ReachabilityAnalysis.jl
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Backward reachability for LTI systems
Instead of computing ΦᵏX(0)
and checking whether the intersection with B
is empty, one can backpropagate B
using (Φᵏ)⁻¹B
. In other words, ΦᵏX(0) ∩ B = ∅ ⟺ X(0) ∩ (Φᵏ)⁻¹B = ∅
.
(Note that Φ
is invertible. This was the description for the homogeneous case, but it can be generalized.)
Possible benefits:
-
B
often has a concise H-representation whileX(0)
depends on the precision. Hence the concrete backpropagation can be more efficient (symbolic propagation does not care). - The abstraction only takes place in
X(0)
, so there is a simple abstraction-refinement algorithm: approximateX(0)
by a box; if the intersection is nonempty, add more template directions (or use a different discretization algorithm). (Again, symbolic representation does not care.) - (Only if there are no inputs:) There is no error because
B
as well as(Φᵏ)⁻¹B
are exact.
Downsides:
-
B
is typically unbounded. - The possible property of
B
being decomposed/sparse cannot be exploited.
Related:
- https://www.cs.ubc.ca/~mitchell/Papers/myHSCC07.pdf
- https://systemanalysisdpt-cmc-msu.github.io/ellipsoids/doc/chap_reach.html