Publications (9)0 Total impact
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Conference Proceeding: Simulation-aided reachability and local gain analysis for nonlinear dynamical systems.
Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008, December 9-11, 2008, Cancún, México; 01/2008 -
Article: Local robust performance analysis for nonlinear dynamical systems
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ABSTRACT: We propose a computational method for local robust performance analysis of nonlinear systems with polynomial dynamics. Specifically, we characterize upper bounds for local L_2 → L_2 input-output gains using polynomial Lyapunov/ storage functions satisfying certain dissipation inequalities and compute safe approximations for these upper bounds via sum-of-squares programming problems. We consider both bounded parametric uncertainties and bounded uncertainties due to unmodeled dynamics -
Article: Help on SOS
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ABSTRACT: In this issue of IEEE Control Systems Magazine, Andy Packard and friends respond to a query on determining the region of attraction using sum-of-squares methods. -
Article: Parameter-Dependent Lyapunov Functions for Linear Systems With Constant Uncertainties
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ABSTRACT: Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties. -
Article: Robust Region-of-Attraction Estimation
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ABSTRACT: We propose a method to compute invariant subsets of the region-of-attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty. Parameter-independent Lyapunov functions are used to characterize invariant subsets of the robust region-of-attraction. A branch-and-bound type refinement procedure reduces the conservatism. We demonstrate the method on an example from the literature and uncertain controlled short-period aircraft dynamics. -
Article: Linearized analysis versus optimization-based nonlinear analysis for nonlinear systems
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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. -
Article: Stability region estimation for systems with unmodeled dynamics
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ABSTRACT: We propose a method to compute invariant subsets of the robust region-of-attraction for the asymptotically stable equilibrium points of systems with polynomial nominal vector fields and unmodeled dynamics. The effects of unmodeled dynamics are accounted for as systems satisfying certain gain relations or dissipation inequalities. The methodology is ex-tended to systems with parametric uncertainties and an in-formal branch-and-bound type refinement procedure to reduce the conservatism is discussed. We demonstrate the method on a polynomial approximation of uncertain controlled short period aircraft dynamics. -
Article: Analytical Validation Tools for Safety Critical Systems
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ABSTRACT: The current practice to validate flight control laws relies on applying linear analysis tools to assess the closed loop stability and performance characteristics about many trim conditions. Nonlinear simulations are then used to provide further confidence in the linear analyses and also to uncover dynamic characteristics, e.g. limit cycles, which are not revealed by the linear analysis. This paper reviews analysis techniques which can be applied to nonlinear systems described by polynomial dynamic equations. The proposed approach is to reduce the analysis problems to a sum-of-squares optimization problem which can then be solved with freely available software. These techniques can fill the gap between linear analysis and nonlinear simulations and hence can be used to provide additional confidence in the flight control law performance. -
Article: Local stability analysis using simulations and sum-of-squares programming
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ABSTRACT: The problem of computing bounds on the region-of-attraction for systems with polynomial vector fields is considered. Invariant subsets of the region-of-attraction are characterized as sublevel sets of Lyapunov functions. Finite-dimensional polynomial parametrizations for Lyapunov functions are used. A methodology utilizing information from simulations to generate Lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed. The suitability of Lyapunov function candidates is assessed solving linear sum-of-squares optimization problems. Qualified candidates are used to compute invariant subsets of the region-of-attraction and to initialize various bilinear search strategies for further optimization. We illustrate the method on small examples from the literature and several control oriented systems.Automatica.
