13,628 research outputs found
Data-driven approximations of dynamical systems operators for control
The Koopman and Perron Frobenius transport operators are fundamentally
changing how we approach dynamical systems, providing linear representations
for even strongly nonlinear dynamics. Although there is tremendous potential
benefit of such a linear representation for estimation and control, transport
operators are infinite-dimensional, making them difficult to work with
numerically. Obtaining low-dimensional matrix approximations of these operators
is paramount for applications, and the dynamic mode decomposition has quickly
become a standard numerical algorithm to approximate the Koopman operator.
Related methods have seen rapid development, due to a combination of an
increasing abundance of data and the extensibility of DMD based on its simple
framing in terms of linear algebra. In this chapter, we review key innovations
in the data-driven characterization of transport operators for control,
providing a high-level and unified perspective. We emphasize important recent
developments around sparsity and control, and discuss emerging methods in big
data and machine learning.Comment: 37 pages, 4 figure
Stability of Polynomial Differential Equations: Complexity and Converse Lyapunov Questions
We consider polynomial differential equations and make a number of
contributions to the questions of (i) complexity of deciding stability, (ii)
existence of polynomial Lyapunov functions, and (iii) existence of sum of
squares (sos) Lyapunov functions.
(i) We show that deciding local or global asymptotic stability of cubic
vector fields is strongly NP-hard. Simple variations of our proof are shown to
imply strong NP-hardness of several other decision problems: testing local
attractivity of an equilibrium point, stability of an equilibrium point in the
sense of Lyapunov, invariance of the unit ball, boundedness of trajectories,
convergence of all trajectories in a ball to a given equilibrium point,
existence of a quadratic Lyapunov function, local collision avoidance, and
existence of a stabilizing control law.
(ii) We present a simple, explicit example of a globally asymptotically
stable quadratic vector field on the plane which does not admit a polynomial
Lyapunov function (joint work with M. Krstic). For the subclass of homogeneous
vector fields, we conjecture that asymptotic stability implies existence of a
polynomial Lyapunov function, but show that the minimum degree of such a
Lyapunov function can be arbitrarily large even for vector fields in fixed
dimension and degree. For the same class of vector fields, we further establish
that there is no monotonicity in the degree of polynomial Lyapunov functions.
(iii) We show via an explicit counterexample that if the degree of the
polynomial Lyapunov function is fixed, then sos programming may fail to find a
valid Lyapunov function even though one exists. On the other hand, if the
degree is allowed to increase, we prove that existence of a polynomial Lyapunov
function for a planar or a homogeneous vector field implies existence of a
polynomial Lyapunov function that is sos and that the negative of its
derivative is also sos.Comment: 30 pages. arXiv admin note: substantial text overlap with
arXiv:1112.0741, arXiv:1210.742
Decomposition of Nonlinear Dynamical Networks via Comparison Systems
In analysis and control of large-scale nonlinear dynamical systems, a
distributed approach is often an attractive option due to its computational
tractability and usually low communication requirements. Success of the
distributed control design relies on the separability of the network into
weakly interacting subsystems such that minimal information exchange between
subsystems is sufficient to achieve satisfactory control performance. While
distributed analysis and control design for dynamical network have been well
studied, decomposition of nonlinear networks into weakly interacting subsystems
has not received as much attention. In this article we propose a vector
Lyapunov functions based approach to quantify the energy-flow in a dynamical
network via a model of a comparison system. Introducing a notion of power and
energy flow in a dynamical network, we use sum-of-squares programming tools to
partition polynomial networks into weakly interacting subsystems. Examples are
provided to illustrate the proposed method of decomposition.Comment: to be presented at ECC 201
Vector Contraction Analysis for Nonlinear Dynamical Systems
This paper derives new results for the analysis of nonlinear systems by
extending contraction theory in the framework of vector distances. A new tool,
vector contraction analysis utilizing a notion of the vector-valued norm which
evidently induces a vector distance between any pair of trajectories of the
system, offers an amenable framework as each component of vector-valued norm
function satisfies fewer strict conditions as that of standard contraction
analysis. Particularly, every element of vector-valued norm derivative need not
be strictly negative definite for convergence of any pair of trajectories of
the system. Moreover, vector-valued norm derivative satisfies a component-wise
inequality employing some comparison system. In fact, the convergence analysis
is performed by comparing the relative distances between any pair of the
trajectories of the original nonlinear system and the comparison system.
Comparison results are derived by utilizing the concepts of quasi-monotonicity
property of the function and vector differential inequalities. Moreover, the
results are also derived in the framework of the cone ordering instead of
utilizing component-wise inequalities between vectors. In addition, the
proposed framework is illustrated by examples
A Local Characterization of Lyapunov Functions and Robust Stability of Perturbed Systems on Riemannian Manifolds
This paper proposes several Converse Lyapunov Theorems for nonlinear
dynamical systems defined on smooth connected Riemannian manifolds and
characterizes properties of corresponding Lyapunov functions in a normal
neighborhood of an equilibrium. We extend the methods of constructing of
Lyapunov functions for ordinary differential equations on \mathds{R}^{n} to
dynamical systems defined on Riemannian manifolds by employing the differential
geometry. By employing the derived properties of Lyapunov functions, we
obtained the stability of perturbed dynamical systems on Riemannian manifolds.
The results are obtained by employing the notions of normal neighborhoods, the
injectivity radius on Riemannian manifolds and existence of bump functions on
manifolds
On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous Vector Fields
It is well-known that asymptotic stability (AS) of homogeneous polynomial
vector fields of degree one (i.e., linear systems) can be decided in polynomial
time e.g. by searching for a quadratic Lyapunov function. Since homogeneous
vector fields of even degree can never be AS, the next interesting degree to
consider is equal to three. In this paper, we prove that deciding AS of
homogeneous cubic vector fields is strongly NP-hard and pose the question of
determining whether it is even decidable. As a byproduct of the reduction that
establishes our NP-hardness result, we obtain a Lyapunov-inspired technique for
proving positivity of forms. We also show that for asymptotically stable
homogeneous cubic vector fields in as few as two variables, the minimum degree
of a polynomial Lyapunov function can be arbitrarily large. Finally, we show
that there is no monotonicity in the degree of polynomial Lyapunov functions
that prove AS; i.e., a homogeneous cubic vector field with no homogeneous
polynomial Lyapunov function of some degree can very well have a
homogeneous polynomial Lyapunov function of degree less than .Comment: 7 page
Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems
The notion of Lyapunov function plays a key role in design and verification
of dynamical systems, as well as hybrid and cyber-physical systems. In this
paper, to analyze the asymptotic stability of a dynamical system, we generalize
standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by
considering higher order Lie derivatives of certain functions along the
system's vector field. Furthermore, we present a complete method to
automatically discovering polynomial RLFs for polynomial dynamical systems
(PDSs). Our method is complete in the sense that it is able to discover all
polynomial RLFs by enumerating all polynomial templates for any PDS.Comment: 6 pages, one algorith
Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions
We analyze stability properties of monotone nonlinear systems via
max-separable Lyapunov functions, motivated by the following observations:
first, recent results have shown that asymptotic stability of a monotone
nonlinear system implies the existence of a max-separable Lyapunov function on
a compact set; second, for monotone linear systems, asymptotic stability
implies the stronger properties of D-stability and insensitivity to
time-delays. This paper establishes that for monotone nonlinear systems,
equivalence holds between asymptotic stability, the existence of a
max-separable Lyapunov function, D-stability, and insensitivity to bounded and
unbounded time-varying delays. In particular, a new and general notion of
D-stability for monotone nonlinear systems is discussed and a set of necessary
and sufficient conditions for delay-independent stability are derived. Examples
show how the results extend the state-of-the-art
Global Stability Results for Systems Under Sampled-Data Control
In this work sufficient conditions expressed by means of single and vector
Lyapunov functions of Uniform Input-to-Output Stability (UIOS) and Uniform
Input-to-State Stability (UISS) are given for finite-dimensional systems under
feedback control with zero order hold.Comment: Submitted for possible publication to the International Journal of
Robust and Nonlinear Contro
Data-driven feedback stabilization of nonlinear systems: Koopman-based model predictive control
In this work, a predictive control framework is presented for feedback
stabilization of nonlinear systems. To achieve this, we integrate Koopman
operator theory with Lyapunov-based model predictive control (LMPC). The main
idea is to transform nonlinear dynamics from state-space to function space
using Koopman eigenfunctions - for control affine systems this results in a
bilinear model in the (lifted) function space. Then, a predictive controller is
formulated in Koopman eigenfunction coordinates which uses an auxiliary Control
Lyapunov Function (CLF) based bounded controller as a constraint to ensure
stability of the Koopman system in the function space. Provided there exists a
continuously differentiable inverse mapping between the original state-space
and (lifted) function space, we show that the designed controller is capable of
translating the feedback stabilizability of the Koopman bilinear system to the
original nonlinear system. Remarkably, the feedback control design proposed in
this work remains completely data-driven and does not require any explicit
knowledge of the original system. Furthermore, due to the bilinear structure of
the Koopman model, seeking a CLF is no longer a bottleneck for LMPC. Benchmark
numerical examples demonstrate the utility of the proposed feedback control
design.Comment: 11 pages, corrected typos, removed commented out line
- …