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Register a new userNumerical Verification of Affine Systems with up to a Billion Dimensions
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Added by
Stanley Bak
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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.
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This paper focuses on finding reinforcement learning policies for control systems with hard state and action constraints. Despite its success in many domains, reinforcement learning is challenging to apply to problems with hard constraints, especially if both the state variables and actions are constrained. Previous works seeking to ensure constraint satisfaction, or safety, have focused on adding a projection step to a learned policy. Yet, this approach requires solving an optimization problem at every policy execution step, which can lead to significant computational costs. To tackle this problem, this paper proposes a new approach, termed Vertex Networks (VNs), with guarantees on safety during exploration and on learned control policies by incorporating the safety constraints into the policy network architecture. Leveraging the geometric property that all points within a convex set can be represented as the convex combination of its vertices, the proposed algorithm first learns the convex combination weights and then uses these weights along with the pre-calculated vertices to output an action. The output action is guaranteed to be safe by construction. Numerical examples illustrate that the proposed VN algorithm outperforms vanilla reinforcement learning in a variety of benchmark control tasks.
In this paper, we propose an incremental abstraction method for dynamically over-approximating nonlinear systems in a bounded domain by solving a sequence of linear programs, resulting in a sequence of affine upper and lower hyperplanes with expanding operating regions. Although the affine abstraction problem can be solved offline using a single linear program, existing approaches suffer from a computation space complexity that grows exponentially with the state dimension. Hence, the motivation for incremental abstraction is to reduce the space complexity for high-dimensional systems, but at the cost of yielding potentially worse abstractions/overapproximations. Specifically, we start with an operating region that is a subregion of the state space and compute two affine hyperplanes that bracket the nonlinear function locally. Then, by incrementally expanding the operating region, we dynamically update the two affine hyperplanes such that we eventually yield hyperplanes that are guaranteed to over-approximate the nonlinear system over the entire domain. Finally, the effectiveness of the proposed approach is demonstrated using numerical examples of high-dimensional nonlinear systems.
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The security in information-flow has become a major concern for cyber-physical systems (CPSs). In this work, we focus on the analysis of an information-flow security property, called opacity. Opacity characterizes the plausible deniability of a systems secret in the presence of a malicious outside intruder. We propose a methodology of checking a notion of opacity, called approximate initial-state opacity, for networks of discrete-time switched systems. Our framework relies on compositional constructions of finite abstractions for networks of switched systems and their so-called approximate initial-state opacity-preserving simulation functions (InitSOPSFs). Those functions characterize how close concrete networks and their finite abstractions are in terms of the satisfaction of approximate initial-state opacity. We show that such InitSOPSFs can be obtained compositionally by assuming some small-gain type conditions and composing so-called local InitSOPSFs constructed for each subsystem separately. Additionally, assuming certain stability property of switched systems, we also provide a technique on constructing their finite abstractions together with the corresponding local InitSOPSFs. Finally, we illustrate the effectiveness of our results through an example.
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