2,852 research outputs found
Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programming
A barrier certificate often serves as an inductive invariant that isolates an
unsafe region from the reachable set of states, and hence is widely used in
proving safety of hybrid systems possibly over an infinite time horizon. We
present a novel condition on barrier certificates, termed the invariant
barrier-certificate condition, that witnesses unbounded-time safety of
differential dynamical systems. The proposed condition is the weakest possible
one to attain inductive invariance. We show that discharging the invariant
barrier-certificate condition -- thereby synthesizing invariant barrier
certificates -- can be encoded as solving an optimization problem subject to
bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm
based on difference-of-convex programming, which approaches a local optimum of
the BMI problem via solving a series of convex optimization problems. This
algorithm is incorporated in a branch-and-bound framework that searches for the
global optimum in a divide-and-conquer fashion. We present a weak completeness
result of our method, namely, a barrier certificate is guaranteed to be found
(under some mild assumptions) whenever there exists an inductive invariant (in
the form of a given template) that suffices to certify safety of the system.
Experimental results on benchmarks demonstrate the effectiveness and efficiency
of our approach.Comment: To be published in Inf. Comput. arXiv admin note: substantial text
overlap with arXiv:2105.1431
Abstraction of Elementary Hybrid Systems by Variable Transformation
Elementary hybrid systems (EHSs) are those hybrid systems (HSs) containing
elementary functions such as exp, ln, sin, cos, etc. EHSs are very common in
practice, especially in safety-critical domains. Due to the non-polynomial
expressions which lead to undecidable arithmetic, verification of EHSs is very
hard. Existing approaches based on partition of state space or
over-approximation of reachable sets suffer from state explosion or inflation
of numerical errors. In this paper, we propose a symbolic abstraction approach
that reduces EHSs to polynomial hybrid systems (PHSs), by replacing all
non-polynomial terms with newly introduced variables. Thus the verification of
EHSs is reduced to the one of PHSs, enabling us to apply all the
well-established verification techniques and tools for PHSs to EHSs. In this
way, it is possible to avoid the limitations of many existing methods. We
illustrate the abstraction approach and its application in safety verification
of EHSs by several real world examples
The Structure of Differential Invariants and Differential Cut Elimination
The biggest challenge in hybrid systems verification is the handling of
differential equations. Because computable closed-form solutions only exist for
very simple differential equations, proof certificates have been proposed for
more scalable verification. Search procedures for these proof certificates are
still rather ad-hoc, though, because the problem structure is only understood
poorly. We investigate differential invariants, which define an induction
principle for differential equations and which can be checked for invariance
along a differential equation just by using their differential structure,
without having to solve them. We study the structural properties of
differential invariants. To analyze trade-offs for proof search complexity, we
identify more than a dozen relations between several classes of differential
invariants and compare their deductive power. As our main results, we analyze
the deductive power of differential cuts and the deductive power of
differential invariants with auxiliary differential variables. We refute the
differential cut elimination hypothesis and show that, unlike standard cuts,
differential cuts are fundamental proof principles that strictly increase the
deductive power. We also prove that the deductive power increases further when
adding auxiliary differential variables to the dynamics
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