A Geometric Method for Generating Discrete Trace Transition System of a Polyhedral Invariant Hybrid Automaton

Affiliation(s)

Power and Water Institute, Kermanshah, Iran.

Electrical and Computer Engineering Department, Isfahan University of Technology, Isfahan, Iran.

Power and Water Institute, Kermanshah, Iran.

Electrical and Computer Engineering Department, Isfahan University of Technology, Isfahan, Iran.

ABSTRACT

Supervisory control and fault diagnosis of hybrid systems need to have complete information about the discrete states transitions of the underling system. From this point of view, the hybrid system should be abstracted to a Discrete Trace Transition System (DTTS) and represented by a discrete mode transition graph. In this paper an effective method is proposed for generating discrete mode transition graph of a hybrid system. This method can be used for a general class of industrial hybrid plants which are defined by Polyhedral Invariant Hybrid Automata (PIHA). In these automata there are no resetting maps, while invariant sets are defined by linear inequalities. Therefore, based on the continuity property of the state trajectories in a PIHA, the problem is reduced to finding possible transitions between all two adjacent discrete modes. In the presented method, the possibility and the direction of such transitions are detected only by computing the angle between the vector field and the normal vector of the switching surfaces. Thus, unlike the most other reachability methods, there is no need to solve differential equations and to do mapping computations. In addition, the proposed method, with some modifications can be applied for extracting Stochastic or Timed Discrete Trace Transition Systems.

Supervisory control and fault diagnosis of hybrid systems need to have complete information about the discrete states transitions of the underling system. From this point of view, the hybrid system should be abstracted to a Discrete Trace Transition System (DTTS) and represented by a discrete mode transition graph. In this paper an effective method is proposed for generating discrete mode transition graph of a hybrid system. This method can be used for a general class of industrial hybrid plants which are defined by Polyhedral Invariant Hybrid Automata (PIHA). In these automata there are no resetting maps, while invariant sets are defined by linear inequalities. Therefore, based on the continuity property of the state trajectories in a PIHA, the problem is reduced to finding possible transitions between all two adjacent discrete modes. In the presented method, the possibility and the direction of such transitions are detected only by computing the angle between the vector field and the normal vector of the switching surfaces. Thus, unlike the most other reachability methods, there is no need to solve differential equations and to do mapping computations. In addition, the proposed method, with some modifications can be applied for extracting Stochastic or Timed Discrete Trace Transition Systems.

Cite this paper

S. Baniardalani and J. Askari, "A Geometric Method for Generating Discrete Trace Transition System of a Polyhedral Invariant Hybrid Automaton,"*Intelligent Control and Automation*, Vol. 3 No. 2, 2012, pp. 197-206. doi: 10.4236/ica.2012.32022.

S. Baniardalani and J. Askari, "A Geometric Method for Generating Discrete Trace Transition System of a Polyhedral Invariant Hybrid Automaton,"

References

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[2] W. Wang, D. H. Zhou and Z. Li, “Robust State Estimation and Fault Diagnosis for Uncertain Hybrid Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 65, No. 12, 2006, pp.2193-2215. doi:10.1016/j.na.2006.02.047

[3] R. Alur, T. A. Henzinger and E. D. Zontag, “Hybrid Systems III. Verification and Control (Lecture Notes in Computer Science),” Springer, Berlin, 1996.

[4] J. Lunze, “Fault Diagnosis of Discretely Controlled Continuous Systems by Means of Discrete Event-Models,” Discrete Event Dynamic Systems, Vol. 18, No. 2, 2008, pp. 181-210. doi:10.1007/s10626-007-0022-3

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[6] S. Baniardalani, “Design and Analysis of Discrete Event Model Based Fault Diagnosis for a Hybrid System,” Ph.D. Thesis, Isfahan University of Technology, Isfahan, 2011.

[7] R. Alur and T. A. Henzinger, “The Algorithmic Analysis of Hybrid System,” Theoretical Computer Science, Vol. 138, No. 1, 1995, pp. 3-34. doi:10.1016/0304-3975(94)00202-T

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[11] S. Ratschan and Z. She, “Safety Verification of Hybrid Systems by Constraint Propagation Based Abstraction Refinement,” Hybrid Systems: Computation and Control, Vol. 3414, 2005, pp. 573-589. doi:10.1007/978-3-540-31954-2_37

[12] G. Frehse, “Algorithmic Verification of Hybrid System Past HyTech,” International Journal on Software Tools for Technology Transfer, Vol. 10, No. 3, 2005, pp. 258273. doi:10.1007/s10009-007-0062-x

[13] M. Blanke, M. Kinnaret; J. Lunze and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, 2nd Edition, Springer, Berlin, 2006.

[14] C. G. Cassandras and S. Lafortune, “Introduction to Discrete Event Systems,” 2nd Eddition, Kluwer Academic Publishers, Dordrecht, 2008.

[15] Askari-Marnani, B. Heiming and J. Lunze, “Control reconfiguration: The cosy Benchmark Problem and Its Solution by Means of a Qualitative Model Part of Chapter 21 of the Final Report of the Project,” Control of Complex Systems (cosy), 2011.

[16] J. Lunze, J. Askari, et al., “Three-Tank Reconfiguration Control, Control of Complex Systems,” 2001, pp. 24-283.

[17] J. Pan and S. Hashtrudi-Zad, “Diagnosability Analysis and Sensor Selection in Discrete-Event Systems with Permanent Failures,” Proceedings of the 3rd IEEE Conference on Automation Science and Engineering, Scottsdale, 22-25 September 2007, pp. 869-874.

[18] M. Sampath, R. Segupta, S. Lafortune; S. K. Sinnamohideen and D. Tenketizis, “Diagnosability of Discrete event Systems,” IEEE Transactions on Automatic Control, Vol. 40, No. 9, pp. 1555-1575. doi:10.1109/9.412626

[19] P. Supavatanakul and J. Lunze, “Diagnosis of Timed Automata: Theory and Application to the DAMADICS Actuator Benchmark Problem,” Control Engineering Practice, Vol. 14, No. 6, 2006, pp. 609-619. doi:10.1016/j.conengprac.2005.03.028

[1] J. Lygeros, “Lecture Notes on Hybrid System,” Department of Electrical and Computer Engineering University of Patras, Patras, 2004.

[2] W. Wang, D. H. Zhou and Z. Li, “Robust State Estimation and Fault Diagnosis for Uncertain Hybrid Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 65, No. 12, 2006, pp.2193-2215. doi:10.1016/j.na.2006.02.047

[3] R. Alur, T. A. Henzinger and E. D. Zontag, “Hybrid Systems III. Verification and Control (Lecture Notes in Computer Science),” Springer, Berlin, 1996.

[4] J. Lunze, “Fault Diagnosis of Discretely Controlled Continuous Systems by Means of Discrete Event-Models,” Discrete Event Dynamic Systems, Vol. 18, No. 2, 2008, pp. 181-210. doi:10.1007/s10626-007-0022-3

[5] A. Chutinan and B. H. Krogh, “Computational Techniques for Hybrid System Verification,” IEEE Transactions on Automatic Control, Vol. 48. No. 1, 2003, pp. 6375. doi:10.1109/TAC.2002.806655

[6] S. Baniardalani, “Design and Analysis of Discrete Event Model Based Fault Diagnosis for a Hybrid System,” Ph.D. Thesis, Isfahan University of Technology, Isfahan, 2011.

[7] R. Alur and T. A. Henzinger, “The Algorithmic Analysis of Hybrid System,” Theoretical Computer Science, Vol. 138, No. 1, 1995, pp. 3-34. doi:10.1016/0304-3975(94)00202-T

[8] J. Lunze and B. Nixdorf, “Discrete Reachability of Hybrid Systems,” International Journal of Control, Vol. 76, No. 14, 2003, pp. 1453-1468. doi:10.1080/0020717031000151309

[9] J. Schr?der, “Modeling, State Observation, and Diagnosis of Quantized Systems,” Springer-Verlag, Berlin, 2003.

[10] D. Forstner and J. Lunze, “Discrete-Event Models of Quantized System for Diagnosis,” International Journal of Control, Vol. 74, No. 7, 2001, pp. 690-700. doi:10.1080/00207170010025276

[11] S. Ratschan and Z. She, “Safety Verification of Hybrid Systems by Constraint Propagation Based Abstraction Refinement,” Hybrid Systems: Computation and Control, Vol. 3414, 2005, pp. 573-589. doi:10.1007/978-3-540-31954-2_37

[12] G. Frehse, “Algorithmic Verification of Hybrid System Past HyTech,” International Journal on Software Tools for Technology Transfer, Vol. 10, No. 3, 2005, pp. 258273. doi:10.1007/s10009-007-0062-x

[13] M. Blanke, M. Kinnaret; J. Lunze and M. Staroswiecki, Diagnosis and Fault-Tolerant Control, 2nd Edition, Springer, Berlin, 2006.

[14] C. G. Cassandras and S. Lafortune, “Introduction to Discrete Event Systems,” 2nd Eddition, Kluwer Academic Publishers, Dordrecht, 2008.

[15] Askari-Marnani, B. Heiming and J. Lunze, “Control reconfiguration: The cosy Benchmark Problem and Its Solution by Means of a Qualitative Model Part of Chapter 21 of the Final Report of the Project,” Control of Complex Systems (cosy), 2011.

[16] J. Lunze, J. Askari, et al., “Three-Tank Reconfiguration Control, Control of Complex Systems,” 2001, pp. 24-283.

[17] J. Pan and S. Hashtrudi-Zad, “Diagnosability Analysis and Sensor Selection in Discrete-Event Systems with Permanent Failures,” Proceedings of the 3rd IEEE Conference on Automation Science and Engineering, Scottsdale, 22-25 September 2007, pp. 869-874.

[18] M. Sampath, R. Segupta, S. Lafortune; S. K. Sinnamohideen and D. Tenketizis, “Diagnosability of Discrete event Systems,” IEEE Transactions on Automatic Control, Vol. 40, No. 9, pp. 1555-1575. doi:10.1109/9.412626

[19] P. Supavatanakul and J. Lunze, “Diagnosis of Timed Automata: Theory and Application to the DAMADICS Actuator Benchmark Problem,” Control Engineering Practice, Vol. 14, No. 6, 2006, pp. 609-619. doi:10.1016/j.conengprac.2005.03.028