Introduction To Optimization Methods Part 1: Mathematical Review

The following is a review of some basic definitions, notations and relations from linear algebra, geometry, and calculus that will be used frequently throughout this book

The Creation and Propagation of Radiation: Fields Inside and Outside of Sources

We present a new algorithm for computing the electromagnetic fields of currents inside and outside of finite current sources, for arbitrary time variations in the currents. Unexpectedly, we find that our solutions for these fields are free of the concepts of differential calculus, in that our solutions only involve the currents and their time integrals, and do not involve the time derivatives of the currents. As examples, we give the solutions for two configurations of current: a planar solenoid and a rotating spherical shell carrying a uniform charge density. For slow time variations in the currents, we show that our general solutions reduce to the standard expressions for the fields in classic magnetic dipole radiation. In the limit of extremely fast turn-on of the currents, we show that for our general solutions the amount of energy radiated is exactly equal to the magnetic energy stored in the static fields a long time after current creation. We give three associated problem statements which can be used in courses at the undergraduate level, and one problem statement suitable for courses at the graduate level. These problems are of physical interest because: (1) they show that current systems of finite extent can radiate even during time intervals when the currents are constant; (2) they explicitly display transit time delays across a source associated with its finite dimensions; and (3) they allow students to see directly the origin of the reaction forces for time-varying systemsComment: 25 pages, 5 figure

The theory of the state: An economic perspective

The principle of laissez-faire, so closely associated with Adam Smith and the classical economists, should certainly not be considered an endorsement of anarchy as the ideal form of social order. Despite the theological overtones of divine providence in the imagery of the "invisible hand", Smith and his followers did not regard the market and the price mechanism as a spontaneous form of natural order that would prevail in any social group. Political organization in some form is necessary to provide the framework of law and order within which justice could be maintained and contracts enforced. Thus even one of their harshest critics, Thomas Carlyle, described their system not as anarchy, but as "anarchy plus the constable". The necessity of the "state" in the sense of the institution that claims a monopoly of the legitimate use of force over a given territory, or as Max Weber (1964, p. 154) defined it, for the proper functioning of "the market" and indeed of all forms of civilized human endeavor, can be traced back to the seminal influence of The Leviathan, the foundation of modern political thought laid by Thomas Hobbes (1651)

Tax-collection costs, public welfare and the predatory state

The collection of taxes, in any economic system, clearly requires the use of resources. In modern democratic states tax legislation is almost always controversial, and subject to extensive lobbying. In developing counties the wealthy often successfully avoid payment of taxes and the burden has to be borne by relatively impoverished rural classes, who are themselves not easy to tax directly because of poor record -keeping and difficulty of communications. In earlier times kings and princes often lacked the necessary means of direct taxation and were forced to rely on decentralized institutions such as feudalism. To convince the skeptical reader that the issue of tax-collection costs is neither trivial nor obvious, we pose the following question. What is the effect of greater efficiency in tax collection on the welfare of the tax-paying public? If the government is benign, taxing only to defray socially necessary public expenditure, a reduction in the costs of collecting these minimal taxes would clearly be a 'good thing'. What, however, if the state is inherently "predatory" in nature, as argued by Brennan and Buchanan (1980) and a number of others? In this case the state taxes not only to pay for public services but also to raise revenue for its own, possibly nefarious, purposes. Would an increase in the efficiency if tax-collection be undesirable under this alternative scenario

The Control of Discrete-Time Uncertain Dynamical Systems

In this project we use the second method of Lyapunov to develop several controllers to stabilize discrete-time dynamical systems with or without parameter uncertainties and/or external disturbances. We also use the notion of a sliding mode on a preferred hyperplane, previously developed for continuous-time variable structure control systems, to stabilize discrete- time dynamical systems. In particular, feedback controllers are proposed that: (i) stabilize discrete systems with no uncertainties by forcing their state trajectories onto prespecified hyperplanes; (ii) provide a needed level of stability robustness to discrete systems with uncertainties which are modeled by cone bounded functions; (iii) robustly stabilize discrete uncertain systems

State-feedback control of non-linear systems

A design method for state-feedback controllers for single-input non-linear systems is proposed. The method makes use of the transformations of the non-linear system into ‘controllable-like’ canonical forms. The resulting non-linear state feedback is designed in such a way that the eigenvalues of the linearized closed-loop model are invariant with respect to any constant operating point. The method constitutes an alternative approach to the design methodology recently proposed by Baumann and Rugh. Also a review of different transformation methods for non-linear systems is presented. An example and simulation results of different control strategies are provided to illustrate the design technique

The creation and propagation of radiation: Fields inside and outside of sources

We present an algorithm for computing the electromagnetic fields due to currents inside and outside of finite sources with a high degree of spatial symmetry for arbitrary time-dependent currents. The solutions for these fields do not involve the time derivatives of the currents but involve only the currents and their time integrals. We give solutions for moving planar sheets of charge, and a rotating spherical shell carrying a uniform charge density. We show that the general solutions reduce to the standard expressions for magnetic dipole radiation for slow time variations of the currents. If the currents are turned on very quickly, the general solutions show that the amount of energy radiated equals the magnetic energy stored in the static fields a long time after current creation. We give three problems which can be used in undergraduate courses and one problem suitable for graduate courses. These problems illustrate that because the generation of radiation depends on what has happened in the past, a system of currents can radiate even during time intervals when the currents are constant due to radiation associated with earlier acceleration

Control of Dynamic Systems via Neural Networks

This report is devoted to the problem of controlling a class of linear time-invariant dynamic systems via controllers based on additive neural network models. In particular, the tracking and stabilization problems are considered. First, we show how to transform the problem of tracking a reference signal by a control system into the stabilization problem. Then, some concepts from the variable structure control theory are utilized to construct stabilizing controllers. In order to facilitate the stability analysis of the closed-loop systems we employ a special state space transformation. This transformation allows us also to reveal connections between the proposed controllers and the additive neural network models