On the construction of Lyapunov functions using the sum of squares decomposition

By Antonis Papachristodoulou, Stephen Prajna. 2002

529 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Numerical Analysis

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Citations in the corpus

Certified Defenses against Adversarial Examples (2018, methodology): “Such methods typically use sum-of-squares veriﬁcation (Papachristodoulou & Prajna, 2002; 2005; Parrilo, 2003) and are restricted to relatively low-dimensional dynamical systems, but could plausibly scale to larger settings.”
Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach (2004, methodology): “Recently, a computational relaxation based on the sum of squares decomposition [15] and semidefinite programming [1, 25] has been proposed for algorithmic construction of Lyapunov functions [15, 14].”
Learning Stable Deep Dynamics Models (2020, methodology): “Although some general techniques such as sum-of-squares certification [16, 15] provide general methods for certifying stability of e.”
Invariant Funnels around Trajectories using Sum-of-Squares Programming (2010, background): “ested which recovers almost identical funnels and is substantially faster to compute. Recently, a great deal of research has been dedicated to region-of-attraction analysis on polynomial vector elds (Papachristodoulou and Prajna (2002); Parrilo (2003); Topcu et al. (2008); Tan and Packard (2008)) an”
LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification (2010, background): “…and the associated interior point method solvers, has had a notable impact on stability verification see, e.g., Parrilo (2000), Rantzer (2001), Papachristodoulou and Prajna (2002), Prajna and Rantzer (2007), Tan and Packard (2008), Topcu et al. (2008), Papachristodoulou et al. (2009) and many…”
Lyapunov Based Analysis and Controller Synthesis for Polynomial Systems using Sum-of-Squares Optimization (2003, background): “These algorithms extend the early results of [20] and [13].”
Analysis of Non-polynomial Systems using the Sum of Squares Decomposition (2005, background): “Recent advances in the areas of semidefinite programming and the use of the sum of squares decomposition to efficiently check nonnegativity have allowed an algorithmic procedure for systems analysis, something that was not possible before [10, 9].”
Nonlinear control synthesis by convex optimization (2004, unknown): “For instance, these tools can be used in the computation of Lyapunov functions for proving that a nonlinear system is stable [12], [11].”
Feedback control of quantum state reduction (2004, methodology): “In fact, the semidefinite programming approach of [53] and [54] even allows us to search for polynomial such that (65) is satisfied; hence we can search numerically for a global stability proof using a computer program.”
A tutorial on sum of squares techniques for systems analysis (2005, methodology): “Just as in the case of ODEs, this is a powerful theorem as it answers questions about stability without requiring a solution to (36).”
Safety Verification of Hybrid Systems Using Barrier Certificates (2004, methodology): “Even though the computational approach discussed in this section assumes that the system is described by polynomials, non-polynomial descriptions can be handled (although possibly with some conservatism) and non-polynomial barrier certificates can be constructed by recasting of variables as proposed”
Learning dynamic polynomial proofs (2019, unknown): “For example, deciding the stability of a control system can be reduced to proving the nonnegativity of a polynomial [PP02].”
LMI Techniques for Optimization Over Polynomials in Control: A Survey (2010, unknown): “A similar strategy can be adopted in the case that the system (55) linearized in the origin is only marginally stable, see [40] for details.”
Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics (2014, background): “Recently, Lyapunov functions have seen a further surge of interest in the nonlinear systems community through powerful tools for obtaining them as sum of squares [31], [50].”
A tutorial review of economic model predictive control methods (2014, methodology): “Some methods exist for the design of Lyapunov functions for nonlinear systems with constraints which include techniques based on Zubov’s method [46] and based on the sum of squares decomposition [134].”
Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints (2014, background): “Non-quadratic Lyapunov functions There has been substantial work in the past decade on efficient algorithms to search over non-quadratic Lyapunov functions [29, 30].”