Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

Bak, Stanley; Bogomolov, Sergiy; Duggirala, Parasara Sridhar; Gerlach, Adam R.; Potomkin, Kostiantyn

doi:10.1016/j.ifacol.2021.08.507

Cited by 16 publications

(31 citation statements)

References 26 publications

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“…We also observed that both these techniques improve the accuracy of the reachable sets for different benchamrks. In future, we intend to investigate Koopman linearization techniques for computing the template directions [4].…”

Section: Discussionmentioning

confidence: 99%

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Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems

Kim

Bak

Duggirala

2021

Preprint

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Reachable set computation is an important technique for the verification of safety properties of dynamical systems. In this paper, we investigate parallelotope bundle based reachable set computation for discrete nonlinear systems. The algorithm relies on computing an upper bound on the supremum of a nonlinear function over a rectangular domain, traditionally done using Bernstein polynomials. We strive to remove the manual step of parallelotope template selection to make the method fully automatic. Further, we show that changing templates dynamically during computations can improve accuracy. 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 Principle 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 NASA's 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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Section: Discussionmentioning

confidence: 99%

“…Consequently, the diagonal parallelotopes will be precisely be defined by unique pairs of these directions, giving us a total 4 2 = 6 diagonal parallelotopes. COVID Supermodel: We further benchmark our dynamic strategies with the recently introduced COVID supermodel [3], [24].…”

Section: Model Dynamicsmentioning

confidence: 99%

Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems

Kim

Bak

Duggirala

2021

Preprint

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“…Instead of using an SMT solver to reason over non-convex initial sets, we propose combining Taylor models with polynomial zonotope refinement. A comparison on the same nonlinear system benchmarks used in the earlier Koopman verification work [5] demonstrates both the improved accuracy and the improved verification speed.…”

Section: Introductionmentioning

confidence: 87%

“…To the best of our knowledge only two approaches exist for far: The first approach [13] utilizes the error bounds for quadratic systems [24] to compute an enclosure of the reachable set for weakly nonlinear systems based on a finite Carleman linearization, where interval arithmetic [17] is applied to enclose the image of the initial set through the observables. The second approach [5], which represents the work closest to our method, presents two different verification strategies: 1) Direct encoding of the nonlinear transformation defined by the observables using a SMT solver, and 2) zonotope domain splitting, where the initial set is recursively split into smaller sets until the specification can be verified or falsified.…”

Section: Related Workmentioning

confidence: 99%

“…Koopman operator techniques are also well-suited for data-driven approaches since the Koopman linearized system can be directly created from measurements, bypassing a potentially complex modeling step. The Koopman framework has been successfully applied to many applications, including control [26,28], state estimation [31] and recently, formal verification [5].…”

Section: Introductionmentioning

confidence: 99%

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Reachability of Koopman Linearized Systems Using Random Fourier Feature Observables and Polynomial Zonotope Refinement

Bak

Bogomolov

Hencey

et al. 2022

Computer Aided Verification

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Koopman operator linearization approximates nonlinear systems of differential equations with higher-dimensional linear systems. For formal verification using reachability analysis, this is an attractive conversion, as highly scalable methods exist to compute reachable sets for linear systems. However, two main challenges are present with this approach, both of which are addressed in this work. First, the approximation must be sufficiently accurate for the result to be meaningful, which is controlled by the choice of observable functions during Koopman operator linearization. By using random Fourier features as observable functions, the process becomes more systematic than earlier work, while providing a higher-accuracy approximation. Second, although the higher-dimensional system is linear, simple convex initial sets in the original space can become complex non-convex initial sets in the linear system. We overcome this using a combination of Taylor model arithmetic and polynomial zonotope refinement. Compared with prior work, the result is more efficient, more systematic and more accurate.

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Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems

Kim

Bak

Duggirala

2021

Lecture Notes in Computer Science

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Reachability of Black-Box Nonlinear Systems after Koopman Operator Linearization

Cited by 16 publications

References 26 publications

Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems

Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems

Reachability of Koopman Linearized Systems Using Random Fourier Feature Observables and Polynomial Zonotope Refinement

Automatic Dynamic Parallelotope Bundles for Reachability Analysis of Nonlinear Systems

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