From Data to Reduced-Order Models via Generalized Balanced Truncation

Abstract

This article proposes a data-driven model reduction approach on the basis of noisy data with a known noise model. Firstl, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin projection to all systems explaining the data characterized in a largedimensional quadratic matrix inequality (QMI) can again be characterized in a lower-dimensional QMI. Next, we develop a data-driven generalized balanced truncation method that relies on two steps. First, we provide necessary and sufficient conditions such that systems explaining the data have common generalized Gramians. Second, these common generalized Gramians are used to construct matrices that allow to characterize a class of reduced-order models via generalized balanced truncation in terms of a lowerdimensional QMI by applying the data reduction concept. Additionally, we present alternative procedures to compute a priori and a posteriori upper bounds with respect to the true system generating the data. Finally, the proposed techniques are illustrated by means of application to an example of a system of a cart with a double-pendulum.

Improved Oversampling Algorithm for Imbalanced Data Based on K-Nearest Neighbor and Interpolation Process Optimization

“…In order to more intuitively show the advantages of the GI-SMOTE algorithm in synthesizing new samples, we visualize the characteristics of data distribution after applying no-sampling, SMOTE, and GI-SMOTE to three different datasets. Since the attribute of the dataset itself is high-dimensional, the principal component analysis (PCA) [23,24] technique is used as a dimension reduction tool to reduce the distribution of these datasets from high-dimensional to two-dimensional [25]. The distributions of these samples are shown in Figure 2, where orange dots represent the minority samples, blue dots represent the majority samples, and green dots represent the synthetic samples.…”

The Informativity Approach: To Data-Driven Analysis and Control

“…Then, the system ( , , , ) A B C D can be seen to satisfy the dissipation inequality (90) if and only if…”

“…The subset of R n n # consisting of all symmetric matrices is denoted by S n . For vectors x and y, we denote x y State feedback stabilization E-IS [68], [70], [71], and [72] Deadbeat controller E-IS [68] LQR E-IS [68] Suboptimal LQR E-IS [73] Suboptimal H2 E-IS [73] Synchronization E-IS [74] Robust MPC E-IS [75] Dynamic feedback stabilization E-ISO [68] Dynamic feedback stabilization E-IO [68] and [76] Dissipativity E-ISO [77] Tracking and regulation E-IS [78] Model reduction (moment matching) E-IO [79] Reachability (conic constraints) E-IO [80] Stability N-S [81] Stabilizability N-IS [81] State feedback stabilization N-IS [70], [72], and [82] Stabilization for switched system N-IS [83], [84], and [85] Control with input saturation N-IS [86] Control of positive systems N-IS [87] State feedback H2 control N-IS [72] and [82] Dynamic feedback H2 control N-IO [88] State feedback H3 control N-IS [72] and [82] Dynamic feedback H3 control N-IO [88] Stability N-IO [89] Dynamic feedback stabilization N-IO [88] and [89] Dissipativity N-ISO [77] Model reduction (balancing) N-ISO [90] Structural properties N-ISO [91] Absolute stabilization Luré systems N-ISO [72...…”

Frequency-Limited Model Reduction for Linear Positive Systems: A Successive Optimization Method

“…Therefore, it is crucial to simplify models of dynamical systems by exploring lower-order models that approximate the original high-order ones according to certain criteria [14,15]. Various model reduction methods, such as balanced truncation [16], Hankel-norm reduction [17], and moment matching [18] have been developed.…”