Invariant Funnels around Trajectories using Sum-of-Squares Programming
By Mark M. Tobenkin, Ian R. Manchester, Russ Tedrake. 2010
81 citations recorded by the source corpus.
Abstract This paper presents numerical methods for computing regions of finite-time invariance (funnels) around solutions of polynomial differential equations. The methods are compared on stabilized trajectories of a six-state model of a satellite. First, we present a method which exactly certifies sufficient conditions for invariance despite relying on approximate trajectories from numerical integration. Our second method relaxes the constraints of the first by sampling in time. On the model system, this recovered almost identical funnels but was much faster to compute. In both cases, funnels are verified using Sum-of-Squares programming to search over time-varying quadratic Lyapunov functions. We examine both time-varying rescalings of quadratic forms computed from linearizations and searching directly over time-varying quadratic Lyapunov functions. On the model system the latter provided larger funnels at the cost of increased computation time.
Subfields: Computational Theory and Mathematics, Numerical Analysis
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Citations in the corpus
- Certified Defenses against Adversarial Examples (2018, background): “We can think of the activations in a neural network as the evolution of a time-varying dynamical system, and attempt to prove stability around a trajectory of this system (Tedrake et al., 2010; Tobenkin et al., 2011).”
- Finite-time regional verification of stochastic non-linear systems (2011, unknown): “Keywords verification, control, stochastic, sum-of-squares, supermartingale, non-linear, convex optimization”