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A mathematical formulation for the control of the Spacecraft Control Laboratory Experiment (SCOLE) configuration is given. Two equivalent approaches, one using a functional equation and the other an abstract wave equation, are illustrated. Such a formulation can help in digital computer simulation to evaluate control laws, providing insight, and generating control laws.
This paper formulates the general methodology for estimating the bias error distribution of a device in a measuring domain from less accurate measurements when a minimal number of standard values (typically two values) are available. A new perspective is that the bias error distribution can be found as a solution of an intrinsic functional equation in a domain. Based on this theory, the scaling- and translation-based methods for determining the bias error distribution arc developed. These methods are virtually applicable to any device as long as the bias error distribution of the device can be sufficiently described by a power series (a polynomial) or a Fourier series in a domain. These methods have been validated through computational simulations and laboratory calibration experiments for a number of different devices.
In this study, we report on the evolution of particle size distributions (PSDs) and habits as measured during slow, Lagrangian-type spiral descents through deep subtropical and tropical cloud layers in Florida, Brazil, and Kwajalein, Marshall Islands, most of which were precipitating. The objective of the flight patterns was to learn more about how the PSDs evolved in the vertical and to obtain information of the vertical structure of microphysical properties. New instrumentation yielding better information on the concentrations of particles in the size (D) range between 0.2 and 2 cm, as well as improved particle imagery, produced more comprehensive observations for tropical stratiform precipitation regions and anvils than have been available previously. Collocated radar observations provided additional information on the vertical structure of the cloud layers sampled. Most of the spirals began at cloud top, with temperatures (T) as low as -50 C, and ended at cloud base or below the melting layer (ML). The PSDs broadened from cloud top towards cloud base, with the largest particles increasing in size from several millimeters at cloud top to one centimeter or larger towards cloud base. Some continued growth was noted in the upper part of the ML. Concentrations of particles less than 1 mm in size decreased with decreasing height. The result was a consistent change in the PSDs in the vertical. Similarly, systematic changes in the size dependence of the particle cross-sectional area was noted with decreasing height. Aggregation-as ascertained from both the changes in the PSDs and evolution of particle habits as observed in high detail with the cloud particle imager (CPI) probe-was responsible for these trends. The PSDs were generally well-represented by gamma distributions of the form N = N0 gamma D microns e- lambda gamma D that were fitted to the PSDs over 1-km horizontal intervals throughout the spirals. The intercept (N0 gamma), slope (lambda gamma), and dispersion (microns) values were derived for each PSD. Exponential curves (N = N0e- lambdaD; micron = 0) were also fitted to the distributions. The lambda gamma values for given spirals varied systematically with temperature as did the values of lambda (exponential), and the data generally conformed to values found in previous studies involving exponential fits to size distributions in mid-latitude frontal and cirrus layers. Considerable variability often noted in the PSD properties during the loops of individual spirals was manifested primarily in large changes in N0 gamma and N0, but micron, lambda gamma and lambda remained fairly stable. Temperature is not found to be the sole factor controlling lambda gamma or lambda but is a primary one. Direct relationships were found between lambda gamma and N0 gamma or lambda gamma and micron for the gamma distributions and lambda and N0 for the exponential. The latter relationship was not found as distinctly in earlier studies; observed PSDs in this study had better fidelity with less scatter. The micron values changed monotonically with T over the range of temperatures and were directly related to N0 gamma or lambda gamma, thereby reducing the number of variables in the PSD functional equation to two. In the upper part of the ML, N0 and lambda continued to decrease, and in the lower part these values began to increase as the largest particles melted. We developed general expressions relating various bulk microphysical, radar, and radiative transfer-related variables to N0 gamma and lambda gamma, useful for both tropical and mid-latitude clouds. These relationships facilitate the specification of a number of bulk properties in cloud and climate models. The results presented in this paper apply best to temperatures between 0 and -40 C, for which the measured radar reflectivities fall in the range of 0 to 25 dBZe.
An improved methodology for designing feedback control systems has been developed based on systematically shaping the loop gain of the system to meet performance requirements such as stability margins, disturbance attenuation, and transient response, while taking into account the actuation system limitations such as actuation rates and range. Loop-shaping for controls design is not new, but past techniques do not directly address how to systematically design the controller to maximize its performance. As a result, classical feedback control systems are designed predominantly using ad hoc control design approaches such as proportional integral derivative (PID), normally satisfied when a workable solution is achieved, without a good understanding of how to maximize the effectiveness of the control design in terms of competing performance requirements, in relation to the limitations of the plant design. The conception of this improved methodology was motivated by challenges in designing control systems of the types needed for supersonic propulsion. But the methodology is generally applicable to any classical control-system design where the transfer function of the plant is known or can be evaluated. In the case of a supersonic aerospace vehicle, a major challenge is to design the system to attenuate anticipated external and internal disturbances, using such actuators as fuel injectors and valves, bypass doors, and ramps, all of which are subject to limitations in actuator response, rates, and ranges. Also, for supersonic vehicles, with long slim type of structures, coupling between the engine and the structural dynamics can produce undesirable effects that could adversely affect vehicle stability and ride quality. In order to design distributed controls that can suppress these potential adverse effects, within the full capabilities of the actuation system, it is important to employ a systematic control design methodology such as this that can maximize the effectiveness of the control design in a methodical and quantifiable way. The emphasis is in generating simple but rather powerful design techniques that will allow even designers with a layman s knowledge in controls to develop effective feedback control designs. Unlike conventional ad hoc methodologies of feedback control design, in this approach actuator rates are incorporated into the design right from the start: The relation between actuator speeds and the desired control bandwidth of the system is established explicitly. The technique developed is demonstrated via design examples in a step-by-step tutorial way. Given the actuation system rates and range limits together with design specifications in terms of stability margins, disturbance rejection, and transient response, the procedure involves designing the feedback loop gain to meet the requirements and maximizing the control system effectiveness, without exceeding the actuation system limits and saturating the controller. Then knowing the plant transfer function, the procedure involves designing the controller so that the controller transfer function together with the plant transfer function equate to the designed loop gain. The technique also shows what the limitations of the controller design are and how to trade competing design requirements such as stability margins and disturbance rejection. Finally, the technique is contrasted against other more familiar control design techniques, like PID control, to show its advantages.
A neutral functional differential equation is defined as d/dt(Dx sub t) = f(x sub t), with D linear, continuous, atomic at zero. The solution generally is no smoother than the initial data after any finite number of steps. A more restrictive class of D-operators for which some smoothing takes place after an infinite number of steps is defined. This result indicates that a solution can be in an omega-limit set only if it corresponds to initial data which are smooth. A space which can be considered as a Banach space with the topology of uniform convergence is also defined.
Two methods are established for identifying constant-coefficient, C to the 2n power type of noise-free linear systems if the time response data of the input-output or of all states are known. 2n response data are required to identify an nth-order transfer function or state equation for an unknown linear system. The order of the unknown system can be identified by checking a sequence of determinants. The Z transform and its inversion are mainly used.
A new stream function formulation is developed for the solution of Euler's equations in the transonic flow region. The stream function and the density are the dependent variables in this method, while the governing equations for adiabatic flow are the momentum equations which are solved in the strong conservation law form. The application of this method does not require a knowledge of the vorticity. The algorithm is combined with the automatic grid solver (GRAPE) of Steger and Sorenson (1979) in order to study arbitrary geometries. Results of the application of this method are presented for the NACA 0012 airfoil at various Mach numbers and angles of attack, and cylinders. In addition, detailed comparisons are made with other solutions of the Euler equations.
For lidar systems with long laser pulses the unusual behavior of the near-range signals causes serious difficulties and large errors in reduction. The commonly used lidar equation is no longer applicable since the convolution of the laser pulse with the atmospheric parameter distributions should be taken into account. It is important to give more insight into this problem and find the solution techniques. Starting from the original equation, a general form is suggested for the single scattering lidar equation where a correction function Cr is introduced. The correction Function Cr(R) derived from the original equation indicates the departure from the normal lidar equation. Examples of Cr(R) for a coaxial CO2 lidar system are presented. The Differential Absorption Lidar (DIAL) errors caused by the differences of Cr(R) for H2O measurements are plotted against height.
Solution of differential equations by series of exponential functions
Liapunov functions from auxiliary exact difference equations for nonlinear difference equation stability analysis
Difference equations and Fourier constants for solving general periodic functions problems
A discrete probability function (DPF) method for the equation of radiative transfer is derived. The DPF is defined as the integral of the probability density function (PDF) over a discrete interval. The derivation allows the evaluation of the PDF of intensities leaving desired radiation paths including turbulence-radiation interactions without the use of computer intensive stochastic methods. The DPF method has a distinct advantage over conventional PDF methods since the creation of a partial differential equation from the equation of transfer is avoided. Further, convergence of all moments of intensity is guaranteed at the basic level of simulation unlike the stochastic method where the number of realizations for convergence of higher order moments increases rapidly. The DPF method is described for a representative path with approximately integral-length scale-sized spatial discretization. The results show good agreement with measurements in a propylene/air flame except for the effects of intermittency resulting from highly correlated realizations. The method can be extended to the treatment of spatial correlations as described in the Appendix. However, information regarding spatial correlations in turbulent flames is needed prior to the execution of this extension.
Moment equations that model plasma transport require an ansatz distribution function to close the system of equations. The resulting transport is sensitive to the specific closure used, and several options have been proposed in the literature. Two different 8-moment distribution functions can be generalized to form a single-parameter family of distribution functions. The transport coefficients resulting from this generalized distribution function can be expressed in terms of this free parameter. This provides the flexibility of matching the 8-moment model to some validating result at a given magnetization value, such as Braginskii’s transport, or the more recent results of Davies et al. [Physics of Plasma, 28, 012305 (2021)]. Here, this process can be thought of as solving for the 8-moment distribution function that matches the value of a transport coefficient given by a Chapman-Enskog expansion, while retaining the improved physical properties, such as finite propagation speeds and time dependence which belong to the hyperbolic moment models. Since the presented generalized distribution function only has a single free parameter, only a single transport coefficient can be matched at a time. However, this generalization process may be extended to provide multiple free parameters. The focus of this Brief Communication is on the dramatically improved thermal conductivity of the proposed model compared to the two base moment models.
Differential-integral equation for dissipation function and derivation of conservation relationship connecting reflection, transmission and dissipation functions of finite slab
An equation of state for material in stellar envelopes, subject to the limits of temperature less than about 10 to the 7th K and density less than about .01 g/cu cm is presented. The equation makes it possible to express free energy as the sum of several terms representing effects such as partial degeneracy of the electron, Coulomb interactions among charged particles, finite-volume, hard sphere repulsion, and van der Waals attraction. An occupation probability formalism is used to represent the effects of the plasma in establishing a finite partition function. It is shown that the use of the static screened Coulomb potential to calculate level shifts and to estimate the cutoff of the internal partition function is invalid. For most of the parameter space relevant to stellar envelopes, perturbations arising from the plasma ions are shown to be dominant in establishing the internal partition function.
A generalised kinetic equation for the equilibrium distribution function in a finite beta, arbitrary tokamak plasma is derived. The equation is correct to second order in ρ/L (ρ is the particle Larmor radius and L is the system size). Resolving finite Larmor radius length scales with no restriction on the ratio of poloidal to total equilibrium magnetic field, B $\vartheta$ /B, it generalises the drift kinetic theory of [Hazeltine Phys. Plasmas 15 (1973) 77] to the limit of B $\vartheta$ /B ~ 1 (e.g. to ensure validity for spherical tokamaks). Two cases are considered. The first provides the equilibrium distribution function, consistent with the generalised gyrokinetic formalism of [Dudkovskaia et al. Plasma Phys. Control. Fusion 65 (2023) 045010], derived specifically to capture neoclassical equilibrium currents in gyrokinetic stability analyses in strong gradient regions. The second assumes short length scales in the direction perpendicular to the magnetic field, which can occur as a result of small coherent magnetic structures in the plasma, such as neoclassical tearing mode magnetic islands close to threshold. This then extends the drift island equations of [Dudkovskaia et al. Nucl. Fusion 63 (2023) 016020] for the plasma response to magnetic islands to a spherical tokamak plasma configuration. In conclusion, resolving ρ ~ ρ $\vartheta$ (or B $\vartheta$ ~ B), where ρ $\vartheta$ is the particle poloidal Larmor radius, is also expected to influence calculations of the magnetic island propagation frequency and the associated contributions to the island onset conditions.
A perturbed functional iteration has been developed to solve nonlinear systems. It adds at each iteration level, unique perturbation parameters to nonlinear Gauss-Seidel iterates which enhances its convergence properties. As convergence is approached these parameters are damped out. Local linearization along the diagonal has been used to compute these parameters. The method requires no computation of Jacobian or factorization of matrices. Analysis of convergence depends on properties of certain contraction-type mappings, known as D-mappings. In this article, application of this method to solve an implicit finite difference approximation of Euler's equation is studied. Some representative results for the well known shock tube problem and compressible flows in a nozzle are given.
Simulation tools for the large deployable reflector (LDR) are discussed. These tools are often the transfer function variety equations. However, transfer functions are inadequate to represent time-varying systems for multiple control systems with overlapping bandwidths characterized by multi-input, multi-output features. Frequency domain approaches are the useful design tools, but a full-up simulation is needed. Because of the need for a dedicated computer for high frequency multi degree of freedom components encountered, non-real time smulation is preferred. Large numerical analysis software programs are useful only to receive inputs and provide output to the next block, and should be kept out of the direct loop of simulation. The following blocks make up the simulation. The thermal model block is a classical heat transfer program. It is a non-steady state program. The quasistatic block deals with problems associated with rigid body control of reflector segments. The steady state block assembles data into equations of motion and dynamics. A differential raytrace is obtained to establish a change in wave aberrations. The observation scene is described. The focal plane module converts the photon intensity impinging on it into electron streams or into permanent film records.