Dimension Reduction in Structured Dynamical Systems: Optimal-$mathcal{H}_2$ Approximation, Data-Driven Balancing, and Real-Time Monitoring}

Abstract

This dissertation considers a variety of problems pertaining to the model-order reduction, data-driven reduced-order modeling, and real-time monitoring of large-scale and structured dynamical systems. In the first part, balancing-based methods for system-theoretic model reduction of linear time-invariant systems are considered. We generalize conditions for which the balanced truncation $mathcal{H}_{infty}$ error bound is known to hold with equality. Specifically, we show that the bound is tight for single-input, single-output systems for which the truncated part of the model is a mild generalization of state-space symmetric. After this, we develop data-driven reformulations of various kinds of balancing-based model reduction for linear first-order and second-order systems. The variants considered are balanced stochastic truncation, positive-real, bounded-real, bounded-real balanced truncation, frequency-weighted balanced truncation, and position-velocity balanced truncation. For each variant, we show how to approximately construct the balanced truncation reduced model from various kinds of input-output invariant frequency response data. In the second part of this dissertation, we consider the $mathcal{H}_2$-optimal model reduction problem for linear dynamical systems with quadratic-output functions. As the significant theoretical contributions of this portion, we establish the Sylvester equation-based (Wilson) and interpolation-based (Meier-Luenberger) $mathcal{H}_2$-optimality frameworks for this class of systems. These frameworks are based on two independent sets of first-order necessary conditions for optimality. We additionally show how to enforce the established necessary optimality conditions using a Petrov-Galerkin projection, and prove that the Wilson optimality conditions imply the interpolation-based optimality conditions under some mild assumptions. Based on the theoretical optimality frameworks, two iterative algorithms for the optimal-$mathcal{H}_2$ approximation of linear quadratic-output systems are proposed. In the final portion, we investigate low-rank interpolatory matrix decompositions for reducing the dimensionality of large matrices of Phasor Measurement Unit (PMU) data in the real-time monitoring of electrical power networks. We propose a theoretical framework for analyzing sparse reconstructions of PMU data during online operations. Drawing upon the numerical linear algebra literature, this framework allows us to state a rigorous, computable upper bound for the interpolatory reconstruction error. This bound can be used to certify whether a collection of PMUs or time instances truly captures the low-rank character of the data, and can be leveraged toward various operational functions. Specifically, we propose a data-driven algorithm for real-time disturbance monitoring that is based upon interpolatory approximations and the discrete empirical interpolation method. Numerical results are included in each portion to validate the theoretical results of the dissertation.