119910ℎ

119909120596

0

119872

42= minus119868

minus1

119910ℎ

119911

119872

45= 119868

minus1

119910ℎ

119911120596

0

119872

46= 119868

minus1

119910ℎ

119909

119872

62= minus119868

minus1

119911[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

65= 119868

minus1

119911[(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910]

(14)

Matrixes 119861 and 119862 in (1) can be determined accordingto the detailed configuration of the system For example asystem with two infrared earth sensors three orthogonalgyros and one main backup thruster can be described as

119906 (119905) = [119879

1199091119879

1199092119879

1199101119879

1199102119879

1199111119879

1199112]

119879

119910 (119905) = [120593

ℎ1120579

ℎ1120593

ℎ2120579

ℎ2119892

119909119892

119910119892

119911]

119879

6 Mathematical Problems in Engineering

119861 =

[

[

[

[

[

[

[

[

0 0 0 0 0 0

119868

minus1

119909119868

minus1

1199090 0 0 0

0 0 0 0 0 0

0 0 119868

minus1

119910119868

minus1

1199100 0

0 0 0 0 0 0

0 0 0 0 119868

minus1

119911119868

minus1

119911

]

]

]

]

]

]

]

]

119862 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

]

]

]

]

(15)

Considering a spacecraft system described by (1) whenfaults appear the premise of achieving system reconfigura-bility is that the remaining of the system is observable andcontrollable The corresponding criterion is given by (2)According to engineering experience one can assume that119868

119909= 119868

119910= 119868

119911and 120596

0= 0 Consider the following

(1) Only one infrared earth sensor is employed forattitude determination as

119862

1= [

1 0 0 0 0 0

0 0 1 0 0 0

] rank[

[

[

[

[

119862

1

119862

1119860

119862

1119860

5

]

]

]

]

]

= 6 (16)

(2) Three gyros are employed for attitude determinationas

119862

2=

[

[

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

rank[

[

[

[

[

119862

2

119862

2119860

119862

2119860

5

]

]

]

]

]

= 5 (17)

(3) One infrared earth sensor and three gyros areemployed for attitude determination as

119862

3=

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

rank[

[

[

[

[

119862

3

119862

3119860

119862

3119860

5

]

]

]

]

]

= 6 (18)

From (16) to (18) the attitude can be measured in thefollowing two ways

M1 by infrared earth sensorsM2 by infrared earth sensors and gyros

In addition it is assumed that two infrared earth sensorsshare one power supply and three gyros share another powersupply then Table 4 lists the MRUs and their correspondingsubfunctions

Table 4 MRUs and their corresponding functions

MRU FunctionsInfrared earth sensor power(ESP)

Power supply for infrared earthsensor (PS for ES)

Infrared earth sensor 1 (ES1) 120593 and 120579measureInfrared earth sensor 2 (ES2) 120593 and 120579measure

Gyro power (GPower) Power supply for gyros(PS for gyro)

Gyro 119909(119866119909) measure 120596

119909

Gyro 119910 (119866119910) measure 120596

119910

Gyro 119911 (119866119911) measure 120596

119911

Table 5 Results of reconfigurability analysis

MRU 120574 IESPower 0 1

ES1 1 05ES2 1 05

GPower 1 0119866

1199091 0

119866

1199101 0

119866

1199111 0

Figure 6 illustrates the function tree constructed by thereconfigurability modeling process The MCS family and theMPS family could be derived by analyzing the function treein Figure 6 as

C = ESPES1 ESPES2

R = ESP ES1ES2 (19)

Thus reconfigurability indexes can be calculated by (7)to (10) Table 5 lists the FRD and IDMEU of all the MRUsFurthermore suppose that the severity and occurrence pos-sibility for all MRUs are the same then 119908

119894= 1 119903 = 67 and

119879 = 0According to the analysis results of IDMRU and SFTD

of all MRUs the weakest link of this system is the power ofinfrared earth sensors Consequently it is better to store abackup in this link

7 Conclusion

To involve reconfigurability in spacecraft design phase forpotential faults a novel reconfigurability analysis method isinvestigated in this paper First on the basis of observabilityand controllability the reconfigurability criterion is givenfor spacecraft that is considered as a rigid body systemThen the function tree is built formodeling reconfigurabilityand evaluation indexes are proposed After that accordingto minimal cut set and minimal path set of the functiontree a quantitative evaluation method for reconfigurabilityindexes and an approach for determining system weak links

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Power supply 1

Power supply 2

Data processing IO Gyro

sensor

RedundantGyro

MRU1 MRU2 MRU3

Structure decomposition

DeterminingMRU

Figure 2 Structure decomposition of gyro

Angle velocity measure

Power supply Measure and data process

Power supply 1

Power supply 2

Dataprocessing IO Gyro

sensor

Functiondecomposition

CorrespondingMRU

(MRU1)(MRU3)

(MRU2)

Figure 3 Function decomposition of gyro

To get a better understanding a gyro system is utilized asan example to illustrate this procedure A gyro can be decom-posed to several modules such as power supply module dataprocessing module IO module and gyro sensor module Ifthe power supply module is redundant while others are notany single power supply module can be considered as MRUand the rest can be treated as MRU Consequently 119865MRUof a gyro is 119901119900119908119890119903 119904119906119901119901119897119910 119898119890119886119904119906119903119890 119886119899119889 119889119886119905119886 119901119903119900119888119890119904119904Figure 2 shows the decomposition structure

Step 3 From the system function decompose higher levelfunctions into lower level functions (or subfunctions) untilthe functions are contained in 119865MRU

Return to the example of gyro ldquoAngle velocity measurerdquois the function of a gyro It can be decomposed into twosubfunctions ldquopower supplyrdquo and ldquomeasure and data processrdquoThen the decomposition process can be terminated becauseldquopower supplyrdquo and ldquomeasure and data processrdquo belong to119865MRU The decomposition process is illustrated in Figure 3

Step 4 Build a function tree by AND gate and OR gateThe vertex of this function tree is the system functionthe branches are the subfunctions and the roots are theMRUs AND gate and OR gate connect the higher layers andthe lower layers according to the relationship between thesubfunctions

AND gate and OR gate in function trees are depictedin Figure 4 The AND gate in Figure 4(a) shows that theupper level function 119884 can only be achieved when all thesubfunctions 119909

119894have been realized 119894 = 1 2 119899 while for

OR gate in Figure 4(b) it can be concluded that the upperlevel function 119884 can be realized when any single or multipleor all subfunctions 119909

119894are achieved 119894 = 1 2 119899

Y

x1 x2 xnmiddot middot middot

(a) AND gate

Y

x1 x2 xnmiddot middot middot

(b) OR gate

Figure 4 AND gate and OR gate

Angle velocity measure

Power supply Measure and data process

Vertex

Branches

RootsMRU1 MRU2 MRU3

Figure 5 Function tree of gyro

According to the stepsmentioned above the function treeof a gyro can be formed which is shown in Figure 5

In order to analyze the reconfigurability quantitativelythe MCS andMPS of function tree should be obtained firstly

Let C119894(119909

119895) denote the ith MCS for the jth level

function 119909

119895 and let C(119884) denote the CS family for the upper

level function 119884 For AND gate

C (119884) = C119894(119909

1) cup C119895(119909

2) cup sdot sdot sdot cup C

119896(119909

119899)

119894 isin (1 2

1003816

1003816

1003816

1003816

C (119909

1)

1003816

1003816

1003816

1003816

)

119895 isin (1 2 sdot sdot sdot

1003816

1003816

1003816

1003816

C (119909

2)

1003816

1003816

1003816

1003816

)

119896 isin (1 2 sdot sdot sdot

1003816

1003816

1003816

1003816

C (119909

119899)

1003816

1003816

1003816

1003816

)

(3)

For OR gate

C (119884) = C (119909

1) cup C (119909

2) cup sdot sdot sdot cup C (119909

119899) (4)

where |C(119909

119894)| 119894 = 1 2 119899 is the cardinal number of C(119909

119894)

which indicates MCS number in the MCS family for thesubfunction 119909

119894

Let R119894(119909

119895) be the 119894th MPS for the 119895th level function 119909

119895

and let R(119884) be the PS family of the upper level function 119884For AND gate

R (119884) = R (119909

1) cupR (119909

2) cup sdot sdot sdot cupR (119909

119899) (5)

For OR gate

R (119884) = R119894(119909

1) cup R119895(119909

2) cup sdot sdot sdot cup R

119896(119909

119899)

119894 isin (1 2

1003816

1003816

1003816

1003816

R (119909

1)

1003816

1003816

1003816

1003816

)

119895 isin (1 2

1003816

1003816

1003816

1003816

R (119909

2)

1003816

1003816

1003816

1003816

)

119896 isin (1 2

1003816

1003816

1003816

1003816

R (119909

119899)

1003816

1003816

1003816

1003816

)

(6)

4 Mathematical Problems in Engineering

where|R(119909

119894)| 119894 = 1 2 119899 is the cardinal number of R(119909

119894)

which corresponds to theMPS number of theMPS family forthe subfunction 119909

119894

Although C(119884) or R(119884) derived by (3) to (6) may not beMCS family or MPS family the MCS and MPS are neededin the upper level function analysis according to (3) to (6)Consequently the MCS and MPS of function 119884 can becalculated by the following steps

Step 1 Initialize Cmin(119884) or Rmin(119884) to be a null set

Step 2 ChooseCmin(119884) orRmin(119884)with a minimum cardinalnumber in all sets in C(119884) or R(119884) and transform it intoCmin(119884) or Rmin(119884)

Step 3 Check all remaining sets in C(119884) orR(119884) If there is aset containing all the MRUs in Cmin(119884) or Rmin(119884) delete itfrom C(119884) or R(119884) and go back to Step 2 otherwise

Step 4 Execute Steps 2 and 3 repeatedly until C(119884) or R(119884)

turns to a null set Then elements C119894(119884) or R

119894(119884) in Cmin(119884)

or Rmin(119884) are the expected MCS or MPS

4 Reconfigurability Evaluation Indexes

Based on the reconfigurability model constructed in thepreceding section reconfigurability evaluation indexes forspacecrafts are given as follows

41 Fault Reconfigurable Degree (FRD) FRD describeswhether the system has available resources and methods forreconfigurations after certain faults as

120574 =

1 fault is reconfigurable0 fault is unreconfigurable

(7)

When certain faults emerge the MCS family shouldbe activated by deleting all the MCSs including the faultreconfigurable units Consider 120574 = 0 if the MCS family isempty consider 120574 = 1 otherwise

42 System Reconfigurable Rate (SRR) SRR indicates the rateof reconfigurable faults with respect to all faults in the system

119903 =

sum

119898

119894=1119908

119894120574

119894

sum

119898

119894=1119908

119894

(8)

where 120574

119894is the FRD of the 119894th fault 119891

119894 119898 is the number

of all the system fault modes and 119908

119894is the weight of fault

119891

119894according to its severity and occurrence probability The

major fault has a bigger weight than aminor one and the faultwith high occurrence probability has a bigger weight thanthe one with low occurrence probability If the fault severitycan be defined as four levels as listed in Table 1 and theoccurrence probability can be divided into five levels as listedin Table 2 then119908

119894can be determined from Table 3 119878 denotes

the fault severity level and 119875 indicates the fault occurrenceprobability in Table 3

Table 1 Fault severity level definition

Level DefinitionI System function is lost or service life is shortened seriously

II System function is degraded seriously or service life isreduced by 14 to 12

III System function is degraded partially or service life isreduced below 14

IV There is little affection in system function and service life

Table 2 Fault occurrence probability definition

Level DefinitionA MRU fault probability ge 20 times total fault probability

B 20 times total fault probability gtMRU fault probability ge

10 times total fault probability

C 10 times total fault probability gtMRU fault probability ge 1times total fault probability

D 1 times total fault probability gtMRU fault probability ge 01times total fault probability

E MRU fault probability lt 01 times total fault probability

Table 3 119908119894matrix

119875

119878

I II III IVA 1 13 17 113B 12 15 19 116C 14 16 111 118D 18 110 114 119E 112 115 117 120

5 Weak Link Analysis inReconfigurability Design

For better reconfigurability the reconfiguration weak linksshould be improved in the design phase of a spacecraft Basedon the established configurability model the following twoindexes are proposed to determine weak links in reconfigu-ration

51 Importance Degree of MRU (IDMRU) IDMRU denotesthe rate of the number of MCSs that includes the MRU withrespect to the number of all MCSs as

119868

119872=

119873

119872

119873

119879

(9)

where 119868

119872is the IDMRU of MRU 119872 119873

119872is the number of

MCSs that comprise the MRU and 119873

119879is the number of all

MCSsFor any system the MRU with maximal IDMRU con-

tributes most in system function realization Consequentlynecessary redundancy or special reliability design should beconsidered for this MRU

52 System Fault Tolerance Degree (SFTD) SFTD representsthe maximal number of failure MRUs that the system can

Mathematical Problems in Engineering 5

tolerate without loss of system functions SFTD reflects thesystem reconfigurability as

119879 = min (

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

) minus 1

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

isin R 119894 = 1 2 |R| (10)

where 119879 denotes SFTD R119894is the 119894th minimal path set of the

function tree |R119894| is the cardinal number of R

119894

In a system the path set with the minimum numberof MPSs is the weakest link And for this part necessaryredundancy or special reliability design should be consideredaccording to the subfunctions of MRUs in the MPS

The four indexes proposed above are closely connectedto each other Let 119891

119894be a fault whose corresponding recon-

figurable degree is equal to zero 120574119894= 0 namely the corre-

sponding MRU cannot be reconfigured then the importancedegree 119868

119872of the MRU will be equal to one and the system

fault tolerance degree 119879 will become zero Otherwise if allfault reconfigurable degrees are one namely all theMRU canbe reconfigured thenwe can conclude that all the importancedegrees will be less than one the system fault tolerance degreewill be not less than one and the system reconfigurable ratewill be equal to 100

6 Empirical Results

In this section we focus on the practical performance ofthe proposed method Our experiment is presented for thereconfigurability analysis of an attitude measuring system ina spacecraft The dynamic functions regarding momentumdevices are shown in (11)The spacecraft is considered as rigidbody systems and the body coordinate system coincides withthe principle axes of inertia as

119868

119909

119909minus (119868

119910minus 119868

119911) 120596

119910120596

119911minus ℎ

119910120596

119911+ ℎ

119911120596

119910= minus

ℎ

119909+ 119879

119909

119868

119910

119910minus (119868

119911minus 119868

119909) 120596

119911120596

119909minus ℎ

119911120596

119909+ ℎ

119909120596

119911= minus

ℎ

119910+ 119879

119910

119868

119911

119911minus (119868

119909minus 119868

119910) 120596

119909120596

119910minus ℎ

119909120596

119910+ ℎ

119910120596

119909= minus

ℎ

119911+ 119879

119911

(11)

where 119868

119909 119868119910and 119868

119911are moments of inertia along axes 119874119909

119874119910 and 119874119911 respectively 120596 = [120596

119909 120596

119910 120596

119911]

119879 is the angularvelocity vector h = [ℎ

119909 ℎ

119910 ℎ

119911]

119879 is the synthesizing angularmomentum vector of all the momentum devices T =

[119879

119909 119879

119910 119879

119911]

119879 is the control torque vector applied on thespacecraft except for the torque from themomentumdevicesTherefore the control torque vector T = [119879

119909 119879

119910 119879

119911]

119879 in(11) includes torques from thrusters other space torques anddisturbing torques

If all attitudes vary in a small scale the dynamic functionscan be simplified as

120596

119909= minus 120596

0120595

120596

119910=

120579 minus 120596

0

120596

119911=

120595 + 120596

0120593

(12)

where 120593 120579 and 120595 are Euler angles 120596

0denotes the orbit

angular velocity with which the spacecraft circles around thecenter body

Then the linearization form of the attitude dynamicfunction can be derived based on (11) and (12) as

119868

119909 + [(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910] 120593

+ [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

120595

= minus

ℎ

119909+ 120596

0ℎ

119911+ 119879

119909

119868

119910

120579 + ℎ

119909(

120595 + 120596

0120593) minus ℎ

119911( minus 120596

0120595) = minus

ℎ

119910+ 119879

119910

119868

119909

120595 + [(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910] 120595

minus [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

= minus

ℎ

119911minus 120596

0ℎ

119909+ 119879

119911

(13)

Accordingly the dynamic function of the spacecraft canbe expressed by a state space form as shown in (1) with thefollowing notations

119909 = [120593 120579

120579 120595

120595]

119879

119860 =

[

[

[

[

[

[

[

[

0 1 0 0 0 0

119872

210 0 0 0 119872

26

0 0 0 1 0 0

119872

41119872

420 0 119872

45119872

46

0 0 0 0 0 1

0 119872

620 0 119872

650

]

]

]

]

]

]

]

]

119872

21= 119868

minus1

119909[(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910]

119872

26= 119868

minus1

119909[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

41= 119868

minus1

119910ℎ

119909120596

0

119872

42= minus119868

minus1

119910ℎ

119911

119872

45= 119868

minus1

119910ℎ

119911120596

0

119872

46= 119868

minus1

119910ℎ

119909

119872

62= minus119868

minus1

119911[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

65= 119868

minus1

119911[(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910]

(14)

Matrixes 119861 and 119862 in (1) can be determined accordingto the detailed configuration of the system For example asystem with two infrared earth sensors three orthogonalgyros and one main backup thruster can be described as

119906 (119905) = [119879

1199091119879

1199092119879

1199101119879

1199102119879

1199111119879

1199112]

119879

119910 (119905) = [120593

ℎ1120579

ℎ1120593

ℎ2120579

ℎ2119892

119909119892

119910119892

119911]

119879

6 Mathematical Problems in Engineering

119861 =

[

[

[

[

[

[

[

[

0 0 0 0 0 0

119868

minus1

119909119868

minus1

1199090 0 0 0

0 0 0 0 0 0

0 0 119868

minus1

119910119868

minus1

1199100 0

0 0 0 0 0 0

0 0 0 0 119868

minus1

119911119868

minus1

119911

]

]

]

]

]

]

]

]

119862 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

]

]

]

]

(15)

Considering a spacecraft system described by (1) whenfaults appear the premise of achieving system reconfigura-bility is that the remaining of the system is observable andcontrollable The corresponding criterion is given by (2)According to engineering experience one can assume that119868

119909= 119868

119910= 119868

119911and 120596

0= 0 Consider the following

(1) Only one infrared earth sensor is employed forattitude determination as

119862

1= [

1 0 0 0 0 0

0 0 1 0 0 0

] rank[

[

[

[

[

119862

1

119862

1119860

119862

1119860

5

]

]

]

]

]

= 6 (16)

(2) Three gyros are employed for attitude determinationas

119862

2=

[

[

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

rank[

[

[

[

[

119862

2

119862

2119860

119862

2119860

5

]

]

]

]

]

= 5 (17)

(3) One infrared earth sensor and three gyros areemployed for attitude determination as

119862

3=

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

rank[

[

[

[

[

119862

3

119862

3119860

119862

3119860

5

]

]

]

]

]

= 6 (18)

From (16) to (18) the attitude can be measured in thefollowing two ways

M1 by infrared earth sensorsM2 by infrared earth sensors and gyros

In addition it is assumed that two infrared earth sensorsshare one power supply and three gyros share another powersupply then Table 4 lists the MRUs and their correspondingsubfunctions

Table 4 MRUs and their corresponding functions

MRU FunctionsInfrared earth sensor power(ESP)

Power supply for infrared earthsensor (PS for ES)

Infrared earth sensor 1 (ES1) 120593 and 120579measureInfrared earth sensor 2 (ES2) 120593 and 120579measure

Gyro power (GPower) Power supply for gyros(PS for gyro)

Gyro 119909(119866119909) measure 120596

119909

Gyro 119910 (119866119910) measure 120596

119910

Gyro 119911 (119866119911) measure 120596

119911

Table 5 Results of reconfigurability analysis

MRU 120574 IESPower 0 1

ES1 1 05ES2 1 05

GPower 1 0119866

1199091 0

119866

1199101 0

119866

1199111 0

Figure 6 illustrates the function tree constructed by thereconfigurability modeling process The MCS family and theMPS family could be derived by analyzing the function treein Figure 6 as

C = ESPES1 ESPES2

R = ESP ES1ES2 (19)

Thus reconfigurability indexes can be calculated by (7)to (10) Table 5 lists the FRD and IDMEU of all the MRUsFurthermore suppose that the severity and occurrence pos-sibility for all MRUs are the same then 119908

119894= 1 119903 = 67 and

119879 = 0According to the analysis results of IDMRU and SFTD

of all MRUs the weakest link of this system is the power ofinfrared earth sensors Consequently it is better to store abackup in this link

7 Conclusion

To involve reconfigurability in spacecraft design phase forpotential faults a novel reconfigurability analysis method isinvestigated in this paper First on the basis of observabilityand controllability the reconfigurability criterion is givenfor spacecraft that is considered as a rigid body systemThen the function tree is built formodeling reconfigurabilityand evaluation indexes are proposed After that accordingto minimal cut set and minimal path set of the functiontree a quantitative evaluation method for reconfigurabilityindexes and an approach for determining system weak links

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where|R(119909

119894)| 119894 = 1 2 119899 is the cardinal number of R(119909

119894)

which corresponds to theMPS number of theMPS family forthe subfunction 119909

119894

Although C(119884) or R(119884) derived by (3) to (6) may not beMCS family or MPS family the MCS and MPS are neededin the upper level function analysis according to (3) to (6)Consequently the MCS and MPS of function 119884 can becalculated by the following steps

Step 1 Initialize Cmin(119884) or Rmin(119884) to be a null set

Step 2 ChooseCmin(119884) orRmin(119884)with a minimum cardinalnumber in all sets in C(119884) or R(119884) and transform it intoCmin(119884) or Rmin(119884)

Step 3 Check all remaining sets in C(119884) orR(119884) If there is aset containing all the MRUs in Cmin(119884) or Rmin(119884) delete itfrom C(119884) or R(119884) and go back to Step 2 otherwise

Step 4 Execute Steps 2 and 3 repeatedly until C(119884) or R(119884)

turns to a null set Then elements C119894(119884) or R

119894(119884) in Cmin(119884)

or Rmin(119884) are the expected MCS or MPS

4 Reconfigurability Evaluation Indexes

Based on the reconfigurability model constructed in thepreceding section reconfigurability evaluation indexes forspacecrafts are given as follows

41 Fault Reconfigurable Degree (FRD) FRD describeswhether the system has available resources and methods forreconfigurations after certain faults as

120574 =

1 fault is reconfigurable0 fault is unreconfigurable

(7)

When certain faults emerge the MCS family shouldbe activated by deleting all the MCSs including the faultreconfigurable units Consider 120574 = 0 if the MCS family isempty consider 120574 = 1 otherwise

42 System Reconfigurable Rate (SRR) SRR indicates the rateof reconfigurable faults with respect to all faults in the system

119903 =

sum

119898

119894=1119908

119894120574

119894

sum

119898

119894=1119908

119894

(8)

where 120574

119894is the FRD of the 119894th fault 119891

119894 119898 is the number

of all the system fault modes and 119908

119894is the weight of fault

119891

119894according to its severity and occurrence probability The

major fault has a bigger weight than aminor one and the faultwith high occurrence probability has a bigger weight thanthe one with low occurrence probability If the fault severitycan be defined as four levels as listed in Table 1 and theoccurrence probability can be divided into five levels as listedin Table 2 then119908

119894can be determined from Table 3 119878 denotes

the fault severity level and 119875 indicates the fault occurrenceprobability in Table 3

Table 1 Fault severity level definition

Level DefinitionI System function is lost or service life is shortened seriously

II System function is degraded seriously or service life isreduced by 14 to 12

III System function is degraded partially or service life isreduced below 14

IV There is little affection in system function and service life

Table 2 Fault occurrence probability definition

Level DefinitionA MRU fault probability ge 20 times total fault probability

B 20 times total fault probability gtMRU fault probability ge

10 times total fault probability

C 10 times total fault probability gtMRU fault probability ge 1times total fault probability

D 1 times total fault probability gtMRU fault probability ge 01times total fault probability

E MRU fault probability lt 01 times total fault probability

Table 3 119908119894matrix

119875

119878

I II III IVA 1 13 17 113B 12 15 19 116C 14 16 111 118D 18 110 114 119E 112 115 117 120

5 Weak Link Analysis inReconfigurability Design

For better reconfigurability the reconfiguration weak linksshould be improved in the design phase of a spacecraft Basedon the established configurability model the following twoindexes are proposed to determine weak links in reconfigu-ration

51 Importance Degree of MRU (IDMRU) IDMRU denotesthe rate of the number of MCSs that includes the MRU withrespect to the number of all MCSs as

119868

119872=

119873

119872

119873

119879

(9)

where 119868

119872is the IDMRU of MRU 119872 119873

119872is the number of

MCSs that comprise the MRU and 119873

119879is the number of all

MCSsFor any system the MRU with maximal IDMRU con-

tributes most in system function realization Consequentlynecessary redundancy or special reliability design should beconsidered for this MRU

52 System Fault Tolerance Degree (SFTD) SFTD representsthe maximal number of failure MRUs that the system can

Mathematical Problems in Engineering 5

tolerate without loss of system functions SFTD reflects thesystem reconfigurability as

119879 = min (

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

) minus 1

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

isin R 119894 = 1 2 |R| (10)

where 119879 denotes SFTD R119894is the 119894th minimal path set of the

function tree |R119894| is the cardinal number of R

119894

In a system the path set with the minimum numberof MPSs is the weakest link And for this part necessaryredundancy or special reliability design should be consideredaccording to the subfunctions of MRUs in the MPS

The four indexes proposed above are closely connectedto each other Let 119891

119894be a fault whose corresponding recon-

figurable degree is equal to zero 120574119894= 0 namely the corre-

sponding MRU cannot be reconfigured then the importancedegree 119868

119872of the MRU will be equal to one and the system

fault tolerance degree 119879 will become zero Otherwise if allfault reconfigurable degrees are one namely all theMRU canbe reconfigured thenwe can conclude that all the importancedegrees will be less than one the system fault tolerance degreewill be not less than one and the system reconfigurable ratewill be equal to 100

6 Empirical Results

In this section we focus on the practical performance ofthe proposed method Our experiment is presented for thereconfigurability analysis of an attitude measuring system ina spacecraft The dynamic functions regarding momentumdevices are shown in (11)The spacecraft is considered as rigidbody systems and the body coordinate system coincides withthe principle axes of inertia as

119868

119909

119909minus (119868

119910minus 119868

119911) 120596

119910120596

119911minus ℎ

119910120596

119911+ ℎ

119911120596

119910= minus

ℎ

119909+ 119879

119909

119868

119910

119910minus (119868

119911minus 119868

119909) 120596

119911120596

119909minus ℎ

119911120596

119909+ ℎ

119909120596

119911= minus

ℎ

119910+ 119879

119910

119868

119911

119911minus (119868

119909minus 119868

119910) 120596

119909120596

119910minus ℎ

119909120596

119910+ ℎ

119910120596

119909= minus

ℎ

119911+ 119879

119911

(11)

where 119868

119909 119868119910and 119868

119911are moments of inertia along axes 119874119909

119874119910 and 119874119911 respectively 120596 = [120596

119909 120596

119910 120596

119911]

119879 is the angularvelocity vector h = [ℎ

119909 ℎ

119910 ℎ

119911]

119879 is the synthesizing angularmomentum vector of all the momentum devices T =

[119879

119909 119879

119910 119879

119911]

119879 is the control torque vector applied on thespacecraft except for the torque from themomentumdevicesTherefore the control torque vector T = [119879

119909 119879

119910 119879

119911]

119879 in(11) includes torques from thrusters other space torques anddisturbing torques

If all attitudes vary in a small scale the dynamic functionscan be simplified as

120596

119909= minus 120596

0120595

120596

119910=

120579 minus 120596

0

120596

119911=

120595 + 120596

0120593

(12)

where 120593 120579 and 120595 are Euler angles 120596

0denotes the orbit

angular velocity with which the spacecraft circles around thecenter body

Then the linearization form of the attitude dynamicfunction can be derived based on (11) and (12) as

119868

119909 + [(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910] 120593

+ [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

120595

= minus

ℎ

119909+ 120596

0ℎ

119911+ 119879

119909

119868

119910

120579 + ℎ

119909(

120595 + 120596

0120593) minus ℎ

119911( minus 120596

0120595) = minus

ℎ

119910+ 119879

119910

119868

119909

120595 + [(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910] 120595

minus [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

= minus

ℎ

119911minus 120596

0ℎ

119909+ 119879

119911

(13)

Accordingly the dynamic function of the spacecraft canbe expressed by a state space form as shown in (1) with thefollowing notations

119909 = [120593 120579

120579 120595

120595]

119879

119860 =

[

[

[

[

[

[

[

[

0 1 0 0 0 0

119872

210 0 0 0 119872

26

0 0 0 1 0 0

119872

41119872

420 0 119872

45119872

46

0 0 0 0 0 1

0 119872

620 0 119872

650

]

]

]

]

]

]

]

]

119872

21= 119868

minus1

119909[(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910]

119872

26= 119868

minus1

119909[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

41= 119868

minus1

119910ℎ

119909120596

0

119872

42= minus119868

minus1

119910ℎ

119911

119872

45= 119868

minus1

119910ℎ

119911120596

0

119872

46= 119868

minus1

119910ℎ

119909

119872

62= minus119868

minus1

119911[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

65= 119868

minus1

119911[(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910]

(14)

Matrixes 119861 and 119862 in (1) can be determined accordingto the detailed configuration of the system For example asystem with two infrared earth sensors three orthogonalgyros and one main backup thruster can be described as

119906 (119905) = [119879

1199091119879

1199092119879

1199101119879

1199102119879

1199111119879

1199112]

119879

119910 (119905) = [120593

ℎ1120579

ℎ1120593

ℎ2120579

ℎ2119892

119909119892

119910119892

119911]

119879

6 Mathematical Problems in Engineering

119861 =

[

[

[

[

[

[

[

[

0 0 0 0 0 0

119868

minus1

119909119868

minus1

1199090 0 0 0

0 0 0 0 0 0

0 0 119868

minus1

119910119868

minus1

1199100 0

0 0 0 0 0 0

0 0 0 0 119868

minus1

119911119868

minus1

119911

]

]

]

]

]

]

]

]

119862 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

]

]

]

]

(15)

Considering a spacecraft system described by (1) whenfaults appear the premise of achieving system reconfigura-bility is that the remaining of the system is observable andcontrollable The corresponding criterion is given by (2)According to engineering experience one can assume that119868

119909= 119868

119910= 119868

119911and 120596

0= 0 Consider the following

(1) Only one infrared earth sensor is employed forattitude determination as

119862

1= [

1 0 0 0 0 0

0 0 1 0 0 0

] rank[

[

[

[

[

119862

1

119862

1119860

119862

1119860

5

]

]

]

]

]

= 6 (16)

(2) Three gyros are employed for attitude determinationas

119862

2=

[

[

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

rank[

[

[

[

[

119862

2

119862

2119860

119862

2119860

5

]

]

]

]

]

= 5 (17)

(3) One infrared earth sensor and three gyros areemployed for attitude determination as

119862

3=

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

rank[

[

[

[

[

119862

3

119862

3119860

119862

3119860

5

]

]

]

]

]

= 6 (18)

From (16) to (18) the attitude can be measured in thefollowing two ways

M1 by infrared earth sensorsM2 by infrared earth sensors and gyros

In addition it is assumed that two infrared earth sensorsshare one power supply and three gyros share another powersupply then Table 4 lists the MRUs and their correspondingsubfunctions

Table 4 MRUs and their corresponding functions

MRU FunctionsInfrared earth sensor power(ESP)

Power supply for infrared earthsensor (PS for ES)

Infrared earth sensor 1 (ES1) 120593 and 120579measureInfrared earth sensor 2 (ES2) 120593 and 120579measure

Gyro power (GPower) Power supply for gyros(PS for gyro)

Gyro 119909(119866119909) measure 120596

119909

Gyro 119910 (119866119910) measure 120596

119910

Gyro 119911 (119866119911) measure 120596

119911

Table 5 Results of reconfigurability analysis

MRU 120574 IESPower 0 1

ES1 1 05ES2 1 05

GPower 1 0119866

1199091 0

119866

1199101 0

119866

1199111 0

Figure 6 illustrates the function tree constructed by thereconfigurability modeling process The MCS family and theMPS family could be derived by analyzing the function treein Figure 6 as

C = ESPES1 ESPES2

R = ESP ES1ES2 (19)

Thus reconfigurability indexes can be calculated by (7)to (10) Table 5 lists the FRD and IDMEU of all the MRUsFurthermore suppose that the severity and occurrence pos-sibility for all MRUs are the same then 119908

119894= 1 119903 = 67 and

119879 = 0According to the analysis results of IDMRU and SFTD

of all MRUs the weakest link of this system is the power ofinfrared earth sensors Consequently it is better to store abackup in this link

7 Conclusion

To involve reconfigurability in spacecraft design phase forpotential faults a novel reconfigurability analysis method isinvestigated in this paper First on the basis of observabilityand controllability the reconfigurability criterion is givenfor spacecraft that is considered as a rigid body systemThen the function tree is built formodeling reconfigurabilityand evaluation indexes are proposed After that accordingto minimal cut set and minimal path set of the functiontree a quantitative evaluation method for reconfigurabilityindexes and an approach for determining system weak links

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

tolerate without loss of system functions SFTD reflects thesystem reconfigurability as

119879 = min (

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

) minus 1

1003816

1003816

1003816

1003816

R119894

1003816

1003816

1003816

1003816

isin R 119894 = 1 2 |R| (10)

where 119879 denotes SFTD R119894is the 119894th minimal path set of the

function tree |R119894| is the cardinal number of R

119894

In a system the path set with the minimum numberof MPSs is the weakest link And for this part necessaryredundancy or special reliability design should be consideredaccording to the subfunctions of MRUs in the MPS

The four indexes proposed above are closely connectedto each other Let 119891

119894be a fault whose corresponding recon-

figurable degree is equal to zero 120574119894= 0 namely the corre-

sponding MRU cannot be reconfigured then the importancedegree 119868

119872of the MRU will be equal to one and the system

fault tolerance degree 119879 will become zero Otherwise if allfault reconfigurable degrees are one namely all theMRU canbe reconfigured thenwe can conclude that all the importancedegrees will be less than one the system fault tolerance degreewill be not less than one and the system reconfigurable ratewill be equal to 100

6 Empirical Results

In this section we focus on the practical performance ofthe proposed method Our experiment is presented for thereconfigurability analysis of an attitude measuring system ina spacecraft The dynamic functions regarding momentumdevices are shown in (11)The spacecraft is considered as rigidbody systems and the body coordinate system coincides withthe principle axes of inertia as

119868

119909

119909minus (119868

119910minus 119868

119911) 120596

119910120596

119911minus ℎ

119910120596

119911+ ℎ

119911120596

119910= minus

ℎ

119909+ 119879

119909

119868

119910

119910minus (119868

119911minus 119868

119909) 120596

119911120596

119909minus ℎ

119911120596

119909+ ℎ

119909120596

119911= minus

ℎ

119910+ 119879

119910

119868

119911

119911minus (119868

119909minus 119868

119910) 120596

119909120596

119910minus ℎ

119909120596

119910+ ℎ

119910120596

119909= minus

ℎ

119911+ 119879

119911

(11)

where 119868

119909 119868119910and 119868

119911are moments of inertia along axes 119874119909

119874119910 and 119874119911 respectively 120596 = [120596

119909 120596

119910 120596

119911]

119879 is the angularvelocity vector h = [ℎ

119909 ℎ

119910 ℎ

119911]

119879 is the synthesizing angularmomentum vector of all the momentum devices T =

[119879

119909 119879

119910 119879

119911]

119879 is the control torque vector applied on thespacecraft except for the torque from themomentumdevicesTherefore the control torque vector T = [119879

119909 119879

119910 119879

119911]

119879 in(11) includes torques from thrusters other space torques anddisturbing torques

If all attitudes vary in a small scale the dynamic functionscan be simplified as

120596

119909= minus 120596

0120595

120596

119910=

120579 minus 120596

0

120596

119911=

120595 + 120596

0120593

(12)

where 120593 120579 and 120595 are Euler angles 120596

0denotes the orbit

angular velocity with which the spacecraft circles around thecenter body

Then the linearization form of the attitude dynamicfunction can be derived based on (11) and (12) as

119868

119909 + [(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910] 120593

+ [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

120595

= minus

ℎ

119909+ 120596

0ℎ

119911+ 119879

119909

119868

119910

120579 + ℎ

119909(

120595 + 120596

0120593) minus ℎ

119911( minus 120596

0120595) = minus

ℎ

119910+ 119879

119910

119868

119909

120595 + [(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910] 120595

minus [(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

= minus

ℎ

119911minus 120596

0ℎ

119909+ 119879

119911

(13)

Accordingly the dynamic function of the spacecraft canbe expressed by a state space form as shown in (1) with thefollowing notations

119909 = [120593 120579

120579 120595

120595]

119879

119860 =

[

[

[

[

[

[

[

[

0 1 0 0 0 0

119872

210 0 0 0 119872

26

0 0 0 1 0 0

119872

41119872

420 0 119872

45119872

46

0 0 0 0 0 1

0 119872

620 0 119872

650

]

]

]

]

]

]

]

]

119872

21= 119868

minus1

119909[(119868

119910minus 119868

119911) 120596

2

0minus 120596

0ℎ

119910]

119872

26= 119868

minus1

119909[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

41= 119868

minus1

119910ℎ

119909120596

0

119872

42= minus119868

minus1

119910ℎ

119911

119872

45= 119868

minus1

119910ℎ

119911120596

0

119872

46= 119868

minus1

119910ℎ

119909

119872

62= minus119868

minus1

119911[(119868

119910minus 119868

119911minus 119868

119909) 120596

0minus ℎ

119910]

119872

65= 119868

minus1

119911[(119868

119910minus 119868

119909) 120596

2

0minus 120596

0ℎ

119910]

(14)

Matrixes 119861 and 119862 in (1) can be determined accordingto the detailed configuration of the system For example asystem with two infrared earth sensors three orthogonalgyros and one main backup thruster can be described as

119906 (119905) = [119879

1199091119879

1199092119879

1199101119879

1199102119879

1199111119879

1199112]

119879

119910 (119905) = [120593

ℎ1120579

ℎ1120593

ℎ2120579

ℎ2119892

119909119892

119910119892

119911]

119879

6 Mathematical Problems in Engineering

119861 =

[

[

[

[

[

[

[

[

0 0 0 0 0 0

119868

minus1

119909119868

minus1

1199090 0 0 0

0 0 0 0 0 0

0 0 119868

minus1

119910119868

minus1

1199100 0

0 0 0 0 0 0

0 0 0 0 119868

minus1

119911119868

minus1

119911

]

]

]

]

]

]

]

]

119862 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

]

]

]

]

(15)

Considering a spacecraft system described by (1) whenfaults appear the premise of achieving system reconfigura-bility is that the remaining of the system is observable andcontrollable The corresponding criterion is given by (2)According to engineering experience one can assume that119868

119909= 119868

119910= 119868

119911and 120596

0= 0 Consider the following

(1) Only one infrared earth sensor is employed forattitude determination as

119862

1= [

1 0 0 0 0 0

0 0 1 0 0 0

] rank[

[

[

[

[

119862

1

119862

1119860

119862

1119860

5

]

]

]

]

]

= 6 (16)

(2) Three gyros are employed for attitude determinationas

119862

2=

[

[

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

rank[

[

[

[

[

119862

2

119862

2119860

119862

2119860

5

]

]

]

]

]

= 5 (17)

(3) One infrared earth sensor and three gyros areemployed for attitude determination as

119862

3=

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

rank[

[

[

[

[

119862

3

119862

3119860

119862

3119860

5

]

]

]

]

]

= 6 (18)

From (16) to (18) the attitude can be measured in thefollowing two ways

M1 by infrared earth sensorsM2 by infrared earth sensors and gyros

In addition it is assumed that two infrared earth sensorsshare one power supply and three gyros share another powersupply then Table 4 lists the MRUs and their correspondingsubfunctions

Table 4 MRUs and their corresponding functions

MRU FunctionsInfrared earth sensor power(ESP)

Power supply for infrared earthsensor (PS for ES)

Infrared earth sensor 1 (ES1) 120593 and 120579measureInfrared earth sensor 2 (ES2) 120593 and 120579measure

Gyro power (GPower) Power supply for gyros(PS for gyro)

Gyro 119909(119866119909) measure 120596

119909

Gyro 119910 (119866119910) measure 120596

119910

Gyro 119911 (119866119911) measure 120596

119911

Table 5 Results of reconfigurability analysis

MRU 120574 IESPower 0 1

ES1 1 05ES2 1 05

GPower 1 0119866

1199091 0

119866

1199101 0

119866

1199111 0

Figure 6 illustrates the function tree constructed by thereconfigurability modeling process The MCS family and theMPS family could be derived by analyzing the function treein Figure 6 as

C = ESPES1 ESPES2

R = ESP ES1ES2 (19)

Thus reconfigurability indexes can be calculated by (7)to (10) Table 5 lists the FRD and IDMEU of all the MRUsFurthermore suppose that the severity and occurrence pos-sibility for all MRUs are the same then 119908

119894= 1 119903 = 67 and

119879 = 0According to the analysis results of IDMRU and SFTD

of all MRUs the weakest link of this system is the power ofinfrared earth sensors Consequently it is better to store abackup in this link

7 Conclusion

To involve reconfigurability in spacecraft design phase forpotential faults a novel reconfigurability analysis method isinvestigated in this paper First on the basis of observabilityand controllability the reconfigurability criterion is givenfor spacecraft that is considered as a rigid body systemThen the function tree is built formodeling reconfigurabilityand evaluation indexes are proposed After that accordingto minimal cut set and minimal path set of the functiontree a quantitative evaluation method for reconfigurabilityindexes and an approach for determining system weak links

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

119861 =

[

[

[

[

[

[

[

[

0 0 0 0 0 0

119868

minus1

119909119868

minus1

1199090 0 0 0

0 0 0 0 0 0

0 0 119868

minus1

119910119868

minus1

1199100 0

0 0 0 0 0 0

0 0 0 0 119868

minus1

119911119868

minus1

119911

]

]

]

]

]

]

]

]

119862 =

[

[

[

[

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

]

]

]

]

(15)

Considering a spacecraft system described by (1) whenfaults appear the premise of achieving system reconfigura-bility is that the remaining of the system is observable andcontrollable The corresponding criterion is given by (2)According to engineering experience one can assume that119868

119909= 119868

119910= 119868

119911and 120596

0= 0 Consider the following

(1) Only one infrared earth sensor is employed forattitude determination as

119862

1= [

1 0 0 0 0 0

0 0 1 0 0 0

] rank[

[

[

[

[

119862

1

119862

1119860

119862

1119860

5

]

]

]

]

]

= 6 (16)

(2) Three gyros are employed for attitude determinationas

119862

2=

[

[

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

rank[

[

[

[

[

119862

2

119862

2119860

119862

2119860

5

]

]

]

]

]

= 5 (17)

(3) One infrared earth sensor and three gyros areemployed for attitude determination as

119862

3=

[

[

[

[

[

[

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 minus120596

00

0 0 0 1 0 0

120596

00 0 0 0 1

]

]

]

]

]

]

rank[

[

[

[

[

119862

3

119862

3119860

119862

3119860

5

]

]

]

]

]

= 6 (18)

From (16) to (18) the attitude can be measured in thefollowing two ways

M1 by infrared earth sensorsM2 by infrared earth sensors and gyros

In addition it is assumed that two infrared earth sensorsshare one power supply and three gyros share another powersupply then Table 4 lists the MRUs and their correspondingsubfunctions

Table 4 MRUs and their corresponding functions

MRU FunctionsInfrared earth sensor power(ESP)

Power supply for infrared earthsensor (PS for ES)

Infrared earth sensor 1 (ES1) 120593 and 120579measureInfrared earth sensor 2 (ES2) 120593 and 120579measure

Gyro power (GPower) Power supply for gyros(PS for gyro)

Gyro 119909(119866119909) measure 120596

119909

Gyro 119910 (119866119910) measure 120596

119910

Gyro 119911 (119866119911) measure 120596

119911

Table 5 Results of reconfigurability analysis

MRU 120574 IESPower 0 1

ES1 1 05ES2 1 05

GPower 1 0119866

1199091 0

119866

1199101 0

119866

1199111 0

Figure 6 illustrates the function tree constructed by thereconfigurability modeling process The MCS family and theMPS family could be derived by analyzing the function treein Figure 6 as

C = ESPES1 ESPES2

R = ESP ES1ES2 (19)

Thus reconfigurability indexes can be calculated by (7)to (10) Table 5 lists the FRD and IDMEU of all the MRUsFurthermore suppose that the severity and occurrence pos-sibility for all MRUs are the same then 119908

119894= 1 119903 = 67 and

119879 = 0According to the analysis results of IDMRU and SFTD

of all MRUs the weakest link of this system is the power ofinfrared earth sensors Consequently it is better to store abackup in this link

7 Conclusion

To involve reconfigurability in spacecraft design phase forpotential faults a novel reconfigurability analysis method isinvestigated in this paper First on the basis of observabilityand controllability the reconfigurability criterion is givenfor spacecraft that is considered as a rigid body systemThen the function tree is built formodeling reconfigurabilityand evaluation indexes are proposed After that accordingto minimal cut set and minimal path set of the functiontree a quantitative evaluation method for reconfigurabilityindexes and an approach for determining system weak links

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Attitudemeasure

PS for ES Φ and 120579

Φ and 120579

measure

measure measure measure measure

measure

ESP

ES1

ES1

ES2

ES2

ESP

PS for ES PS for gyro 120596y120596x 120596z

Gpower

Gyro

M1

M1

M2

Gx Gy Gz

Figure 6 Function tree for attitude determinations

are summarized Theoretical research and empirical studyboth illustrate the benefit of the constructedmethodology forspacecraft reconfigurability design on reliability criterions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous reviewers andthe editor for their critical and constructive comments onthis paper This study was supported by the National NaturalScience Foundation of China (Grant nos 61203093 and11202011)

References

[1] T Kreider and J Ross ldquoRe-configurable spacecraft softwaredemands and solutionrdquo in Proceedings of the IEEE AerospaceConference Proceedings pp 2364ndash2369 March 2004

[2] W D Nadir I-Y Kim D Hauser and O L De WeckldquoMultidisciplinary structural truss topology optimization forreconfigurabilityrdquo in Proceedings of the 10th AIAAISSMOMul-tidisciplinary Analysis and Optimization Conference pp 472ndash487 New York NY USA September 2004

[3] Y Zhang and J Jiang ldquoBibliographical review on reconfigurablefault-tolerant control systemsrdquo Annual Reviews in Control vol32 no 2 pp 229ndash252 2008

[4] D U Campos-Delgado and K Zhou ldquoReconfigurable fault-tolerant control using GIMC structurerdquo IEEE Transactions onAutomatic Control vol 48 no 5 pp 832ndash838 2003

[5] K Zhou and Z Ren ldquoA new controller architecture for high per-formance robust and fault-tolerant controlrdquo IEEE Transactionson Automatic Control vol 46 no 10 pp 1613ndash1618 2001

[6] Z Mao and B Jiang ldquoFault identification and fault-tolerantcontrol for a class of networked control systemsrdquo InternationalJournal of Innovative Computing Information and Control vol3 no 5 pp 1121ndash1130 2007

[7] L Meng and B Jiang ldquoRobust active fault-tolerant control fora class of uncertain nonlinear systems with actuator faultsrdquoInternational Journal of Innovative Computing Information andControl vol 6 no 6 pp 2637ndash2644 2010

[8] X Gao K Lay Teo and G Duan ldquoAn optimal control approachto robust control of nonlinear spacecraft rendezvous systemwith 120579-D techniquerdquo International Journal of Innovative Com-puting Information and Control vol 9 no 5 pp 2099ndash21102013

[9] R Qi L Zhu and B Jiang ldquoFault-tolerant reconfigurablecontrol for MIMO system using online fuzzy identificationrdquoInternational Journal of Innovative Computing Information andControl vol 9 no 10 pp 3915ndash3928 2013

[10] S P Joshi Z Tidwell W A Crossley and S RamakrishnanldquoComparison of morphing wing strategies based upon aircraftperformance impactsrdquo in Proceedings of the 45th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference AIAA-2004-1722 pp 2348ndash2354 Palm SpringsCalif USA April 2004

[11] C W Frei F J Kraus and M Blanke ldquoRecoverability viewedas a system propertyrdquo in Proceedings of the European ControlConference (ECC rsquo99) Karlsruhe Germany 1999

[12] N E Wu K Zhou and G Salomon ldquoControl reconfigurabilityof linear time-invariant systemsrdquoAutomatica vol 36 no 11 pp1767ndash1771 2000

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

[13] M Staroswiecki ldquoOn reconfigurability with respect to actuatorfailuresrdquo in Proceedings of the 15th Triennial World CongressBarcelona Spain 2002

[14] A Siddiqi Reconfigurability in Space Systems ArchitectingFramework and Case Studies Massachusetts Institute of Tech-nology 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of