858 research outputs found
Composite control Lyapunov functions for robust stabilization of constrained uncertain dynamical systems
This work presents innovative scientific results on the robust stabilization of constrained uncertain dynamical systems via Lyapunov-based state feedback control.
Given two control Lyapunov functions, a novel class of smooth composite control Lyapunov functions is presented. This class, which is based on the R-functions theory, is universal for the stabilizability of linear differential inclusions and has the following property. Once a desired controlled invariant set is fixed, the shape of the inner level sets can be made arbitrary close to any given ones, in a smooth and non-homothetic way. This procedure is an example of ``merging'' two control Lyapunov functions.
In general, a merging function consists in a control Lyapunov function whose gradient is a continuous combination of the gradients of the two parents control Lyapunov functions.
The problem of merging two control Lyapunov functions, for instance a global control Lyapunov function with a large controlled domain of attraction and a local one with a guaranteed local performance, is considered important for several control applications. The main reason is that when simultaneously concerning constraints, robustness and optimality, a single Lyapunov function is usually suitable for just one of these goals, but ineffective for the others.
For nonlinear control-affine systems, both equations and inclusions, some equivalence properties are shown between the control-sharing property, namely the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given control Lyapunov functions, and the existence of merging control Lyapunov functions.
Even for linear systems, the control-sharing property does not always hold, with the remarkable exception of planar systems.
For the class of linear differential inclusions, linear programs and linear matrix inequalities conditions are given for the the control-sharing property to hold.
The proposed Lyapunov-based control laws are illustrated and simulated on benchmark case studies, with positive numerical results
Constructive necessary and sufficient condition for the stability of quasi-periodic linear impulsive systems
International audienceThe paper provides a computation-oriented necessary and sufficient condition for the global exponential stability of linear impulsive systems, whose impulsions are assumed to occur quasi-periodically. Based on the set-theoretic conditions for robust stability of uncertain linear systems, the existence of polyhedral Lyapunov functions is proved to be necessary and sufficient for global exponential stability of quasi-periodic linear impulsive systems. A constructive method is developed for testing the stability of the system and for computing set-induced polyhedral Lyapunov functions. The method leads to an algorithm whose complexity is similar to the standard algorithm related to discrete-time parametric uncertain systems with the state matrix belonging to a convex polytopic set
Robust Stability Analysis of Nonlinear Hybrid Systems
We present a methodology for robust stability analysis of nonlinear hybrid systems, through the algorithmic construction of polynomial and piecewise polynomial Lyapunov-like functions using convex optimization and in particular the sum of squares decomposition of multivariate polynomials. Several improvements compared to previous approaches are discussed, such as treating in a unified way polynomial switching surfaces and robust stability analysis for nonlinear hybrid systems
State feedback policies for robust receding horizon control: uniqueness, continuity, and stability
Published versio
Data-driven computation of invariant sets of discrete time-invariant black-box systems
We consider the problem of computing the maximal invariant set of
discrete-time black-box nonlinear systems without analytic dynamical models.
Under the assumption that the system is asymptotically stable, the maximal
invariant set coincides with the domain of attraction. A data-driven framework
relying on the observation of trajectories is proposed to compute
almost-invariant sets, which are invariant almost everywhere except a small
subset. Based on these observations, scenario optimization problems are
formulated and solved. We show that probabilistic invariance guarantees on the
almost-invariant sets can be established. To get explicit expressions of such
sets, a set identification procedure is designed with a verification step that
provides inner and outer approximations in a probabilistic sense. The proposed
data-driven framework is illustrated by several numerical examples.Comment: A shorter version with the title "Scenario-based set invariance
verification for black-box nonlinear systems" is published in the IEEE
Control Systems Letters (L-CSS
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