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Results for “linear time-invariant dynamical systems”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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9 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

A compute-bound formulation of Galerkin model reduction for linear time-invariant dynamical systems

This work aims to advance computational methods for projection-based reduced-order models (ROMs) of linear time-invariant (LTI) dynamical systems. For such systems, current practice relies on ROM formulations expressing the state as a rank-1 tensor (i.e., a vector), leading to computational kernels that are memory bandwidth bound and, therefore, ill-suited for scalable performance on modern architectures. This weakness can be particularly limiting when tackling many-query studies, where one needs to run a large number of simulations. This work introduces a reformulation, called rank-2 Galerkin, of the Galerkin ROM for LTI dynamical systems which converts the nature of the ROM problem from memory bandwidth to compute bound. We present the details of the formulation and its implementation, and demonstrate its utility through numerical experiments using, as a test case, the simulation of elastic seismic shear waves in an axisymmetric domain. We quantify and analyze performance and scaling results for varying numbers of threads and problem sizes. In conclusion, we present an end-to-end demonstration of using the rank-2 Galerkin ROM for a Monte Carlo sampling study. We show that the rank-2 Galerkin ROM is one order of magnitude more efficient than the rank-1 Galerkin ROM (the current practice) and about 970 times more efficient than the full-order model, while maintaining accuracy in both the mean and statistics of the field.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Fast and Scalable FFT-Based GPU-Accelerated Algorithms for Block-Triangular Toeplitz Matrices with Application to Linear Inverse Problems Governed by Autonomous Dynamical Systems

In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Reinforcement Learning of Structured Stabilizing Control for Linear Systems With Unknown State Matrix

This paper delves into designing feedback control gains for a continuous-time linear quadratic regulator (LQR) problem that is constrained to certain predefined structure with unknown state matrix. We bring forth the ideas from reinforcement learning (RL) in conjunction with sufficient stability and performance guarantees in order to design these structured gains using the trajectory measurements of states and controls. Here we first formulate a model-based framework using dynamic programming (DP) to embed the structural constraint to the LQR gain computation in the continuous-time setting, and then subsequently, formulate a policy iteration RL algorithm that can alleviate the requirement of known state matrix in conjunction with maintaining the feedback gain structure. The design enables a distributed learning control design which is necessary for many large-scale cyber-physical systems. Theoretical guarantees are provided for stability and convergence of the structured reinforcement learning (SRL) algorithm. We validate our theoretical results with numerical simulations on a multi-agent networked linear time-invariant (LTI) dynamic system.

42 ENGINEERING↗
DOE OSTI2026

Goal-oriented real-time Bayesian inference for linear autonomous dynamical systems with application to digital twins for tsunami early warning

We present a goal-oriented framework for constructing digital twins with the following properties: (1) they employ discretizations of high-fidelity partial differential equation (PDE) models governed by autonomous dynamical systems, leading to large-scale forward problems; (2) they solve a linear inverse problem to assimilate observational data to infer uncertain model components followed by a forward prediction of the evolving dynamics; and (3) the entire end-to-end, data-to-inference-to-prediction computation is carried out without approximation and in real time through a Bayesian framework that rigorously accounts for uncertainties. Several challenges must be overcome to realize this framework, including the large scale of the forward problem, the high dimensionality of the parameter space, and for a class of problems including those we target, the slow decay of the singular values of the parameter-to-observable map. Here we introduce a methodology to overcome these challenges by exploiting the autonomous structure of the forward model to decompose the solution of the inverse problem into a one-time-only offline phase in which the PDE model is solved a limited number of times (equal to the number of sensors), and an online phase that maps well onto GPUs and computes the parameter inference and prediction of quantities of interest in real time, given observational data. Our ultimate goal is to apply this framework to construct digital twins for subduction zones, including Cascadia, to provide early warning for tsunamis generated by megathrust earthquakes. To this end, we demonstrate how our methodology can be used to employ seafloor pressure observations, along with the coupled acoustic–gravity wave equations, to infer the earthquake-induced spatiotemporal seafloor motion (discretized with $\mathscr{O}$ (10 9 ) parameters) and forward predict the tsunami propagation. We present results of an end-to-end inference, prediction, and uncertainty quantification for a representative test problem with $\mathscr{O}$ (10 8 ) inversion parameters for which goal-oriented Bayesian inference is accomplished exactly and in real time, that is, in a matter of seconds.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Efficient Topology Design Algorithms for Power Grid Stability

The dynamic response of power grids to small disturbances influences their overall stability. This letter examines the effect of network topology on the linearized time-invariant dynamics of electric power systems. The proposed framework utilizes H 2 -norm based stability metrics to study the optimal placement of lines on existing networks as well as the topology design of new networks. The design task is first posed as an NP-hard mixed-integer nonlinear program (MINLP) that is exactly reformulated as a mixed-integer linear program (MILP) using McCormick linearization. To improve computation time, graph-theoretic properties are exploited to derive valid inequalities (cuts) and tighten bounds on the continuous optimization variables. Moreover, a cutting plane generation procedure is put forth that is able to interject the MILP solver and augment additional constraints to the problem on-the-fly. Finally, the efficacy of our approach in designing optimal grid topologies is demonstrated through numerical tests on the IEEE 39-bus network.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

A Secure Learning Control Strategy via Dynamic Camouflaging for Unknown Dynamical Systems under Attacks

This paper presents a secure reinforcement learning (RL) based control method for unknown linear time-invariant cyber-physical systems (CPSs) that are subjected to compositional attacks such as eavesdropping and covert attack. We consider the attack scenario where the attacker learns about the dynamic model during the exploration phase of the learning conducted by the designer to learn a linear quadratic regulator (LQR), and thereafter, use such information to conduct a covert attack on the dynamic system, which we refer to as doubly learning-based control and attack (DLCA) framework. We propose a dynamic camouflaging based attack-resilient reinforcement learning (ARRL) algorithm which can learn the desired optimal controller for the dynamic system, and at the same time, can inject sufficient misinformation in the estimation of system dynamics by the attacker. The algorithm is accompanied by theoretical guarantees and extensive numerical experiments on a consensus multi-agent system and on a benchmark power grid model.

Mukherjee, Sayak↗
DOE OSTI2021

Model-based and Model-free Designs for an Extended Continuous-time LQR with Exogenous Inputs

We present an extended linear quadratic regulator (LQR) design for continuous-time linear time-invariant (LTI) systems in the presence of exogenous inputs. We first propose a model-based solution with cost minimization guarantees for states and inputs using dynamic programming (DP). The control law consists of a combination of the optimal state feedback and an additional optimal term dependent on the exogenous inputs. The control gains for the two components are obtained by solving a set of matrix differential equations. We provide these solutions for both finite horizons and steady-state cases. In the second part of the paper, we formulate a reinforcement learning (RL) based algorithm which does not need any model information except the input matrix, and can compute an approximate steady-state LQR gain using measurements of the states, the control inputs, and the exogenous inputs. Both model-based and data-driven optimal control algorithms are tested with a numerical example under different exogenous inputs showcasing the effectiveness of the designs.

Mukherjee, Sayak↗
DOE OSTI2023

Implementation of a feature selection algorithm in FARM to identify important state variables and time-invariant matrices

The FARM (Feasible Actuator Range Modifier) software module is a component of the RAVEN-based FORCE framework for analysis of Integrated Energy Systems (IES). FARM aids the HERON software module in the evaluation of the optimal dispatch for the different IES components. Set-point trajectories are required to meet limits on both production variables (i.e., the variables to be optimized such as the electrical power, the hydrogen production rate, etc.) and process variables tied to the service life of equipment (e.g., steam flowrate, vessel pressure, turbine firing temperature, etc.). To evaluate the feasibility of HERON generated set-points and to do so in an acceptable time, FARM employs reduced order models to represent the dynamic behavior of the systems to be dispatched. These surrogate models take the form of a linear dynamic system with sets of Linear Parameter Varying (LPV) matrices that are mapped to the system operating space. These matrices are derived from the trajectories of system state variables and system output variables during transients. The accuracy of LPV matrices depends on the selection of state variables. In previous reports, state variables were selected by adopting a complicated workflow requiring multiple software licenses and an advanced level of user expertise. In this report, a new workflow that automates the state variable selection process is presented. It significantly reduces the frequency of user interventions and does not require multiple software licenses. Each module in the new workflow is described in detail, and the input / output examples in each step of the workflow are provided. It was demonstrated that this workflow can greatly reduce the complexity of the state variable selection process, and that the updated FARM-Gamma and FARM-Delta validators can benefit from this workflow when solving the power dispatch problem of a representative IES test case. Finally, some code improvements that can further enhance the efficiency are suggested.

42 ENGINEERING↗