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Register a new userEllipsotopes: Combining Ellipsoids and Zonotopes for Reachability Analysis and Fault Detection
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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.
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This article presents a new set representation named the hybrid zonotope. The hybrid zonotope is shown to be equivalent to $2^N$ constrained zonotopes through the addition of $N$ binary zonotope factors and is well-suited for the analysis of hybrid systems with both continuous and discrete states and inputs. The major contribution of this manuscript is a closed-form solution for exact forward reachable sets of linear mixed logical dynamical systems. This is given by a simple identity and does not require solving any optimization programs or taking set approximations. The proposed approach captures the worst-case exponential growth in the number of convex sets required to represent the nonconvex reachable set of a hybrid system while exhibiting only linear growth in the complexity of the hybrid zonotope set representation. To reduce both set representation complexity and the computational burden of reachability analysis, a binary tree is used to store only the combinations of binary factors of the hybrid zonotope that map to nonempty convex sets. The proposed approach is applied to an established benchmark example where the exact reachable set of a discrete-time hybrid system with six continuous and two discrete states is given by a single hybrid zonotope equivalent to the union of 657 constrained zonotopes, and is represented using only 283 continuous factors, 29 binary factors, and 177 linear equality constraints. Furthermore, the hybrid zonotope is closed under linear mappings, Minkowski sums, generalized intersections, and halfspace intersections.
This paper presents methods for using zonotopes and constrained zonotopes to improve the practicality of a wide variety of set-based operations commonly used in control theory. The proposed methods extend the use of constrained zonotopes to represent sets resulting from operations including halfspace intersections, convex hulls, robust positively invariant sets, and Pontryagin differences. Order reduction techniques are also presented that provide lower-complexity inner-approximations of zonotopes and constrained zonotopes. Numerical examples are used to demonstrate the efficacy and computational advantages of using zonotope-based set representations for dynamic system analysis and control.
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.
This paper considers the problem of fault detection and localization in active distribution networks using PMUs. The proposed algorithm consists in computing a set of weighted least squares state estimates whose results are used to detect, characterize and localize the occurrence of a fault. Moreover, a criteria to minimize the number of PMUs required to correctly perform the proposed algorithm is defined. Such a criteria, based on system observability conditions, allows the design of an optimization problem to set the positions of PMUs along the grid, in order to get the desired fault localization resolution. The performances of the strategy are tested via simulations on a benchmark distribution system.
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.
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