Christian Himpe

Cross-Gramian-Based Combined State and Parameter Reduction for Large-Scale Control Systems

This work introduces the empirical cross-gramian for multiple-input-multiple-output systems. The cross-gramian is a tool for reducing the state space of control systems, by conjoining controllability and observability information into a single matrix and does not require balancing. Its empirical gramian variant extends the applicability of the cross-gramian to nonlinear systems. Furthermore, for parametrized systems, the empirical gramians can also be utilized for sensitivity analysis or parameter identification and thus for parameter reduction. This work also introduces the empirical joint gramian, which is derived from the empirical cross-gramian. The joint gramian allows not only a reduction of the parameter space but also the combined state and parameter space reduction, which is tested on a linear and a nonlinear control system. Controllability- and observability-based combined reduction methods are also presented, which are benchmarked against the joint gramian.

A Unified Software Framework for Empirical Gramians

A common approach in model reduction is balanced truncation, which is based on Gramian matrices classifying certain attributes of states or parameters of a given dynamic system. Initially restricted to linear systems, the empirical Gramians not only extended this concept to nonlinear systems but also provided a uniform computational method. This work introduces a unified software framework supplying routines for six types of empirical Gramians. The Gramian types will be discussed and applied in a model reduction framework for multiple-input multiple-output systems.

A note on the cross Gramian for non-symmetric systems

ABSTRACT The cross Gramian matrix is a tool for model reduction and system identification, but it is only applicable to square control systems. For symmetric systems, the cross Gramian possesses a useful relation to the systems associated Hankel singular values. Yet, many real-life models are neither square nor symmetric. In this work, concepts from decentralized control are used to approximate a cross Gramian for non-symmetric and non-square systems. To illustrate this new non-symmetric cross Gramian, it is applied in the context of model order reduction.

Model reduction for complex hyperbolic networks

We recently introduced the joint gramian for combined state and parameter reduction [C. Himpe and M. Ohlberger. Cross-Gramian-Based Combined State and Parameter Reduction for Large-Scale Control Systems. arXiv:1302.0634, 2013], which is applied in this work to reduce a parametrized linear time-varying control system modeling a hyperbolic network. The reduction encompasses the dimension of nodes and parameters of the underlying control system. Networks with a hyperbolic structure have many applications as models for large-scale systems. A prominent example is the brain, for which a network structure of the various regions is often assumed to model propagation of information. Networks with many nodes, and parametrized, uncertain or even unknown connectivity require many and individually computationally costly simulations. The presented model order reduction enables vast simulations of surrogate networks exhibiting almost the same dynamics with a small error compared to full order model.

Cross-Gramian-Based Model Reduction: A Comparison

As an alternative to the popular balanced truncation method, the cross Gramian matrix induces a class of balancing model reduction techniques. Besides the classical computation of the cross Gramian by a Sylvester matrix equation, an empirical cross Gramian can be computed based on simulated trajectories. This work assesses the cross Gramian and its empirical Gramian variant for state-space reduction on a procedural benchmark based on the cross Gramian itself.

Data-driven combined state and parameter reduction for inverse problems

In this contribution we present an accelerated optimization-based approach for combined state and parameter reduction of a parametrized forward model, which is used to construct a surrogate model in a Bayesian inverse problem setting. Following the ideas presented in Lieberman et al. (SIAM J. Sci. Comput. 32(5), 2523–2542, 2010), our approach is based on a generalized data-driven optimization functional in the construction process of the reduced order model and the usage of a Monte-Carlo basis enrichment strategy that results in an additional speed-up of the overall method. In principal, the model reduction procedure is based on the offline construction of appropriate low-dimensional state and parameter spaces and an online inversion step using the resulting surrogate model that is obtained through projection of the underlying forward model onto the reduced spaces. The generalizations and enhancements presented in this work are shown to decrease overall computational time and thus allow an application to large-scale problems. Numerical experiments for a generic model and a fMRI connectivity model are presented in order to compare the computational efficiency of our improved method with the original approach.

The Versatile Cross Gramian for System Theoretic Model Reduction and More

Read Me: C. Himpe and M. Ohlberger. “A note on the non-symmetric cross gramian”. arXiv Preprint, math.OC(1501.05519):1–6, 2015. C. Himpe and M. Ohlberger. “Cross-gramian based combined state and parameter reduction for large-scale control systems”. Mathematical Problems in Engineering, 2014:1–13, 2014. L.A. Mironovskii and T.N. Solvéva. “Analysis of multiplicity of hankel singular values of control systems”. Automation and Remote Control, 76(2):205–218, 2015. T.C. Ionescu, K. Fujimoto, and J.M.A. Scherpen. “Singular value analysis of non-linear symmetric systems”. Transactions on Automatic Control of the IEEE, 56(9):2073–2086, 2011. Abstract: For input-output systems, the cross gramian matrix encodes controllability and observability information into a single matrix, which are essential to system-theoretic applications. This system gramian can be used, in example, for model order reduction, sensitivity analysis, system identification, decentralized control and parameter identification. Beyond linear symmetric systems, the cross gramian is also available for parametric, non-symmetric, non-square and nonlinear systems.