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A new technique for projecting high-dimensional data to low-dimensional spaces, called locally linear embedding (LLE) has recently been introduced.
The objective of this paper is to demonstrate the feasibility of a Nonlinear Generalized Predictive Control algorithm by showing real-time adaptive control on a plant with relatively fast time-constants. Generalized Predictive Control has classically been used in process control where linear control laws were formulated for plants with relatively slow time-constants. The plant of interest for this paper is a magnetic levitation device that is nonlinear and open-loop unstable. In this application, the reference model of the plant is a neural network that has an embedded nominal linear model in the network weights. The control based on the linear model provides initial stability at the beginning of network training. In using a neural network the control laws are nonlinear and online adaptation of the model is possible to capture unmodeled or time-varying dynamics. Newton-Raphson is the minimization algorithm. Newton-Raphson requires the calculation of the Hessian, but even with this computational expense the low iteration rate make this a viable algorithm for real-time control.
Space-filling curves is a popular approach based on a geometric embedding for linearizing computational meshes. We present a new O(n log n) combinatorial algorithm for constructing a self avoiding walk through a two dimensional mesh containing n triangles. We show that for hierarchical adaptive meshes, the algorithm can be locally adapted and easily parallelized by taking advantage of the regularity of the refinement rules. The proposed approach should be very useful in the runtime partitioning and load balancing of adaptive unstructured grids.
A document presents the concept of a curved telescope primary reflector structure, made mostly of silicon, that would have an areal mass density 1 kg/m2 and would be deployed in outer space, where it would be operated at a temperature in the cryogenic range. The concept provides for adjustment of the shape of the mirror to maintain the required precise optical surface figure despite the flexibility inherent in the ultra-lightweight design. The structure would include a thin mirror layer divided into hexagonal segments supported by flexure hinges on a lightweight two-layer backing structure. Each segment would also be supported at three points by sets of piezoelectric linear microactuators that could impose small displacements along the optical axis. The excitations applied to the aforementioned microactuators would be chosen to effect fine adjustments of the axial positions and the orientations of the segments relative to the supporting structure. Other piezoelectric linear microactuators embedded in the backing structure would enable control of the displacements of the segments along the hexagonal axes; they would also enable control of the curvature of the backing structure and, thus, additional control of the curvature of the reflector.
Differential dynamic logic is a formal framework to specify and reason about hybrid programs (HPs). The core of dL is a proof calculus that contains a collection of axioms and rules for the rigorous verification of properties of HPs. Recently, dL has been embedded within the theorem prover Prototype Verification System (PVS) resulting in the tool Plaidypvs2. The integration of dL into PVS expands its expressive power; user defined functions, such as trigonometric and other transcendental functions, can be used inside the dL framework, and meta-reasoning about HPs can be performed, including reasoning about entire classes of HPs, specified using dependent types in PVS. The differential temporal dynamic logic (dTL2) extends dL with temporal logic operators to reason about all the states reachable during the execution of an HP. This paper presents a work in progress focusing on embedding dTL2 in PVS as an extension of Plaidypvs. Plaidypvs is expanded with the formalization of a trace semantics for HPs, the definition of the LTL temporal operators eventually and globally, and the implementation of the proof calculus for dTL2. This new embedding has the same capabilties as Plaidypvs, which allows user defined functions and meta-reasoning of properties of HPs. To the best of the authors’ knowledge this is the first implementation of dTL2.
The equations of radiative transfer in anisotropically scattering media are reformulated as linear operator equations in a single independent variable. The resulting equations are suitable for solution by a variety of standard mathematical techniques. The operators appearing in the resulting equations are in general nonsymmetric; however, it is shown that every bounded linear operator equation can be embedded in a symmetric linear operator equation and a variational solution can be obtained in a straightforward way. For purposes of demonstration, a Rayleigh-Ritz variational method is applied to three problems involving simple phase functions. It is to be noted that the variational technique demonstrated is of general applicability and permits simple solutions for a wide range of otherwise difficult mathematical problems in physics.
Some recent results obtained by the authors are summarized concerning the stability of transversely isotropic flat panels whose materials exhibit a viscoelastic behavior and whose edges are subjected to in-plane biaxial compressive loads. Two transversely isotropic type materials, largely used in advanced technology, are considered: (1) the pyrolytic-graphite type, used in the thermal protection of aerospace vehicles, and (2) the type corresponding to unidirectional fiber-reinforced composites. In the former case, the planes of isotropy are parallel at each point to the midplane of the plate. In the latter case, they are normal to the fiber directions. The micromechanical relations developed by Aboudi (1984, 1986, 1987) are considered in conjunction with the correspondence principle of linear viscoelastic theory in order to predict the macroscopic viscoelastic properties of a material composed of uniaxial elastic fibers embedded in a linear viscoelastic matrix.
This study analyzes the stability behavior of unidirectional fiber-reinforced composite plates with viscoelastic material behavior subject to in-plane biaxial compressive edge loads. To predict the effective time-dependent material properties, elastic fibers embedded in a linearly viscoelastic matrix are examined. The micromechanical relations developed for a transversely isotropic medium are discussed along with the correspondence principle of linear viscoelasticity. It is concluded that the stability boundary obtained for a viscoelastic plate is lower (more critical) than its elastic counterpart, and the transverse shear deformation effects are more pronounced in viscoelastic plates than in their elastic counterparts.
This note follows up on a recent study by Shie et al. (2005) and extends the investigation of the domain-averaged moisture-temperature (Q-T) relationship from the Tropics (i.e., the previous study) to the tropical Pacific, Atlantic and Indian Oceans. The Q and T data examined in this study are obtained from the GEOS-3 [Goddard Earth Observing System Version-3] global re-analysis monthly products. Similar to what was found earlier in the Tropics, Q is also found to increase with T over the entire oceanic region; however, Q increases faster with T over oceans than over the Tropics. The Q-T distribution for the Tropics is in a quasi-linear relationship, which is embedded in a global Q-T distribution that is, however, in a more complex curvilinear relationship. The Q-T distribution over the oceanic regions seems to fall within the lower bound (ie., the relatively colder and driver regime) of the tropical Q-T distribution. T over oceans is also found increasing with SST (sea surface temperature), which seemingly implies that an air mass might have gained heat more readily from a warmer ocean as compared to a colder ocean. Q is also found to increase with SST in a manner that quantitatively resembles an earlier finding by Stevens (1990). We also found that relative humidity exhibits similar behaviors for oceanic and tropical regions, respectively, i.e., it increases with both SST and T over oceans and increases with T in the Tropics (Shie et al. 2005). All these similar features found between oceanic and tropical regions seem to inform us that oceans occupy most of the Tropics and so play a key role in determining what have happened in the Tropics.
This paper describes a methodology for the extraction of a linear time invariant model from a nonlinear helicopter model, and followed by an examination of the interactions of the Higher Harmonic Control (HHC) and the Automatic Flight Control System (AFCS). This new method includes an embedded harmonic analyzer inside a linear time invariant model, which allows the periodicity of the helicopter response to be captured. The: coupled high-order model provides the needed level of dynamic fidelity to permit an analysis and optimization of the AFCS and HHC loops. Results of this study indicate that the closed-loop HHC system has little influence on the AFCS or on the vehicle handling qualities, which indicates that the AFCS does not need modification to work with the HHC system. The results also show that the vibration response to maneuvers must be considered during the HHC design process, which leads to much higher required HHC loop crossover frequencies. This research also demonstrates that the transient vibration response during maneuvers can be reduced by optimizing the closed-loop higher harmonic control laws using conventional control system analyses.
The problem of determining unsteady airloads on a thin, three-dimensional, planar wing oscillating with infinitesimal amplitude in a transonic flow is considered. The flow is assumed to be governed by the transonic small disturbance equation. The unsteady disturbance is taken to be a small perturbation superposed on a given steady mean flowfield. The equations governing the unsteady field, allowing for induced oscillations of any embedded shocks, are obtained. The linearization is shown to fail, locally, at the intersection of a shock with the wing surface, although the failure has little influence on the sectional characteristics of the wing.
The Green's Function Method for linearized 3D unsteady potential flow (embedded in the computer code SOUSSA P) is extended to include the time-domain analysis as well as the nonlinear term retained in the transonic small disturbance equation. The differential-delay equations in time, as obtained by applying the Green's Function Method (in a generalized sense) and the finite-element technique to the transonic equation, are solved directly in the time domain. Comparisons are made with both linearized frequency-domain calculations and existing nonlinear results.
Each supersonic jet contains a long segment of transonic flow, and it is known through acoustic experiments that the strongest noise production region is embedded in this segment. The linearized convected wave equation breaks down in the transonic regime and a nonlinear equation must be used. Amplitude, directivity, and the transonic similarity rule for noise radiation can be established through proper matching of the displacement amplitude between the source and the wave field. This paper is a first step towards establishing a general theory of transonic aeroacoustics.
A steady, axisymmetric, quasi-radial, global model is developed for thermally driven stellar winds with embedded magnetic fields. The asymptotic, linear results are presented for 0(1) latitudinal variations in the radial magnetic field, mass-loss rate, and radial velocity of the wind. The MHD equations are solved for the latitudinal dependence of the rotational velocity and magnetic field. They are driven by the meridional flows that develop naturally from internal magnetic stresses. Most flows open flux tubes in the stars equatorial plane, redistributing mass and magnetic flux as a function of stellar latitude. The plasma spins up to conserve angular momentum in fields and plasma.
The effect of a parasitic wire array on the scan admittance of a slot array has been investigated. Structures considered can consist of an infinite slot array and an arbitrary number of parasitic infinte arrays of piecewise linear wires, all arrays being embedded in a stratified medium. These include, as particular cases, phased arrays of Clavin elements. Expressing the fields from the arrays as plane waves, a procedure similar to the periodic moment method for infinite periodic structures is set up to obtain the the scan admittance of the slot array. Scan admittances are presented for a slot array with monopole arrays in free space, and a slot array with a tilted dipole array in a stratified medium. Blind spots at which the incident energy is mainly reflected rather than transmitted were found. Results obtained indicate the possibility of using parasitic wire arrays for scan compensation of active slot arrays.
There are many real-world problems whose mathematical models turn out to be linear systems Ax = b , where A is an m by x n matrix. Each equation of the linear system is an information. An information, in a physical problem, such as 4 mangoes, 6 bananas, and 5 oranges cost $10, is mathematically modeled as 4x(sub 1) + 6x(sub 2) + 5x (sub 3) = 10, where x(sub 1), x(sub 2), x(sub 3) are each cost of one mango, that of one banana, and that of one orange, respectively. All the information put together in a specified context, constitutes the physical problem and need not be all distinct. Some of these could be redundant, which cannot be readily identified by inspection. The resulting mathematical model will thus have equations corresponding to this redundant information and hence are linearly dependent and thus superfluous. Consequently, these equations once identified should be better pruned in the process of solving the system. The benefits are (i) less computation and hence less error and consequently a better quality of solution and (ii) reduced storage requirements. In literature, the pruning concept is not in vogue so far although it is most desirable. In a numerical linear system, the system could be slightly inconsistent or inconsistent of varying degree. If the system is too inconsistent, then we should fall back on to the physical problem (PP), check the correctness of the PP derived from the material universe, modify it, if necessary, and then check the corresponding mathematical model (MM) and correct it. In nature/material universe, inconsistency is completely nonexistent. If the MM becomes inconsistent, it could be due to error introduced by the concerned measuring device and/or due to assumptions made on the PP to obtain an MM which is relatively easily solvable or simply due to human error. No measuring device can usually measure a quantity with an accuracy greater that 0.005% or, equivalently with a relative error less than 0.005%. Hence measurement error is unavoidable in a numerical linear system when the quantities are continuous (or even discrete with extremely large number). Assumptions, though not desirable, are usually made when we find the problem sufficiently difficult to be solved within the available means/tools/resources and hence distort the PP and the corresponding MM. The error thus introduced in the system could (not always necessarily though) make the system somewhat inconsistent. If the inconsistency (contradiction) is too much then one should definitely not proceed to solve the system in terms of getting a least-squares solution or a minimum norm solution or the minimum-norm least-squares solution. All these solutions will be invariably of no real-world use. If, on the other hand, inconsistency is reasonably low, i.e. the system is near-consistent or, equivalently, has near-linearly-dependent rows, then the foregoing solutions are useful. Pruning in such a near-consistent system should be performed based on the desired accuracy and on the definition of near-linear dependence. In this article, we discuss pruning over various kinds of linear systems and strongly suggest its use as a pre-processor or as a part of an algorithm. Ideally pruning should (i) be a part of the solution process (algorithm) of the system, (ii) reduce both computational error and complexity of the process, and (iii) take into account the numerical zero defined in the context. These are precisely what we achieve through our proposed O(mn2) algorithm presented in Matlab, that uses a subprogram of solving a single linear equation and that has embedded in it the pruning.
There are many real-world problems whose mathematical models turn out to be linear systems Ax = b, where A is an m x n matrix. Each equation of the linear system is an information. An information, in a physical problem, such as 4 mangoes, 6 bananas, and 5 oranges cost $10, is mathematically modeled as an equation 4x(sub 1) + 6x(sub 2) + 5x(sub 3) = 10 , where x(sub 1), x(sub 2), x(sub 3) are each cost of one mango, that of one banana, and that of one orange, respectively. All the information put together in a specified context, constitutes the physical problem and need not be all distinct. Some of these could be redundant, which cannot be readily identified by inspection. The resulting mathematical model will thus have equations corresponding to this redundant information and hence are linearly dependent and thus superfluous. Consequently, these equations once identified should be better pruned in the process of solving the system. The benefits are (i) less computation and hence less error and consequently a better quality of solution and (ii) reduced storage requirements. In literature, the pruning concept is not in vogue so far although it is most desirable. It is assumed that at least one information, i.e. one equation is known to be correct and which will be our first equation. In a numerical linear system, the system could be slightly inconsistent or inconsistent of varying degree. If the system is too inconsistent, then we should fall back on to the physical problem (PP), check the correctness of the PP derived from the material universe, modify it, if necessary, and then check the corresponding mathematical model (MM) and correct it. In nature/material universe, inconsistency is completely nonexistent. If the MM becomes inconsistent, it could be due to error introduced by the concerned measuring device and/or due to assumptions made on the PP to obtain an MM which is relatively easily solvable or simply due to human error. No measuring device can usually measure a quantity with an accuracy greater that 0.005% or, equivalently with a relative error less than 0.005%. Hence measurement error is unavoidable in a numerical linear system when the quantities are continuous (or even discrete with extremely large number). Assumptions, though not desirable, are usually made when we find the problem sufficiently difficult to be solved within the available means/tools/resources and hence distort the PP and the corresponding MM. The . error thus introduced in the system could (not always necessarily though) make the system somewhat inconsistent. If the inconsistency (contradiction) is too much then one should definitely not proceed to solve the system in terms of getting a least-squares solution or the minimum-norm least-squares solution. All these solutions will be invariably of no real-world use. If, on the other hand, inconsistency is reasonably low, i.e. the system is near-consistent or, equivalently, has near-linearly-dependent rows, then the foregoing solutions are useful. Pruning in such a near-consistent system should be performed based on the desired accuracy and on the definition of near-linear dependence. In this article, we discuss pruning over various kinds of linear systems and strongly suggest its use as a pre-processor or as a part of an algorithm. Ideally pruning should (i) be a part of the solution process (algorithm) of the system, (ii) reduce both computational error and complexity of the process, and (iii) take into account the numerical zero defined in the context. These are precisely what we achieve through our proposed O(mn2) algorithm presented in Matlab, that uses a subprogram of solving a single linear equation and that has embedded in it the pruning.
The design, fabrication, and performance of K-band high-efficiency, linear power pHEMT amplifiers implemented in Embedded Transmission Line (ETL) MMIC configuration with unthinned GaAs substrate and topside grounding are reported. A three-stage amplifier achieved a power-added efficiency of 40.5% with 264 mW output at 20.2 GHz. The linear gain is 28.5 dB with 1-dB gain compression output power of 200 mW and 31% power-added efficiency. The carrier-to-third-order intermodulation ratio is approx. 20 dBc at the 1-dB compression point. A RF functional yield of more than 90% has been achieved.