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After showing that a complex dynamic system with a competitive structure has highly reliable stability, a class of noncompetitive dynamic systems for which competitive models can be constructed is defined. It is shown that such a construction is possible in the context of the hierarchic stability analysis. The scheme is based on the comparison principle and vector Liapunov functions.
This work proposes a space--time least-squares Petrov--Galerkin (ST-LSPG) projection method for model reduction of nonlinear dynamical systems. In contrast to typical nonlinear model-reduction methods that first apply (Petrov--)Galerkin projection in the spatial dimension and subsequently apply time integration to numerically resolve the resulting low-dimensional dynamical system, the proposed method applies projection in space and time simultaneously. To accomplish this, the method first introduces a low-dimensional space--time trial subspace, which can be obtained by computing tensor decompositions of state-snapshot data. The method then computes discrete-optimal approximations in this space--time trial subspace by minimizing the residual arising after time discretization over all space and time in a weighted $\ell^2$-norm. This norm can be defined to enable complexity reduction (i.e., hyper-reduction) in time, which leads to space--time collocation and space--time Gauss--Newton with Approximated Tensors (GNAT) variants of the ST-LSPG method. Advantages of the approach relative to typical spatial-projection-based nonlinear model reduction methods such as Galerkin projection and least-squares Petrov--Galerkin projection include a reduction of both the spatial and temporal dimensions of the dynamical system, and a priori error bounds that bound the solution error by the best space--time approximation error and whose stability constants exhibit slower growth in time. Numerical examples performed on model problems in fluid dynamics demonstrate the ability of the method to generate orders-of-magnitude computational savings relative to spatial-projection-based reduced-order models without sacrificing accuracy for a fixed spatio-temporal discretization.
The Dynamic Isotope Power Systems (DIPS) demonstration program, sponsored by the U.S. Department of Energy (DOE) with support funding from the National Aeronautics and Space Administration (NASA), is currently focused on the development of a standardized 2.5 kWe portable generator for multiple applications on the lunar or Martian surface. A variety of potential remote and mobile applications has been identified by NASA including surface rovers for both short and extended duration missions, remote power to science packages, and backup to central base power. Recent work focused on refining the 2.5 kWe design including assessing compatibility with the Martian environment to assure the design is suitable while imposing only a minor mass penalty on lunar operations. Additional work included a study performed to compare the DIPS with regenerative fuel cell systems for lunar mobile and remote power systems. Power requirements were reviewed and a modular system chosen for the comparison. Finally, a plan and cycle schematic were generated for an early demonstration of a prototypic isotope power Brayton system using the Antarctic as the test bed.
We propose a method for the investigation of periodic solutions of autonomous dynamical systems described by ordinary differential equations with phase and integral restrictions. We formulate the general problem of periodic solutions as a boundary-value problem with restrictions. By introducing a fictitious control, we transform the boundary-value problem into a control problem for dynamical systems with phase and integral restrictions. The control problem is solved by reducing it to an integral Fredholm equation of the first kind. We establish necessary and sufficient conditions for the existence of periodic solutions and propose an algorithm for finding periodic solution according to the limit points of the minimizing sequences.
The problem of acquiring a simple but sufficiently accurate model of a dynamic system is made more difficult when the dynamic system of interest is a multibody system comprised of several components. A low order system model may be created by reducing the order of the component models and making use of various available multibody dynamics programs to assemble them into a system model. The difficulty is in choosing the reduced order component models to meet system level requirements. The projection and assembly method, proposed originally by Eke, solves this difficulty by forming the full order system model, performing model reduction at the the system level using system level requirements, and then projecting the desired modes onto the components for component level model reduction. The projection and assembly method is analyzed to show the conditions under which the desired modes are captured exactly; to the numerical precision of the algorithm.
Purpose: To evaluate the lateral beam penumbra in pencil beam scanning proton therapy delivered using a dynamic collimator device capable of trimming a portion of the primary beam in close proximity to the patient. Methods: Monte Carlo simulations of pencil beams were performed using MCNPX. Each simulation transported a 125 MeV proton pencil beam through a range shifter, past acollimator, and into a water phantom. Two parameters were varied among the simulations, the source beam size (sigma in air from 3 to 9 mm), and the position of the edge of the collimator (placed from 0 to 30 mm from the central axis of the beam). Proton flux was tallied at the phantom surface to determine the effective beam sizefor all combinations of source beam size and collimator edge position. Results: Quantifying beam size at the phantom surface provides a useful measure tocompare performance among varying source beam sizes and collimation conditions. For arelatively large source beam size (9 mm) entering the range shifter, sigma at thesurface was found to be 10 mm without collimation versus 4 mm with collimation. Additionally, sigma at the surface achievable with collimation was found to be smallerthan for any uncollimated beam, even for very small source beam sizes. Finally, thelateral penumbra achievable with collimation was determined to be largely independentof the source beam size. Conclusion: Collimation can significantly reduce proton pencil beam lateral penumbra.Given the known dosimetric disadvantages resulting from large beam spot sizes,employing a dynamic collimation system can significantly improve lateral tissuesparing in spot-scanned dose distributions.
We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This definition generalizes the construction of Hamiltonian systems by a Hamiltonian and a symplectic form. Some low dimensional examples are given, as well as a discretized field theory with a large number of degrees of freedom and a local nearest neighbor interaction. We also evaluate unequal-time correlations of these systems without direct numerical simulation, by Padé approximants of a short-time expansion. We briefly speculate on the possibility of constructing chaotic dynamical systems with non-integer dimension and exactly known statistics. In this case there is no probability density, suggesting an alternative construction in terms of a Hopf characteristic function and a two-form.
The Advanced Closed Loop System (ACLS) is currently under development by Airbus Defense and Space and is slated for launch to the International Space Station (ISS) in 2017. The addition of new hardware into an already complex system such as the ISS life support system (LSS) always poses operational risks. It is therefore important to understand the impacts ACLS will have on the existing systems to ensure smooth operations for the ISS. This analysis can be done by using dynamic computer simulations and one possible tool for such a simulation is Virtual Habitat (V-HAB). Based on Matlab (Registered Trademark) V-HAB has been under development at the Institute of Astronautics of the Technical University Munich (TUM) since 2006 and in the past has been successfully used to simulate the ISS life support systems. The existing V-HAB ISS simulation model treated the interior volume of the space station as one large ideally-stirred container. This model was improved to allow the calculation of the atmospheric composition inside the individual modules of the ISS by splitting it into ten distinct volumes. The virtual volumes are connected by a simulation of the inter-module ventilation flows. This allows for a combined simulation of the LSS hardware and the atmospheric composition aboard the ISS. A dynamic model of ACLS is added to the ISS simulation and different operating modes for both ACLS and the existing ISS life support systems are studied to determine the impacts of ACLS on the rest of the system. The results suggest that the US, Russian and ACLS CO2 systems can operate at the same time without impeding each other. Furthermore, based on the results of this analysis, the US and ACLS Sabatier systems can be operated in parallel as well to achieve the highest possible CO2 recycling together with a low CO2 concentration.
The Advanced Closed Loop System (ACLS) is currently under development by Airbus Defense and Space and is slated for launch to the International Space Station (ISS) in 2017. The addition of new hardware into an already complex system such as the ISS life support system (LSS) always poses operational risks. It is therefore important to understand the impacts ACLS will have on the existing systems to ensure smooth operations for the ISS. This analysis can be done by using dynamic computer simulations and one possible tool for such a simulation is the Virtual Habitat (V-HAB). Based on MATLAB, V-HAB has been under development at the Institute of Astronautics of the Technical University of Munich (TUM) since 2004 and in the past has been successfully used to simulate the ISS life support systems. The existing V-HAB ISS simulation model treated the interior volume of the space station as one large, ideally-stirred container. This model was improved to allow the calculation of the atmospheric composition inside individual modules of the ISS by splitting it into twelve distinct volumes. The virtual volumes are connected by a simulation of the inter-module ventilation flows. This allows for a combined simulation of the LSS hardware and the atmospheric composition aboard the ISS. A dynamic model of ACLS is added to the ISS Simulation and several different operating modes for both ACLS and the existing ISS life support systems are studied and the impacts of ACLS on the rest of the system are determined. The results suggest that the US, Russian and ACLS CO2 systems can operate at the same time without impeding each other. Furthermore, based on the results of this analysis, the US and ACLS Sabatier systems can be operated in parallel as well to a achieve a very low CO2 concentration in the cabin atmosphere.
In this paper, a deep learning-based multiple model estimation framework is presented for the state estimation of hybrid dynamical systems from high dimensional observations such as camera images. A low dimensional vector which represents the measurement of the latent dynamical system and its corresponding variance are learned using a deep encoder neural network. An Interacting Multiple Model (IMM) filter is used to generate the latent state estimates and covariances using multiple dynamical models, which can be learned using backpropagation through time. The state estimates of the dynamical system and the corresponding covariance matrix are generated from the latent state estimates and covariance using a deep decoder neural network. The whole network is trained in an end-to-end manner using a loss function which minimizes the negative log-likelihood of the neural network parameters. Simulation results are presented using a 2D bouncing ball example and estimation error statistics are computed which demonstrates the accuracy and consistency of the estimation.
In this paper we address the problem of computing the numerical solution to kinetic partial differential equations involving many phase variables. These types of equations arise naturally in many different areas of mathematical physics, e.g., in particle systems (Liouville and Boltzmann equations), stochastic dynamical systems (Fokker–Planck and Dostupov–Pugachev equations), random wave theory (Malakhov–Saichev equations) and coarse-grained stochastic systems (Mori–Zwanzig equations). We propose three different classes of new algorithms addressing high-dimensionality: The first one is based on separated series expansions resulting in a sequence of low-dimensional problems that can be solved recursively and in parallel by using alternating direction methods. The second class of algorithms relies on truncation of interaction in low-orders that resembles the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) framework of kinetic gas theory and it yields a hierarchy of coupled probability density function equations. The third class of algorithms is based on high-dimensional model representations, e.g., the ANOVA method and probabilistic collocation methods. A common feature of all these approaches is that they are reducible to the problem of computing the solution to high-dimensional equations via a sequence of low-dimensional problems. The effectiveness of the new algorithms is demonstrated in numerical examples involving nonlinear stochastic dynamical systems and partial differential equations, with up to 120 variables.
The paper presents a computer program developed to model the steady-state performance of the tapered artery heat pipe for use in the radiator of the solar dynamic power system of the NASA Space Station. The program solves six governing equations to ascertain which one is limiting the maximum heat transfer rate of the heat pipe. The present model appeared to be slightly better than the LTV model in matching the 1-g data for the standard 15-ft test heat pipe.
The enhancement of images that are characterized only by statistical data where the picture contains additive noise is considered. The random process representing the output of the scanner is characterized by the output of a dynamic system with white noise input. The dynamic system describes a first-order vector-Markov process. The procedure of Kalman filtering is then utilized to recursively determine the minimum mean-square error estimate of the image. The result is then extended to obtain the smoothing of the data.
Taylor's formula shows how to approximate a certain class of functions by polynomials. The approximations are arbitrarily good in some neighborhood whenever the function is analytic and they are easy to compute. The main goal is to give an efficient algorithm to approximate a neighborhood of the configuration space of a dynamical system by a nilpotent, explicitly integrable dynamical system. The major areas covered include: an approximating map; the generalized Baker-Campbell-Hausdorff formula; the Picard-Taylor method; the main theorem; simultaneous integration of trajectories; and examples.
Air cargo companies and aircraft manufacturers are making significant investments to enable the movement of cargo via various levels of automated aircraft, such as aircraft with simplified operations requiring a pilot, remotely monitored or piloted aircraft, and fully autonomous aircraft. These investments will enable greater utilization of aircraft while unlocking new air markets traditionally served by ground transportation only. Many cargo companies and aerospace experts envision an operating environment where a single pilot can remotely pilot numerous aircraft for significant increases in aircraft utilization. The future operating environment is also expected to include air cargo companies flying smaller aircraft from airports and distribution centers outside of major U.S. cities directly to city centers, avoiding congested roads and increasing the velocity of cargo shipments, particularly those that are high-value, time-sensitive, and security sensitive (e.g., pharmaceuticals). These are just a few benefits and use cases cargo companies and aerospace experts see with the advancement of automated aircraft. To better understand industry’s direction, NASA asked the LMI team to research the forecasted market, timeline, risks, and opportunities for integrating unmanned air cargo vehicles into the National Airspace System (NAS) for the development and prioritization of the NASA Air Traffic Management Exploration’s research portfolio. To begin the market assessment, we gathered data via numerous interviews with key stakeholders and subject matter experts and literature reviews. We then incorporated the data into a custom-developed systems dynamics model and visualization dashboard. The systems dynamics model classifies the size of the market (e.g., overall fleet size of automated aircraft) for four distinct use cases over the next 20 years. The model projects the year in which various types of automated aircraft will enter the commercial cargo market based on our team’s collective research on when the aircraft will become viable due to manufacturing and certification timelines and the lifespan of current, traditional aircraft in service, to name a few factors. While this report defines our team’s estimated timeline of entry and growth, the model is dynamic—it enables NASA users to change variables based on future-year events. If the necessary technology does not mature in accordance with our assumptions, then NASA can change the entry of service point to a future year to evaluate the changes in market size in the out years. This flexibility will be key to deciding when and how NASA should invest in various areas.
The fractional dynamic process of a quantum system is a novel and complicated problem. The establishment of a fractional dynamic model is a significant attempt that is expected to reveal the mechanism of fractional quantum system. In this paper, a generalized time fractional Schrödinger equation is proposed. To study the fractional dynamics of quantum systems, we take the two-level system as an example and derive the time fractional equations of motion. The basic properties of the system are investigated by solving this set of equations in the absence of light field analytically. Then, when the system is subject to the light field, the equations are solved numerically. It shows that the two-level system described by the time fractional Schrödinger equation we proposed is a confirmable system.
The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD.
This report summarizes the activities of this project, which aims at design and testing a scalable, robust, and online framework that provides secure monitoring of photovoltaic generation in the face of potential cyber-attacks.