Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes

Jagtap, Pushpak; Pappas, George J.

doi:10.1109/cdc42340.2020.9303847

Cited by 59 publications

(33 citation statements)

References 28 publications

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“…Proposition 1 assumes that the noise on the measurements is R-sub-Gaussian, which is more general than Gaussian noise. It depends on the constants f κ and γ d κ which have been described as challenging to deal with [34], and are often picked according to heuristics without guarantees [8,28]. In this work, we derive formal upper bounds for each of these terms in Section 4.2.2.…”

Section: Reproducing Kernel Hilbert Spacesmentioning

confidence: 99%

“…Each of the previous studies considers the safety problem with a simple objective, whereas our focus is on verification against complex (temporal logic) specifications. Control barrier functions have been applied to a system with polynomial dynamics learned via GP regression for control generation subject to LTL specifications [28], although the structural assumption is restrictive and there is no guarantee that a control barrier function can be found if one exists. In addition, many methods that use GPs employ parameter approximations, e.g., Reproducing Kernel Hilbert Space (RKHS) constants, that subject final guarantees to the approximation correctness.…”

Section: Related Workmentioning

confidence: 99%

“…For unknown linear dynamics, specifically, techniques such as Bayesian inference along with trajectory sampling and chance-constrained have been explored for behavior and stability analysis [20,29]. For more general systems, estimation methods such as piecewisepolynomial functions and GPs coupled with barrier functions have been developed [2,28]. While these approaches provide means for analysis, they often lack the required formalism of the learning uncertainty or do not allow the use of full power of model-based frameworks.…”

Section: Introductionmentioning

confidence: 99%

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Safety Verification of Unknown Dynamical Systems via Gaussian Process Regression

Jackson

Laurenti

Frew

et al. 2020

2020 59th IEEE Conference on Decision and Control (CDC)

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Leveraging autonomous systems in safety-critical scenarios requires verifying their behaviors in the presence of uncertainties and black-box components that influence the system dynamics. In this article, we develop a framework for verifying partiallyobservable, discrete-time dynamical systems with unmodelled dynamics against temporal logic specifications from a given inputoutput dataset. The verification framework employs Gaussian process (GP) regression to learn the unknown dynamics from the dataset and abstract the continuous-space system as a finite-state, uncertain Markov decision process (MDP). This abstraction relies on space discretization and transition probability intervals that capture the uncertainty due to the error in GP regression by using reproducible kernel Hilbert space analysis as well as the uncertainty induced by discretization. The framework utilizes existing model checking tools for verification of the uncertain MDP abstraction against a given temporal logic specification. We establish the correctness of extending the verification results on the abstraction to the underlying partially-observable system. We show that the computational complexity of the framework is polynomial in the size of the dataset and discrete abstraction. The complexity analysis illustrates a trade-off between the quality of the verification results and the computational burden to handle larger datasets and finer abstractions. Finally, we demonstrate the efficacy of our learning and verification framework on several case studies with linear, nonlinear, and switched dynamical systems.

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Section: Reproducing Kernel Hilbert Spacesmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Safety Verification of Unknown Dynamical Systems via Gaussian Process Regression

Jackson

Laurenti

Frew

et al. 2020

2020 59th IEEE Conference on Decision and Control (CDC)

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“…A sub-linear algorithm is developed in [32] for the barrierbased data-driven model validation of dynamical systems which computes the barrier function using a large dataset of trajectories. In [33], a two-step procedure is proposed to synthesize a controller for an unknown nonlinear system, where the first step is to learn a Gaussian process as a replacement of the unknown dynamics, and the second step is to construct the control barrier function for the learned dynamics.…”

Section: Introductionmentioning

confidence: 99%

Data-driven verification and synthesis of stochastic systems through barrier certificates

Salamati¹,

Lavaei²,

Soudjani³

2021

Preprint

Self Cite

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In this work, we study verification and synthesis problems for safety specifications over unknown discrete-time stochastic systems. When a model of the system is available, barrier certificates have been successfully applied for ensuring the satisfaction of safety specifications. In this work, we formulate the computation of barrier certificates as a robust convex program (RCP). Solving the acquired RCP is hard in general because the model of the system that appears in one of the constraints of the RCP is unknown. We propose a data-driven approach that replaces the uncountable number of constraints in the RCP with a finite number of constraints by taking finitely many random samples from the trajectories of the system. We thus replace the original RCP with a scenario convex program (SCP) and show how to relate their optimizers. We guarantee that the solution of the SCP is a solution of the RCP with a priori guaranteed confidence when the number of samples is larger than a pre-computed value. This provides a lower bound on the safety probability of the original unknown system together with a controller in the case of synthesis. We also discuss an extension of our verification approach to a case where the associated robust program is non-convex and show how a similar methodology can be applied. Finally, the applicability of our proposed approach is illustrated through three case studies.

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“…Although there are some model identification techniques available in the relevant literature to learn the model from data, e.g., [1,2,3,4,5] to name a few, acquiring an accurate model for complex systems is always complicated, time-consuming and expensive. As the second difficulty, providing safety certification and guaranteeing correctness of the design of such autonomous vehicles in a formal as well as time-and cost-effective way have been always the major obstacles in their successful deployment.…”

Section: Introductionmentioning

confidence: 99%

Formal Estimation of Collision Risks for Autonomous Vehicles: A Compositional Data-Driven Approach

Lavaei¹,

Lillo²,

Censi³

2021

Preprint

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In this work, we propose a compositional data-driven approach for the formal estimation of collision risks for autonomous vehicles (AVs) with black-box dynamics acting in a stochastic multi-agent framework. The proposed approach is based on the construction of sub-barrier certificates for each stochastic agent via a set of data collected from its trajectories while providing a-priori guaranteed confidence on the datadriven estimation. In our proposed setting, we first cast the original collision risk problem for each agent as a robust optimization program (ROP). Solving the acquired ROP is not tractable due to an unknown model that appears in one of its constraints. To tackle this difficulty, we collect finite numbers of data from trajectories of each agent and provide a scenario optimization program (SOP) corresponding to the original ROP. We then establish a probabilistic bridge between the optimal value of SOP and that of ROP, and accordingly, we formally construct the sub-barrier certificate for each unknown agent based on the number of data and a required level of confidence. We then propose a compositional technique based on small-gain reasoning to quantify the collision risk for multi-agent AVs with some desirable confidence based on sub-barrier certificates of individual agents constructed from data. For the case that the proposed compositionality conditions are not satisfied, we provide a relaxed version of compositional results without requiring any compositionality conditions but at the cost of providing a potentially conservative collision risk. Eventually, we develop our approaches for non-stochastic multi-agent AVs. We demonstrate the effectiveness of our proposed results by applying them to a vehicle platooning consisting of 100 vehicles with 1 leader and 99 followers. We formally estimate the collision risk for the whole network by collecting sampled data from trajectories of each agent.

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Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes

Cited by 59 publications

References 28 publications

Safety Verification of Unknown Dynamical Systems via Gaussian Process Regression

Safety Verification of Unknown Dynamical Systems via Gaussian Process Regression

Data-driven verification and synthesis of stochastic systems through barrier certificates

Formal Estimation of Collision Risks for Autonomous Vehicles: A Compositional Data-Driven Approach

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