Comparing a current-carrying circular wire with polygons of equal perimeter; Magnetic field versus magnetic flux

We compare the magnetic field at the center of and the self-magnetic flux through a current-carrying circular loop, with those obtained for current-carrying polygons with the same perimeter. As the magnetic field diverges at the position of the wires, we compare the self-fluxes utilizing several regularization procedures. The calculation is best performed utilizing the vector potential, thus highlighting its usefulness in practical applications. Our analysis answers some of the intuition challenges students face when they encounter a related simple textbook example. These results can be applied directly to the determination of mutual inductances in a variety of situations.Comment: 9 pages, 4 figure

Auxiliary field method and analytical solutions of the Schr\"{o}dinger equation with exponential potentials

The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies and eigenvectors of the Schr\"{o}dinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schr\"{o}dinger equation with exponential potentials of the form $-\alpha r^\lambda \exp(-\beta r)$ can also be analytically solved by using the auxiliary field method. Formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn on the Yukawa potential and the pure exponential one

Duality relations in the auxiliary field method

The eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$ of a system of $N$ identical particles with a mass $m$ are functions of the various radial quantum numbers $n_i$ and orbital quantum numbers $l_i$. Approximations $E^{(N)}(m;Q)$ of these eigenenergies, depending on a principal quantum number $Q(\{n_i,l_i\})$, can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities $E^{(N)}(m;Q)$ and $E^{(p)}(m';Q')$ for various forms of the potentials (independent of $m$ and $N$) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies $\epsilon^{(N)}(m;\{n_i,l_i\})$

Full-vector analysis of a realistic photonic crystal fiber

We analyze the guiding problem in a realistic photonic crystal fiber using a novel full-vector modal technique, a biorthogonal modal method based on the nonselfadjoint character of the electromagnetic propagation in a fiber. Dispersion curves of guided modes for different fiber structural parameters are calculated along with the 2D transverse intensity distribution of the fundamental mode. Our results match those achieved in recent experiments, where the feasibility of this type of fiber was shown.Comment: 3 figures, submitted to Optics Letter

Self-Triggered and Event-Triggered Set-Valued Observers

This paper addresses the problem of reducing the required network load and computational power for the implementation of Set-Valued Observers (SVOs) in Networked Control System (NCS). Event- and self-triggered strategies for NCS, modeled as discrete-time Linear Parameter-Varying (LPV) systems, are studied by showing how the triggering condition can be selected. The methodology provided can be applied to determine when it is required to perform a full (``classical'') computation of the SVOs, while providing low-complexity state overbounds for the remaining time, at the expenses of temporarily reducing the estimation accuracy. As part of the procedure, an algorithm is provided to compute a suitable centrally symmetric polytope that allows to find hyper-parallelepiped and ellipsoidal overbounds to the exact set-valued state estimates calculated by the SVOs. By construction, the proposed triggering techniques do not influence the convergence of the SVOs, as at some subsequent time instants, set-valued estimates are computed using the \emph{conventional} SVOs. Results are provided for the triggering frequency of the self-triggered strategy and two interesting cases: distributed systems when the dynamics of all nodes are equal up to a reordering of the matrix; and when the probability distribution of the parameters influencing the dynamics is known. The performance of the proposed algorithm is demonstrated in simulation by using a time-sensitive example

Finite-time average consensus in a Byzantine environment using Set-Valued Observers

This paper addresses the problem of consensus in the presence of Byzantine faults, modeled by an attacker injecting a perturbation in the state of the nodes of a network. It is firstly shown that Set-Valued Observers (SVOs) attain finite-time consensus, even in the case where the state estimates are not shared between nodes, at the expenses of requiring large horizons, thus rendering the computation problem intractable in the general case. A novel algorithm is therefore proposed that achieves finite-time consensus, even if the aforementioned requirement is dropped, by intersecting the set-valued state estimates of neighboring nodes, making it suitable for practical applications and enabling nodes to determine a stopping time. This is in contrast with the standard iterative solutions found in the literature, for which the algorithms typically converge asymptotically and without any guarantees regarding the maximum error of the final consensus value, under faulty environments. The algorithm suggested is evaluated in simulation, illustrating, in particular, the finite-time consensus property

Distributed Fault Detection Using Relative Information in Linear Multi-Agent Networks

This paper addresses the problem of fault detection in the context of a collection of agents performing a shared task and exchanging relative information over a communication network. We resort to techniques in the literature to construct a meaningful observable system and overcome the issue that the system of systems is not observable. A solution involving Set-Valued Observers (SVOs) is proposed to estimate the state in a distributed fashion and a proof of convergence of the estimates is given under mild assumptions. The performance of the proposed algorithm is assessed through simulations

Fault detection for LPV systems using Set-Valued Observers: A coprime factorization approach

This paper addresses the problem of fault detection for linear parameter-varying systems in the presence of measurement noise and exogenous disturbances. The applicability of current methods is limited in the sense that, to increase accuracy, the detection requires a large number of past measurements and the boundedness of the set-valued estimates is only guaranteed for stable systems. In order to widen the class of systems to be modeled and also to reduce the associated computational cost, the aforementioned issues must be addressed. A solution involving left-coprime factorization and deadbeat observers is proposed in order to reduce the required number of past measurements without compromising accuracy and allowing the design of Set-Valued Observers (SVOs) for fault detection of unstable systems by using the resulting stable subsystems of the coprime factorization. The algorithm is shown to produce bounded set-valued estimates and an example is provided. Performance is assessed through simulations, illustrating, in particular that small-magnitude faults (compared to exogenous disturbances) can be detected under mild assumptions