532 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
Reachability computation for polynomial dynamical systems
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
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
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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