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NASA NTRS1988

A study of helicopter stability and control including blade dynamics

A linearized model of rotorcraft dynamics has been developed through the use of symbolic automatic equation generating techniques. The dynamic model has been formulated in a unique way such that it can be used to analyze a variety of rotor/body coupling problems including a rotor mounted on a flexible shaft with a number of modes as well as free-flight stability and control characteristics. Direct comparison of the time response to longitudinal, lateral and directional control inputs at various trim conditions shows that the linear model yields good to very good correlation with flight test. In particular it is shown that a dynamic inflow model is essential to obtain good time response correlation, especially for the hover trim condition. It also is shown that the main rotor wake interaction with the tail rotor and fixed tail surfaces is a significant contributor to the response at translational flight trim conditions. A relatively simple model for the downwash and sidewash at the tail surfaces based on flat vortex wake theory is shown to produce good agreement. Then, the influence of rotor flap and lag dynamics on automatic control systems feedback gain limitations is investigated with the model. It is shown that the blade dynamics, especially lagging dynamics, can severly limit the useable values of the feedback gain for simple feedback control and that multivariable optimal control theory is a powerful tool to design high gain augmentation control system. The frequency-shaped optimal control design can offer much better flight dynamic characteristics and a stable margin for the feedback system without need to model the lagging dynamics.

Zhao, Xin↗

NASA NTRS1990

A reduced-dynamic technique for precise orbit determination

Observations of the Global Positioning System (GPS) will enable a reduced-dynamic technique for achieving subdecimeter orbit determination of earth-orbiting satellites. With this technique, information on the transition between satellite states at different observing times is furnished by both a formal dynamic model and observed satellite positional change (which is inferred kinematically from continuous GPS carrier-phase data). The relative weighting of dynamic and kinematic information can be freely varied. Covariance studies show that in situations where observing geometry is poor and the dynamic model is good, the model dominates determination of the state transition; where the dynamic model is poor and the geometry strong, carrier phase governs the determination of the transition. When neither kinematic nor dynamic information is clearly superior, the reduced-dynamic combination of the two can substantially improve the orbit-determination solution. Guidelines are given here for selecting a near-optimal weighting for the reduced-dynamic solution, and sensitivity of solution accuracy to this weighting is examined.

Wu, S. C.↗

NASA NTRS1989

A continuum model for dynamic analysis of the Space Station

Dynamic analysis of the International Space Station using MSC/NASTRAN had 1312 rod elements, 62 beam elements, 489 nodes and 1473 dynamic degrees of freedom. A realtime, man-in-the-loop simulation of such a model is impractical. This paper discusses the mathematical model for realtime dynamic simulation of the Space Station. Several key questions in structures and structural dynamics are addressed. First, to achieve a significant reduction in the number of dynamic degrees of freedom, a continuum equivalent representation of the Space Station truss structure which accounted for the unsymmetry of the basic configuration and resulted in the coupling of extensional and transverse deformation, is developed. Next, dynamic equations for the continuum equivalent of the Space Station truss structure are formulated using a matrix version of Kane's dynamical equations. Flexibility is accounted for by using a theory that accommodates extension, bending in two principal planes and shear displacement. Finally, constraint equations suitable for dynamic analysis of flexible bodies with closed loop configuration are developed and solution of the resulting system of equations is based on the zero eigenvalue theorem.

Thomas, Segun↗

NASA NTRS1990

A simple dynamic engine model for use in a real-time aircraft simulation with thrust vectoring

A simple dynamic engine model was developed at the NASA Ames Research Center, Dryden Flight Research Facility, for use in thrust vectoring control law development and real-time aircraft simulation. The simple dynamic engine model of the F404-GE-400 engine (General Electric, Lynn, Massachusetts) operates within the aircraft simulator. It was developed using tabular data generated from a complete nonlinear dynamic engine model supplied by the manufacturer. Engine dynamics were simulated using a throttle rate limiter and low-pass filter. Included is a description of a method to account for axial thrust loss resulting from thrust vectoring. In addition, the development of the simple dynamic engine model and its incorporation into the F-18 high alpha research vehicle (HARV) thrust vectoring simulation. The simple dynamic engine model was evaluated at Mach 0.2, 35,000 ft altitude and at Mach 0.7, 35,000 ft altitude. The simple dynamic engine model is within 3 percent of the steady state response, and within 25 percent of the transient response of the complete nonlinear dynamic engine model.

Johnson, Steven A.↗

NASA NTRS1989

On the dynamic singularities in the control of free-floating space manipulators

It is shown that free-floating space manipulator systems have configurations which are dynamically singular. At a dynamically singular position, the manipulator is unable to move its end effector in some direction. This problem appears in any free-floating space manipulator system that permits the vehicle to move in response to manipulator motion without correction from the vehicle's attitude control system. Dynamic singularities are functions of the dynamic properties of the system; their existence and locations cannot be predicted solely from the kinematic structure of the manipulator, unlike the singularities for fixed base manipulators. It is also shown that the location of these dynamic singularities in the workplace is dependent upon the path taken by the manipulator in reaching them. Dynamic singularities must be considered in the control, planning and design of free-floating space manipulator systems. A method for calculating these dynamic singularities is presented, and it is shown that the system parameters can be selected to reduce the effect of dynamic singularities on a system's performance.

Papadopoulos, E.↗

NASA NTRS1990

Recursive dynamics for flexible multibody systems using spatial operators

Due to their structural flexibility, spacecraft and space manipulators are multibody systems with complex dynamics and possess a large number of degrees of freedom. Here the spatial operator algebra methodology is used to develop a new dynamics formulation and spatially recursive algorithms for such flexible multibody systems. A key feature of the formulation is that the operator description of the flexible system dynamics is identical in form to the corresponding operator description of the dynamics of rigid multibody systems. A significant advantage of this unifying approach is that it allows ideas and techniques for rigid multibody systems to be easily applied to flexible multibody systems. The algorithms use standard finite-element and assumed modes models for the individual body deformation. A Newton-Euler Operator Factorization of the mass matrix of the multibody system is first developed. It forms the basis for recursive algorithms such as for the inverse dynamics, the computation of the mass matrix, and the composite body forward dynamics for the system. Subsequently, an alternative Innovations Operator Factorization of the mass matrix, each of whose factors is invertible, is developed. It leads to an operator expression for the inverse of the mass matrix, and forms the basis for the recursive articulated body forward dynamics algorithm for the flexible multibody system. For simplicity, most of the development here focuses on serial chain multibody systems. However, extensions of the algorithms to general topology flexible multibody systems are described. While the computational cost of the algorithms depends on factors such as the topology and the amount of flexibility in the multibody system, in general, it appears that in contrast to the rigid multibody case, the articulated body forward dynamics algorithm is the more efficient algorithm for flexible multibody systems containing even a small number of flexible bodies. The variety of algorithms described here permits a user to choose the algorithm which is optimal for the multibody system at hand. The availability of a number of algorithms is even more important for real-time applications, where implementation on parallel processors or custom computing hardware is often necessary to maximize speed.

Jain, A.↗

NASA NTRS1994

Axis switching and spreading of an asymmetric jet: Role of vorticity dynamics

The effects of vortex generators and periodic excitation on vorticity dynamics and the phenomenon of axis switching in a free asymmetric jet are studied experimentally. Most of the data reported are for a 3:1 rectangular jet at a Reynolds number of 450,000 and a Mach number of 0.31. The vortex generators are in the form of 'delta tabs', triangular shaped protrusions into the flow, placed at the nozzle exit. With suitable placement of the tabs, axis switching could be either stopped or augmented. Two mechanisms are identified governing the phenomenon. One, as described by previous researchers and referred to here as the omega(sub theta)-induced dynamics, is due to difference in induced velocities for different segments of a rolled up azimuthal vortical structure. The other, omega(sub x)-induced dynamics, is due to the induced velocities of streamwise vortex pairs in the flow. Both dynamics can be active in a natural asymmetric jet; the tendency for axis switching caused by the omega(sub theta)-induced dynamics may be, depending on the streamwise vorticity distribution, either resisted or enhanced by the omega(sub x)-induced dynamics. While this simple framework qualitatively explains the various observations made on axis switching, mechanisms actually in play may be much more complex. The two dynamics are not independent as the flow field is replete with both azimuthal and streamwise vortical structures which continually interact. Phase averaged flow field data for a periodically forced case, over a volume of the flow field, are presented and discussed in an effort to gain insight into the dynamics of these vortical structures.

Zaman, K. B. M. Q.↗

NASA NTRS1999

Automated Flight Dynamics Product Generation for the EOS AM-1 Spacecraft

As part of NASA's Earth Science Enterprise, the Earth Observing System (EOS) AM-1 spacecraft is designed to monitor long-term, global, environmental changes. Because of the complexity of the AM-1 spacecraft, the mission operations center requires more than 80 distinct flight dynamics products (reports). To create these products, the AM-1 Flight Dynamics Team (FDT) will use a combination of modified commercial software packages (e.g., Analytical Graphic's Satellite ToolKit) and NASA-developed software applications. While providing the most cost-effective solution to meeting the mission requirements, the integration of these software applications raises several operational concerns: (1) Routine product generation requires knowledge of multiple applications executing on variety of hardware platforms. (2) Generating products is a highly interactive process requiring a user to interact with each application multiple times to generate each product. (3) Routine product generation requires several hours to complete. (4) User interaction with each application introduces the potential for errors, since users are required to manually enter filenames and input parameters as well as run applications in the correct sequence. Generating products requires some level of flight dynamics expertise to determine the appropriate inputs and sequencing. To address these issues, the FDT developed an automation software tool called AutoProducts, which runs on a single hardware platform and provides all necessary coordination and communication among the various flight dynamics software applications. AutoProducts, autonomously retrieves necessary files, sequences and executes applications with correct input parameters, and deliver the final flight dynamics products to the appropriate customers. Although AutoProducts will normally generate pre-programmed sets of routine products, its graphical interface allows for easy configuration of customized and one-of-a-kind products. Additionally, AutoProducts has been designed as a mission-independent tool, and can be easily reconfigured to support other missions or incorporate new flight dynamics software packages. After the AM-1 launch, AutoProducts will run automatically at pre-determined time intervals . The AutoProducts tool reduces many of the concerns associated with the flight dynamics product generation. Although AutoProducts required a significant effort to develop because of the complexity of the interfaces involved, its use will provide significant cost savings through reduced operator time and maximum product reliability. In addition, user satisfaction is significantly improved and flight dynamics experts have more time to perform valuable analysis work. This paper will describe the evolution of the AutoProducts tool, highlighting the cost savings and customer satisfaction resulting from its development. It will also provide details about the tool including its graphical interface and operational capabilities.

Matusow, Carla↗

NASA NTRS2001

Dynamic Response Assessment for the MEMS Accelerometer Under Severe Shock Loads

NASA Goddard Space Flight Center (GSFC) has evaluated the dynamic response of a commercial-off-the-shelf (COTS) microelectromechanical systems (MEMS) device made by Analog Device, Inc. The device is designated as ADXL250 and is designed mainly for sensing dynamic acceleration. It is also used to measure the tilting angle of any system or component from its original level position. The device has been in commercial use (e.g., in automobile airbag deployment system as a dual-axial accelerometer and in the electronic game play-station as a tilting sensor) with success, but NASA needs an in-depth assessment of its performance under severe dynamic shock environments. It was realized while planning this evaluation task that two assessments would be beneficial to NASA's missions: (1) severe dynamic shock response under nominal thermal environments; and (2) general dynamic performance under cryogenic environments. The first evaluation aims at obtaining a good understanding of its micromachined structure within a framework of brittle fracture dynamics, while the second evaluation focuses on the structure integrity under cryogenic temperature conditions. The information we gathered from the manufacturer indicated that the environmental stresses under NASA's evaluation program have been far beyond what the device has experienced with commercial applications, for which the device was designed. Thus NASA needs the outcome of this evaluation in order to make the selection for possible use for its missions. This paper provides details of the first evaluation the dynamic response under severe multi-axial single-pulse shock load. It was performed using finite element tools with nonlinear dynamics procedures.

Fan, Mark S.↗

NASA NTRS2007

Flight Dynamics Performances of the MetOp A Satellite during the First Months of Operations

The 19th of October 2006 at 16:28 UTC the first MetOp satellite (MetOp A) was successfully launched from the Baykonur cosmodrome by a Soyuz/Fregat launcher. After only three days of LEOP operations, performed by ESOC, the satellite was handed over to EUMETSAT, who is since then taking care of all satellite operations. MetOp A is the first European operational satellite for meteorology flying in a Low Earth Orbit (LEO), all previous satellites operated by EUMETSAT, belonging to the METEOSAT family, being located in the Geo-stationary orbit. To ensure safe operations for a LEO satellite accurate and continuous commanding from ground of the on-board AOCS is required. That makes the operational transition at the end of the LEOP quite challenging, as the continuity of the Flight Dynamics operations is to be maintained. That means that the main functions of the Flight Dynamics have to be fully validated on-flight during the LEOP, before taking over the operational responsibility on the spacecraft, and continuously monitored during the entire mission. Due to the nature of a meteorological operational mission, very stringent requirements in terms of overall service availability (99 % of the collected data), timeliness of processing of the observation data (3 hours after sensing) and accuracy of the geo-location of the meteorological products (1 km) are to be fulfilled. That translates in tight requirements imposed to the Flight Dynamics facility (FDF) in terms of accuracy, timeliness and availability of the generated orbit and clock solutions; a detailed monitoring of the quality of these products is thus mandatory. Besides, being the accuracy of the image geo-location strongly related with the pointing performance of the platform and with the on-board timing stability, monitoring from ground of the behaviour of the on-board sensors and clock is needed. This paper presents an overview of the Flight Dynamics operations performed during the different phases of the MetOp A mission up to routine. The activities performed to validate all the Flight Dynamics functions, characterize the behaviour of the satellite and monitor the performances of the Flight Dynamics facility will be highlighted. The MetOp Flight Dynamics Operations team is led by Anders Meier Soerensen and composed by Pier Luigi Righetti, Francisco Sancho, Antimo Damiano and David Lazaro. The team is supported by Hilda Meixner, responsible for all Flight Dynamics validation activities.

Righetti, Pier Luigi↗

NASA NTRS2010

4D Dynamic RNP Annual Interim Report-Year 1

This Annual Interim Report summarizes the activities led by Raytheon, in collaboration with GE Aviation and SAIC, and presents the results obtained during the first year of this research effort to expand the RNP concept to 4 dimensions relative to a dynamic frame of reference. Joint Program Development Office (JPDO)Concepts of Operations for the Next Generation Air Transportation System (NextGen) considers 4 Dimension Trajectory (4DT) procedures a key enabler to Trajectory Based Operations (TBO). The JPDO defines 4DT as a precise description of an aircraft path in space and time . While NextGen assumes that this path is defined within an Earth-reference frame, many 4DT procedure implementations will require an aircraft to precisely navigate relative to a moving reference such as another aircraft to form aggregate flows or a weather cell to allow for flows to shift. Current methods of implementing routes and flight paths rely on aircraft meeting a Required Navigation Performance (RNP) specification and being equipped with a monitoring and alerting capability to annunciate when the aircraft system is unable to meet the performance specification required for the operation. Since all aircraft today operate within the NAS relative to fixed reference points, the current RNP definition is deemed satisfactory. However, it is not well understood how the current RNP construct will support NextGen 4DT procedures where aircraft operate relative to each other or to other dynamic frames of reference. The objective of this research effort is to analyze candidate 4DT procedures from both an Air Navigation Service Provider (ANSP) and aircraft perspective, to identify their specific navigational requirements, assess the shortcomings of the current RNP construct to meet these requirements, to propose an extended 4 Dimensional Dynamic RNP (4D Dynamic RNP) construct that accounts for the dynamic spatial and temporal nature of the selected 4DT procedures, and finally, to design an experiment using the Airspace and Traffic Operations Simulation (ATOS) system to validate the 4D Dynamic RNP construct. This Annual Interim Report summarizes the activities led by Raytheon, in collaboration with GE Aviation and SAIC, and presents the results obtained during the first year of this research effort to expand the RNP concept to 4 dimensions relative to a dynamic frame of reference. A comprehensive assessment of the state-of-the-art international implementation of current RNP was completed and presented in the Contractor Report RNP State-of-the-Art Assessment, Version 4, 17 December 2008 . The team defined in detail two 4DT operations, Airborne Precision Spacing and Self-Separation, that are ideally suited to be supported by 4D Dynamic RNP and developed their respective conceptual frameworks, Required Interval Management Performance (RIMP) Version 1.1, 13 April 2009 and Required Self Separation Performance (RSSP) Version 1.1, 13 April 2009 . Finally, the team started the development of a mathematical model and simulation tool for RIMP and RSSP scheduled to be delivered during the second year of this research effort.

Finkelsztein, Daniel M.↗

NASA NTRS2011

Constraint Embedding Technique for Multibody System Dynamics

Multibody dynamics play a critical role in simulation testbeds for space missions. There has been a considerable interest in the development of efficient computational algorithms for solving the dynamics of multibody systems. Mass matrix factorization and inversion techniques and the O(N) class of forward dynamics algorithms developed using a spatial operator algebra stand out as important breakthrough on this front. Techniques such as these provide the efficient algorithms and methods for the application and implementation of such multibody dynamics models. However, these methods are limited only to tree-topology multibody systems. Closed-chain topology systems require different techniques that are not as efficient or as broad as those for tree-topology systems. The closed-chain forward dynamics approach consists of treating the closed-chain topology as a tree-topology system subject to additional closure constraints. The resulting forward dynamics solution consists of: (a) ignoring the closure constraints and using the O(N) algorithm to solve for the free unconstrained accelerations for the system; (b) using the tree-topology solution to compute a correction force to enforce the closure constraints; and (c) correcting the unconstrained accelerations with correction accelerations resulting from the correction forces. This constraint-embedding technique shows how to use direct embedding to eliminate local closure-loops in the system and effectively convert the system back to a tree-topology system. At this point, standard tree-topology techniques can be brought to bear on the problem. The approach uses a spatial operator algebra approach to formulating the equations of motion. The operators are block-partitioned around the local body subgroups to convert them into aggregate bodies. Mass matrix operator factorization and inversion techniques are applied to the reformulated tree-topology system. Thus in essence, the new technique allows conversion of a system with closure-constraints into an equivalent tree-topology system, and thus allows one to take advantage of the host of techniques available to the latter class of systems. This technology is highly suitable for the class of multibody systems where the closure-constraints are local, i.e., where they are confined to small groupings of bodies within the system. Important examples of such local closure-constraints are constraints associated with four-bar linkages, geared motors, differential suspensions, etc. One can eliminate these closure-constraints and convert the system into a tree-topology system by embedding the constraints directly into the system dynamics and effectively replacing the body groupings with virtual aggregate bodies. Once eliminated, one can apply the well-known results and algorithms for tree-topology systems to solve the dynamics of such closed-chain system.

Woo, Simon S.↗

NASA NTRS2016

Dynamic Mesh CFD Simulations of Orion Parachute Pendulum Motion During Atmospheric Entry

This paper demonstrates the usage of computational fluid dynamics to study the effects of pendulum motion dynamics of the NASAs Orion Multi-Purpose Crew Vehicle parachute system on the stability of the vehicles atmospheric entry and decent. Significant computational fluid dynamics testing has already been performed at NASAs Johnson Space Center, but this study sought to investigate the effect of bulk motion of the parachute, such as pitching, on the induced aerodynamic forces. Simulations were performed with a moving grid geometry oscillating according to the parameters observed in flight tests. As with the previous simulations, OVERFLOW computational fluid dynamics tool is used with the assumption of rigid, non-permeable geometry. Comparison to parachute wind tunnel tests is included for a preliminary validation of the dynamic mesh model. Results show qualitative differences in the flow fields of the static and dynamic simulations and quantitative differences in the induced aerodynamic forces, suggesting that dynamic mesh modeling of the parachute pendulum motion may uncover additional dynamic effects.

Halstrom, Logan D.↗

NASA NTRS2017

Implicit Formulation of Muscle Dynamics in OpenSim

Astronauts lose bone and muscle mass during spaceflight. Exercise countermeasure is the primary method for counteracting bone and muscle mass loss in space. New spacecraft exercise device concepts are currently being developed for the NASAs new crew exploration vehicle. The NASA Digital Astronaut Project (DAP) uses computational modeling to help determine if the new exercise devices will be effective as countermeasures. The NASA Digital Astronaut Project is developing the ability to utilize predictive simulation to provide insight into the change in kinematics and kinetics with a change in device and gravitational environment (1-g versus 0-g). For example, in space exercise the subject's body weight is applied in addition to the loads prescribed for musculoskeletal maintenance. How and where these loads are applied obviously directly impacts bone and tissue loads. Additionally, due to space vehicle structural requirements, exercise devices are often placed on vibration isolation systems. This changes the apparent impedance or stiffness of the device as seen by the user. Data collection under these conditions is often impractical and limited. Predictive modeling provides a means to have a virtual subject to test hypotheses. Predictive simulation provides a virtual subject for which we are able to perform studies such as sensitivity to device loading and vibration isolation without the need for laboratory kinematic or kinetic test data.Direct Collocation optimization provides an efficient means to perform task based optimization and predictive modeling. It is relatively straight forward to structure a physical exercise task in a Direct Collocation mathematical formulation: perform a motion such that you start at an initial pose, achieve a given amount of deflection i.e a squat, return to the initial pose, and minimize muscle activation cost. Direct Collocation is advantageous in that it does not require numerical integration to evaluate the objective function. Instead, the system dynamics are transformed to discrete time and the optimizer is constrained such that the solution is not considered to be a valid unless the dynamic equations are satisfied at all time points. The simulation and optimization are effectively done simultaneously. Due to the implicit integration, time steps can be more coarse than in a differential equation solver. In a gait scenario this means that that the model constraints and cost function are evaluated at 100 nodes in the gait cycle versus 10,000 integration steps in a variable-step forward dynamic simulation. Furthermore, no time is wasted on accurate simulations of movements that are far from the optimum. Constrained optimization algorithms require a Jacobian matrix that contains the partial derivatives of each of the dynamic constraints with respect to of each of the state and control variables at all time points. This is a large but sparse matrix. An implicit dynamics formulation requires computation of the dynamic residuals f as a function of the states x and their derivatives, and controls u:f(x, dxdt, u) 0If the dynamics of musculoskeletal system are formulated implicitly, the Jacobian elements are often available analytically, eliminating the need for numerical differentiation; this is obviously computationally advantageous. Additionally, implicit formulation of musculoskeletal dynamics do not suffer from singularities from low mass bodies, zero muscle activation, or other stiff system or

physical exercise↗

DOE OSTI2017

NetFlow Dynamics

NetFlow Dynamics is a web-accessible analysis environment for simulating dynamic flows of materials on model networks. Performing a simulation requires both the NetFlow Dynamics application and a network model which is a description of the structure of the nodes and edges of a network including the flow capacity of each edge and the storage capacity of each node, and the sources and sinks of the material flowing on the network. NetFlow Dynamics consists of databases for storing network models, algorithms to calculate flows on networks, and a GIS-based graphical interface for performing simulations and viewing simulation results. Simulated flows are dynamic in the sense that flows on each edge of the network and inventories at each node change with time and can be out of equilibrium with boundary conditions. Any number of network models could be simulated using Net Flow Dynamics. To date, the models simulated have been models of petroleum infrastructure. The main model has been the National Transportation Fuels Model (NTFM), a network of U.S. oil fields, transmission pipelines, rail lines, refineries, tank farms, and distribution terminals. NetFlow Dynamics supports two different flow algorithms, the Gradient Flow algorithm and the Inventory Control algorithm, that were developed specifically for the NetFlow Dynamics application. The intent is to add additional algorithms in the future as needed. The ability to select from multiple algorithms is desirable because a single algorithm never covers all analysis needs. The current algorithms use a demand-driven capacity-constrained formulation which means that the algorithms strive to use all available capacity and stored inventory to meet desired flows to sinks, subject to the capacity constraints of each network component. The current flow algorithms are best suited for problems in which a material flows on a capacity-constrained network representing a supply chain in which the material supplied can be stored at each node of the network. In the petroleum models, the flowing materials are crude oil and refined products that can be stored at tank farms, refineries, or terminals (i.e. the nodes of the network). Examples of other network models that could be simulated are currency flowing in a financial network, agricultural products moving to market, or natural gas flowing on a pipeline network.

Corbet Jr., Thomas F↗

DOE OSTI2020

Data-driven modeling and control of dynamical systems using Koopman and Perron-Frobenius operators

This dissertation studies the data-driven modeling and control problem of nonlinear systems by exploiting the linear operator theoretic framework involving Koopman and Perro-Frobenius operator. A systematic linear-operator based controller design procedure has been established, which can be used to solve a variety of nonlinear control problems, including feedback stabilization using control Lyapunov functions, optimal quadratic regulation using Koopman eigenfunctions and convex optimization formulation of optimal control problem using P-F and Koopman operator approximation. As the core of data-driven modeling, we first propose a new algorithm for the finite-dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time-series data. We argue that the existing approach for the finite-dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose preserves these two properties. We call the proposed algorithm as naturally structured DMD (NSDMD) since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However, preserving positivity property is critical for capturing the real transient dynamics of the system. This positivity property of the transfer operators and it's finite-dimensional approximation play an important role for controller and estimator design of nonlinear systems. To solve the feedback stabilization problem for nonlinear control systems, we tried to take advantage of the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this dissertation is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Next, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite-dimensional bilinear system. The optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite-dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. We applied a variety of examples and compared the simulation results between our framework and conventional LQR control using linearized model. For more general optimal control problems, we provide a density-function based convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the stability theory of a dynamical system involving density function and Perron-Frobenius operator. The optimal control problem is formulated as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the data-driven computation of the Koopman operator, which is dual to the Perron-Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.

Huang, Bowen↗

DOE OSTI2025

On the Recoverability of Reactor Dynamics from Point Kinetics Data using SINDYc

Data driven models for reactor dynamics tend to face a few notable challenges. First, the broad range of timescales present across the various feedback mechanisms. Next, high standards for safety and importance of model performance in all operational domains. Finally, the correlations between state variables observed in most transients leads to difficulties when methods attempt to attribute certain dynamic phenomena to a particular cause. The current paper seeks to support efforts towards incorporating physics knowledge into one particular data-driven method for finding reactor dynamics called ``Sparse Identification of Nonlinear Dynamics with Control (SINDYc). The incorporation of physical knowledge into SINDYc will help address the challenges listed above by giving the model an initial understanding of the system being modeled. The demonstration of an exact representation of a set of point reactor dynamics equations in SINDYc is provided. Then, this allows for further discussion with mathematical justification as to why SINDYc, and other data-driven methods, may be difficult to apply to nuclear reactor dynamics due to correlations in the state variables. A particularly useful result of this work is the set of candidate functions required in SINDYc to exactly represent the reactor dynamics under a point approximation. In the future, SINDYc can be integrated with point models for reactor dynamics before being applied to the physical system to yield higher accuracy.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2023

gLaSDI: Parametric physics-informed greedy latent space dynamics identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. Here, an interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a -nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors.

97 MATHEMATICS AND COMPUTING↗