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A relaxation method is demonstrated which reliably solves the nonlinear two point boundary value problem which arises when optimal control theory is applied to determination of large angle maneuvers of flexible spacecraft. The basic ideas are summarized and several idealized maneuvers are determined. The emphasis is upon demonstrating the basic ideas and practical aspects of the methodology.
The effect of nonlinear feedback on nonlinear systems is discussed for problems where the controls are entered linearly. The invariance of certain quantities under feedback are established, and it is shown that these quantities contain enough information to determine if the system can be linearized using feedback and change of coordinates. Attention is given to scalar input systems emphasizing a new F-invariant property.
The parameter identification problem of general discrete time, nonlinear, multiple input/multiple output dynamic systems with Gaussian white distributed measurement errors is considered. The knowledge of the system parameterization was assumed to be known. Concepts of local parameter identifiability and local constrained maximum likelihood parameter identifiability were established. A set of sufficient conditions for the existence of a region of parameter identifiability was derived. A computation procedure employing interval arithmetic was provided for finding the regions of parameter identifiability. If the vector of the true parameters is locally constrained maximum likelihood (CML) identifiable, then with probability one, the vector of true parameters is a unique maximal point of the maximum likelihood function in the region of parameter identifiability and the constrained maximum likelihood estimation sequence will converge to the vector of true parameters.
Two main areas of application of adaptive state estimation theory are presented. Following a review of the basic estimation approach, its application to both the control of nonlinear plants and to the problem of tracking maneuvering targets is presented. Results are brought together from these two areas of investigation to provide insight into the wide range of possible applications of the general estimation method.
Dual-shaped reflectors have been used for many years. Thus, these reflectors have been used as high gain antennas on Voyagers 1 and 2. The objectives of the geometrical optics (GO) dual shaped synthesis are considered. Concerning the synthesis of dual shaped reflectors, it has been shown for circular symmetric reflectors that an exact GO solution can be found to the problem of transforming, by two reflections, any feed pattern into any aperture distribution. This problem involves solving two simultaneous nonlinear ordinary differential equations. The same approach for offset geometry leads to two simultaneous nonlinear partial differential equations. It is shown that these equations could also be integrated numerically, except that in general these equations are not total and therefore, in general, they do not have a 'smooth' solution. It is further shown that the offset partial differentials often very nearly form a total differential in many cases of practical importance.
A computational technique, based on the integral solution of the full potential equation, has been developed for the solution of three-dimensional high-subsonic flows past wings at high angles of attack. The problem includes two sources of nonlinearities: a boundary oriented nonlinearity (separated flow roll up) and a region oriented nonlinearity (flow compressibility). The former is represented by a nonlinear vortex lattice while the latter is represented by a source distribution inside a finite volume. The solution is obtained by using double iteration cycles; a separated flow (wake) iteration cycle and a compressibility iteration cycle. The computational technique is applied to a delta wing. The results show that the technique is very promising and efficient.
The development of integrated optical circuits (IOC) for numerical-computation applications is reviewed, with a focus on the use of systolic architectures. The basic architecture criteria for optical processors are shown to be the same as those proposed by Kung (1982) for VLSI design, and the advantages of IOCs over bulk techniques are indicated. The operation and fabrication of electrooptic grating structures are outlined, and the application of IOCs of this type to an existing 32-bit, 32-Mbit/sec digital correlator, a proposed matrix multiplier, and a proposed pipeline processor for polynomial evaluation is discussed. The problems arising from the inherent nonlinearity of electrooptic gratings are considered. Diagrams and drawings of the application concepts are provided.
For transonic flow calculations, a combination of the successive line over-relaxation (SLOR) and the preconditioned conjugate gradient (CG) method has been suggested by Wong and Hafez (1981). This paper studies the method of minimal residual (MR) which avoids a combined iteration. This method is closely related to the CG method, may be regarded as a first-order gradient method, and is applicable to symmetric and nonsymmetric matrices. The problem is formulated as a nonlinear mixed elliptic-hyperbolic partial differential equation which includes an artificial viscosity and a switching function which is zero in subsonic regions and nonzero in supersonic regions. Alternatives to the SLOR method which provide faster convergence rates are introduced. The preconditioned MR algorithm is developed, and transonic potential flows around NACA 0012 airfoil are calculated for different Mach numbers and angles of attack. Preliminary results are presented, demonstrating that the MR algorithm requires no parameter estimation and rapidly converges for subsonic flows.
The design of a composite panel requires some way of finding the minimum thickness laminate which will withstand the load requirements without failure. The mathematical complexity of this problem dictates the use of nonlinear optimization techniques. Specialized laminate optimization programs were developed which are compact and efficient enough to run on microcomputers. Only stresses at a point and inplane loads and deflections are considered. The programs are simple to use and require no knowledge of optimization. Techniques are developed which find minimum thickness laminates with either ply ratios or ply angles as design variables. A method is presented for finding the optimum orientation for the axis of symmetry of an orthotropic laminate. The orthotropic laminate program uses an approximate failure theory, which speed up computations dramatically.
Discrete attitude commands have become a standard task for flying qualities evaluation and control system testing. Much pilot opinion data is now available for ground-based and in-flight simulations, but adequate performance measures and prediction methods have not been established. The Step Target Tracking Prediction method, introduced in 1978, correlated time-on-target and rms tracking data with NT-33 in-flight longitudinal simulations, but did not employ parameters easily measured in manned flight and simulation. Recent application of the Step Target Tracking Prediction method to lateral flying qualities analysis has led to a new measure of performance. This quantity, called Maximum Normalized Rate (MNR), reflects the greatest attitude rate a pilot can employ during a discrete maneuver without excessive overshoot and oscillation. MNR correlates NT-33 lateral pilot opinion ratings well, and is easily measured during flight test or simulation. Futhermore, the Step Target MNR method can be used to analyze large amplitude problems concerning rate limiting and nonlinear aerodynamics.
A model of the solar corona during sunspot minimum was constructed. It was suggested that pole-equator asymmetry is caused by interaction of global azimuthal electric currents in the corona and potential magnetic fields originating below the corona. The problem was reduced to a nonlinear ordinary differential equation for a structural function. Taking the difference in electron density between the equatorial plane and the polar direction from observations, the basic equation was solved and the global electric current distribution around the Sun along with the magnetic and thermodynamic structures of the solar corona was derived. The asymmetry between the two hemispheres, which can be due to the magnetic quadrupole, is considered in detail. It is shown that an additional quadrupole term in the magnetic field representation affects the position of polar coronal hole boundaries differently in the two hemispheres. As a result, the outflow from the two polar regions is found to be different.
Least-squares-type algorithms for reducing the order of linear systems in the frequency domain and simplifying nonlinear systems in time domain are developed and demonstrated. The possible model structures are represented as nodes in a tree, and costs along the branches are assigned using the repeated-Gram-Schmidt orthogonalization procedure of Desrochers and Saridis (1980), permitting identification of the optimal n-term model by searching the tree to depth n, with no need for parameter identification. The efficiency and flexibility of the algorithms is shown in applications to the eighth-order linear system studied by Hsia (1972), a three-state eight-nonlinear-term aircraft-dynamics problem, and the related linear-controller problem (Garrard and Jordan, 1977).
The gradient projection method of numerical optimization which is applied to problems having linear constraints but nonlinear objective functions is described and analyzed. The algorithm is found to be efficient and thorough for small systems, but requires the addition of auxiliary methods and programming for large scale systems with severe nonlinearities. In order to verify the theoretical results a digital computer is used to simulate the algorithm.
The design concept of a control for rapid torque-limited slewing of a rigid-mast version of the NASA SCOLE configuration is presented and demonstrated by means of numerical simulation. The time-optimal control problem for the system is decomposed into separate single-axis problems, expanding analytically the implicit nonlinear transcendental expression for the SCOLE line-of-sight error, and the final Euler attitude angles and slew angles are determined. The simulation results are presented in tables and graphs, and it is found that bang-bang or bang-pause-bang slew maneuvers with control moment applied to the Shuttle and control force applied to the reflector, and with a 5-deg/s slew-rate limit, produce the best pointing accuracy and the shortest slew times, although the specified line-of-sight error of 0.02 deg cannot be achieved using such open-loop single-axis maneuvers.
The potential role of symbolic manipulation programs in development and solution of the governing equations for rotorcraft dynamics problems is discussed and illustrated. Nonlinear equations of motion for a helicopter rotor blade represented by a rotating beam are developed making use of the computerized symbolic manipulation program MACSYMA. The use of computerized symbolic manipulation allows the analyst to concentrate on more meaningful tasks, such as establishment of physical assumptions, without being sidetracked by the tedious and trivial details of the algebraic manipulations. Furthermore, the resulting equations can be produced, if necessary, in a format suitable for numerical solution. A perturbation-type solution for the resulting dynamical equations is shown to be possible with a combination of symbolic manipulation and standard numerical techniques. This should ultimately lead to a greater physical understanding of the behavior of the solution than is possible with purely numerical techniques. The perturbation analysis of the flapping motion of a rigid rotor blade in forward flight is presented, for illustrative purposes, via computerized symbolic manipulation with a method that bypasses Floquet theory.
A new approximation method for dealing with stress constraints in structural synthesis is presented. The finite element nodal forces are approximated and these are used to create an explicit, but often nonlinear, approximation to the original problem. The principal motivation is to create the best approximation possible, in order to reduce the number of detailed finite element analyses needed to reach the optimum. Examples are offered and compared with published results, to demonstrate the efficiency and reliability of the proposed method.
Until a decade ago most solar physicists thought of a sunspot as the upper end of a giant flux tube floating vertically. The existence of umbral dots and penumbral grains has been known for several decades. On the basis of available observations, they seem to be regions of photospheric intensity with upflowing gas motion and magnetic fields much weaker than in the surrounding sunspot surface. It has also been suggested that the differences in the appearances of umbral dots and granular cells are caused by the highly nonlinear nature of the convection problem in the presense of strong magnetic fields. The main ideas are presented here without any equations. It can be shown that a pocket of field free gas surrounded by a vertical magnetic field in the presence of gravity takes up the shape of a tapering column ending at a vertex at the top. Some convection is expected to take place in the trapped field free gas, whereas the magnetic field around it makes those regions stable against convection. Eventually the apex of the tapering column reaches the photospheric surface where the bulging of the magnetic field makes the field no longer able to close on the field free gas and trap it underneath.
A two-level design optimization metholology is described. A progress report of its application to Printed Wiring Board (PWB) assembly examples is given. The design of PWB assemblies is a complex task which is generally conducted as a sequential process. Individual PWBs are usually designed first, followed by the composition of the PWBs into an assembly. As a result, optimizing design considerations such as assembly reliability cannot be accomplished. This study showed that a two-level decomposition method can be employed to optimize for reliability at both the PWB- and the assembly-level in a coupled manner. The two-level decomposition method also resolved the mixed-integer nonlinear programming nature of the problem rather easily.