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Filtering for nonlinear dynamical systems.
Filtering for nonlinear dynamical systems with white Gaussian noise processes
Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations
Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.
Data Systems Dynamic Simulator
The Data System Dynamic Simulator (DSDS) is a discrete event simulation tool. It was developed for NASA for the specific purpose of evaluating candidate architectures for data systems of the Space Station era. DSDS provides three methods for meeting this requirement. First, the user has access to a library of standard pre-programmed elements. These elements represent tailorable components of NASA data systems and can be connected in any logical manner. Secondly, DSDS supports the development of additional elements. This allows the more sophisticated DSDS user the option of extending the standard element set. Thirdly, DSDS supports the use of data streams simulation. Data streams is the name given to a technique that ignores packet boundaries, but is sensitive to rate changes. Because rate changes are rare compared to packet arrivals in a typical NASA data system, data stream simulations require a fraction of the CPU run time. Additionally, the data stream technique is considerably more accurate than another commonly-used optimization technique.
Determination of eigenvalues of dynamical systems by symbolic computation
A symbolic computation technique for determining the eigenvalues of dynamical systems is described wherein algebraic operations, symbolic differentiation, matrix formulation and inversion, etc., can be performed on a digital computer equipped with a formula-manipulation compiler. An example is included that demonstrates the facility with which the system dynamics matrix and the control distribution matrix from the state space formulation of the equations of motion can be processed to obtain eigenvalue loci as a function of a system parameter. The example chosen to demonstrate the technique is a fourth-order system representing the longitudinal response of a DC 8 aircraft to elevator inputs. This simplified system has two dominant modes, one of which is lightly damped and the other well damped. The loci may be used to determine the value of the controlling parameter that satisfied design requirements. The results were obtained using the MACSYMA symbolic manipulation system.
Solving differential‐algebraic equations in power system dynamic analysis with quantum computing
Abstract Power system dynamics are generally modeled by high dimensional non‐linear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system non‐linearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.
Vulnerability of dynamic systems
Directed graphs are associated with dynamic systems in order to determine in any given system if each state can be reached by at least one input (input reachability), or can each state reach at least one output (output reachability). Then, the structural perturbations of a dynamic system are identified as lines or points removals from the corresponding digraph, and a system is considered vulnerable at those lines or points of the digraph whose removal destroys its input or output reachability. A suitable framework is formulated for resolving the problems of reachability and vulnerability which applies to both linear and nonlinear systems alike.
Sensitivity-Driven Experimental Design to Facilitate Control of Dynamical Systems
Control of nonlinear dynamical systems is a complex and multifaceted process. Essential elements of many engineering systems include high-fidelity physics-based modeling, offline trajectory planning, feedback control design, and data acquisition strategies to reduce uncertainties. Here this article proposes an optimization-centric perspective which couples these elements in a cohesive framework. We introduce a novel use of hyper-differential sensitivity analysis to understand the sensitivity of feedback controllers to parametric uncertainty in physics-based models used for trajectory planning. These sensitivities provide a foundation to define an optimal experimental design which seeks to acquire data most relevant in reducing demand on the feedback controller. Our proposed framework is illustrated on the Zermelo navigation problem and a hypersonic trajectory control problem using data from NASA’s X-43 hypersonic flight tests.
Feasibility study of a tungsten water moderated nuclear rocket. 7 - System dynamics and control analysis
System dynamics and control analysis of tungsten water moderated nuclear rocket
A matrix approach to parameter analysis of dynamical systems
Parameter analysis of dynamical systems by matrix approach, using n-dimensional parameter space
The investigation of tethered satellite system dynamics
Progress in tethered satellite system dynamics research is reported. A retrieval rate control law with no angular feedback to investigate the system's dynamic response was studied. The initial conditions for the computer code which simulates the satellite's rotational dynamics were extended to a generic orbit. The model of the satellite thrusters was modified to simulate a pulsed thrust, by making the SKYHOOK integrator suitable for dealing with delta functions without loosing computational efficiency. Tether breaks were simulated with the high resolution computer code SLACK3. Shuttle's maneuvers were tested. The electric potential around a severed conductive tether with insulator, in the case of a tether breakage at 20 km from the Shuttle, was computed. The electrodynamic hazards due to the breakage of the TSS electrodynamic tether in a plasma are evaluated.
The investigation of tethered satellite system dynamics
Tethered satellite system (TSS) dynamics were studied. The dynamic response of the TSS during the entire stationkeeping phase for the first electrodynamic mission was investigated. An out of plane swing amplitude and the tether's bowing were observed. The dynamics of the slack tether was studied and computer code, SLACK2, was improved both in capabilities and computational speed. Speed hazard related to tether breakage or plasma contactor failure was examined. Preliminary values of the potential difference after the failure and of the drop of the electric field along the tether axis have been computed. The update of the satellite rotational dynamics model is initiated.
Probabilistic evolution of stochastic dynamical systems: A meso-scale perspective
Stochastic dynamical systems arise naturally across nearly all areas of science and engineering. Typically, a dynamical system model is based on some prior knowledge about the underlying dynamics of interest in which probabilistic features are used to quantify and propagate uncertainties associated with the initial conditions, external excitations, etc. From a probabilistic modeling standing point, two broad classes of methods exist, i.e. macro-scale methods and micro-scale methods. Classically, macro-scale methods such as statistical moments-based strategies are usually too coarse to capture the multi-mode shape or tails of a non-Gaussian distribution. Micro-scale methods such as random samples-based approaches, on the other hand, become computationally very challenging in dealing with high-dimensional stochastic systems. In view of these potential limitations, a meso-scale scheme is proposed here that utilizes a meso-scale statistical structure to describe the dynamical evolution from a probabilistic perspective. The significance of this statistical structure is twofold. First, it can be tailored to any arbitrary random space. Second, it not only maintains the probability evolution around sample trajectories but also requires fewer meso-scale components than the micro-scale samples. To demonstrate the efficacy of the proposed meso-scale scheme, a set of examples of increasing complexity are provided. Connections to the benchmark stochastic models as conservative and Markov models along with practical implementation guidelines are presented.
Application of dynamical systems theory to the high angle of attack dynamics of the F-14
Dynamical systems theory has been used to study the nonlinear dynamics of the F-14. An eight degree of freedom model that does not include the control system present in operational F-14s has been analyzed. The aerodynamic model, supplied by NASA, includes nonlinearities as functions of the angles of attack and sideslip, the rotation rate, and the elevator deflection. A continuation method has been used to calculate the steady states of the F-14 as continuous functions of the control surface deflections. Bifurcations of these steady states have been used to predict the onset of wing rock, spiral divergence, and jump phenomena which cause the aircraft to enter a spin. A simple feedback control system was designed to eliminate the wing rock and spiral divergence instabilities. The predictions were verified with numerical simulations.
Optimal post-experiment estimation of poorly modeled dynamic systems
Recently, a novel strategy for post-experiment state estimation of discretely-measured dynamic systems has been developed. The method accounts for errors in the system dynamic model equations in a more general and rigorous manner than do filter-smoother algorithms. The dynamic model error terms do not require the usual process noise assumptions of zero-mean, symmetrically distributed random disturbances. Instead, the model error terms require no prior assumptions other than piecewise continuity. The resulting state estimates are more accurate than filters for applications in which the dynamic model error clearly violates the typical process noise assumptions, and the available measurements are sparse and/or noisy. Estimates of the dynamic model error, in addition to the states, are obtained as part of the solution of a two-point boundary value problem, and may be exploited for numerous reasons. In this paper, the basic technique is explained, and several example applications are given. Included among the examples are both state estimation and exploitation of the model error estimates.
Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. In this work, we describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.
The dynamics of software development project management: An integrative systems dynamic perspective
Rather than continuing to focus on software development projects per se, the system dynamics modeling approach outlined is extended to investigate a broader set of issues pertaining to the software development organization. Rather than trace the life cycle(s) of one or more software projects, the focus is on the operations of a software development department as a continuous stream of software products are developed, placed into operation, and maintained. A number of research questions are ""ripe'' for investigating including: (1) the efficacy of different organizational structures in different software development environments, (2) personnel turnover, (3) impact of management approaches such as management by objectives, and (4) the organizational/environmental determinants of productivity.
Invariant hyperplanes for linear dynamical systems
Invariant hyperplanes for linear dynamical systems