arXiv subjects

Joel A. Rosenfeld

Publications and source records attributed to Joel A. Rosenfeld.

24 records · Page 2Linked to original sources

arXiv2017

Invariance-like results for Nonautonomous Switched Systems

This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A candidate common Lyapunov function that has a negative semidefinite derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle-Yoshizawa results for the switched system. Results for regular and non-regular candidate Lyapunov functions are presented using an appropriate generalization of the time derivative. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.

eess.SY↗

arXiv2015

State Following (StaF) Kernel Functions for Function Approximation

A function approximation method is developed that aims to approximate a function in a small neighborhood of a state that travels within a compact set. The development is based on the theory of universal reproducing kernel Hilbert spaces over the $n$-dimensional Euclidean space. Several theorems are introduced that support the development of this State Following (StaF) method. In particular, it is shown that there is a bound on the number of kernel functions required for the maintenance of an accurate function approximation as a state moves through a compact set. Additionally, a weight update law, based on gradient descent, is introduced where arbitrarily close accuracy can be achieved provided the weight update law is iterated at a sufficient frequency, as detailed in Theorem 6.1. To illustrate the advantage, the impact of the StaF method is that for some applications the number of basis functions can be reduced. The StaF method is applied to an adaptive dynamic programming (ADP) application to demonstrate that stability is maintained with a reduced number of basis functions. Simulation results demonstrate the utility of the StaF methodology for the maintenance of accurate function approximation as well as solving an infinite horizon optimal regulation problem through ADP. The results of the simulation indicate that fewer basis functions are required to guarantee stability and approximate optimality than are required when a global approximation approach is used.

math.OC↗

arXiv2015

Introducing the Polylogarithmic Hardy Space

In this paper we investigate the reproducing kernel Hilbert space where the polylogarithm appears as kernel functions. This investigation begins with the properties of functions in this space, and here a connection to the classical Hardy space is shown through the Bose-Einstein integral equation. Next we consider function theoretic operators over the polylogarithmic Hardy space, such as multiplication and Toeplitz operators. It is determined that there are only trivial densely defined multiplication operators (and therefore only trivial bounded multipliers) over this space, which makes this space the first for which this has been found to be true. In the case of Toeplitz operators, a connection between a certain subset of these operators and the number theoretic divisor function is found. Finally, the paper concludes with an operator theoretic proof of the divisibility of the divisor function.

math.FA↗

arXiv2015

Efficient model-based reinforcement learning for approximate online optimal

In this paper the infinite horizon optimal regulation problem is solved online for a deterministic control-affine nonlinear dynamical system using the state following (StaF) kernel method to approximate the value function. Unlike traditional methods that aim to approximate a function over a large compact set, the StaF kernel method aims to approximate a function in a small neighborhood of a state that travels within a compact set. Simulation results demonstrate that stability and approximate optimality of the control system can be achieved with significantly fewer basis functions than may be required for global approximation methods.

eess.SY↗

arXiv2014

Densely Defined Multiplication on the Sobolev Space

Following Sarason's classification of the densely defined multiplication operators over the Hardy space, we classify the densely defined multipliers over the Sobolev space, $W^{1,2}[0,1]$. In this paper we find that the collection of such multipliers for the Sobolev space is exactly the Sobolev space itself. This sharpens a result of Shields concerning bounded multipliers. The densely defined multiplication operators over the subspace $W_0 = \{f \in W^{1,2}[0,1] : f(0)=f(1)=0 \}$ are also classified. In this case the densely defined multiplication operators can be written as a ratio of functions in $W_0$ where the denominator is non-vanishing. This is proved using a contructive argument.

math.FA↗

arXiv2013

Decentralized formation control with connectivity maintenance and collision avoidance under limited and intermittent sensing

A decentralized switched controller is developed for dynamic agents to perform global formation configuration convergence while maintaining network connectivity and avoiding collision within agents and between stationary obstacles, using only local feedback under limited and intermittent sensing. Due to the intermittent sensing, constant position feedback may not be available for agents all the time. Intermittent sensing can also lead to a disconnected network or collisions between agents. Using a navigation function framework, a decentralized switched controller is developed to navigate the agents to the desired positions while ensuring network maintenance and collision avoidance.

eess.SY↗