36,421 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
Model Predictive Regulation
We show how optimal nonlinear regulation can be achieved in a model
predictive control fashion
Stability Optimization of Positive Semi-Markov Jump Linear Systems via Convex Optimization
In this paper, we study the problem of optimizing the stability of positive
semi-Markov jump linear systems. We specifically consider the problem of tuning
the coefficients of the system matrices for maximizing the exponential decay
rate of the system under a budget-constraint. By using a result from the matrix
theory on the log-log convexity of the spectral radius of nonnegative matrices,
we show that the stability optimization problem reduces to a convex
optimization problem under certain regularity conditions on the system matrices
and the cost function. We illustrate the validity and effectiveness of the
proposed results by using an example from the population biology
On dual Schur domain decomposition method for linear first-order transient problems
This paper addresses some numerical and theoretical aspects of dual Schur
domain decomposition methods for linear first-order transient partial
differential equations. In this work, we consider the trapezoidal family of
schemes for integrating the ordinary differential equations (ODEs) for each
subdomain and present four different coupling methods, corresponding to
different algebraic constraints, for enforcing kinematic continuity on the
interface between the subdomains.
Method 1 (d-continuity) is based on the conventional approach using
continuity of the primary variable and we show that this method is unstable for
a lot of commonly used time integrators including the mid-point rule. To
alleviate this difficulty, we propose a new Method 2 (Modified d-continuity)
and prove its stability for coupling all time integrators in the trapezoidal
family (except the forward Euler). Method 3 (v-continuity) is based on
enforcing the continuity of the time derivative of the primary variable.
However, this constraint introduces a drift in the primary variable on the
interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte
stabilization to limit this drift and we derive bounds for the stabilization
parameter to ensure stability.
Our stability analysis is based on the ``energy'' method, and one of the main
contributions of this paper is the extension of the energy method (which was
previously introduced in the context of numerical methods for ODEs) to assess
the stability of numerical formulations for index-2 differential-algebraic
equations (DAEs).Comment: 22 Figures, 49 pages (double spacing using amsart
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