236 research outputs found
Computation of Polytopic Invariants for Polynomial Dynamical Systems using Linear Programming
This paper deals with the computation of polytopic invariant sets for
polynomial dynamical systems. An invariant set of a dynamical system is a
subset of the state space such that if the state of the system belongs to the
set at a given instant, it will remain in the set forever in the future.
Polytopic invariants for polynomial systems can be verified by solving a set of
optimization problems involving multivariate polynomials on bounded polytopes.
Using the blossoming principle together with properties of multi-affine
functions on rectangles and Lagrangian duality, we show that certified lower
bounds of the optimal values of such optimization problems can be computed
effectively using linear programs. This allows us to propose a method based on
linear programming for verifying polytopic invariant sets of polynomial
dynamical systems. Additionally, using sensitivity analysis of linear programs,
one can iteratively compute a polytopic invariant set. Finally, we show using a
set of examples borrowed from biological applications, that our approach is
effective in practice
Robust Region-of-Attraction Estimation
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
Constructing Attractors of Nonlinear Dynamical Systems
In a previous work, we have shown how to generate attractor sets of affine hybrid systems using a method of state space decomposition. We show here how to adapt the method to polynomial dynamics systems by approximating them as switched affine systems. We show the practical
interest of the method on standard examples of the literature
The Maximal Positively Invariant Set: Polynomial Setting
This note considers the maximal positively invariant set for polynomial
discrete time dynamics subject to constraints specified by a basic
semialgebraic set. The note utilizes a relatively direct, but apparently
overlooked, fact stating that the related preimage map preserves basic
semialgebraic structure. In fact, this property propagates to underlying
set--dynamics induced by the associated restricted preimage map in general and
to its maximal trajectory in particular. The finite time convergence of the
corresponding maximal trajectory to the maximal positively invariant set is
verified under reasonably mild conditions. The analysis is complemented with a
discussion of computational aspects and a prototype implementation based on
existing toolboxes for polynomial optimization
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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