Neural Lyapunov Control

Chang, Ya-Chien; Roohi, Nima; Gao, Sicun

doi:10.48550/arxiv.2005.00611

Cited by 36 publications

(54 citation statements)

References 11 publications

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“…In light of the latter objective, it is desirable to increase the number of stable ReLUs in f N N (x), i.e., ReLUs that always output zero or one. For these ReLUs, we do not need to introduce binary variables in the MIQP (13). As a result, any training method that can increase the number of stable ReLUs can significantly improve the complexity of the verification.…”

Section: Neural Network Trainingmentioning

confidence: 99%

“…Since the approximation error bound (4) is written in terms of the ∞ norm, the training loss function is set to the mean absolute error. Furthermore, in order to alleviate the practical computational cost of the MIQP (13) in the analysis phase, we add 1 regularization with weight 10 −4 during training.…”

Section: -D Rational Systemmentioning

confidence: 99%

“…For comparison, the convex relaxation based methods [10] easily become excessively conservative as the size of the neural network or the model uncertainty increases. The counterexample-guided Lyapunov function synthesis methods in [11][12][13] do not have finite-step termination guarantees which means that the number of calls to the computationally expensive verifier cannot be bounded. Our method is inspired by [9] which proposes a sampling-based method for stability analysis of autonomous hybrid systems.…”

Section: Introductionmentioning

confidence: 99%

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Learning Region of Attraction for Nonlinear Systems

Chen¹,

Fazlyab²,

Morari³

et al. 2021

Preprint

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Estimating the region of attraction (ROA) of general nonlinear autonomous systems remains a challenging problem and requires a case-by-case analysis. Leveraging the universal approximation property of neural networks, in this paper, we propose a counterexample-guided method to estimate the ROA of general nonlinear dynamical systems provided that they can be approximated by piecewise linear neural networks and that the approximation error can be bounded. Specifically, our method searches for robust Lyapunov functions using counterexamples, i.e., the states at which the Lyapunov conditions fail. We generate the counterexamples using Mixed-Integer Quadratic Programming. Our method is guaranteed to find a robust Lyapunov function in the parameterized function class, if exists, after collecting a finite number of counterexamples. We illustrate our method through numerical examples.

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Section: Neural Network Trainingmentioning

confidence: 99%

Section: -D Rational Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Learning Region of Attraction for Nonlinear Systems

Chen¹,

Fazlyab²,

Morari³

et al. 2021

Preprint

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show abstract

“…Meanwhile, in the control theory, there exist studies about provable energy-function-based safety guarantee of dynamic systems called the safety certificate, or safety index (Wieland and Allgöwer, 2007;Ames et al, 2014;Chang et al, 2020). These methods first synthesize an energy function such that the safe states are with low energy, and then design control laws satisfying the safe action constraint to make the system dissipate energy (Wei and Liu, 2019).…”

Section: Introductionmentioning

confidence: 99%

“…If there exists feasible control for all states in a safe set to satisfy the safe action constraint dissipating the energy, then the system will never leave the safe set (i.e., forward invariance). Despite its soundness, the safety index synthesis (SIS) by hand is extremely hard for complicated or unknown systems, which stimulates a rapidly growing interest in learning-based SIS (Chang et al, 2020;Saveriano and Lee, 2019;Srinivasan et al, 2020;Qin et al, 2021). These studies assume to know the dynamical models (either white-box or black-box).…”

Section: Introductionmentioning

confidence: 99%

Joint Synthesis of Safety Certificate and Safe Control Policy using Constrained Reinforcement Learning

Ma¹,

Liu²,

Li³

et al. 2021

Preprint

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Safety is the major consideration in controlling complex dynamical systems using reinforcement learning (RL), where the safety certificate can provide provable safety guarantee. A valid safety certificate is an energy function indicating that safe states are with low energy, and there exists a corresponding safe control policy that allows the energy function to always dissipate. The safety certificate and the safe control policy are closely related to each other and both challenging to synthesize. Therefore, existing learning-based studies treat either of them as prior knowledge to learn the other, which limits their applicability with general unknown dynamics. This paper proposes a novel approach that simultaneously synthesizes the energy-function-based safety certificate and learns the safe control policy with CRL. We do not rely on the prior knowledge about either an available model-based controller or a perfect safety certificate. In particular, we formulate a loss function to optimize the safety certificate parameters by minimizing the occurrence of energy increases. By adding this optimization procedure as an outer loop to the Lagrangian-based constrained reinforcement learning (CRL), we jointly update the policy and safety certificate parameters, and prove that they will converge to their respective local optima, the optimal safe policy and a valid safety certificate. We evaluate our algorithms on multiple safety-critical benchmark environments. The results show that the proposed algorithm learns provably safe policies with no constraint violation. The validity, or feasibility of synthesized safety certificate is also verified numerically.

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Machine‐learning‐based construction of barrier functions and models for safe model predictive control

Chen

Christofides

2021

AIChE Journal

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In this paper, we propose a control Lyapunov‐barrier function‐based model predictive control method utilizing a feed‐forward neural network specified control barrier function (CBF) and a recurrent neural network (RNN) predictive model to stabilize nonlinear processes with input constraints, and to guarantee that safety requirements are met for all times. The nonlinear system is first modeled using RNN techniques, and a CBF is characterized by constructing a feed‐forward neural network (FNN) model with unique structures and properties. The FNN model for the CBF is trained based on data samples collected from safe and unsafe operating regions, and the resulting FNN model is verified to demonstrate that the safety properties of the CBF are satisfied. Given sufficiently small bounded modeling errors for both the FNN and the RNN models, the proposed control system is able to guarantee closed‐loop stability while preventing the closed‐loop states from entering unsafe regions in state‐space under sample‐and‐hold control action implementation. We provide the theoretical analysis for bounded unsafe sets in state‐space, and demonstrate the effectiveness of the proposed control strategy using a nonlinear chemical process example with a bounded unsafe region.

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Neural Lyapunov Control

Cited by 36 publications

References 11 publications

Learning Region of Attraction for Nonlinear Systems

Learning Region of Attraction for Nonlinear Systems

Joint Synthesis of Safety Certificate and Safe Control Policy using Constrained Reinforcement Learning

Machine‐learning‐based construction of barrier functions and models for safe model predictive control

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