Balanced Truncation via Tangential Interpolation

TL;DR

This work tackles reducing high-order stable LTI systems by preserving dominant energy-transfer characteristics, framing balanced truncation (BT) as a tangential interpolation problem when truncated states are weakly controllable/observable. It develops two fully automatic algorithms, ALRS-LYAP (for low-rank Lyapunov solves) and ATIA-BT (for automatic tangential interpolation in BT), which iteratively identify interpolation data and the reduced order $r$ by exploiting low-rank approximations of the controllability/observability Gramians and the Hankel singular values. The methods are demonstrated on benchmark models, showing that the resulting reduced-order models preserve the $r$ largest Hankel singular values with interpolation at mirror images of the poles along residual directions, and scale to very large systems (including a $10^7$-state heat-transfer model). The results indicate that the proposed adaptive interpolatory approaches can match BT accuracy while avoiding explicit, large Lyapunov solves, offering automatic order selection and applicability to large-scale MOR problems.

Abstract

This paper examines the construction of rth-order truncated balanced realizations via tangential interpolation at r specified interpolation points. It is demonstrated that when the truncated Hankel singular values are negligible-that is, when the discarded states are nearly uncontrollable and unobservable-balanced truncation simplifies to a bi-tangential Hermite interpolation problem at r interpolation points. In such cases, the resulting truncated balanced realization is nearly H2-optimal and thus interpolates the original model at the mirror images of its poles along its residual directions. Like standard H2-optimal model reduction, where the interpolation points and tangential directions that yield a local optimum are not known, in balanced truncation as well, the interpolation points and tangential directions required to produce a truncated balanced realization remain unknown. To address this, we propose an iterative tangential interpolation-based algorithm for balanced truncation. Upon convergence, the algorithm yields a low-rank truncated balanced realization that accurately preserves the r largest Hankel singular values of the original system. An adaptive scheme to automatically select the order r of the reduced model is also proposed. The algorithm is fully automatic, choosing both the interpolation data and the model order without user intervention. Additionally, an adaptive low-rank solver for Lyapunov equations based on tangential interpolation is proposed, automatically selecting both the interpolation data and the rank without user intervention. The performance of the proposed algorithms is evaluated on benchmark models, confirming their efficacy.

Figures (14)

• Figure 1: Decay in singular values of $P$ and the relative error $\frac{||P-V_rP_rV_r^T||_2}{||P||_2}$
• Figure 2: Decay in Hankel singular values of $H(s)$ and the relative error $\frac{||PQ-V_rP_rV_r^TW_rQ_rW_r^T||_2}{||PQ||_2}$
• Figure 3: Singular values of $P$ and $V_rP_rV_r^T$
• Figure 4: Singular values of $Q$ and $W_rQ_rW_r^T$
• Figure 5: Hankel Singular Values of $16^{th}$-order $H_r(s)$

Theorems & Definitions (2)

• Remark 1
• Remark 2