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Geometric and Numerical Foundations of Movements pp 89–112Cite as

Some Recent Directions in Algebraic Methods for Optimization and Lyapunov Analysis

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Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 117))

Abstract

Exciting recent developments at the interface of optimization and control have shown that several fundamental problems in dynamics and control, such as stability, collision avoidance, robust performance, and controller synthesis can be addressed by a synergy of classical tools from Lyapunov theory and modern computational techniques from algebraic optimization. In this chapter, we give a brief overview of our recent research efforts (with various coauthors) to (i) enhance the scalability of the algorithms in this field, and (ii) understand their worst case performance guarantees as well as fundamental limitations. The topics covered include the concepts of “dsos and sdsos optimization”, path-complete and non-monotonic Lyapunov functions, and some lower bounds and complexity results for Lyapunov analysis of polynomial vector fields and hybrid systems. In each case, our relevant papers are tersely surveyed and the challenges/opportunities that lie ahead are stated.

This chapter is a revised and expanded version of the conference paper in [13], which was presented as a tutorial talk at the 53rd annual Conference on Decision and Control.

Amir Ali Ahmadi—His research is partially supported by an NSF CAREER Award, an AFOSR Young Investigator Program Award, and a Google Research Award.

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Notes

  1. 1.

    The familiar reader may safely skip this section. For a more comprehensive introductary exposition, see: https://blogs.princeton.edu/imabandit/guest-posts/.

  2. 2.

    Here, we are assuming a strictly feasible solution to the SDP. Indeed a strictly feasible solution to (7) is required to get the strict inequalities in (6). Luckily, unless the SDP has an empty interior, a strictly feasible solution will automatically be returned by the interior point solver. See the discussion in [1, p. 41].

  3. 3.

    Once again, strict feasibility of the constraints in (8) is required to rule out trivial solutions and lead to the strict inequalities that we would like to impose on V.

  4. 4.

    https://github.com/spot-toolbox/spotless.

  5. 5.

    A homogeneous polynomial vector field is one where all monomials have the same degree. Linear systems are an example.

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Acknowledgements

We are grateful to Anirudha Majumdar for his contributions to the work presented in Sect. 4 and to Russ Tedrake for the robotics applications and many insightful discussions.

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, USA

    Amir Ali Ahmadi

  2. Laboratory for Information and Decision Systems and The Department of Electrical Engineering and Computer Science, MIT, Cambridge, MA, USA

    Pablo A. Parrilo

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  1. Amir Ali Ahmadi
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Correspondence to Amir Ali Ahmadi .

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  1. LAAS-CNRS , Toulouse, France

    Jean-Paul Laumond

  2. LAAS-CNRS , Toulouse, France

    Nicolas Mansard

  3. LAAS-CNRS , Toulouse, France

    Jean-Bernard Lasserre

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Ahmadi, A.A., Parrilo, P.A. (2017). Some Recent Directions in Algebraic Methods for Optimization and Lyapunov Analysis. In: Laumond, JP., Mansard, N., Lasserre, JB. (eds) Geometric and Numerical Foundations of Movements . Springer Tracts in Advanced Robotics, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-51547-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-51547-2_5

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