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NASA NTRS1991

Formulation of the linear model from the nonlinear simulation for the F18 HARV

The F-18 HARV is a modified F-18 Aircraft which is capable of flying in the post-stall regime in order to achieve superagility. The onset of aerodynamic stall, and continued into the post-stall region, is characterized by nonlinearities in the aerodynamic coefficients. These aerodynamic coefficients are not expressed as analytic functions, but rather in the form of tabular data. The nonlinearities in the aerodynamic coefficients yield a nonlinear model of the aircraft's dynamics. Nonlinear system theory has made many advances, but this area is not sufficiently developed to allow its application to this problem, since many of the theorems are existance theorems and that the systems are composed of analytic functions. Thus, the feedback matrices and the state estimators are obtained from linear system theory techniques. It is important, in order to obtain the correct feedback matrices and state estimators, that the linear description of the nonlinear flight dynamics be as accurate as possible. A nonlinear simulation is run under the Advanced Continuous Simulation Language (ACSL). The ACSL simulation uses FORTRAN subroutines to interface to the look-up tables for the aerodynamic data. ACSL has commands to form the linear representation for the system. Other aspects of this investigation are discussed.

Hall, Charles E., Jr.↗

NASA NTRS1985

Difference methods for problems with different time scales

The use of difference methods for weakly nonlinear systems of ordinary differential equations with rapidly oscillating solutions is considered, and a general approach which depends only on the smoothness of the coefficients and the nonlinearities is developed. In particular, one is led to a strategy which is suitable for the detection and resolution of turning points in such systems. A computational example is presented.

Scheid, R. E., Jr.↗

DOE OSTI2024

Microscopic Theory of Nonlinear Hall Effect in Three-Dimensional Magnetic Systems

Abstract Nonlinear Hall effect (NLHE) has been detected in various of condensed matter systems. Unlike linear Hall effect, NLHE may exist in physical systems with broken inversion symmetry in crystals. On the other hand, real space spin texture may also break inversion symmetry and result in NLHE. We employ the Feynman diagrammatic technique to calculate non-linear Hall conductivity (NLHC) in three-dimensional magnetic systems. The results connect NLHE with the physical quantity of emergent electrodynamics which originates from the magnetic texture. The leading order contribution of NLHC, χabb , is proportional to the emergent toroidal moment T a e , which reflects how the spin textures wind in three dimensions.

Hou 侯, Wen-Tao 文涛↗

NASA NTRS1984

Second-order analytic solution for aerocapture and ballistic fly-through trajectories

A generalized Yaroshevskii's system of equations is derived for analyzing ballistic entry at supercircular speeds. By an artificial introduction of a small parameter, the nonlinear system can then be integrated by Poincare's method. It is pointed out that the second-order theory displays explicitly the influence of the ballistic coefficient, entry speed, and entry angle on exit conditions. The analytic solution is found to be in excellent agreement with the numerical solution. Using an explicit formula, the critical entry angle at which the vehicle fails to skip out can be predicted to within one hundredth of a degree.

Vinh, N. X.↗

NASA NTRS1990

Large scale nonlinear numerical optimal control for finite element models of flexible structures

This paper discusses the development of large scale numerical optimal control algorithms for nonlinear systems and their application to finite element models of structures. This work is based on our expansion of the optimal control algorithm (DDP) in the following steps: improvement of convergence for initial policies in non-convex regions, development of a numerically accurate penalty function method approach for constrained DDP problems, and parallel processing on supercomputers. The expanded constrained DDP algorithm was applied to the control of a four-bay, two dimensional truss with 12 soft members, which generates geometric nonlinearities. Using an explicit finite element model to describe the structural system requires 32 state variables and 10,000 time steps. Our numerical results indicate that for constrained or unconstrained structural problems with nonlinear dynamics, the results obtained by our expanded constrained DDP are significantly better than those obtained using linear-quadratic feedback control.

Shoemaker, Christine A.↗

NASA NTRS1985

Nonlinear optimal control with tensors - Some computational issues

Some computational issues associated with the calcualtion of optimal feedback controls for nonlinear systems in a tensor setting are described. The specific issues addressed pertain to the combinatorial nature of the loading of the elements into tensors used to represent the system, cost, and feedback, and the subsequent calculations involving these elements. Particular attention is given to: the symmetric tensor algebra which is a natural setting for representing polynomials; the conversions between symmetric and nonsymmetric tensors; the general nature of the calculations required; and the solution equation for nonlinear optimal feedback control. It is concluded that nonlinear tensor feedback can improve performance both in terms of system responses and in terms of system stability region.

Osullivan, J. A.↗

DOE OSTI2026

Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving Hénon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts.

43 PARTICLE ACCELERATORS↗

NASA NTRS1977

A numerical scheme to solve unstable boundary value problems

The considered scheme makes it possible to determine an unstable steady state solution in cases in which, because of lack of symmetry, such a solution cannot be obtained analytically, and other time integration or relaxation schemes, because of instability, fail to converge. The iterative solution of a single complex equation is discussed and a nonlinear system of equations is considered. Described applications of the scheme are related to a steady state solution with shear instability, an unstable nonlinear Ekman boundary layer, and the steady state solution of a baroclinic atmosphere with asymmetric forcing. The scheme makes use of forward and backward time integrations of the original spatial differential operators and of an approximation of the adjoint operators. Only two computations of the time derivative per iteration are required.

Kalnay-Rivas, E.↗

NASA NTRS1984

Application of nonlinear transformations to automatic flight control

The theory of transformations of nonlinear systems to linear ones is applied to the design of an automatic flight controller for the UH-1H helicopter. The helicopter mathematical model is described and it is shown to satisfy the necessary and sufficient conditions for transformability. The mapping is constructed, taking the nonlinear model to canonical form. The performance of the automatic control system in a detailed simulation on the flight computer is summarized.

Meyer, G.↗

NASA NTRS1974

Periodic solutions in a class of nonlinear dual-mass systems

A class of nonlinear spring-mass systems with two degrees of freedom is studied. A numerical method is developed for determining the periodic solutions in such systems. The method grows out of a recent paper of Yen (1974a) in which properties of such periodic solutions have been investigated theoretically. Numerical results, obtained for some example systems involving linear, cubic and quintic spring forces, support the findings of Yen and reveal further details of the properties of such periodic solutions.

Yen, D. H. Y.↗

NASA NTRS1988

A nonlinear filtering process diagnostic system for the Space Station

A nonlinear filtering process diagnostic system, terrestrial simulation and real time implementation studies is presented. Possible applications to Space Station subsystem elements are discussed. A process diagnostic system using model based nonlinear filtering for systems with random structure was shown to provide improvements in stability, robustness, and overall performance in comparison to linear filter based systems. A suboptimal version of the nonlinear filter (zero order approximation filter, or ZOA filter) was used in simulation studies, initially, with a pressurized water reactor model and then with water/steam heat exchanger models. Finally, a real time implementation for leak detection in a water/steam heat exchanger was conducted using the ZOA filter and heat exchanger models.

Yoel, Raymond R.↗

DOE OSTI2022

Squeeze Film Damper Experimental and Numerical Correlation: Test Setup Description and Parameter Identification of Dry System

The process of characterizing a damper involves both experimental techniques and model simulation. Experiments characterize dynamic behavior of a system through empirical modal analysis using impact and shaker testing. The layout of the test setup is crucial for a low-mass, nonlinear system. Physical arrangement, orientation of the system, and instrumentation affect the results that are discussed in this paper. These results can be used to validate a numerical simulation of the same system. With a validated model, important physical input parameters and modeling techniques can be determined and implemented in future designs. In other words, this modeling can be used with confidence to design alternative dampers that may be difficult, costly, and time-consuming to characterize experimentally.

Cook, Jason↗

NASA NTRS1973

Determination and evaluation of performance limit criteria of fast-response electrohydraulic servosystems

Limit criteria for determining the dynamic performance capabilities of high performance, fast-response (greater than 100 Hz) electrohydraulic servosystems are presented. A detailed analysis of the maximum load locus of these systems is used as a basis for the derivation of improved limit criteria. These criteria predict the maximum performance limits caused by system nonlinearities and physical limitations. The criteria are applied to experimental data to verify their validity. Design criteria which assist in the selection of system components for optimal performance are also discussed.

Zeller, J. R.↗

NASA NTRS1999

Reduced Order Models Based on Linear and Nonlinear Aerodynamic Impulse Responses

This paper discusses a method for the identification and application of reduced-order models based on linear and nonlinear aerodynamic impulse responses. The Volterra theory of nonlinear systems and an appropriate kernel identification technique are described. Insight into the nature of kernels is provided by applying the method to the nonlinear Riccati equation in a non-aerodynamic application. The method is then applied to a nonlinear aerodynamic model of an RAE 2822 supercritical airfoil undergoing plunge motions using the CFL3D Navier-Stokes flow solver with the Spalart-Allmaras turbulence model. Results demonstrate the computational efficiency of the technique.

Silva, Walter A.↗

NASA NTRS1999

Reduced-Order Models Based on Linear and Nonlinear Aerodynamic Impulse Responses

This paper discusses a method for the identification and application of reduced-order models based on linear and nonlinear aerodynamic impulse responses. The Volterra theory of nonlinear systems and an appropriate kernel identification technique are described. Insight into the nature of kernels is provided by applying the method to the nonlinear Riccati equation in a non-aerodynamic application. The method is then applied to a nonlinear aerodynamic model of RAE 2822 supercritical airfoil undergoing plunge motions using the CFL3D Navier-Stokes flow solver with the Spalart-Allmaras turbulence model. Results demonstrate the computational efficiency of the technique.

Silva, Walter A.↗

NASA NTRS1975

Dynamic analysis of a flexible spacecraft with rotating components. Volume 1: Analytical developments

Analytical procedures and digital computer code are presented for the dynamic analysis of a flexible spacecraft with rotating components. Topics, considered include: (1) nonlinear response in the time domain, and (2) linear response in the frequency domain. The spacecraft is assumed to consist of an assembly of connected rigid or flexible subassemblies. The total system is not restricted to a topological connection arrangement and may be acting under the influence of passive or active control systems and external environments. The analytics and associated digital code provide the user with the capability to establish spacecraft system nonlinear total response for specified initial conditions, linear perturbation response about a calculated or specified nominal motion, general frequency response and graphical display, and spacecraft system stability analysis.

Bodley, C. S.↗

DOE OSTI2024

Optimal Control of SOEC-Based Hydrogen Production Systems for Demand Response Using Deep Reinforcement Learning in Smart Grids

Solid oxide electrolysis cell (SOEC) hydrogen production technology can range in size from small, appliance-size equipment to large-scale, central production facilities that can be tied directly to renewable or non-greenhouse-gas-emitting forms of electricity production, making it an ideal resource for demand response (DR). The SOEC hydrogen production system is a complex integrated system that encompasses fluid dynamics, electrical dynamics, and electrochemical and thermal dynamics, all of which involve non-linearity and non-convexity. Proper control of the SOEC hydrogen production system is crucial to enable its participation in the DR program. Here, to overcome the difficulty of designing an explicit control law for such nonlinear systems with nonconvex optimization features in DR applications, deep reinforcement learning (DRL) is explored to achieve the optimal control of the SOEC system for DR participation. Specifically, a twin delayed deterministic policy gradient (TD3) control framework is applied to achieve optimal response performance during DR events by considering power tracking error and hydrogen production efficiency with a suitable reward function. Two case studies with grid connections for tracking different DR commands were investigated. The first case study involved operating conditions reaching the boundaries, while the second involved operating conditions within the boundaries. The results showed that the proposed DRL-based control for SOEC can track the DR signal in a timely manner while maintaining high energy efficiency.

08 HYDROGEN↗

NASA NTRS1981

Construction of composite three dimensional grids from subregion grids generated by elliptic systems

A technique is presented for constructing a boundary-conforming grid throughout a general three-dimensional flow region as a composite of subregion grids. Each subregion grid is generated numerically by solving a nonlinear system of elliptic equations. The boundary values represent nodal points in a quasi-two-dimensional grid that covers the curved surface bounding the subregion, and are generated numerically by a modified elliptic system. A primary feature of the technique is that the composite three-dimensional grid remains both continuous and smooth across the surface of juncture between any two adjoining subregions. Sample results are given for a wing-body combination.

Thomas, P. D.↗