168,759 research outputs found
Optimal Stabilization using Lyapunov Measures
Numerical solutions for the optimal feedback stabilization of discrete time
dynamical systems is the focus of this paper. Set-theoretic notion of almost
everywhere stability introduced by the Lyapunov measure, weaker than
conventional Lyapunov function-based stabilization methods, is used for optimal
stabilization. The linear Perron-Frobenius transfer operator is used to pose
the optimal stabilization problem as an infinite dimensional linear program.
Set-oriented numerical methods are used to obtain the finite dimensional
approximation of the linear program. We provide conditions for the existence of
stabilizing feedback controls and show the optimal stabilizing feedback control
can be obtained as a solution of a finite dimensional linear program. The
approach is demonstrated on stabilization of period two orbit in a controlled
standard map
A Region-Dependent Gain Condition for Asymptotic Stability
A sufficient condition for the stability of a system resulting from the
interconnection of dynamical systems is given by the small gain theorem.
Roughly speaking, to apply this theorem, it is required that the gains
composition is continuous, increasing and upper bounded by the identity
function. In this work, an alternative sufficient condition is presented for
the case in which this criterion fails due to either lack of continuity or the
bound of the composed gain is larger than the identity function. More
precisely, the local (resp. non-local) asymptotic stability of the origin
(resp. global attractivity of a compact set) is ensured by a region-dependent
small gain condition. Under an additional condition that implies convergence of
solutions for almost all initial conditions in a suitable domain, the almost
global asymptotic stability of the origin is ensured. Two examples illustrate
and motivate this approach
Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions
This paper proposes a notion of viscosity weak supersolutions to build a
bridge between stochastic Lyapunov stability theory and viscosity solution
theory. Different from ordinary differential equations, stochastic differential
equations can have the origins being stable despite having no smooth stochastic
Lyapunov functions (SLFs). The feature naturally requires that the related
Lyapunov equations are illustrated via viscosity solution theory, which deals
with non-smooth solutions to partial differential equations. This paper claims
that stochastic Lyapunov stability theory needs a weak extension of viscosity
supersolutions, and the proposed viscosity weak supersolutions describe
non-smooth SLFs ensuring a large class of the origins being noisily
(asymptotically) stable and (asymptotically) stable in probability. The
contribution of the non-smooth SLFs are confirmed by a few examples;
especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled
systems have the origins being noisily asymptotically stable for any additive
noises
Output-input stability and minimum-phase nonlinear systems
This paper introduces and studies the notion of output-input stability, which
represents a variant of the minimum-phase property for general smooth nonlinear
control systems. The definition of output-input stability does not rely on a
particular choice of coordinates in which the system takes a normal form or on
the computation of zero dynamics. In the spirit of the ``input-to-state
stability'' philosophy, it requires the state and the input of the system to be
bounded by a suitable function of the output and derivatives of the output,
modulo a decaying term depending on initial conditions. The class of
output-input stable systems thus defined includes all affine systems in global
normal form whose internal dynamics are input-to-state stable and also all
left-invertible linear systems whose transmission zeros have negative real
parts. As an application, we explain how the new concept enables one to develop
a natural extension to nonlinear systems of a basic result from linear adaptive
control.Comment: Revised version, to appear in IEEE Transactions on Automatic Control.
See related work in http://www.math.rutgers.edu/~sontag and
http://black.csl.uiuc.edu/~liberzo
On the Relation of Delay Equations to First-Order Hyperbolic Partial Differential Equations
This paper establishes the equivalence between systems described by a single
first-order hyperbolic partial differential equation and systems described by
integral delay equations. System-theoretic results are provided for both
classes of systems (among them converse Lyapunov results). The proposed
framework can allow the study of discontinuous solutions for nonlinear systems
described by a single first-order hyperbolic partial differential equation
under the effect of measurable inputs acting on the boundary and/or on the
differential equation. An illustrative example shows that the conversion of a
system described by a single first-order hyperbolic partial differential
equation to an integral delay system can simplify considerably the solution of
the corresponding robust feedback stabilization problem.Comment: 32 pages, submitted for possible publication to ESAIM COC
Input-Output-to-State Stability
This work explores Lyapunov characterizations of the input-output-to-state
stability (IOSS) property for nonlinear systems. The notion of IOSS is a
natural generalization of the standard zero-detectability property used in the
linear case. The main contribution of this work is to establish a complete
equivalence between the input-output-to-state stability property and the
existence of a certain type of smooth Lyapunov function. As corollaries, one
shows the existence of ``norm-estimators'', and obtains characterizations of
nonlinear detectability in terms of relative stability and of finite-energy
estimates.Comment: Many related papers can be found in:
http://www.math.rutgers.edu/~sonta
Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control
The control of the spread of dengue fever by introduction of the
intracellular parasitic bacterium Wolbachia in populations of the vector Aedes
aegypti, is presently one of the most promising tools for eliminating dengue,
in the absence of an efficient vaccine. The success of this operation requires
locally careful planning to determine the adequate number of individuals
carrying the Wolbachia parasite that need to be introduced into the natural
population. The introduced mosquitoes are expected to eventually replace the
Wolbachia-free population and guarantee permanent protection against the
transmission of dengue to human.
In this study, we propose and analyze a model describing the fundamental
aspects of the competition between mosquitoes carrying Wolbachia and mosquitoes
free of the parasite. We then use feedback control techniques to devise an
introduction protocol which is proved to guarantee that the population
converges to a stable equilibrium where the totality of mosquitoes carry
Wolbachia.Comment: 24 pages, 5 figure
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