Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userData-Driven Reachability Analysis with Christoffel Functions
90
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We present an algorithm for data-driven reachability analysis that estimates finite-horizon forward reachable sets for general nonlinear systems using level sets of a certain class of polynomials known as Christoffel functions. The level sets of Christoffel functions are known empirically to provide good approximations to the support of probability distributions: the algorithm uses this property for reachability analysis by solving a probabilistic relaxation of the reachable set computation problem. We also provide a guarantee that the output of the algorithm is an accurate reachable set approximation in a probabilistic sense, provided that a certain sample size is attained. We also investigate three numerical examples to demonstrate the algorithms capabilities, such as providing non-convex reachable set approximations and detecting holes in the reachable set.
rate research
Read More
This paper presents algorithms for performing data-driven reachability analysis under temporal logic side information. In certain scenarios, the data-driven reachable sets of a robot can be prohibitively conservative due to the inherent noise in the robots historical measurement data. In the same scenarios, we often have side information about the robots expected motion (e.g., limits on how much a robot can move in a one-time step) that could be useful for further specifying the reachability analysis. In this work, we show that if we can model this side information using a signal temporal logic (STL) fragment, we can constrain the data-driven reachability analysis and safely limit the conservatism of the computed reachable sets. Moreover, we provide formal guarantees that, even after incorporating side information, the computed reachable sets still properly over-approximate the robots future states. Lastly, we empirically validate the practicality of the over-approximation by computing constrained, data-driven reachable sets for the Small-Vehicles-for-Autonomy (SVEA) hardware platform in two driving scenarios.
Interval approaches for the reachability analysis of initial value problems for sets of classical ordinary differential equations have been investigated and implemented by many researchers during the last decades. However, there exist numerous applications in computational science and engineering, where continuous-time system dynamics cannot be described adequately by integer-order differential equations. Especially in cases in which long-term memory effects are observed, fractional-order system representations are promising to describe the dynamics, on the one hand, with sufficient accuracy and, on the other hand, to limit the number of required state variables and parameters to a reasonable amount. Real-life applications for such fractional-order models can, among others, be found in the field of electrochemistry, where methods for impedance spectroscopy are typically used to identify fractional-order models for the charging/discharging behavior of batteries or for the dynamic relation between voltage and current in fuel cell systems if operated in a non-stationary state. This paper aims at presenting an iterative method for reachability analysis of fractional-order systems that is based on an interval arithmetic extension of Mittag-Leffler functions. An illustrating example, inspired by a low-order model of battery systems concludes this contribution.
Reachable set computation is an important technique for the verification of safety properties of dynamical systems. In this paper, we investigate reachable set computation for discrete nonlinear systems based on parallelotope bundles. The algorithm relies on computing an upper bound on the supremum of a nonlinear function over a rectangular domain, which has been traditionally done using Bernstein polynomials. We strive to remove the manual step of parallelotope template selection to make the method fully automatic. Furthermore, we show that changing templates dynamically during computations cans improve accuracy. To this end, we investigate two techniques for generating the template directions. The first technique approximates the dynamics as a linear transformation and generates templates using this linear transformation. The second technique uses Principal Component Analysis (PCA) of sample trajectories for generating templates. We have implemented our approach in a Python-based tool called Kaa and improve its performance by two main enhancements. The tool is modular and use two types of global optimization solvers, the first using Bernstein polynomials and the second using NASAs Kodiak nonlinear optimization library. Second, we leverage the natural parallelism of the reachability algorithm and parallelize the Kaa implementation. We demonstrate the improved accuracy of our approach on several standard nonlinear benchmark systems.
In this paper we study a discrete-time SIS (susceptible-infected-susceptible) model, where the infection and healing parameters and the underlying network may change over time. We provide conditions for the model to be well-defined and study its stability. For systems with homogeneous infection rates over symmetric graphs,we provide a sufficient condition for global exponential stability (GES) of the healthy state, that is, where the virus is eradicated. For systems with heterogeneous virus spread over directed graphs, provided that the variation is not too fast, a sufficient condition for GES of the healthy state is established.
Ellipsoids are a common representation for reachability analysis because they are closed under affine maps and allow conservative approximation of Minkowski sums; this enables one to incorporate uncertainty and linearization error in a dynamical system by exapnding the size of the reachable set. Zonotopes, a type of symmetric, convex polytope, are similarly frequently used due to efficient numerical implementation of affine maps and exact Minkowski sums. Both of these representations also enable efficient, convex collision detection for fault detection or formal verification tasks, wherein one checks if the reachable set of a system collides (i.e., intersects) with an unsafe set. However, both representations often result in conservative representations for reachable sets of arbitrary systems, and neither is closed under intersection. Recently, constrained zonotopes and constrained polynomial zonotopes have been shown to overcome some of these conservatism challenges, and are closed under intersection. However, constrained zonotopes can not represent shapes with smooth boundaries such as ellipsoids, and constrained polynomial zonotopes can require solving a non-convex program for collision checking (i.e., fault detection). This paper introduces ellipsotopes, a set representation that is closed under affine maps, Minkowski sums, and intersections. Ellipsotopes combine the advantages of ellipsoids and zonotopes, and enable convex collision checking at the expense of more conservative reachable sets than constrained polynomial zonotopes. The utility of this representation is demonstrated on several examples.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
