S. Inversement and . Qu, Considérons V Z = {z j , z k }, d'après le Théorème 5.4.3, V Z est une solution pour leprobì eme de RPRM. Dans ce cas, la suppression de z j ne change pas la solubilité duprobì eme de RPRM parce que {V Z \z j } = {z k } est une solution

. Dans-cet-exemple, nous avons I * = {x 2 } et la perturbation d affecte seulement x 2 ? I * . D'après la Définition 5.4.2, nous trouvons D d = ?. Sans aucune mesure, on a J * = {x 1 } donc I * ? J * = X alors leprobì eme de RPRM est soluble sans aucune mesure, En effet

. Dans-cet-exemple, nous avons I * = {x 1

J. {x, x 8 }, alors leprobì eme RPRM est soluble d'après le Théorème 5.2.5. Nous avons F I * = {x 2

Y. Ali and S. Narasimhan, Sensor network design for maximizing reliability of linear processes, AIChE Journal, vol.39, issue.5, pp.2563-2575, 1996.
DOI : 10.1002/aic.690390510

M. Assad, Analyse de la tolérance aux fautes : Approches fonctionnelle et structurelle, 2002.

M. J. Bagajewicz, Design and retrofit of sensor networks in process plants, AIChE Journal, vol.43, issue.9, pp.2300-2306, 1997.
DOI : 10.1002/aic.690430915

M. Bahar, J. Jantzen, C. Commault, J. M. Dion, and V. Hovelaque, Structural decoupling and disturbance rejection in a distillation column, World Congress IFAC'96, 1996.

C. Berge, Graphes et hypergraphes, 1973.

M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and faulttolerant control, 2003.

B. O. Bouamama, K. Medjaher, M. Bayart, A. K. Samantaray, C. et al., Fault detection and isolation of smart actuators using bond graphs and external models, Control Engineering Practice, vol.13, issue.2, pp.159-175, 2005.
DOI : 10.1016/j.conengprac.2004.03.003

T. Boukhobza and F. Hamelin, Observability analysis for structured bilinear systems : A graph-theoretic approach, Automatica, issue.11, pp.431968-1974, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00159428

T. Boukhobza, F. Hamelin, and D. Sauter, Observability of structured linear systems in descriptor form: A graph-theoretic approach, Automatica, vol.42, issue.4, pp.629-635, 2006.
DOI : 10.1016/j.automatica.2005.12.001
URL : https://hal.archives-ouvertes.fr/hal-00120927

T. Boukhobza, F. Hamelin, and D. Sauter, Uniform Observability Analysis for Structured Bilinear Systems A Graph-theoretic Approach, European Journal of Control, vol.12, issue.5, pp.505-518, 2006.
DOI : 10.3166/ejc.12.505-518
URL : https://hal.archives-ouvertes.fr/hal-00120933

B. Cassar, J. Staroswiecki, and M. , A structural approach for the design of failure detection and identification systems, IFAC, IFIP,IMACS Conference on Control of Industrial Systems, pp.329-334, 1997.
URL : https://hal.archives-ouvertes.fr/hal-01403803

A. Chamseddine, H. Noura, R. , and T. , Optimal Sensor Network Design for Observability of Complex Systems, 2007 American Control Conference, pp.1705-1710, 2007.
DOI : 10.1109/ACC.2007.4282689

J. Chen and R. Patton, Robust model-based fault diagnosis for dynamic systems, 1999.
DOI : 10.1007/978-1-4615-5149-2

D. J. Chmielewski, T. Palmer, and V. Manousiouthakis, On the theory of optimal sensor placement, AIChE Journal, vol.10, issue.5, pp.1001-1012, 2002.
DOI : 10.1002/aic.690480510

C. Commault, J. M. Dion, and M. Benahcène, Output feedback disturbance decoupling graph interpretation for structured systems, Automatica, vol.29, issue.6, pp.1463-1472, 1993.
DOI : 10.1016/0005-1098(93)90010-Q

C. Commault, J. M. Dion, and V. Hovelaque, A geometric approach for structured systems: Application to disturbance decoupling, Automatica, vol.33, issue.3, pp.403-409, 1997.
DOI : 10.1016/S0005-1098(96)00186-0

C. Commault, J. M. Dion, and A. Perez, Disturbance rejection for structured systems, IEEE Transactions on Automatic Control, vol.36, issue.7, pp.884-887, 1991.
DOI : 10.1109/9.85072

C. Commault, J. M. Dion, O. Sename, and R. Motyeian, Observer-based fault detection and isolation for structured systems, IEEE Transactions on Automatic Control, vol.47, issue.12, pp.472074-2079, 2002.
DOI : 10.1109/TAC.2002.805673

C. Commault, J. M. Dion, T. , and D. H. , Observability recovering by additional sensor implementation in linear structured systems, Proceedings of the 44th IEEE Conference on Decision and Control, pp.7193-7197, 2005.
DOI : 10.1109/CDC.2005.1583321
URL : https://hal.archives-ouvertes.fr/hal-00363445

C. Commault, J. M. Dion, T. , and D. H. , Observability preservation under sensor failure, IEEE CDC Conf, pp.7193-7197, 2006.
DOI : 10.1109/cdc.2006.377111
URL : https://hal.archives-ouvertes.fr/hal-00320200

C. Commault, J. M. Dion, T. , and D. H. , Observability Preservation Under Sensor Failure, IEEE Transactions on Automatic Control, vol.53, issue.6, pp.1554-1559, 2008.
DOI : 10.1109/TAC.2008.928307
URL : https://hal.archives-ouvertes.fr/hal-00320200

C. Commault, J. M. Dion, and J. Van-der-woude, Characterization of generic properties of linear structured systems for efficient computations, Kybernetika, vol.38, issue.5, pp.503-520, 2002.

C. Commault, J. M. Dion, and S. Yacoub-agha, Location of additional sensors for FDI, 4th Workshop on Advanced Control and Diagnosis, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00113032

B. Commault, C. Dion, J. M. Yacoub-agha, and S. , Structural analysis for the sensor location problem in fault detection and isolation, IFAC Safe Process, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00301490

C. Commault, J. M. Dion, and S. Yacoub-agha, SENSOR CLASSIFICATION FOR THE FAULT DETECTION AND ISOLATION PROBLEM, IFAC Symp. SSC 07, 2007.
DOI : 10.3182/20071017-3-BR-2923.00108
URL : https://hal.archives-ouvertes.fr/hal-00149494

G. Dauphin-tanguy, Les Bond Graphs. Systèmes automatisés, 2000.

E. J. Davison, Connectability and structural controllability of composite systems, Automatica, vol.13, issue.2, pp.109-123, 1977.
DOI : 10.1016/0005-1098(77)90036-X

J. M. Dion and C. Commault, Feedback decoupling of structured systems, IEEE Transactions on Automatic Control, vol.38, issue.7, pp.1132-1135, 1993.
DOI : 10.1109/9.231471

J. M. Dion, C. Commault, T. , and D. H. , Sensor classification for the disturbance rejection by measurement feedback problem, IFAC WC 2008, 2008.
DOI : 10.3182/20080706-5-KR-1001.01914
URL : https://hal.archives-ouvertes.fr/hal-00199401

J. M. Dion, C. Commault, T. , and D. H. , Sensor network design for the disturbance rejection by measurement feedback problem, IEEE CDC Conf, 2008.

J. M. Dion, C. Commault, and J. Van-der-woude, Generic properties and control of linear structured systems: a survey, Automatica, vol.39, issue.7, pp.391125-1144, 2003.
DOI : 10.1016/S0005-1098(03)00104-3

T. H. Do, Défaillance des actionneurs et commandabilité des systèmes structurés, 2008.

J. Fraden, Handbook of Modern Sensors, 2004.

P. Frank, Analytical and Qualitative Model-based Fault Diagnosis ??? A Survey and Some New Results, European Journal of Control, vol.2, issue.1, pp.6-28, 1996.
DOI : 10.1016/S0947-3580(96)70024-9

E. Frisk and M. Krysander, Sensor placement for maximum fault isolability, 18th International Workshop on Principles of Diagnosis (DX-07), 2007.

M. Garey and D. Johnson, Computers and intractability : A guide to the theory of NP-Completeness, 1999.

K. Glover and L. Silverman, Characterization of structural controllability, IEEE Transactions on Automatic Control, vol.21, issue.4, pp.534-537, 1976.
DOI : 10.1109/TAC.1976.1101257

J. Hopcroft and R. Karp, An $n^{5/2} $ Algorithm for Maximum Matchings in Bipartite Graphs, SIAM Journal on Computing, vol.2, issue.4, pp.225-231, 1973.
DOI : 10.1137/0202019

S. Hosoe and K. Matsumoto, On the irreducibility condition in the structural controllability theorem, IEEE Transactions on Automatic Control, vol.24, issue.6, pp.963-966, 1979.
DOI : 10.1109/TAC.1979.1102192

V. Hovelaque, C. Commault, D. , and J. M. , Analysis of linear structured systems using a primal-dual algorithm, Systems & Control Letters, vol.27, issue.2, pp.73-85, 1996.
DOI : 10.1016/0167-6911(95)00039-9

R. D. Johnston, G. W. Barton, and M. L. Brisk, Determination of the generic rank of structural matrices, International Journal of Control, vol.4, issue.2, pp.257-264, 1984.
DOI : 10.1080/00207178408933271

T. Kailath, Linear Systems, 1980.

N. Karcanias, Integrated process design: A generic control theory/design based framework, Computers in Industry, vol.26, issue.3, pp.291-301, 1995.
DOI : 10.1016/0166-3615(95)00044-5

N. Karcanias, Control problems in global process instrumentation: A structural approach, Computers & Chemical Engineering, vol.20, pp.1101-1106, 1996.
DOI : 10.1016/0098-1354(96)00191-3

A. Kretsovalis and R. S. Mah, Effect of redundancy on estimation accuracy in process data reconciliation, Chemical Engineering Science, vol.42, issue.9, pp.2115-2121, 1987.
DOI : 10.1016/0009-2509(87)85032-7

C. Lin, Structural controllability, IEEE Trans. Automat. Control, vol.19, issue.3, pp.201-208, 1974.

C. Lin, System strcture and minimal structure controllability, IEEE Trans. Automat. Control, vol.22, issue.5, pp.855-862, 1977.

A. Linnemann, Decoupling of structured systems, Systems & Control Letters, vol.1, issue.2, pp.71-86, 1981.
DOI : 10.1016/S0167-6911(81)80040-0

A. Linnemann, A further simplification in the proof of the structural controllability theorem, IEEE Transactions on Automatic Control, vol.31, issue.7, pp.31638-639, 1986.
DOI : 10.1109/TAC.1986.1104350

F. Madron and V. Veverka, Optimal selection of measuring points in complex plants by linear models, AIChE Journal, vol.38, issue.2, pp.227-236, 1993.
DOI : 10.1002/aic.690380208

H. Mayeda, On structural controllability theorem, IEEE Transactions on Automatic Control, vol.26, issue.3, pp.795-798, 1981.
DOI : 10.1109/TAC.1981.1102707

M. Meyer, J. M. Le-lann, B. Koehret, and M. Enjalbert, Optimal selection of sensor location on a complex plant, using a graph oriented approach, Computers & Chemical Engineering, vol.18, pp.535-540, 1994.
DOI : 10.1016/0098-1354(94)80087-1

K. Murota, Systems Analysis by Graphs and Matroids, volume 3 of Algorithms and Combinatorics, 1987.

K. Murota, Matrices and Matroids for Systems Analysis. Algorithms and Combinatorics 20, 2000.

Y. Ohta and S. Kodama, Structural invertibility of transfer functions, IEEE Transactions on Automatic Control, vol.30, issue.8, pp.818-819, 1985.
DOI : 10.1109/TAC.1985.1104043

R. Bibliographie-raghuraj, M. Bhushan, R. , and R. , Locating sensors in complex chemical plants based on fault diagnostic observability criteria, AIChE Journal, vol.9, issue.2, pp.310-322, 1999.
DOI : 10.1002/aic.690450212

A. Rahmani, Bond Graph Approach for Structural Study of Linear Systems, 1993.

A. Rahmani and G. Dauphin-tanguy, Structural analysis of switching systems modelled by bond graph, Mathematical and Computer Modelling of Dynamical Systems, vol.26, issue.2-3, pp.235-247, 2006.
DOI : 10.1080/1383950500068344

A. Rahmani, C. Sueur, and G. Dauphin-tanguy, On the infinite structure of systems modelled by bond graph: feedback decoupling, 1996 IEEE International Conference on Systems, Man and Cybernetics. Information Intelligence and Systems (Cat. No.96CH35929), pp.1617-1622, 1996.
DOI : 10.1109/ICSMC.1996.565337

K. Reinschke, Multivariable control : A graph-theoretic approach, Lect. Notes in Control and Information Sciences, vol.108, 1988.
DOI : 10.1007/BFb0051585

J. Schumacher, Compensator synthesis using (C,A,B)-pairs, IEEE Transactions on Automatic Control, vol.25, issue.6, pp.1133-1138, 1980.
DOI : 10.1109/TAC.1980.1102515

R. Shields and J. Pearson, Structural controllability of multiinput linear systems, IEEE Transactions on Automatic Control, vol.21, issue.2, pp.203-212, 1976.
DOI : 10.1109/TAC.1976.1101198

S. Spanache, T. Escobet, and L. Travé-massuyès, Sensor placement optimisation using genetic algorithms, 15th International Workshop on Principles of Diagnosis, pp.179-183, 2004.

M. Staroswiecki, Observability, monitorability and fault tolerant estimation by structural analysis, DDP'03, 2003.

M. Staroswiecki, Advances in Control Theory and Applications, volume 353 of Lecture Notes in Control and Information Sciences, chapter Observability and the Design of Fault Tolerant Estimation Using Structural Analysis, pp.257-278, 2007.

M. Staroswiecki and J. Cassar, Approche structurelle pour la conception des systèmes de surveillance, 1996.

M. Staroswiecki, J. P. Cassar, and P. Declerck, Issues of fault diagnosis for dynamic systems, chapter A structural framework for the design of FDI system in large scale industrial plants, pp.245-283, 2000.

M. Staroswiecki, G. Hoblos, and A. Aitouche, Sensor network design for fault tolerant estimation, International Journal of Adaptive Control and Signal Processing, vol.18, issue.1, pp.55-72, 2004.
DOI : 10.1002/acs.781

B. Suda, N. Wan, B. Ueno, and I. , The Orders of Infinite Zeros of Structured Systems, Transactions of the Society of Instrument and Control Engineers, vol.25, issue.10, pp.346-348, 1989.
DOI : 10.9746/sicetr1965.25.1062

C. Sueur and G. Dauphin-tanguy, Structural controllability/observability of linear systems represented by bond graphs, Journal of the Franklin Institute, vol.326, issue.6, pp.869-883, 1989.
DOI : 10.1016/0016-0032(89)90009-4

C. Sueur and G. Dauphin-tanguy, Bond-graph approach for structural analysis of MIMO linear systems, Journal of the Franklin Institute, vol.328, issue.1, pp.55-70, 1991.
DOI : 10.1016/0016-0032(91)90006-O

L. Travé-massuyès, T. Escobet, and R. Milne, Model-based diagnosability and sensor placement application to a frame 6 gas turbine subsystem, 12th Int.Workshop on principles of diagnosis, pp.205-212, 2001.

D. H. Trinh, Préservation de l'observabilité en présence de défauts de capteurs. e-STA, 4(Spécial JD-JN-MACS 07 ; Cette article a ´ eté présentés lorsque JDMACS'07, 2007.

D. H. Trinh, C. Commault, D. , and J. M. , Useless sensors for the disturbance rejection by measurement feedback problem, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00363400

J. Van-der-woude, A graph-theoretic characterization for the rank of the transfer matrix of a structured system, Mathematics of Control, Signals and Systems, vol.27, issue.1, pp.33-40, 1991.
DOI : 10.1007/BF02551379

J. Van-der-woude, On the structure at infinity of a structured system, Linear Algebra and its Applications, vol.148, pp.145-169, 1991.
DOI : 10.1016/0024-3795(91)90091-A

J. Van-der-woude, On the structure at infinity of a structured system, Linear Algebra and its Applications, vol.148, pp.145-169, 1991.
DOI : 10.1016/0024-3795(91)90091-A

J. Van-der-woude, Disturbance decoupling by measurement feedback for structured systems : a graph theoretic approach, Proc. 2nd Europ. Cont. Conf. ECC'93, pp.1132-1137, 1993.

J. Van-der-woude, Graph theoretic methods for the computation of disturbance decoupling feedback matrices for structured systems, Linear Algebra and its Applications, vol.196, pp.119-162, 1994.
DOI : 10.1016/0024-3795(94)90320-4

J. Van-der-woude, The Generic Number of Invariant Zeros of a Structured Linear System, SIAM Journal on Control and Optimization, vol.38, issue.1, pp.1-21, 2000.
DOI : 10.1137/S0363012996310119

V. Václavek and M. Loucka, Selection of measurements necessary to achieve multicomponent mass balances in chemical plant, Chemical Engineering Science, vol.31, issue.12, pp.311199-1205, 1976.
DOI : 10.1016/0009-2509(76)85029-4

G. Verghese, Infinite frequency behavior in generalized dynamical systems, 1978.

J. Bibliographie-willems and C. Commault, Disturbance Decoupling by Measurement Feedback with Stability or Pole Placement, SIAM Journal on Control and Optimization, vol.19, issue.4, pp.490-504, 1981.
DOI : 10.1137/0319029

W. Wonham, Linear multivariable control : a geometric approach, 1974.
DOI : 10.1007/978-1-4684-0068-7

W. Wonham, Linear multivariable control : a geometric approach, 1979.
DOI : 10.1007/978-1-4684-0068-7

S. Yacoub-agha, Application des systèemes structurésstructurésà l'´ etude du diagnostic : Localisation de capteurs, 2007.

T. Yamada, Comments on "On structural controllability theorem", IEEE Transactions on Automatic Control, vol.29, issue.12, pp.1130-1131, 1984.
DOI : 10.1109/TAC.1984.1103449

A. A. Yassine, S. Ploix, and J. Flaus, New results for sensor placement with diagnosability purpose, 18th International Workshop on Principles of Diagnosis (DX-07), 2007.
URL : https://hal.archives-ouvertes.fr/hal-00381328