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This paper demonstrates continuous optimization on the differentiable manifold formed by continuous constraint functions. The first order tensor geodesic differential equation is solved on the manifold in both numerical and closed analytic form for simple nonlinear programs. Advantages and disadvantages with respect to conventional optimization techniques are discussed.
This paper demonstrates the capability to obtain and dynamically track eigenvalues and eigenvectors using continuous optimization on constraint manifolds.
Minimum average cost ordering policies for two product inventory systems with joint setup costs
Low thrust trajectories for minimum time rendezvous between continuous thrust interceptor and passive target vehicle, comparing calculus of variations and steepest ascent analyses
During the last decades thin film ferroelectrics have been investigated in great detail because of their versatility and advantages for tunable devices. Many reports on tunable components such as resonators, filters, oscillators and phase shifters have been published by researchers all around world. These efforts have shed light on the primary advantages as well as the current shortcomings of these materials that should be considered for the development of practical microwave devices and systems. In this paper, we intend to discuss the most relevant outcomes of current collaborations taking place at the NASA Glenn Research Center aimed at improving the quality of ferroelectric microwave components (e.g., phase shifters) via careful processing and evaluation of the materials properties of the thin films. To the extent that our applications allow, we have worked with fixed circuit configurations to determine which material modifications are of most impact to the performance of a given circuit. Examples on the improvements on the performance of these circuits will be provided.
Continuous parameter stochastic optimization principle using abstract variational theory
Stable high radiance in visible and near-ultraviolet wavelengths is desirable for radiometric calibration sources. In this work, newly available electrodeless radio-frequency (RF) driven plasma light sources were combined with research grade, low-noise power supplies and coupled to an integrating sphere to produce a uniform radiance source. The stock light sources consist of a 28 VDC power supply, RF driver, and a resonant RF cavity. The RF cavity includes a small bulb with a fill gas that is ionized by the electric field and emits light. This assembly is known as the emitter. The RF driver supplies a source of RF energy to the emitter. In commercial form, embedded electronics within the RF driver perform a continual optimization routine to maximize energy transfer to the emitter. This optimization routine continually varies the light output sinusoidally by approximately 2% over a several-second period. Modifying to eliminate this optimization eliminates the sinusoidal variation but allows the output to slowly drift over time. This drift can be minimized by allowing sufficient warm-up time to achieve thermal equilibrium. It was also found that supplying the RF driver with a low-noise source of DC electrical power improves the stability of the lamp output. Finally, coupling the light into an integrating sphere reduces the effect of spatial fluctuations, and decreases noise at the output port of the sphere.
A Proximate Time-Optimal Servo (PTOS) is developed, along with conditions for its stability. An algorithm is proposed for adapting the PTOS (APTOS) to improve performance in the face of uncertain plant parameters. Under ideal conditions APTOS is shown to be uniformly asymptotically stable. Simulation results demonstrate the predicted performance.
The paper examines some questions relating to the optimal design of continuous one-dimensional structures driven by harmonically oscillating loads. Optimal-control methods are applied to a cantilever bar driven sinusoidally by an axial force at its tip to illustrate the minimum-weight design of one-dimensional structures under dynamic excitation. Realistic constraints are imposed during the optimizations, including a maximum allowable stress amplitude at any point along the bar and a minimum cross-sectional area. It is shown that in the absence of damping, the design space may contain many disjoint feasible regions, and multiple optima can exist. Detailed solutions are obtained for continuous bars with an excitation frequency less than, and then greater than, the fundamental free-vibration frequency. It is found that above a certain excitation frequency, two or more arcs with different constraints characterize the optimal designs. It is concluded that when more than two different constrained arcs characterize the optimal solution, the continuum approach may be impractical, and finite-element approximations may offer the only alternative.
A continuation method is applied to a singular perturbation parameter to obtain a new numerical method for computing optimal trajectories. This method allows one to use simply calculated reduced order approximations as starting solutions and continue the perturbation parameter until the optimal full-order solution is obtained. The method does not require the calculation of higher order correction terms nor does it require the perturbation parameter to be small - thus, it has potentially superior convergence properties compared to conventional asymptotic expansions when the perturbation parameter is large. A simple trajectory optimization problem is considered to illustrate the method.
In contrast to binary filters, continuously variable optical filters offer an ability to conpensate for certain imperfections in the optics of a hybrid correlator. Arbitrary static phase errors are introduced into a model of a phase-only filtering hybrid correlator, and a method of discovering a correction for them simulated. By a recursive technique a first approximation to the impulse's matched filter is adjusted (allowed to relax) so as to produce successively more localized distribution of the output in the correlation plane. The method is motivated by the development of continuously-variable phase-only spatial light modulators, but it is applicable to amplitude modulators and -with appropriate modification -- to binary modulators as well. The technique is robust against the form of the system's departure from ideal behavior.
The effectiveness of viscous elements in introducing damping in a structure is a function of several variables including their number, their location in the structure, and their physical properties. In this paper, the optimal damper placement and tuning problem is posed to optimize these variables. Both discrete and continuous optimization problems are formulated and solved corresponding, respectively, to the problems of placement of passive elements and to the tuning of their parameters. The paper particularly emphasizes the critical computational issues resulting from the optimization formulations. Numerical results involving a lightly damped testbed structure are presented.
Abstract variational theory application to continuous parameter stochastic optimization problems to derive maximum principles in linear programming
The performance seeking control algorithm is designed to continuously optimize the performance of propulsion systems. The performance seeking control algorithm uses a nominal model of the propulsion system and estimates, in flight, the engine deviation parameters characterizing the engine deviations with respect to nominal conditions. In practice, because of measurement biases and/or model uncertainties, the estimated engine deviation parameters may not reflect the engine's actual off-nominal condition. This factor has a necessary impact on the overall performance seeking control scheme exacerbated by the open-loop character of the algorithm. The effects produced by unknown measurement biases over the estimation algorithm are evaluated. This evaluation allows for identification of the most critical measurements for application of the performance seeking control algorithm to an F100 engine. An equivalence relation between the biases and engine deviation parameters stems from an observability study; therefore, it is undecided whether the estimated engine deviation parameters represent the actual engine deviation or whether they simply reflect the measurement biases. A new algorithm, based on the engine's (steady-state) optimization model, is proposed and tested with flight data. When compared with previous Kalman filter schemes, based on local engine dynamic models, the new algorithm is easier to design and tune and it reduces the computational burden of the onboard computer.
The performance seeking control (PSC) algorithm is designed to continuously optimize the performance of propulsion systems. The PSC algorithm uses a nominal propulsion system model and estimates, in flight, the engine deviation parameters (EDPs) characterizing the engine deviations with respect to nominal conditions. In practice, because of measurement biases and/or model uncertainties, the estimated EDPs may not reflect the engine's actual off-nominal condition. This factor has a direct impact on the PSC scheme exacerbated by the open-loop character of the algorithm. In this paper, the effects produced by unknown measurement biases over the estimation algorithm are evaluated. This evaluation allows for identification of the most critical measurements for application of the PSC algorithm to an F100 engine. An equivalence relation between the biases and EDPs stems from the analysis; therefore, it is undecided whether the estimated EDPs represent the actual engine deviation or whether they simply reflect the measurement biases. A new algorithm, based on the engine's (steady-state) optimization model, is proposed and tested with flight data. When compared with previous Kalman filter schemes, based on local engine dynamic models, the new algorithm is easier to design and tune and it reduces the computational burden of the onboard computer.
Recent developments in spatial light modulators have led to devices which are capable of continuous phase modulation, even if only over a limited range. One of these devices, the deformable mirror device is used, to compare the relative merits of binary and partially-continuous phase filters in a specific problem of pattern recognition by optical correlation. Each filter was physically limited to only about a radiation of modulation. Researchers have predicted that for low input noise levels, continuous phase-only filters should have a higher absolute correlator peak output than the corresponding binary filters, as well as having a larger SNR. When continuous and binary filters were implemented on the DMD and they exhibited the same performance; an ad hoc filter optimization procedure was developed for use in the laboratory. The optimized continuous filter gave higher correlation peaks than did an independently optimized binary filter. Background behavior in the correlation plane was similar for the two filters, and thus the SNR showed the same improvement for the continuous filter. A phasor diagram analysis and computer simulation have explained part of the optimization procedure's success.
The sampled-data optimal linear regulator problem provides a means whereby a control designer can use an understanding of continuous optimal regulator design to produce a digital state variable feedback control law which satisfies continuous system performance specifications. A basic difficulty in applying the sampled-data regulator theory is the requirement that certain digital performance index weighting matrices, expressed as complicated functions of system matrices, be computed. Infinite series representations are presented for the weighting matrices of the time-invariant version of the optimal linear sampled-data regulator problem. Error bounds are given for estimating the effect of truncating the series expressions after a finite number of terms, and a method is described for their computer implementation. A numerical example is given to illustrate the results.
Research on optimization-based design is discussed. Illustrative examples are given for cases involving continuous optimization with discrete variables and optimization with tolerances. Approximation of computationally expensive and noisy functions, electromechanical actuator/control system design using decomposition and application of knowledge-based systems and optimization for the design of a valve anti-cavitation device are among the topics covered.