An official website of the United States government
The Koopman operator theory provides a global linearization framework for general nonlinear dynamics, offering significant advantages for system analysis and control. However, practical applications typically involve approximating the infinite-dimensional Koopman operator in a lifted space spanned by a finite set of observable functions. The accuracy of this approximation is the key to effective Koopman operator-based analysis and control methods, generally improving as the dimension of the observables increases. Nonetheless, this increase in dimensionality significantly escalates both storage requirements and computational complexity, particularly for high-dimensional systems, thereby limiting the applicability of these methods in real-world problems. In this paper, we address this problem by reformulating the Koopman operator in tensor format to break the curse of dimensionality associated with its approximation through tensor decomposition techniques. This effective reduction in complexity enables the selection of high-dimensional observable functions and the handling of large-scale datasets, which leads to a precise linear prediction model utilizing the tensor-based Koopman operator. Furthermore, we propose an optimal control framework with the tensor-based Koopman operator, which adeptly addresses the nonlinear dynamics and constraints by linear reformulation in the lifted space and significantly reduces the computational complexity through separated representation of the tensor structure.
more »
« less
Model Order Reduction for Water Quality Dynamics
Abstract A state‐space representation of water quality (WQ) dynamics describing disinfectant (e.g., chlorine) transport dynamics in drinking water distribution networks has been recently proposed. Such representation is a byproduct of space‐ and time‐discretization of the partial differential equations modeling transport dynamics. This results in a large state‐space dimension even for small networks with tens of nodes. Although such a state‐space model provides a model‐driven approach to predict WQ dynamics, incorporating it into model‐based control algorithms or state estimators for large networks is challenging and at times intractable. To that end, this paper investigates model order reduction (MOR) methods for WQ dynamics with the objective of performing post‐reduction feedback control. The presented investigation focuses on reducing state‐dimension by orders of magnitude, the stability of the MOR methods, and the application of these methods to model predictive control.
more »
« less
- PAR ID:
- 10418966
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Water Resources Research
- Volume:
- 58
- Issue:
- 4
- ISSN:
- 0043-1397
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Real‐time water quality control (WQC) in water distribution networks (WDN), the problem of regulating disinfectant levels, is challenging due to lack of (i) a proper control‐oriented modeling considering complicated components (junctions, reservoirs, tanks, pipes, pumps, and valves) for water quality modeling in WDN and (ii) a corresponding scalable control algorithm that performs real‐time water quality regulation. In this paper, we solve the WQC problem by (a) proposing a novel state‐space representation of the WQC problem that provides an explicit relationship between inputs (chlorine dosage at booster stations) and states/outputs (chlorine concentrations in the entire network) and (b) designing a highly scalable model predictive control (MPC) algorithm that showcases fast response time and resilience against some sources of uncertainty.more » « less
-
Chambers, Erin W.; Gudmundsson, Joachim (Ed.)Datasets with non-trivial large scale topology can be hard to embed in low-dimensional Euclidean space with existing dimensionality reduction algorithms. We propose to model topologically complex datasets using vector bundles, in such a way that the base space accounts for the large scale topology, while the fibers account for the local geometry. This allows one to reduce the dimensionality of the fibers, while preserving the large scale topology. We formalize this point of view and, as an application, we describe a dimensionality reduction algorithm based on topological inference for vector bundles. The algorithm takes as input a dataset together with an initial representation in Euclidean space, assumed to recover part of its large scale topology, and outputs a new representation that integrates local representations obtained through local linear dimensionality reduction. We demonstrate this algorithm on examples coming from dynamical systems and chemistry. In these examples, our algorithm is able to learn topologically faithful embeddings of the data in lower target dimension than various well known metric-based dimensionality reduction algorithms.more » « less
-
Abstract Model reduction methods usually focus on the error performance analysis; however, in presence of uncertainties, it is important to analyze the robustness properties of the error in model reduction as well. This problem is particularly relevant for engineered biological systems that need to function in a largely unknown and uncertain environment. We give robustness guarantees for structured model reduction of linear and nonlinear dynamical systems under parametric uncertainties. We consider a model reduction problem where the states in the reduced model are a strict subset of the states of the full model, and the dynamics for all of the other states are collapsed to zero (similar to quasi‐steady‐state approximation). We show two approaches to compute a robustness guarantee metric for any such model reduction—a direct linear analysis method for linear dynamics and a sensitivity analysis based approach that also works for nonlinear dynamics. Using the robustness guarantees with an error metric and an input‐output mapping metric, we propose an automated model reduction method to determine the best possible reduced model for a given detailed system model. We apply our method for the (1) design space exploration of a gene expression system that leads to a new mathematical model that accounts for the limited resources in the system and (2) model reduction of a population control circuit in bacterial cells.more » « less
-
In this work we determine the time-domain dynamics of a complex mechanical network of integer-order components, e.g., springs and dampers, with an overall transfer function described by implicitly defined operators. This type of transfer functions can be used to describe very large scale dynamics of robot formations, multi-agent systems or viscoelastic phenomena. Such large-scale integrated systems are becoming increasingly important in modern engineering systems, and an accurate model of their dynamics is very important to achieve their control. We give a time domain representation for the dynamics of the system by using a complex variable analysis to find its impulse response. Furthermore, we validate how our infinite order model can be used to describe dynamics of finite order networks, which can be useful as a model reduction method.more » « less
