Balanced Truncation Model Reduction with A Priori Error Bounds for LTI Systems with Nonzero Initial Value

TL;DR

This paper introduces a new balanced truncation method for linear time-invariant systems with nonzero initial conditions, providing a priori error bounds and flexible reduction strategies for different system components.

Contribution

It proposes a novel balancing procedure with a shift transformation that yields a priori error bounds and allows separate reduction of input-dependent and initial-value-dependent parts.

Findings

Derived a joint projection reduced-order model with proven error bounds

Developed a separate projection method enabling different reduction orders

Validated the methods through numerical experiments comparing with existing approaches

Abstract

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 joint projection reduced-order model in which the part of the system depending only on the input and the one depending only on the initial value are reduced at once and we prove an a priori error bound. With this result at hand, we derive a separate projection procedure in which the two parts are reduced separately. This gives the freedom to choose different reduction orders for the different subsystems. Moreover, we discuss how the reduced-order models can be constructed in practice. Since the error bounds are parameter-dependent we show how they can be optimized efficiently. We conclude this paper by comparing our results with the ones from the literature by a series of numerical experiments.