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Partial differential equations finite difference approximation, applying convergence theorems with probabilistic method
Poiseuille pipe flow stability from finite difference equations approximation to nonlinear axisymmetric Navier-Stokes equations under stream function perturbation
Multistage two player zero-sum games with state determined by difference equations, deriving optimal strategy pair satisfying saddle point condition
Digital phase locked loop for FM signals demodulation, considering system nonlinear difference equation
A digital simulation of the continuous error of the localized beam of a conventional instrument landing system is discussed. The digital simulation was developed during the analysis of space shuttle navigation capabilities. A discrete mathematical model for use on a digital computer is described. The model generates an output random sequence which is equivalent, for simulation purposes, to the desired random process. The model is a system of difference equations driven by a zero-mean Gaussian random sequence.
A control model describing a network flow hindered by multiple pure time (or transport) delays is formulated. Feedbacks connect each desired output with a single control sector situated at the origin. The dynamic formulation invokes the use of differential difference equations. This causes the characteristic equation of the model to consist of transcendental functions instead of a common algebraic polynomial. A general graphical criterion is developed to evaluate the stability of such a problem. A digital computer simulation confirms the validity of such criterion. An optimal decision making process with multiple delays is presented.
Demonstration that approximate solutions to certain classes of differential and difference equations can be expressed in form of finite state machines. Based on this result, a finite-state machine model of an adaptive gain changer in an aircraft stability augmentation system is developed. Results of simulated flights using the finite-state machine gain changer are presented.
Investigation of the feasibility of using a Kalman filter estimator for the identification of unknown parameters in nonlinear dynamic systems with a time delay. The problem considered is the application of estimation theory to determine the parameters of a family of pilot models containing delayed states. In particular, the pilot-plant dynamics are described by differential-difference equations of the retarded type. The pilot delay, included as one of the unknown parameters to be determined, is kept in pure form as opposed to the Pade approximations generally used for these systems. Problem areas associated with processing real pilot response data are included in the discussion.
Experimental study of the problem of the forced adjustment of the wind field to the height field with the Mintz-Arakawa two-level atmospheric general circulation model. In all but one of the experiments, the height field was assumed to be perfectly observed at 6-hr intervals, over a time period of one day or less, and from these height data the vector wind field was computed by forced dynamical adjustment. In one experiment, the temperature alone was prescribed. The winds computed in these experiments were compared with the 'control' winds of the general circulation simulation. The best agreement between the computed and the control winds was obtained when the time-differencing scheme in the governing finite-difference equations of motion had a large rate of damping of high-frequency motions. This damping rate also determined the optimum fraction and frequency of restoration of the height (or temperature) fields. With strong damping, total restoration at every time step gave the most rapid rate of wind error reduction and the smallest asymptotic limit of the wind error.
The problem of estimation of state in nonlinear dynamical systems containing time delays is studied. The plant is specified by a set of nonlinear differential-difference equations. Observations are a nonlinear function of current and/or delayed states. Both contain additive disturbances. The criterion used for the optimal estimates is the integral of the weighted squared error. Using the theory of the calculus of variations, equations are developed for the estimation. They are first expressed in the form of a split boundary value problem, which is then converted to an initial value problem for on-line estimation. The result yields a sequential estimation scheme in which filtered and smoothed estimates are computed in a sequential manner. The applicability of the procedure is demonstrated by a practical example.
A numerical method based on reference plane characteristics has been developed for the calculation of highly complex supersonic nozzle-exhaust flow fields. The difference equations have been developed for three coordinate systems. Local reference plane orientations are employed using the three coordinate systems concurrently thus catering to a wide class of flow geometries. Discontinuities such as the underexpansion shock and contact surfaces are computed explicitly for nonuniform vehicle external flows. The nozzles considered may have irregular cross-sections with swept throats and may be stacked in modules using the vehicle undersurface for additional expansion. Results are presented for several nozzle configurations.
A finite difference formulation is presented for sound propagation in a two-dimensional straight soft-walled duct with uniform flow. The difference analysis is developed in terms of complex notation. The governing acoustic difference equations and the appropriate displacement boundary conditions associated with uniform flow are presented for the sound attenuation in straight hard and soft-walled ducts. At present the finite Mach number case is solved only for the one-dimensional hard walled duct.
A finite difference formulation is presented for sound propagation in a two-dimensional straight soft-walled duct with uniform flow. The difference analysis is developed in terms of complex notation. The governing acoustic difference equations and the appropriate displacement boundary conditions associated with uniform flow are presented. Example calculations are presented for the sound attenuation in straight hard and soft-walled ducts. At present the finite Mach number case is solved only for the one-dimensional hard walled duct.
The finite difference formulation and method of solution is presented for a wide variety of fluid flow problems with associated heat transfer. Only a few direct results from these formulations are given as examples, since the book is intended primarily to serve a discussion of the techniques and as a starting point for further investigations; however, the formulations are sufficiently complete that a workable computer program may be written from them. In the appendixes a number of topics are discussed which are of interest with respect to the finite difference equations presented. These include a very rapid method for solving certain sets of linear algebraic equations, a discussion of numerical stability, the inherent error in flow rate for confined flow problems, and a method for obtaining high accuracy with a relatively small number of mesh points.
A midcourse guidance and navigation system for continuous low thrust vehicles is described. A set of orbit elements, known as the equinoctial elements, are selected as the state variables. The uncertainties are modelled statistically by random vector and stochastic processes. The motion of the vehicle and the measurements are described by nonlinear stochastic differential and difference equations respectively. A minimum time nominal trajectory is defined and the equation of motion and the measurement equation are linearized about this nominal trajectory. An exponential cost criterion is constructed and a linear feedback guidance law is derived to control the thrusting direction of the engine. Using this guidance law, the vehicle will fly in a trajectory neighboring the nominal trajectory. The extended Kalman filter is used for state estimation. Finally a short mission using this system is simulated. The results indicate that this system is very efficient for short missions.
A finite difference formulation is presented for sound propagation in a rectangular two-dimensional duct without steady flow for plane wave input. Before the difference equations are formulated, the governing Helmholtz equation is first transformed to a form whose solution does not oscillate along the length of the duct. This transformation reduces the required number of grid points by an order of magnitude, and the number of grid points becomes independent of the sound frequency. Physically, the transformed pressure represents the amplitude of the conventional sound wave. Example solutions are presented for sound propagation in a one-dimensional straight hard-wall duct and in a two-dimensional straight soft-wall duct without steady flow. The numerical solutions show evidence of the existence along the duct wall of a developing acoustic pressure diffusion boundary layer which is similar in nature to the conventional viscous flow boundary layer. In order to better illustrate this concept, the wave equation and boundary conditions are written such that the frequency no longer appears explicitly in them. The frequency effects in duct propagation can be visualized solely as an expansion and stretching of the suppressor duct.
Several techniques for modeling the disturbances to a spacecraft's attitude caused by moving crew members are presented. These disturbances can be the largest moments acting on a manned spacecraft, and knowledge of their effect is important in the sizing, design, and analysis/simulation of spacecraft attitude control systems. The modeling techniques are identified as two principal types: deterministic and stochastic. Three techniques of each type are presented. The deterministic models include point-mass motion derivatives and a discussion on dynamic models of moving crew members. The stochastic techniques are highlighted by a Fourier transform method and the representation of long-term crew disturbance activities as outputs from appropriately designed filters. A z-transform technique is developed to obtain a difference-equation form of stochastic models for use on digital computers. An appendix derives spacecraft equations of motion which can be used with many of the models discussed.
An all digital phase locked loop which tracks the phase of the incoming sinusoidal signal once per carrier cycle is proposed. The different elements and their functions and the phase lock operation are explained in detail. The nonlinear difference equations which govern the operation of the digital loop when the incoming signal is embedded in white Gaussian noise are derived, and a suitable model is specified. The performance of the digital loop is considered for the synchronization of a sinusoidal signal. For this, the noise term is suitably modelled which allows specification of the output probabilities for the two level quantizer in the loop at any given phase error. The loop filter considered increases the probability of proper phase correction. The phase error states in modulo two-pi forms a finite state Markov chain which enables the calculation of steady state probabilities, RMS phase error, transient response and mean time for cycle skipping.