Sum-of-Squares methods for controlled invariant sets with applications to model-predictive control

Legat, Benoît; Tabuada, Paulo; Jungers, Raphaël M.

doi:10.1016/j.nahs.2020.100858

Cited by 12 publications

(14 citation statements)

References 21 publications

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“…SOS programming solutions [14,16,29] are not quite comparable to this work. Foremostly, they are not sound, i.e.…”

Section: Introductioncontrasting

confidence: 57%

“…SOS-based procedures [16,29], have required high-order polynomial templates, which has suggested the use of alternative activation functions. The BC, shown in Fig.…”

Section: Case Studies and Experimental Resultsmentioning

confidence: 99%

“…Background and Related Work A few techniques have been developed to synthesise BC. For polynomial models, sum-of-squares (SOS) and semi-definite programming relaxations [14,16,29] convert the BC synthesis problem into constraints expressed as linear or bilinear matrix inequalities: these are numerically solved as a convex optimisation problem, however unsoundly. To increase scalability and to enhance expressiveness, numerous barrier formats have been considered: BC based on exponential conditions are presented in [14]; BC based on Darboux polynomials are outlined in [33]; [30] newly introduces a multi-dimensional generalisation of BC, thus broadening their scope and applicability.…”

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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“…SOS programming solutions [14,16,29] are not quite comparable to this work. Foremostly, they are not sound, i.e.…”

Section: Introductioncontrasting

confidence: 57%

“…SOS-based procedures [16,29], have required high-order polynomial templates, which has suggested the use of alternative activation functions. The BC, shown in Fig.…”

Section: Case Studies and Experimental Resultsmentioning

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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“…The invariance condition of a set S for the linear autonomous system (7) can be written as AS ⊆ S. (15) We see with Proposition 7 that this is equivalent to…”

Section: A Autonomous Systemsmentioning

confidence: 98%

Piecewise Semi-Ellipsoidal Control Invariant Sets

Legat

Raković

Jungers

2021

IEEE Control Syst. Lett.

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Computing control invariant sets is paramount in many applications. The families of sets commonly used for computations are ellipsoids and polyhedra. However, searching for a control invariant set over the family of ellipsoids is conservative for systems more complex than unconstrained linear time invariant systems. Moreover, even if the control invariant set may be approximated arbitrarily closely by polyhedra, the complexity of the polyhedra may grow rapidly in certain directions. An attractive generalization of these two families are piecewise semiellipsoids. We provide in this paper a convex programming approach for computing control invariant sets of this family.

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“…This paper genralizes [16], [15] and [17] into a framework for computing convex controlled invariant sets for linear hybrid control systems. In [16], the authors treat the particular case where the continuous dynamic at each mode (see definition 1) is trivial, i.e., ẋ = 0. In [15], the authors extends [16] to piecewise semi-ellipsoids.…”

Section: Introductionmentioning

confidence: 99%

Geometric control of hybrid systems

Legat¹,

Jungers²

2021

Preprint

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In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched linear systems. We reformulate the invariance of a set as an inequality for its support function that is valid for any convex set. This produces novel algebraic conditions for the invariance of sets with polynomial or piecewise quadratic support function.

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Sum-of-Squares methods for controlled invariant sets with applications to model-predictive control

Cited by 12 publications

References 21 publications

Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Automated and Formal Synthesis of Neural Barrier Certificates for Dynamical Models

Piecewise Semi-Ellipsoidal Control Invariant Sets

Geometric control of hybrid systems

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