Outer Approximations of The Minimal Disturbance Invariant Set

2006 ◽  
Author(s):  
C.J. Ong ◽  
E.G. Gilbert
Keyword(s):  
Invariant Set ◽  
Outer Approximations

The minimal disturbance invariant set: Outer approximations via its partial sums

Automatica ◽  
2006 ◽  
Vol 42 (9) ◽  
pp. 1563-1568 ◽  
Author(s):  
Chong-Jin Ong ◽  
Elmer G. Gilbert
Keyword(s):  
Invariant Set ◽  
Partial Sums ◽  
Outer Approximations

Robust Motion Planning for Uncertain Systems With Disturbances Using the Invariant-Set Motion Planner

2020 ◽  
Vol 65 (10) ◽  
pp. 4456-4463
Author(s):  
Claus Danielson ◽  
Karl Berntorp ◽  
Avishai Weiss ◽  
Stefano Di Cairano
Keyword(s):  
Motion Planning ◽  
Uncertain Systems ◽  
Invariant Set

scholarly journals The Invariant Set Postulate: a new geometric framework for the foundations of quantum theory and the role played by gravity

2009 ◽  
Vol 465 (2110) ◽  
pp. 3165-3185 ◽  
Author(s):  
T. N. Palmer
Keyword(s):  
Quantum Theory ◽  
Science Fiction ◽  
State Space ◽  
Fractal Geometry ◽  
Quantum Physics ◽  
Nonlinear Dynamical Systems ◽  
Physical Reality ◽  
Invariant Set ◽  
Geometric Framework

A new law of physics is proposed, defined on the cosmological scale but with significant implications for the microscale. Motivated by nonlinear dynamical systems theory and black-hole thermodynamics, the Invariant Set Postulate proposes that cosmological states of physical reality belong to a non-computable fractal state-space geometry I , invariant under the action of some subordinate deterministic causal dynamics D I . An exploratory analysis is made of a possible causal realistic framework for quantum physics based on key properties of I . For example, sparseness is used to relate generic counterfactual states to points p ∉ I of unreality, thus providing a geometric basis for the essential contextuality of quantum physics and the role of the abstract Hilbert Space in quantum theory. Also, self-similarity, described in a symbolic setting, provides a possible realistic perspective on the essential role of complex numbers and quaternions in quantum theory. A new interpretation is given to the standard ‘mysteries’ of quantum theory: superposition, measurement, non-locality, emergence of classicality and so on. It is proposed that heterogeneities in the fractal geometry of I are manifestations of the phenomenon of gravity. Since quantum theory is inherently blind to the existence of such state-space geometries, the analysis here suggests that attempts to formulate unified theories of physics within a conventional quantum-theoretic framework are misguided, and that a successful quantum theory of gravity should unify the causal non-Euclidean geometry of space–time with the atemporal fractal geometry of state space. The task is not to make sense of the quantum axioms by heaping more structure, more definitions, more science fiction imagery on top of them, but to throw them away wholesale and start afresh. We should be relentless in asking ourselves: From what deep physical principles might we derive this exquisite structure? These principles should be crisp, they should be compelling. They should stir the soul. Chris Fuchs ( Gilder 2008 , p. 335)

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

SET STABILIZATION OF A MODIFIED CHUA'S CIRCUIT

2005 ◽  
Vol 15 (02) ◽  
pp. 567-604 ◽  
Author(s):  
SHIHUA LI ◽  
YU-PING TIAN
Keyword(s):  
Equilibrium Point ◽  
Lyapunov Stability ◽  
Closed Loop ◽  
Invariant Set ◽  
Linear Feedback ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Linear Feedback Controller ◽  
Simulation Results ◽  
First Time

In this paper, we develop a simple linear feedback controller, which employs only one of the states of the system, to stabilize the modified Chua's circuit to an invariant set which consists of its nontrivial equilibria. Moreover, we show for the first time that the closed loop modified Chua's circuit satisfies set stability which can be considered as a generalization of common Lyapunov stability of an equilibrium point. Simulation results are presented to verify our method.

scholarly journals Low Model Analysis and Synchronous Simulation of the Wave Mechanics

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wenyuan Duan ◽  
Heyuan Wang ◽  
Meng Kan
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Chaotic System ◽  
Invariant Set ◽  
Positive Definite ◽  
Chaotic Attractors ◽  
Wave Mechanics ◽  
Wave Dynamics ◽  
Simulation Results ◽  
Positive Invariant Set

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

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