122 research outputs found
Stochastic Stability Analysis of Discrete Time System Using Lyapunov Measure
In this paper, we study the stability problem of a stochastic, nonlinear,
discrete-time system. We introduce a linear transfer operator-based Lyapunov
measure as a new tool for stability verification of stochastic systems. Weaker
set-theoretic notion of almost everywhere stochastic stability is introduced
and verified, using Lyapunov measure-based stochastic stability theorems.
Furthermore, connection between Lyapunov functions, a popular tool for
stochastic stability verification, and Lyapunov measures is established. Using
the duality property between the linear transfer Perron-Frobenius and Koopman
operators, we show the Lyapunov measure and Lyapunov function used for the
verification of stochastic stability are dual to each other. Set-oriented
numerical methods are proposed for the finite dimensional approximation of the
Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability
results in finite dimensional approximation space are also presented. Finite
dimensional approximation is shown to introduce further weaker notion of
stability referred to as coarse stochastic stability. The results in this paper
extend our earlier work on the use of Lyapunov measures for almost everywhere
stability verification of deterministic dynamical systems ("Lyapunov Measure
for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol.
53, No. 1, Feb. 2008).Comment: Proceedings of American Control Conference, Chicago IL, 201
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
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