153 research outputs found
A Contractive Approach to Separable Lyapunov Functions for Monotone Systems
Monotone systems preserve a partial ordering of states along system
trajectories and are often amenable to separable Lyapunov functions that are
either the sum or the maximum of a collection of functions of a scalar
argument. In this paper, we consider constructing separable Lyapunov functions
for monotone systems that are also contractive, that is, the distance between
any pair of trajectories exponentially decreases. The distance is defined in
terms of a possibly state-dependent norm. When this norm is a weighted
one-norm, we obtain conditions which lead to sum-separable Lyapunov functions,
and when this norm is a weighted infinity-norm, symmetric conditions lead to
max-separable Lyapunov functions. In addition, we consider two classes of
Lyapunov functions: the first class is separable along the system's state, and
the second class is separable along components of the system's vector field.
The latter case is advantageous for many practically motivated systems for
which it is difficult to measure the system's state but easier to measure the
system's velocity or rate of change. In addition, we present an algorithm based
on sum-of-squares programming to compute such separable Lyapunov functions. We
provide several examples to demonstrate our results.Comment: arXiv admin note: text overlap with arXiv:1609.0625
Diffusion with nonlocal Dirichlet boundary conditions on unbounded domains
We consider a second order differential operator on an
(typically unbounded) open and Dirichlet regular set and subject to nonlocal Dirichlet boundary conditions of the form
u(z) = \int_\Omega u(x)\mu (z, dx) \quad \mbox{ for } z\in \partial \Omega.
Here, is a -continuous map taking values in the
probability measures on . Under suitable assumptions on the
coefficients in , which may be unbounded, we prove that a
realization of subject to the nonlocal boundary
condition, generates a (not strongly continuous) semigroup on
. We also establish a sufficient condition for this semigroup
to be Markovian and prove that in this case, it enjoys the strong Feller
property. We also study the asymptotic behavior of the semigroup.Comment: 27 pages, no figures. This is a revision based on the comments of the
referee
Stability Theory of Stochastic Models in Opinion Dynamics
We consider a certain class of nonlinear maps that preserve the probability
simplex, i.e., stochastic maps, that are inspired by the DeGroot-Friedkin model
of belief/opinion propagation over influence networks. The corresponding
dynamical models describe the evolution of the probability distribution of
interacting species. Such models where the probability transition mechanism
depends nonlinearly on the current state are often referred to as {\em
nonlinear Markov chains}. In this paper we develop stability results and study
the behavior of representative opinion models. The stability certificates are
based on the contractivity of the nonlinear evolution in the -metric.
We apply the theory to two types of opinion models where the adaptation of the
transition probabilities to the current state is exponential and linear,
respectively--both of these can display a wide range of behaviors. We discuss
continuous-time and other generalizations.Comment: 11 pages, 6 figure
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
Distributed Nonlinear Control Design using Separable Control Contraction Metrics
This paper gives convex conditions for synthesis of a distributed control
system for large-scale networked nonlinear dynamic systems. It is shown that
the technique of control contraction metrics (CCMs) can be extended to this
problem by utilizing separable metric structures, resulting in controllers that
only depend on information from local sensors and communications from immediate
neighbours. The conditions given are pointwise linear matrix inequalities, and
are necessary and sufficient for linear positive systems and certain monotone
nonlinear systems. Distributed synthesis methods for systems on chordal graphs
are also proposed based on SDP decompositions. The results are illustrated on a
problem of vehicle platooning with heterogeneous vehicles, and a network of
nonlinear dynamic systems with over 1000 states that is not feedback
linearizable and has an uncontrollable linearizationComment: Conditionally accepted to IEEE Transactions on Control of Networked
System
Non-Euclidean Contraction Theory for Monotone and Positive Systems
In this note we study strong contractivity of monotone systems and
equilibrium-contractivity of positive systems with respect to non-Euclidean
norms. We first introduce the notion of conic matrix measure and study its
properties. Using conic matrix measures and weak semi-inner products, we
characterize strongly contracting monotone systems in non-Euclidean spaces. We
apply this framework to i) study stability of monotone separable systems, ii)
establish strong contractivity of excitatory Hopfield neural networks, and iii)
obtain a novel version of the Matrosov-Bellman comparison lemma. We also
characterize equilibrium-contracting positive systems in non-Euclidean spaces
and provide a sufficient condition for equilibrium-contractivity using conic
measures. We apply this framework to i) study contractivity of positive
separable systems, and ii) develop a novel comparison-based framework for
studying interconnected systems
Performance guarantees for model-based Approximate Dynamic Programming in continuous spaces
We study both the value function and Q-function formulation of the Linear
Programming approach to Approximate Dynamic Programming. The approach is
model-based and optimizes over a restricted function space to approximate the
value function or Q-function. Working in the discrete time, continuous space
setting, we provide guarantees for the fitting error and online performance of
the policy. In particular, the online performance guarantee is obtained by
analyzing an iterated version of the greedy policy, and the fitting error
guarantee by analyzing an iterated version of the Bellman inequality. These
guarantees complement the existing bounds that appear in the literature. The
Q-function formulation offers benefits, for example, in decentralized
controller design, however it can lead to computationally demanding
optimization problems. To alleviate this drawback, we provide a condition that
simplifies the formulation, resulting in improved computational times.Comment: 18 pages, 5 figures, journal pape
Graph-Theoretic Stability Conditions for Metzler Matrices and Monotone Systems
This paper studies the graph-theoretic conditions for stability of positive
monotone systems. Using concepts from input-to-state stability and network
small-gain theory, we first establish necessary and sufficient conditions for
the stability of linear positive systems described by Metzler matrices.
Specifically, we derive two sets of stability conditions based on two forms of
input-to-state stability gains for Metzler systems, namely max-interconnection
gains and sum-interconnection gains. Based on the max-interconnection gains, we
show that the cyclic small-gain theorem becomes necessary and sufficient for
the stability of Metzler systems; based on the sum-interconnection gains, we
obtain novel graph-theoretic conditions for the stability of Metzler systems.
All these conditions highlight the role of cycles in the interconnection graph
and unveil how the structural properties of the graph affect stability.
Finally, we extend our results to the nonlinear monotone system and obtain
similar sufficient conditions for global asymptotic stability
Payoff Dynamics Model and Evolutionary Dynamics Model: Feedback and Convergence to Equilibria
This tutorial article puts forth a framework to analyze the noncooperative
strategic interactions among the members of a large population of bounded
rationality agents. Our approach hinges on, unifies and generalizes existing
methods and models predicated in evolutionary and population games. It does so
by adopting a system-theoretic formalism that is well-suited for a broad
engineering audience familiar with the basic tenets of nonlinear dynamical
systems, Lyapunov stability, storage functions, and passivity. The framework is
pertinent for engineering applications in which a large number of agents have
the authority to select and repeatedly revise their strategies. A mechanism
that is inherent to the problem at hand or is designed and implemented by a
coordinator ascribes a payoff to each possible strategy. Typically, the agents
will prioritize switching to strategies whose payoff is either higher than the
current one or exceeds the population average. The article puts forth a
systematic methodology to characterize the stability of the dynamical system
that results from the feedback interaction between the payoff mechanism and the
revision process. This is important because the set of stable equilibria is an
accurate predictor of the population's long-term behavior. The article includes
rigorous proofs and examples of application of the stability results, which
also extend the state of the art because, unlike previously published work,
they allow for a rather general class of dynamical payoff mechanisms. The new
results and concepts proposed here are thoroughly compared to previous work,
methods and applications of evolutionary and population games
Subadditive and Multiplicative Ergodic Theorems
A result for subadditive ergodic cocycles is proved that provides more
delicate information than Kingman's subadditive ergodic theorem. As an
application we deduce a multiplicative ergodic theorem generalizing an earlier
result of Karlsson-Ledrappier, showing that the growth of a random product of
semi-contractions is always directed by some horofunction. We discuss
applications of this result to ergodic cocycles of bounded linear operators,
holomorphic maps and topical operators, as well as a random mean ergodic
theorem.Comment: 20 page
- …