Linearized analysis versus optimization-based nonlinear analysis for nonlinear systems

Authors:
Ufuk Topcu;Andrew Packard
Affiliations:
Control and Dynamical Systems, California Institute of Technology, Pasadena, CA;Department of Mechanical Engineering, The University of California, Berkeley, CA
Venue:
ACC'09 Proceedings of the 2009 conference on American Control Conference
Year:
2009

Citing 5
Cited 0

Nonlinear systems analysis (2nd ed.)

Nonlinear systems analysis (2nd ed.)
Robust adaptive control

Robust adaptive control
Global Optimization with Polynomials and the Problem of Moments

SIAM Journal on Optimization
Brief paper: Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

Automatica (Journal of IFAC)
Brief paper: Local stability analysis using simulations and sum-of-squares programming

Automatica (Journal of IFAC)

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Abstract

For autonomous nonlinear systems stability and input-output properties in small enough (infinitesimally small) neighborhoods of (linearly) asymptotically stable equilibrium points can be inferred from the properties of the linearized dynamics. On the other hand, generalizations of the S-procedure and sum-of-squares programming promise a framework potentially capable of generating certificates valid over quantifiable, finite size neighborhoods of the equilibrium points. However, this procedure involves multiple relaxations (unidirectional implications). Therefore, it is not obvious if the sum-of-squares programming based nonlinear analysis can return a feasible answer whenever linearization based analysis does. Here, we prove that, for a restricted but practically useful class of systems, conditions in sum-of-squares programming based region-of-attraction, reachability, and input-output gain analyses are feasible whenever linearization based analysis is conclusive. Besides the theoretical interest, such results may lead to computationally less demanding, potentially more conservative nonlinear (compared to direct use of sum-of-squares formulations) analysis tools.