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Verifying Safety and Persistence in Hybrid Systems Using Flowpipes and Continuous Invariants

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Abstract

We describe a method for verifying the temporal property of persistence in non-linear hybrid systems. Given some system and an initial set of states, the method establishes that system trajectories always eventually evolve into some specified target subset of the states of one of the discrete modes of the system, and always remain within this target region. The method also computes a time-bound within which the target region is always reached. The approach combines flowpipe computation with deductive reasoning about invariants and is more general than each technique alone. We illustrate the method with a case study showing that potentially destructive stick-slip oscillations of an oil-well drill eventually die away for a certain choice of drill control parameters. The case study demonstrates how just using flowpipes or just reasoning about invariants alone can be insufficient and shows the richness of systems that one can handle with the proposed method, since the systems features modes with non-polynomial ODEs. We also propose an alternative method for proving persistence that relies solely on flowpipe computation.

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Notes

  1. Metric Temporal Logic; see e.g. [35].

  2. The system exhibits sliding behaviour on a portion of this surface known as the sliding set. See [49].

  3. The Lyapunov function in fact has rational coefficients, but their representation is too bulky to be given here exactly.

  4. Here \(\nabla \) denotes the gradient of V, i.e. the vector of partial derivatives \((\frac{\partial V}{\partial x_1},\dots ,\frac{\partial V}{\partial x_n})\).

  5. E.g. those featured in the right-hand side of the ODE, i.e. \(f(\mathbf {x})\).

  6. The interested reader may find a superbly readable elementary introduction to the CAD algorithm in [31] and a good overview of current state of the art in [12].

  7. Intel i5-2520M CPU @ 2.50GHz, 4GB RAM, running Arch Linux kernel 4.2.5-1.

  8. e.g numerical solution computation with “qualitative” features, such as invariance of certain regions.

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Acknowledgements

The authors wish to thank the anonymous reviewers for their careful reading and valuable suggestions for improving this work and extend special thanks to Dr. E.M. Navarro-López for pointing out the highly relevant work on deadness [50] before it appeared in print.

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Authors and Affiliations

  1. Carnegie Mellon University, GHC 9116, 5000 Forbes Ave., Pittsburgh, PA, 15213, USA

    Andrew Sogokon

  2. University of Edinburgh, 2.12 Informatics Forum, 10 Crichton Street, Edinburgh, EH8 9AB, UK

    Paul B. Jackson

  3. Vanderbilt University, Room 300, 1025 16th Ave. S., Nashville, TN, 37212, USA

    Taylor T. Johnson

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This material is based upon work supported by the UK Engineering and Physical Sciences Research Council under Grants EPSRC EP/I010335/1 and EP/J001058/1, the National Science Foundation (NSF) under Grant Numbers CNS 1464311 and CCF 1527398, the Air Force Research Laboratory (AFRL) through Contract Number FA8750-15-1-0105, and the Air Force Office of Scientific Research (AFOSR) under Contract Number FA9550-15-1-0258.

Appendix A

Appendix A

See Figs. 10 and 11.

Fig. 10
figure 10

Flowpipes from an initial box S (in green) around a stable equilibrium of an oscillator under different damping. Taylor model order: 10; time bound: 15. Flowpipe convergence is easier to show for larger damping factors. a\(d=0.05\) and b\(d=0.005\). (Color figure online)

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Fig. 11
figure 11

Bounds on flowpipes from an initial box S. a Bounds on the \(x_1\) component; \(d=0.05\), b bounds on the \(x_1\) component; \(d=0.005\), c bounds on the \(x_2\) component; \(d=0.05\) and d bounds on the \(x_2\) component; \(d=0.005\). (Color figure online)

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Sogokon, A., Jackson, P.B. & Johnson, T.T. Verifying Safety and Persistence in Hybrid Systems Using Flowpipes and Continuous Invariants. J Autom Reasoning 63, 1005–1029 (2019). https://doi.org/10.1007/s10817-018-9497-x

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