Optimal Control and Spatial Heterogeneity: Pattern Formation in Economic-Ecological Models

This paper extends Turing analysis to standard recursive optimal control frameworks in economics and applies it to dynamic bioeconomic problems where the interaction of coupled economic and ecological dynamics under optimal control over space creates (or destroys) spatial heterogeneity. We show how our approach reduces the analysis to a tractable extension of linearization methods applied to the spatial analog of the well known costate/state dynamics. We explicitly show the existence of a non-empty Turing space of diffusive instability by developing a linear-quadratic approximation of the original non-linear problem. We apply our method to a bioeconomic problem, but the method has more general economic applications where spatial considerations and pattern formation are important. We believe that the extension of Turing analysis and the theory associated with the dispersion relationship to recursive infinite horizon optimal control settings is new.Spatial analysis, Pattern formation, Turing mechanism, Turing space, Pontryagin’s principle, Bioeconomics

Optimal control and identification of stochastic systems using differential game theory

This dissertation deals with linear systems subjected to stochastic disturbances. The class of stochastic processes considered is the class of second order stochastic processes characterized by having finite continuous covariance. The properties of the covariance provide means to formulate optimization problems without the difficulties present when the covariance is not finite or continuous. The first aspect studied was several classes of optimal control problems. The effects of the stochastic processes were approximated by the effects of its first two moments. This procedure resulted in allowing optimal system controls to be found whatever the first two moments of the stochastic input were, or worst case optimal controls were found. Differential game theory was used to solve the worst case problem. Then, a model reference adaptive control system was employed to permit simultaneous parameter identification and control to be obtained in an on-line environment. The parameter identification was accomplished using gradient or steepest descent techniques. The control inputs were updated as the parameters were changed yielding sub-optimal control of the physical system. In addition, minimum error covariance estimation of linear systems with second order stochastic disturbances was developed

Meshless methods for shear-deformable beams and plates based on mixed weak forms

Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin plate theories have seen wide use throughout engineering practice to simulate the response of structures with planar dimensions far larger than their thickness dimension. Meshless methods have been applied to construct numerical methods to solve the shear deformable theories. Similarly to the finite element method, meshless methods must be carefully designed to overcome the well-known shear-locking problem. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed weak form. In the mixed weak form the shear stresses are treated as an independent variational quantity in addition to the usual displacement variables. We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam problem that converges to the stable first-order/zero-order finite element method in the local limit when using maximum entropy meshless basis functions. The resulting formulation is free from the effects shear-locking. We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking. Finally we consider the construction of a generalised displacement method where the shear stresses are eliminated prior to the solution of the final linear system of equations. We implement an existing technique in the literature for the Stokes problem called the nodal volume averaging technique. To ensure stability we split the shear energy between a part calculated using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from the effects of shear-locking.Open Acces

Exploration of a Scalable Holomorphic Embedding Method Formulation for Power System Analysis Applications

abstract: The holomorphic embedding method (HEM) applied to the power-flow problem (HEPF) has been used in the past to obtain the voltages and flows for power systems. The incentives for using this method over the traditional Newton-Raphson based nu-merical methods lie in the claim that the method is theoretically guaranteed to converge to the operable solution, if one exists. In this report, HEPF will be used for two power system analysis purposes: a. Estimating the saddle-node bifurcation point (SNBP) of a system b. Developing reduced-order network equivalents for distribution systems. Typically, the continuation power flow (CPF) is used to estimate the SNBP of a system, which involves solving multiple power-flow problems. One of the advantages of HEPF is that the solution is obtained as an analytical expression of the embedding parameter, and using this property, three of the proposed HEPF-based methods can es-timate the SNBP of a given power system without solving multiple power-flow prob-lems (if generator VAr limits are ignored). If VAr limits are considered, the mathemat-ical representation of the power-flow problem changes and thus an iterative process would have to be performed in order to estimate the SNBP of the system. This would typically still require fewer power-flow problems to be solved than CPF in order to estimate the SNBP. Another proposed application is to develop reduced order network equivalents for radial distribution networks that retain the nonlinearities of the eliminated portion of the network and hence remain more accurate than traditional Ward-type reductions (which linearize about the given operating point) when the operating condition changes. Different ways of accelerating the convergence of the power series obtained as a part of HEPF, are explored and it is shown that the eta method is the most efficient of all methods tested. The local-measurement-based methods of estimating the SNBP are studied. Non-linear Thévenin-like networks as well as multi-bus networks are built using model data to estimate the SNBP and it is shown that the structure of these networks can be made arbitrary by appropriately modifying the nonlinear current injections, which can sim-plify the process of building such networks from measurements.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Numerical optimization of isolation systems for reciprocating engines

The use of numerical optimization methods to select reciprocating engine anti-vibration characteristics is investigated. A rigid body power train model coupled through an arbitrary array of vibration isolators to a rigid supporting structure forms the basis of the dynamic model. By calculating the forced response of the power train to its internally generated excitation, the strain energy summed over the isolators may be determined. This energy, which is indicative of the efficiency of the vibration isolative mounts, is used as the objective function in the optimization procedure. The method is expected to be useful in preliminary design studies of front wheel drive vehicles where traditional methods of mounting automotive engines are not necessarily applicable. [Continues.

Optimal design of unmodeled linear systems using control-based continuation

This thesis describes the use of control-based continuation for design optimization, in the presence of constraints and without access to a model, of the response of a linear system to harmonic input. A proof of concept of this paradigm is presented in the context of an armature-controlled DC motor. Specifically, three design problems are formulated with the objective function equal to the maximum angular velocity response to a harmonic torque disturbance, and a constraint that is imposed on each of three distinct stability margins, respectively. The analysis shows that the simulation model for the DC motor may be treated analogously to an actual experiment with all information drawn from real-time measurements, rather than from the model itself. The control-based continuation paradigm is formulated in terms of a non-invasive, yet locally stabilizing control scheme, which can be tuned to accelerate convergence to the steady state response. The numerical analysis uses the matlab-compatible continuation platform coco to determine the implicit relationship between model parameters that results from the constraint, and to evaluate the objective function along the corresponding constraint manifold. A comparison between a scheme that relies on finite differences for approximating the problem Jacobian and an algorithm based on the Broyden update is also included

Evaluating Projects and Assessing Sustainable Development in Imperfect Economies

We are interested in three related questions: (1) How should accounting prices be estimated? (2) How should we evaluate policy change in an imperfect economy? (3) How can we check whether intergenerational well-being will be sustained along a projected economic programme? We do not presume that the economy is convex, nor do we assume that the government optimizes on behalf of its citizens. We show that the same set of accounting prices should be used both for policy evaluation and for assessing whether or not intergenerational welfare along a given economic path will be sustained. We also show that a comprehensive measure of wealth, computed in terms of the accounting prices, can be used as an index for problems (2) and (3) above. The remainder of the paper is concerned with rules for estimating the accounting prices of several specific environmental natural resources, transacted in a few well known economic institutions.Sustainable development, Imperfect economies

Mean Performance Optimization of an Orbiting Distributed Aperture by Warped Aperture Image Plane Comparisons

This work models the aggregate performance of satellite receiver formations functioning as orbiting interferometers as compared to filled apertures of similar geometries. These models facilitate selecting initial conditions for formations such that their control-free dynamics yield interferometry performance with minimal errors as compared to the filled apertures. The solution method draws on the dynamic models of an orbiting planar satellite formation to define the size and shape of a reference aperture and to define the degrees of freedom for the formation members. The paths of formation elements yield geometries for which the aggregate performance of the array of discrete receivers may be calculated. The objective of the optimization process is therefore minimizing the time-average square of the difference between the filled aperture\u27s intensity map and that generated by the discrete receiver array. This yields a formation whose configuration offers minimum errors for imaging processes beginning at any arbitrary start time. The problem as posed is non-convex, and requires implementation of a global search method. Genetic algorithms are used. The solution method includes a new analytic solution for the intensity map of an elliptical aperture and a technique for generalizing this solution to include the effect of non-ideal viewing geometries