695 research outputs found
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
Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method
The dynamic complexity of robots and mechatronic systems often pertains to
the hybrid nature of dynamics, where governing equations consist of
heterogenous equations that are switched depending on the state of the system.
Legged robots and manipulator robots experience contact-noncontact discrete
transitions, causing switching of governing equations. Analysis of these
systems have been a challenge due to the lack of a global, unified model that
is amenable to analysis of the global behaviors. Composition operator theory
has the potential to provide a global, unified representation by converting
them to linear dynamical systems in a lifted space. The current work presents a
method for encoding nonlinear heterogenous dynamics into a high dimensional
space of observables in the form of Koopman operator. First, a new formula is
established for representing the Koopman operator in a Hilbert space by using
inner products of observable functions and their composition with the governing
state transition function. This formula, called Direct Encoding, allows for
converting a class of heterogenous systems directly to a global, unified linear
model. Unlike prevalent data-driven methods, where results can vary depending
on numerical data, the proposed method is globally valid, not requiring
numerical simulation of the original dynamics. A simple example validates the
theoretical results, and the method is applied to a multi-cable suspension
system.Comment: 12 pages, 7 figure
Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity
This paper introduces the Koopman Control Family (KCF), a mathematical
framework for modeling general discrete-time nonlinear control systems with the
aim of providing a solid theoretical foundation for the use of Koopman-based
methods in systems with inputs. We demonstrate that the concept of KCF can
completely capture the behavior of nonlinear control systems on a (potentially
infinite-dimensional) function space. By employing a generalized notion of
subspace invariance under the KCF, we establish a universal form for
finite-dimensional models, which encompasses the commonly used linear,
bilinear, and linear switched models as specific instances. In cases where the
subspace is not invariant under the KCF, we propose a method for approximating
models in general form and characterize the model's accuracy using the concept
of invariance proximity. The proposed framework naturally lends itself to the
incorporation of data-driven methods in modeling and control.Comment: 16 page
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