Abstract
This paper extends the application of the Cantor metric as a mathematical tool for defining causalities from pure discrete models to mixed-signal and hybrid models. Using the Cantor metric, which maps timed signals, continuous or discrete, into a metric space, we define causality as contractive properties of processes operating on these signals. Thus, the Banach fixed point theorem can be applies to establish conditions for the existence, uniqueness, and liveness of the behaviors for mixed-signal and hybrid systems. The results also provide theoretical foundations for the simulation technologies for such systems, including the time-marching strategy, evaluation of feedback loops, and the necessity of supporting rollback.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Henzinger, T.A.: The theory of hybrid automata. In: Proceedings of the 11th Annual Symposium on Logic in Computer Science (LICS). IEEE Computer Society Press (1996) 278–292
Bornot, S., Sifakis, J.: On the composition of hybrid systems. In: Hybrid Systems: Computation and Control, HSCC’98, Berkeley, CA, April, 1998. Volume 1386 of Lecture Notes in Computer Science. Springer (1998) 49–63
Lygeros, J., Johansson, K.H., Simic, S.N., Zhang, J., Sastry, S.: Dynamical properties of hybrid automata. IEEE Transactions on Automatic Control 48 (2003), to appear.
van der Schaft, A., Schumacher, J.: Complementarity modeling of hybrid systems. IEEE Transactions on Automatic Control 43 (1998) 483–490
Lemmon, M.: On the existence of solutions to controlled hybrid automata. In: Hybrid Systems: Computation and Control, HSCC’00, Pittsburgh, PA, USA, March, 2000. Volume 1790 of Lecture Notes in Computer Science. Springer (2000) 229-242
Benveniste, A.: Compositional and uniform modeling of hybrid systems. IEEE Transactions on Automatic Control 43 (1998) 579–584
IEEE 1076.1 Working Group: VHDL 1076.1-1999: Analog and mixed-signal extensions to VHDL (1999)
Harman, T.L., Dabney, J.B.: Mastering Simulink 4. Prentice Hall (2001)
Lee, E.A., Sangiovanni-Vincentelli, A.: A framework for comparing models of computation. IEEE Transactions on CAD 17 (1998) 1217–1229
Lee, E.A.: Modeling concurrent real-time processes using discrete events. Annals of Software Engineering 7 (1999) 25–45
Royden, H.: Real Analysis. 3rd edn. Prentice Hall, Englewood Cliffs, NJ (1988)
Sastry, S.: Nonlinear Systems, Analysis, Stability and Control. Springer-Verlag (1999)
Zhang, J., Johansson, K.H., Lygeros, J., Sastry, S.: Dynamical systems revisited: Hybrid systems with Zeno executions. In Krogh, B., Lynch, N., eds.: Hybrid Systems: Computation and Control, LNCS 1790. Springer-Verlag (2000)
Bryant, V.: Metric Spaces. Cambridge University Press (1985)
Johansson, K., Egerstedt, M., Lygeros, J., Sastry, S.: On the regularization of Zeno hybrid automata. Systems and Control Letters 38 (1999) 141–150
Liu, J., Lee, E.A.: Component-based hierarchical modeling of systems with continuous and discrete dynamics,. In: IEEE Symposium on Computer-Aided Control System Design (CACSD’00), Anchorage, Alaska, USA. (2000) 95–100
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Liu, J., Lee, E.A. (2003). On the Causality of Mixed-Signal and Hybrid Models. In: Maler, O., Pnueli, A. (eds) Hybrid Systems: Computation and Control. HSCC 2003. Lecture Notes in Computer Science, vol 2623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36580-X_25
Download citation
DOI: https://doi.org/10.1007/3-540-36580-X_25
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00913-9
Online ISBN: 978-3-540-36580-8
eBook Packages: Springer Book Archive