Quadratic Zonotopes:An extension of Zonotopes to Quadratic Arithmetics

Affine forms are a common way to represent convex sets of $\mathbb{R}$ using a base of error terms $\epsilon \in [-1, 1]^m$. Quadratic forms are an extension of affine forms enabling the use of quadratic error terms $\epsilon_i \epsilon_j$. In static analysis, the zonotope domain, a relational abstract domain based on affine forms has been used in a wide set of settings, e.g. set-based simulation for hybrid systems, or floating point analysis, providing relational abstraction of functions with a cost linear in the number of errors terms. In this paper, we propose a quadratic version of zonotopes. We also present a new algorithm based on semi-definite programming to project a quadratic zonotope, and therefore quadratic forms, to intervals. All presented material has been implemented and applied on representative examples.Comment: 17 pages, 5 figures, 1 tabl

Set-based state estimation and fault diagnosis of linear discrete-time descriptor systems using constrained zonotopes

This paper presents new methods for set-valued state estimation and active fault diagnosis of linear descriptor systems. The algorithms are based on constrained zonotopes, a generalization of zonotopes capable of describing strongly asymmetric convex sets, while retaining the computational advantages of zonotopes. Additionally, unlike other set representations like intervals, zonotopes, ellipsoids, paralletopes, among others, linear static constraints on the state variables, typical of descriptor systems, can be directly incorporated in the mathematical description of constrained zonotopes. Therefore, the proposed methods lead to more accurate results in state estimation in comparison to existing methods based on the previous sets without requiring rank assumptions on the structure of the descriptor system and with a fair trade-off between accuracy and efficiency. These advantages are highlighted in two numerical examples.Comment: This paper was accepted and presented in the 1st IFAC Virtual World Congress, 202

Set-based state estimation and fault diagnosis using constrained zonotopes and applications

This doctoral thesis develops new methods for set-based state estimation and active fault diagnosis (AFD) of (i) nonlinear discrete-time systems, (ii) discrete-time nonlinear systems whose trajectories satisfy nonlinear equality constraints (called invariants), (iii) linear descriptor systems, and (iv) joint state and parameter estimation of nonlinear descriptor systems. Set-based estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. To address this issue, the present doctoral thesis proposes new methods for efficiently propagating constrained zonotopes (CZs) through nonlinear mappings. Besides, this thesis improves the standard prediction-update framework for systems with invariants using new algorithms for refining CZs based on nonlinear constraints. In addition, this thesis introduces a new approach for set-based AFD of a class of nonlinear discrete-time systems. An affine parametrization of the reachable sets is obtained for the design of an optimal input for set-based AFD. In addition, this thesis presents new methods based on CZs for set-valued state estimation and AFD of linear descriptor systems. Linear static constraints on the state variables can be directly incorporated into CZs. Moreover, this thesis proposes a new representation for unbounded sets based on zonotopes, which allows to develop methods for state estimation and AFD also of unstable linear descriptor systems, without the knowledge of an enclosure of all the trajectories of the system. This thesis also develops a new method for set-based joint state and parameter estimation of nonlinear descriptor systems using CZs in a unified framework. Lastly, this manuscript applies the proposed set-based state estimation and AFD methods using CZs to unmanned aerial vehicles, water distribution networks, and a lithium-ion cell.Comment: My PhD Thesis from Federal University of Minas Gerais, Brazil. Most of the research work has already been published in DOIs 10.1109/CDC.2018.8618678, 10.23919/ECC.2018.8550353, 10.1016/j.automatica.2019.108614, 10.1016/j.ifacol.2020.12.2484, 10.1016/j.ifacol.2021.08.308, 10.1016/j.automatica.2021.109638, 10.1109/TCST.2021.3130534, 10.1016/j.automatica.2022.11042

Toward a Standard Benchmark Format and Suite for Floating-Point Analysis

We introduce FPBench, a standard benchmark format for validation and optimization of numerical accuracy in floating-point computations. FPBench is a first step toward addressing an increasing need in our community for comparisons and combinations of tools from different application domains. To this end, FPBench provides a basic floating-point benchmark format and accuracy measures for comparing different tools. The FPBench format and measures allow comparing and composing different floating-point tools. We describe the FPBench format and measures and show that FPBench expresses benchmarks from recent papers in the literature, by building an initial benchmark suite drawn from these papers. We intend for FPBench to grow into a standard benchmark suite for the members of the floating-point tools research community

Static Analysis for Efficient Affine Arithmetic on GPUs

Range arithmetic is a way of calculating with variables that hold ranges of real values. This ability to manage uncertainty during computation has many applications. Examples in graphics include rendering and surface modeling, and there are more general applications like global optimization and solving systems of nonlinear equations. This thesis focuses on affine arithmetic, one kind of range arithmetic. The main drawbacks of affine arithmetic are that it taxes processors with heavy use of floating point arithmetic and uses expensive sparse vectors to represent noise symbols. Stream processors like graphics processing units (GPUs) excel at intense computation, since they were originally designed for high throughput media applications. Heavy control flow and irregular data structures pose problems though, so the conventional implementation of affine arithmetic with dynamically managed sparse vectors runs slowly at best. The goal of this thesis is to map affine arithmetic efficiently onto GPUs by turning sparse vectors into shorter dense vectors at compile time using static analysis. In addition, we look at how to improve efficiency further during the static analysis using unique symbol condensation. We demonstrate our implementation and performance of the condensation on several graphics applications

Data-driven Reachability of Non-linear Systems via Optimization of Chen-Fliess Series

A reachable set is the set of all possible states produced by applying a set of inputs, initial states, and parameters. The fundamental problem of reachability is checking if a set of states is reached provided a set of inputs, initial states, and parameters, typically, in a finite time. In the engineering field, reachability analysis is used to test the guarantees of the operation’s safety of a system. In the present work, the reachability analysis of nonlinear control affine systems is studied by means of the Chen-Fliess series. Different perspectives for addressing the reachability problem, such as interval arithmetic, mixed-monotonicity, and optimization, are used in this dissertation. The first two provide, in general, an overestimation of the reachable set that is not guaranteed to be the smallest. To improve these methods and obtainthe minimum bounding box of the reachable set, the derivative-based optimization of Chen-Fliess series is developed. To achieve this, the closed form of the Gâteaux and Fréchet derivatives of Chen-Fliess series and several other tools from analysis are obtained. To provide a representation of these tools practically and systematically, an abstract algebraic derivative acting on words of a monoid is defined. Three nonconvex optimization algorithms are implemented for Chen-Fliess series. The problem of computing an inner approximation of the reachable set via Chen-Fliess series is also solved by means of convex analysis tools. Furthermore, a method for the computation of the backward reachable set of an output set is also provided. In this case, different from forward reachability analysis, the feasibility problem represents a challenge and requires using the Positivestellensatz. Examples and simulations are provided for every method presented. The application of control barrier functions via Chen-Fliess series is outlined. Finally, the future work and conclusions are stated in the last chapter