Formal Synthesis of Lyapunov Neural Networks

Abate, Alessandro; Ahmed, Daniele; Giacobbe, Mirco; Peruffo, Andrea

doi:10.1109/lcsys.2020.3005328

Cited by 77 publications

(68 citation statements)

References 18 publications

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“…Consequently, we have passed them to the Z3 SMT solver, which should easily handle polynomial formulae: only one of them ('Hybrid Model') has been successfully verified; instead, the candidate for the 'Polynomial Model' has been invalidated (namely Z3 has found a counterexample for it), whereas the verification of the remaining BC candidates has run out of time. For the latter instances, we have experienced that SOSTOOLS generally returns numerically ill-conditioned expressions, namely candidates with coefficients of rather different magnitude, with many decimal digits: even after rounding, expressions with this structure are known to be hardly handled by SMT solvers [2,5], which results in long time needed to return an answer -this explains the experienced timeouts. These experiments suggest that the use of SOS programs within a CEGIS loop appears hardly attainable.…”

Section: Case Studies and Experimental Resultsmentioning

confidence: 99%

“…Ongoing work is porting presented and related [5,2] theoretical results into a software tool [1]. Towards increased automation, future work includes the development of an automated selection of activation functions that are tailored to the dynamical models of interest.…”

Section: Discussionmentioning

confidence: 99%

“…The network is then translated into a logical formula in an appropriate theory, by evaluating it with symbolic inputs, instead of canonical floating point numbers. The details of this conversion are outlined in [2]. This encoded candidate is passed to the second component, a Verifier, which acts as an oracle: either it proves that the solution is valid, or it finds one (or more) instance (called a counter-example) where the candidate BC does not comply with required conditions.…”

Section: Synthesis Of Neural Barrier Certificates Via Learning and Vementioning

confidence: 99%

“…Formal methods for Lyapunov synthesis are introduced in [5], together with a counterexample-based approach using polynomial candidates. The work is later extended in [2], which employs NN as candidates over polynomial dynamical models. The generation of control Lyapunov functions using counterexample-based NN is similarly considered in [9], however this is done by means of differing architectural details and does not extend to BC synthesis.…”

Section: Introductionmentioning

confidence: 99%

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Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Peruffo

Ahmed

Abate

2021

Lecture Notes in Computer Science

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We introduce an automated, formal, counterexample-based approach to synthesise Barrier Certificates (BC) for the safety verification of continuous and hybrid dynamical models. The approach is underpinned by an inductive framework: this is structured as a sequential loop between a learner, which manipulates a candidate BC structured as a neural network, and a sound verifier, which either certifies the candidate’s validity or generates counter-examples to further guide the learner. We compare the approach against state-of-the-art techniques, over polynomial and non-polynomial dynamical models: the outcomes show that we can synthesise sound BCs up to two orders of magnitude faster, with in particular a stark speedup on the verification engine (up to three orders less), whilst needing a far smaller data set (up to three orders less) for the learning part. Beyond improvements over the state of the art, we further challenge the new approach on a hybrid dynamical model and on larger-dimensional models, and showcase the numerical robustness of our algorithms and codebase.

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Section: Case Studies and Experimental Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Synthesis Of Neural Barrier Certificates Via Learning and Vementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Peruffo

Ahmed

Abate

2021

Lecture Notes in Computer Science

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“…There are several techniques to verify certain properties of neural network outputs for all inputs within a range. These techniques can be categorized by whether the verification is exact, e.g., using Satisfiability Modulo Theories (SMT) solvers [29,11,1] or mixed-integer programs (MIP) solvers [10,52,14,12], versus inexact verification by solving a relaxed convex problem [7,17,57,45]. Another important distinction among these techniques is the activation functions used in the neural networks.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov-stable neural-network control

Dai

Benoit

Yang

et al. 2021

Robotics: Science and Systems XVII

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Deep learning has had a far reaching impact in robotics. Specifically, deep reinforcement learning algorithms have been highly effective in synthesizing neural-network controllers for a wide range of tasks. However, despite this empirical success, these controllers still lack theoretical guarantees on their performance, such as Lyapunov stability (i.e., all trajectories of the closed-loop system are guaranteed to converge to a goal state under the control policy). This is in stark contrast to traditional model-based controller design, where principled approaches (like LQR) can synthesize stable controllers with provable guarantees. To address this gap, we propose a generic method to synthesize a Lyapunov-stable neural-network controller, together with a neural-network Lyapunov function to simultaneously certify its stability. Our approach formulates the Lyapunov condition verification as a mixed-integer linear program (MIP). Our MIP verifier either certifies the Lyapunov condition, or generates counter examples that can help improve the candidate controller and the Lyapunov function. We also present an optimization program to compute an inner approximation of the region of attraction for the closed-loop system. We apply our approach to robots including an inverted pendulum, a 2D and a 3D quadrotor, and showcase that our neural-network controller outperforms a baseline LQR controller. The code is open sourced at https://github.com/StanfordASL/neural-network-lyapunov.

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Circuit theory approach to stability and passivity analysis of nonlinear dynamical systems

Ateş

2021

Circuit Theory & Apps

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In this paper, we address the problem of global asymptotic stability and strong passivity analysis of nonlinear time-varying systems controlled by a secondorder vector differential equation. First, we obtain this equation from a nonlinear time varying network of the circuit theory. Then, we construct the Lyapunov candidate function directly from the physical meaning of the given system. By the way, we review a number of previous results from the point view of Lyapunov's direct method. Our system with its real energy function generalize and improve upon some well-known studies. The new concept facilitates the formulation of the energy (Lyapunov) function from the powerenergy relationship of the given system. Then, we also realized that the time derivative of the Lyapunov function for a given dynamical systems is the negative value of the power dissipated in the system. Therefore, with the proposed approach, one can inspect the result of the time derivative of the energy function for a given physical system. Finally, two examples (one with simulations) are used to illustrate the superiority and validity of the obtained results.

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Formal Synthesis of Lyapunov Neural Networks

Cited by 77 publications

References 18 publications

Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Lyapunov-stable neural-network control

Circuit theory approach to stability and passivity analysis of nonlinear dynamical systems

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