This paper presents data-driven implementations of balanced truncation and several of its generalizations that rely exclusively on transfer function samples on the imaginary axis. Rather than implicitly approximating the Gramians via numerical quadrature, the proposed approach approximates them implicitly through projection. This enables multiple members of the balanced truncation family to be implemented non-intrusively using practically measurable data, without requiring spectral factorizations. Using this projection-based framework, data-driven implementations are developed for standard balanced truncation, frequency-limited balanced truncation, time-limited balanced truncation, self-weighted balanced truncation, LQG balanced truncation, H-infinity balanced truncation, positive-real balanced truncation, bounded-real balanced truncation, and stochastic balanced truncation. Numerical results demonstrate that the proposed non-intrusive implementations achieve performance comparable to their intrusive counterparts and accurately capture the dominant Hankel singular values.
Compression and Distillation of Data Quadruplets in Non-intrusive Reduced-order Modeling
Yangtze University（长江大学）；Hubei Normal University（湖北师范大学）；Wuhan University of Science and Technology（武汉科技大学）；Shanxi University（山西大学）；The University of Western Australia（西澳大学）
Balanced truncation (BT) is a model reduction method that utilizes a coordinate transformation to retain eigen-directions that are highly observable and reachable. To address realizability and scalability of BT applied to highly stiff and lightly-damped systems, a non-intrusive data-driven method is developed for balancing discrete-time systems via the eigensystem realization algorithm (ERA). The advantage of ERA for balancing transformation makes full-state outputs tractable. Further, ERA enables balancing despite stiffness, by eliminating computation of balancing modes and adjoint simulation…
In this work, we present a nonlinear model reduction approach for reducing two commonly used nonlinear dynamical models of power grids: the effective network (EN) model and the synchronous motor (SM) model. Such models are essential in real-time security assessments of power grids. However, as power grids are often large-scale, it is necessary to reduce the models in order to utilize them in real-time. We reformulate the nonlinear power grid models as quadratic systems and reduce them using balanced truncation based on approximations of the reachability and observability Gramians. Finally, we …
We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.
The paper presents a model reduction framework geared towards the analysis and design of systems that switch and oscillate. While such phenomena are ubiquitous in nature and engineering, model reduction methods are not well developed for non-equilibrium behaviors. The proposed framework addresses this need by exploiting recent advances on dominance theory. Classical balanced truncation for linear time-invariant systems is used to develop a dominance-preserving model reduction method for Lure systems, i.e. systems that can be decomposed as the feedback interconnection of a linear system and a s…
Balanced Truncation of Linear Time-Invariant Systems over Finite-frequency Ranges
Max Planck Institute for Dynamics of Complex Technical Systems（马普复杂技术系统动力学研究所）；School of Mechatronic Engineering and Automation, Shanghai University（上海大学机电工程与自动化学院）
Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. Two computational challenges have so far prevented its deployment on large-scale systems: (a) the energy functions required for characterization of controllability and observability are solutions of high-dimensional Hamilton-Jacobi-(Bellman) equations, which have been computationally intractable and (b) the transformations to construct the reduced-order models (ROMs) …
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL), UMR CNRS 9189, CNRS, Ecole Centrale de Lille（里尔信息、信号与自动化研究中心 (CRIStAL)，“CNRS联合研究单位9189”，法国国家科学研究中心，里尔中央理工学院）；Department of Electrical and Computer Engineering, Rice University（莱斯大学电气与计算机工程系）；CNRS CRIStAL UMR 9189, Université de Lille 1, Sciences et Technologies（法国国家科学研究中心 CRIStAL “联合研究单位9189”，里尔第一大学科学与技术学院）
In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a …
In this paper, we explore the role of tensor algebra in balanced truncation (BT) based model reduction/identification for high-dimensional multilinear/linear time invariant systems. In particular, we employ tensor train decomposition (TTD), which provides a good compromise between numerical stability and level of compression, and has an associated algebra that facilitates computations. Using TTD, we propose a new BT approach which we refer to as higher-order balanced truncation, and consider different data-driven variations including higher-order empirical gramians, higher-order balanced prope…
We present a novel reformulation of balanced truncation, a classical model reduction method. The principal innovation that we introduce comes through the use of system response data that has been either measured or computed, without reference to any prescribed realization of the original model. Data are represented by sampled values of the transfer function {or the impulse response} corresponding to the original model. We discuss parallels that our approach bears with the Loewner framework, another popular data-driven model reduction method. We illustrate our approach numerically in both conti…
Kernel Methods for the Approximation of Nonlinear Systems
Laboratory for Computational and Statistical Learning, Massachusetts Institute of Technology（麻省理工学院计算与统计学习实验室）；Department of Mathematics, Duke University（杜克大学数学系）；Department of Mathematics, Imperial College London（帝国理工学院伦敦数学系）
