136 research outputs found
Numerical Verification of Affine Systems with up to a Billion Dimensions
Affine systems reachability is the basis of many verification methods. With
further computation, methods exist to reason about richer models with inputs,
nonlinear differential equations, and hybrid dynamics. As such, the scalability
of affine systems verification is a prerequisite to scalable analysis for more
complex systems. In this paper, we improve the scalability of affine systems
verification, in terms of the number of dimensions (variables) in the system.
The reachable states of affine systems can be written in terms of the matrix
exponential, and safety checking can be performed at specific time steps with
linear programming. Unfortunately, for large systems with many state variables,
this direct approach requires an intractable amount of memory while using an
intractable amount of computation time. We overcome these challenges by
combining several methods that leverage common problem structure. Memory is
reduced by exploiting initial states that are not full-dimensional and safety
properties (outputs) over a few linear projections of the state variables.
Computation time is saved by using numerical simulations to compute only
projections of the matrix exponential relevant for the verification problem.
Since large systems often have sparse dynamics, we use Krylov-subspace
simulation approaches based on the Arnoldi or Lanczos iterations. Our method
produces accurate counter-examples when properties are violated and, in the
extreme case with sufficient problem structure, can analyze a system with one
billion real-valued state variables
Model Reduction by Moment Matching for Linear Switched Systems
Two moment-matching methods for model reduction of linear switched systems
(LSSs) are presented. The methods are similar to the Krylov subspace methods
used for moment matching for linear systems. The more general one of the two
methods, is based on the so called "nice selection" of some vectors in the
reachability or observability space of the LSS. The underlying theory is
closely related to the (partial) realization theory of LSSs. In this paper, the
connection of the methods to the realization theory of LSSs is provided, and
algorithms are developed for the purpose of model reduction. Conditions for
applicability of the methods for model reduction are stated and finally the
results are illustrated on numerical examples.Comment: Sent for publication in IEEE TAC, on October 201
Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets
Reachability analysis is a formal method to guarantee safety of dynamical
systems under the influence of uncertainties. A major bottleneck of all
reachability algorithms is the requirement to adequately tune certain algorithm
parameters such as the time step size, which requires expert knowledge. In this
work, we solve this issue with a fully-automated reachability algorithm that
tunes all algorithm parameters internally such that the reachable set enclosure
satisfies a user-defined accuracy in terms of distance to the exact reachable
set. Knowing the distance to the exact reachable set, an inner-approximation of
the reachable set can be efficiently extracted from the outer-approximation
using the Minkowski difference. Finally, we propose a novel verification
algorithm that automatically refines the accuracy of the outer- and
inner-approximation until specifications given by time-varying safe and unsafe
sets can either be verified or falsified. The numerical evaluation demonstrates
that our verification algorithm successfully verifies or falsifies benchmarks
from different domains without any requirement for manual tuning.Comment: 16 page
Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices
Approximating the set of reachable states of a dynamical system is an
algorithmic yet mathematically rigorous way to reason about its safety.
Although progress has been made in the development of efficient algorithms for
affine dynamical systems, available algorithms still lack scalability to ensure
their wide adoption in the industrial setting. While modern linear algebra
packages are efficient for matrices with tens of thousands of dimensions,
set-based image computations are limited to a few hundred. We propose to
decompose reach set computations such that set operations are performed in low
dimensions, while matrix operations like exponentiation are carried out in the
full dimension. Our method is applicable both in dense- and discrete-time
settings. For a set of standard benchmarks, it shows a speed-up of up to two
orders of magnitude compared to the respective state-of-the art tools, with
only modest losses in accuracy. For the dense-time case, we show an experiment
with more than 10.000 variables, roughly two orders of magnitude higher than
possible with previous approaches
Reachability analysis of linear hybrid systems via block decomposition
Reachability analysis aims at identifying states reachable by a system within
a given time horizon. This task is known to be computationally expensive for
linear hybrid systems. Reachability analysis works by iteratively applying
continuous and discrete post operators to compute states reachable according to
continuous and discrete dynamics, respectively. In this paper, we enhance both
of these operators and make sure that most of the involved computations are
performed in low-dimensional state space. In particular, we improve the
continuous-post operator by performing computations in high-dimensional state
space only for time intervals relevant for the subsequent application of the
discrete-post operator. Furthermore, the new discrete-post operator performs
low-dimensional computations by leveraging the structure of the guard and
assignment of a considered transition. We illustrate the potential of our
approach on a number of challenging benchmarks.Comment: Accepted at EMSOFT 202
Guaranteed optimal reachability control of reaction-diffusion equations using one-sided Lipschitz constants and model reduction
We show that, for any spatially discretized system of reaction-diffusion, the
approximate solution given by the explicit Euler time-discretization scheme
converges to the exact time-continuous solution, provided that diffusion
coefficient be sufficiently large. By "sufficiently large", we mean that the
diffusion coefficient value makes the one-sided Lipschitz constant of the
reaction-diffusion system negative. We apply this result to solve a finite
horizon control problem for a 1D reaction-diffusion example. We also explain
how to perform model reduction in order to improve the efficiency of the
method
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