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Reachability computation for polynomial dynamical systems

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Abstract

This paper is concerned with the problem of computing the bounded time reachable set of a polynomial discrete-time dynamical system. The problem is well-known for being difficult when nonlinear systems are considered. In this regard, we propose three reachability methods that differ in the set representation. The proposed algorithms adopt boxes, parallelotopes, and parallelotope bundles to construct flowpipes that contain the actual reachable sets. The latter is a new data structure for the symbolic representation of polytopes. Our methods exploit the Bernstein expansion of polynomials to bound the images of sets. The scalability and precision of the presented methods are analyzed on a number of dynamical systems, in comparison with other existing approaches.

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Notes

  1. A set of nonempty subsets of X whose union contains the given set X is called a cover of X.

  2. http://tommasodreossi.github.io/sapo/.

  3. http://www.ginac.de.

  4. https://www.gnu.org/software/glpk/.

  5. https://www.bitcraze.io/.

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Author information

Authors and Affiliations

  1. University of California, 253 Cory Hall #1770, Berkeley, CA, 94720-1770, USA

    Tommaso Dreossi

  2. Verimag, Centre Equation, 2, Avenue de Vignate, 38610, Gières, France

    Thao Dang

  3. University of Udine, Via Delle Scienze, 206, 33100, Udine, Italy

    Carla Piazza

Authors
  1. Tommaso Dreossi
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  2. Thao Dang
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  3. Carla Piazza
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Corresponding author

Correspondence to Tommaso Dreossi.

Additional information

This work is partially supported by Toyota under the CHESS center, DARPA BRASS project, ANR MALTHY project (ANR-12-INSE-003), and INdAM GNCS.

Appendix: Experiment details

Appendix: Experiment details

1.1 Van der Pol

$$\begin{aligned} D = \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 1 \\ -1 &{}\quad 1 \\ 1 &{}\quad 1 \\ \end{pmatrix} \quad T = \begin{pmatrix} 0 &{}\quad 1 \\ 2 &{}\quad 3 \\ 0 &{}\quad 2 \\ 1 &{}\quad 3 \\ 0 &{}\quad 3 \\ 1 &{}\quad 2 \\ \end{pmatrix} \end{aligned}$$
(54)

1.2 Rössler attractor

$$\begin{aligned} D = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0.5 &{}\quad 0 \\ 0.5 &{}\quad 0 &{}\quad 0.5 \\ \end{pmatrix} \quad T = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 3\\ 2 &{}\quad 3 &{}\quad 4\\ \end{pmatrix} \end{aligned}$$
(55)

1.3 SIR epidemi model

$$\begin{aligned} D = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 \\ 1 &{}\quad 0.5 &{}\quad 0 \\ 0.5 &{}\quad 0 &{}\quad 0.5 \\ \end{pmatrix} \quad T = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 2 \\ 1 &{}\quad 2 &{}\quad 3\\ 2 &{}\quad 3 &{}\quad 4\\ \end{pmatrix} \end{aligned}$$
(56)

1.4 Generalized Lotka–Volterra model

$$\begin{aligned} D = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ -1 &{}\quad 0 &{}\quad 0 &{}\quad -1 &{}\quad 1\\ \end{pmatrix} \quad T = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 2 &{}\quad 3 &{}\quad 4\\ 1 &{}\quad 2 &{}\quad 3 &{}\quad 5 &{}\quad 6\\ 2 &{}\quad 3 &{}\quad 4 &{}\quad 5 &{}\quad 6\\ \end{pmatrix} \end{aligned}$$
(57)

1.5 Phosphorelay systems

$$\begin{aligned} D = \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 0\\ \end{pmatrix} \quad T = \begin{pmatrix} 0 &{}\quad 1 &{}\quad 2 &{}\quad 3 &{}\quad 4 &{}\quad 5 &{}\quad 6\\ 0 &{}\quad 1 &{}\quad 2 &{}\quad 4 &{}\quad 5 &{}\quad 6 &{}\quad 7\\ 0 &{}\quad 1 &{}\quad 2 &{}\quad 5 &{}\quad 6 &{}\quad 7 &{}\quad 8\\ 0 &{}\quad 1 &{}\quad 2 &{}\quad 5 &{}\quad 6 &{}\quad 7 &{}\quad 9\\ \end{pmatrix} \end{aligned}$$
(58)

1.6 Quadcopter drone

$$\begin{aligned} D_{(i,i)} = i \quad D_{(17,j)} = (0\ 0\ 0.5\ 0\ 0\ 0.5\ 0.5\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0.25) \end{aligned}$$
(59)

for \(i,j=0,1,\dots ,16\) and

$$\begin{aligned} T_{(i,j)} = {\left\{ \begin{array}{ll} 17 &{}\quad \text {if i=1 and j=5}\\ i &{}\quad \text {otherwise}\\ \end{array}\right. } \end{aligned}$$
(60)

for \(i=0,1,\dots ,16\) and \(j=0,1\).

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Dreossi, T., Dang, T. & Piazza, C. Reachability computation for polynomial dynamical systems. Form Methods Syst Des 50, 1–38 (2017). https://doi.org/10.1007/s10703-016-0266-3

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